%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP251^1 : 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 : n017.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 09:11:29 AM UTC 2026
% Result : Theorem 145.00s 25.60s
% Output : Refutation 145.00s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 8
% Syntax : Number of formulae : 49 ( 28 unt; 0 typ; 0 def)
% Number of atoms : 195 ( 55 equ; 0 cnn)
% Maximal formula atoms : 3 ( 3 avg)
% Number of connectives : 815 ( 28 ~; 15 |; 0 &; 763 @)
% ( 0 <=>; 9 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 5 avg)
% Number of types : 88 ( 87 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 986 ( 983 usr; 59 con; 0-8 aty)
% Number of variables : 67 ( 7 ^; 60 !; 0 ?; 67 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
rat: $tType ).
thf(type_def_6,type,
int: $tType ).
thf(type_def_8,type,
real: $tType ).
thf(type_def_9,type,
nat: $tType ).
thf(type_def_10,type,
product_prod_int_int: $tType ).
thf(type_def_11,type,
num: $tType ).
thf(type_def_12,type,
option_num: $tType ).
thf(type_def_13,type,
product_prod_num_num: $tType ).
thf(type_def_14,type,
code_integer: $tType ).
thf(type_def_15,type,
produc6271795597528267376eger_o: $tType ).
thf(type_def_16,type,
produc8923325533196201883nteger: $tType ).
thf(type_def_17,type,
set_real: $tType ).
thf(type_def_18,type,
set_int: $tType ).
thf(type_def_19,type,
complex: $tType ).
thf(type_def_20,type,
filter_real: $tType ).
thf(type_def_21,type,
product_prod_nat_nat: $tType ).
thf(type_def_22,type,
filter_nat: $tType ).
thf(type_def_23,type,
set_o: $tType ).
thf(type_def_24,type,
set_complex: $tType ).
thf(type_def_25,type,
set_list_nat: $tType ).
thf(type_def_26,type,
set_nat: $tType ).
thf(type_def_27,type,
set_Product_unit: $tType ).
thf(type_def_28,type,
set_char: $tType ).
thf(type_def_29,type,
set_list_o: $tType ).
thf(type_def_30,type,
set_list_complex: $tType ).
thf(type_def_31,type,
set_list_int: $tType ).
thf(type_def_32,type,
set_list_VEBT_VEBT: $tType ).
thf(type_def_33,type,
set_num: $tType ).
thf(type_def_34,type,
set_rat: $tType ).
thf(type_def_35,type,
set_set_complex: $tType ).
thf(type_def_36,type,
set_set_int: $tType ).
thf(type_def_37,type,
set_set_nat: $tType ).
thf(type_def_38,type,
set_VEBT_VEBT: $tType ).
thf(type_def_39,type,
char: $tType ).
thf(type_def_40,type,
list_nat: $tType ).
thf(type_def_41,type,
extended_enat: $tType ).
thf(type_def_42,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(type_def_43,type,
list_o: $tType ).
thf(type_def_44,type,
list_int: $tType ).
thf(type_def_45,type,
option_nat: $tType ).
thf(type_def_46,type,
vEBT_VEBT: $tType ).
thf(type_def_47,type,
set_Pr958786334691620121nt_int: $tType ).
thf(type_def_48,type,
list_complex: $tType ).
thf(type_def_49,type,
list_P6011104703257516679at_nat: $tType ).
thf(type_def_50,type,
list_real: $tType ).
thf(type_def_51,type,
list_VEBT_VEBT: $tType ).
thf(type_def_52,type,
list_Code_integer: $tType ).
thf(type_def_53,type,
list_num: $tType ).
thf(type_def_54,type,
list_P4002435161011370285od_o_o: $tType ).
thf(type_def_55,type,
product_prod_o_o: $tType ).
thf(type_def_56,type,
list_P3795440434834930179_o_int: $tType ).
thf(type_def_57,type,
product_prod_o_int: $tType ).
thf(type_def_58,type,
list_P6285523579766656935_o_nat: $tType ).
thf(type_def_59,type,
product_prod_o_nat: $tType ).
thf(type_def_60,type,
list_P7495141550334521929T_VEBT: $tType ).
thf(type_def_61,type,
produc2504756804600209347T_VEBT: $tType ).
thf(type_def_62,type,
list_P8526636022914148096eger_o: $tType ).
thf(type_def_63,type,
list_P3744719386663036955um_num: $tType ).
thf(type_def_64,type,
list_P3126845725202233233VEBT_o: $tType ).
thf(type_def_65,type,
produc334124729049499915VEBT_o: $tType ).
thf(type_def_66,type,
list_P4547456442757143711BT_int: $tType ).
thf(type_def_67,type,
produc4894624898956917775BT_int: $tType ).
thf(type_def_68,type,
list_P7037539587688870467BT_nat: $tType ).
thf(type_def_69,type,
produc9072475918466114483BT_nat: $tType ).
thf(type_def_70,type,
list_P7413028617227757229T_VEBT: $tType ).
thf(type_def_71,type,
produc8243902056947475879T_VEBT: $tType ).
thf(type_def_72,type,
list_P7333126701944960589_nat_o: $tType ).
thf(type_def_73,type,
list_P5647936690300460905T_VEBT: $tType ).
thf(type_def_74,type,
option4927543243414619207at_nat: $tType ).
thf(type_def_75,type,
set_Pr448751882837621926eger_o: $tType ).
thf(type_def_76,type,
set_Pr6200539531224447659at_num: $tType ).
thf(type_def_77,type,
set_Pr8218934625190621173um_num: $tType ).
thf(type_def_78,type,
set_Code_integer: $tType ).
thf(type_def_79,type,
produc4953844613479565601on_nat: $tType ).
thf(type_def_80,type,
produc2233624965454879586on_nat: $tType ).
thf(type_def_81,type,
produc7248412053542808358at_nat: $tType ).
thf(type_def_82,type,
produc4471711990508489141at_nat: $tType ).
thf(type_def_83,type,
produc8306885398267862888on_nat: $tType ).
thf(type_def_84,type,
produc2963631642982155120at_num: $tType ).
thf(type_def_85,type,
produc3368934014287244435at_num: $tType ).
thf(type_def_86,type,
produc3447558737645232053on_num: $tType ).
thf(type_def_87,type,
produc7036089656553540234on_num: $tType ).
thf(type_def_88,type,
produc1193250871479095198on_num: $tType ).
thf(type_def_89,type,
produc6121120109295599847at_nat: $tType ).
thf(type_def_90,type,
produc5491161045314408544at_nat: $tType ).
thf(type_def_91,type,
produc5542196010084753463at_nat: $tType ).
thf(type_def_92,type,
product_prod_nat_num: $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,
list_P7413028617227757229T_VEBT: $tType ).
thf(func_def_10,type,
produc3447558737645232053on_num: $tType ).
thf(func_def_11,type,
produc4953844613479565601on_nat: $tType ).
thf(func_def_12,type,
produc2963631642982155120at_num: $tType ).
thf(func_def_13,type,
produc7248412053542808358at_nat: $tType ).
thf(func_def_14,type,
list_P7037539587688870467BT_nat: $tType ).
thf(func_def_15,type,
list_P4547456442757143711BT_int: $tType ).
thf(func_def_16,type,
list_P5647936690300460905T_VEBT: $tType ).
thf(func_def_17,type,
produc8243902056947475879T_VEBT: $tType ).
thf(func_def_18,type,
produc8923325533196201883nteger: $tType ).
thf(func_def_19,type,
list_P3126845725202233233VEBT_o: $tType ).
thf(func_def_20,type,
list_P7495141550334521929T_VEBT: $tType ).
thf(func_def_21,type,
list_P8526636022914148096eger_o: $tType ).
thf(func_def_22,type,
set_Pr448751882837621926eger_o: $tType ).
thf(func_def_23,type,
option4927543243414619207at_nat: $tType ).
thf(func_def_24,type,
list_P3744719386663036955um_num: $tType ).
thf(func_def_25,type,
list_P6011104703257516679at_nat: $tType ).
thf(func_def_26,type,
produc9072475918466114483BT_nat: $tType ).
thf(func_def_27,type,
produc4894624898956917775BT_int: $tType ).
thf(func_def_28,type,
set_Pr8218934625190621173um_num: $tType ).
thf(func_def_29,type,
set_Pr6200539531224447659at_num: $tType ).
thf(func_def_30,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(func_def_31,type,
set_Pr958786334691620121nt_int: $tType ).
thf(func_def_32,type,
list_P7333126701944960589_nat_o: $tType ).
thf(func_def_33,type,
list_P6285523579766656935_o_nat: $tType ).
thf(func_def_34,type,
list_P3795440434834930179_o_int: $tType ).
thf(func_def_35,type,
set_list_VEBT_VEBT: $tType ).
thf(func_def_36,type,
produc334124729049499915VEBT_o: $tType ).
thf(func_def_37,type,
produc2504756804600209347T_VEBT: $tType ).
thf(func_def_38,type,
produc6271795597528267376eger_o: $tType ).
thf(func_def_39,type,
product_prod_num_num: $tType ).
thf(func_def_40,type,
product_prod_nat_num: $tType ).
thf(func_def_41,type,
product_prod_nat_nat: $tType ).
thf(func_def_42,type,
product_prod_int_int: $tType ).
thf(func_def_43,type,
list_P4002435161011370285od_o_o: $tType ).
thf(func_def_44,type,
set_list_complex: $tType ).
thf(func_def_45,type,
set_set_complex: $tType ).
thf(func_def_46,type,
list_VEBT_VEBT: $tType ).
thf(func_def_47,type,
set_list_nat: $tType ).
thf(func_def_48,type,
set_list_int: $tType ).
thf(func_def_49,type,
product_prod_o_nat: $tType ).
thf(func_def_50,type,
product_prod_o_int: $tType ).
thf(func_def_51,type,
list_Code_integer: $tType ).
thf(func_def_52,type,
set_VEBT_VEBT: $tType ).
thf(func_def_53,type,
set_set_nat: $tType ).
thf(func_def_54,type,
set_set_int: $tType ).
thf(func_def_55,type,
set_Code_integer: $tType ).
thf(func_def_56,type,
set_Product_unit: $tType ).
thf(func_def_57,type,
list_complex: $tType ).
thf(func_def_58,type,
set_list_o: $tType ).
thf(func_def_59,type,
product_prod_o_o: $tType ).
thf(func_def_60,type,
set_complex: $tType ).
thf(func_def_61,type,
filter_real: $tType ).
thf(func_def_62,type,
option_num: $tType ).
thf(func_def_63,type,
option_nat: $tType ).
thf(func_def_64,type,
filter_nat: $tType ).
thf(func_def_65,type,
set_char: $tType ).
thf(func_def_66,type,
list_real: $tType ).
thf(func_def_67,type,
set_real: $tType ).
thf(func_def_68,type,
list_num: $tType ).
thf(func_def_69,type,
list_nat: $tType ).
thf(func_def_70,type,
list_int: $tType ).
thf(func_def_71,type,
vEBT_VEBT: $tType ).
thf(func_def_72,type,
set_rat: $tType ).
thf(func_def_73,type,
set_num: $tType ).
thf(func_def_74,type,
set_nat: $tType ).
thf(func_def_75,type,
set_int: $tType ).
thf(func_def_76,type,
code_integer: $tType ).
thf(func_def_77,type,
extended_enat: $tType ).
thf(func_def_78,type,
list_o: $tType ).
thf(func_def_79,type,
complex: $tType ).
thf(func_def_80,type,
set_o: $tType ).
thf(func_def_81,type,
char: $tType ).
thf(func_def_82,type,
real: $tType ).
thf(func_def_83,type,
rat: $tType ).
thf(func_def_84,type,
num: $tType ).
thf(func_def_85,type,
nat: $tType ).
thf(func_def_86,type,
int: $tType ).
thf(func_def_87,type,
archim2889992004027027881ng_rat: rat > int ).
thf(func_def_88,type,
archim7802044766580827645g_real: real > int ).
thf(func_def_89,type,
archim3151403230148437115or_rat: rat > int ).
thf(func_def_90,type,
archim6058952711729229775r_real: real > int ).
thf(func_def_91,type,
archim7778729529865785530nd_rat: rat > int ).
thf(func_def_92,type,
archim8280529875227126926d_real: real > int ).
thf(func_def_93,type,
binomial: nat > nat > nat ).
thf(func_def_94,type,
bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_95,type,
bit_and_not_num: num > num > option_num ).
thf(func_def_96,type,
bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_97,type,
bit_concat_bit: nat > int > int > int ).
thf(func_def_98,type,
bit_or_not_num_neg: num > num > num ).
thf(func_def_99,type,
bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_100,type,
bit_ri7919022796975470100ot_int: int > int ).
thf(func_def_101,type,
bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).
thf(func_def_102,type,
bit_ri631733984087533419it_int: nat > int > int ).
thf(func_def_103,type,
bit_se725231765392027082nd_int: int > int > int ).
thf(func_def_104,type,
bit_se727722235901077358nd_nat: nat > nat > nat ).
thf(func_def_105,type,
bit_se8568078237143864401it_int: nat > int > int ).
thf(func_def_106,type,
bit_se8570568707652914677it_nat: nat > nat > nat ).
thf(func_def_107,type,
bit_se1345352211410354436nteger: nat > code_integer > code_integer ).
thf(func_def_108,type,
bit_se2159334234014336723it_int: nat > int > int ).
thf(func_def_109,type,
bit_se2161824704523386999it_nat: nat > nat > nat ).
thf(func_def_110,type,
bit_se2000444600071755411sk_int: nat > int ).
thf(func_def_111,type,
bit_se2002935070580805687sk_nat: nat > nat ).
thf(func_def_112,type,
bit_se1409905431419307370or_int: int > int > int ).
thf(func_def_113,type,
bit_se1412395901928357646or_nat: nat > nat > nat ).
thf(func_def_114,type,
bit_se545348938243370406it_int: nat > int > int ).
thf(func_def_115,type,
bit_se547839408752420682it_nat: nat > nat > nat ).
thf(func_def_116,type,
bit_se2793503036327961859nteger: nat > code_integer > code_integer ).
thf(func_def_117,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(func_def_118,type,
bit_se7882103937844011126it_nat: nat > nat > nat ).
thf(func_def_119,type,
bit_se2923211474154528505it_int: nat > int > int ).
thf(func_def_120,type,
bit_se2925701944663578781it_nat: nat > nat > nat ).
thf(func_def_121,type,
bit_se8260200283734997820nteger: nat > code_integer > code_integer ).
thf(func_def_122,type,
bit_se4203085406695923979it_int: nat > int > int ).
thf(func_def_123,type,
bit_se4205575877204974255it_nat: nat > nat > nat ).
thf(func_def_124,type,
bit_se6526347334894502574or_int: int > int > int ).
thf(func_def_125,type,
bit_se6528837805403552850or_nat: nat > nat > nat ).
thf(func_def_126,type,
bit_se1146084159140164899it_int: int > nat > $o ).
thf(func_def_127,type,
bit_se1148574629649215175it_nat: nat > nat > $o ).
thf(func_def_128,type,
bit_take_bit_num: nat > num > option_num ).
thf(func_def_129,type,
bit_un1837492267222099188nd_num: num > num > option_num ).
thf(func_def_130,type,
bit_un5425074673868309765um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_131,type,
bit_un6178654185764691216or_num: num > num > option_num ).
thf(func_def_132,type,
bit_un3595099601533988841um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_133,type,
bit_un7362597486090784418nd_num: num > num > option_num ).
thf(func_def_134,type,
bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_135,type,
bit_un2480387367778600638or_num: num > num > option_num ).
thf(func_def_136,type,
bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_137,type,
code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).
thf(func_def_138,type,
code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_139,type,
code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_140,type,
code_int_of_integer: code_integer > int ).
thf(func_def_141,type,
code_integer_of_int: int > code_integer ).
thf(func_def_142,type,
code_integer_of_num: num > code_integer ).
thf(func_def_143,type,
code_nat_of_integer: code_integer > nat ).
thf(func_def_144,type,
code_negative: num > code_integer ).
thf(func_def_145,type,
code_num_of_integer: code_integer > num ).
thf(func_def_146,type,
code_positive: num > code_integer ).
thf(func_def_147,type,
code_Target_negative: num > int ).
thf(func_def_148,type,
code_Target_positive: num > int ).
thf(func_def_149,type,
comple4887499456419720421f_real: set_real > real ).
thf(func_def_150,type,
complete_Sup_Sup_int: set_int > int ).
thf(func_def_151,type,
comple1385675409528146559p_real: set_real > real ).
thf(func_def_152,type,
arg: complex > real ).
thf(func_def_153,type,
cis: real > complex ).
thf(func_def_154,type,
cnj: complex > complex ).
thf(func_def_155,type,
complex2: real > real > complex ).
thf(func_def_156,type,
im: complex > real ).
thf(func_def_157,type,
re: complex > real ).
thf(func_def_158,type,
csqrt: complex > complex ).
thf(func_def_159,type,
imaginary_unit: complex ).
thf(func_def_160,type,
differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).
thf(func_def_161,type,
has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).
thf(func_def_162,type,
has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).
thf(func_def_163,type,
adjust_div: product_prod_int_int > int ).
thf(func_def_164,type,
adjust_mod: int > int > int ).
thf(func_def_165,type,
divmod_nat: nat > nat > product_prod_nat_nat ).
thf(func_def_166,type,
eucl_rel_int: int > int > product_prod_int_int > $o ).
thf(func_def_167,type,
unique5706413561485394159nteger: produc8923325533196201883nteger > $o ).
thf(func_def_168,type,
unique6319869463603278526ux_int: product_prod_int_int > $o ).
thf(func_def_169,type,
unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).
thf(func_def_170,type,
unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).
thf(func_def_171,type,
unique5052692396658037445od_int: num > num > product_prod_int_int ).
thf(func_def_172,type,
unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).
thf(func_def_173,type,
unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_174,type,
unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).
thf(func_def_175,type,
unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_176,type,
comm_s8582702949713902594nteger: code_integer > nat > code_integer ).
thf(func_def_177,type,
comm_s2602460028002588243omplex: complex > nat > complex ).
thf(func_def_178,type,
comm_s4660882817536571857er_int: int > nat > int ).
thf(func_def_179,type,
comm_s4663373288045622133er_nat: nat > nat > nat ).
thf(func_def_180,type,
comm_s4028243227959126397er_rat: rat > nat > rat ).
thf(func_def_181,type,
comm_s7457072308508201937r_real: real > nat > real ).
thf(func_def_182,type,
semiri3624122377584611663nteger: nat > code_integer ).
thf(func_def_183,type,
semiri5044797733671781792omplex: nat > complex ).
thf(func_def_184,type,
semiri1406184849735516958ct_int: nat > int ).
thf(func_def_185,type,
semiri1408675320244567234ct_nat: nat > nat ).
thf(func_def_186,type,
semiri773545260158071498ct_rat: nat > rat ).
thf(func_def_187,type,
semiri2265585572941072030t_real: nat > real ).
thf(func_def_188,type,
invers8013647133539491842omplex: complex > complex ).
thf(func_def_189,type,
inverse_inverse_rat: rat > rat ).
thf(func_def_190,type,
inverse_inverse_real: real > real ).
thf(func_def_191,type,
at_bot_real: filter_real ).
thf(func_def_192,type,
at_top_nat: filter_nat ).
thf(func_def_193,type,
at_top_real: filter_real ).
thf(func_def_194,type,
eventually_nat: ( nat > $o ) > filter_nat > $o ).
thf(func_def_195,type,
filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
thf(func_def_196,type,
filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
thf(func_def_197,type,
filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
thf(func_def_198,type,
finite_card_o: set_o > nat ).
thf(func_def_199,type,
finite_card_complex: set_complex > nat ).
thf(func_def_200,type,
finite_card_int: set_int > nat ).
thf(func_def_201,type,
finite_card_list_nat: set_list_nat > nat ).
thf(func_def_202,type,
finite_card_nat: set_nat > nat ).
thf(func_def_203,type,
finite410649719033368117t_unit: set_Product_unit > nat ).
thf(func_def_204,type,
finite_card_char: set_char > nat ).
thf(func_def_205,type,
finite_finite_o: set_o > $o ).
thf(func_def_206,type,
finite3207457112153483333omplex: set_complex > $o ).
thf(func_def_207,type,
finite_finite_int: set_int > $o ).
thf(func_def_208,type,
finite_finite_list_o: set_list_o > $o ).
thf(func_def_209,type,
finite8712137658972009173omplex: set_list_complex > $o ).
thf(func_def_210,type,
finite3922522038869484883st_int: set_list_int > $o ).
thf(func_def_211,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(func_def_212,type,
finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o ).
thf(func_def_213,type,
finite_finite_nat: set_nat > $o ).
thf(func_def_214,type,
finite_finite_num: set_num > $o ).
thf(func_def_215,type,
finite_finite_rat: set_rat > $o ).
thf(func_def_216,type,
finite_finite_real: set_real > $o ).
thf(func_def_217,type,
finite6551019134538273531omplex: set_set_complex > $o ).
thf(func_def_218,type,
finite6197958912794628473et_int: set_set_int > $o ).
thf(func_def_219,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(func_def_220,type,
finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o ).
thf(func_def_221,type,
bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
thf(func_def_222,type,
bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).
thf(func_def_223,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(func_def_224,type,
comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_225,type,
comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_226,type,
comp_C3531382070062128313er_num: ( code_integer > code_integer ) > ( num > code_integer ) > num > code_integer ).
thf(func_def_227,type,
comp_int_int_num: ( int > int ) > ( num > int ) > num > int ).
thf(func_def_228,type,
comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(func_def_229,type,
comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).
thf(func_def_230,type,
id_o: $o > $o ).
thf(func_def_231,type,
id_nat: nat > nat ).
thf(func_def_232,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(func_def_233,type,
inj_on_nat_char: ( nat > char ) > set_nat > $o ).
thf(func_def_234,type,
inj_on_real_real: ( real > real ) > set_real > $o ).
thf(func_def_235,type,
map_fu434086159418415080_int_o: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > $o ) > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > int > int > $o ).
thf(func_def_236,type,
map_fu4960017516451851995nt_int: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > int > int > int ).
thf(func_def_237,type,
map_fu4826362097070443709at_o_o: ( int > product_prod_nat_nat ) > ( $o > $o ) > ( product_prod_nat_nat > $o ) > int > $o ).
thf(func_def_238,type,
map_fu2345160673673942751at_nat: ( int > product_prod_nat_nat ) > ( nat > nat ) > ( product_prod_nat_nat > nat ) > int > nat ).
thf(func_def_239,type,
map_fu3667384564859982768at_int: ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int ) > ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ).
thf(func_def_240,type,
map_fu4333342158222067775at_rat: ( rat > product_prod_int_int ) > ( ( product_prod_int_int > product_prod_int_int ) > rat > rat ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > rat > rat > rat ).
thf(func_def_241,type,
map_fu898904425404107465nt_o_o: ( rat > product_prod_int_int ) > ( $o > $o ) > ( product_prod_int_int > $o ) > rat > $o ).
thf(func_def_242,type,
map_fu5673905371560938248nt_rat: ( rat > product_prod_int_int ) > ( product_prod_int_int > rat ) > ( product_prod_int_int > product_prod_int_int ) > rat > rat ).
thf(func_def_243,type,
strict1292158309912662752at_nat: ( nat > nat ) > set_nat > $o ).
thf(func_def_244,type,
the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).
thf(func_def_245,type,
gcd_Gcd_int: set_int > int ).
thf(func_def_246,type,
gcd_Gcd_nat: set_nat > nat ).
thf(func_def_247,type,
bezw: nat > nat > product_prod_int_int ).
thf(func_def_248,type,
bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_249,type,
gcd_gcd_int: int > int > int ).
thf(func_def_250,type,
gcd_gcd_nat: nat > nat > nat ).
thf(func_def_251,type,
gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_252,type,
abs_abs_Code_integer: code_integer > code_integer ).
thf(func_def_253,type,
abs_abs_complex: complex > complex ).
thf(func_def_254,type,
abs_abs_int: int > int ).
thf(func_def_255,type,
abs_abs_rat: rat > rat ).
thf(func_def_256,type,
abs_abs_real: real > real ).
thf(func_def_257,type,
minus_8727706125548526216plex_o: ( complex > $o ) > ( complex > $o ) > complex > $o ).
thf(func_def_258,type,
minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).
thf(func_def_259,type,
minus_1139252259498527702_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).
thf(func_def_260,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(func_def_261,type,
minus_2270307095948843157_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(func_def_262,type,
minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).
thf(func_def_263,type,
minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).
thf(func_def_264,type,
minus_minus_complex: complex > complex > complex ).
thf(func_def_265,type,
minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_266,type,
minus_minus_int: int > int > int ).
thf(func_def_267,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_268,type,
minus_minus_rat: rat > rat > rat ).
thf(func_def_269,type,
minus_minus_real: real > real > real ).
thf(func_def_270,type,
minus_811609699411566653omplex: set_complex > set_complex > set_complex ).
thf(func_def_271,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(func_def_272,type,
minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(func_def_273,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_274,type,
minus_1356011639430497352at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_275,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(func_def_276,type,
one_one_Code_integer: code_integer ).
thf(func_def_277,type,
one_one_complex: complex ).
thf(func_def_278,type,
one_on7984719198319812577d_enat: extended_enat ).
thf(func_def_279,type,
one_one_int: int ).
thf(func_def_280,type,
one_one_nat: nat ).
thf(func_def_281,type,
one_one_rat: rat ).
thf(func_def_282,type,
one_one_real: real ).
thf(func_def_283,type,
plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).
thf(func_def_284,type,
plus_plus_complex: complex > complex > complex ).
thf(func_def_285,type,
plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_286,type,
plus_plus_int: int > int > int ).
thf(func_def_287,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_288,type,
plus_plus_num: num > num > num ).
thf(func_def_289,type,
plus_plus_rat: rat > rat > rat ).
thf(func_def_290,type,
plus_plus_real: real > real > real ).
thf(func_def_291,type,
sgn_sgn_Code_integer: code_integer > code_integer ).
thf(func_def_292,type,
sgn_sgn_complex: complex > complex ).
thf(func_def_293,type,
sgn_sgn_int: int > int ).
thf(func_def_294,type,
sgn_sgn_rat: rat > rat ).
thf(func_def_295,type,
sgn_sgn_real: real > real ).
thf(func_def_296,type,
times_3573771949741848930nteger: code_integer > code_integer > code_integer ).
thf(func_def_297,type,
times_times_complex: complex > complex > complex ).
thf(func_def_298,type,
times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_299,type,
times_times_int: int > int > int ).
thf(func_def_300,type,
times_times_nat: nat > nat > nat ).
thf(func_def_301,type,
times_times_num: num > num > num ).
thf(func_def_302,type,
times_times_rat: rat > rat > rat ).
thf(func_def_303,type,
times_times_real: real > real > real ).
thf(func_def_304,type,
uminus1351360451143612070nteger: code_integer > code_integer ).
thf(func_def_305,type,
uminus1482373934393186551omplex: complex > complex ).
thf(func_def_306,type,
uminus_uminus_int: int > int ).
thf(func_def_307,type,
uminus_uminus_rat: rat > rat ).
thf(func_def_308,type,
uminus_uminus_real: real > real ).
thf(func_def_309,type,
uminus1532241313380277803et_int: set_int > set_int ).
thf(func_def_310,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(func_def_311,type,
uminus612125837232591019t_real: set_real > set_real ).
thf(func_def_312,type,
zero_z3403309356797280102nteger: code_integer ).
thf(func_def_313,type,
zero_zero_complex: complex ).
thf(func_def_314,type,
zero_z5237406670263579293d_enat: extended_enat ).
thf(func_def_315,type,
zero_zero_int: int ).
thf(func_def_316,type,
zero_zero_nat: nat ).
thf(func_def_317,type,
zero_zero_rat: rat ).
thf(func_def_318,type,
zero_zero_real: real ).
thf(func_def_319,type,
groups6621422865394947399nteger: ( complex > code_integer ) > set_complex > code_integer ).
thf(func_def_320,type,
groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).
thf(func_def_321,type,
groups5690904116761175830ex_int: ( complex > int ) > set_complex > int ).
thf(func_def_322,type,
groups5693394587270226106ex_nat: ( complex > nat ) > set_complex > nat ).
thf(func_def_323,type,
groups5058264527183730370ex_rat: ( complex > rat ) > set_complex > rat ).
thf(func_def_324,type,
groups5808333547571424918x_real: ( complex > real ) > set_complex > real ).
thf(func_def_325,type,
groups7873554091576472773nteger: ( int > code_integer ) > set_int > code_integer ).
thf(func_def_326,type,
groups3049146728041665814omplex: ( int > complex ) > set_int > complex ).
thf(func_def_327,type,
groups4538972089207619220nt_int: ( int > int ) > set_int > int ).
thf(func_def_328,type,
groups4541462559716669496nt_nat: ( int > nat ) > set_int > nat ).
thf(func_def_329,type,
groups3906332499630173760nt_rat: ( int > rat ) > set_int > rat ).
thf(func_def_330,type,
groups8778361861064173332t_real: ( int > real ) > set_int > real ).
thf(func_def_331,type,
groups7501900531339628137nteger: ( nat > code_integer ) > set_nat > code_integer ).
thf(func_def_332,type,
groups2073611262835488442omplex: ( nat > complex ) > set_nat > complex ).
thf(func_def_333,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
thf(func_def_334,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_335,type,
groups2906978787729119204at_rat: ( nat > rat ) > set_nat > rat ).
thf(func_def_336,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(func_def_337,type,
groups6381953495645901045omplex: ( product_prod_nat_nat > complex ) > set_Pr1261947904930325089at_nat > complex ).
thf(func_def_338,type,
groups977919841031483927at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > nat ).
thf(func_def_339,type,
groups4567486121110086003t_real: ( product_prod_nat_nat > real ) > set_Pr1261947904930325089at_nat > real ).
thf(func_def_340,type,
groups7713935264441627589nteger: ( real > code_integer ) > set_real > code_integer ).
thf(func_def_341,type,
groups5754745047067104278omplex: ( real > complex ) > set_real > complex ).
thf(func_def_342,type,
groups1932886352136224148al_int: ( real > int ) > set_real > int ).
thf(func_def_343,type,
groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).
thf(func_def_344,type,
groups1300246762558778688al_rat: ( real > rat ) > set_real > rat ).
thf(func_def_345,type,
groups8097168146408367636l_real: ( real > real ) > set_real > real ).
thf(func_def_346,type,
groups1705073143266064639nt_int: ( int > int ) > set_int > int ).
thf(func_def_347,type,
groups705719431365010083at_int: ( nat > int ) > set_nat > int ).
thf(func_def_348,type,
groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_349,type,
groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).
thf(func_def_350,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(func_def_351,type,
the_int: ( int > $o ) > int ).
thf(func_def_352,type,
the_real: ( real > $o ) > real ).
thf(func_def_353,type,
if_Code_integer: $o > code_integer > code_integer > code_integer ).
thf(func_def_354,type,
if_complex: $o > complex > complex > complex ).
thf(func_def_355,type,
if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).
thf(func_def_356,type,
if_int: $o > int > int > int ).
thf(func_def_357,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(func_def_358,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(func_def_359,type,
if_nat: $o > nat > nat > nat ).
thf(func_def_360,type,
if_num: $o > num > num > num ).
thf(func_def_361,type,
if_option_nat: $o > option_nat > option_nat > option_nat ).
thf(func_def_362,type,
if_option_num: $o > option_num > option_num > option_num ).
thf(func_def_363,type,
if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).
thf(func_def_364,type,
if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_365,type,
if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).
thf(func_def_366,type,
if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_367,type,
if_rat: $o > rat > rat > rat ).
thf(func_def_368,type,
if_real: $o > real > real > real ).
thf(func_def_369,type,
if_set_int: $o > set_int > set_int > set_int ).
thf(func_def_370,type,
if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_371,type,
abs_Integ: product_prod_nat_nat > int ).
thf(func_def_372,type,
rep_Integ: int > product_prod_nat_nat ).
thf(func_def_373,type,
int_ge_less_than: int > set_Pr958786334691620121nt_int ).
thf(func_def_374,type,
int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
thf(func_def_375,type,
nat2: int > nat ).
thf(func_def_376,type,
power_int_real: real > int > real ).
thf(func_def_377,type,
ring_1_Ints_real: set_real ).
thf(func_def_378,type,
ring_18347121197199848620nteger: int > code_integer ).
thf(func_def_379,type,
ring_17405671764205052669omplex: int > complex ).
thf(func_def_380,type,
ring_1_of_int_int: int > int ).
thf(func_def_381,type,
ring_1_of_int_rat: int > rat ).
thf(func_def_382,type,
ring_1_of_int_real: int > real ).
thf(func_def_383,type,
inf_in1870772243966228564d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_384,type,
inf_inf_nat: nat > nat > nat ).
thf(func_def_385,type,
semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(func_def_386,type,
sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_387,type,
sup_sup_nat: nat > nat > nat ).
thf(func_def_388,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_389,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(func_def_390,type,
bfun_nat_real: ( nat > real ) > filter_nat > $o ).
thf(func_def_391,type,
at_infinity_real: filter_real ).
thf(func_def_392,type,
append_int: list_int > list_int > list_int ).
thf(func_def_393,type,
append_nat: list_nat > list_nat > list_nat ).
thf(func_def_394,type,
drop_nat: nat > list_nat > list_nat ).
thf(func_def_395,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(func_def_396,type,
cons_int: int > list_int > list_int ).
thf(func_def_397,type,
cons_nat: nat > list_nat > list_nat ).
thf(func_def_398,type,
nil_int: list_int ).
thf(func_def_399,type,
nil_nat: list_nat ).
thf(func_def_400,type,
hd_nat: list_nat > nat ).
thf(func_def_401,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(func_def_402,type,
set_o2: list_o > set_o ).
thf(func_def_403,type,
set_complex2: list_complex > set_complex ).
thf(func_def_404,type,
set_int2: list_int > set_int ).
thf(func_def_405,type,
set_nat2: list_nat > set_nat ).
thf(func_def_406,type,
set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_407,type,
set_real2: list_real > set_real ).
thf(func_def_408,type,
set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
thf(func_def_409,type,
size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
thf(func_def_410,type,
tl_nat: list_nat > list_nat ).
thf(func_def_411,type,
list_update_o: list_o > nat > $o > list_o ).
thf(func_def_412,type,
list_update_complex: list_complex > nat > complex > list_complex ).
thf(func_def_413,type,
list_update_int: list_int > nat > int > list_int ).
thf(func_def_414,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(func_def_415,type,
list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
thf(func_def_416,type,
list_update_real: list_real > nat > real > list_real ).
thf(func_def_417,type,
list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_418,type,
nth_o: list_o > nat > $o ).
thf(func_def_419,type,
nth_Code_integer: list_Code_integer > nat > code_integer ).
thf(func_def_420,type,
nth_complex: list_complex > nat > complex ).
thf(func_def_421,type,
nth_int: list_int > nat > int ).
thf(func_def_422,type,
nth_nat: list_nat > nat > nat ).
thf(func_def_423,type,
nth_num: list_num > nat > num ).
thf(func_def_424,type,
nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).
thf(func_def_425,type,
nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).
thf(func_def_426,type,
nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat ).
thf(func_def_427,type,
nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).
thf(func_def_428,type,
nth_Pr8522763379788166057eger_o: list_P8526636022914148096eger_o > nat > produc6271795597528267376eger_o ).
thf(func_def_429,type,
nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).
thf(func_def_430,type,
nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num ).
thf(func_def_431,type,
nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
thf(func_def_432,type,
nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).
thf(func_def_433,type,
nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).
thf(func_def_434,type,
nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
thf(func_def_435,type,
nth_real: list_real > nat > real ).
thf(func_def_436,type,
nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_437,type,
product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
thf(func_def_438,type,
product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
thf(func_def_439,type,
product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
thf(func_def_440,type,
product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
thf(func_def_441,type,
produc3607205314601156340eger_o: list_Code_integer > list_o > list_P8526636022914148096eger_o ).
thf(func_def_442,type,
product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
thf(func_def_443,type,
produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
thf(func_def_444,type,
product_num_num: list_num > list_num > list_P3744719386663036955um_num ).
thf(func_def_445,type,
product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
thf(func_def_446,type,
produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
thf(func_def_447,type,
produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
thf(func_def_448,type,
produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
thf(func_def_449,type,
replicate_o: nat > $o > list_o ).
thf(func_def_450,type,
replicate_complex: nat > complex > list_complex ).
thf(func_def_451,type,
replicate_int: nat > int > list_int ).
thf(func_def_452,type,
replicate_nat: nat > nat > list_nat ).
thf(func_def_453,type,
replic4235873036481779905at_nat: nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
thf(func_def_454,type,
replicate_real: nat > real > list_real ).
thf(func_def_455,type,
replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_456,type,
sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
thf(func_def_457,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(func_def_458,type,
take_nat: nat > list_nat > list_nat ).
thf(func_def_459,type,
upt: nat > nat > list_nat ).
thf(func_def_460,type,
upto: int > int > list_int ).
thf(func_def_461,type,
upto_aux: int > int > list_int > list_int ).
thf(func_def_462,type,
upto_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_463,type,
suc: nat > nat ).
thf(func_def_464,type,
compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(func_def_465,type,
case_nat_o: $o > ( nat > $o ) > nat > $o ).
thf(func_def_466,type,
case_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(func_def_467,type,
case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
thf(func_def_468,type,
pred: nat > nat ).
thf(func_def_469,type,
semiri4939895301339042750nteger: nat > code_integer ).
thf(func_def_470,type,
semiri8010041392384452111omplex: nat > complex ).
thf(func_def_471,type,
semiri4216267220026989637d_enat: nat > extended_enat ).
thf(func_def_472,type,
semiri1314217659103216013at_int: nat > int ).
thf(func_def_473,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(func_def_474,type,
semiri681578069525770553at_rat: nat > rat ).
thf(func_def_475,type,
semiri5074537144036343181t_real: nat > real ).
thf(func_def_476,type,
semiri2816024913162550771omplex: ( complex > complex ) > nat > complex > complex ).
thf(func_def_477,type,
semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).
thf(func_def_478,type,
semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).
thf(func_def_479,type,
semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).
thf(func_def_480,type,
semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).
thf(func_def_481,type,
size_size_list_o: list_o > nat ).
thf(func_def_482,type,
size_s3445333598471063425nteger: list_Code_integer > nat ).
thf(func_def_483,type,
size_s3451745648224563538omplex: list_complex > nat ).
thf(func_def_484,type,
size_size_list_int: list_int > nat ).
thf(func_def_485,type,
size_size_list_nat: list_nat > nat ).
thf(func_def_486,type,
size_size_list_num: list_num > nat ).
thf(func_def_487,type,
size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).
thf(func_def_488,type,
size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).
thf(func_def_489,type,
size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).
thf(func_def_490,type,
size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).
thf(func_def_491,type,
size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).
thf(func_def_492,type,
size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).
thf(func_def_493,type,
size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat ).
thf(func_def_494,type,
size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).
thf(func_def_495,type,
size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).
thf(func_def_496,type,
size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).
thf(func_def_497,type,
size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).
thf(func_def_498,type,
size_size_list_real: list_real > nat ).
thf(func_def_499,type,
size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
thf(func_def_500,type,
size_size_num: num > nat ).
thf(func_def_501,type,
size_size_option_nat: option_nat > nat ).
thf(func_def_502,type,
size_size_option_num: option_num > nat ).
thf(func_def_503,type,
size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).
thf(func_def_504,type,
size_size_VEBT_VEBT: vEBT_VEBT > nat ).
thf(func_def_505,type,
nat_list_encode: list_nat > nat ).
thf(func_def_506,type,
nat_list_encode_rel: list_nat > list_nat > $o ).
thf(func_def_507,type,
nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
thf(func_def_508,type,
nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_509,type,
nat_prod_encode: product_prod_nat_nat > nat ).
thf(func_def_510,type,
nat_set_decode: nat > set_nat ).
thf(func_def_511,type,
nat_set_encode: set_nat > nat ).
thf(func_def_512,type,
nat_triangle: nat > nat ).
thf(func_def_513,type,
root: nat > real > real ).
thf(func_def_514,type,
sqrt: real > real ).
thf(func_def_515,type,
bitM: num > num ).
thf(func_def_516,type,
inc: num > num ).
thf(func_def_517,type,
neg_nu8804712462038260780nteger: code_integer > code_integer ).
thf(func_def_518,type,
neg_nu7009210354673126013omplex: complex > complex ).
thf(func_def_519,type,
neg_numeral_dbl_int: int > int ).
thf(func_def_520,type,
neg_numeral_dbl_rat: rat > rat ).
thf(func_def_521,type,
neg_numeral_dbl_real: real > real ).
thf(func_def_522,type,
neg_nu7757733837767384882nteger: code_integer > code_integer ).
thf(func_def_523,type,
neg_nu6511756317524482435omplex: complex > complex ).
thf(func_def_524,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(func_def_525,type,
neg_nu3179335615603231917ec_rat: rat > rat ).
thf(func_def_526,type,
neg_nu6075765906172075777c_real: real > real ).
thf(func_def_527,type,
neg_nu5831290666863070958nteger: code_integer > code_integer ).
thf(func_def_528,type,
neg_nu8557863876264182079omplex: complex > complex ).
thf(func_def_529,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(func_def_530,type,
neg_nu5219082963157363817nc_rat: rat > rat ).
thf(func_def_531,type,
neg_nu8295874005876285629c_real: real > real ).
thf(func_def_532,type,
neg_numeral_sub_int: num > num > int ).
thf(func_def_533,type,
bit0: num > num ).
thf(func_def_534,type,
bit1: num > num ).
thf(func_def_535,type,
one: num ).
thf(func_def_536,type,
case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
thf(func_def_537,type,
size_num: num > nat ).
thf(func_def_538,type,
num_of_nat: nat > num ).
thf(func_def_539,type,
numera6620942414471956472nteger: num > code_integer ).
thf(func_def_540,type,
numera6690914467698888265omplex: num > complex ).
thf(func_def_541,type,
numera1916890842035813515d_enat: num > extended_enat ).
thf(func_def_542,type,
numeral_numeral_int: num > int ).
thf(func_def_543,type,
numeral_numeral_nat: num > nat ).
thf(func_def_544,type,
numeral_numeral_rat: num > rat ).
thf(func_def_545,type,
numeral_numeral_real: num > real ).
thf(func_def_546,type,
pow: num > num > num ).
thf(func_def_547,type,
pred_numeral: num > nat ).
thf(func_def_548,type,
sqr: num > num ).
thf(func_def_549,type,
none_nat: option_nat ).
thf(func_def_550,type,
none_num: option_num ).
thf(func_def_551,type,
none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
thf(func_def_552,type,
some_nat: nat > option_nat ).
thf(func_def_553,type,
some_num: num > option_num ).
thf(func_def_554,type,
some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
thf(func_def_555,type,
case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
thf(func_def_556,type,
case_option_int_num: int > ( num > int ) > option_num > int ).
thf(func_def_557,type,
case_option_num_num: num > ( num > num ) > option_num > num ).
thf(func_def_558,type,
case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).
thf(func_def_559,type,
map_option_num_num: ( num > num ) > option_num > option_num ).
thf(func_def_560,type,
size_option_nat: ( nat > nat ) > option_nat > nat ).
thf(func_def_561,type,
size_option_num: ( num > nat ) > option_num > nat ).
thf(func_def_562,type,
size_o8335143837870341156at_nat: ( product_prod_nat_nat > nat ) > option4927543243414619207at_nat > nat ).
thf(func_def_563,type,
the_nat: option_nat > nat ).
thf(func_def_564,type,
the_num: option_num > num ).
thf(func_def_565,type,
the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat ).
thf(func_def_566,type,
bot_bo4731626569425807221er_o_o: code_integer > $o > $o ).
thf(func_def_567,type,
bot_bot_int_int_o: int > int > $o ).
thf(func_def_568,type,
bot_bot_nat_nat_o: nat > nat > $o ).
thf(func_def_569,type,
bot_bot_nat_num_o: nat > num > $o ).
thf(func_def_570,type,
bot_bot_num_num_o: num > num > $o ).
thf(func_def_571,type,
bot_bo4199563552545308370d_enat: extended_enat ).
thf(func_def_572,type,
bot_bot_nat: nat ).
thf(func_def_573,type,
bot_bot_set_complex: set_complex ).
thf(func_def_574,type,
bot_bot_set_int: set_int ).
thf(func_def_575,type,
bot_bot_set_nat: set_nat ).
thf(func_def_576,type,
bot_bot_set_num: set_num ).
thf(func_def_577,type,
bot_bo5379713665208646970eger_o: set_Pr448751882837621926eger_o ).
thf(func_def_578,type,
bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).
thf(func_def_579,type,
bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).
thf(func_def_580,type,
bot_bo7038385379056416535at_num: set_Pr6200539531224447659at_num ).
thf(func_def_581,type,
bot_bo9056780473022590049um_num: set_Pr8218934625190621173um_num ).
thf(func_def_582,type,
bot_bot_set_rat: set_rat ).
thf(func_def_583,type,
bot_bot_set_real: set_real ).
thf(func_def_584,type,
bot_bot_set_set_int: set_set_int ).
thf(func_def_585,type,
ord_le6747313008572928689nteger: code_integer > code_integer > $o ).
thf(func_def_586,type,
ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
thf(func_def_587,type,
ord_less_int: int > int > $o ).
thf(func_def_588,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_589,type,
ord_less_num: num > num > $o ).
thf(func_def_590,type,
ord_less_rat: rat > rat > $o ).
thf(func_def_591,type,
ord_less_real: real > real > $o ).
thf(func_def_592,type,
ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o ).
thf(func_def_593,type,
ord_less_set_complex: set_complex > set_complex > $o ).
thf(func_def_594,type,
ord_less_set_int: set_int > set_int > $o ).
thf(func_def_595,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(func_def_596,type,
ord_less_set_num: set_num > set_num > $o ).
thf(func_def_597,type,
ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_598,type,
ord_less_set_rat: set_rat > set_rat > $o ).
thf(func_def_599,type,
ord_less_set_real: set_real > set_real > $o ).
thf(func_def_600,type,
ord_less_set_set_int: set_set_int > set_set_int > $o ).
thf(func_def_601,type,
ord_le2162486998276636481er_o_o: ( code_integer > $o > $o ) > ( code_integer > $o > $o ) > $o ).
thf(func_def_602,type,
ord_le4573692005234683329plex_o: ( complex > $o ) > ( complex > $o ) > $o ).
thf(func_def_603,type,
ord_le6741204236512500942_int_o: ( int > int > $o ) > ( int > int > $o ) > $o ).
thf(func_def_604,type,
ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_605,type,
ord_le6558929396352911974_nat_o: ( list_nat > list_nat > $o ) > ( list_nat > list_nat > $o ) > $o ).
thf(func_def_606,type,
ord_le1520216061033275535_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).
thf(func_def_607,type,
ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(func_def_608,type,
ord_le3404735783095501756_num_o: ( nat > num > $o ) > ( nat > num > $o ) > $o ).
thf(func_def_609,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_610,type,
ord_le6124364862034508274_num_o: ( num > num > $o ) > ( num > num > $o ) > $o ).
thf(func_def_611,type,
ord_le1598226405681992910_int_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > $o ).
thf(func_def_612,type,
ord_le8369615600986905444_int_o: ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
thf(func_def_613,type,
ord_le5604493270027003598_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).
thf(func_def_614,type,
ord_le704812498762024988_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
thf(func_def_615,type,
ord_le2556027599737686990_num_o: ( product_prod_num_num > product_prod_num_num > $o ) > ( product_prod_num_num > product_prod_num_num > $o ) > $o ).
thf(func_def_616,type,
ord_le2239182809043710856_num_o: ( product_prod_num_num > $o ) > ( product_prod_num_num > $o ) > $o ).
thf(func_def_617,type,
ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(func_def_618,type,
ord_le3102999989581377725nteger: code_integer > code_integer > $o ).
thf(func_def_619,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
thf(func_def_620,type,
ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).
thf(func_def_621,type,
ord_le4104064031414453916r_real: filter_real > filter_real > $o ).
thf(func_def_622,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_623,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_624,type,
ord_less_eq_num: num > num > $o ).
thf(func_def_625,type,
ord_less_eq_rat: rat > rat > $o ).
thf(func_def_626,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_627,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(func_def_628,type,
ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o ).
thf(func_def_629,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(func_def_630,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(func_def_631,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(func_def_632,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(func_def_633,type,
ord_less_eq_set_num: set_num > set_num > $o ).
thf(func_def_634,type,
ord_le8980329558974975238eger_o: set_Pr448751882837621926eger_o > set_Pr448751882837621926eger_o > $o ).
thf(func_def_635,type,
ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).
thf(func_def_636,type,
ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_637,type,
ord_le8085105155179020875at_num: set_Pr6200539531224447659at_num > set_Pr6200539531224447659at_num > $o ).
thf(func_def_638,type,
ord_le880128212290418581um_num: set_Pr8218934625190621173um_num > set_Pr8218934625190621173um_num > $o ).
thf(func_def_639,type,
ord_less_eq_set_rat: set_rat > set_rat > $o ).
thf(func_def_640,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(func_def_641,type,
ord_le4403425263959731960et_int: set_set_int > set_set_int > $o ).
thf(func_def_642,type,
ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).
thf(func_def_643,type,
ord_max_Code_integer: code_integer > code_integer > code_integer ).
thf(func_def_644,type,
ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_645,type,
ord_max_int: int > int > int ).
thf(func_def_646,type,
ord_max_nat: nat > nat > nat ).
thf(func_def_647,type,
ord_max_num: num > num > num ).
thf(func_def_648,type,
ord_max_rat: rat > rat > rat ).
thf(func_def_649,type,
ord_max_real: real > real > real ).
thf(func_def_650,type,
ord_max_set_int: set_int > set_int > set_int ).
thf(func_def_651,type,
ord_mi8085742599997312461d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_652,type,
ord_min_nat: nat > nat > nat ).
thf(func_def_653,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(func_def_654,type,
order_9091379641038594480t_real: ( nat > real ) > $o ).
thf(func_def_655,type,
order_mono_nat_nat: ( nat > nat ) > $o ).
thf(func_def_656,type,
order_mono_nat_real: ( nat > real ) > $o ).
thf(func_def_657,type,
order_5726023648592871131at_nat: ( nat > nat ) > $o ).
thf(func_def_658,type,
top_top_set_o: set_o ).
thf(func_def_659,type,
top_top_set_nat: set_nat ).
thf(func_def_660,type,
top_to1996260823553986621t_unit: set_Product_unit ).
thf(func_def_661,type,
top_top_set_real: set_real ).
thf(func_def_662,type,
top_top_set_char: set_char ).
thf(func_def_663,type,
power_8256067586552552935nteger: code_integer > nat > code_integer ).
thf(func_def_664,type,
power_power_complex: complex > nat > complex ).
thf(func_def_665,type,
power_power_int: int > nat > int ).
thf(func_def_666,type,
power_power_nat: nat > nat > nat ).
thf(func_def_667,type,
power_power_rat: rat > nat > rat ).
thf(func_def_668,type,
power_power_real: real > nat > real ).
thf(func_def_669,type,
produc4035269172776083154on_nat: ( nat > nat > $o ) > produc4953844613479565601on_nat > produc2233624965454879586on_nat ).
thf(func_def_670,type,
produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_nat ).
thf(func_def_671,type,
produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).
thf(func_def_672,type,
produc851828971589881931at_num: ( nat > num > num ) > produc2963631642982155120at_num > produc3368934014287244435at_num ).
thf(func_def_673,type,
produc3576312749637752826on_num: ( num > num > $o ) > produc3447558737645232053on_num > produc7036089656553540234on_num ).
thf(func_def_674,type,
produc5778274026573060048on_num: ( num > num > num ) > produc3447558737645232053on_num > produc1193250871479095198on_num ).
thf(func_def_675,type,
produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).
thf(func_def_676,type,
produc2899441246263362727at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > produc6121120109295599847at_nat > produc5542196010084753463at_nat ).
thf(func_def_677,type,
product_Pair_o_o: $o > $o > product_prod_o_o ).
thf(func_def_678,type,
product_Pair_o_int: $o > int > product_prod_o_int ).
thf(func_def_679,type,
product_Pair_o_nat: $o > nat > product_prod_o_nat ).
thf(func_def_680,type,
produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).
thf(func_def_681,type,
produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).
thf(func_def_682,type,
produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_683,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(func_def_684,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(func_def_685,type,
product_Pair_nat_num: nat > num > product_prod_nat_num ).
thf(func_def_686,type,
produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).
thf(func_def_687,type,
produc1195630363706982562at_num: nat > product_prod_nat_num > produc2963631642982155120at_num ).
thf(func_def_688,type,
product_Pair_num_num: num > num > product_prod_num_num ).
thf(func_def_689,type,
produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat ).
thf(func_def_690,type,
produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num ).
thf(func_def_691,type,
produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat ).
thf(func_def_692,type,
produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).
thf(func_def_693,type,
produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).
thf(func_def_694,type,
produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
thf(func_def_695,type,
produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).
thf(func_def_696,type,
produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_697,type,
produc7828578312038201481er_o_o: ( code_integer > $o > $o ) > produc6271795597528267376eger_o > $o ).
thf(func_def_698,type,
produc1043322548047392435omplex: ( code_integer > $o > set_complex ) > produc6271795597528267376eger_o > set_complex ).
thf(func_def_699,type,
produc1253318751659547953et_int: ( code_integer > $o > set_int ) > produc6271795597528267376eger_o > set_int ).
thf(func_def_700,type,
produc5431169771168744661et_nat: ( code_integer > $o > set_nat ) > produc6271795597528267376eger_o > set_nat ).
thf(func_def_701,type,
produc242741666403216561t_real: ( code_integer > $o > set_real ) > produc6271795597528267376eger_o > set_real ).
thf(func_def_702,type,
produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
thf(func_def_703,type,
produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
thf(func_def_704,type,
produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).
thf(func_def_705,type,
produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
thf(func_def_706,type,
produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_707,type,
produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
thf(func_def_708,type,
produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
thf(func_def_709,type,
produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_710,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_711,type,
produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
thf(func_def_712,type,
produc1917071388513777916omplex: ( nat > nat > complex ) > product_prod_nat_nat > complex ).
thf(func_def_713,type,
produc6840382203811409530at_int: ( nat > nat > int ) > product_prod_nat_nat > int ).
thf(func_def_714,type,
produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
thf(func_def_715,type,
produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_716,type,
produc6207742614233964070at_rat: ( nat > nat > rat ) > product_prod_nat_nat > rat ).
thf(func_def_717,type,
produc1703576794950452218t_real: ( nat > nat > real ) > product_prod_nat_nat > real ).
thf(func_def_718,type,
produc4927758841916487424_num_o: ( nat > num > $o ) > product_prod_nat_num > $o ).
thf(func_def_719,type,
produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
thf(func_def_720,type,
produc6231982587499038204omplex: ( nat > num > set_complex ) > product_prod_nat_num > set_complex ).
thf(func_def_721,type,
produc1435849484188172666t_real: ( nat > num > set_real ) > product_prod_nat_num > set_real ).
thf(func_def_722,type,
produc5703948589228662326_num_o: ( num > num > $o ) > product_prod_num_num > $o ).
thf(func_def_723,type,
produc2866383454006189126omplex: ( num > num > set_complex ) > product_prod_num_num > set_complex ).
thf(func_def_724,type,
produc6406642877701697732et_int: ( num > num > set_int ) > product_prod_num_num > set_int ).
thf(func_def_725,type,
produc1361121860356118632et_nat: ( num > num > set_nat ) > product_prod_num_num > set_nat ).
thf(func_def_726,type,
produc8296048397933160132t_real: ( num > num > set_real ) > product_prod_num_num > set_real ).
thf(func_def_727,type,
product_fst_int_int: product_prod_int_int > int ).
thf(func_def_728,type,
product_fst_nat_nat: product_prod_nat_nat > nat ).
thf(func_def_729,type,
product_snd_int_int: product_prod_int_int > int ).
thf(func_def_730,type,
product_snd_nat_nat: product_prod_nat_nat > nat ).
thf(func_def_731,type,
abs_Rat: product_prod_int_int > rat ).
thf(func_def_732,type,
fract: int > int > rat ).
thf(func_def_733,type,
frct: product_prod_int_int > rat ).
thf(func_def_734,type,
rep_Rat: rat > product_prod_int_int ).
thf(func_def_735,type,
field_5140801741446780682s_real: set_real ).
thf(func_def_736,type,
field_7254667332652039916t_real: rat > real ).
thf(func_def_737,type,
normalize: product_prod_int_int > product_prod_int_int ).
thf(func_def_738,type,
of_int: int > rat ).
thf(func_def_739,type,
positive: rat > $o ).
thf(func_def_740,type,
quotient_of: rat > product_prod_int_int ).
thf(func_def_741,type,
ratrel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_742,type,
real_V2521375963428798218omplex: set_complex ).
thf(func_def_743,type,
real_V5970128139526366754l_real: ( real > real ) > $o ).
thf(func_def_744,type,
real_V3694042436643373181omplex: complex > complex > real ).
thf(func_def_745,type,
real_V975177566351809787t_real: real > real > real ).
thf(func_def_746,type,
real_V1022390504157884413omplex: complex > real ).
thf(func_def_747,type,
real_V7735802525324610683m_real: real > real ).
thf(func_def_748,type,
real_V4546457046886955230omplex: real > complex ).
thf(func_def_749,type,
real_V1803761363581548252l_real: real > real ).
thf(func_def_750,type,
real_V2046097035970521341omplex: real > complex > complex ).
thf(func_def_751,type,
real_V1485227260804924795R_real: real > real > real ).
thf(func_def_752,type,
divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
thf(func_def_753,type,
divide1717551699836669952omplex: complex > complex > complex ).
thf(func_def_754,type,
divide_divide_int: int > int > int ).
thf(func_def_755,type,
divide_divide_nat: nat > nat > nat ).
thf(func_def_756,type,
divide_divide_rat: rat > rat > rat ).
thf(func_def_757,type,
divide_divide_real: real > real > real ).
thf(func_def_758,type,
dvd_dvd_Code_integer: code_integer > code_integer > $o ).
thf(func_def_759,type,
dvd_dvd_complex: complex > complex > $o ).
thf(func_def_760,type,
dvd_dvd_int: int > int > $o ).
thf(func_def_761,type,
dvd_dvd_nat: nat > nat > $o ).
thf(func_def_762,type,
dvd_dvd_rat: rat > rat > $o ).
thf(func_def_763,type,
dvd_dvd_real: real > real > $o ).
thf(func_def_764,type,
modulo364778990260209775nteger: code_integer > code_integer > code_integer ).
thf(func_def_765,type,
modulo_modulo_int: int > int > int ).
thf(func_def_766,type,
modulo_modulo_nat: nat > nat > nat ).
thf(func_def_767,type,
zero_n356916108424825756nteger: $o > code_integer ).
thf(func_def_768,type,
zero_n1201886186963655149omplex: $o > complex ).
thf(func_def_769,type,
zero_n2684676970156552555ol_int: $o > int ).
thf(func_def_770,type,
zero_n2687167440665602831ol_nat: $o > nat ).
thf(func_def_771,type,
zero_n2052037380579107095ol_rat: $o > rat ).
thf(func_def_772,type,
zero_n3304061248610475627l_real: $o > real ).
thf(func_def_773,type,
suminf_complex: ( nat > complex ) > complex ).
thf(func_def_774,type,
suminf_int: ( nat > int ) > int ).
thf(func_def_775,type,
suminf_nat: ( nat > nat ) > nat ).
thf(func_def_776,type,
suminf_real: ( nat > real ) > real ).
thf(func_def_777,type,
summable_complex: ( nat > complex ) > $o ).
thf(func_def_778,type,
summable_int: ( nat > int ) > $o ).
thf(func_def_779,type,
summable_nat: ( nat > nat ) > $o ).
thf(func_def_780,type,
summable_real: ( nat > real ) > $o ).
thf(func_def_781,type,
sums_complex: ( nat > complex ) > complex > $o ).
thf(func_def_782,type,
sums_int: ( nat > int ) > int > $o ).
thf(func_def_783,type,
sums_nat: ( nat > nat ) > nat > $o ).
thf(func_def_784,type,
sums_real: ( nat > real ) > real > $o ).
thf(func_def_785,type,
collect_Code_integer: ( code_integer > $o ) > set_Code_integer ).
thf(func_def_786,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(func_def_787,type,
collect_int: ( int > $o ) > set_int ).
thf(func_def_788,type,
collect_list_o: ( list_o > $o ) > set_list_o ).
thf(func_def_789,type,
collect_list_complex: ( list_complex > $o ) > set_list_complex ).
thf(func_def_790,type,
collect_list_int: ( list_int > $o ) > set_list_int ).
thf(func_def_791,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(func_def_792,type,
collec5608196760682091941T_VEBT: ( list_VEBT_VEBT > $o ) > set_list_VEBT_VEBT ).
thf(func_def_793,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_794,type,
collect_num: ( num > $o ) > set_num ).
thf(func_def_795,type,
collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
thf(func_def_796,type,
collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
thf(func_def_797,type,
collect_rat: ( rat > $o ) > set_rat ).
thf(func_def_798,type,
collect_real: ( real > $o ) > set_real ).
thf(func_def_799,type,
collect_set_complex: ( set_complex > $o ) > set_set_complex ).
thf(func_def_800,type,
collect_set_int: ( set_int > $o ) > set_set_int ).
thf(func_def_801,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(func_def_802,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(func_def_803,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(func_def_804,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_805,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(func_def_806,type,
image_nat_char: ( nat > char ) > set_nat > set_char ).
thf(func_def_807,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(func_def_808,type,
image_char_nat: ( char > nat ) > set_char > set_nat ).
thf(func_def_809,type,
image_VEBT_VEBT_nat: ( vEBT_VEBT > nat ) > set_VEBT_VEBT > set_nat ).
thf(func_def_810,type,
insert_int: int > set_int > set_int ).
thf(func_def_811,type,
insert_nat: nat > set_nat > set_nat ).
thf(func_def_812,type,
insert_real: real > set_real > set_real ).
thf(func_def_813,type,
insert_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).
thf(func_def_814,type,
set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).
thf(func_def_815,type,
set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).
thf(func_def_816,type,
set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
thf(func_def_817,type,
set_fo8365102181078989356at_num: ( nat > num > num ) > nat > nat > num > num ).
thf(func_def_818,type,
set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).
thf(func_def_819,type,
set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).
thf(func_def_820,type,
set_fo3699595496184130361el_nat: produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ).
thf(func_def_821,type,
set_fo256927282339908995el_num: produc3368934014287244435at_num > produc3368934014287244435at_num > $o ).
thf(func_def_822,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(func_def_823,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(func_def_824,type,
set_or7049704709247886629st_num: num > num > set_num ).
thf(func_def_825,type,
set_or633870826150836451st_rat: rat > rat > set_rat ).
thf(func_def_826,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(func_def_827,type,
set_or370866239135849197et_int: set_int > set_int > set_set_int ).
thf(func_def_828,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(func_def_829,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(func_def_830,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(func_def_831,type,
set_ord_atLeast_real: real > set_real ).
thf(func_def_832,type,
set_ord_atMost_int: int > set_int ).
thf(func_def_833,type,
set_ord_atMost_nat: nat > set_nat ).
thf(func_def_834,type,
set_ord_atMost_num: num > set_num ).
thf(func_def_835,type,
set_ord_atMost_rat: rat > set_rat ).
thf(func_def_836,type,
set_ord_atMost_real: real > set_real ).
thf(func_def_837,type,
set_or58775011639299419et_int: set_int > set_set_int ).
thf(func_def_838,type,
set_or6656581121297822940st_int: int > int > set_int ).
thf(func_def_839,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(func_def_840,type,
set_or5832277885323065728an_int: int > int > set_int ).
thf(func_def_841,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(func_def_842,type,
set_or1633881224788618240n_real: real > real > set_real ).
thf(func_def_843,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(func_def_844,type,
set_or5849166863359141190n_real: real > set_real ).
thf(func_def_845,type,
set_ord_lessThan_int: int > set_int ).
thf(func_def_846,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(func_def_847,type,
set_ord_lessThan_num: num > set_num ).
thf(func_def_848,type,
set_ord_lessThan_rat: rat > set_rat ).
thf(func_def_849,type,
set_or5984915006950818249n_real: real > set_real ).
thf(func_def_850,type,
ascii_of: char > char ).
thf(func_def_851,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(func_def_852,type,
comm_s629917340098488124ar_nat: char > nat ).
thf(func_def_853,type,
integer_of_char: char > code_integer ).
thf(func_def_854,type,
unique3096191561947761185of_nat: nat > char ).
thf(func_def_855,type,
topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
thf(func_def_856,type,
topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
thf(func_def_857,type,
topolo4899668324122417113eq_int: ( nat > int ) > $o ).
thf(func_def_858,type,
topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).
thf(func_def_859,type,
topolo1459490580787246023eq_num: ( nat > num ) > $o ).
thf(func_def_860,type,
topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).
thf(func_def_861,type,
topolo6980174941875973593q_real: ( nat > real ) > $o ).
thf(func_def_862,type,
topolo3100542954746470799et_int: ( nat > set_int ) > $o ).
thf(func_def_863,type,
topolo2177554685111907308n_real: real > set_real > filter_real ).
thf(func_def_864,type,
topolo2815343760600316023s_real: real > filter_real ).
thf(func_def_865,type,
topolo4055970368930404560y_real: ( nat > real ) > $o ).
thf(func_def_866,type,
arccos: real > real ).
thf(func_def_867,type,
arcosh_real: real > real ).
thf(func_def_868,type,
arcsin: real > real ).
thf(func_def_869,type,
arctan: real > real ).
thf(func_def_870,type,
arsinh_real: real > real ).
thf(func_def_871,type,
artanh_real: real > real ).
thf(func_def_872,type,
cos_complex: complex > complex ).
thf(func_def_873,type,
cos_real: real > real ).
thf(func_def_874,type,
cos_coeff: nat > real ).
thf(func_def_875,type,
cosh_real: real > real ).
thf(func_def_876,type,
cot_real: real > real ).
thf(func_def_877,type,
diffs_complex: ( nat > complex ) > nat > complex ).
thf(func_def_878,type,
diffs_int: ( nat > int ) > nat > int ).
thf(func_def_879,type,
diffs_rat: ( nat > rat ) > nat > rat ).
thf(func_def_880,type,
diffs_real: ( nat > real ) > nat > real ).
thf(func_def_881,type,
exp_complex: complex > complex ).
thf(func_def_882,type,
exp_real: real > real ).
thf(func_def_883,type,
ln_ln_real: real > real ).
thf(func_def_884,type,
log: real > real > real ).
thf(func_def_885,type,
pi: real ).
thf(func_def_886,type,
powr_real: real > real > real ).
thf(func_def_887,type,
sin_complex: complex > complex ).
thf(func_def_888,type,
sin_real: real > real ).
thf(func_def_889,type,
sin_coeff: nat > real ).
thf(func_def_890,type,
sinh_real: real > real ).
thf(func_def_891,type,
tan_complex: complex > complex ).
thf(func_def_892,type,
tan_real: real > real ).
thf(func_def_893,type,
tanh_complex: complex > complex ).
thf(func_def_894,type,
tanh_real: real > real ).
thf(func_def_895,type,
transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_896,type,
transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_897,type,
vEBT_T_i_n_s_e_r_t: vEBT_VEBT > nat > nat ).
thf(func_def_898,type,
vEBT_T_i_n_s_e_r_t2: vEBT_VEBT > nat > nat ).
thf(func_def_899,type,
vEBT_T5076183648494686801_t_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_900,type,
vEBT_T9217963907923527482_t_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_901,type,
vEBT_T_m_a_x_t: vEBT_VEBT > nat ).
thf(func_def_902,type,
vEBT_T_m_a_x_t_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_903,type,
vEBT_T_m_e_m_b_e_r: vEBT_VEBT > nat > nat ).
thf(func_def_904,type,
vEBT_T_m_e_m_b_e_r2: vEBT_VEBT > nat > nat ).
thf(func_def_905,type,
vEBT_T8099345112685741742_r_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_906,type,
vEBT_T5837161174952499735_r_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_907,type,
vEBT_T_m_i_n_N_u_l_l: vEBT_VEBT > nat ).
thf(func_def_908,type,
vEBT_T5462971552011256508_l_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_909,type,
vEBT_T_m_i_n_t: vEBT_VEBT > nat ).
thf(func_def_910,type,
vEBT_T_m_i_n_t_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_911,type,
vEBT_Leaf: $o > $o > vEBT_VEBT ).
thf(func_def_912,type,
vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_913,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(func_def_914,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(func_def_915,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(func_def_916,type,
vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
thf(func_def_917,type,
vEBT_VEBT_low: nat > nat > nat ).
thf(func_def_918,type,
vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
thf(func_def_919,type,
vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_920,type,
vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
thf(func_def_921,type,
vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_922,type,
vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
thf(func_def_923,type,
vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_924,type,
vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
thf(func_def_925,type,
vEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(func_def_926,type,
vEBT_vebt_buildup: nat > vEBT_VEBT ).
thf(func_def_927,type,
vEBT_v4011308405150292612up_rel: nat > nat > $o ).
thf(func_def_928,type,
vEBT_VEBT_height: vEBT_VEBT > nat ).
thf(func_def_929,type,
vEBT_VEBT_height_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_930,type,
vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_931,type,
vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_932,type,
vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
thf(func_def_933,type,
vEBT_VEBT_minNull: vEBT_VEBT > $o ).
thf(func_def_934,type,
vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_935,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(func_def_936,type,
vEBT_vebt_member: vEBT_VEBT > nat > $o ).
thf(func_def_937,type,
vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_938,type,
vEBT_VEBT_add: option_nat > option_nat > option_nat ).
thf(func_def_939,type,
vEBT_VEBT_greater: option_nat > option_nat > $o ).
thf(func_def_940,type,
vEBT_VEBT_less: option_nat > option_nat > $o ).
thf(func_def_941,type,
vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
thf(func_def_942,type,
vEBT_VEBT_max_in_set: set_nat > nat > $o ).
thf(func_def_943,type,
vEBT_VEBT_min_in_set: set_nat > nat > $o ).
thf(func_def_944,type,
vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
thf(func_def_945,type,
vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
thf(func_def_946,type,
vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).
thf(func_def_947,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_948,type,
vEBT_V3895251965096974666el_nat: produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ).
thf(func_def_949,type,
vEBT_V452583751252753300el_num: produc1193250871479095198on_num > produc1193250871479095198on_num > $o ).
thf(func_def_950,type,
vEBT_V7235779383477046023at_nat: produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ).
thf(func_def_951,type,
vEBT_VEBT_power: option_nat > option_nat > option_nat ).
thf(func_def_952,type,
vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
thf(func_def_953,type,
vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_954,type,
vEBT_vebt_mint: vEBT_VEBT > option_nat ).
thf(func_def_955,type,
vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_956,type,
vEBT_is_pred_in_set: set_nat > nat > nat > $o ).
thf(func_def_957,type,
vEBT_vebt_pred: vEBT_VEBT > nat > option_nat ).
thf(func_def_958,type,
vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_959,type,
vEBT_is_succ_in_set: set_nat > nat > nat > $o ).
thf(func_def_960,type,
vEBT_vebt_succ: vEBT_VEBT > nat > option_nat ).
thf(func_def_961,type,
vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_962,type,
accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
thf(func_def_963,type,
accp_nat: ( nat > nat > $o ) > nat > $o ).
thf(func_def_964,type,
accp_P6019419558468335806at_nat: ( produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ) > produc4471711990508489141at_nat > $o ).
thf(func_def_965,type,
accp_P5496254298877145759on_nat: ( produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ) > produc8306885398267862888on_nat > $o ).
thf(func_def_966,type,
accp_P4916641582247091100at_num: ( produc3368934014287244435at_num > produc3368934014287244435at_num > $o ) > produc3368934014287244435at_num > $o ).
thf(func_def_967,type,
accp_P7605991808943153877on_num: ( produc1193250871479095198on_num > produc1193250871479095198on_num > $o ) > produc1193250871479095198on_num > $o ).
thf(func_def_968,type,
accp_P3267385326087170368at_nat: ( produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ) > produc5542196010084753463at_nat > $o ).
thf(func_def_969,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(func_def_970,type,
accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(func_def_971,type,
accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
thf(func_def_972,type,
accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
thf(func_def_973,type,
accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
thf(func_def_974,type,
measure_int: ( int > nat ) > set_Pr958786334691620121nt_int ).
thf(func_def_975,type,
measure_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat ).
thf(func_def_976,type,
measure_num: ( num > nat ) > set_Pr8218934625190621173um_num ).
thf(func_def_977,type,
pred_nat: set_Pr1261947904930325089at_nat ).
thf(func_def_978,type,
fChoice_real: ( real > $o ) > real ).
thf(func_def_979,type,
member_o: $o > set_o > $o ).
thf(func_def_980,type,
member_complex: complex > set_complex > $o ).
thf(func_def_981,type,
member_int: int > set_int > $o ).
thf(func_def_982,type,
member_list_o: list_o > set_list_o > $o ).
thf(func_def_983,type,
member_list_int: list_int > set_list_int > $o ).
thf(func_def_984,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(func_def_985,type,
member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).
thf(func_def_986,type,
member_nat: nat > set_nat > $o ).
thf(func_def_987,type,
member_num: num > set_num > $o ).
thf(func_def_988,type,
member1379723562493234055eger_o: produc6271795597528267376eger_o > set_Pr448751882837621926eger_o > $o ).
thf(func_def_989,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(func_def_990,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_991,type,
member9148766508732265716at_num: product_prod_nat_num > set_Pr6200539531224447659at_num > $o ).
thf(func_def_992,type,
member7279096912039735102um_num: product_prod_num_num > set_Pr8218934625190621173um_num > $o ).
thf(func_def_993,type,
member_rat: rat > set_rat > $o ).
thf(func_def_994,type,
member_real: real > set_real > $o ).
thf(func_def_995,type,
member_set_int: set_int > set_set_int > $o ).
thf(func_def_996,type,
member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
thf(func_def_997,type,
deg: nat ).
thf(func_def_998,type,
m: nat ).
thf(func_def_999,type,
ma: nat ).
thf(func_def_1000,type,
mi: nat ).
thf(func_def_1001,type,
na: nat ).
thf(func_def_1002,type,
summary: vEBT_VEBT ).
thf(func_def_1003,type,
treeList: list_VEBT_VEBT ).
thf(func_def_1004,type,
xa: nat ).
thf(func_def_1023,type,
sP0: nat > nat > nat > $o ).
thf(func_def_1024,type,
sP1: nat > nat > nat > $o ).
thf(func_def_1025,type,
sP2: nat > nat > nat > $o ).
thf(func_def_1026,type,
sP3: vEBT_VEBT > nat > $o ).
thf(func_def_1027,type,
sP4: vEBT_VEBT > nat > $o ).
thf(func_def_1028,type,
sK5: ( nat > nat > $o ) > nat ).
thf(func_def_1029,type,
sK6: ( nat > nat > $o ) > nat ).
thf(func_def_1030,type,
sK7: ( nat > nat > $o ) > nat ).
thf(func_def_1031,type,
sK8: ( nat > nat > $o ) > nat ).
thf(func_def_1032,type,
sK9: ( nat > nat > $o ) > nat ).
thf(func_def_1033,type,
sK10: nat > nat > nat ).
thf(func_def_1034,type,
sK11: ( nat > $o ) > nat ).
thf(func_def_1035,type,
sK12: ( nat > nat ) > nat ).
thf(func_def_1036,type,
sK13: ( nat > nat ) > nat ).
thf(func_def_1037,type,
sK14: product_prod_nat_nat > nat ).
thf(func_def_1038,type,
sK15: product_prod_nat_nat > nat ).
thf(func_def_1039,type,
sK16: ( product_prod_nat_nat > $o ) > nat ).
thf(func_def_1040,type,
sK17: ( product_prod_nat_nat > $o ) > nat ).
thf(func_def_1041,type,
sK18: nat > nat > nat ).
thf(func_def_1042,type,
sK19: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1043,type,
sK20: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1044,type,
sK21: vEBT_VEBT > option4927543243414619207at_nat ).
thf(func_def_1045,type,
sK22: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1046,type,
sK23: vEBT_VEBT > option4927543243414619207at_nat ).
thf(func_def_1047,type,
sK24: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1048,type,
sK25: vEBT_VEBT > nat ).
thf(func_def_1049,type,
sK26: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1050,type,
sK27: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1051,type,
sK28: vEBT_VEBT > nat > nat > nat ).
thf(func_def_1052,type,
sK29: vEBT_VEBT > nat > nat > nat ).
thf(func_def_1053,type,
sK30: vEBT_VEBT > nat > nat > list_VEBT_VEBT ).
thf(func_def_1054,type,
sK31: vEBT_VEBT > nat > nat > nat ).
thf(func_def_1055,type,
sK32: vEBT_VEBT > nat > nat > vEBT_VEBT ).
thf(func_def_1056,type,
sK33: vEBT_VEBT > $o ).
thf(func_def_1057,type,
sK34: vEBT_VEBT > $o ).
thf(func_def_1058,type,
sK35: nat > ( nat > $o ) > nat ).
thf(func_def_1059,type,
sK36: nat > nat > nat ).
thf(func_def_1060,type,
sK37: ( nat > $o ) > nat ).
thf(func_def_1061,type,
sK38: nat > ( nat > $o ) > nat ).
thf(func_def_1062,type,
sK39: product_prod_nat_nat > nat ).
thf(func_def_1063,type,
sK40: product_prod_nat_nat > nat ).
thf(func_def_1064,type,
sK41: nat > nat ).
thf(func_def_1065,type,
sK42: ( nat > $o ) > nat ).
thf(func_def_1066,type,
sK43: vEBT_VEBT > nat ).
thf(func_def_1067,type,
sK44: nat > nat > nat ).
thf(func_def_1068,type,
sK45: product_prod_nat_nat > nat ).
thf(func_def_1069,type,
sK46: product_prod_nat_nat > nat ).
thf(func_def_1070,type,
sK47: ( nat > nat > $o ) > nat ).
thf(func_def_1071,type,
sK48: ( nat > nat > $o ) > nat ).
thf(func_def_1072,type,
sK49: ( nat > nat > $o ) > nat ).
thf(func_def_1073,type,
sK50: ( nat > nat > $o ) > nat ).
thf(func_def_1074,type,
sK51: ( nat > nat ) > nat ).
thf(func_def_1075,type,
sK52: ( nat > nat ) > nat ).
thf(func_def_1076,type,
sK53: nat > ( nat > $o ) > nat ).
thf(func_def_1077,type,
sK54: nat > ( nat > $o ) > nat ).
thf(func_def_1078,type,
sK55: nat > nat ).
thf(func_def_1079,type,
sK56: ( nat > $o ) > nat > nat ).
thf(func_def_1080,type,
sK57: ( nat > $o ) > nat ).
thf(func_def_1081,type,
sK58: nat > nat ).
thf(func_def_1082,type,
sK59: nat > nat ).
thf(func_def_1083,type,
sK60: nat > nat ).
thf(func_def_1084,type,
sK61: nat > nat > nat ).
thf(func_def_1085,type,
sK62: nat > nat ).
thf(func_def_1086,type,
sK63: nat > nat > nat ).
thf(func_def_1087,type,
ph64:
!>[X0: $tType] : X0 ).
thf(f37737,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f37729,f24653]) ).
thf(f24653,plain,
( $true
= ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_eq_nat @ ( plus_plus_nat @ xa @ ( numeral_numeral_nat @ one ) ) @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( set_VEBT_VEBT2 @ treeList ) ) ),
inference(beta_eta_normalization,[status(thm)],[f24499]) ).
thf(f24499,plain,
( $true
= ( member_VEBT_VEBT
@ ( nth_VEBT_VEBT @ treeList
@ ( vEBT_VEBT_high
@ ( if_nat
@ ( ^ [Y0: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y0 @ ( numeral_numeral_nat @ one ) ) )
@ xa
@ mi )
@ mi
@ xa )
@ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ ( set_VEBT_VEBT2 @ treeList ) ) ),
inference(definition_unfolding,[status(thm)],[f24293,f24391]) ).
thf(f24391,plain,
( ord_less_nat
= ( ^ [Y0: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y0 @ ( numeral_numeral_nat @ one ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f24279,f24091]) ).
thf(f24091,plain,
( one_one_nat
= ( numeral_numeral_nat @ one ) ),
inference(cnf_transformation,[status(thm)],[f463]) ).
thf(f463,axiom,
( one_one_nat
= ( numeral_numeral_nat @ one ) ),
file('/export/starexec/sandbox/tmp/tmp.jDIxpPE1PX/DTF2THF_16103.p',fact_462_numerals_I1_J) ).
thf(f24279,plain,
( ord_less_nat
= ( ^ [Y0: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y0 @ one_one_nat ) ) ) ),
inference(cnf_transformation,[status(thm)],[f13162]) ).
thf(f13162,plain,
( ord_less_nat
= ( ^ [Y0: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y0 @ one_one_nat ) ) ) ),
inference(fool_elimination,[status(thm)],[f13161]) ).
thf(f13161,plain,
( ord_less_nat
= ( ^ [X0: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X0 @ one_one_nat ) ) ) ),
inference(rectify,[status(thm)],[f2639]) ).
thf(f2639,axiom,
( ord_less_nat
= ( ^ [X52: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X52 @ one_one_nat ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.jDIxpPE1PX/DTF2THF_16103.p',fact_2638_discrete) ).
thf(f24293,plain,
( ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( set_VEBT_VEBT2 @ treeList ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f18526]) ).
thf(f18526,plain,
( ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( set_VEBT_VEBT2 @ treeList ) )
= $true ),
inference(fool_elimination,[status(thm)],[f18525]) ).
thf(f18525,plain,
member_VEBT_VEBT @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( set_VEBT_VEBT2 @ treeList ),
inference(rectify,[status(thm)],[f10]) ).
thf(f10,axiom,
member_VEBT_VEBT @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( set_VEBT_VEBT2 @ treeList ),
file('/export/starexec/sandbox/tmp/tmp.jDIxpPE1PX/DTF2THF_16103.p',fact_9__092_060open_062treeList_A_B_Ahigh_A_Iif_Ax_A_060_Ami_Athen_Ami_Aelse_Ax_J_A_Ideg_Adiv_A2_J_A_092_060in_062_Aset_AtreeList_092_060close_062) ).
thf(f37729,plain,
( $true
!= ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_eq_nat @ ( plus_plus_nat @ xa @ ( numeral_numeral_nat @ one ) ) @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( set_VEBT_VEBT2 @ treeList ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f37712]) ).
thf(f37712,plain,
( ( $true
!= ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_eq_nat @ ( plus_plus_nat @ xa @ ( numeral_numeral_nat @ one ) ) @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( set_VEBT_VEBT2 @ treeList ) ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f25572,f24394]) ).
thf(f24394,plain,
! [X2: vEBT_VEBT,X3: list_VEBT_VEBT,X0: vEBT_VEBT,X1: nat,X4: option4927543243414619207at_nat] :
( ( $true
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ one ) @ ( vEBT_VEBT_height @ X0 ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ X4 @ X1 @ X3 @ X2 ) ) ) )
| ( $true
!= ( member_VEBT_VEBT @ X0 @ ( set_VEBT_VEBT2 @ X3 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f24079,f24091]) ).
thf(f24079,plain,
! [X2: vEBT_VEBT,X3: list_VEBT_VEBT,X0: vEBT_VEBT,X1: nat,X4: option4927543243414619207at_nat] :
( ( ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ X0 ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ X4 @ X1 @ X3 @ X2 ) ) )
= $true )
| ( $true
!= ( member_VEBT_VEBT @ X0 @ ( set_VEBT_VEBT2 @ X3 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f23797]) ).
thf(f23797,plain,
! [X0: vEBT_VEBT,X1: nat,X2: vEBT_VEBT,X3: list_VEBT_VEBT,X4: option4927543243414619207at_nat] :
( ( ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ X0 ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ X4 @ X1 @ X3 @ X2 ) ) )
= $true )
| ( $true
!= ( member_VEBT_VEBT @ X0 @ ( set_VEBT_VEBT2 @ X3 ) ) ) ),
inference(ennf_transformation,[status(thm)],[f16090]) ).
thf(f16090,plain,
! [X0: vEBT_VEBT,X1: nat,X2: vEBT_VEBT,X3: list_VEBT_VEBT,X4: option4927543243414619207at_nat] :
( ( $true
= ( member_VEBT_VEBT @ X0 @ ( set_VEBT_VEBT2 @ X3 ) ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ X0 ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ X4 @ X1 @ X3 @ X2 ) ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f16089]) ).
thf(f16089,plain,
! [X0: vEBT_VEBT,X1: nat,X2: vEBT_VEBT,X3: list_VEBT_VEBT,X4: option4927543243414619207at_nat] :
( ( member_VEBT_VEBT @ X0 @ ( set_VEBT_VEBT2 @ X3 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ X0 ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ X4 @ X1 @ X3 @ X2 ) ) ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,axiom,
! [X8: vEBT_VEBT,X5: nat,X3: vEBT_VEBT,X6: list_VEBT_VEBT,X4: option4927543243414619207at_nat] :
( ( member_VEBT_VEBT @ X8 @ ( set_VEBT_VEBT2 @ X6 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ X8 ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ X4 @ X5 @ X6 @ X3 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.jDIxpPE1PX/DTF2THF_16103.p',fact_5_height__compose__child) ).
thf(f25572,plain,
( $true
!= ( ord_less_eq_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ one ) @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_eq_nat @ ( plus_plus_nat @ xa @ ( numeral_numeral_nat @ one ) ) @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f25547]) ).
thf(f25547,plain,
( ( $true != $true )
| ( $true
!= ( ord_less_eq_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ one ) @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_eq_nat @ ( plus_plus_nat @ xa @ ( numeral_numeral_nat @ one ) ) @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) ) ) ) ),
inference(superposition,[status(thm)],[f24954,f24665]) ).
thf(f24665,plain,
( $true
= ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ one ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ one ) @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_eq_nat @ ( plus_plus_nat @ xa @ ( numeral_numeral_nat @ one ) ) @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f24473]) ).
thf(f24473,plain,
( $true
= ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
@ ( plus_plus_nat @ ( numeral_numeral_nat @ one )
@ ( plus_plus_nat @ ( numeral_numeral_nat @ one )
@ ( vEBT_VEBT_height
@ ( nth_VEBT_VEBT @ treeList
@ ( vEBT_VEBT_high
@ ( if_nat
@ ( ^ [Y0: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y0 @ ( numeral_numeral_nat @ one ) ) )
@ xa
@ mi )
@ mi
@ xa )
@ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f24233,f24091,f24091,f24391]) ).
thf(f24233,plain,
( ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f22869]) ).
thf(f22869,plain,
( ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f22868]) ).
thf(f22868,plain,
ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f7]) ).
thf(f7,axiom,
ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.jDIxpPE1PX/DTF2THF_16103.p',fact_6__092_060open_062T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_092_060le_062_A1_A_L_A_I1_A_L_Aheight_A_ItreeList_A_B_Ahigh_A_Iif_Ax_A_060_Ami_Athen_Ami_Aelse_Ax_J_A_Ideg_Adiv_A2_J_J_J_092_060close_062) ).
thf(f24954,plain,
! [X0: nat] :
( ( ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ one ) @ X0 ) )
!= $true )
| ( ( ord_less_eq_nat @ X0 @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) ) )
!= $true ) ),
inference(superposition,[status(thm)],[f24703,f24327]) ).
thf(f24327,plain,
! [X2: nat,X0: nat,X1: nat] :
( ( ord_less_eq_nat @ X2 @ X1 )
= ( ord_less_eq_nat @ ( plus_plus_nat @ X0 @ X2 ) @ ( plus_plus_nat @ X0 @ X1 ) ) ),
inference(cnf_transformation,[status(thm)],[f23747]) ).
thf(f23747,plain,
! [X0: nat,X1: nat,X2: nat] :
( ( ord_less_eq_nat @ X2 @ X1 )
= ( ord_less_eq_nat @ ( plus_plus_nat @ X0 @ X2 ) @ ( plus_plus_nat @ X0 @ X1 ) ) ),
inference(rectify,[status(thm)],[f11]) ).
thf(f11,axiom,
! [X11: nat,X13: nat,X12: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ X11 @ X12 ) @ ( plus_plus_nat @ X11 @ X13 ) )
= ( ord_less_eq_nat @ X12 @ X13 ) ),
file('/export/starexec/sandbox/tmp/tmp.jDIxpPE1PX/DTF2THF_16103.p',fact_10_nat__add__left__cancel__le) ).
thf(f24703,plain,
! [X0: nat] :
( ( $true
!= ( ord_less_eq_nat @ X0 @ ( plus_plus_nat @ ( numeral_numeral_nat @ one ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) ) ) ) )
| ( ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ X0 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f24696]) ).
thf(f24696,plain,
! [X0: nat] :
( ( $true
!= ( ord_less_eq_nat @ X0 @ ( plus_plus_nat @ ( numeral_numeral_nat @ one ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) ) ) ) )
| ( $true != $true )
| ( ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ X0 )
!= $true ) ),
inference(superposition,[status(thm)],[f24454,f24083]) ).
thf(f24083,plain,
! [X2: nat,X0: nat,X1: nat] :
( ( $true
= ( ord_less_eq_nat @ X2 @ X0 ) )
| ( ( ord_less_eq_nat @ X2 @ X1 )
!= $true )
| ( ( ord_less_eq_nat @ X1 @ X0 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f23985]) ).
thf(f23985,plain,
! [X0: nat,X1: nat,X2: nat] :
( ( $true
= ( ord_less_eq_nat @ X2 @ X0 ) )
| ( ( ord_less_eq_nat @ X1 @ X0 )
!= $true )
| ( ( ord_less_eq_nat @ X2 @ X1 )
!= $true ) ),
inference(flattening,[status(thm)],[f23984]) ).
thf(f23984,plain,
! [X0: nat,X1: nat,X2: nat] :
( ( $true
= ( ord_less_eq_nat @ X2 @ X0 ) )
| ( ( ord_less_eq_nat @ X1 @ X0 )
!= $true )
| ( ( ord_less_eq_nat @ X2 @ X1 )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f13792]) ).
thf(f13792,plain,
! [X0: nat,X1: nat,X2: nat] :
( ( ( ord_less_eq_nat @ X2 @ X1 )
= $true )
=> ( ( ( ord_less_eq_nat @ X1 @ X0 )
= $true )
=> ( $true
= ( ord_less_eq_nat @ X2 @ X0 ) ) ) ),
inference(fool_elimination,[status(thm)],[f13791]) ).
thf(f13791,plain,
! [X0: nat,X1: nat,X2: nat] :
( ( ord_less_eq_nat @ X2 @ X1 )
=> ( ( ord_less_eq_nat @ X1 @ X0 )
=> ( ord_less_eq_nat @ X2 @ X0 ) ) ),
inference(rectify,[status(thm)],[f174]) ).
thf(f174,axiom,
! [X11: nat,X45: nat,X44: nat] :
( ( ord_less_eq_nat @ X44 @ X45 )
=> ( ( ord_less_eq_nat @ X45 @ X11 )
=> ( ord_less_eq_nat @ X44 @ X11 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.jDIxpPE1PX/DTF2THF_16103.p',fact_173_le__trans) ).
thf(f24454,plain,
( $true
!= ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ one ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f24197,f24091]) ).
thf(f24197,plain,
( ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f23708]) ).
thf(f23708,plain,
( ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) ) ) )
!= $true ),
inference(flattening,[status(thm)],[f14818]) ).
thf(f14818,plain,
( ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) ) ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f14817]) ).
thf(f14817,plain,
~ ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) ) ) ),
inference(rectify,[status(thm)],[f10249]) ).
thf(f10249,negated_conjecture,
~ ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) ) ) ),
inference(negated_conjecture,[status(cth)],[f10248]) ).
thf(f10248,conjecture,
ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) ) ),
file('/export/starexec/sandbox/tmp/tmp.jDIxpPE1PX/DTF2THF_16103.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP251^1 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n017.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 00:31:56 EST 2026
% 0.16/0.35 % CPUTime :
% 0.16/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.47 ---- Original DTF file ---
% 0.30/0.47 thf(spec,logic,$$dhol).
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 % File : ITP251^1 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.30/0.47 % Domain : Interactive Theorem Proving
% 0.30/0.47 % Problem : Sledgehammer problem VEBT_Bounds 00983_054636
% 0.30/0.47 % Version : [Des22] axioms.
% 0.30/0.47 % English :
% 0.30/0.47
% 0.30/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.47 % : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% 0.30/0.47 % Source : [Des22]
% 0.30/0.47 % Names : 0071_VEBT_Bounds_00983_054636 [Des22]
% 0.30/0.47
% 0.30/0.47 % Status : Theorem
% 0.30/0.47 % Rating : ? v9.2.1
% 0.30/0.47 % Syntax : Number of formulae : 11253 (5829 unt;1005 typ; 0 def)
% 0.30/0.47 % Number of atoms : 28332 (12311 equ; 0 cnn)
% 0.30/0.47 % Maximal formula atoms : 71 ( 2 avg)
% 0.30/0.47 % Number of connectives : 112315 (2845 ~; 534 |;1714 &;96677 @)
% 0.30/0.47 % ( 0 <=>;10545 =>; 0 <=; 0 <~>)
% 0.30/0.47 % Maximal formula depth : 39 ( 6 avg)
% 0.30/0.47 % Number of types : 88 ( 87 usr)
% 0.30/0.47 % Number of type conns : 3756 (3756 >; 0 *; 0 +; 0 <<)
% 0.30/0.47 % Number of symbols : 921 ( 918 usr; 59 con; 0-8 aty)
% 0.30/0.47 % Number of variables : 25586 (1817 ^;22963 !; 806 ?;25586 :)
% 0.30/0.47 % SPC : TH0_THM_EQU_NAR
% 0.30/0.47
% 0.30/0.47 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.30/0.47 % from the van Emde Boas Trees session in the Archive of Formal
% 0.30/0.47 % proofs -
% 0.30/0.47 % www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 0.30/0.47 % 2022-02-18 03:11:27.428
% 0.30/0.47 % Bugfixes : Added ()s around negated sides of equations.
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 % Could-be-implicit typings (87)
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% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
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% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 0.30/0.48 nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.30/0.48 nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Onth_001t__Real__Oreal,type,
% 0.30/0.48 nth_real: list_real > nat > real ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.48 nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
% 0.30/0.48 product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
% 0.30/0.48 product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
% 0.30/0.48 product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.48 product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001_Eo,type,
% 0.30/0.48 produc3607205314601156340eger_o: list_Code_integer > list_o > list_P8526636022914148096eger_o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
% 0.30/0.48 product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.48 produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oproduct_001t__Num__Onum_001t__Num__Onum,type,
% 0.30/0.48 product_num_num: list_num > list_num > list_P3744719386663036955um_num ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 0.30/0.48 produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 0.30/0.48 produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.48 produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oreplicate_001_Eo,type,
% 0.30/0.48 replicate_o: nat > $o > list_o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oreplicate_001t__Complex__Ocomplex,type,
% 0.30/0.48 replicate_complex: nat > complex > list_complex ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
% 0.30/0.48 replicate_int: nat > int > list_int ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
% 0.30/0.48 replicate_nat: nat > nat > list_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.30/0.48 replic4235873036481779905at_nat: nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
% 0.30/0.48 replicate_real: nat > real > list_real ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.48 replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
% 0.30/0.48 sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
% 0.30/0.48 sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Otake_001t__Nat__Onat,type,
% 0.30/0.48 take_nat: nat > list_nat > list_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oupt,type,
% 0.30/0.48 upt: nat > nat > list_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oupto,type,
% 0.30/0.48 upto: int > int > list_int ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oupto__aux,type,
% 0.30/0.48 upto_aux: int > int > list_int > list_int ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_List_Oupto__rel,type,
% 0.30/0.48 upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_OSuc,type,
% 0.30/0.48 suc: nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.30/0.48 compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 0.30/0.48 case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 0.30/0.48 case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.30/0.48 case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Onat_Opred,type,
% 0.30/0.48 pred: nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
% 0.30/0.48 semiri4939895301339042750nteger: nat > code_integer ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
% 0.30/0.48 semiri8010041392384452111omplex: nat > complex ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
% 0.30/0.48 semiri4216267220026989637d_enat: nat > extended_enat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 0.30/0.48 semiri1314217659103216013at_int: nat > int ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 0.30/0.48 semiri1316708129612266289at_nat: nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
% 0.30/0.48 semiri681578069525770553at_rat: nat > rat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 0.30/0.48 semiri5074537144036343181t_real: nat > real ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
% 0.30/0.48 semiri2816024913162550771omplex: ( complex > complex ) > nat > complex > complex ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
% 0.30/0.48 semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
% 0.30/0.48 semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
% 0.30/0.48 semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
% 0.30/0.48 semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
% 0.30/0.48 size_size_list_o: list_o > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
% 0.30/0.48 size_s3445333598471063425nteger: list_Code_integer > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 0.30/0.48 size_s3451745648224563538omplex: list_complex > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
% 0.30/0.48 size_size_list_int: list_int > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 0.30/0.48 size_size_list_nat: list_nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
% 0.30/0.48 size_size_list_num: list_num > nat ).
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% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 0.30/0.48 size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).
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% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
% 0.30/0.48 size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).
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% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
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% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.30/0.48 size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).
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% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
% 0.30/0.48 size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.30/0.48 size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).
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% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.30/0.48 size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
% 0.30/0.48 size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
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% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
% 0.30/0.48 size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).
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% 0.30/0.48 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,
% 0.30/0.48 size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
% 0.30/0.48 size_size_list_real: list_real > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 0.30/0.48 size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
% 0.30/0.48 size_size_num: num > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
% 0.30/0.48 size_size_option_nat: option_nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.30/0.48 size_size_option_num: option_num > nat ).
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% 0.30/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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% 0.30/0.48 size_size_VEBT_VEBT: vEBT_VEBT > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat__Bijection_Olist__encode,type,
% 0.30/0.48 nat_list_encode: list_nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
% 0.30/0.48 nat_list_encode_rel: list_nat > list_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 0.30/0.48 nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 0.30/0.48 nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 0.30/0.48 nat_prod_encode: product_prod_nat_nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat__Bijection_Oset__decode,type,
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% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat__Bijection_Oset__encode,type,
% 0.30/0.48 nat_set_encode: set_nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Nat__Bijection_Otriangle,type,
% 0.30/0.48 nat_triangle: nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_NthRoot_Oroot,type,
% 0.30/0.48 root: nat > real > real ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_NthRoot_Osqrt,type,
% 0.30/0.48 sqrt: real > real ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Num_OBitM,type,
% 0.30/0.48 bitM: num > num ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Num_Oinc,type,
% 0.30/0.48 inc: num > num ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
% 0.30/0.48 neg_nu8804712462038260780nteger: code_integer > code_integer ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
% 0.30/0.48 neg_nu7009210354673126013omplex: complex > complex ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
% 0.30/0.48 neg_numeral_dbl_int: int > int ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
% 0.30/0.48 neg_numeral_dbl_rat: rat > rat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
% 0.30/0.48 neg_numeral_dbl_real: real > real ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
% 0.30/0.48 neg_nu7757733837767384882nteger: code_integer > code_integer ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
% 0.30/0.48 neg_nu6511756317524482435omplex: complex > complex ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
% 0.30/0.48 neg_nu3811975205180677377ec_int: int > int ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
% 0.30/0.48 neg_nu3179335615603231917ec_rat: rat > rat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
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% 0.30/0.48 exp_real: real > real ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
% 0.30/0.48 ln_ln_real: real > real ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Transcendental_Olog,type,
% 0.30/0.48 log: real > real > real ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Transcendental_Opi,type,
% 0.30/0.48 pi: real ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 0.30/0.48 powr_real: real > real > real ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
% 0.30/0.48 sin_complex: complex > complex ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
% 0.30/0.48 sin_real: real > real ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Transcendental_Osin__coeff,type,
% 0.30/0.48 sin_coeff: nat > real ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
% 0.30/0.48 sinh_real: real > real ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
% 0.30/0.48 tan_complex: complex > complex ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
% 0.30/0.48 tan_real: real > real ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
% 0.30/0.48 tanh_complex: complex > complex ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
% 0.30/0.48 tanh_real: real > real ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
% 0.30/0.48 transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
% 0.30/0.48 transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
% 0.30/0.48 vEBT_T_i_n_s_e_r_t: vEBT_VEBT > nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
% 0.30/0.48 vEBT_T_i_n_s_e_r_t2: vEBT_VEBT > nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
% 0.30/0.48 vEBT_T5076183648494686801_t_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
% 0.30/0.48 vEBT_T9217963907923527482_t_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
% 0.30/0.48 vEBT_T_m_a_x_t: vEBT_VEBT > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
% 0.30/0.48 vEBT_T_m_a_x_t_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
% 0.30/0.48 vEBT_T_m_e_m_b_e_r: vEBT_VEBT > nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
% 0.30/0.48 vEBT_T_m_e_m_b_e_r2: vEBT_VEBT > nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
% 0.30/0.48 vEBT_T8099345112685741742_r_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
% 0.30/0.48 vEBT_T5837161174952499735_r_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
% 0.30/0.48 vEBT_T_m_i_n_N_u_l_l: vEBT_VEBT > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l__rel,type,
% 0.30/0.48 vEBT_T5462971552011256508_l_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
% 0.30/0.48 vEBT_T_m_i_n_t: vEBT_VEBT > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
% 0.30/0.48 vEBT_T_m_i_n_t_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
% 0.30/0.48 vEBT_Leaf: $o > $o > vEBT_VEBT ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
% 0.30/0.48 vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
% 0.30/0.48 vEBT_size_VEBT: vEBT_VEBT > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
% 0.30/0.48 vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
% 0.30/0.48 vEBT_VEBT_high: nat > nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
% 0.30/0.48 vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
% 0.30/0.48 vEBT_VEBT_low: nat > nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
% 0.30/0.48 vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
% 0.30/0.48 vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
% 0.30/0.48 vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
% 0.30/0.48 vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
% 0.30/0.48 vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
% 0.30/0.48 vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
% 0.30/0.48 vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Definitions_Oset__vebt,type,
% 0.30/0.48 vEBT_set_vebt: vEBT_VEBT > set_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
% 0.30/0.48 vEBT_vebt_buildup: nat > vEBT_VEBT ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
% 0.30/0.48 vEBT_v4011308405150292612up_rel: nat > nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
% 0.30/0.48 vEBT_VEBT_height: vEBT_VEBT > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
% 0.30/0.48 vEBT_VEBT_height_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Insert_Ovebt__insert,type,
% 0.30/0.48 vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
% 0.30/0.48 vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
% 0.30/0.48 vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
% 0.30/0.48 vEBT_VEBT_minNull: vEBT_VEBT > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
% 0.30/0.48 vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
% 0.30/0.48 vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Member_Ovebt__member,type,
% 0.30/0.48 vEBT_vebt_member: vEBT_VEBT > nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
% 0.30/0.48 vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
% 0.30/0.48 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_Ooption__shift__rel_001t__Nat__Onat,type,
% 0.30/0.48 vEBT_V3895251965096974666el_nat: produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Num__Onum,type,
% 0.30/0.48 vEBT_V452583751252753300el_num: produc1193250871479095198on_num > produc1193250871479095198on_num > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.30/0.48 vEBT_V7235779383477046023at_nat: produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ).
% 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__List__Olist_It__Nat__Onat_J,type,
% 0.30/0.48 accp_list_nat: ( list_nat > list_nat > $o ) > list_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_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.30/0.48 accp_P6019419558468335806at_nat: ( produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ) > produc4471711990508489141at_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Wellfounded_Oaccp_001t__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,
% 0.30/0.48 accp_P5496254298877145759on_nat: ( produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ) > produc8306885398267862888on_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J_J,type,
% 0.30/0.48 accp_P4916641582247091100at_num: ( produc3368934014287244435at_num > produc3368934014287244435at_num > $o ) > produc3368934014287244435at_num > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Wellfounded_Oaccp_001t__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,
% 0.30/0.48 accp_P7605991808943153877on_num: ( produc1193250871479095198on_num > produc1193250871479095198on_num > $o ) > produc1193250871479095198on_num > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Wellfounded_Oaccp_001t__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,
% 0.30/0.48 accp_P3267385326087170368at_nat: ( produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ) > produc5542196010084753463at_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__Num__Onum_Mt__Num__Onum_J,type,
% 0.30/0.48 accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $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_Wellfounded_Omeasure_001t__Int__Oint,type,
% 0.30/0.48 measure_int: ( int > nat ) > set_Pr958786334691620121nt_int ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Wellfounded_Omeasure_001t__Nat__Onat,type,
% 0.30/0.48 measure_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Wellfounded_Omeasure_001t__Num__Onum,type,
% 0.30/0.48 measure_num: ( num > nat ) > set_Pr8218934625190621173um_num ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Wellfounded_Opred__nat,type,
% 0.30/0.48 pred_nat: set_Pr1261947904930325089at_nat ).
% 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__Nat__Onat_J,type,
% 0.30/0.48 member_list_nat: list_nat > set_list_nat > $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__Code____Numeral__Ointeger_M_Eo_J,type,
% 0.30/0.48 member1379723562493234055eger_o: produc6271795597528267376eger_o > set_Pr448751882837621926eger_o > $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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.30/0.48 member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 0.30/0.48 member9148766508732265716at_num: product_prod_nat_num > set_Pr6200539531224447659at_num > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_member_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 0.30/0.48 member7279096912039735102um_num: product_prod_num_num > set_Pr8218934625190621173um_num > $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__Int__Oint_J,type,
% 0.30/0.48 member_set_int: set_int > set_set_int > $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_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_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_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 % Relevant facts (10209)
% 0.30/0.48 thf(fact_0__C5_Ohyps_C_I7_J,axiom,
% 0.30/0.48 ord_less_eq_nat @ mi @ ma ).
% 0.30/0.48
% 0.30/0.48 % "5.hyps"(7)
% 0.30/0.48 thf(fact_1_max__in__set__def,axiom,
% 0.30/0.48 ( vEBT_VEBT_max_in_set
% 0.30/0.48 = ( ^ [Xs: set_nat,X: nat] :
% 0.30/0.48 ( ( member_nat @ X @ Xs )
% 0.30/0.48 & ! [Y: nat] :
% 0.30/0.48 ( ( member_nat @ Y @ Xs )
% 0.30/0.48 => ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % max_in_set_def
% 0.30/0.48 thf(fact_2_min__in__set__def,axiom,
% 0.30/0.48 ( vEBT_VEBT_min_in_set
% 0.30/0.48 = ( ^ [Xs: set_nat,X: nat] :
% 0.30/0.48 ( ( member_nat @ X @ Xs )
% 0.30/0.48 & ! [Y: nat] :
% 0.30/0.48 ( ( member_nat @ Y @ Xs )
% 0.30/0.48 => ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % min_in_set_def
% 0.30/0.48 thf(fact_3_height__compose__summary,axiom,
% 0.30/0.48 ! [Summary: vEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ Summary ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % height_compose_summary
% 0.30/0.48 thf(fact_4__C5_OIH_C_I2_J,axiom,
% 0.30/0.48 ! [X2: nat] : ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ summary @ X2 ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ summary ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % "5.IH"(2)
% 0.30/0.48 thf(fact_5_height__compose__child,axiom,
% 0.30/0.48 ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,Summary: vEBT_VEBT] :
% 0.30/0.48 ( ( member_VEBT_VEBT @ T @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.48 => ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % height_compose_child
% 0.30/0.48 thf(fact_6__092_060open_062T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_092_060le_062_A1_A_L_A_I1_A_L_Aheight_A_ItreeList_A_B_Ahigh_A_Iif_Ax_A_060_Ami_Athen_Ami_Aelse_Ax_J_A_Ideg_Adiv_A2_J_J_J_092_060close_062,axiom,
% 0.30/0.48 ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t' (Node (Some (mi, ma)) deg treeList summary) x \<le> 1 + (1 + height (treeList ! high (if x < mi then mi else x) (deg div 2)))\<close>
% 0.30/0.48 thf(fact_7__C5_Ohyps_C_I6_J,axiom,
% 0.30/0.48 ( ( mi = ma )
% 0.30/0.48 => ! [X3: vEBT_VEBT] :
% 0.30/0.48 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ treeList ) )
% 0.30/0.48 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % "5.hyps"(6)
% 0.30/0.48 thf(fact_8__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,axiom,
% 0.30/0.48 ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).
% 0.30/0.48
% 0.30/0.48 % \<open>2 \<le> deg\<close>
% 0.30/0.48 thf(fact_9__092_060open_062treeList_A_B_Ahigh_A_Iif_Ax_A_060_Ami_Athen_Ami_Aelse_Ax_J_A_Ideg_Adiv_A2_J_A_092_060in_062_Aset_AtreeList_092_060close_062,axiom,
% 0.30/0.48 member_VEBT_VEBT @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( set_VEBT_VEBT2 @ treeList ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>treeList ! high (if x < mi then mi else x) (deg div 2) \<in> set treeList\<close>
% 0.30/0.48 thf(fact_10_nat__add__left__cancel__le,axiom,
% 0.30/0.48 ! [K: nat,M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.30/0.48 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_add_left_cancel_le
% 0.30/0.48 thf(fact_11_add__le__cancel__left,axiom,
% 0.30/0.48 ! [C: real,A: real,B: real] :
% 0.30/0.48 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.30/0.48 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_le_cancel_left
% 0.30/0.48 thf(fact_12_add__le__cancel__left,axiom,
% 0.30/0.48 ! [C: rat,A: rat,B: rat] :
% 0.30/0.48 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.30/0.48 = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_le_cancel_left
% 0.30/0.48 thf(fact_13_add__le__cancel__left,axiom,
% 0.30/0.48 ! [C: nat,A: nat,B: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.30/0.48 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_le_cancel_left
% 0.30/0.48 thf(fact_14_add__le__cancel__left,axiom,
% 0.30/0.48 ! [C: int,A: int,B: int] :
% 0.30/0.48 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.30/0.48 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_le_cancel_left
% 0.30/0.48 thf(fact_15_add__le__cancel__right,axiom,
% 0.30/0.48 ! [A: real,C: real,B: real] :
% 0.30/0.48 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.30/0.48 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_le_cancel_right
% 0.30/0.48 thf(fact_16_add__le__cancel__right,axiom,
% 0.30/0.48 ! [A: rat,C: rat,B: rat] :
% 0.30/0.48 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.48 = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_le_cancel_right
% 0.30/0.48 thf(fact_17_add__le__cancel__right,axiom,
% 0.30/0.48 ! [A: nat,C: nat,B: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.30/0.48 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_le_cancel_right
% 0.30/0.48 thf(fact_18_add__le__cancel__right,axiom,
% 0.30/0.48 ! [A: int,C: int,B: int] :
% 0.30/0.48 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.30/0.48 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_le_cancel_right
% 0.30/0.48 thf(fact_19__C5_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 % "5.hyps"(4)
% 0.30/0.48 thf(fact_20_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
% 0.30/0.48 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.30/0.48 ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 0.30/0.48 = one_one_nat ) ).
% 0.30/0.48
% 0.30/0.48 % T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
% 0.30/0.48 thf(fact_21_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
% 0.30/0.48 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.30/0.48 ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 0.30/0.48 = one_one_nat ) ).
% 0.30/0.48
% 0.30/0.48 % T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
% 0.30/0.48 thf(fact_22_VEBT_Oinject_I1_J,axiom,
% 0.30/0.48 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
% 0.30/0.48 ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 0.30/0.48 = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
% 0.30/0.48 = ( ( X11 = Y11 )
% 0.30/0.48 & ( X12 = Y12 )
% 0.30/0.48 & ( X13 = Y13 )
% 0.30/0.48 & ( X14 = Y14 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % VEBT.inject(1)
% 0.30/0.48 thf(fact_23_option_Oinject,axiom,
% 0.30/0.48 ! [X22: product_prod_nat_nat,Y2: product_prod_nat_nat] :
% 0.30/0.48 ( ( ( some_P7363390416028606310at_nat @ X22 )
% 0.30/0.48 = ( some_P7363390416028606310at_nat @ Y2 ) )
% 0.30/0.48 = ( X22 = Y2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % option.inject
% 0.30/0.48 thf(fact_24_option_Oinject,axiom,
% 0.30/0.48 ! [X22: nat,Y2: nat] :
% 0.30/0.48 ( ( ( some_nat @ X22 )
% 0.30/0.48 = ( some_nat @ Y2 ) )
% 0.30/0.48 = ( X22 = Y2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % option.inject
% 0.30/0.48 thf(fact_25_option_Oinject,axiom,
% 0.30/0.48 ! [X22: num,Y2: num] :
% 0.30/0.48 ( ( ( some_num @ X22 )
% 0.30/0.48 = ( some_num @ Y2 ) )
% 0.30/0.48 = ( X22 = Y2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % option.inject
% 0.30/0.48 thf(fact_26_add__right__cancel,axiom,
% 0.30/0.48 ! [B: real,A: real,C: real] :
% 0.30/0.48 ( ( ( plus_plus_real @ B @ A )
% 0.30/0.48 = ( plus_plus_real @ C @ A ) )
% 0.30/0.48 = ( B = C ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_right_cancel
% 0.30/0.48 thf(fact_27_add__right__cancel,axiom,
% 0.30/0.48 ! [B: rat,A: rat,C: rat] :
% 0.30/0.48 ( ( ( plus_plus_rat @ B @ A )
% 0.30/0.48 = ( plus_plus_rat @ C @ A ) )
% 0.30/0.48 = ( B = C ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_right_cancel
% 0.30/0.48 thf(fact_28_add__right__cancel,axiom,
% 0.30/0.48 ! [B: nat,A: nat,C: nat] :
% 0.30/0.48 ( ( ( plus_plus_nat @ B @ A )
% 0.30/0.48 = ( plus_plus_nat @ C @ A ) )
% 0.30/0.48 = ( B = C ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_right_cancel
% 0.30/0.48 thf(fact_29_add__right__cancel,axiom,
% 0.30/0.48 ! [B: int,A: int,C: int] :
% 0.30/0.48 ( ( ( plus_plus_int @ B @ A )
% 0.30/0.48 = ( plus_plus_int @ C @ A ) )
% 0.30/0.48 = ( B = C ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_right_cancel
% 0.30/0.48 thf(fact_30_add__left__cancel,axiom,
% 0.30/0.48 ! [A: real,B: real,C: real] :
% 0.30/0.48 ( ( ( plus_plus_real @ A @ B )
% 0.30/0.48 = ( plus_plus_real @ A @ C ) )
% 0.30/0.48 = ( B = C ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_left_cancel
% 0.30/0.48 thf(fact_31_add__left__cancel,axiom,
% 0.30/0.48 ! [A: rat,B: rat,C: rat] :
% 0.30/0.48 ( ( ( plus_plus_rat @ A @ B )
% 0.30/0.48 = ( plus_plus_rat @ A @ C ) )
% 0.30/0.48 = ( B = C ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_left_cancel
% 0.30/0.48 thf(fact_32_add__left__cancel,axiom,
% 0.30/0.48 ! [A: nat,B: nat,C: nat] :
% 0.30/0.48 ( ( ( plus_plus_nat @ A @ B )
% 0.30/0.48 = ( plus_plus_nat @ A @ C ) )
% 0.30/0.48 = ( B = C ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_left_cancel
% 0.30/0.48 thf(fact_33_add__left__cancel,axiom,
% 0.30/0.48 ! [A: int,B: int,C: int] :
% 0.30/0.48 ( ( ( plus_plus_int @ A @ B )
% 0.30/0.48 = ( plus_plus_int @ A @ C ) )
% 0.30/0.48 = ( B = C ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_left_cancel
% 0.30/0.48 thf(fact_34_add__less__cancel__right,axiom,
% 0.30/0.48 ! [A: real,C: real,B: real] :
% 0.30/0.48 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.30/0.48 = ( ord_less_real @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_less_cancel_right
% 0.30/0.48 thf(fact_35_add__less__cancel__right,axiom,
% 0.30/0.48 ! [A: rat,C: rat,B: rat] :
% 0.30/0.48 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.48 = ( ord_less_rat @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_less_cancel_right
% 0.30/0.48 thf(fact_36_add__less__cancel__right,axiom,
% 0.30/0.48 ! [A: nat,C: nat,B: nat] :
% 0.30/0.48 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.30/0.48 = ( ord_less_nat @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_less_cancel_right
% 0.30/0.48 thf(fact_37_add__less__cancel__right,axiom,
% 0.30/0.48 ! [A: int,C: int,B: int] :
% 0.30/0.48 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.30/0.48 = ( ord_less_int @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_less_cancel_right
% 0.30/0.48 thf(fact_38_add__less__cancel__left,axiom,
% 0.30/0.48 ! [C: real,A: real,B: real] :
% 0.30/0.48 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.30/0.48 = ( ord_less_real @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_less_cancel_left
% 0.30/0.48 thf(fact_39_add__less__cancel__left,axiom,
% 0.30/0.48 ! [C: rat,A: rat,B: rat] :
% 0.30/0.48 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.30/0.48 = ( ord_less_rat @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_less_cancel_left
% 0.30/0.48 thf(fact_40_add__less__cancel__left,axiom,
% 0.30/0.48 ! [C: nat,A: nat,B: nat] :
% 0.30/0.48 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.30/0.48 = ( ord_less_nat @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_less_cancel_left
% 0.30/0.48 thf(fact_41_add__less__cancel__left,axiom,
% 0.30/0.48 ! [C: int,A: int,B: int] :
% 0.30/0.48 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.30/0.48 = ( ord_less_int @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_less_cancel_left
% 0.30/0.48 thf(fact_42_nat__add__left__cancel__less,axiom,
% 0.30/0.48 ! [K: nat,M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.30/0.48 = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_add_left_cancel_less
% 0.30/0.48 thf(fact_43_lesseq__shift,axiom,
% 0.30/0.48 ( ord_less_eq_nat
% 0.30/0.48 = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lesseq_shift
% 0.30/0.48 thf(fact_44__C5_Ohyps_C_I3_J,axiom,
% 0.30/0.48 ( m
% 0.30/0.48 = ( suc @ na ) ) ).
% 0.30/0.48
% 0.30/0.48 % "5.hyps"(3)
% 0.30/0.48 thf(fact_45_False,axiom,
% 0.30/0.48 ~ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % False
% 0.30/0.48 thf(fact_46__C5_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 % "5.hyps"(8)
% 0.30/0.48 thf(fact_47__C5_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 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ X4 ) )
% 0.30/0.48 = ( vEBT_V8194947554948674370ptions @ summary @ I ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % "5.hyps"(5)
% 0.30/0.48 thf(fact_48__C5_Ohyps_C_I1_J,axiom,
% 0.30/0.48 vEBT_invar_vebt @ summary @ m ).
% 0.30/0.48
% 0.30/0.48 % "5.hyps"(1)
% 0.30/0.48 thf(fact_49_True,axiom,
% 0.30/0.48 ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) )
% 0.30/0.48 & ~ ( ( xa = mi )
% 0.30/0.48 | ( xa = ma ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % True
% 0.30/0.48 thf(fact_50__C5_OIH_C_I1_J,axiom,
% 0.30/0.48 ! [X3: vEBT_VEBT] :
% 0.30/0.48 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ treeList ) )
% 0.30/0.48 => ( ( vEBT_invar_vebt @ X3 @ na )
% 0.30/0.48 & ! [Xa: nat] : ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ X3 @ Xa ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ X3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % "5.IH"(1)
% 0.30/0.48 thf(fact_51_nat__neq__iff,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( M != N )
% 0.30/0.48 = ( ( ord_less_nat @ M @ N )
% 0.30/0.48 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_neq_iff
% 0.30/0.48 thf(fact_52_less__not__refl,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ~ ( ord_less_nat @ N @ N ) ).
% 0.30/0.48
% 0.30/0.48 % less_not_refl
% 0.30/0.48 thf(fact_53_less__not__refl2,axiom,
% 0.30/0.48 ! [N: nat,M: nat] :
% 0.30/0.48 ( ( ord_less_nat @ N @ M )
% 0.30/0.48 => ( M != N ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_not_refl2
% 0.30/0.48 thf(fact_54_less__not__refl3,axiom,
% 0.30/0.48 ! [S: nat,T: nat] :
% 0.30/0.48 ( ( ord_less_nat @ S @ T )
% 0.30/0.48 => ( S != T ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_not_refl3
% 0.30/0.48 thf(fact_55_less__irrefl__nat,axiom,
% 0.30/0.48 ! [N: nat] :
% 0.30/0.48 ~ ( ord_less_nat @ N @ N ) ).
% 0.30/0.48
% 0.30/0.48 % less_irrefl_nat
% 0.30/0.48 thf(fact_56_nat__less__induct,axiom,
% 0.30/0.48 ! [P: nat > $o,N: nat] :
% 0.30/0.48 ( ! [N2: nat] :
% 0.30/0.48 ( ! [M2: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M2 @ N2 )
% 0.30/0.48 => ( P @ M2 ) )
% 0.30/0.48 => ( P @ N2 ) )
% 0.30/0.48 => ( P @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_less_induct
% 0.30/0.48 thf(fact_57_infinite__descent,axiom,
% 0.30/0.48 ! [P: nat > $o,N: nat] :
% 0.30/0.48 ( ! [N2: nat] :
% 0.30/0.48 ( ~ ( P @ N2 )
% 0.30/0.48 => ? [M2: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M2 @ N2 )
% 0.30/0.48 & ~ ( P @ M2 ) ) )
% 0.30/0.48 => ( P @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % infinite_descent
% 0.30/0.48 thf(fact_58_linorder__neqE__nat,axiom,
% 0.30/0.48 ! [X2: nat,Y3: nat] :
% 0.30/0.48 ( ( X2 != Y3 )
% 0.30/0.48 => ( ~ ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.48 => ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linorder_neqE_nat
% 0.30/0.48 thf(fact_59_add__less__imp__less__right,axiom,
% 0.30/0.48 ! [A: real,C: real,B: real] :
% 0.30/0.48 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.30/0.48 => ( ord_less_real @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_less_imp_less_right
% 0.30/0.48 thf(fact_60_add__less__imp__less__right,axiom,
% 0.30/0.48 ! [A: rat,C: rat,B: rat] :
% 0.30/0.48 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.48 => ( ord_less_rat @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_less_imp_less_right
% 0.30/0.48 thf(fact_61_add__less__imp__less__right,axiom,
% 0.30/0.48 ! [A: nat,C: nat,B: nat] :
% 0.30/0.48 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.30/0.48 => ( ord_less_nat @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_less_imp_less_right
% 0.30/0.48 thf(fact_62_add__less__imp__less__right,axiom,
% 0.30/0.48 ! [A: int,C: int,B: int] :
% 0.30/0.48 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.30/0.48 => ( ord_less_int @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_less_imp_less_right
% 0.30/0.48 thf(fact_63_add__less__imp__less__left,axiom,
% 0.30/0.48 ! [C: real,A: real,B: real] :
% 0.30/0.48 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.30/0.48 => ( ord_less_real @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_less_imp_less_left
% 0.30/0.48 thf(fact_64_add__less__imp__less__left,axiom,
% 0.30/0.48 ! [C: rat,A: rat,B: rat] :
% 0.30/0.48 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.30/0.48 => ( ord_less_rat @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_less_imp_less_left
% 0.30/0.48 thf(fact_65_add__less__imp__less__left,axiom,
% 0.30/0.48 ! [C: nat,A: nat,B: nat] :
% 0.30/0.48 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.30/0.48 => ( ord_less_nat @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_less_imp_less_left
% 0.30/0.48 thf(fact_66_add__less__imp__less__left,axiom,
% 0.30/0.48 ! [C: int,A: int,B: int] :
% 0.30/0.48 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.30/0.48 => ( ord_less_int @ A @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_less_imp_less_left
% 0.30/0.48 thf(fact_67_mem__Collect__eq,axiom,
% 0.30/0.48 ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
% 0.30/0.48 ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_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: 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_69_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_70_mem__Collect__eq,axiom,
% 0.30/0.48 ! [A: list_nat,P: list_nat > $o] :
% 0.30/0.48 ( ( member_list_nat @ A @ ( collect_list_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_71_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_72_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_73_Collect__mem__eq,axiom,
% 0.30/0.48 ! [A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.48 ( ( collec3392354462482085612at_nat
% 0.30/0.48 @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ 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_complex] :
% 0.30/0.48 ( ( collect_complex
% 0.30/0.48 @ ^ [X: complex] : ( member_complex @ X @ 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_real] :
% 0.30/0.48 ( ( collect_real
% 0.30/0.48 @ ^ [X: real] : ( member_real @ X @ 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__mem__eq,axiom,
% 0.30/0.48 ! [A2: set_list_nat] :
% 0.30/0.48 ( ( collect_list_nat
% 0.30/0.48 @ ^ [X: list_nat] : ( member_list_nat @ X @ A2 ) )
% 0.30/0.48 = A2 ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_mem_eq
% 0.30/0.48 thf(fact_77_Collect__mem__eq,axiom,
% 0.30/0.48 ! [A2: set_nat] :
% 0.30/0.48 ( ( collect_nat
% 0.30/0.48 @ ^ [X: nat] : ( member_nat @ X @ A2 ) )
% 0.30/0.48 = A2 ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_mem_eq
% 0.30/0.48 thf(fact_78_Collect__mem__eq,axiom,
% 0.30/0.48 ! [A2: set_int] :
% 0.30/0.48 ( ( collect_int
% 0.30/0.48 @ ^ [X: int] : ( member_int @ X @ A2 ) )
% 0.30/0.48 = A2 ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_mem_eq
% 0.30/0.48 thf(fact_79_Collect__cong,axiom,
% 0.30/0.48 ! [P: complex > $o,Q: complex > $o] :
% 0.30/0.48 ( ! [X5: complex] :
% 0.30/0.48 ( ( P @ X5 )
% 0.30/0.48 = ( Q @ X5 ) )
% 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_80_Collect__cong,axiom,
% 0.30/0.48 ! [P: real > $o,Q: real > $o] :
% 0.30/0.48 ( ! [X5: real] :
% 0.30/0.48 ( ( P @ X5 )
% 0.30/0.48 = ( Q @ X5 ) )
% 0.30/0.48 => ( ( collect_real @ P )
% 0.30/0.48 = ( collect_real @ Q ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_cong
% 0.30/0.48 thf(fact_81_Collect__cong,axiom,
% 0.30/0.48 ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.30/0.48 ( ! [X5: list_nat] :
% 0.30/0.48 ( ( P @ X5 )
% 0.30/0.48 = ( Q @ X5 ) )
% 0.30/0.48 => ( ( collect_list_nat @ P )
% 0.30/0.48 = ( collect_list_nat @ Q ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_cong
% 0.30/0.48 thf(fact_82_Collect__cong,axiom,
% 0.30/0.48 ! [P: nat > $o,Q: nat > $o] :
% 0.30/0.48 ( ! [X5: nat] :
% 0.30/0.48 ( ( P @ X5 )
% 0.30/0.48 = ( Q @ X5 ) )
% 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_83_Collect__cong,axiom,
% 0.30/0.48 ! [P: int > $o,Q: int > $o] :
% 0.30/0.48 ( ! [X5: int] :
% 0.30/0.48 ( ( P @ X5 )
% 0.30/0.48 = ( Q @ X5 ) )
% 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_84_add__strict__right__mono,axiom,
% 0.30/0.48 ! [A: real,B: real,C: real] :
% 0.30/0.48 ( ( ord_less_real @ A @ B )
% 0.30/0.48 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_strict_right_mono
% 0.30/0.48 thf(fact_85_add__strict__right__mono,axiom,
% 0.30/0.48 ! [A: rat,B: rat,C: rat] :
% 0.30/0.48 ( ( ord_less_rat @ A @ B )
% 0.30/0.48 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_strict_right_mono
% 0.30/0.48 thf(fact_86_add__strict__right__mono,axiom,
% 0.30/0.48 ! [A: nat,B: nat,C: nat] :
% 0.30/0.48 ( ( ord_less_nat @ A @ B )
% 0.30/0.48 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_strict_right_mono
% 0.30/0.48 thf(fact_87_add__strict__right__mono,axiom,
% 0.30/0.48 ! [A: int,B: int,C: int] :
% 0.30/0.48 ( ( ord_less_int @ A @ B )
% 0.30/0.48 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_strict_right_mono
% 0.30/0.48 thf(fact_88_add__strict__left__mono,axiom,
% 0.30/0.48 ! [A: real,B: real,C: real] :
% 0.30/0.48 ( ( ord_less_real @ A @ B )
% 0.30/0.48 => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_strict_left_mono
% 0.30/0.48 thf(fact_89_add__strict__left__mono,axiom,
% 0.30/0.48 ! [A: rat,B: rat,C: rat] :
% 0.30/0.48 ( ( ord_less_rat @ A @ B )
% 0.30/0.48 => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_strict_left_mono
% 0.30/0.48 thf(fact_90_add__strict__left__mono,axiom,
% 0.30/0.48 ! [A: nat,B: nat,C: nat] :
% 0.30/0.48 ( ( ord_less_nat @ A @ B )
% 0.30/0.48 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_strict_left_mono
% 0.30/0.48 thf(fact_91_add__strict__left__mono,axiom,
% 0.30/0.48 ! [A: int,B: int,C: int] :
% 0.30/0.48 ( ( ord_less_int @ A @ B )
% 0.30/0.48 => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_strict_left_mono
% 0.30/0.48 thf(fact_92_add__strict__mono,axiom,
% 0.30/0.48 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.48 ( ( ord_less_real @ A @ B )
% 0.30/0.48 => ( ( ord_less_real @ C @ D )
% 0.30/0.48 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_strict_mono
% 0.30/0.48 thf(fact_93_add__strict__mono,axiom,
% 0.30/0.48 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.48 ( ( ord_less_rat @ A @ B )
% 0.30/0.48 => ( ( ord_less_rat @ C @ D )
% 0.30/0.48 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_strict_mono
% 0.30/0.48 thf(fact_94_add__strict__mono,axiom,
% 0.30/0.48 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.48 ( ( ord_less_nat @ A @ B )
% 0.30/0.48 => ( ( ord_less_nat @ C @ D )
% 0.30/0.48 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_strict_mono
% 0.30/0.48 thf(fact_95_add__strict__mono,axiom,
% 0.30/0.48 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.48 ( ( ord_less_int @ A @ B )
% 0.30/0.48 => ( ( ord_less_int @ C @ D )
% 0.30/0.48 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_strict_mono
% 0.30/0.48 thf(fact_96_add__mono__thms__linordered__field_I1_J,axiom,
% 0.30/0.48 ! [I2: real,J: real,K: real,L: real] :
% 0.30/0.48 ( ( ( ord_less_real @ I2 @ J )
% 0.30/0.48 & ( K = L ) )
% 0.30/0.48 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_mono_thms_linordered_field(1)
% 0.30/0.48 thf(fact_97_add__mono__thms__linordered__field_I1_J,axiom,
% 0.30/0.48 ! [I2: rat,J: rat,K: rat,L: rat] :
% 0.30/0.48 ( ( ( ord_less_rat @ I2 @ J )
% 0.30/0.48 & ( K = L ) )
% 0.30/0.48 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_mono_thms_linordered_field(1)
% 0.30/0.48 thf(fact_98_add__mono__thms__linordered__field_I1_J,axiom,
% 0.30/0.48 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.30/0.48 ( ( ( ord_less_nat @ I2 @ J )
% 0.30/0.48 & ( K = L ) )
% 0.30/0.48 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_mono_thms_linordered_field(1)
% 0.30/0.48 thf(fact_99_add__mono__thms__linordered__field_I1_J,axiom,
% 0.30/0.48 ! [I2: int,J: int,K: int,L: int] :
% 0.30/0.48 ( ( ( ord_less_int @ I2 @ J )
% 0.30/0.48 & ( K = L ) )
% 0.30/0.48 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_mono_thms_linordered_field(1)
% 0.30/0.48 thf(fact_100_add__mono__thms__linordered__field_I2_J,axiom,
% 0.30/0.48 ! [I2: real,J: real,K: real,L: real] :
% 0.30/0.48 ( ( ( I2 = J )
% 0.30/0.48 & ( ord_less_real @ K @ L ) )
% 0.30/0.48 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_mono_thms_linordered_field(2)
% 0.30/0.48 thf(fact_101_add__mono__thms__linordered__field_I2_J,axiom,
% 0.30/0.48 ! [I2: rat,J: rat,K: rat,L: rat] :
% 0.30/0.48 ( ( ( I2 = J )
% 0.30/0.48 & ( ord_less_rat @ K @ L ) )
% 0.30/0.48 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_mono_thms_linordered_field(2)
% 0.30/0.48 thf(fact_102_add__mono__thms__linordered__field_I2_J,axiom,
% 0.30/0.48 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.30/0.48 ( ( ( I2 = J )
% 0.30/0.48 & ( ord_less_nat @ K @ L ) )
% 0.30/0.48 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_mono_thms_linordered_field(2)
% 0.30/0.48 thf(fact_103_add__mono__thms__linordered__field_I2_J,axiom,
% 0.30/0.48 ! [I2: int,J: int,K: int,L: int] :
% 0.30/0.48 ( ( ( I2 = J )
% 0.30/0.48 & ( ord_less_int @ K @ L ) )
% 0.30/0.48 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_mono_thms_linordered_field(2)
% 0.30/0.48 thf(fact_104_add__mono__thms__linordered__field_I5_J,axiom,
% 0.30/0.48 ! [I2: real,J: real,K: real,L: real] :
% 0.30/0.48 ( ( ( ord_less_real @ I2 @ J )
% 0.30/0.48 & ( ord_less_real @ K @ L ) )
% 0.30/0.48 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_mono_thms_linordered_field(5)
% 0.30/0.48 thf(fact_105_add__mono__thms__linordered__field_I5_J,axiom,
% 0.30/0.48 ! [I2: rat,J: rat,K: rat,L: rat] :
% 0.30/0.48 ( ( ( ord_less_rat @ I2 @ J )
% 0.30/0.48 & ( ord_less_rat @ K @ L ) )
% 0.30/0.48 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_mono_thms_linordered_field(5)
% 0.30/0.48 thf(fact_106_add__mono__thms__linordered__field_I5_J,axiom,
% 0.30/0.48 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.30/0.48 ( ( ( ord_less_nat @ I2 @ J )
% 0.30/0.48 & ( ord_less_nat @ K @ L ) )
% 0.30/0.48 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_mono_thms_linordered_field(5)
% 0.30/0.48 thf(fact_107_add__mono__thms__linordered__field_I5_J,axiom,
% 0.30/0.48 ! [I2: int,J: int,K: int,L: int] :
% 0.30/0.48 ( ( ( ord_less_int @ I2 @ J )
% 0.30/0.48 & ( ord_less_int @ K @ L ) )
% 0.30/0.48 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_mono_thms_linordered_field(5)
% 0.30/0.48 thf(fact_108_less__mono__imp__le__mono,axiom,
% 0.30/0.48 ! [F: nat > nat,I2: nat,J: nat] :
% 0.30/0.48 ( ! [I3: nat,J2: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I3 @ J2 )
% 0.30/0.48 => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
% 0.30/0.48 => ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.48 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_mono_imp_le_mono
% 0.30/0.48 thf(fact_109_le__neq__implies__less,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.48 => ( ( M != N )
% 0.30/0.48 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_neq_implies_less
% 0.30/0.48 thf(fact_110_less__or__eq__imp__le,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ( ord_less_nat @ M @ N )
% 0.30/0.48 | ( M = N ) )
% 0.30/0.48 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_or_eq_imp_le
% 0.30/0.48 thf(fact_111_le__eq__less__or__eq,axiom,
% 0.30/0.48 ( ord_less_eq_nat
% 0.30/0.48 = ( ^ [M3: nat,N3: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M3 @ N3 )
% 0.30/0.48 | ( M3 = N3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_eq_less_or_eq
% 0.30/0.48 thf(fact_112_less__imp__le__nat,axiom,
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M @ N )
% 0.30/0.48 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_imp_le_nat
% 0.30/0.48 thf(fact_113_nat__less__le,axiom,
% 0.30/0.48 ( ord_less_nat
% 0.30/0.48 = ( ^ [M3: nat,N3: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.48 & ( M3 != N3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_less_le
% 0.30/0.48 thf(fact_114_less__add__eq__less,axiom,
% 0.30/0.48 ! [K: nat,L: nat,M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_nat @ K @ L )
% 0.30/0.48 => ( ( ( plus_plus_nat @ M @ L )
% 0.30/0.48 = ( plus_plus_nat @ K @ N ) )
% 0.30/0.48 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_add_eq_less
% 0.30/0.48 thf(fact_115_trans__less__add2,axiom,
% 0.30/0.48 ! [I2: nat,J: nat,M: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.48 => ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % trans_less_add2
% 0.30/0.48 thf(fact_116_trans__less__add1,axiom,
% 0.30/0.48 ! [I2: nat,J: nat,M: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.48 => ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % trans_less_add1
% 0.30/0.48 thf(fact_117_add__less__mono1,axiom,
% 0.30/0.48 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.48 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_less_mono1
% 0.30/0.48 thf(fact_118_not__add__less2,axiom,
% 0.30/0.48 ! [J: nat,I2: nat] :
% 0.30/0.48 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% 0.30/0.48
% 0.30/0.48 % not_add_less2
% 0.30/0.48 thf(fact_119_not__add__less1,axiom,
% 0.30/0.48 ! [I2: nat,J: nat] :
% 0.30/0.48 ~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% 0.30/0.48
% 0.30/0.48 % not_add_less1
% 0.30/0.48 thf(fact_120_add__less__mono,axiom,
% 0.30/0.48 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.48 => ( ( ord_less_nat @ K @ L )
% 0.30/0.48 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_less_mono
% 0.30/0.48 thf(fact_121_add__lessD1,axiom,
% 0.30/0.48 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.48 ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
% 0.30/0.48 => ( ord_less_nat @ I2 @ K ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_lessD1
% 0.30/0.48 thf(fact_122_one__reorient,axiom,
% 0.30/0.48 ! [X2: complex] :
% 0.30/0.48 ( ( one_one_complex = X2 )
% 0.30/0.48 = ( X2 = one_one_complex ) ) ).
% 0.30/0.48
% 0.30/0.48 % one_reorient
% 0.30/0.48 thf(fact_123_one__reorient,axiom,
% 0.30/0.48 ! [X2: real] :
% 0.30/0.48 ( ( one_one_real = X2 )
% 0.30/0.48 = ( X2 = one_one_real ) ) ).
% 0.30/0.48
% 0.30/0.48 % one_reorient
% 0.30/0.48 thf(fact_124_one__reorient,axiom,
% 0.30/0.48 ! [X2: rat] :
% 0.30/0.48 ( ( one_one_rat = X2 )
% 0.30/0.48 = ( X2 = one_one_rat ) ) ).
% 0.30/0.48
% 0.30/0.48 % one_reorient
% 0.30/0.48 thf(fact_125_one__reorient,axiom,
% 0.30/0.48 ! [X2: nat] :
% 0.30/0.48 ( ( one_one_nat = X2 )
% 0.30/0.48 = ( X2 = one_one_nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % one_reorient
% 0.30/0.48 thf(fact_126_one__reorient,axiom,
% 0.30/0.48 ! [X2: int] :
% 0.30/0.48 ( ( one_one_int = X2 )
% 0.30/0.48 = ( X2 = one_one_int ) ) ).
% 0.30/0.48
% 0.30/0.48 % one_reorient
% 0.30/0.48 thf(fact_127_add__right__imp__eq,axiom,
% 0.30/0.48 ! [B: real,A: real,C: real] :
% 0.30/0.48 ( ( ( plus_plus_real @ B @ A )
% 0.30/0.48 = ( plus_plus_real @ C @ A ) )
% 0.30/0.48 => ( B = C ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_right_imp_eq
% 0.30/0.48 thf(fact_128_add__right__imp__eq,axiom,
% 0.30/0.48 ! [B: rat,A: rat,C: rat] :
% 0.30/0.48 ( ( ( plus_plus_rat @ B @ A )
% 0.30/0.48 = ( plus_plus_rat @ C @ A ) )
% 0.30/0.48 => ( B = C ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_right_imp_eq
% 0.30/0.48 thf(fact_129_add__right__imp__eq,axiom,
% 0.30/0.48 ! [B: nat,A: nat,C: nat] :
% 0.30/0.48 ( ( ( plus_plus_nat @ B @ A )
% 0.30/0.48 = ( plus_plus_nat @ C @ A ) )
% 0.30/0.48 => ( B = C ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_right_imp_eq
% 0.30/0.48 thf(fact_130_add__right__imp__eq,axiom,
% 0.30/0.48 ! [B: int,A: int,C: int] :
% 0.30/0.48 ( ( ( plus_plus_int @ B @ A )
% 0.30/0.48 = ( plus_plus_int @ C @ A ) )
% 0.30/0.48 => ( B = C ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_right_imp_eq
% 0.30/0.48 thf(fact_131_add__left__imp__eq,axiom,
% 0.30/0.48 ! [A: real,B: real,C: real] :
% 0.30/0.48 ( ( ( plus_plus_real @ A @ B )
% 0.30/0.48 = ( plus_plus_real @ A @ C ) )
% 0.30/0.48 => ( B = C ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_left_imp_eq
% 0.30/0.48 thf(fact_132_add__left__imp__eq,axiom,
% 0.30/0.48 ! [A: rat,B: rat,C: rat] :
% 0.30/0.48 ( ( ( plus_plus_rat @ A @ B )
% 0.30/0.48 = ( plus_plus_rat @ A @ C ) )
% 0.30/0.48 => ( B = C ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_left_imp_eq
% 0.30/0.49 thf(fact_133_add__left__imp__eq,axiom,
% 0.30/0.49 ! [A: nat,B: nat,C: nat] :
% 0.30/0.49 ( ( ( plus_plus_nat @ A @ B )
% 0.30/0.49 = ( plus_plus_nat @ A @ C ) )
% 0.30/0.49 => ( B = C ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_left_imp_eq
% 0.30/0.49 thf(fact_134_add__left__imp__eq,axiom,
% 0.30/0.49 ! [A: int,B: int,C: int] :
% 0.30/0.49 ( ( ( plus_plus_int @ A @ B )
% 0.30/0.49 = ( plus_plus_int @ A @ C ) )
% 0.30/0.49 => ( B = C ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_left_imp_eq
% 0.30/0.49 thf(fact_135_add_Oleft__commute,axiom,
% 0.30/0.49 ! [B: real,A: real,C: real] :
% 0.30/0.49 ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 0.30/0.49 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add.left_commute
% 0.30/0.49 thf(fact_136_add_Oleft__commute,axiom,
% 0.30/0.49 ! [B: rat,A: rat,C: rat] :
% 0.30/0.49 ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
% 0.30/0.49 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add.left_commute
% 0.30/0.49 thf(fact_137_add_Oleft__commute,axiom,
% 0.30/0.49 ! [B: nat,A: nat,C: nat] :
% 0.30/0.49 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 0.30/0.49 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add.left_commute
% 0.30/0.49 thf(fact_138_add_Oleft__commute,axiom,
% 0.30/0.49 ! [B: int,A: int,C: int] :
% 0.30/0.49 ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 0.30/0.49 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add.left_commute
% 0.30/0.49 thf(fact_139_add_Ocommute,axiom,
% 0.30/0.49 ( plus_plus_real
% 0.30/0.49 = ( ^ [A3: real,B2: real] : ( plus_plus_real @ B2 @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add.commute
% 0.30/0.49 thf(fact_140_add_Ocommute,axiom,
% 0.30/0.49 ( plus_plus_rat
% 0.30/0.49 = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ B2 @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add.commute
% 0.30/0.49 thf(fact_141_add_Ocommute,axiom,
% 0.30/0.49 ( plus_plus_nat
% 0.30/0.49 = ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add.commute
% 0.30/0.49 thf(fact_142_add_Ocommute,axiom,
% 0.30/0.49 ( plus_plus_int
% 0.30/0.49 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add.commute
% 0.30/0.49 thf(fact_143_add_Oright__cancel,axiom,
% 0.30/0.49 ! [B: real,A: real,C: real] :
% 0.30/0.49 ( ( ( plus_plus_real @ B @ A )
% 0.30/0.49 = ( plus_plus_real @ C @ A ) )
% 0.30/0.49 = ( B = C ) ) ).
% 0.30/0.49
% 0.30/0.49 % add.right_cancel
% 0.30/0.49 thf(fact_144_add_Oright__cancel,axiom,
% 0.30/0.49 ! [B: rat,A: rat,C: rat] :
% 0.30/0.49 ( ( ( plus_plus_rat @ B @ A )
% 0.30/0.49 = ( plus_plus_rat @ C @ A ) )
% 0.30/0.49 = ( B = C ) ) ).
% 0.30/0.49
% 0.30/0.49 % add.right_cancel
% 0.30/0.49 thf(fact_145_add_Oright__cancel,axiom,
% 0.30/0.49 ! [B: int,A: int,C: int] :
% 0.30/0.49 ( ( ( plus_plus_int @ B @ A )
% 0.30/0.49 = ( plus_plus_int @ C @ A ) )
% 0.30/0.49 = ( B = C ) ) ).
% 0.30/0.49
% 0.30/0.49 % add.right_cancel
% 0.30/0.49 thf(fact_146_add_Oleft__cancel,axiom,
% 0.30/0.49 ! [A: real,B: real,C: real] :
% 0.30/0.49 ( ( ( plus_plus_real @ A @ B )
% 0.30/0.49 = ( plus_plus_real @ A @ C ) )
% 0.30/0.49 = ( B = C ) ) ).
% 0.30/0.49
% 0.30/0.49 % add.left_cancel
% 0.30/0.49 thf(fact_147_add_Oleft__cancel,axiom,
% 0.30/0.49 ! [A: rat,B: rat,C: rat] :
% 0.30/0.49 ( ( ( plus_plus_rat @ A @ B )
% 0.30/0.49 = ( plus_plus_rat @ A @ C ) )
% 0.30/0.49 = ( B = C ) ) ).
% 0.30/0.49
% 0.30/0.49 % add.left_cancel
% 0.30/0.49 thf(fact_148_add_Oleft__cancel,axiom,
% 0.30/0.49 ! [A: int,B: int,C: int] :
% 0.30/0.49 ( ( ( plus_plus_int @ A @ B )
% 0.30/0.49 = ( plus_plus_int @ A @ C ) )
% 0.30/0.49 = ( B = C ) ) ).
% 0.30/0.49
% 0.30/0.49 % add.left_cancel
% 0.30/0.49 thf(fact_149_add_Oassoc,axiom,
% 0.30/0.49 ! [A: real,B: real,C: real] :
% 0.30/0.49 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.30/0.49 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add.assoc
% 0.30/0.49 thf(fact_150_add_Oassoc,axiom,
% 0.30/0.49 ! [A: rat,B: rat,C: rat] :
% 0.30/0.49 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.30/0.49 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add.assoc
% 0.30/0.49 thf(fact_151_add_Oassoc,axiom,
% 0.30/0.49 ! [A: nat,B: nat,C: nat] :
% 0.30/0.49 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.49 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add.assoc
% 0.30/0.49 thf(fact_152_add_Oassoc,axiom,
% 0.30/0.49 ! [A: int,B: int,C: int] :
% 0.30/0.49 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.49 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add.assoc
% 0.30/0.49 thf(fact_153_group__cancel_Oadd2,axiom,
% 0.30/0.49 ! [B3: real,K: real,B: real,A: real] :
% 0.30/0.49 ( ( B3
% 0.30/0.49 = ( plus_plus_real @ K @ B ) )
% 0.30/0.49 => ( ( plus_plus_real @ A @ B3 )
% 0.30/0.49 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % group_cancel.add2
% 0.30/0.49 thf(fact_154_group__cancel_Oadd2,axiom,
% 0.30/0.49 ! [B3: rat,K: rat,B: rat,A: rat] :
% 0.30/0.49 ( ( B3
% 0.30/0.49 = ( plus_plus_rat @ K @ B ) )
% 0.30/0.49 => ( ( plus_plus_rat @ A @ B3 )
% 0.30/0.49 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % group_cancel.add2
% 0.30/0.49 thf(fact_155_group__cancel_Oadd2,axiom,
% 0.30/0.49 ! [B3: nat,K: nat,B: nat,A: nat] :
% 0.30/0.49 ( ( B3
% 0.30/0.49 = ( plus_plus_nat @ K @ B ) )
% 0.30/0.49 => ( ( plus_plus_nat @ A @ B3 )
% 0.30/0.49 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % group_cancel.add2
% 0.30/0.49 thf(fact_156_group__cancel_Oadd2,axiom,
% 0.30/0.49 ! [B3: int,K: int,B: int,A: int] :
% 0.30/0.49 ( ( B3
% 0.30/0.49 = ( plus_plus_int @ K @ B ) )
% 0.30/0.49 => ( ( plus_plus_int @ A @ B3 )
% 0.30/0.49 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % group_cancel.add2
% 0.30/0.49 thf(fact_157_group__cancel_Oadd1,axiom,
% 0.30/0.49 ! [A2: real,K: real,A: real,B: real] :
% 0.30/0.49 ( ( A2
% 0.30/0.49 = ( plus_plus_real @ K @ A ) )
% 0.30/0.49 => ( ( plus_plus_real @ A2 @ B )
% 0.30/0.49 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % group_cancel.add1
% 0.30/0.49 thf(fact_158_group__cancel_Oadd1,axiom,
% 0.30/0.49 ! [A2: rat,K: rat,A: rat,B: rat] :
% 0.30/0.49 ( ( A2
% 0.30/0.49 = ( plus_plus_rat @ K @ A ) )
% 0.30/0.49 => ( ( plus_plus_rat @ A2 @ B )
% 0.30/0.49 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % group_cancel.add1
% 0.30/0.49 thf(fact_159_group__cancel_Oadd1,axiom,
% 0.30/0.49 ! [A2: nat,K: nat,A: nat,B: nat] :
% 0.30/0.49 ( ( A2
% 0.30/0.49 = ( plus_plus_nat @ K @ A ) )
% 0.30/0.49 => ( ( plus_plus_nat @ A2 @ B )
% 0.30/0.49 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % group_cancel.add1
% 0.30/0.49 thf(fact_160_group__cancel_Oadd1,axiom,
% 0.30/0.49 ! [A2: int,K: int,A: int,B: int] :
% 0.30/0.49 ( ( A2
% 0.30/0.49 = ( plus_plus_int @ K @ A ) )
% 0.30/0.49 => ( ( plus_plus_int @ A2 @ B )
% 0.30/0.49 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % group_cancel.add1
% 0.30/0.49 thf(fact_161_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/0.49 ! [I2: real,J: real,K: real,L: real] :
% 0.30/0.49 ( ( ( I2 = J )
% 0.30/0.49 & ( K = L ) )
% 0.30/0.49 => ( ( plus_plus_real @ I2 @ K )
% 0.30/0.49 = ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_semiring(4)
% 0.30/0.49 thf(fact_162_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/0.49 ! [I2: rat,J: rat,K: rat,L: rat] :
% 0.30/0.49 ( ( ( I2 = J )
% 0.30/0.49 & ( K = L ) )
% 0.30/0.49 => ( ( plus_plus_rat @ I2 @ K )
% 0.30/0.49 = ( plus_plus_rat @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_semiring(4)
% 0.30/0.49 thf(fact_163_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/0.49 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.30/0.49 ( ( ( I2 = J )
% 0.30/0.49 & ( K = L ) )
% 0.30/0.49 => ( ( plus_plus_nat @ I2 @ K )
% 0.30/0.49 = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_semiring(4)
% 0.30/0.49 thf(fact_164_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/0.49 ! [I2: int,J: int,K: int,L: int] :
% 0.30/0.49 ( ( ( I2 = J )
% 0.30/0.49 & ( K = L ) )
% 0.30/0.49 => ( ( plus_plus_int @ I2 @ K )
% 0.30/0.49 = ( plus_plus_int @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_semiring(4)
% 0.30/0.49 thf(fact_165_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/0.49 ! [A: real,B: real,C: real] :
% 0.30/0.49 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.30/0.49 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ab_semigroup_add_class.add_ac(1)
% 0.30/0.49 thf(fact_166_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/0.49 ! [A: rat,B: rat,C: rat] :
% 0.30/0.49 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.30/0.49 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ab_semigroup_add_class.add_ac(1)
% 0.30/0.49 thf(fact_167_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/0.49 ! [A: nat,B: nat,C: nat] :
% 0.30/0.49 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.49 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ab_semigroup_add_class.add_ac(1)
% 0.30/0.49 thf(fact_168_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/0.49 ! [A: int,B: int,C: int] :
% 0.30/0.49 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.49 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ab_semigroup_add_class.add_ac(1)
% 0.30/0.49 thf(fact_169_Nat_Oex__has__greatest__nat,axiom,
% 0.30/0.49 ! [P: nat > $o,K: nat,B: nat] :
% 0.30/0.49 ( ( P @ K )
% 0.30/0.49 => ( ! [Y4: nat] :
% 0.30/0.49 ( ( P @ Y4 )
% 0.30/0.49 => ( ord_less_eq_nat @ Y4 @ B ) )
% 0.30/0.49 => ? [X5: nat] :
% 0.30/0.49 ( ( P @ X5 )
% 0.30/0.49 & ! [Y5: nat] :
% 0.30/0.49 ( ( P @ Y5 )
% 0.30/0.49 => ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Nat.ex_has_greatest_nat
% 0.30/0.49 thf(fact_170_nat__le__linear,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.49 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_le_linear
% 0.30/0.49 thf(fact_171_le__antisym,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.49 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.49 => ( M = N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_antisym
% 0.30/0.49 thf(fact_172_eq__imp__le,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( M = N )
% 0.30/0.49 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % eq_imp_le
% 0.30/0.49 thf(fact_173_le__trans,axiom,
% 0.30/0.49 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.49 => ( ( ord_less_eq_nat @ J @ K )
% 0.30/0.49 => ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_trans
% 0.30/0.49 thf(fact_174_le__refl,axiom,
% 0.30/0.49 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.30/0.49
% 0.30/0.49 % le_refl
% 0.30/0.49 thf(fact_175_add__less__le__mono,axiom,
% 0.30/0.49 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.49 ( ( ord_less_real @ A @ B )
% 0.30/0.49 => ( ( ord_less_eq_real @ C @ D )
% 0.30/0.49 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_less_le_mono
% 0.30/0.49 thf(fact_176_add__less__le__mono,axiom,
% 0.30/0.49 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.49 ( ( ord_less_rat @ A @ B )
% 0.30/0.49 => ( ( ord_less_eq_rat @ C @ D )
% 0.30/0.49 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_less_le_mono
% 0.30/0.49 thf(fact_177_add__less__le__mono,axiom,
% 0.30/0.49 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.49 ( ( ord_less_nat @ A @ B )
% 0.30/0.49 => ( ( ord_less_eq_nat @ C @ D )
% 0.30/0.49 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_less_le_mono
% 0.30/0.49 thf(fact_178_add__less__le__mono,axiom,
% 0.30/0.49 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.49 ( ( ord_less_int @ A @ B )
% 0.30/0.49 => ( ( ord_less_eq_int @ C @ D )
% 0.30/0.49 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_less_le_mono
% 0.30/0.49 thf(fact_179_add__le__less__mono,axiom,
% 0.30/0.49 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.49 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.49 => ( ( ord_less_real @ C @ D )
% 0.30/0.49 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_le_less_mono
% 0.30/0.49 thf(fact_180_add__le__less__mono,axiom,
% 0.30/0.49 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.49 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.49 => ( ( ord_less_rat @ C @ D )
% 0.30/0.49 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_le_less_mono
% 0.30/0.49 thf(fact_181_add__le__less__mono,axiom,
% 0.30/0.49 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.49 => ( ( ord_less_nat @ C @ D )
% 0.30/0.49 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_le_less_mono
% 0.30/0.49 thf(fact_182_add__le__less__mono,axiom,
% 0.30/0.49 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.49 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.49 => ( ( ord_less_int @ C @ D )
% 0.30/0.49 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_le_less_mono
% 0.30/0.49 thf(fact_183_add__mono__thms__linordered__field_I3_J,axiom,
% 0.30/0.49 ! [I2: real,J: real,K: real,L: real] :
% 0.30/0.49 ( ( ( ord_less_real @ I2 @ J )
% 0.30/0.49 & ( ord_less_eq_real @ K @ L ) )
% 0.30/0.49 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_field(3)
% 0.30/0.49 thf(fact_184_add__mono__thms__linordered__field_I3_J,axiom,
% 0.30/0.49 ! [I2: rat,J: rat,K: rat,L: rat] :
% 0.30/0.49 ( ( ( ord_less_rat @ I2 @ J )
% 0.30/0.49 & ( ord_less_eq_rat @ K @ L ) )
% 0.30/0.49 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_field(3)
% 0.30/0.49 thf(fact_185_add__mono__thms__linordered__field_I3_J,axiom,
% 0.30/0.49 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.30/0.49 ( ( ( ord_less_nat @ I2 @ J )
% 0.30/0.49 & ( ord_less_eq_nat @ K @ L ) )
% 0.30/0.49 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_field(3)
% 0.30/0.49 thf(fact_186_add__mono__thms__linordered__field_I3_J,axiom,
% 0.30/0.49 ! [I2: int,J: int,K: int,L: int] :
% 0.30/0.49 ( ( ( ord_less_int @ I2 @ J )
% 0.30/0.49 & ( ord_less_eq_int @ K @ L ) )
% 0.30/0.49 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_field(3)
% 0.30/0.49 thf(fact_187_add__mono__thms__linordered__field_I4_J,axiom,
% 0.30/0.49 ! [I2: real,J: real,K: real,L: real] :
% 0.30/0.49 ( ( ( ord_less_eq_real @ I2 @ J )
% 0.30/0.49 & ( ord_less_real @ K @ L ) )
% 0.30/0.49 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_field(4)
% 0.30/0.49 thf(fact_188_add__mono__thms__linordered__field_I4_J,axiom,
% 0.30/0.49 ! [I2: rat,J: rat,K: rat,L: rat] :
% 0.30/0.49 ( ( ( ord_less_eq_rat @ I2 @ J )
% 0.30/0.49 & ( ord_less_rat @ K @ L ) )
% 0.30/0.49 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_field(4)
% 0.30/0.49 thf(fact_189_add__mono__thms__linordered__field_I4_J,axiom,
% 0.30/0.49 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.30/0.49 ( ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.49 & ( ord_less_nat @ K @ L ) )
% 0.30/0.49 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_field(4)
% 0.30/0.49 thf(fact_190_add__mono__thms__linordered__field_I4_J,axiom,
% 0.30/0.49 ! [I2: int,J: int,K: int,L: int] :
% 0.30/0.49 ( ( ( ord_less_eq_int @ I2 @ J )
% 0.30/0.49 & ( ord_less_int @ K @ L ) )
% 0.30/0.49 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_field(4)
% 0.30/0.49 thf(fact_191_mono__nat__linear__lb,axiom,
% 0.30/0.49 ! [F: nat > nat,M: nat,K: nat] :
% 0.30/0.49 ( ! [M4: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_nat @ M4 @ N2 )
% 0.30/0.49 => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
% 0.30/0.49 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mono_nat_linear_lb
% 0.30/0.49 thf(fact_192_add__le__imp__le__right,axiom,
% 0.30/0.49 ! [A: real,C: real,B: real] :
% 0.30/0.49 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.30/0.49 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_le_imp_le_right
% 0.30/0.49 thf(fact_193_add__le__imp__le__right,axiom,
% 0.30/0.49 ! [A: rat,C: rat,B: rat] :
% 0.30/0.49 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.49 => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_le_imp_le_right
% 0.30/0.49 thf(fact_194_add__le__imp__le__right,axiom,
% 0.30/0.49 ! [A: nat,C: nat,B: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.30/0.49 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_le_imp_le_right
% 0.30/0.49 thf(fact_195_add__le__imp__le__right,axiom,
% 0.30/0.49 ! [A: int,C: int,B: int] :
% 0.30/0.49 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.30/0.49 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_le_imp_le_right
% 0.30/0.49 thf(fact_196_add__le__imp__le__left,axiom,
% 0.30/0.49 ! [C: real,A: real,B: real] :
% 0.30/0.49 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.30/0.49 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_le_imp_le_left
% 0.30/0.49 thf(fact_197_add__le__imp__le__left,axiom,
% 0.30/0.49 ! [C: rat,A: rat,B: rat] :
% 0.30/0.49 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.30/0.49 => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_le_imp_le_left
% 0.30/0.49 thf(fact_198_add__le__imp__le__left,axiom,
% 0.30/0.49 ! [C: nat,A: nat,B: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.30/0.49 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_le_imp_le_left
% 0.30/0.49 thf(fact_199_add__le__imp__le__left,axiom,
% 0.30/0.49 ! [C: int,A: int,B: int] :
% 0.30/0.49 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.30/0.49 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_le_imp_le_left
% 0.30/0.49 thf(fact_200_le__iff__add,axiom,
% 0.30/0.49 ( ord_less_eq_nat
% 0.30/0.49 = ( ^ [A3: nat,B2: nat] :
% 0.30/0.49 ? [C2: nat] :
% 0.30/0.49 ( B2
% 0.30/0.49 = ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_iff_add
% 0.30/0.49 thf(fact_201_add__right__mono,axiom,
% 0.30/0.49 ! [A: real,B: real,C: real] :
% 0.30/0.49 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.49 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_right_mono
% 0.30/0.49 thf(fact_202_add__right__mono,axiom,
% 0.30/0.49 ! [A: rat,B: rat,C: rat] :
% 0.30/0.49 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.49 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_right_mono
% 0.30/0.49 thf(fact_203_add__right__mono,axiom,
% 0.30/0.49 ! [A: nat,B: nat,C: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.49 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_right_mono
% 0.30/0.49 thf(fact_204_add__right__mono,axiom,
% 0.30/0.49 ! [A: int,B: int,C: int] :
% 0.30/0.49 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.49 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_right_mono
% 0.30/0.49 thf(fact_205_less__eqE,axiom,
% 0.30/0.49 ! [A: nat,B: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.49 => ~ ! [C3: nat] :
% 0.30/0.49 ( B
% 0.30/0.49 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_eqE
% 0.30/0.49 thf(fact_206_add__left__mono,axiom,
% 0.30/0.49 ! [A: real,B: real,C: real] :
% 0.30/0.49 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.49 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_left_mono
% 0.30/0.49 thf(fact_207_add__left__mono,axiom,
% 0.30/0.49 ! [A: rat,B: rat,C: rat] :
% 0.30/0.49 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.49 => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_left_mono
% 0.30/0.49 thf(fact_208_add__left__mono,axiom,
% 0.30/0.49 ! [A: nat,B: nat,C: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.49 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_left_mono
% 0.30/0.49 thf(fact_209_add__left__mono,axiom,
% 0.30/0.49 ! [A: int,B: int,C: int] :
% 0.30/0.49 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.49 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_left_mono
% 0.30/0.49 thf(fact_210_add__mono,axiom,
% 0.30/0.49 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.49 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.49 => ( ( ord_less_eq_real @ C @ D )
% 0.30/0.49 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono
% 0.30/0.49 thf(fact_211_add__mono,axiom,
% 0.30/0.49 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.49 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.49 => ( ( ord_less_eq_rat @ C @ D )
% 0.30/0.49 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono
% 0.30/0.49 thf(fact_212_add__mono,axiom,
% 0.30/0.49 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.49 => ( ( ord_less_eq_nat @ C @ D )
% 0.30/0.49 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono
% 0.30/0.49 thf(fact_213_add__mono,axiom,
% 0.30/0.49 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.49 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.49 => ( ( ord_less_eq_int @ C @ D )
% 0.30/0.49 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono
% 0.30/0.49 thf(fact_214_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/0.49 ! [I2: real,J: real,K: real,L: real] :
% 0.30/0.49 ( ( ( ord_less_eq_real @ I2 @ J )
% 0.30/0.49 & ( ord_less_eq_real @ K @ L ) )
% 0.30/0.49 => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_semiring(1)
% 0.30/0.49 thf(fact_215_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/0.49 ! [I2: rat,J: rat,K: rat,L: rat] :
% 0.30/0.49 ( ( ( ord_less_eq_rat @ I2 @ J )
% 0.30/0.49 & ( ord_less_eq_rat @ K @ L ) )
% 0.30/0.49 => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_semiring(1)
% 0.30/0.49 thf(fact_216_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/0.49 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.30/0.49 ( ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.49 & ( ord_less_eq_nat @ K @ L ) )
% 0.30/0.49 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_semiring(1)
% 0.30/0.49 thf(fact_217_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/0.49 ! [I2: int,J: int,K: int,L: int] :
% 0.30/0.49 ( ( ( ord_less_eq_int @ I2 @ J )
% 0.30/0.49 & ( ord_less_eq_int @ K @ L ) )
% 0.30/0.49 => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_semiring(1)
% 0.30/0.49 thf(fact_218_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/0.49 ! [I2: real,J: real,K: real,L: real] :
% 0.30/0.49 ( ( ( I2 = J )
% 0.30/0.49 & ( ord_less_eq_real @ K @ L ) )
% 0.30/0.49 => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_semiring(2)
% 0.30/0.49 thf(fact_219_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/0.49 ! [I2: rat,J: rat,K: rat,L: rat] :
% 0.30/0.49 ( ( ( I2 = J )
% 0.30/0.49 & ( ord_less_eq_rat @ K @ L ) )
% 0.30/0.49 => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_semiring(2)
% 0.30/0.49 thf(fact_220_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/0.49 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.30/0.49 ( ( ( I2 = J )
% 0.30/0.49 & ( ord_less_eq_nat @ K @ L ) )
% 0.30/0.49 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_semiring(2)
% 0.30/0.49 thf(fact_221_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/0.49 ! [I2: int,J: int,K: int,L: int] :
% 0.30/0.49 ( ( ( I2 = J )
% 0.30/0.49 & ( ord_less_eq_int @ K @ L ) )
% 0.30/0.49 => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_semiring(2)
% 0.30/0.49 thf(fact_222_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/0.49 ! [I2: real,J: real,K: real,L: real] :
% 0.30/0.49 ( ( ( ord_less_eq_real @ I2 @ J )
% 0.30/0.49 & ( K = L ) )
% 0.30/0.49 => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_semiring(3)
% 0.30/0.49 thf(fact_223_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/0.49 ! [I2: rat,J: rat,K: rat,L: rat] :
% 0.30/0.49 ( ( ( ord_less_eq_rat @ I2 @ J )
% 0.30/0.49 & ( K = L ) )
% 0.30/0.49 => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_semiring(3)
% 0.30/0.49 thf(fact_224_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/0.49 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.30/0.49 ( ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.49 & ( K = L ) )
% 0.30/0.49 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_semiring(3)
% 0.30/0.49 thf(fact_225_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/0.49 ! [I2: int,J: int,K: int,L: int] :
% 0.30/0.49 ( ( ( ord_less_eq_int @ I2 @ J )
% 0.30/0.49 & ( K = L ) )
% 0.30/0.49 => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_mono_thms_linordered_semiring(3)
% 0.30/0.49 thf(fact_226_nat__le__iff__add,axiom,
% 0.30/0.49 ( ord_less_eq_nat
% 0.30/0.49 = ( ^ [M3: nat,N3: nat] :
% 0.30/0.49 ? [K2: nat] :
% 0.30/0.49 ( N3
% 0.30/0.49 = ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_le_iff_add
% 0.30/0.49 thf(fact_227_trans__le__add2,axiom,
% 0.30/0.49 ! [I2: nat,J: nat,M: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.49 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % trans_le_add2
% 0.30/0.49 thf(fact_228_trans__le__add1,axiom,
% 0.30/0.49 ! [I2: nat,J: nat,M: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.49 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % trans_le_add1
% 0.30/0.49 thf(fact_229_add__le__mono1,axiom,
% 0.30/0.49 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.49 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_le_mono1
% 0.30/0.49 thf(fact_230_add__le__mono,axiom,
% 0.30/0.49 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.49 => ( ( ord_less_eq_nat @ K @ L )
% 0.30/0.49 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_le_mono
% 0.30/0.49 thf(fact_231_le__Suc__ex,axiom,
% 0.30/0.49 ! [K: nat,L: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ K @ L )
% 0.30/0.49 => ? [N2: nat] :
% 0.30/0.49 ( L
% 0.30/0.49 = ( plus_plus_nat @ K @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_Suc_ex
% 0.30/0.49 thf(fact_232_add__leD2,axiom,
% 0.30/0.49 ! [M: nat,K: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.30/0.49 => ( ord_less_eq_nat @ K @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_leD2
% 0.30/0.49 thf(fact_233_add__leD1,axiom,
% 0.30/0.49 ! [M: nat,K: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.30/0.49 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_leD1
% 0.30/0.49 thf(fact_234_le__add2,axiom,
% 0.30/0.49 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_add2
% 0.30/0.49 thf(fact_235_le__add1,axiom,
% 0.30/0.49 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_add1
% 0.30/0.49 thf(fact_236_add__leE,axiom,
% 0.30/0.49 ! [M: nat,K: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.30/0.49 => ~ ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.49 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_leE
% 0.30/0.49 thf(fact_237__C2_C,axiom,
% 0.30/0.49 ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.30/0.49 = ( plus_plus_nat @ one_one_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % "2"
% 0.30/0.49 thf(fact_238_insert__simp__mima,axiom,
% 0.30/0.49 ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.49 ( ( ( X2 = Mi )
% 0.30/0.49 | ( X2 = 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 @ TreeList @ Summary ) @ X2 )
% 0.30/0.49 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % insert_simp_mima
% 0.30/0.49 thf(fact_239_add__self__div__2,axiom,
% 0.30/0.49 ! [M: nat] :
% 0.30/0.49 ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49 = M ) ).
% 0.30/0.49
% 0.30/0.49 % add_self_div_2
% 0.30/0.49 thf(fact_240_succ__min,axiom,
% 0.30/0.49 ! [Deg: nat,X2: nat,Mi: nat,Ma: nat,TreeList: 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 @ X2 @ Mi )
% 0.30/0.49 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 0.30/0.49 = ( some_nat @ Mi ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % succ_min
% 0.30/0.49 thf(fact_241_pred__max,axiom,
% 0.30/0.49 ! [Deg: nat,Ma: nat,X2: nat,Mi: nat,TreeList: 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 @ X2 )
% 0.30/0.49 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 0.30/0.49 = ( some_nat @ Ma ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % pred_max
% 0.30/0.49 thf(fact_242_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_243_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_244_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_245_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_246_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_247_one__plus__numeral,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
% 0.30/0.49 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral
% 0.30/0.49 thf(fact_248_one__plus__numeral,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 0.30/0.49 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral
% 0.30/0.49 thf(fact_249_one__plus__numeral,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 0.30/0.49 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral
% 0.30/0.49 thf(fact_250_one__plus__numeral,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 0.30/0.49 = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral
% 0.30/0.49 thf(fact_251_one__plus__numeral,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 0.30/0.49 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral
% 0.30/0.49 thf(fact_252_numeral__plus__one,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
% 0.30/0.49 = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_plus_one
% 0.30/0.49 thf(fact_253_numeral__plus__one,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 0.30/0.49 = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_plus_one
% 0.30/0.49 thf(fact_254_numeral__plus__one,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 0.30/0.49 = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_plus_one
% 0.30/0.49 thf(fact_255_numeral__plus__one,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 0.30/0.49 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_plus_one
% 0.30/0.49 thf(fact_256_numeral__plus__one,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 0.30/0.49 = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_plus_one
% 0.30/0.49 thf(fact_257_one__less__numeral__iff,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 0.30/0.49 = ( ord_less_num @ one @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_less_numeral_iff
% 0.30/0.49 thf(fact_258_one__less__numeral__iff,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 0.30/0.49 = ( ord_less_num @ one @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_less_numeral_iff
% 0.30/0.49 thf(fact_259_one__less__numeral__iff,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 0.30/0.49 = ( ord_less_num @ one @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_less_numeral_iff
% 0.30/0.49 thf(fact_260_one__less__numeral__iff,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 0.30/0.49 = ( ord_less_num @ one @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_less_numeral_iff
% 0.30/0.49 thf(fact_261_numeral__le__one__iff,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 0.30/0.49 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_le_one_iff
% 0.30/0.49 thf(fact_262_numeral__le__one__iff,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 0.30/0.49 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_le_one_iff
% 0.30/0.49 thf(fact_263_numeral__le__one__iff,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 0.30/0.49 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_le_one_iff
% 0.30/0.49 thf(fact_264_numeral__le__one__iff,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 0.30/0.49 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_le_one_iff
% 0.30/0.49 thf(fact_265_one__eq__numeral__iff,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( one_one_complex
% 0.30/0.49 = ( numera6690914467698888265omplex @ N ) )
% 0.30/0.49 = ( one = N ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_eq_numeral_iff
% 0.30/0.49 thf(fact_266_one__eq__numeral__iff,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( one_one_real
% 0.30/0.49 = ( numeral_numeral_real @ N ) )
% 0.30/0.49 = ( one = N ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_eq_numeral_iff
% 0.30/0.49 thf(fact_267_one__eq__numeral__iff,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( one_one_rat
% 0.30/0.49 = ( numeral_numeral_rat @ N ) )
% 0.30/0.49 = ( one = N ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_eq_numeral_iff
% 0.30/0.49 thf(fact_268_one__eq__numeral__iff,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( one_one_nat
% 0.30/0.49 = ( numeral_numeral_nat @ N ) )
% 0.30/0.49 = ( one = N ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_eq_numeral_iff
% 0.30/0.49 thf(fact_269_one__eq__numeral__iff,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( one_one_int
% 0.30/0.49 = ( numeral_numeral_int @ N ) )
% 0.30/0.49 = ( one = N ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_eq_numeral_iff
% 0.30/0.49 thf(fact_270_numeral__eq__one__iff,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( ( numera6690914467698888265omplex @ N )
% 0.30/0.49 = one_one_complex )
% 0.30/0.49 = ( N = one ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_eq_one_iff
% 0.30/0.49 thf(fact_271_numeral__eq__one__iff,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( ( numeral_numeral_real @ N )
% 0.30/0.49 = one_one_real )
% 0.30/0.49 = ( N = one ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_eq_one_iff
% 0.30/0.49 thf(fact_272_numeral__eq__one__iff,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( ( numeral_numeral_rat @ N )
% 0.30/0.49 = one_one_rat )
% 0.30/0.49 = ( N = one ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_eq_one_iff
% 0.30/0.49 thf(fact_273_numeral__eq__one__iff,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( ( numeral_numeral_nat @ N )
% 0.30/0.49 = one_one_nat )
% 0.30/0.49 = ( N = one ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_eq_one_iff
% 0.30/0.49 thf(fact_274_numeral__eq__one__iff,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( ( numeral_numeral_int @ N )
% 0.30/0.49 = one_one_int )
% 0.30/0.49 = ( N = one ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_eq_one_iff
% 0.30/0.49 thf(fact_275_even__odd__cases,axiom,
% 0.30/0.49 ! [X2: nat] :
% 0.30/0.49 ( ! [N2: nat] :
% 0.30/0.49 ( X2
% 0.30/0.49 != ( plus_plus_nat @ N2 @ N2 ) )
% 0.30/0.49 => ~ ! [N2: nat] :
% 0.30/0.49 ( X2
% 0.30/0.49 != ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % even_odd_cases
% 0.30/0.49 thf(fact_276_deg__deg__n,axiom,
% 0.30/0.49 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
% 0.30/0.49 => ( Deg = N ) ) ).
% 0.30/0.49
% 0.30/0.49 % deg_deg_n
% 0.30/0.49 thf(fact_277_not__min__Null__member,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT] :
% 0.30/0.49 ( ~ ( vEBT_VEBT_minNull @ T )
% 0.30/0.49 => ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 ) ) ).
% 0.30/0.49
% 0.30/0.49 % not_min_Null_member
% 0.30/0.49 thf(fact_278_deg__SUcn__Node,axiom,
% 0.30/0.49 ! [Tree: vEBT_VEBT,N: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
% 0.30/0.49 => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.49 ( Tree
% 0.30/0.49 = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList2 @ S2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % deg_SUcn_Node
% 0.30/0.49 thf(fact_279_inthall,axiom,
% 0.30/0.49 ! [Xs2: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,N: nat] :
% 0.30/0.49 ( ! [X5: product_prod_nat_nat] :
% 0.30/0.49 ( ( member8440522571783428010at_nat @ X5 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
% 0.30/0.49 => ( P @ X5 ) )
% 0.30/0.49 => ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
% 0.30/0.49 => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % inthall
% 0.30/0.49 thf(fact_280_inthall,axiom,
% 0.30/0.49 ! [Xs2: list_complex,P: complex > $o,N: nat] :
% 0.30/0.49 ( ! [X5: complex] :
% 0.30/0.49 ( ( member_complex @ X5 @ ( set_complex2 @ Xs2 ) )
% 0.30/0.49 => ( P @ X5 ) )
% 0.30/0.49 => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 0.30/0.49 => ( P @ ( nth_complex @ Xs2 @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % inthall
% 0.30/0.49 thf(fact_281_inthall,axiom,
% 0.30/0.49 ! [Xs2: list_real,P: real > $o,N: nat] :
% 0.30/0.49 ( ! [X5: real] :
% 0.30/0.49 ( ( member_real @ X5 @ ( set_real2 @ Xs2 ) )
% 0.30/0.49 => ( P @ X5 ) )
% 0.30/0.49 => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 0.30/0.49 => ( P @ ( nth_real @ Xs2 @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % inthall
% 0.30/0.49 thf(fact_282_inthall,axiom,
% 0.30/0.49 ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
% 0.30/0.49 ( ! [X5: vEBT_VEBT] :
% 0.30/0.49 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.30/0.49 => ( P @ X5 ) )
% 0.30/0.49 => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.49 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % inthall
% 0.30/0.49 thf(fact_283_inthall,axiom,
% 0.30/0.49 ! [Xs2: list_o,P: $o > $o,N: nat] :
% 0.30/0.49 ( ! [X5: $o] :
% 0.30/0.49 ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
% 0.30/0.49 => ( P @ X5 ) )
% 0.30/0.49 => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.49 => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % inthall
% 0.30/0.49 thf(fact_284_inthall,axiom,
% 0.30/0.49 ! [Xs2: list_nat,P: nat > $o,N: nat] :
% 0.30/0.49 ( ! [X5: nat] :
% 0.30/0.49 ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
% 0.30/0.49 => ( P @ X5 ) )
% 0.30/0.49 => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.49 => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % inthall
% 0.30/0.49 thf(fact_285_inthall,axiom,
% 0.30/0.49 ! [Xs2: list_int,P: int > $o,N: nat] :
% 0.30/0.49 ( ! [X5: int] :
% 0.30/0.49 ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
% 0.30/0.49 => ( P @ X5 ) )
% 0.30/0.49 => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.49 => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % inthall
% 0.30/0.49 thf(fact_286_power__shift,axiom,
% 0.30/0.49 ! [X2: nat,Y3: nat,Z: nat] :
% 0.30/0.49 ( ( ( power_power_nat @ X2 @ Y3 )
% 0.30/0.49 = Z )
% 0.30/0.49 = ( ( vEBT_VEBT_power @ ( some_nat @ X2 ) @ ( some_nat @ Y3 ) )
% 0.30/0.49 = ( some_nat @ Z ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_shift
% 0.30/0.49 thf(fact_287_bit__split__inv,axiom,
% 0.30/0.49 ! [X2: nat,D: nat] :
% 0.30/0.49 ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X2 @ D ) @ ( vEBT_VEBT_low @ X2 @ D ) @ D )
% 0.30/0.49 = X2 ) ).
% 0.30/0.49
% 0.30/0.49 % bit_split_inv
% 0.30/0.49 thf(fact_288_numeral__eq__iff,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( ( numera6690914467698888265omplex @ M )
% 0.30/0.49 = ( numera6690914467698888265omplex @ N ) )
% 0.30/0.49 = ( M = N ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_eq_iff
% 0.30/0.49 thf(fact_289_numeral__eq__iff,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( ( numeral_numeral_real @ M )
% 0.30/0.49 = ( numeral_numeral_real @ N ) )
% 0.30/0.49 = ( M = N ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_eq_iff
% 0.30/0.49 thf(fact_290_numeral__eq__iff,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( ( numeral_numeral_rat @ M )
% 0.30/0.49 = ( numeral_numeral_rat @ N ) )
% 0.30/0.49 = ( M = N ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_eq_iff
% 0.30/0.49 thf(fact_291_numeral__eq__iff,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( ( numeral_numeral_nat @ M )
% 0.30/0.49 = ( numeral_numeral_nat @ N ) )
% 0.30/0.49 = ( M = N ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_eq_iff
% 0.30/0.49 thf(fact_292_numeral__eq__iff,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( ( numeral_numeral_int @ M )
% 0.30/0.49 = ( numeral_numeral_int @ N ) )
% 0.30/0.49 = ( M = N ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_eq_iff
% 0.30/0.49 thf(fact_293_nat_Oinject,axiom,
% 0.30/0.49 ! [X22: nat,Y2: nat] :
% 0.30/0.49 ( ( ( suc @ X22 )
% 0.30/0.49 = ( suc @ Y2 ) )
% 0.30/0.49 = ( X22 = Y2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat.inject
% 0.30/0.49 thf(fact_294_old_Onat_Oinject,axiom,
% 0.30/0.49 ! [Nat: nat,Nat2: nat] :
% 0.30/0.49 ( ( ( suc @ Nat )
% 0.30/0.49 = ( suc @ Nat2 ) )
% 0.30/0.49 = ( Nat = Nat2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % old.nat.inject
% 0.30/0.49 thf(fact_295__C5_Ohyps_C_I2_J,axiom,
% 0.30/0.49 ( ( size_s6755466524823107622T_VEBT @ treeList )
% 0.30/0.49 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% 0.30/0.49
% 0.30/0.49 % "5.hyps"(2)
% 0.30/0.49 thf(fact_296_pow__sum,axiom,
% 0.30/0.49 ! [A: nat,B: nat] :
% 0.30/0.49 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.49 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).
% 0.30/0.49
% 0.30/0.49 % pow_sum
% 0.30/0.49 thf(fact_297_high__def,axiom,
% 0.30/0.49 ( vEBT_VEBT_high
% 0.30/0.49 = ( ^ [X: nat,N3: nat] : ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % high_def
% 0.30/0.49 thf(fact_298_mi__eq__ma__no__ch,axiom,
% 0.30/0.49 ! [Mi: nat,Ma: nat,Deg: nat,TreeList: 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 @ TreeList @ Summary ) @ Deg )
% 0.30/0.49 => ( ( Mi = Ma )
% 0.30/0.49 => ( ! [X3: vEBT_VEBT] :
% 0.30/0.49 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.49 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
% 0.30/0.49 & ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mi_eq_ma_no_ch
% 0.30/0.49 thf(fact_299_high__bound__aux,axiom,
% 0.30/0.49 ! [Ma: nat,N: nat,M: nat] :
% 0.30/0.49 ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.30/0.49 => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % high_bound_aux
% 0.30/0.49 thf(fact_300_Suc__less__eq,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.49 = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_less_eq
% 0.30/0.49 thf(fact_301_Suc__mono,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_nat @ M @ N )
% 0.30/0.49 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_mono
% 0.30/0.49 thf(fact_302_lessI,axiom,
% 0.30/0.49 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % lessI
% 0.30/0.49 thf(fact_303_Suc__le__mono,axiom,
% 0.30/0.49 ! [N: nat,M: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.30/0.49 = ( ord_less_eq_nat @ N @ M ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_le_mono
% 0.30/0.49 thf(fact_304_add__Suc__right,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 0.30/0.49 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_Suc_right
% 0.30/0.49 thf(fact_305_helpypredd,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49 => ( ( ( vEBT_vebt_pred @ T @ X2 )
% 0.30/0.49 = ( some_nat @ Y3 ) )
% 0.30/0.49 => ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % helpypredd
% 0.30/0.49 thf(fact_306_helpyd,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49 => ( ( ( vEBT_vebt_succ @ T @ X2 )
% 0.30/0.49 = ( some_nat @ Y3 ) )
% 0.30/0.49 => ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % helpyd
% 0.30/0.49 thf(fact_307_valid__insert__both__member__options__pres,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.49 => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.49 => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
% 0.30/0.49 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y3 ) @ X2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % valid_insert_both_member_options_pres
% 0.30/0.49 thf(fact_308_valid__insert__both__member__options__add,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.49 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X2 ) @ X2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % valid_insert_both_member_options_add
% 0.30/0.49 thf(fact_309_set__n__deg__not__0,axiom,
% 0.30/0.49 ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
% 0.30/0.49 ( ! [X5: vEBT_VEBT] :
% 0.30/0.49 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.49 => ( vEBT_invar_vebt @ X5 @ N ) )
% 0.30/0.49 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.30/0.49 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.49 => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % set_n_deg_not_0
% 0.30/0.49 thf(fact_310_mi__ma__2__deg,axiom,
% 0.30/0.49 ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 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_311__C5_Ohyps_C_I9_J,axiom,
% 0.30/0.49 ( ( mi != ma )
% 0.30/0.49 => ! [I: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 0.30/0.49 => ( ( ( ( vEBT_VEBT_high @ ma @ na )
% 0.30/0.49 = I )
% 0.30/0.49 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
% 0.30/0.49 & ! [X3: nat] :
% 0.30/0.49 ( ( ( ( vEBT_VEBT_high @ X3 @ na )
% 0.30/0.49 = I )
% 0.30/0.49 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ ( vEBT_VEBT_low @ X3 @ na ) ) )
% 0.30/0.49 => ( ( ord_less_nat @ mi @ X3 )
% 0.30/0.49 & ( ord_less_eq_nat @ X3 @ ma ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % "5.hyps"(9)
% 0.30/0.49 thf(fact_312_numeral__plus__numeral,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 0.30/0.49 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_plus_numeral
% 0.30/0.49 thf(fact_313_numeral__plus__numeral,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.30/0.49 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_plus_numeral
% 0.30/0.49 thf(fact_314_numeral__plus__numeral,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.30/0.49 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_plus_numeral
% 0.30/0.49 thf(fact_315_numeral__plus__numeral,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.49 = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_plus_numeral
% 0.30/0.49 thf(fact_316_numeral__plus__numeral,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.49 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_plus_numeral
% 0.30/0.49 thf(fact_317_add__numeral__left,axiom,
% 0.30/0.49 ! [V: num,W: num,Z: complex] :
% 0.30/0.49 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 0.30/0.49 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_numeral_left
% 0.30/0.49 thf(fact_318_add__numeral__left,axiom,
% 0.30/0.49 ! [V: num,W: num,Z: real] :
% 0.30/0.49 ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 0.30/0.49 = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_numeral_left
% 0.30/0.49 thf(fact_319_add__numeral__left,axiom,
% 0.30/0.49 ! [V: num,W: num,Z: rat] :
% 0.30/0.49 ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 0.30/0.49 = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_numeral_left
% 0.30/0.49 thf(fact_320_add__numeral__left,axiom,
% 0.30/0.49 ! [V: num,W: num,Z: nat] :
% 0.30/0.49 ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 0.30/0.49 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_numeral_left
% 0.30/0.49 thf(fact_321_add__numeral__left,axiom,
% 0.30/0.49 ! [V: num,W: num,Z: int] :
% 0.30/0.49 ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 0.30/0.49 = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_numeral_left
% 0.30/0.49 thf(fact_322_both__member__options__ding,axiom,
% 0.30/0.49 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X2: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
% 0.30/0.49 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.30/0.49 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.49 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % both_member_options_ding
% 0.30/0.49 thf(fact_323_numeral__le__iff,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.30/0.49 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_le_iff
% 0.30/0.49 thf(fact_324_numeral__le__iff,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.30/0.49 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_le_iff
% 0.30/0.49 thf(fact_325_numeral__le__iff,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.49 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_le_iff
% 0.30/0.49 thf(fact_326_numeral__le__iff,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.49 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_le_iff
% 0.30/0.49 thf(fact_327_numeral__less__iff,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.30/0.49 = ( ord_less_num @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_less_iff
% 0.30/0.49 thf(fact_328_numeral__less__iff,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.30/0.49 = ( ord_less_num @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_less_iff
% 0.30/0.49 thf(fact_329_numeral__less__iff,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.49 = ( ord_less_num @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_less_iff
% 0.30/0.49 thf(fact_330_numeral__less__iff,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.49 = ( ord_less_num @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_less_iff
% 0.30/0.49 thf(fact_331_both__member__options__from__complete__tree__to__child,axiom,
% 0.30/0.49 ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: 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 @ TreeList @ Summary ) @ X2 )
% 0.30/0.49 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.49 | ( X2 = Mi )
% 0.30/0.49 | ( X2 = Ma ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % both_member_options_from_complete_tree_to_child
% 0.30/0.49 thf(fact_332_both__member__options__from__chilf__to__complete__tree,axiom,
% 0.30/0.49 ! [X2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.30/0.49 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.30/0.49 => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 0.30/0.49 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.49 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % both_member_options_from_chilf_to_complete_tree
% 0.30/0.49 thf(fact_333_Suc__numeral,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( suc @ ( numeral_numeral_nat @ N ) )
% 0.30/0.49 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_numeral
% 0.30/0.49 thf(fact_334_add__2__eq__Suc_H,axiom,
% 0.30/0.49 ! [N: nat] :
% 0.30/0.49 ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49 = ( suc @ ( suc @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_2_eq_Suc'
% 0.30/0.49 thf(fact_335_add__2__eq__Suc,axiom,
% 0.30/0.49 ! [N: nat] :
% 0.30/0.49 ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.49 = ( suc @ ( suc @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_2_eq_Suc
% 0.30/0.49 thf(fact_336_Suc__1,axiom,
% 0.30/0.49 ( ( suc @ one_one_nat )
% 0.30/0.49 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_1
% 0.30/0.49 thf(fact_337_div2__Suc__Suc,axiom,
% 0.30/0.49 ! [M: nat] :
% 0.30/0.49 ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49 = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div2_Suc_Suc
% 0.30/0.49 thf(fact_338__C1_C,axiom,
% 0.30/0.49 ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.30/0.49 = ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % "1"
% 0.30/0.49 thf(fact_339__C0_C,axiom,
% 0.30/0.49 ( ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) )
% 0.30/0.49 & ~ ( ( xa = mi )
% 0.30/0.49 | ( xa = ma ) ) )
% 0.30/0.49 => ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.30/0.49 = ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) ) ) )
% 0.30/0.49 & ( ~ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) )
% 0.30/0.49 & ~ ( ( xa = mi )
% 0.30/0.49 | ( xa = ma ) ) )
% 0.30/0.49 => ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.30/0.49 = one_one_nat ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % "0"
% 0.30/0.49 thf(fact_340_Suc__inject,axiom,
% 0.30/0.49 ! [X2: nat,Y3: nat] :
% 0.30/0.49 ( ( ( suc @ X2 )
% 0.30/0.49 = ( suc @ Y3 ) )
% 0.30/0.49 => ( X2 = Y3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_inject
% 0.30/0.49 thf(fact_341_n__not__Suc__n,axiom,
% 0.30/0.49 ! [N: nat] :
% 0.30/0.49 ( N
% 0.30/0.49 != ( suc @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % n_not_Suc_n
% 0.30/0.49 thf(fact_342_size__neq__size__imp__neq,axiom,
% 0.30/0.49 ! [X2: list_VEBT_VEBT,Y3: list_VEBT_VEBT] :
% 0.30/0.49 ( ( ( size_s6755466524823107622T_VEBT @ X2 )
% 0.30/0.49 != ( size_s6755466524823107622T_VEBT @ Y3 ) )
% 0.30/0.49 => ( X2 != Y3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % size_neq_size_imp_neq
% 0.30/0.49 thf(fact_343_size__neq__size__imp__neq,axiom,
% 0.30/0.49 ! [X2: list_o,Y3: list_o] :
% 0.30/0.49 ( ( ( size_size_list_o @ X2 )
% 0.30/0.49 != ( size_size_list_o @ Y3 ) )
% 0.30/0.49 => ( X2 != Y3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % size_neq_size_imp_neq
% 0.30/0.49 thf(fact_344_size__neq__size__imp__neq,axiom,
% 0.30/0.49 ! [X2: list_nat,Y3: list_nat] :
% 0.30/0.49 ( ( ( size_size_list_nat @ X2 )
% 0.30/0.49 != ( size_size_list_nat @ Y3 ) )
% 0.30/0.49 => ( X2 != Y3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % size_neq_size_imp_neq
% 0.30/0.49 thf(fact_345_size__neq__size__imp__neq,axiom,
% 0.30/0.49 ! [X2: list_int,Y3: list_int] :
% 0.30/0.49 ( ( ( size_size_list_int @ X2 )
% 0.30/0.49 != ( size_size_list_int @ Y3 ) )
% 0.30/0.49 => ( X2 != Y3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % size_neq_size_imp_neq
% 0.30/0.49 thf(fact_346_size__neq__size__imp__neq,axiom,
% 0.30/0.49 ! [X2: num,Y3: num] :
% 0.30/0.49 ( ( ( size_size_num @ X2 )
% 0.30/0.49 != ( size_size_num @ Y3 ) )
% 0.30/0.49 => ( X2 != Y3 ) ) ).
% 0.30/0.49
% 0.30/0.49 % size_neq_size_imp_neq
% 0.30/0.49 thf(fact_347_not__less__less__Suc__eq,axiom,
% 0.30/0.49 ! [N: nat,M: nat] :
% 0.30/0.49 ( ~ ( ord_less_nat @ N @ M )
% 0.30/0.49 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.30/0.49 = ( N = M ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % not_less_less_Suc_eq
% 0.30/0.49 thf(fact_348_strict__inc__induct,axiom,
% 0.30/0.49 ! [I2: nat,J: nat,P: nat > $o] :
% 0.30/0.49 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.49 => ( ! [I3: nat] :
% 0.30/0.49 ( ( J
% 0.30/0.49 = ( suc @ I3 ) )
% 0.30/0.49 => ( P @ I3 ) )
% 0.30/0.49 => ( ! [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ J )
% 0.30/0.49 => ( ( P @ ( suc @ I3 ) )
% 0.30/0.49 => ( P @ I3 ) ) )
% 0.30/0.49 => ( P @ I2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % strict_inc_induct
% 0.30/0.49 thf(fact_349_less__Suc__induct,axiom,
% 0.30/0.49 ! [I2: nat,J: nat,P: nat > nat > $o] :
% 0.30/0.49 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.49 => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
% 0.30/0.49 => ( ! [I3: nat,J2: nat,K3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ J2 )
% 0.30/0.49 => ( ( ord_less_nat @ J2 @ K3 )
% 0.30/0.49 => ( ( P @ I3 @ J2 )
% 0.30/0.49 => ( ( P @ J2 @ K3 )
% 0.30/0.49 => ( P @ I3 @ K3 ) ) ) ) )
% 0.30/0.49 => ( P @ I2 @ J ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_Suc_induct
% 0.30/0.49 thf(fact_350_less__trans__Suc,axiom,
% 0.30/0.49 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.49 => ( ( ord_less_nat @ J @ K )
% 0.30/0.49 => ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_trans_Suc
% 0.30/0.49 thf(fact_351_Suc__less__SucD,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.49 => ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_less_SucD
% 0.30/0.49 thf(fact_352_less__antisym,axiom,
% 0.30/0.49 ! [N: nat,M: nat] :
% 0.30/0.49 ( ~ ( ord_less_nat @ N @ M )
% 0.30/0.49 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.30/0.49 => ( M = N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_antisym
% 0.30/0.49 thf(fact_353_Suc__less__eq2,axiom,
% 0.30/0.49 ! [N: nat,M: nat] :
% 0.30/0.49 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.49 = ( ? [M5: nat] :
% 0.30/0.49 ( ( M
% 0.30/0.49 = ( suc @ M5 ) )
% 0.30/0.49 & ( ord_less_nat @ N @ M5 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_less_eq2
% 0.30/0.49 thf(fact_354_All__less__Suc,axiom,
% 0.30/0.49 ! [N: nat,P: nat > $o] :
% 0.30/0.49 ( ( ! [I4: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.30/0.49 => ( P @ I4 ) ) )
% 0.30/0.49 = ( ( P @ N )
% 0.30/0.49 & ! [I4: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I4 @ N )
% 0.30/0.49 => ( P @ I4 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % All_less_Suc
% 0.30/0.49 thf(fact_355_not__less__eq,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ~ ( ord_less_nat @ M @ N ) )
% 0.30/0.49 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % not_less_eq
% 0.30/0.49 thf(fact_356_less__Suc__eq,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.30/0.49 = ( ( ord_less_nat @ M @ N )
% 0.30/0.49 | ( M = N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_Suc_eq
% 0.30/0.49 thf(fact_357_Ex__less__Suc,axiom,
% 0.30/0.49 ! [N: nat,P: nat > $o] :
% 0.30/0.49 ( ( ? [I4: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.30/0.49 & ( P @ I4 ) ) )
% 0.30/0.49 = ( ( P @ N )
% 0.30/0.49 | ? [I4: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I4 @ N )
% 0.30/0.49 & ( P @ I4 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Ex_less_Suc
% 0.30/0.49 thf(fact_358_less__SucI,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_nat @ M @ N )
% 0.30/0.49 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_SucI
% 0.30/0.49 thf(fact_359_less__SucE,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.30/0.49 => ( ~ ( ord_less_nat @ M @ N )
% 0.30/0.49 => ( M = N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_SucE
% 0.30/0.49 thf(fact_360_Suc__lessI,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_nat @ M @ N )
% 0.30/0.49 => ( ( ( suc @ M )
% 0.30/0.49 != N )
% 0.30/0.49 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_lessI
% 0.30/0.49 thf(fact_361_Suc__lessE,axiom,
% 0.30/0.49 ! [I2: nat,K: nat] :
% 0.30/0.49 ( ( ord_less_nat @ ( suc @ I2 ) @ K )
% 0.30/0.49 => ~ ! [J2: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I2 @ J2 )
% 0.30/0.49 => ( K
% 0.30/0.49 != ( suc @ J2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_lessE
% 0.30/0.49 thf(fact_362_Suc__lessD,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 0.30/0.49 => ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_lessD
% 0.30/0.49 thf(fact_363_Nat_OlessE,axiom,
% 0.30/0.49 ! [I2: nat,K: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I2 @ K )
% 0.30/0.49 => ( ( K
% 0.30/0.49 != ( suc @ I2 ) )
% 0.30/0.49 => ~ ! [J2: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I2 @ J2 )
% 0.30/0.49 => ( K
% 0.30/0.49 != ( suc @ J2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Nat.lessE
% 0.30/0.49 thf(fact_364_Suc__leD,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.30/0.49 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_leD
% 0.30/0.49 thf(fact_365_le__SucE,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.49 => ( ~ ( ord_less_eq_nat @ M @ N )
% 0.30/0.49 => ( M
% 0.30/0.49 = ( suc @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_SucE
% 0.30/0.49 thf(fact_366_le__SucI,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.49 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_SucI
% 0.30/0.49 thf(fact_367_Suc__le__D,axiom,
% 0.30/0.49 ! [N: nat,M6: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
% 0.30/0.49 => ? [M4: nat] :
% 0.30/0.49 ( M6
% 0.30/0.49 = ( suc @ M4 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_le_D
% 0.30/0.49 thf(fact_368_le__Suc__eq,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.49 = ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.49 | ( M
% 0.30/0.49 = ( suc @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_Suc_eq
% 0.30/0.49 thf(fact_369_Suc__n__not__le__n,axiom,
% 0.30/0.49 ! [N: nat] :
% 0.30/0.49 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_n_not_le_n
% 0.30/0.49 thf(fact_370_not__less__eq__eq,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 0.30/0.49 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 0.30/0.49
% 0.30/0.49 % not_less_eq_eq
% 0.30/0.49 thf(fact_371_full__nat__induct,axiom,
% 0.30/0.49 ! [P: nat > $o,N: nat] :
% 0.30/0.49 ( ! [N2: nat] :
% 0.30/0.49 ( ! [M2: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
% 0.30/0.49 => ( P @ M2 ) )
% 0.30/0.49 => ( P @ N2 ) )
% 0.30/0.49 => ( P @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % full_nat_induct
% 0.30/0.49 thf(fact_372_nat__induct__at__least,axiom,
% 0.30/0.49 ! [M: nat,N: nat,P: nat > $o] :
% 0.30/0.49 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.49 => ( ( P @ M )
% 0.30/0.49 => ( ! [N2: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.49 => ( ( P @ N2 )
% 0.30/0.49 => ( P @ ( suc @ N2 ) ) ) )
% 0.30/0.49 => ( P @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_induct_at_least
% 0.30/0.49 thf(fact_373_transitive__stepwise__le,axiom,
% 0.30/0.49 ! [M: nat,N: nat,R: nat > nat > $o] :
% 0.30/0.49 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.49 => ( ! [X5: nat] : ( R @ X5 @ X5 )
% 0.30/0.49 => ( ! [X5: nat,Y4: nat,Z2: nat] :
% 0.30/0.49 ( ( R @ X5 @ Y4 )
% 0.30/0.49 => ( ( R @ Y4 @ Z2 )
% 0.30/0.49 => ( R @ X5 @ Z2 ) ) )
% 0.30/0.49 => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
% 0.30/0.49 => ( R @ M @ N ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % transitive_stepwise_le
% 0.30/0.49 thf(fact_374_nat__arith_Osuc1,axiom,
% 0.30/0.49 ! [A2: nat,K: nat,A: nat] :
% 0.30/0.49 ( ( A2
% 0.30/0.49 = ( plus_plus_nat @ K @ A ) )
% 0.30/0.49 => ( ( suc @ A2 )
% 0.30/0.49 = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_arith.suc1
% 0.30/0.49 thf(fact_375_add__Suc,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.30/0.49 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_Suc
% 0.30/0.49 thf(fact_376_add__Suc__shift,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.30/0.49 = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_Suc_shift
% 0.30/0.49 thf(fact_377_Suc__div__le__mono,axiom,
% 0.30/0.49 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_div_le_mono
% 0.30/0.49 thf(fact_378_insertsimp_H,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N: nat,L: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49 => ( ( vEBT_VEBT_minNull @ T )
% 0.30/0.49 => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ L ) @ one_one_nat ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % insertsimp'
% 0.30/0.49 thf(fact_379_Suc__nat__number__of__add,axiom,
% 0.30/0.49 ! [V: num,N: nat] :
% 0.30/0.49 ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
% 0.30/0.49 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_nat_number_of_add
% 0.30/0.49 thf(fact_380_lift__Suc__mono__less__iff,axiom,
% 0.30/0.49 ! [F: nat > real,N: nat,M: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.49 => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 0.30/0.49 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_mono_less_iff
% 0.30/0.49 thf(fact_381_lift__Suc__mono__less__iff,axiom,
% 0.30/0.49 ! [F: nat > rat,N: nat,M: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.49 => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
% 0.30/0.49 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_mono_less_iff
% 0.30/0.49 thf(fact_382_lift__Suc__mono__less__iff,axiom,
% 0.30/0.49 ! [F: nat > num,N: nat,M: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.49 => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
% 0.30/0.49 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_mono_less_iff
% 0.30/0.49 thf(fact_383_lift__Suc__mono__less__iff,axiom,
% 0.30/0.49 ! [F: nat > nat,N: nat,M: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.49 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 0.30/0.49 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_mono_less_iff
% 0.30/0.49 thf(fact_384_lift__Suc__mono__less__iff,axiom,
% 0.30/0.49 ! [F: nat > int,N: nat,M: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.49 => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 0.30/0.49 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_mono_less_iff
% 0.30/0.49 thf(fact_385_lift__Suc__mono__less,axiom,
% 0.30/0.49 ! [F: nat > real,N: nat,N4: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.49 => ( ( ord_less_nat @ N @ N4 )
% 0.30/0.49 => ( ord_less_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_mono_less
% 0.30/0.49 thf(fact_386_lift__Suc__mono__less,axiom,
% 0.30/0.49 ! [F: nat > rat,N: nat,N4: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.49 => ( ( ord_less_nat @ N @ N4 )
% 0.30/0.49 => ( ord_less_rat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_mono_less
% 0.30/0.49 thf(fact_387_lift__Suc__mono__less,axiom,
% 0.30/0.49 ! [F: nat > num,N: nat,N4: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.49 => ( ( ord_less_nat @ N @ N4 )
% 0.30/0.49 => ( ord_less_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_mono_less
% 0.30/0.49 thf(fact_388_lift__Suc__mono__less,axiom,
% 0.30/0.49 ! [F: nat > nat,N: nat,N4: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.49 => ( ( ord_less_nat @ N @ N4 )
% 0.30/0.49 => ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_mono_less
% 0.30/0.49 thf(fact_389_lift__Suc__mono__less,axiom,
% 0.30/0.49 ! [F: nat > int,N: nat,N4: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.49 => ( ( ord_less_nat @ N @ N4 )
% 0.30/0.49 => ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_mono_less
% 0.30/0.49 thf(fact_390_lift__Suc__mono__le,axiom,
% 0.30/0.49 ! [F: nat > set_int,N: nat,N4: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.49 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.49 => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_mono_le
% 0.30/0.49 thf(fact_391_lift__Suc__mono__le,axiom,
% 0.30/0.49 ! [F: nat > rat,N: nat,N4: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.49 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.49 => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_mono_le
% 0.30/0.49 thf(fact_392_lift__Suc__mono__le,axiom,
% 0.30/0.49 ! [F: nat > num,N: nat,N4: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.49 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.49 => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_mono_le
% 0.30/0.49 thf(fact_393_lift__Suc__mono__le,axiom,
% 0.30/0.49 ! [F: nat > nat,N: nat,N4: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.49 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.49 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_mono_le
% 0.30/0.49 thf(fact_394_lift__Suc__mono__le,axiom,
% 0.30/0.49 ! [F: nat > int,N: nat,N4: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.49 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.49 => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_mono_le
% 0.30/0.49 thf(fact_395_lift__Suc__antimono__le,axiom,
% 0.30/0.49 ! [F: nat > set_int,N: nat,N4: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.30/0.49 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.49 => ( ord_less_eq_set_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_antimono_le
% 0.30/0.49 thf(fact_396_lift__Suc__antimono__le,axiom,
% 0.30/0.49 ! [F: nat > rat,N: nat,N4: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.30/0.49 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.49 => ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_antimono_le
% 0.30/0.49 thf(fact_397_lift__Suc__antimono__le,axiom,
% 0.30/0.49 ! [F: nat > num,N: nat,N4: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.30/0.49 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.49 => ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_antimono_le
% 0.30/0.49 thf(fact_398_lift__Suc__antimono__le,axiom,
% 0.30/0.49 ! [F: nat > nat,N: nat,N4: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.30/0.49 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.49 => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_antimono_le
% 0.30/0.49 thf(fact_399_lift__Suc__antimono__le,axiom,
% 0.30/0.49 ! [F: nat > int,N: nat,N4: nat] :
% 0.30/0.49 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.30/0.49 => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.49 => ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % lift_Suc_antimono_le
% 0.30/0.49 thf(fact_400_Suc__leI,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_nat @ M @ N )
% 0.30/0.49 => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_leI
% 0.30/0.49 thf(fact_401_Suc__le__eq,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.30/0.49 = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_le_eq
% 0.30/0.49 thf(fact_402_dec__induct,axiom,
% 0.30/0.49 ! [I2: nat,J: nat,P: nat > $o] :
% 0.30/0.49 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.49 => ( ( P @ I2 )
% 0.30/0.49 => ( ! [N2: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ I2 @ N2 )
% 0.30/0.49 => ( ( ord_less_nat @ N2 @ J )
% 0.30/0.49 => ( ( P @ N2 )
% 0.30/0.49 => ( P @ ( suc @ N2 ) ) ) ) )
% 0.30/0.49 => ( P @ J ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dec_induct
% 0.30/0.49 thf(fact_403_inc__induct,axiom,
% 0.30/0.49 ! [I2: nat,J: nat,P: nat > $o] :
% 0.30/0.49 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.49 => ( ( P @ J )
% 0.30/0.49 => ( ! [N2: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ I2 @ N2 )
% 0.30/0.49 => ( ( ord_less_nat @ N2 @ J )
% 0.30/0.49 => ( ( P @ ( suc @ N2 ) )
% 0.30/0.49 => ( P @ N2 ) ) ) )
% 0.30/0.49 => ( P @ I2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % inc_induct
% 0.30/0.49 thf(fact_404_Suc__le__lessD,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.30/0.49 => ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_le_lessD
% 0.30/0.49 thf(fact_405_le__less__Suc__eq,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.49 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.30/0.49 = ( N = M ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_less_Suc_eq
% 0.30/0.49 thf(fact_406_less__Suc__eq__le,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.30/0.49 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_Suc_eq_le
% 0.30/0.49 thf(fact_407_less__eq__Suc__le,axiom,
% 0.30/0.49 ( ord_less_nat
% 0.30/0.49 = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_eq_Suc_le
% 0.30/0.49 thf(fact_408_le__imp__less__Suc,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.49 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_imp_less_Suc
% 0.30/0.49 thf(fact_409_less__natE,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_nat @ M @ N )
% 0.30/0.49 => ~ ! [Q2: nat] :
% 0.30/0.49 ( N
% 0.30/0.49 != ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_natE
% 0.30/0.49 thf(fact_410_less__add__Suc1,axiom,
% 0.30/0.49 ! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_add_Suc1
% 0.30/0.49 thf(fact_411_less__add__Suc2,axiom,
% 0.30/0.49 ! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_add_Suc2
% 0.30/0.49 thf(fact_412_less__iff__Suc__add,axiom,
% 0.30/0.49 ( ord_less_nat
% 0.30/0.49 = ( ^ [M3: nat,N3: nat] :
% 0.30/0.49 ? [K2: nat] :
% 0.30/0.49 ( N3
% 0.30/0.49 = ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_iff_Suc_add
% 0.30/0.49 thf(fact_413_less__imp__Suc__add,axiom,
% 0.30/0.49 ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_nat @ M @ N )
% 0.30/0.49 => ? [K3: nat] :
% 0.30/0.49 ( N
% 0.30/0.49 = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_imp_Suc_add
% 0.30/0.49 thf(fact_414_Suc__eq__plus1,axiom,
% 0.30/0.49 ( suc
% 0.30/0.49 = ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_eq_plus1
% 0.30/0.49 thf(fact_415_plus__1__eq__Suc,axiom,
% 0.30/0.49 ( ( plus_plus_nat @ one_one_nat )
% 0.30/0.49 = suc ) ).
% 0.30/0.49
% 0.30/0.49 % plus_1_eq_Suc
% 0.30/0.49 thf(fact_416_Suc__eq__plus1__left,axiom,
% 0.30/0.49 ( suc
% 0.30/0.49 = ( plus_plus_nat @ one_one_nat ) ) ).
% 0.30/0.49
% 0.30/0.49 % Suc_eq_plus1_left
% 0.30/0.49 thf(fact_417_invar__vebt_Ointros_I5_J,axiom,
% 0.30/0.49 ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 0.30/0.49 ( ! [X5: vEBT_VEBT] :
% 0.30/0.49 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.49 => ( vEBT_invar_vebt @ X5 @ N ) )
% 0.30/0.49 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.30/0.49 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.30/0.49 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.49 => ( ( M
% 0.30/0.49 = ( suc @ N ) )
% 0.30/0.49 => ( ( Deg
% 0.30/0.49 = ( plus_plus_nat @ N @ M ) )
% 0.30/0.49 => ( ! [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.49 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X4 ) )
% 0.30/0.49 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 0.30/0.49 => ( ( ( Mi = Ma )
% 0.30/0.49 => ! [X5: vEBT_VEBT] :
% 0.30/0.49 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.49 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 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 => ! [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.49 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 0.30/0.49 = I3 )
% 0.30/0.49 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 0.30/0.49 & ! [X5: nat] :
% 0.30/0.49 ( ( ( ( vEBT_VEBT_high @ X5 @ N )
% 0.30/0.49 = I3 )
% 0.30/0.49 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
% 0.30/0.49 => ( ( ord_less_nat @ Mi @ X5 )
% 0.30/0.49 & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
% 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 % invar_vebt.intros(5)
% 0.30/0.49 thf(fact_418_invar__vebt_Ointros_I4_J,axiom,
% 0.30/0.49 ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 0.30/0.49 ( ! [X5: vEBT_VEBT] :
% 0.30/0.49 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.49 => ( vEBT_invar_vebt @ X5 @ N ) )
% 0.30/0.49 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.30/0.49 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.30/0.49 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.49 => ( ( M = N )
% 0.30/0.49 => ( ( Deg
% 0.30/0.49 = ( plus_plus_nat @ N @ M ) )
% 0.30/0.49 => ( ! [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.49 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X4 ) )
% 0.30/0.49 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 0.30/0.49 => ( ( ( Mi = Ma )
% 0.30/0.49 => ! [X5: vEBT_VEBT] :
% 0.30/0.49 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.49 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 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 => ! [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.49 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 0.30/0.49 = I3 )
% 0.30/0.49 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 0.30/0.49 & ! [X5: nat] :
% 0.30/0.49 ( ( ( ( vEBT_VEBT_high @ X5 @ N )
% 0.30/0.49 = I3 )
% 0.30/0.49 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
% 0.30/0.49 => ( ( ord_less_nat @ Mi @ X5 )
% 0.30/0.49 & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
% 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 % invar_vebt.intros(4)
% 0.30/0.49 thf(fact_419_is__num__normalize_I1_J,axiom,
% 0.30/0.49 ! [A: real,B: real,C: real] :
% 0.30/0.49 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.30/0.49 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % is_num_normalize(1)
% 0.30/0.49 thf(fact_420_is__num__normalize_I1_J,axiom,
% 0.30/0.49 ! [A: rat,B: rat,C: rat] :
% 0.30/0.49 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.30/0.49 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % is_num_normalize(1)
% 0.30/0.49 thf(fact_421_is__num__normalize_I1_J,axiom,
% 0.30/0.49 ! [A: int,B: int,C: int] :
% 0.30/0.49 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.49 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % is_num_normalize(1)
% 0.30/0.49 thf(fact_422_add__One__commute,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( plus_plus_num @ one @ N )
% 0.30/0.49 = ( plus_plus_num @ N @ one ) ) ).
% 0.30/0.49
% 0.30/0.49 % add_One_commute
% 0.30/0.49 thf(fact_423_le__num__One__iff,axiom,
% 0.30/0.49 ! [X2: num] :
% 0.30/0.49 ( ( ord_less_eq_num @ X2 @ one )
% 0.30/0.49 = ( X2 = one ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_num_One_iff
% 0.30/0.49 thf(fact_424_insersimp_H,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N: nat,Y3: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 )
% 0.30/0.49 => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ Y3 ) @ one_one_nat ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % insersimp'
% 0.30/0.49 thf(fact_425_height__node,axiom,
% 0.30/0.49 ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 0.30/0.49 => ( ord_less_eq_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % height_node
% 0.30/0.49 thf(fact_426_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_427_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_428_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_429_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_430_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_431_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_432_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_433_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_434_div__le__mono,axiom,
% 0.30/0.49 ! [M: nat,N: nat,K: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.49 => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % div_le_mono
% 0.30/0.49 thf(fact_435_div__le__dividend,axiom,
% 0.30/0.49 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 0.30/0.49
% 0.30/0.49 % div_le_dividend
% 0.30/0.49 thf(fact_436_one__le__numeral,axiom,
% 0.30/0.49 ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_le_numeral
% 0.30/0.49 thf(fact_437_one__le__numeral,axiom,
% 0.30/0.49 ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_le_numeral
% 0.30/0.49 thf(fact_438_one__le__numeral,axiom,
% 0.30/0.49 ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_le_numeral
% 0.30/0.49 thf(fact_439_one__le__numeral,axiom,
% 0.30/0.49 ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_le_numeral
% 0.30/0.49 thf(fact_440_not__numeral__less__one,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% 0.30/0.49
% 0.30/0.49 % not_numeral_less_one
% 0.30/0.49 thf(fact_441_not__numeral__less__one,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% 0.30/0.49
% 0.30/0.49 % not_numeral_less_one
% 0.30/0.49 thf(fact_442_not__numeral__less__one,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% 0.30/0.49
% 0.30/0.49 % not_numeral_less_one
% 0.30/0.49 thf(fact_443_not__numeral__less__one,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% 0.30/0.49
% 0.30/0.49 % not_numeral_less_one
% 0.30/0.49 thf(fact_444_numeral__Bit0,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 0.30/0.49 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_Bit0
% 0.30/0.49 thf(fact_445_numeral__Bit0,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 0.30/0.49 = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_Bit0
% 0.30/0.49 thf(fact_446_numeral__Bit0,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 0.30/0.49 = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_Bit0
% 0.30/0.49 thf(fact_447_numeral__Bit0,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 0.30/0.49 = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_Bit0
% 0.30/0.49 thf(fact_448_numeral__Bit0,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 0.30/0.49 = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_Bit0
% 0.30/0.49 thf(fact_449_one__plus__numeral__commute,axiom,
% 0.30/0.49 ! [X2: num] :
% 0.30/0.49 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X2 ) )
% 0.30/0.49 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X2 ) @ one_one_complex ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral_commute
% 0.30/0.49 thf(fact_450_one__plus__numeral__commute,axiom,
% 0.30/0.49 ! [X2: num] :
% 0.30/0.49 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X2 ) )
% 0.30/0.49 = ( plus_plus_real @ ( numeral_numeral_real @ X2 ) @ one_one_real ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral_commute
% 0.30/0.49 thf(fact_451_one__plus__numeral__commute,axiom,
% 0.30/0.49 ! [X2: num] :
% 0.30/0.49 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X2 ) )
% 0.30/0.49 = ( plus_plus_rat @ ( numeral_numeral_rat @ X2 ) @ one_one_rat ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral_commute
% 0.30/0.49 thf(fact_452_one__plus__numeral__commute,axiom,
% 0.30/0.49 ! [X2: num] :
% 0.30/0.49 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X2 ) )
% 0.30/0.49 = ( plus_plus_nat @ ( numeral_numeral_nat @ X2 ) @ one_one_nat ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral_commute
% 0.30/0.49 thf(fact_453_one__plus__numeral__commute,axiom,
% 0.30/0.49 ! [X2: num] :
% 0.30/0.49 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X2 ) )
% 0.30/0.49 = ( plus_plus_int @ ( numeral_numeral_int @ X2 ) @ one_one_int ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral_commute
% 0.30/0.49 thf(fact_454_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_455_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_456_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_457_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_458_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_459_divide__numeral__1,axiom,
% 0.30/0.49 ! [A: complex] :
% 0.30/0.49 ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
% 0.30/0.49 = A ) ).
% 0.30/0.49
% 0.30/0.49 % divide_numeral_1
% 0.30/0.49 thf(fact_460_divide__numeral__1,axiom,
% 0.30/0.49 ! [A: real] :
% 0.30/0.49 ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
% 0.30/0.49 = A ) ).
% 0.30/0.49
% 0.30/0.49 % divide_numeral_1
% 0.30/0.49 thf(fact_461_divide__numeral__1,axiom,
% 0.30/0.49 ! [A: rat] :
% 0.30/0.49 ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
% 0.30/0.49 = A ) ).
% 0.30/0.49
% 0.30/0.49 % divide_numeral_1
% 0.30/0.49 thf(fact_462_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_463_numeral__Bit0__div__2,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49 = ( numeral_numeral_nat @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_Bit0_div_2
% 0.30/0.49 thf(fact_464_numeral__Bit0__div__2,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.49 = ( numeral_numeral_int @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_Bit0_div_2
% 0.30/0.49 thf(fact_465_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_466_in__children__def,axiom,
% 0.30/0.49 ( vEBT_V5917875025757280293ildren
% 0.30/0.49 = ( ^ [N3: nat,TreeList3: list_VEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X @ N3 ) ) @ ( vEBT_VEBT_low @ X @ N3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % in_children_def
% 0.30/0.49 thf(fact_467_member__inv,axiom,
% 0.30/0.49 ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 0.30/0.49 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 0.30/0.49 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.49 & ( ( X2 = Mi )
% 0.30/0.49 | ( X2 = Ma )
% 0.30/0.49 | ( ( ord_less_nat @ X2 @ Ma )
% 0.30/0.49 & ( ord_less_nat @ Mi @ X2 )
% 0.30/0.49 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.30/0.49 & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % member_inv
% 0.30/0.49 thf(fact_468_power__increasing__iff,axiom,
% 0.30/0.49 ! [B: real,X2: nat,Y3: nat] :
% 0.30/0.49 ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.49 => ( ( ord_less_eq_real @ ( power_power_real @ B @ X2 ) @ ( power_power_real @ B @ Y3 ) )
% 0.30/0.49 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_increasing_iff
% 0.30/0.49 thf(fact_469_power__increasing__iff,axiom,
% 0.30/0.49 ! [B: rat,X2: nat,Y3: nat] :
% 0.30/0.49 ( ( ord_less_rat @ one_one_rat @ B )
% 0.30/0.49 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X2 ) @ ( power_power_rat @ B @ Y3 ) )
% 0.30/0.49 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_increasing_iff
% 0.30/0.49 thf(fact_470_power__increasing__iff,axiom,
% 0.30/0.49 ! [B: nat,X2: nat,Y3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ one_one_nat @ B )
% 0.30/0.49 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X2 ) @ ( power_power_nat @ B @ Y3 ) )
% 0.30/0.49 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_increasing_iff
% 0.30/0.49 thf(fact_471_power__increasing__iff,axiom,
% 0.30/0.49 ! [B: int,X2: nat,Y3: nat] :
% 0.30/0.49 ( ( ord_less_int @ one_one_int @ B )
% 0.30/0.49 => ( ( ord_less_eq_int @ ( power_power_int @ B @ X2 ) @ ( power_power_int @ B @ Y3 ) )
% 0.30/0.49 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_increasing_iff
% 0.30/0.49 thf(fact_472_less__shift,axiom,
% 0.30/0.49 ( ord_less_nat
% 0.30/0.49 = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_shift
% 0.30/0.49 thf(fact_473_greater__shift,axiom,
% 0.30/0.49 ( ord_less_nat
% 0.30/0.49 = ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % greater_shift
% 0.30/0.49 thf(fact_474_misiz,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N: nat,M: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49 => ( ( ( some_nat @ M )
% 0.30/0.49 = ( vEBT_vebt_mint @ T ) )
% 0.30/0.49 => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % misiz
% 0.30/0.49 thf(fact_475_power__strict__increasing__iff,axiom,
% 0.30/0.49 ! [B: real,X2: nat,Y3: nat] :
% 0.30/0.49 ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.49 => ( ( ord_less_real @ ( power_power_real @ B @ X2 ) @ ( power_power_real @ B @ Y3 ) )
% 0.30/0.49 = ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_strict_increasing_iff
% 0.30/0.49 thf(fact_476_power__strict__increasing__iff,axiom,
% 0.30/0.49 ! [B: rat,X2: nat,Y3: nat] :
% 0.30/0.49 ( ( ord_less_rat @ one_one_rat @ B )
% 0.30/0.49 => ( ( ord_less_rat @ ( power_power_rat @ B @ X2 ) @ ( power_power_rat @ B @ Y3 ) )
% 0.30/0.49 = ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_strict_increasing_iff
% 0.30/0.49 thf(fact_477_power__strict__increasing__iff,axiom,
% 0.30/0.49 ! [B: nat,X2: nat,Y3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ one_one_nat @ B )
% 0.30/0.49 => ( ( ord_less_nat @ ( power_power_nat @ B @ X2 ) @ ( power_power_nat @ B @ Y3 ) )
% 0.30/0.49 = ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_strict_increasing_iff
% 0.30/0.49 thf(fact_478_power__strict__increasing__iff,axiom,
% 0.30/0.49 ! [B: int,X2: nat,Y3: nat] :
% 0.30/0.49 ( ( ord_less_int @ one_one_int @ B )
% 0.30/0.49 => ( ( ord_less_int @ ( power_power_int @ B @ X2 ) @ ( power_power_int @ B @ Y3 ) )
% 0.30/0.49 = ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_strict_increasing_iff
% 0.30/0.49 thf(fact_479_pred__list__to__short,axiom,
% 0.30/0.49 ! [Deg: nat,X2: nat,Ma: nat,TreeList: 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 @ X2 @ Ma )
% 0.30/0.49 => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X2 @ ( 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 @ TreeList @ Summary ) @ X2 )
% 0.30/0.49 = none_nat ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % pred_list_to_short
% 0.30/0.49 thf(fact_480_succ__list__to__short,axiom,
% 0.30/0.49 ! [Deg: nat,Mi: nat,X2: nat,TreeList: 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 @ X2 )
% 0.30/0.49 => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X2 @ ( 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 @ TreeList @ Summary ) @ X2 )
% 0.30/0.49 = none_nat ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % succ_list_to_short
% 0.30/0.49 thf(fact_481_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 => ? [N2: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 0.30/0.49 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ex_power_ivl1
% 0.30/0.49 thf(fact_482_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 => ? [N2: nat] :
% 0.30/0.49 ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 0.30/0.49 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ex_power_ivl2
% 0.30/0.49 thf(fact_483_semiring__norm_I76_J,axiom,
% 0.30/0.49 ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(76)
% 0.30/0.49 thf(fact_484_min__Null__member,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,X2: nat] :
% 0.30/0.49 ( ( vEBT_VEBT_minNull @ T )
% 0.30/0.49 => ~ ( vEBT_vebt_member @ T @ X2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % min_Null_member
% 0.30/0.49 thf(fact_485_both__member__options__equiv__member,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49 => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
% 0.30/0.49 = ( vEBT_vebt_member @ T @ X2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % both_member_options_equiv_member
% 0.30/0.49 thf(fact_486_valid__member__both__member__options,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49 => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
% 0.30/0.49 => ( vEBT_vebt_member @ T @ X2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % valid_member_both_member_options
% 0.30/0.49 thf(fact_487_minNullmin,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT] :
% 0.30/0.49 ( ( vEBT_VEBT_minNull @ T )
% 0.30/0.49 => ( ( vEBT_vebt_mint @ T )
% 0.30/0.49 = none_nat ) ) ).
% 0.30/0.49
% 0.30/0.49 % minNullmin
% 0.30/0.49 thf(fact_488_minminNull,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT] :
% 0.30/0.49 ( ( ( vEBT_vebt_mint @ T )
% 0.30/0.49 = none_nat )
% 0.30/0.49 => ( vEBT_VEBT_minNull @ T ) ) ).
% 0.30/0.49
% 0.30/0.49 % minminNull
% 0.30/0.49 thf(fact_489_semiring__norm_I87_J,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( ( bit0 @ M )
% 0.30/0.49 = ( bit0 @ N ) )
% 0.30/0.49 = ( M = N ) ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(87)
% 0.30/0.49 thf(fact_490_mint__member,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49 => ( ( ( vEBT_vebt_mint @ T )
% 0.30/0.49 = ( some_nat @ Maxi ) )
% 0.30/0.49 => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mint_member
% 0.30/0.49 thf(fact_491_mint__corr__help,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N: nat,Mini: nat,X2: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49 => ( ( ( vEBT_vebt_mint @ T )
% 0.30/0.49 = ( some_nat @ Mini ) )
% 0.30/0.49 => ( ( vEBT_vebt_member @ T @ X2 )
% 0.30/0.49 => ( ord_less_eq_nat @ Mini @ X2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mint_corr_help
% 0.30/0.49 thf(fact_492_member__bound,axiom,
% 0.30/0.49 ! [Tree: vEBT_VEBT,X2: nat,N: nat] :
% 0.30/0.49 ( ( vEBT_vebt_member @ Tree @ X2 )
% 0.30/0.49 => ( ( vEBT_invar_vebt @ Tree @ N )
% 0.30/0.49 => ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % member_bound
% 0.30/0.49 thf(fact_493_geqmaxNone,axiom,
% 0.30/0.49 ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X2: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 0.30/0.49 => ( ( ord_less_eq_nat @ Ma @ X2 )
% 0.30/0.49 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 0.30/0.49 = none_nat ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % geqmaxNone
% 0.30/0.49 thf(fact_494_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_495_semiring__norm_I83_J,axiom,
% 0.30/0.49 ! [N: num] :
% 0.30/0.49 ( one
% 0.30/0.49 != ( bit0 @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(83)
% 0.30/0.49 thf(fact_496_power__one,axiom,
% 0.30/0.49 ! [N: nat] :
% 0.30/0.49 ( ( power_power_rat @ one_one_rat @ N )
% 0.30/0.49 = one_one_rat ) ).
% 0.30/0.49
% 0.30/0.49 % power_one
% 0.30/0.49 thf(fact_497_power__one,axiom,
% 0.30/0.49 ! [N: nat] :
% 0.30/0.49 ( ( power_power_nat @ one_one_nat @ N )
% 0.30/0.49 = one_one_nat ) ).
% 0.30/0.49
% 0.30/0.49 % power_one
% 0.30/0.49 thf(fact_498_power__one,axiom,
% 0.30/0.49 ! [N: nat] :
% 0.30/0.49 ( ( power_power_real @ one_one_real @ N )
% 0.30/0.49 = one_one_real ) ).
% 0.30/0.49
% 0.30/0.49 % power_one
% 0.30/0.49 thf(fact_499_power__one,axiom,
% 0.30/0.49 ! [N: nat] :
% 0.30/0.49 ( ( power_power_int @ one_one_int @ N )
% 0.30/0.49 = one_one_int ) ).
% 0.30/0.49
% 0.30/0.49 % power_one
% 0.30/0.49 thf(fact_500_power__one,axiom,
% 0.30/0.49 ! [N: nat] :
% 0.30/0.49 ( ( power_power_complex @ one_one_complex @ N )
% 0.30/0.49 = one_one_complex ) ).
% 0.30/0.49
% 0.30/0.49 % power_one
% 0.30/0.49 thf(fact_501_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_502_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_503_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_504_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_505_not__Some__eq,axiom,
% 0.30/0.49 ! [X2: option4927543243414619207at_nat] :
% 0.30/0.49 ( ( ! [Y: product_prod_nat_nat] :
% 0.30/0.49 ( X2
% 0.30/0.49 != ( some_P7363390416028606310at_nat @ Y ) ) )
% 0.30/0.49 = ( X2 = none_P5556105721700978146at_nat ) ) ).
% 0.30/0.49
% 0.30/0.49 % not_Some_eq
% 0.30/0.49 thf(fact_506_not__Some__eq,axiom,
% 0.30/0.49 ! [X2: option_nat] :
% 0.30/0.49 ( ( ! [Y: nat] :
% 0.30/0.49 ( X2
% 0.30/0.49 != ( some_nat @ Y ) ) )
% 0.30/0.49 = ( X2 = none_nat ) ) ).
% 0.30/0.49
% 0.30/0.49 % not_Some_eq
% 0.30/0.49 thf(fact_507_not__Some__eq,axiom,
% 0.30/0.49 ! [X2: option_num] :
% 0.30/0.49 ( ( ! [Y: num] :
% 0.30/0.49 ( X2
% 0.30/0.49 != ( some_num @ Y ) ) )
% 0.30/0.49 = ( X2 = none_num ) ) ).
% 0.30/0.49
% 0.30/0.49 % not_Some_eq
% 0.30/0.49 thf(fact_508_not__None__eq,axiom,
% 0.30/0.49 ! [X2: option4927543243414619207at_nat] :
% 0.30/0.49 ( ( X2 != none_P5556105721700978146at_nat )
% 0.30/0.49 = ( ? [Y: product_prod_nat_nat] :
% 0.30/0.49 ( X2
% 0.30/0.49 = ( some_P7363390416028606310at_nat @ Y ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % not_None_eq
% 0.30/0.49 thf(fact_509_not__None__eq,axiom,
% 0.30/0.49 ! [X2: option_nat] :
% 0.30/0.49 ( ( X2 != none_nat )
% 0.30/0.49 = ( ? [Y: nat] :
% 0.30/0.49 ( X2
% 0.30/0.49 = ( some_nat @ Y ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % not_None_eq
% 0.30/0.49 thf(fact_510_not__None__eq,axiom,
% 0.30/0.49 ! [X2: option_num] :
% 0.30/0.49 ( ( X2 != none_num )
% 0.30/0.49 = ( ? [Y: num] :
% 0.30/0.49 ( X2
% 0.30/0.49 = ( some_num @ Y ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % not_None_eq
% 0.30/0.49 thf(fact_511_post__member__pre__member,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.49 => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.49 => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X2 ) @ Y3 )
% 0.30/0.49 => ( ( vEBT_vebt_member @ T @ Y3 )
% 0.30/0.49 | ( X2 = Y3 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % post_member_pre_member
% 0.30/0.49 thf(fact_512_member__correct,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49 => ( ( vEBT_vebt_member @ T @ X2 )
% 0.30/0.49 = ( member_nat @ X2 @ ( vEBT_set_vebt @ T ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % member_correct
% 0.30/0.49 thf(fact_513_semiring__norm_I6_J,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.49 = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(6)
% 0.30/0.49 thf(fact_514_semiring__norm_I71_J,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.49 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(71)
% 0.30/0.49 thf(fact_515_semiring__norm_I78_J,axiom,
% 0.30/0.49 ! [M: num,N: num] :
% 0.30/0.49 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.49 = ( ord_less_num @ M @ N ) ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(78)
% 0.30/0.49 thf(fact_516_semiring__norm_I68_J,axiom,
% 0.30/0.49 ! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(68)
% 0.30/0.49 thf(fact_517_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_518_power__inject__exp,axiom,
% 0.30/0.49 ! [A: real,M: nat,N: 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 @ N ) )
% 0.30/0.49 = ( M = N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_inject_exp
% 0.30/0.49 thf(fact_519_power__inject__exp,axiom,
% 0.30/0.49 ! [A: rat,M: nat,N: 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 @ N ) )
% 0.30/0.49 = ( M = N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_inject_exp
% 0.30/0.49 thf(fact_520_power__inject__exp,axiom,
% 0.30/0.49 ! [A: nat,M: nat,N: 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 @ N ) )
% 0.30/0.49 = ( M = N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_inject_exp
% 0.30/0.49 thf(fact_521_power__inject__exp,axiom,
% 0.30/0.49 ! [A: int,M: nat,N: 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 @ N ) )
% 0.30/0.49 = ( M = N ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_inject_exp
% 0.30/0.49 thf(fact_522_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_523_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_524_local_Opower__def,axiom,
% 0.30/0.49 ( vEBT_VEBT_power
% 0.30/0.49 = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).
% 0.30/0.49
% 0.30/0.49 % local.power_def
% 0.30/0.49 thf(fact_525_combine__options__cases,axiom,
% 0.30/0.49 ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 0.30/0.49 ( ( ( X2 = none_P5556105721700978146at_nat )
% 0.30/0.49 => ( P @ X2 @ Y3 ) )
% 0.30/0.49 => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 0.30/0.49 => ( P @ X2 @ Y3 ) )
% 0.30/0.49 => ( ! [A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
% 0.30/0.49 ( ( X2
% 0.30/0.49 = ( some_P7363390416028606310at_nat @ A4 ) )
% 0.30/0.49 => ( ( Y3
% 0.30/0.49 = ( some_P7363390416028606310at_nat @ B4 ) )
% 0.30/0.49 => ( P @ X2 @ Y3 ) ) )
% 0.30/0.49 => ( P @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % combine_options_cases
% 0.30/0.49 thf(fact_526_combine__options__cases,axiom,
% 0.30/0.49 ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y3: option_nat] :
% 0.30/0.49 ( ( ( X2 = none_P5556105721700978146at_nat )
% 0.30/0.49 => ( P @ X2 @ Y3 ) )
% 0.30/0.49 => ( ( ( Y3 = none_nat )
% 0.30/0.49 => ( P @ X2 @ Y3 ) )
% 0.30/0.49 => ( ! [A4: product_prod_nat_nat,B4: nat] :
% 0.30/0.49 ( ( X2
% 0.30/0.49 = ( some_P7363390416028606310at_nat @ A4 ) )
% 0.30/0.49 => ( ( Y3
% 0.30/0.49 = ( some_nat @ B4 ) )
% 0.30/0.49 => ( P @ X2 @ Y3 ) ) )
% 0.30/0.49 => ( P @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % combine_options_cases
% 0.30/0.49 thf(fact_527_combine__options__cases,axiom,
% 0.30/0.49 ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y3: option_num] :
% 0.30/0.49 ( ( ( X2 = none_P5556105721700978146at_nat )
% 0.30/0.49 => ( P @ X2 @ Y3 ) )
% 0.30/0.49 => ( ( ( Y3 = none_num )
% 0.30/0.49 => ( P @ X2 @ Y3 ) )
% 0.30/0.49 => ( ! [A4: product_prod_nat_nat,B4: num] :
% 0.30/0.49 ( ( X2
% 0.30/0.49 = ( some_P7363390416028606310at_nat @ A4 ) )
% 0.30/0.49 => ( ( Y3
% 0.30/0.49 = ( some_num @ B4 ) )
% 0.30/0.49 => ( P @ X2 @ Y3 ) ) )
% 0.30/0.49 => ( P @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % combine_options_cases
% 0.30/0.49 thf(fact_528_combine__options__cases,axiom,
% 0.30/0.49 ! [X2: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 0.30/0.49 ( ( ( X2 = none_nat )
% 0.30/0.49 => ( P @ X2 @ Y3 ) )
% 0.30/0.49 => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 0.30/0.49 => ( P @ X2 @ Y3 ) )
% 0.30/0.49 => ( ! [A4: nat,B4: product_prod_nat_nat] :
% 0.30/0.49 ( ( X2
% 0.30/0.49 = ( some_nat @ A4 ) )
% 0.30/0.49 => ( ( Y3
% 0.30/0.49 = ( some_P7363390416028606310at_nat @ B4 ) )
% 0.30/0.49 => ( P @ X2 @ Y3 ) ) )
% 0.30/0.49 => ( P @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % combine_options_cases
% 0.30/0.49 thf(fact_529_combine__options__cases,axiom,
% 0.30/0.49 ! [X2: option_nat,P: option_nat > option_nat > $o,Y3: option_nat] :
% 0.30/0.49 ( ( ( X2 = none_nat )
% 0.30/0.49 => ( P @ X2 @ Y3 ) )
% 0.30/0.49 => ( ( ( Y3 = none_nat )
% 0.30/0.49 => ( P @ X2 @ Y3 ) )
% 0.30/0.49 => ( ! [A4: nat,B4: nat] :
% 0.30/0.49 ( ( X2
% 0.30/0.49 = ( some_nat @ A4 ) )
% 0.30/0.49 => ( ( Y3
% 0.30/0.49 = ( some_nat @ B4 ) )
% 0.30/0.49 => ( P @ X2 @ Y3 ) ) )
% 0.30/0.49 => ( P @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % combine_options_cases
% 0.30/0.49 thf(fact_530_combine__options__cases,axiom,
% 0.30/0.49 ! [X2: option_nat,P: option_nat > option_num > $o,Y3: option_num] :
% 0.30/0.49 ( ( ( X2 = none_nat )
% 0.30/0.49 => ( P @ X2 @ Y3 ) )
% 0.30/0.49 => ( ( ( Y3 = none_num )
% 0.30/0.49 => ( P @ X2 @ Y3 ) )
% 0.30/0.49 => ( ! [A4: nat,B4: num] :
% 0.30/0.49 ( ( X2
% 0.30/0.49 = ( some_nat @ A4 ) )
% 0.30/0.49 => ( ( Y3
% 0.30/0.49 = ( some_num @ B4 ) )
% 0.30/0.49 => ( P @ X2 @ Y3 ) ) )
% 0.30/0.49 => ( P @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % combine_options_cases
% 0.30/0.49 thf(fact_531_combine__options__cases,axiom,
% 0.30/0.49 ! [X2: option_num,P: option_num > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 0.30/0.49 ( ( ( X2 = none_num )
% 0.30/0.49 => ( P @ X2 @ Y3 ) )
% 0.30/0.49 => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 0.30/0.49 => ( P @ X2 @ Y3 ) )
% 0.30/0.49 => ( ! [A4: num,B4: product_prod_nat_nat] :
% 0.30/0.49 ( ( X2
% 0.30/0.49 = ( some_num @ A4 ) )
% 0.30/0.49 => ( ( Y3
% 0.30/0.49 = ( some_P7363390416028606310at_nat @ B4 ) )
% 0.30/0.49 => ( P @ X2 @ Y3 ) ) )
% 0.30/0.49 => ( P @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % combine_options_cases
% 0.30/0.49 thf(fact_532_combine__options__cases,axiom,
% 0.30/0.49 ! [X2: option_num,P: option_num > option_nat > $o,Y3: option_nat] :
% 0.30/0.49 ( ( ( X2 = none_num )
% 0.30/0.49 => ( P @ X2 @ Y3 ) )
% 0.30/0.49 => ( ( ( Y3 = none_nat )
% 0.30/0.49 => ( P @ X2 @ Y3 ) )
% 0.30/0.49 => ( ! [A4: num,B4: nat] :
% 0.30/0.49 ( ( X2
% 0.30/0.49 = ( some_num @ A4 ) )
% 0.30/0.49 => ( ( Y3
% 0.30/0.49 = ( some_nat @ B4 ) )
% 0.30/0.49 => ( P @ X2 @ Y3 ) ) )
% 0.30/0.49 => ( P @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % combine_options_cases
% 0.30/0.49 thf(fact_533_combine__options__cases,axiom,
% 0.30/0.49 ! [X2: option_num,P: option_num > option_num > $o,Y3: option_num] :
% 0.30/0.49 ( ( ( X2 = none_num )
% 0.30/0.49 => ( P @ X2 @ Y3 ) )
% 0.30/0.49 => ( ( ( Y3 = none_num )
% 0.30/0.49 => ( P @ X2 @ Y3 ) )
% 0.30/0.49 => ( ! [A4: num,B4: num] :
% 0.30/0.49 ( ( X2
% 0.30/0.49 = ( some_num @ A4 ) )
% 0.30/0.49 => ( ( Y3
% 0.30/0.49 = ( some_num @ B4 ) )
% 0.30/0.49 => ( P @ X2 @ Y3 ) ) )
% 0.30/0.49 => ( P @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % combine_options_cases
% 0.30/0.49 thf(fact_534_split__option__all,axiom,
% 0.30/0.49 ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 0.30/0.49 ! [X6: option4927543243414619207at_nat] : ( P2 @ X6 ) )
% 0.30/0.49 = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 0.30/0.49 ( ( P3 @ none_P5556105721700978146at_nat )
% 0.30/0.49 & ! [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % split_option_all
% 0.30/0.49 thf(fact_535_split__option__all,axiom,
% 0.30/0.49 ( ( ^ [P2: option_nat > $o] :
% 0.30/0.49 ! [X6: option_nat] : ( P2 @ X6 ) )
% 0.30/0.49 = ( ^ [P3: option_nat > $o] :
% 0.30/0.49 ( ( P3 @ none_nat )
% 0.30/0.49 & ! [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % split_option_all
% 0.30/0.49 thf(fact_536_split__option__all,axiom,
% 0.30/0.49 ( ( ^ [P2: option_num > $o] :
% 0.30/0.49 ! [X6: option_num] : ( P2 @ X6 ) )
% 0.30/0.49 = ( ^ [P3: option_num > $o] :
% 0.30/0.49 ( ( P3 @ none_num )
% 0.30/0.49 & ! [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % split_option_all
% 0.30/0.49 thf(fact_537_split__option__ex,axiom,
% 0.30/0.49 ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 0.30/0.49 ? [X6: option4927543243414619207at_nat] : ( P2 @ X6 ) )
% 0.30/0.49 = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 0.30/0.49 ( ( P3 @ none_P5556105721700978146at_nat )
% 0.30/0.49 | ? [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % split_option_ex
% 0.30/0.49 thf(fact_538_split__option__ex,axiom,
% 0.30/0.49 ( ( ^ [P2: option_nat > $o] :
% 0.30/0.49 ? [X6: option_nat] : ( P2 @ X6 ) )
% 0.30/0.49 = ( ^ [P3: option_nat > $o] :
% 0.30/0.49 ( ( P3 @ none_nat )
% 0.30/0.49 | ? [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % split_option_ex
% 0.30/0.49 thf(fact_539_split__option__ex,axiom,
% 0.30/0.49 ( ( ^ [P2: option_num > $o] :
% 0.30/0.49 ? [X6: option_num] : ( P2 @ X6 ) )
% 0.30/0.49 = ( ^ [P3: option_num > $o] :
% 0.30/0.49 ( ( P3 @ none_num )
% 0.30/0.49 | ? [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % split_option_ex
% 0.30/0.49 thf(fact_540_option_Oexhaust,axiom,
% 0.30/0.49 ! [Y3: option4927543243414619207at_nat] :
% 0.30/0.49 ( ( Y3 != none_P5556105721700978146at_nat )
% 0.30/0.49 => ~ ! [X23: product_prod_nat_nat] :
% 0.30/0.49 ( Y3
% 0.30/0.49 != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % option.exhaust
% 0.30/0.49 thf(fact_541_option_Oexhaust,axiom,
% 0.30/0.49 ! [Y3: option_nat] :
% 0.30/0.49 ( ( Y3 != none_nat )
% 0.30/0.49 => ~ ! [X23: nat] :
% 0.30/0.49 ( Y3
% 0.30/0.49 != ( some_nat @ X23 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % option.exhaust
% 0.30/0.49 thf(fact_542_option_Oexhaust,axiom,
% 0.30/0.49 ! [Y3: option_num] :
% 0.30/0.49 ( ( Y3 != none_num )
% 0.30/0.49 => ~ ! [X23: num] :
% 0.30/0.49 ( Y3
% 0.30/0.49 != ( some_num @ X23 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % option.exhaust
% 0.30/0.49 thf(fact_543_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 ) )
% 0.30/0.50 => ( Option != none_P5556105721700978146at_nat ) ) ).
% 0.30/0.50
% 0.30/0.50 % option.discI
% 0.30/0.50 thf(fact_544_option_OdiscI,axiom,
% 0.30/0.50 ! [Option: option_nat,X22: nat] :
% 0.30/0.50 ( ( Option
% 0.30/0.50 = ( some_nat @ X22 ) )
% 0.30/0.50 => ( Option != none_nat ) ) ).
% 0.30/0.50
% 0.30/0.50 % option.discI
% 0.30/0.50 thf(fact_545_option_OdiscI,axiom,
% 0.30/0.50 ! [Option: option_num,X22: num] :
% 0.30/0.50 ( ( Option
% 0.30/0.50 = ( some_num @ X22 ) )
% 0.30/0.50 => ( Option != none_num ) ) ).
% 0.30/0.50
% 0.30/0.50 % option.discI
% 0.30/0.50 thf(fact_546_option_Odistinct_I1_J,axiom,
% 0.30/0.50 ! [X22: product_prod_nat_nat] :
% 0.30/0.50 ( none_P5556105721700978146at_nat
% 0.30/0.50 != ( some_P7363390416028606310at_nat @ X22 ) ) ).
% 0.30/0.50
% 0.30/0.50 % option.distinct(1)
% 0.30/0.50 thf(fact_547_option_Odistinct_I1_J,axiom,
% 0.30/0.50 ! [X22: nat] :
% 0.30/0.50 ( none_nat
% 0.30/0.50 != ( some_nat @ X22 ) ) ).
% 0.30/0.50
% 0.30/0.50 % option.distinct(1)
% 0.30/0.50 thf(fact_548_option_Odistinct_I1_J,axiom,
% 0.30/0.50 ! [X22: num] :
% 0.30/0.50 ( none_num
% 0.30/0.50 != ( some_num @ X22 ) ) ).
% 0.30/0.50
% 0.30/0.50 % option.distinct(1)
% 0.30/0.50 thf(fact_549_power__divide,axiom,
% 0.30/0.50 ! [A: complex,B: complex,N: nat] :
% 0.30/0.50 ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N )
% 0.30/0.50 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_divide
% 0.30/0.50 thf(fact_550_power__divide,axiom,
% 0.30/0.50 ! [A: real,B: real,N: nat] :
% 0.30/0.50 ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
% 0.30/0.50 = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_divide
% 0.30/0.50 thf(fact_551_power__divide,axiom,
% 0.30/0.50 ! [A: rat,B: rat,N: nat] :
% 0.30/0.50 ( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N )
% 0.30/0.50 = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_divide
% 0.30/0.50 thf(fact_552_one__le__power,axiom,
% 0.30/0.50 ! [A: real,N: nat] :
% 0.30/0.50 ( ( ord_less_eq_real @ one_one_real @ A )
% 0.30/0.50 => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % one_le_power
% 0.30/0.50 thf(fact_553_one__le__power,axiom,
% 0.30/0.50 ! [A: rat,N: nat] :
% 0.30/0.50 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.30/0.50 => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % one_le_power
% 0.30/0.50 thf(fact_554_one__le__power,axiom,
% 0.30/0.50 ! [A: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.30/0.50 => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % one_le_power
% 0.30/0.50 thf(fact_555_one__le__power,axiom,
% 0.30/0.50 ! [A: int,N: nat] :
% 0.30/0.50 ( ( ord_less_eq_int @ one_one_int @ A )
% 0.30/0.50 => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % one_le_power
% 0.30/0.50 thf(fact_556_power__one__over,axiom,
% 0.30/0.50 ! [A: complex,N: nat] :
% 0.30/0.50 ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
% 0.30/0.50 = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_one_over
% 0.30/0.50 thf(fact_557_power__one__over,axiom,
% 0.30/0.50 ! [A: real,N: nat] :
% 0.30/0.50 ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
% 0.30/0.50 = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_one_over
% 0.30/0.50 thf(fact_558_power__one__over,axiom,
% 0.30/0.50 ! [A: rat,N: nat] :
% 0.30/0.50 ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
% 0.30/0.50 = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_one_over
% 0.30/0.50 thf(fact_559_power__gt1,axiom,
% 0.30/0.50 ! [A: real,N: 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 @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_gt1
% 0.30/0.50 thf(fact_560_power__gt1,axiom,
% 0.30/0.50 ! [A: rat,N: 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 @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_gt1
% 0.30/0.50 thf(fact_561_power__gt1,axiom,
% 0.30/0.50 ! [A: nat,N: 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 @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_gt1
% 0.30/0.50 thf(fact_562_power__gt1,axiom,
% 0.30/0.50 ! [A: int,N: 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 @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_gt1
% 0.30/0.50 thf(fact_563_power__less__imp__less__exp,axiom,
% 0.30/0.50 ! [A: real,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.50 => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 0.30/0.50 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_less_imp_less_exp
% 0.30/0.50 thf(fact_564_power__less__imp__less__exp,axiom,
% 0.30/0.50 ! [A: rat,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.50 => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 0.30/0.50 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_less_imp_less_exp
% 0.30/0.50 thf(fact_565_power__less__imp__less__exp,axiom,
% 0.30/0.50 ! [A: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.50 => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.30/0.50 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_less_imp_less_exp
% 0.30/0.50 thf(fact_566_power__less__imp__less__exp,axiom,
% 0.30/0.50 ! [A: int,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.50 => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.30/0.50 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_less_imp_less_exp
% 0.30/0.50 thf(fact_567_power__strict__increasing,axiom,
% 0.30/0.50 ! [N: nat,N5: nat,A: real] :
% 0.30/0.50 ( ( ord_less_nat @ N @ N5 )
% 0.30/0.50 => ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.50 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_strict_increasing
% 0.30/0.50 thf(fact_568_power__strict__increasing,axiom,
% 0.30/0.50 ! [N: nat,N5: nat,A: rat] :
% 0.30/0.50 ( ( ord_less_nat @ N @ N5 )
% 0.30/0.50 => ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.50 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_strict_increasing
% 0.30/0.50 thf(fact_569_power__strict__increasing,axiom,
% 0.30/0.50 ! [N: nat,N5: nat,A: nat] :
% 0.30/0.50 ( ( ord_less_nat @ N @ N5 )
% 0.30/0.50 => ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.50 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_strict_increasing
% 0.30/0.50 thf(fact_570_power__strict__increasing,axiom,
% 0.30/0.50 ! [N: nat,N5: nat,A: int] :
% 0.30/0.50 ( ( ord_less_nat @ N @ N5 )
% 0.30/0.50 => ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.50 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_strict_increasing
% 0.30/0.50 thf(fact_571_power__increasing,axiom,
% 0.30/0.50 ! [N: nat,N5: nat,A: real] :
% 0.30/0.50 ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.50 => ( ( ord_less_eq_real @ one_one_real @ A )
% 0.30/0.50 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_increasing
% 0.30/0.50 thf(fact_572_power__increasing,axiom,
% 0.30/0.50 ! [N: nat,N5: nat,A: rat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.50 => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.30/0.50 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_increasing
% 0.30/0.50 thf(fact_573_power__increasing,axiom,
% 0.30/0.50 ! [N: nat,N5: nat,A: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.50 => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.30/0.50 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_increasing
% 0.30/0.50 thf(fact_574_power__increasing,axiom,
% 0.30/0.50 ! [N: nat,N5: nat,A: int] :
% 0.30/0.50 ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.50 => ( ( ord_less_eq_int @ one_one_int @ A )
% 0.30/0.50 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_increasing
% 0.30/0.50 thf(fact_575_power__le__imp__le__exp,axiom,
% 0.30/0.50 ! [A: real,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.50 => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 0.30/0.50 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_le_imp_le_exp
% 0.30/0.50 thf(fact_576_power__le__imp__le__exp,axiom,
% 0.30/0.50 ! [A: rat,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.50 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 0.30/0.50 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_le_imp_le_exp
% 0.30/0.50 thf(fact_577_power__le__imp__le__exp,axiom,
% 0.30/0.50 ! [A: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.50 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.30/0.50 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_le_imp_le_exp
% 0.30/0.50 thf(fact_578_power__le__imp__le__exp,axiom,
% 0.30/0.50 ! [A: int,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.50 => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.30/0.50 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_le_imp_le_exp
% 0.30/0.50 thf(fact_579_one__power2,axiom,
% 0.30/0.50 ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = one_one_rat ) ).
% 0.30/0.50
% 0.30/0.50 % one_power2
% 0.30/0.50 thf(fact_580_one__power2,axiom,
% 0.30/0.50 ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = one_one_nat ) ).
% 0.30/0.50
% 0.30/0.50 % one_power2
% 0.30/0.50 thf(fact_581_one__power2,axiom,
% 0.30/0.50 ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = one_one_real ) ).
% 0.30/0.50
% 0.30/0.50 % one_power2
% 0.30/0.50 thf(fact_582_one__power2,axiom,
% 0.30/0.50 ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = one_one_int ) ).
% 0.30/0.50
% 0.30/0.50 % one_power2
% 0.30/0.50 thf(fact_583_one__power2,axiom,
% 0.30/0.50 ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = one_one_complex ) ).
% 0.30/0.50
% 0.30/0.50 % one_power2
% 0.30/0.50 thf(fact_584_less__exp,axiom,
% 0.30/0.50 ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_exp
% 0.30/0.50 thf(fact_585_self__le__ge2__pow,axiom,
% 0.30/0.50 ! [K: nat,M: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.30/0.50 => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % self_le_ge2_pow
% 0.30/0.50 thf(fact_586_power2__nat__le__eq__le,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_nat_le_eq_le
% 0.30/0.50 thf(fact_587_power2__nat__le__imp__le,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 0.30/0.50 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_nat_le_imp_le
% 0.30/0.50 thf(fact_588_mint__corr,axiom,
% 0.30/0.50 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.30/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.50 => ( ( ( vEBT_vebt_mint @ T )
% 0.30/0.50 = ( some_nat @ X2 ) )
% 0.30/0.50 => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mint_corr
% 0.30/0.50 thf(fact_589_mint__sound,axiom,
% 0.30/0.50 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.30/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.50 => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 )
% 0.30/0.50 => ( ( vEBT_vebt_mint @ T )
% 0.30/0.50 = ( some_nat @ X2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mint_sound
% 0.30/0.50 thf(fact_590_mintlistlength,axiom,
% 0.30/0.50 ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.30/0.50 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 0.30/0.50 => ( ( Mi != Ma )
% 0.30/0.50 => ( ( ord_less_nat @ Mi @ Ma )
% 0.30/0.50 & ? [M4: nat] :
% 0.30/0.50 ( ( ( some_nat @ M4 )
% 0.30/0.50 = ( vEBT_vebt_mint @ Summary ) )
% 0.30/0.50 & ( ord_less_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mintlistlength
% 0.30/0.50 thf(fact_591_maxt__corr__help,axiom,
% 0.30/0.50 ! [T: vEBT_VEBT,N: nat,Maxi: nat,X2: nat] :
% 0.30/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.50 => ( ( ( vEBT_vebt_maxt @ T )
% 0.30/0.50 = ( some_nat @ Maxi ) )
% 0.30/0.50 => ( ( vEBT_vebt_member @ T @ X2 )
% 0.30/0.50 => ( ord_less_eq_nat @ X2 @ Maxi ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % maxt_corr_help
% 0.30/0.50 thf(fact_592_maxt__member,axiom,
% 0.30/0.50 ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 0.30/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.50 => ( ( ( vEBT_vebt_maxt @ T )
% 0.30/0.50 = ( some_nat @ Maxi ) )
% 0.30/0.50 => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % maxt_member
% 0.30/0.50 thf(fact_593_div__exp__eq,axiom,
% 0.30/0.50 ! [A: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.50 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_exp_eq
% 0.30/0.50 thf(fact_594_div__exp__eq,axiom,
% 0.30/0.50 ! [A: int,M: nat,N: nat] :
% 0.30/0.50 ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.50 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_exp_eq
% 0.30/0.50 thf(fact_595_field__less__half__sum,axiom,
% 0.30/0.50 ! [X2: real,Y3: real] :
% 0.30/0.50 ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.50 => ( ord_less_real @ X2 @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % field_less_half_sum
% 0.30/0.50 thf(fact_596_field__less__half__sum,axiom,
% 0.30/0.50 ! [X2: rat,Y3: rat] :
% 0.30/0.50 ( ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.50 => ( ord_less_rat @ X2 @ ( divide_divide_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % field_less_half_sum
% 0.30/0.50 thf(fact_597_two__powr__height__bound__deg,axiom,
% 0.30/0.50 ! [T: vEBT_VEBT,N: nat] :
% 0.30/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.50 => ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_VEBT_height @ T ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % two_powr_height_bound_deg
% 0.30/0.50 thf(fact_598_bit__concat__def,axiom,
% 0.30/0.50 ( vEBT_VEBT_bit_concat
% 0.30/0.50 = ( ^ [H: nat,L2: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % bit_concat_def
% 0.30/0.50 thf(fact_599_low__inv,axiom,
% 0.30/0.50 ! [X2: nat,N: nat,Y3: nat] :
% 0.30/0.50 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.50 => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X2 ) @ N )
% 0.30/0.50 = X2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % low_inv
% 0.30/0.50 thf(fact_600_high__inv,axiom,
% 0.30/0.50 ! [X2: nat,N: nat,Y3: nat] :
% 0.30/0.50 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.50 => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X2 ) @ N )
% 0.30/0.50 = Y3 ) ) ).
% 0.30/0.50
% 0.30/0.50 % high_inv
% 0.30/0.50 thf(fact_601_maxbmo,axiom,
% 0.30/0.50 ! [T: vEBT_VEBT,X2: nat] :
% 0.30/0.50 ( ( ( vEBT_vebt_maxt @ T )
% 0.30/0.50 = ( some_nat @ X2 ) )
% 0.30/0.50 => ( vEBT_V8194947554948674370ptions @ T @ X2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % maxbmo
% 0.30/0.50 thf(fact_602_set__vebt__set__vebt_H__valid,axiom,
% 0.30/0.50 ! [T: vEBT_VEBT,N: nat] :
% 0.30/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.50 => ( ( vEBT_set_vebt @ T )
% 0.30/0.50 = ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % set_vebt_set_vebt'_valid
% 0.30/0.50 thf(fact_603_maxt__sound,axiom,
% 0.30/0.50 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.30/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.50 => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 )
% 0.30/0.50 => ( ( vEBT_vebt_maxt @ T )
% 0.30/0.50 = ( some_nat @ X2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % maxt_sound
% 0.30/0.50 thf(fact_604_maxt__corr,axiom,
% 0.30/0.50 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.30/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.50 => ( ( ( vEBT_vebt_maxt @ T )
% 0.30/0.50 = ( some_nat @ X2 ) )
% 0.30/0.50 => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % maxt_corr
% 0.30/0.50 thf(fact_605_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_606_numeral__times__numeral,axiom,
% 0.30/0.50 ! [M: num,N: num] :
% 0.30/0.50 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 0.30/0.50 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % numeral_times_numeral
% 0.30/0.50 thf(fact_607_numeral__times__numeral,axiom,
% 0.30/0.50 ! [M: num,N: num] :
% 0.30/0.50 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.30/0.50 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % numeral_times_numeral
% 0.30/0.50 thf(fact_608_numeral__times__numeral,axiom,
% 0.30/0.50 ! [M: num,N: num] :
% 0.30/0.50 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.30/0.50 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % numeral_times_numeral
% 0.30/0.50 thf(fact_609_numeral__times__numeral,axiom,
% 0.30/0.50 ! [M: num,N: num] :
% 0.30/0.50 ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.50 = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % numeral_times_numeral
% 0.30/0.50 thf(fact_610_numeral__times__numeral,axiom,
% 0.30/0.50 ! [M: num,N: num] :
% 0.30/0.50 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.50 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % numeral_times_numeral
% 0.30/0.50 thf(fact_611_mult__numeral__left__semiring__numeral,axiom,
% 0.30/0.50 ! [V: num,W: num,Z: complex] :
% 0.30/0.50 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 0.30/0.50 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_numeral_left_semiring_numeral
% 0.30/0.50 thf(fact_612_mult__numeral__left__semiring__numeral,axiom,
% 0.30/0.50 ! [V: num,W: num,Z: real] :
% 0.30/0.50 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 0.30/0.50 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_numeral_left_semiring_numeral
% 0.30/0.50 thf(fact_613_mult__numeral__left__semiring__numeral,axiom,
% 0.30/0.50 ! [V: num,W: num,Z: rat] :
% 0.30/0.50 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 0.30/0.50 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_numeral_left_semiring_numeral
% 0.30/0.50 thf(fact_614_mult__numeral__left__semiring__numeral,axiom,
% 0.30/0.50 ! [V: num,W: num,Z: nat] :
% 0.30/0.50 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 0.30/0.50 = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_numeral_left_semiring_numeral
% 0.30/0.50 thf(fact_615_mult__numeral__left__semiring__numeral,axiom,
% 0.30/0.50 ! [V: num,W: num,Z: int] :
% 0.30/0.50 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 0.30/0.50 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_numeral_left_semiring_numeral
% 0.30/0.50 thf(fact_616_mult_Oright__neutral,axiom,
% 0.30/0.50 ! [A: complex] :
% 0.30/0.50 ( ( times_times_complex @ A @ one_one_complex )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult.right_neutral
% 0.30/0.50 thf(fact_617_mult_Oright__neutral,axiom,
% 0.30/0.50 ! [A: real] :
% 0.30/0.50 ( ( times_times_real @ A @ one_one_real )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult.right_neutral
% 0.30/0.50 thf(fact_618_mult_Oright__neutral,axiom,
% 0.30/0.50 ! [A: rat] :
% 0.30/0.50 ( ( times_times_rat @ A @ one_one_rat )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult.right_neutral
% 0.30/0.50 thf(fact_619_mult_Oright__neutral,axiom,
% 0.30/0.50 ! [A: nat] :
% 0.30/0.50 ( ( times_times_nat @ A @ one_one_nat )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult.right_neutral
% 0.30/0.50 thf(fact_620_mult_Oright__neutral,axiom,
% 0.30/0.50 ! [A: int] :
% 0.30/0.50 ( ( times_times_int @ A @ one_one_int )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult.right_neutral
% 0.30/0.50 thf(fact_621_mult__1,axiom,
% 0.30/0.50 ! [A: complex] :
% 0.30/0.50 ( ( times_times_complex @ one_one_complex @ A )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult_1
% 0.30/0.50 thf(fact_622_mult__1,axiom,
% 0.30/0.50 ! [A: real] :
% 0.30/0.50 ( ( times_times_real @ one_one_real @ A )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult_1
% 0.30/0.50 thf(fact_623_mult__1,axiom,
% 0.30/0.50 ! [A: rat] :
% 0.30/0.50 ( ( times_times_rat @ one_one_rat @ A )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult_1
% 0.30/0.50 thf(fact_624_mult__1,axiom,
% 0.30/0.50 ! [A: nat] :
% 0.30/0.50 ( ( times_times_nat @ one_one_nat @ A )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult_1
% 0.30/0.50 thf(fact_625_mult__1,axiom,
% 0.30/0.50 ! [A: int] :
% 0.30/0.50 ( ( times_times_int @ one_one_int @ A )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult_1
% 0.30/0.50 thf(fact_626_add__diff__cancel,axiom,
% 0.30/0.50 ! [A: real,B: real] :
% 0.30/0.50 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel
% 0.30/0.50 thf(fact_627_add__diff__cancel,axiom,
% 0.30/0.50 ! [A: rat,B: rat] :
% 0.30/0.50 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel
% 0.30/0.50 thf(fact_628_add__diff__cancel,axiom,
% 0.30/0.50 ! [A: int,B: int] :
% 0.30/0.50 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel
% 0.30/0.50 thf(fact_629_diff__add__cancel,axiom,
% 0.30/0.50 ! [A: real,B: real] :
% 0.30/0.50 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add_cancel
% 0.30/0.50 thf(fact_630_diff__add__cancel,axiom,
% 0.30/0.50 ! [A: rat,B: rat] :
% 0.30/0.50 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add_cancel
% 0.30/0.50 thf(fact_631_diff__add__cancel,axiom,
% 0.30/0.50 ! [A: int,B: int] :
% 0.30/0.50 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add_cancel
% 0.30/0.50 thf(fact_632_add__diff__cancel__left,axiom,
% 0.30/0.50 ! [C: real,A: real,B: real] :
% 0.30/0.50 ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.30/0.50 = ( minus_minus_real @ A @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_left
% 0.30/0.50 thf(fact_633_add__diff__cancel__left,axiom,
% 0.30/0.50 ! [C: rat,A: rat,B: rat] :
% 0.30/0.50 ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.30/0.50 = ( minus_minus_rat @ A @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_left
% 0.30/0.50 thf(fact_634_add__diff__cancel__left,axiom,
% 0.30/0.50 ! [C: nat,A: nat,B: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.30/0.50 = ( minus_minus_nat @ A @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_left
% 0.30/0.50 thf(fact_635_add__diff__cancel__left,axiom,
% 0.30/0.50 ! [C: int,A: int,B: int] :
% 0.30/0.50 ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.30/0.50 = ( minus_minus_int @ A @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_left
% 0.30/0.50 thf(fact_636_add__diff__cancel__left_H,axiom,
% 0.30/0.50 ! [A: real,B: real] :
% 0.30/0.50 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 0.30/0.50 = B ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_left'
% 0.30/0.50 thf(fact_637_add__diff__cancel__left_H,axiom,
% 0.30/0.50 ! [A: rat,B: rat] :
% 0.30/0.50 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
% 0.30/0.50 = B ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_left'
% 0.30/0.50 thf(fact_638_add__diff__cancel__left_H,axiom,
% 0.30/0.50 ! [A: nat,B: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 0.30/0.50 = B ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_left'
% 0.30/0.50 thf(fact_639_add__diff__cancel__left_H,axiom,
% 0.30/0.50 ! [A: int,B: int] :
% 0.30/0.50 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 0.30/0.50 = B ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_left'
% 0.30/0.50 thf(fact_640_add__diff__cancel__right,axiom,
% 0.30/0.50 ! [A: real,C: real,B: real] :
% 0.30/0.50 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.30/0.50 = ( minus_minus_real @ A @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_right
% 0.30/0.50 thf(fact_641_add__diff__cancel__right,axiom,
% 0.30/0.50 ! [A: rat,C: rat,B: rat] :
% 0.30/0.50 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.50 = ( minus_minus_rat @ A @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_right
% 0.30/0.50 thf(fact_642_add__diff__cancel__right,axiom,
% 0.30/0.50 ! [A: nat,C: nat,B: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.30/0.50 = ( minus_minus_nat @ A @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_right
% 0.30/0.50 thf(fact_643_add__diff__cancel__right,axiom,
% 0.30/0.50 ! [A: int,C: int,B: int] :
% 0.30/0.50 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.30/0.50 = ( minus_minus_int @ A @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_right
% 0.30/0.50 thf(fact_644_add__diff__cancel__right_H,axiom,
% 0.30/0.50 ! [A: real,B: real] :
% 0.30/0.50 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_right'
% 0.30/0.50 thf(fact_645_add__diff__cancel__right_H,axiom,
% 0.30/0.50 ! [A: rat,B: rat] :
% 0.30/0.50 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_right'
% 0.30/0.50 thf(fact_646_add__diff__cancel__right_H,axiom,
% 0.30/0.50 ! [A: nat,B: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_right'
% 0.30/0.50 thf(fact_647_add__diff__cancel__right_H,axiom,
% 0.30/0.50 ! [A: int,B: int] :
% 0.30/0.50 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_cancel_right'
% 0.30/0.50 thf(fact_648_bits__div__by__1,axiom,
% 0.30/0.50 ! [A: nat] :
% 0.30/0.50 ( ( divide_divide_nat @ A @ one_one_nat )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % bits_div_by_1
% 0.30/0.50 thf(fact_649_bits__div__by__1,axiom,
% 0.30/0.50 ! [A: int] :
% 0.30/0.50 ( ( divide_divide_int @ A @ one_one_int )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % bits_div_by_1
% 0.30/0.50 thf(fact_650_diff__Suc__Suc,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.50 = ( minus_minus_nat @ M @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_Suc_Suc
% 0.30/0.50 thf(fact_651_Suc__diff__diff,axiom,
% 0.30/0.50 ! [M: nat,N: nat,K: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 0.30/0.50 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_diff_diff
% 0.30/0.50 thf(fact_652_diff__diff__cancel,axiom,
% 0.30/0.50 ! [I2: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ I2 @ N )
% 0.30/0.50 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
% 0.30/0.50 = I2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_diff_cancel
% 0.30/0.50 thf(fact_653_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_654_nat__1__eq__mult__iff,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( one_one_nat
% 0.30/0.50 = ( times_times_nat @ M @ N ) )
% 0.30/0.50 = ( ( M = one_one_nat )
% 0.30/0.50 & ( N = one_one_nat ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_1_eq_mult_iff
% 0.30/0.50 thf(fact_655_nat__mult__eq__1__iff,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( ( times_times_nat @ M @ N )
% 0.30/0.50 = one_one_nat )
% 0.30/0.50 = ( ( M = one_one_nat )
% 0.30/0.50 & ( N = one_one_nat ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_mult_eq_1_iff
% 0.30/0.50 thf(fact_656_distrib__left__numeral,axiom,
% 0.30/0.50 ! [V: num,B: complex,C: complex] :
% 0.30/0.50 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
% 0.30/0.50 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % distrib_left_numeral
% 0.30/0.50 thf(fact_657_distrib__left__numeral,axiom,
% 0.30/0.50 ! [V: num,B: real,C: real] :
% 0.30/0.50 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
% 0.30/0.50 = ( plus_plus_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 % distrib_left_numeral
% 0.30/0.50 thf(fact_658_distrib__left__numeral,axiom,
% 0.30/0.50 ! [V: num,B: rat,C: rat] :
% 0.30/0.50 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.50 = ( plus_plus_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 % distrib_left_numeral
% 0.30/0.50 thf(fact_659_distrib__left__numeral,axiom,
% 0.30/0.50 ! [V: num,B: nat,C: nat] :
% 0.30/0.50 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
% 0.30/0.50 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % distrib_left_numeral
% 0.30/0.50 thf(fact_660_distrib__left__numeral,axiom,
% 0.30/0.50 ! [V: num,B: int,C: int] :
% 0.30/0.50 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
% 0.30/0.50 = ( plus_plus_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 % distrib_left_numeral
% 0.30/0.50 thf(fact_661_distrib__right__numeral,axiom,
% 0.30/0.50 ! [A: complex,B: complex,V: num] :
% 0.30/0.50 ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 0.30/0.50 = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % distrib_right_numeral
% 0.30/0.50 thf(fact_662_distrib__right__numeral,axiom,
% 0.30/0.50 ! [A: real,B: real,V: num] :
% 0.30/0.50 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.50 = ( plus_plus_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 % distrib_right_numeral
% 0.30/0.50 thf(fact_663_distrib__right__numeral,axiom,
% 0.30/0.50 ! [A: rat,B: rat,V: num] :
% 0.30/0.50 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.50 = ( plus_plus_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 % distrib_right_numeral
% 0.30/0.50 thf(fact_664_distrib__right__numeral,axiom,
% 0.30/0.50 ! [A: nat,B: nat,V: num] :
% 0.30/0.50 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.50 = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % distrib_right_numeral
% 0.30/0.50 thf(fact_665_distrib__right__numeral,axiom,
% 0.30/0.50 ! [A: int,B: int,V: num] :
% 0.30/0.50 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.50 = ( plus_plus_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 % distrib_right_numeral
% 0.30/0.50 thf(fact_666_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_667_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_668_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_669_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_670_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_671_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_672_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_673_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_674_mult__Suc__right,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( times_times_nat @ M @ ( suc @ N ) )
% 0.30/0.50 = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_Suc_right
% 0.30/0.50 thf(fact_675_diff__Suc__1,axiom,
% 0.30/0.50 ! [N: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 0.30/0.50 = N ) ).
% 0.30/0.50
% 0.30/0.50 % diff_Suc_1
% 0.30/0.50 thf(fact_676_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_677_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_678_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_679_divide__le__eq__numeral1_I1_J,axiom,
% 0.30/0.50 ! [B: real,W: num,A: real] :
% 0.30/0.50 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 0.30/0.50 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % divide_le_eq_numeral1(1)
% 0.30/0.50 thf(fact_680_divide__le__eq__numeral1_I1_J,axiom,
% 0.30/0.50 ! [B: rat,W: num,A: rat] :
% 0.30/0.50 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 0.30/0.50 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % divide_le_eq_numeral1(1)
% 0.30/0.50 thf(fact_681_le__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.50 ! [A: real,B: real,W: num] :
% 0.30/0.50 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.50 = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_divide_eq_numeral1(1)
% 0.30/0.50 thf(fact_682_le__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.50 ! [A: rat,B: rat,W: num] :
% 0.30/0.50 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 0.30/0.50 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_divide_eq_numeral1(1)
% 0.30/0.50 thf(fact_683_less__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.50 ! [A: real,B: real,W: num] :
% 0.30/0.50 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.50 = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_divide_eq_numeral1(1)
% 0.30/0.50 thf(fact_684_less__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.50 ! [A: rat,B: rat,W: num] :
% 0.30/0.50 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 0.30/0.50 = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_divide_eq_numeral1(1)
% 0.30/0.50 thf(fact_685_divide__less__eq__numeral1_I1_J,axiom,
% 0.30/0.50 ! [B: real,W: num,A: real] :
% 0.30/0.50 ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 0.30/0.50 = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % divide_less_eq_numeral1(1)
% 0.30/0.50 thf(fact_686_divide__less__eq__numeral1_I1_J,axiom,
% 0.30/0.50 ! [B: rat,W: num,A: rat] :
% 0.30/0.50 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 0.30/0.50 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % divide_less_eq_numeral1(1)
% 0.30/0.50 thf(fact_687_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_688_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_689_power__add__numeral,axiom,
% 0.30/0.50 ! [A: complex,M: num,N: num] :
% 0.30/0.50 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.30/0.50 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_add_numeral
% 0.30/0.50 thf(fact_690_power__add__numeral,axiom,
% 0.30/0.50 ! [A: real,M: num,N: num] :
% 0.30/0.50 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.30/0.50 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_add_numeral
% 0.30/0.50 thf(fact_691_power__add__numeral,axiom,
% 0.30/0.50 ! [A: rat,M: num,N: num] :
% 0.30/0.50 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.30/0.50 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_add_numeral
% 0.30/0.50 thf(fact_692_power__add__numeral,axiom,
% 0.30/0.50 ! [A: nat,M: num,N: num] :
% 0.30/0.50 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.30/0.50 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_add_numeral
% 0.30/0.50 thf(fact_693_power__add__numeral,axiom,
% 0.30/0.50 ! [A: int,M: num,N: num] :
% 0.30/0.50 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.30/0.50 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_add_numeral
% 0.30/0.50 thf(fact_694_power__add__numeral2,axiom,
% 0.30/0.50 ! [A: complex,M: num,N: num,B: complex] :
% 0.30/0.50 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.30/0.50 = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_add_numeral2
% 0.30/0.50 thf(fact_695_power__add__numeral2,axiom,
% 0.30/0.50 ! [A: real,M: num,N: num,B: real] :
% 0.30/0.50 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.30/0.50 = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_add_numeral2
% 0.30/0.50 thf(fact_696_power__add__numeral2,axiom,
% 0.30/0.50 ! [A: rat,M: num,N: num,B: rat] :
% 0.30/0.50 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.30/0.50 = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_add_numeral2
% 0.30/0.50 thf(fact_697_power__add__numeral2,axiom,
% 0.30/0.50 ! [A: nat,M: num,N: num,B: nat] :
% 0.30/0.50 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.30/0.50 = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_add_numeral2
% 0.30/0.50 thf(fact_698_power__add__numeral2,axiom,
% 0.30/0.50 ! [A: int,M: num,N: num,B: int] :
% 0.30/0.50 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.30/0.50 = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_add_numeral2
% 0.30/0.50 thf(fact_699_pred__member,axiom,
% 0.30/0.50 ! [T: vEBT_VEBT,X2: nat,Y3: nat] :
% 0.30/0.50 ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Y3 )
% 0.30/0.50 = ( ( vEBT_vebt_member @ T @ Y3 )
% 0.30/0.50 & ( ord_less_nat @ Y3 @ X2 )
% 0.30/0.50 & ! [Z3: nat] :
% 0.30/0.50 ( ( ( vEBT_vebt_member @ T @ Z3 )
% 0.30/0.50 & ( ord_less_nat @ Z3 @ X2 ) )
% 0.30/0.50 => ( ord_less_eq_nat @ Z3 @ Y3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % pred_member
% 0.30/0.50 thf(fact_700_succ__member,axiom,
% 0.30/0.50 ! [T: vEBT_VEBT,X2: nat,Y3: nat] :
% 0.30/0.50 ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Y3 )
% 0.30/0.50 = ( ( vEBT_vebt_member @ T @ Y3 )
% 0.30/0.50 & ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.50 & ! [Z3: nat] :
% 0.30/0.50 ( ( ( vEBT_vebt_member @ T @ Z3 )
% 0.30/0.50 & ( ord_less_nat @ X2 @ Z3 ) )
% 0.30/0.50 => ( ord_less_eq_nat @ Y3 @ Z3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % succ_member
% 0.30/0.50 thf(fact_701_pred__corr,axiom,
% 0.30/0.50 ! [T: vEBT_VEBT,N: nat,X2: nat,Px: nat] :
% 0.30/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.50 => ( ( ( vEBT_vebt_pred @ T @ X2 )
% 0.30/0.50 = ( some_nat @ Px ) )
% 0.30/0.50 = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Px ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % pred_corr
% 0.30/0.50 thf(fact_702_succ__corr,axiom,
% 0.30/0.50 ! [T: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
% 0.30/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.50 => ( ( ( vEBT_vebt_succ @ T @ X2 )
% 0.30/0.50 = ( some_nat @ Sx ) )
% 0.30/0.50 = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % succ_corr
% 0.30/0.50 thf(fact_703_pred__correct,axiom,
% 0.30/0.50 ! [T: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
% 0.30/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.50 => ( ( ( vEBT_vebt_pred @ T @ X2 )
% 0.30/0.50 = ( some_nat @ Sx ) )
% 0.30/0.50 = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % pred_correct
% 0.30/0.50 thf(fact_704_succ__correct,axiom,
% 0.30/0.50 ! [T: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
% 0.30/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.50 => ( ( ( vEBT_vebt_succ @ T @ X2 )
% 0.30/0.50 = ( some_nat @ Sx ) )
% 0.30/0.50 = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % succ_correct
% 0.30/0.50 thf(fact_705_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.30/0.50 ! [A: real,B: real,C: real] :
% 0.30/0.50 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.30/0.50 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ab_semigroup_mult_class.mult_ac(1)
% 0.30/0.50 thf(fact_706_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.30/0.50 ! [A: rat,B: rat,C: rat] :
% 0.30/0.50 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.30/0.50 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ab_semigroup_mult_class.mult_ac(1)
% 0.30/0.50 thf(fact_707_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.30/0.50 ! [A: nat,B: nat,C: nat] :
% 0.30/0.50 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.30/0.50 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ab_semigroup_mult_class.mult_ac(1)
% 0.30/0.50 thf(fact_708_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.30/0.50 ! [A: int,B: int,C: int] :
% 0.30/0.50 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 0.30/0.50 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ab_semigroup_mult_class.mult_ac(1)
% 0.30/0.50 thf(fact_709_diff__eq__diff__eq,axiom,
% 0.30/0.50 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.50 ( ( ( minus_minus_real @ A @ B )
% 0.30/0.50 = ( minus_minus_real @ C @ D ) )
% 0.30/0.50 => ( ( A = B )
% 0.30/0.50 = ( C = D ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_eq_diff_eq
% 0.30/0.50 thf(fact_710_diff__eq__diff__eq,axiom,
% 0.30/0.50 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.50 ( ( ( minus_minus_rat @ A @ B )
% 0.30/0.50 = ( minus_minus_rat @ C @ D ) )
% 0.30/0.50 => ( ( A = B )
% 0.30/0.50 = ( C = D ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_eq_diff_eq
% 0.30/0.50 thf(fact_711_diff__eq__diff__eq,axiom,
% 0.30/0.50 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.50 ( ( ( minus_minus_int @ A @ B )
% 0.30/0.50 = ( minus_minus_int @ C @ D ) )
% 0.30/0.50 => ( ( A = B )
% 0.30/0.50 = ( C = D ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_eq_diff_eq
% 0.30/0.50 thf(fact_712_mult_Oassoc,axiom,
% 0.30/0.50 ! [A: real,B: real,C: real] :
% 0.30/0.50 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.30/0.50 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult.assoc
% 0.30/0.50 thf(fact_713_mult_Oassoc,axiom,
% 0.30/0.50 ! [A: rat,B: rat,C: rat] :
% 0.30/0.50 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.30/0.50 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult.assoc
% 0.30/0.50 thf(fact_714_mult_Oassoc,axiom,
% 0.30/0.50 ! [A: nat,B: nat,C: nat] :
% 0.30/0.50 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.30/0.50 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult.assoc
% 0.30/0.50 thf(fact_715_mult_Oassoc,axiom,
% 0.30/0.50 ! [A: int,B: int,C: int] :
% 0.30/0.50 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 0.30/0.50 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult.assoc
% 0.30/0.50 thf(fact_716_mult_Ocommute,axiom,
% 0.30/0.50 ( times_times_real
% 0.30/0.50 = ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult.commute
% 0.30/0.50 thf(fact_717_mult_Ocommute,axiom,
% 0.30/0.50 ( times_times_rat
% 0.30/0.50 = ( ^ [A3: rat,B2: rat] : ( times_times_rat @ B2 @ A3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult.commute
% 0.30/0.50 thf(fact_718_mult_Ocommute,axiom,
% 0.30/0.50 ( times_times_nat
% 0.30/0.50 = ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult.commute
% 0.30/0.50 thf(fact_719_mult_Ocommute,axiom,
% 0.30/0.50 ( times_times_int
% 0.30/0.50 = ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult.commute
% 0.30/0.50 thf(fact_720_mult_Oleft__commute,axiom,
% 0.30/0.50 ! [B: real,A: real,C: real] :
% 0.30/0.50 ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 0.30/0.50 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult.left_commute
% 0.30/0.50 thf(fact_721_mult_Oleft__commute,axiom,
% 0.30/0.50 ! [B: rat,A: rat,C: rat] :
% 0.30/0.50 ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
% 0.30/0.50 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult.left_commute
% 0.30/0.50 thf(fact_722_mult_Oleft__commute,axiom,
% 0.30/0.50 ! [B: nat,A: nat,C: nat] :
% 0.30/0.50 ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 0.30/0.50 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult.left_commute
% 0.30/0.50 thf(fact_723_mult_Oleft__commute,axiom,
% 0.30/0.50 ! [B: int,A: int,C: int] :
% 0.30/0.50 ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
% 0.30/0.50 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult.left_commute
% 0.30/0.50 thf(fact_724_diff__right__commute,axiom,
% 0.30/0.50 ! [A: real,C: real,B: real] :
% 0.30/0.50 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 0.30/0.50 = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_right_commute
% 0.30/0.50 thf(fact_725_diff__right__commute,axiom,
% 0.30/0.50 ! [A: rat,C: rat,B: rat] :
% 0.30/0.50 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
% 0.30/0.50 = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_right_commute
% 0.30/0.50 thf(fact_726_diff__right__commute,axiom,
% 0.30/0.50 ! [A: nat,C: nat,B: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.30/0.50 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_right_commute
% 0.30/0.50 thf(fact_727_diff__right__commute,axiom,
% 0.30/0.50 ! [A: int,C: int,B: int] :
% 0.30/0.50 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 0.30/0.50 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_right_commute
% 0.30/0.50 thf(fact_728_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_729_diff__mult__distrib,axiom,
% 0.30/0.50 ! [M: nat,N: nat,K: nat] :
% 0.30/0.50 ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 0.30/0.50 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_mult_distrib
% 0.30/0.50 thf(fact_730_diff__mult__distrib2,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.50 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_mult_distrib2
% 0.30/0.50 thf(fact_731_mult__diff__mult,axiom,
% 0.30/0.50 ! [X2: real,Y3: real,A: real,B: real] :
% 0.30/0.50 ( ( minus_minus_real @ ( times_times_real @ X2 @ Y3 ) @ ( times_times_real @ A @ B ) )
% 0.30/0.50 = ( plus_plus_real @ ( times_times_real @ X2 @ ( minus_minus_real @ Y3 @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X2 @ A ) @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_diff_mult
% 0.30/0.50 thf(fact_732_mult__diff__mult,axiom,
% 0.30/0.50 ! [X2: rat,Y3: rat,A: rat,B: rat] :
% 0.30/0.50 ( ( minus_minus_rat @ ( times_times_rat @ X2 @ Y3 ) @ ( times_times_rat @ A @ B ) )
% 0.30/0.50 = ( plus_plus_rat @ ( times_times_rat @ X2 @ ( minus_minus_rat @ Y3 @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X2 @ A ) @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_diff_mult
% 0.30/0.50 thf(fact_733_mult__diff__mult,axiom,
% 0.30/0.50 ! [X2: int,Y3: int,A: int,B: int] :
% 0.30/0.50 ( ( minus_minus_int @ ( times_times_int @ X2 @ Y3 ) @ ( times_times_int @ A @ B ) )
% 0.30/0.50 = ( plus_plus_int @ ( times_times_int @ X2 @ ( minus_minus_int @ Y3 @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X2 @ A ) @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_diff_mult
% 0.30/0.50 thf(fact_734_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_735_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_736_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_737_nat__eq__add__iff1,axiom,
% 0.30/0.50 ! [J: nat,I2: nat,U: nat,M: nat,N: 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 ) @ N ) )
% 0.30/0.50 = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M )
% 0.30/0.50 = N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_eq_add_iff1
% 0.30/0.50 thf(fact_738_nat__eq__add__iff2,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,U: nat,M: nat,N: 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 ) @ N ) )
% 0.30/0.50 = ( M
% 0.30/0.50 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_eq_add_iff2
% 0.30/0.50 thf(fact_739_nat__le__add__iff1,axiom,
% 0.30/0.50 ! [J: nat,I2: nat,U: nat,M: nat,N: 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 ) @ N ) )
% 0.30/0.50 = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_le_add_iff1
% 0.30/0.50 thf(fact_740_nat__le__add__iff2,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,U: nat,M: nat,N: 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 ) @ N ) )
% 0.30/0.50 = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_le_add_iff2
% 0.30/0.50 thf(fact_741_nat__diff__add__eq1,axiom,
% 0.30/0.50 ! [J: nat,I2: nat,U: nat,M: nat,N: 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 ) @ N ) )
% 0.30/0.50 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_diff_add_eq1
% 0.30/0.50 thf(fact_742_nat__diff__add__eq2,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,U: nat,M: nat,N: 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 ) @ N ) )
% 0.30/0.50 = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_diff_add_eq2
% 0.30/0.50 thf(fact_743_diff__mono,axiom,
% 0.30/0.50 ! [A: real,B: real,D: real,C: real] :
% 0.30/0.50 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.50 => ( ( ord_less_eq_real @ D @ C )
% 0.30/0.50 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_mono
% 0.30/0.50 thf(fact_744_diff__mono,axiom,
% 0.30/0.50 ! [A: rat,B: rat,D: rat,C: rat] :
% 0.30/0.50 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.50 => ( ( ord_less_eq_rat @ D @ C )
% 0.30/0.50 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_mono
% 0.30/0.50 thf(fact_745_diff__mono,axiom,
% 0.30/0.50 ! [A: int,B: int,D: int,C: int] :
% 0.30/0.50 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.50 => ( ( ord_less_eq_int @ D @ C )
% 0.30/0.50 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_mono
% 0.30/0.50 thf(fact_746_diff__left__mono,axiom,
% 0.30/0.50 ! [B: real,A: real,C: real] :
% 0.30/0.50 ( ( ord_less_eq_real @ B @ A )
% 0.30/0.50 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_left_mono
% 0.30/0.50 thf(fact_747_diff__left__mono,axiom,
% 0.30/0.50 ! [B: rat,A: rat,C: rat] :
% 0.30/0.50 ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.50 => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_left_mono
% 0.30/0.50 thf(fact_748_diff__left__mono,axiom,
% 0.30/0.50 ! [B: int,A: int,C: int] :
% 0.30/0.50 ( ( ord_less_eq_int @ B @ A )
% 0.30/0.50 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_left_mono
% 0.30/0.50 thf(fact_749_diff__right__mono,axiom,
% 0.30/0.50 ! [A: real,B: real,C: real] :
% 0.30/0.50 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.50 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_right_mono
% 0.30/0.50 thf(fact_750_diff__right__mono,axiom,
% 0.30/0.50 ! [A: rat,B: rat,C: rat] :
% 0.30/0.50 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.50 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_right_mono
% 0.30/0.50 thf(fact_751_diff__right__mono,axiom,
% 0.30/0.50 ! [A: int,B: int,C: int] :
% 0.30/0.50 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.50 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_right_mono
% 0.30/0.50 thf(fact_752_diff__eq__diff__less__eq,axiom,
% 0.30/0.50 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.50 ( ( ( minus_minus_real @ A @ B )
% 0.30/0.50 = ( minus_minus_real @ C @ D ) )
% 0.30/0.50 => ( ( ord_less_eq_real @ A @ B )
% 0.30/0.50 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_eq_diff_less_eq
% 0.30/0.50 thf(fact_753_diff__eq__diff__less__eq,axiom,
% 0.30/0.50 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.50 ( ( ( minus_minus_rat @ A @ B )
% 0.30/0.50 = ( minus_minus_rat @ C @ D ) )
% 0.30/0.50 => ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.50 = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_eq_diff_less_eq
% 0.30/0.50 thf(fact_754_diff__eq__diff__less__eq,axiom,
% 0.30/0.50 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.50 ( ( ( minus_minus_int @ A @ B )
% 0.30/0.50 = ( minus_minus_int @ C @ D ) )
% 0.30/0.50 => ( ( ord_less_eq_int @ A @ B )
% 0.30/0.50 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_eq_diff_less_eq
% 0.30/0.50 thf(fact_755_diff__strict__right__mono,axiom,
% 0.30/0.50 ! [A: real,B: real,C: real] :
% 0.30/0.50 ( ( ord_less_real @ A @ B )
% 0.30/0.50 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_strict_right_mono
% 0.30/0.50 thf(fact_756_diff__strict__right__mono,axiom,
% 0.30/0.50 ! [A: rat,B: rat,C: rat] :
% 0.30/0.50 ( ( ord_less_rat @ A @ B )
% 0.30/0.50 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_strict_right_mono
% 0.30/0.50 thf(fact_757_diff__strict__right__mono,axiom,
% 0.30/0.50 ! [A: int,B: int,C: int] :
% 0.30/0.50 ( ( ord_less_int @ A @ B )
% 0.30/0.50 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_strict_right_mono
% 0.30/0.50 thf(fact_758_diff__strict__left__mono,axiom,
% 0.30/0.50 ! [B: real,A: real,C: real] :
% 0.30/0.50 ( ( ord_less_real @ B @ A )
% 0.30/0.50 => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_strict_left_mono
% 0.30/0.50 thf(fact_759_diff__strict__left__mono,axiom,
% 0.30/0.50 ! [B: rat,A: rat,C: rat] :
% 0.30/0.50 ( ( ord_less_rat @ B @ A )
% 0.30/0.50 => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_strict_left_mono
% 0.30/0.50 thf(fact_760_diff__strict__left__mono,axiom,
% 0.30/0.50 ! [B: int,A: int,C: int] :
% 0.30/0.50 ( ( ord_less_int @ B @ A )
% 0.30/0.50 => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_strict_left_mono
% 0.30/0.50 thf(fact_761_diff__eq__diff__less,axiom,
% 0.30/0.50 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.50 ( ( ( minus_minus_real @ A @ B )
% 0.30/0.50 = ( minus_minus_real @ C @ D ) )
% 0.30/0.50 => ( ( ord_less_real @ A @ B )
% 0.30/0.50 = ( ord_less_real @ C @ D ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_eq_diff_less
% 0.30/0.50 thf(fact_762_diff__eq__diff__less,axiom,
% 0.30/0.50 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.50 ( ( ( minus_minus_rat @ A @ B )
% 0.30/0.50 = ( minus_minus_rat @ C @ D ) )
% 0.30/0.50 => ( ( ord_less_rat @ A @ B )
% 0.30/0.50 = ( ord_less_rat @ C @ D ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_eq_diff_less
% 0.30/0.50 thf(fact_763_diff__eq__diff__less,axiom,
% 0.30/0.50 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.50 ( ( ( minus_minus_int @ A @ B )
% 0.30/0.50 = ( minus_minus_int @ C @ D ) )
% 0.30/0.50 => ( ( ord_less_int @ A @ B )
% 0.30/0.50 = ( ord_less_int @ C @ D ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_eq_diff_less
% 0.30/0.50 thf(fact_764_diff__strict__mono,axiom,
% 0.30/0.50 ! [A: real,B: real,D: real,C: real] :
% 0.30/0.50 ( ( ord_less_real @ A @ B )
% 0.30/0.50 => ( ( ord_less_real @ D @ C )
% 0.30/0.50 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_strict_mono
% 0.30/0.50 thf(fact_765_diff__strict__mono,axiom,
% 0.30/0.50 ! [A: rat,B: rat,D: rat,C: rat] :
% 0.30/0.50 ( ( ord_less_rat @ A @ B )
% 0.30/0.50 => ( ( ord_less_rat @ D @ C )
% 0.30/0.50 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_strict_mono
% 0.30/0.50 thf(fact_766_diff__strict__mono,axiom,
% 0.30/0.50 ! [A: int,B: int,D: int,C: int] :
% 0.30/0.50 ( ( ord_less_int @ A @ B )
% 0.30/0.50 => ( ( ord_less_int @ D @ C )
% 0.30/0.50 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_strict_mono
% 0.30/0.50 thf(fact_767_group__cancel_Osub1,axiom,
% 0.30/0.50 ! [A2: real,K: real,A: real,B: real] :
% 0.30/0.50 ( ( A2
% 0.30/0.50 = ( plus_plus_real @ K @ A ) )
% 0.30/0.50 => ( ( minus_minus_real @ A2 @ B )
% 0.30/0.50 = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % group_cancel.sub1
% 0.30/0.50 thf(fact_768_group__cancel_Osub1,axiom,
% 0.30/0.50 ! [A2: rat,K: rat,A: rat,B: rat] :
% 0.30/0.50 ( ( A2
% 0.30/0.50 = ( plus_plus_rat @ K @ A ) )
% 0.30/0.50 => ( ( minus_minus_rat @ A2 @ B )
% 0.30/0.50 = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % group_cancel.sub1
% 0.30/0.50 thf(fact_769_group__cancel_Osub1,axiom,
% 0.30/0.50 ! [A2: int,K: int,A: int,B: int] :
% 0.30/0.50 ( ( A2
% 0.30/0.50 = ( plus_plus_int @ K @ A ) )
% 0.30/0.50 => ( ( minus_minus_int @ A2 @ B )
% 0.30/0.50 = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % group_cancel.sub1
% 0.30/0.50 thf(fact_770_diff__eq__eq,axiom,
% 0.30/0.50 ! [A: real,B: real,C: real] :
% 0.30/0.50 ( ( ( minus_minus_real @ A @ B )
% 0.30/0.50 = C )
% 0.30/0.50 = ( A
% 0.30/0.50 = ( plus_plus_real @ C @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_eq_eq
% 0.30/0.50 thf(fact_771_diff__eq__eq,axiom,
% 0.30/0.50 ! [A: rat,B: rat,C: rat] :
% 0.30/0.50 ( ( ( minus_minus_rat @ A @ B )
% 0.30/0.50 = C )
% 0.30/0.50 = ( A
% 0.30/0.50 = ( plus_plus_rat @ C @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_eq_eq
% 0.30/0.50 thf(fact_772_diff__eq__eq,axiom,
% 0.30/0.50 ! [A: int,B: int,C: int] :
% 0.30/0.50 ( ( ( minus_minus_int @ A @ B )
% 0.30/0.50 = C )
% 0.30/0.50 = ( A
% 0.30/0.50 = ( plus_plus_int @ C @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_eq_eq
% 0.30/0.50 thf(fact_773_eq__diff__eq,axiom,
% 0.30/0.50 ! [A: real,C: real,B: real] :
% 0.30/0.50 ( ( A
% 0.30/0.50 = ( minus_minus_real @ C @ B ) )
% 0.30/0.50 = ( ( plus_plus_real @ A @ B )
% 0.30/0.50 = C ) ) ).
% 0.30/0.50
% 0.30/0.50 % eq_diff_eq
% 0.30/0.50 thf(fact_774_eq__diff__eq,axiom,
% 0.30/0.50 ! [A: rat,C: rat,B: rat] :
% 0.30/0.50 ( ( A
% 0.30/0.50 = ( minus_minus_rat @ C @ B ) )
% 0.30/0.50 = ( ( plus_plus_rat @ A @ B )
% 0.30/0.50 = C ) ) ).
% 0.30/0.50
% 0.30/0.50 % eq_diff_eq
% 0.30/0.50 thf(fact_775_eq__diff__eq,axiom,
% 0.30/0.50 ! [A: int,C: int,B: int] :
% 0.30/0.50 ( ( A
% 0.30/0.50 = ( minus_minus_int @ C @ B ) )
% 0.30/0.50 = ( ( plus_plus_int @ A @ B )
% 0.30/0.50 = C ) ) ).
% 0.30/0.50
% 0.30/0.50 % eq_diff_eq
% 0.30/0.50 thf(fact_776_add__diff__eq,axiom,
% 0.30/0.50 ! [A: real,B: real,C: real] :
% 0.30/0.50 ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.30/0.50 = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_eq
% 0.30/0.50 thf(fact_777_add__diff__eq,axiom,
% 0.30/0.50 ! [A: rat,B: rat,C: rat] :
% 0.30/0.50 ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.30/0.50 = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_eq
% 0.30/0.50 thf(fact_778_add__diff__eq,axiom,
% 0.30/0.50 ! [A: int,B: int,C: int] :
% 0.30/0.50 ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.30/0.50 = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_eq
% 0.30/0.50 thf(fact_779_diff__diff__eq2,axiom,
% 0.30/0.50 ! [A: real,B: real,C: real] :
% 0.30/0.50 ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.30/0.50 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_diff_eq2
% 0.30/0.50 thf(fact_780_diff__diff__eq2,axiom,
% 0.30/0.50 ! [A: rat,B: rat,C: rat] :
% 0.30/0.50 ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.30/0.50 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_diff_eq2
% 0.30/0.50 thf(fact_781_diff__diff__eq2,axiom,
% 0.30/0.50 ! [A: int,B: int,C: int] :
% 0.30/0.50 ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.30/0.50 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_diff_eq2
% 0.30/0.50 thf(fact_782_diff__add__eq,axiom,
% 0.30/0.50 ! [A: real,B: real,C: real] :
% 0.30/0.50 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.30/0.50 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add_eq
% 0.30/0.50 thf(fact_783_diff__add__eq,axiom,
% 0.30/0.50 ! [A: rat,B: rat,C: rat] :
% 0.30/0.50 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.30/0.50 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add_eq
% 0.30/0.50 thf(fact_784_diff__add__eq,axiom,
% 0.30/0.50 ! [A: int,B: int,C: int] :
% 0.30/0.50 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.50 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add_eq
% 0.30/0.50 thf(fact_785_diff__add__eq__diff__diff__swap,axiom,
% 0.30/0.50 ! [A: real,B: real,C: real] :
% 0.30/0.50 ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.30/0.50 = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add_eq_diff_diff_swap
% 0.30/0.50 thf(fact_786_diff__add__eq__diff__diff__swap,axiom,
% 0.30/0.50 ! [A: rat,B: rat,C: rat] :
% 0.30/0.50 ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.50 = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add_eq_diff_diff_swap
% 0.30/0.50 thf(fact_787_diff__add__eq__diff__diff__swap,axiom,
% 0.30/0.50 ! [A: int,B: int,C: int] :
% 0.30/0.50 ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.30/0.50 = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add_eq_diff_diff_swap
% 0.30/0.50 thf(fact_788_add__implies__diff,axiom,
% 0.30/0.50 ! [C: real,B: real,A: real] :
% 0.30/0.50 ( ( ( plus_plus_real @ C @ B )
% 0.30/0.50 = A )
% 0.30/0.50 => ( C
% 0.30/0.50 = ( minus_minus_real @ A @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_implies_diff
% 0.30/0.50 thf(fact_789_add__implies__diff,axiom,
% 0.30/0.50 ! [C: rat,B: rat,A: rat] :
% 0.30/0.50 ( ( ( plus_plus_rat @ C @ B )
% 0.30/0.50 = A )
% 0.30/0.50 => ( C
% 0.30/0.50 = ( minus_minus_rat @ A @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_implies_diff
% 0.30/0.50 thf(fact_790_add__implies__diff,axiom,
% 0.30/0.50 ! [C: nat,B: nat,A: nat] :
% 0.30/0.50 ( ( ( plus_plus_nat @ C @ B )
% 0.30/0.50 = A )
% 0.30/0.50 => ( C
% 0.30/0.50 = ( minus_minus_nat @ A @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_implies_diff
% 0.30/0.50 thf(fact_791_add__implies__diff,axiom,
% 0.30/0.50 ! [C: int,B: int,A: int] :
% 0.30/0.50 ( ( ( plus_plus_int @ C @ B )
% 0.30/0.50 = A )
% 0.30/0.50 => ( C
% 0.30/0.50 = ( minus_minus_int @ A @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_implies_diff
% 0.30/0.50 thf(fact_792_diff__diff__eq,axiom,
% 0.30/0.50 ! [A: real,B: real,C: real] :
% 0.30/0.50 ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.30/0.50 = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_diff_eq
% 0.30/0.50 thf(fact_793_diff__diff__eq,axiom,
% 0.30/0.50 ! [A: rat,B: rat,C: rat] :
% 0.30/0.50 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.30/0.50 = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_diff_eq
% 0.30/0.50 thf(fact_794_diff__diff__eq,axiom,
% 0.30/0.50 ! [A: nat,B: nat,C: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 0.30/0.50 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_diff_eq
% 0.30/0.50 thf(fact_795_diff__diff__eq,axiom,
% 0.30/0.50 ! [A: int,B: int,C: int] :
% 0.30/0.50 ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.50 = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_diff_eq
% 0.30/0.50 thf(fact_796_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 => ( ! [N2: nat] :
% 0.30/0.50 ( ( P @ ( suc @ N2 ) )
% 0.30/0.50 => ( P @ N2 ) )
% 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_797_comm__monoid__mult__class_Omult__1,axiom,
% 0.30/0.50 ! [A: complex] :
% 0.30/0.50 ( ( times_times_complex @ one_one_complex @ A )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % comm_monoid_mult_class.mult_1
% 0.30/0.50 thf(fact_798_comm__monoid__mult__class_Omult__1,axiom,
% 0.30/0.50 ! [A: real] :
% 0.30/0.50 ( ( times_times_real @ one_one_real @ A )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % comm_monoid_mult_class.mult_1
% 0.30/0.50 thf(fact_799_comm__monoid__mult__class_Omult__1,axiom,
% 0.30/0.50 ! [A: rat] :
% 0.30/0.50 ( ( times_times_rat @ one_one_rat @ A )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % comm_monoid_mult_class.mult_1
% 0.30/0.50 thf(fact_800_comm__monoid__mult__class_Omult__1,axiom,
% 0.30/0.50 ! [A: nat] :
% 0.30/0.50 ( ( times_times_nat @ one_one_nat @ A )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % comm_monoid_mult_class.mult_1
% 0.30/0.50 thf(fact_801_comm__monoid__mult__class_Omult__1,axiom,
% 0.30/0.50 ! [A: int] :
% 0.30/0.50 ( ( times_times_int @ one_one_int @ A )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % comm_monoid_mult_class.mult_1
% 0.30/0.50 thf(fact_802_mult_Ocomm__neutral,axiom,
% 0.30/0.50 ! [A: complex] :
% 0.30/0.50 ( ( times_times_complex @ A @ one_one_complex )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult.comm_neutral
% 0.30/0.50 thf(fact_803_mult_Ocomm__neutral,axiom,
% 0.30/0.50 ! [A: real] :
% 0.30/0.50 ( ( times_times_real @ A @ one_one_real )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult.comm_neutral
% 0.30/0.50 thf(fact_804_mult_Ocomm__neutral,axiom,
% 0.30/0.50 ! [A: rat] :
% 0.30/0.50 ( ( times_times_rat @ A @ one_one_rat )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult.comm_neutral
% 0.30/0.50 thf(fact_805_mult_Ocomm__neutral,axiom,
% 0.30/0.50 ! [A: nat] :
% 0.30/0.50 ( ( times_times_nat @ A @ one_one_nat )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult.comm_neutral
% 0.30/0.50 thf(fact_806_mult_Ocomm__neutral,axiom,
% 0.30/0.50 ! [A: int] :
% 0.30/0.50 ( ( times_times_int @ A @ one_one_int )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult.comm_neutral
% 0.30/0.50 thf(fact_807_less__imp__diff__less,axiom,
% 0.30/0.50 ! [J: nat,K: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_nat @ J @ K )
% 0.30/0.50 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_imp_diff_less
% 0.30/0.50 thf(fact_808_diff__less__mono2,axiom,
% 0.30/0.50 ! [M: nat,N: nat,L: nat] :
% 0.30/0.50 ( ( ord_less_nat @ M @ N )
% 0.30/0.50 => ( ( ord_less_nat @ M @ L )
% 0.30/0.50 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_less_mono2
% 0.30/0.50 thf(fact_809_power__commuting__commutes,axiom,
% 0.30/0.50 ! [X2: complex,Y3: complex,N: nat] :
% 0.30/0.50 ( ( ( times_times_complex @ X2 @ Y3 )
% 0.30/0.50 = ( times_times_complex @ Y3 @ X2 ) )
% 0.30/0.50 => ( ( times_times_complex @ ( power_power_complex @ X2 @ N ) @ Y3 )
% 0.30/0.50 = ( times_times_complex @ Y3 @ ( power_power_complex @ X2 @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_commuting_commutes
% 0.30/0.50 thf(fact_810_power__commuting__commutes,axiom,
% 0.30/0.50 ! [X2: real,Y3: real,N: nat] :
% 0.30/0.50 ( ( ( times_times_real @ X2 @ Y3 )
% 0.30/0.50 = ( times_times_real @ Y3 @ X2 ) )
% 0.30/0.50 => ( ( times_times_real @ ( power_power_real @ X2 @ N ) @ Y3 )
% 0.30/0.50 = ( times_times_real @ Y3 @ ( power_power_real @ X2 @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_commuting_commutes
% 0.30/0.50 thf(fact_811_power__commuting__commutes,axiom,
% 0.30/0.50 ! [X2: rat,Y3: rat,N: nat] :
% 0.30/0.50 ( ( ( times_times_rat @ X2 @ Y3 )
% 0.30/0.50 = ( times_times_rat @ Y3 @ X2 ) )
% 0.30/0.50 => ( ( times_times_rat @ ( power_power_rat @ X2 @ N ) @ Y3 )
% 0.30/0.50 = ( times_times_rat @ Y3 @ ( power_power_rat @ X2 @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_commuting_commutes
% 0.30/0.50 thf(fact_812_power__commuting__commutes,axiom,
% 0.30/0.50 ! [X2: nat,Y3: nat,N: nat] :
% 0.30/0.50 ( ( ( times_times_nat @ X2 @ Y3 )
% 0.30/0.50 = ( times_times_nat @ Y3 @ X2 ) )
% 0.30/0.50 => ( ( times_times_nat @ ( power_power_nat @ X2 @ N ) @ Y3 )
% 0.30/0.50 = ( times_times_nat @ Y3 @ ( power_power_nat @ X2 @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_commuting_commutes
% 0.30/0.50 thf(fact_813_power__commuting__commutes,axiom,
% 0.30/0.50 ! [X2: int,Y3: int,N: nat] :
% 0.30/0.50 ( ( ( times_times_int @ X2 @ Y3 )
% 0.30/0.50 = ( times_times_int @ Y3 @ X2 ) )
% 0.30/0.50 => ( ( times_times_int @ ( power_power_int @ X2 @ N ) @ Y3 )
% 0.30/0.50 = ( times_times_int @ Y3 @ ( power_power_int @ X2 @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_commuting_commutes
% 0.30/0.50 thf(fact_814_power__mult__distrib,axiom,
% 0.30/0.50 ! [A: complex,B: complex,N: nat] :
% 0.30/0.50 ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
% 0.30/0.50 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_mult_distrib
% 0.30/0.50 thf(fact_815_power__mult__distrib,axiom,
% 0.30/0.50 ! [A: real,B: real,N: nat] :
% 0.30/0.50 ( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
% 0.30/0.50 = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_mult_distrib
% 0.30/0.50 thf(fact_816_power__mult__distrib,axiom,
% 0.30/0.50 ! [A: rat,B: rat,N: nat] :
% 0.30/0.50 ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
% 0.30/0.50 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_mult_distrib
% 0.30/0.50 thf(fact_817_power__mult__distrib,axiom,
% 0.30/0.50 ! [A: nat,B: nat,N: nat] :
% 0.30/0.50 ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
% 0.30/0.50 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_mult_distrib
% 0.30/0.50 thf(fact_818_power__mult__distrib,axiom,
% 0.30/0.50 ! [A: int,B: int,N: nat] :
% 0.30/0.50 ( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
% 0.30/0.50 = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_mult_distrib
% 0.30/0.50 thf(fact_819_power__commutes,axiom,
% 0.30/0.50 ! [A: complex,N: nat] :
% 0.30/0.50 ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
% 0.30/0.50 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_commutes
% 0.30/0.50 thf(fact_820_power__commutes,axiom,
% 0.30/0.50 ! [A: real,N: nat] :
% 0.30/0.50 ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
% 0.30/0.50 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_commutes
% 0.30/0.50 thf(fact_821_power__commutes,axiom,
% 0.30/0.50 ! [A: rat,N: nat] :
% 0.30/0.50 ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
% 0.30/0.50 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_commutes
% 0.30/0.50 thf(fact_822_power__commutes,axiom,
% 0.30/0.50 ! [A: nat,N: nat] :
% 0.30/0.50 ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
% 0.30/0.50 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_commutes
% 0.30/0.50 thf(fact_823_power__commutes,axiom,
% 0.30/0.50 ! [A: int,N: nat] :
% 0.30/0.50 ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
% 0.30/0.50 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_commutes
% 0.30/0.50 thf(fact_824_Suc__mult__cancel1,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 0.30/0.50 = ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.30/0.50 = ( M = N ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_mult_cancel1
% 0.30/0.50 thf(fact_825_eq__diff__iff,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.50 => ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.50 => ( ( ( minus_minus_nat @ M @ K )
% 0.30/0.50 = ( minus_minus_nat @ N @ K ) )
% 0.30/0.50 = ( M = N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % eq_diff_iff
% 0.30/0.50 thf(fact_826_le__diff__iff,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.50 => ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.50 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.30/0.50 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_diff_iff
% 0.30/0.50 thf(fact_827_Nat_Odiff__diff__eq,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.50 => ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.50 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.30/0.50 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Nat.diff_diff_eq
% 0.30/0.50 thf(fact_828_diff__le__mono,axiom,
% 0.30/0.50 ! [M: nat,N: nat,L: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.50 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_le_mono
% 0.30/0.50 thf(fact_829_diff__le__self,axiom,
% 0.30/0.50 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.30/0.50
% 0.30/0.50 % diff_le_self
% 0.30/0.50 thf(fact_830_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_831_diff__le__mono2,axiom,
% 0.30/0.50 ! [M: nat,N: nat,L: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.50 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_le_mono2
% 0.30/0.50 thf(fact_832_power__mult,axiom,
% 0.30/0.50 ! [A: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
% 0.30/0.50 = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_mult
% 0.30/0.50 thf(fact_833_power__mult,axiom,
% 0.30/0.50 ! [A: real,M: nat,N: nat] :
% 0.30/0.50 ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
% 0.30/0.50 = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_mult
% 0.30/0.50 thf(fact_834_power__mult,axiom,
% 0.30/0.50 ! [A: int,M: nat,N: nat] :
% 0.30/0.50 ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
% 0.30/0.50 = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_mult
% 0.30/0.50 thf(fact_835_power__mult,axiom,
% 0.30/0.50 ! [A: complex,M: nat,N: nat] :
% 0.30/0.50 ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
% 0.30/0.50 = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_mult
% 0.30/0.50 thf(fact_836_Nat_Odiff__cancel,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.30/0.50 = ( minus_minus_nat @ M @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % Nat.diff_cancel
% 0.30/0.50 thf(fact_837_diff__cancel2,axiom,
% 0.30/0.50 ! [M: nat,K: nat,N: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 0.30/0.50 = ( minus_minus_nat @ M @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_cancel2
% 0.30/0.50 thf(fact_838_diff__add__inverse,axiom,
% 0.30/0.50 ! [N: nat,M: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 0.30/0.50 = M ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add_inverse
% 0.30/0.50 thf(fact_839_diff__add__inverse2,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 0.30/0.50 = M ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add_inverse2
% 0.30/0.50 thf(fact_840_le__cube,axiom,
% 0.30/0.50 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_cube
% 0.30/0.50 thf(fact_841_le__square,axiom,
% 0.30/0.50 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_square
% 0.30/0.50 thf(fact_842_mult__le__mono,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,K: nat,L: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.50 => ( ( ord_less_eq_nat @ K @ L )
% 0.30/0.50 => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_le_mono
% 0.30/0.50 thf(fact_843_mult__le__mono1,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 @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_le_mono1
% 0.30/0.50 thf(fact_844_mult__le__mono2,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 @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_le_mono2
% 0.30/0.50 thf(fact_845_add__mult__distrib,axiom,
% 0.30/0.50 ! [M: nat,N: nat,K: nat] :
% 0.30/0.50 ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 0.30/0.50 = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_mult_distrib
% 0.30/0.50 thf(fact_846_add__mult__distrib2,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.50 = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_mult_distrib2
% 0.30/0.50 thf(fact_847_left__add__mult__distrib,axiom,
% 0.30/0.50 ! [I2: nat,U: nat,J: nat,K: nat] :
% 0.30/0.50 ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 0.30/0.50 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U ) @ K ) ) ).
% 0.30/0.50
% 0.30/0.50 % left_add_mult_distrib
% 0.30/0.50 thf(fact_848_nat__mult__1,axiom,
% 0.30/0.50 ! [N: nat] :
% 0.30/0.50 ( ( times_times_nat @ one_one_nat @ N )
% 0.30/0.50 = N ) ).
% 0.30/0.50
% 0.30/0.50 % nat_mult_1
% 0.30/0.50 thf(fact_849_nat__mult__1__right,axiom,
% 0.30/0.50 ! [N: nat] :
% 0.30/0.50 ( ( times_times_nat @ N @ one_one_nat )
% 0.30/0.50 = N ) ).
% 0.30/0.50
% 0.30/0.50 % nat_mult_1_right
% 0.30/0.50 thf(fact_850_div__mult2__eq,axiom,
% 0.30/0.50 ! [M: nat,N: nat,Q3: nat] :
% 0.30/0.50 ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q3 ) )
% 0.30/0.50 = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q3 ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_mult2_eq
% 0.30/0.50 thf(fact_851_nat__less__add__iff1,axiom,
% 0.30/0.50 ! [J: nat,I2: nat,U: nat,M: nat,N: 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 ) @ N ) )
% 0.30/0.50 = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_less_add_iff1
% 0.30/0.50 thf(fact_852_nat__less__add__iff2,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,U: nat,M: nat,N: 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 ) @ N ) )
% 0.30/0.50 = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_less_add_iff2
% 0.30/0.50 thf(fact_853_diff__le__eq,axiom,
% 0.30/0.50 ! [A: real,B: real,C: real] :
% 0.30/0.50 ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.30/0.50 = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_le_eq
% 0.30/0.50 thf(fact_854_diff__le__eq,axiom,
% 0.30/0.50 ! [A: rat,B: rat,C: rat] :
% 0.30/0.50 ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.30/0.50 = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_le_eq
% 0.30/0.50 thf(fact_855_diff__le__eq,axiom,
% 0.30/0.50 ! [A: int,B: int,C: int] :
% 0.30/0.50 ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.50 = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_le_eq
% 0.30/0.50 thf(fact_856_le__diff__eq,axiom,
% 0.30/0.50 ! [A: real,C: real,B: real] :
% 0.30/0.50 ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 0.30/0.50 = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_diff_eq
% 0.30/0.50 thf(fact_857_le__diff__eq,axiom,
% 0.30/0.50 ! [A: rat,C: rat,B: rat] :
% 0.30/0.50 ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 0.30/0.50 = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_diff_eq
% 0.30/0.50 thf(fact_858_le__diff__eq,axiom,
% 0.30/0.50 ! [A: int,C: int,B: int] :
% 0.30/0.50 ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.30/0.50 = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_diff_eq
% 0.30/0.50 thf(fact_859_diff__add,axiom,
% 0.30/0.50 ! [A: nat,B: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.50 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 0.30/0.50 = B ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add
% 0.30/0.50 thf(fact_860_le__add__diff,axiom,
% 0.30/0.50 ! [A: nat,B: nat,C: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.50 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_add_diff
% 0.30/0.50 thf(fact_861_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.30/0.50 ! [A: nat,B: nat,C: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.50 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.30/0.50 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.30/0.50 thf(fact_862_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.30/0.50 ! [A: nat,B: nat,C: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.50 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.30/0.50 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.30/0.50 thf(fact_863_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.30/0.50 ! [A: nat,B: nat,C: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.50 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 0.30/0.50 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.30/0.50 thf(fact_864_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.30/0.50 ! [A: nat,B: nat,C: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.50 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 0.30/0.50 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.30/0.50 thf(fact_865_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.30/0.50 ! [A: nat,B: nat,C: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.50 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 0.30/0.50 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.30/0.50 thf(fact_866_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.30/0.50 ! [A: nat,B: nat,C: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.50 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.30/0.50 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.30/0.50 thf(fact_867_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.30/0.50 ! [A: nat,B: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.50 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 0.30/0.50 = B ) ) ).
% 0.30/0.50
% 0.30/0.50 % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.30/0.50 thf(fact_868_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.30/0.50 ! [A: nat,B: nat,C: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.50 => ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.50 => ( ( ( minus_minus_nat @ B @ A )
% 0.30/0.50 = C )
% 0.30/0.50 = ( B
% 0.30/0.50 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.30/0.50 thf(fact_869_diff__less__eq,axiom,
% 0.30/0.50 ! [A: real,B: real,C: real] :
% 0.30/0.50 ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.30/0.50 = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_less_eq
% 0.30/0.50 thf(fact_870_diff__less__eq,axiom,
% 0.30/0.50 ! [A: rat,B: rat,C: rat] :
% 0.30/0.50 ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.30/0.50 = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_less_eq
% 0.30/0.50 thf(fact_871_diff__less__eq,axiom,
% 0.30/0.50 ! [A: int,B: int,C: int] :
% 0.30/0.50 ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.50 = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_less_eq
% 0.30/0.50 thf(fact_872_less__diff__eq,axiom,
% 0.30/0.50 ! [A: real,C: real,B: real] :
% 0.30/0.50 ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
% 0.30/0.50 = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_diff_eq
% 0.30/0.50 thf(fact_873_less__diff__eq,axiom,
% 0.30/0.50 ! [A: rat,C: rat,B: rat] :
% 0.30/0.50 ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 0.30/0.50 = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_diff_eq
% 0.30/0.50 thf(fact_874_less__diff__eq,axiom,
% 0.30/0.50 ! [A: int,C: int,B: int] :
% 0.30/0.50 ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.30/0.50 = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_diff_eq
% 0.30/0.50 thf(fact_875_mult__numeral__1__right,axiom,
% 0.30/0.50 ! [A: complex] :
% 0.30/0.50 ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult_numeral_1_right
% 0.30/0.50 thf(fact_876_mult__numeral__1__right,axiom,
% 0.30/0.50 ! [A: real] :
% 0.30/0.50 ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult_numeral_1_right
% 0.30/0.50 thf(fact_877_mult__numeral__1__right,axiom,
% 0.30/0.50 ! [A: rat] :
% 0.30/0.50 ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult_numeral_1_right
% 0.30/0.50 thf(fact_878_mult__numeral__1__right,axiom,
% 0.30/0.50 ! [A: nat] :
% 0.30/0.50 ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult_numeral_1_right
% 0.30/0.50 thf(fact_879_mult__numeral__1__right,axiom,
% 0.30/0.50 ! [A: int] :
% 0.30/0.50 ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult_numeral_1_right
% 0.30/0.50 thf(fact_880_mult__numeral__1,axiom,
% 0.30/0.50 ! [A: complex] :
% 0.30/0.50 ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult_numeral_1
% 0.30/0.50 thf(fact_881_mult__numeral__1,axiom,
% 0.30/0.50 ! [A: real] :
% 0.30/0.50 ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult_numeral_1
% 0.30/0.50 thf(fact_882_mult__numeral__1,axiom,
% 0.30/0.50 ! [A: rat] :
% 0.30/0.50 ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult_numeral_1
% 0.30/0.50 thf(fact_883_mult__numeral__1,axiom,
% 0.30/0.50 ! [A: nat] :
% 0.30/0.50 ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult_numeral_1
% 0.30/0.50 thf(fact_884_mult__numeral__1,axiom,
% 0.30/0.50 ! [A: int] :
% 0.30/0.50 ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
% 0.30/0.50 = A ) ).
% 0.30/0.50
% 0.30/0.50 % mult_numeral_1
% 0.30/0.50 thf(fact_885_diff__less__Suc,axiom,
% 0.30/0.50 ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_less_Suc
% 0.30/0.50 thf(fact_886_Suc__diff__Suc,axiom,
% 0.30/0.50 ! [N: nat,M: nat] :
% 0.30/0.50 ( ( ord_less_nat @ N @ M )
% 0.30/0.50 => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 0.30/0.50 = ( minus_minus_nat @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_diff_Suc
% 0.30/0.50 thf(fact_887_left__right__inverse__power,axiom,
% 0.30/0.50 ! [X2: complex,Y3: complex,N: nat] :
% 0.30/0.50 ( ( ( times_times_complex @ X2 @ Y3 )
% 0.30/0.50 = one_one_complex )
% 0.30/0.50 => ( ( times_times_complex @ ( power_power_complex @ X2 @ N ) @ ( power_power_complex @ Y3 @ N ) )
% 0.30/0.50 = one_one_complex ) ) ).
% 0.30/0.50
% 0.30/0.50 % left_right_inverse_power
% 0.30/0.50 thf(fact_888_left__right__inverse__power,axiom,
% 0.30/0.50 ! [X2: real,Y3: real,N: nat] :
% 0.30/0.50 ( ( ( times_times_real @ X2 @ Y3 )
% 0.30/0.50 = one_one_real )
% 0.30/0.50 => ( ( times_times_real @ ( power_power_real @ X2 @ N ) @ ( power_power_real @ Y3 @ N ) )
% 0.30/0.50 = one_one_real ) ) ).
% 0.30/0.50
% 0.30/0.50 % left_right_inverse_power
% 0.30/0.50 thf(fact_889_left__right__inverse__power,axiom,
% 0.30/0.50 ! [X2: rat,Y3: rat,N: nat] :
% 0.30/0.50 ( ( ( times_times_rat @ X2 @ Y3 )
% 0.30/0.50 = one_one_rat )
% 0.30/0.50 => ( ( times_times_rat @ ( power_power_rat @ X2 @ N ) @ ( power_power_rat @ Y3 @ N ) )
% 0.30/0.50 = one_one_rat ) ) ).
% 0.30/0.50
% 0.30/0.50 % left_right_inverse_power
% 0.30/0.50 thf(fact_890_left__right__inverse__power,axiom,
% 0.30/0.50 ! [X2: nat,Y3: nat,N: nat] :
% 0.30/0.50 ( ( ( times_times_nat @ X2 @ Y3 )
% 0.30/0.50 = one_one_nat )
% 0.30/0.50 => ( ( times_times_nat @ ( power_power_nat @ X2 @ N ) @ ( power_power_nat @ Y3 @ N ) )
% 0.30/0.50 = one_one_nat ) ) ).
% 0.30/0.50
% 0.30/0.50 % left_right_inverse_power
% 0.30/0.50 thf(fact_891_left__right__inverse__power,axiom,
% 0.30/0.50 ! [X2: int,Y3: int,N: nat] :
% 0.30/0.50 ( ( ( times_times_int @ X2 @ Y3 )
% 0.30/0.50 = one_one_int )
% 0.30/0.50 => ( ( times_times_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y3 @ N ) )
% 0.30/0.50 = one_one_int ) ) ).
% 0.30/0.50
% 0.30/0.50 % left_right_inverse_power
% 0.30/0.50 thf(fact_892_Suc__diff__le,axiom,
% 0.30/0.50 ! [N: nat,M: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.50 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.30/0.50 = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_diff_le
% 0.30/0.50 thf(fact_893_power__Suc2,axiom,
% 0.30/0.50 ! [A: complex,N: nat] :
% 0.30/0.50 ( ( power_power_complex @ A @ ( suc @ N ) )
% 0.30/0.50 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc2
% 0.30/0.50 thf(fact_894_power__Suc2,axiom,
% 0.30/0.50 ! [A: real,N: nat] :
% 0.30/0.50 ( ( power_power_real @ A @ ( suc @ N ) )
% 0.30/0.50 = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc2
% 0.30/0.50 thf(fact_895_power__Suc2,axiom,
% 0.30/0.50 ! [A: rat,N: nat] :
% 0.30/0.50 ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.30/0.50 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc2
% 0.30/0.50 thf(fact_896_power__Suc2,axiom,
% 0.30/0.50 ! [A: nat,N: nat] :
% 0.30/0.50 ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.30/0.50 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc2
% 0.30/0.50 thf(fact_897_power__Suc2,axiom,
% 0.30/0.50 ! [A: int,N: nat] :
% 0.30/0.50 ( ( power_power_int @ A @ ( suc @ N ) )
% 0.30/0.50 = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc2
% 0.30/0.50 thf(fact_898_power__Suc,axiom,
% 0.30/0.50 ! [A: complex,N: nat] :
% 0.30/0.50 ( ( power_power_complex @ A @ ( suc @ N ) )
% 0.30/0.50 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc
% 0.30/0.50 thf(fact_899_power__Suc,axiom,
% 0.30/0.50 ! [A: real,N: nat] :
% 0.30/0.50 ( ( power_power_real @ A @ ( suc @ N ) )
% 0.30/0.50 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc
% 0.30/0.50 thf(fact_900_power__Suc,axiom,
% 0.30/0.50 ! [A: rat,N: nat] :
% 0.30/0.50 ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.30/0.50 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc
% 0.30/0.50 thf(fact_901_power__Suc,axiom,
% 0.30/0.50 ! [A: nat,N: nat] :
% 0.30/0.50 ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.30/0.50 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc
% 0.30/0.50 thf(fact_902_power__Suc,axiom,
% 0.30/0.50 ! [A: int,N: nat] :
% 0.30/0.50 ( ( power_power_int @ A @ ( suc @ N ) )
% 0.30/0.50 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc
% 0.30/0.50 thf(fact_903_less__diff__iff,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.50 => ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.50 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.30/0.50 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_diff_iff
% 0.30/0.50 thf(fact_904_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_905_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_906_add__diff__inverse__nat,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ~ ( ord_less_nat @ M @ N )
% 0.30/0.50 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.50 = M ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_inverse_nat
% 0.30/0.50 thf(fact_907_diff__Suc__eq__diff__pred,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.30/0.50 = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_Suc_eq_diff_pred
% 0.30/0.50 thf(fact_908_Suc__mult__less__cancel1,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.30/0.50 = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_mult_less_cancel1
% 0.30/0.50 thf(fact_909_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_910_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_911_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_912_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_913_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_914_power__add,axiom,
% 0.30/0.50 ! [A: complex,M: nat,N: nat] :
% 0.30/0.50 ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.50 = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_add
% 0.30/0.50 thf(fact_915_power__add,axiom,
% 0.30/0.50 ! [A: real,M: nat,N: nat] :
% 0.30/0.50 ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.50 = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_add
% 0.30/0.50 thf(fact_916_power__add,axiom,
% 0.30/0.50 ! [A: rat,M: nat,N: nat] :
% 0.30/0.50 ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.50 = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_add
% 0.30/0.50 thf(fact_917_power__add,axiom,
% 0.30/0.50 ! [A: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.50 = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_add
% 0.30/0.50 thf(fact_918_power__add,axiom,
% 0.30/0.50 ! [A: int,M: nat,N: nat] :
% 0.30/0.50 ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.50 = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_add
% 0.30/0.50 thf(fact_919_Suc__mult__le__cancel1,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.30/0.50 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_mult_le_cancel1
% 0.30/0.50 thf(fact_920_mult__Suc,axiom,
% 0.30/0.50 ! [M: nat,N: nat] :
% 0.30/0.50 ( ( times_times_nat @ ( suc @ M ) @ N )
% 0.30/0.50 = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_Suc
% 0.30/0.50 thf(fact_921_less__mult__imp__div__less,axiom,
% 0.30/0.50 ! [M: nat,I2: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_nat @ M @ ( times_times_nat @ I2 @ N ) )
% 0.30/0.50 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_mult_imp_div_less
% 0.30/0.50 thf(fact_922_div__times__less__eq__dividend,axiom,
% 0.30/0.50 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% 0.30/0.50
% 0.30/0.50 % div_times_less_eq_dividend
% 0.30/0.50 thf(fact_923_times__div__less__eq__dividend,axiom,
% 0.30/0.50 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% 0.30/0.50
% 0.30/0.50 % times_div_less_eq_dividend
% 0.30/0.50 thf(fact_924_power__odd__eq,axiom,
% 0.30/0.50 ! [A: complex,N: nat] :
% 0.30/0.50 ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.50 = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_odd_eq
% 0.30/0.50 thf(fact_925_power__odd__eq,axiom,
% 0.30/0.50 ! [A: real,N: nat] :
% 0.30/0.50 ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.50 = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_odd_eq
% 0.30/0.50 thf(fact_926_power__odd__eq,axiom,
% 0.30/0.50 ! [A: rat,N: nat] :
% 0.30/0.50 ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.50 = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_odd_eq
% 0.30/0.50 thf(fact_927_power__odd__eq,axiom,
% 0.30/0.50 ! [A: nat,N: nat] :
% 0.30/0.50 ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.50 = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_odd_eq
% 0.30/0.50 thf(fact_928_power__odd__eq,axiom,
% 0.30/0.50 ! [A: int,N: nat] :
% 0.30/0.50 ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.50 = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_odd_eq
% 0.30/0.50 thf(fact_929_power2__diff,axiom,
% 0.30/0.50 ! [X2: complex,Y3: complex] :
% 0.30/0.50 ( ( power_power_complex @ ( minus_minus_complex @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_diff
% 0.30/0.50 thf(fact_930_power2__diff,axiom,
% 0.30/0.50 ! [X2: real,Y3: real] :
% 0.30/0.50 ( ( power_power_real @ ( minus_minus_real @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_diff
% 0.30/0.50 thf(fact_931_power2__diff,axiom,
% 0.30/0.50 ! [X2: rat,Y3: rat] :
% 0.30/0.50 ( ( power_power_rat @ ( minus_minus_rat @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_diff
% 0.30/0.50 thf(fact_932_power2__diff,axiom,
% 0.30/0.50 ! [X2: int,Y3: int] :
% 0.30/0.50 ( ( power_power_int @ ( minus_minus_int @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_diff
% 0.30/0.50 thf(fact_933_power__less__power__Suc,axiom,
% 0.30/0.50 ! [A: real,N: nat] :
% 0.30/0.50 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.50 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_less_power_Suc
% 0.30/0.50 thf(fact_934_power__less__power__Suc,axiom,
% 0.30/0.50 ! [A: rat,N: nat] :
% 0.30/0.50 ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.50 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_less_power_Suc
% 0.30/0.50 thf(fact_935_power__less__power__Suc,axiom,
% 0.30/0.50 ! [A: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.50 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_less_power_Suc
% 0.30/0.50 thf(fact_936_power__less__power__Suc,axiom,
% 0.30/0.50 ! [A: int,N: nat] :
% 0.30/0.50 ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.50 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_less_power_Suc
% 0.30/0.50 thf(fact_937_power__gt1__lemma,axiom,
% 0.30/0.50 ! [A: real,N: nat] :
% 0.30/0.50 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.50 => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_gt1_lemma
% 0.30/0.50 thf(fact_938_power__gt1__lemma,axiom,
% 0.30/0.50 ! [A: rat,N: nat] :
% 0.30/0.50 ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.50 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_gt1_lemma
% 0.30/0.50 thf(fact_939_power__gt1__lemma,axiom,
% 0.30/0.50 ! [A: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.50 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_gt1_lemma
% 0.30/0.50 thf(fact_940_power__gt1__lemma,axiom,
% 0.30/0.50 ! [A: int,N: nat] :
% 0.30/0.50 ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.50 => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_gt1_lemma
% 0.30/0.50 thf(fact_941_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_942_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
% 0.30/0.50 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.30/0.50 ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 0.30/0.50 = one_one_nat ) ).
% 0.30/0.50
% 0.30/0.50 % T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
% 0.30/0.50 thf(fact_943_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
% 0.30/0.50 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.30/0.50 ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 0.30/0.50 = one_one_nat ) ).
% 0.30/0.50
% 0.30/0.50 % T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
% 0.30/0.50 thf(fact_944_power2__commute,axiom,
% 0.30/0.50 ! [X2: complex,Y3: complex] :
% 0.30/0.50 ( ( power_power_complex @ ( minus_minus_complex @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( power_power_complex @ ( minus_minus_complex @ Y3 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_commute
% 0.30/0.50 thf(fact_945_power2__commute,axiom,
% 0.30/0.50 ! [X2: real,Y3: real] :
% 0.30/0.50 ( ( power_power_real @ ( minus_minus_real @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( power_power_real @ ( minus_minus_real @ Y3 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_commute
% 0.30/0.50 thf(fact_946_power2__commute,axiom,
% 0.30/0.50 ! [X2: rat,Y3: rat] :
% 0.30/0.50 ( ( power_power_rat @ ( minus_minus_rat @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( power_power_rat @ ( minus_minus_rat @ Y3 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_commute
% 0.30/0.50 thf(fact_947_power2__commute,axiom,
% 0.30/0.50 ! [X2: int,Y3: int] :
% 0.30/0.50 ( ( power_power_int @ ( minus_minus_int @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( power_power_int @ ( minus_minus_int @ Y3 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_commute
% 0.30/0.50 thf(fact_948_left__add__twice,axiom,
% 0.30/0.50 ! [A: complex,B: complex] :
% 0.30/0.50 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
% 0.30/0.50 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % left_add_twice
% 0.30/0.50 thf(fact_949_left__add__twice,axiom,
% 0.30/0.50 ! [A: real,B: real] :
% 0.30/0.50 ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.30/0.50 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % left_add_twice
% 0.30/0.50 thf(fact_950_left__add__twice,axiom,
% 0.30/0.50 ! [A: rat,B: rat] :
% 0.30/0.50 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.30/0.50 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % left_add_twice
% 0.30/0.50 thf(fact_951_left__add__twice,axiom,
% 0.30/0.50 ! [A: nat,B: nat] :
% 0.30/0.50 ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.30/0.50 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % left_add_twice
% 0.30/0.50 thf(fact_952_left__add__twice,axiom,
% 0.30/0.50 ! [A: int,B: int] :
% 0.30/0.50 ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.30/0.50 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.30/0.50
% 0.30/0.50 % left_add_twice
% 0.30/0.50 thf(fact_953_mult__2__right,axiom,
% 0.30/0.50 ! [Z: complex] :
% 0.30/0.50 ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( plus_plus_complex @ Z @ Z ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_2_right
% 0.30/0.50 thf(fact_954_mult__2__right,axiom,
% 0.30/0.50 ! [Z: real] :
% 0.30/0.50 ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( plus_plus_real @ Z @ Z ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_2_right
% 0.30/0.50 thf(fact_955_mult__2__right,axiom,
% 0.30/0.50 ! [Z: rat] :
% 0.30/0.50 ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( plus_plus_rat @ Z @ Z ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_2_right
% 0.30/0.50 thf(fact_956_mult__2__right,axiom,
% 0.30/0.50 ! [Z: nat] :
% 0.30/0.50 ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( plus_plus_nat @ Z @ Z ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_2_right
% 0.30/0.50 thf(fact_957_mult__2__right,axiom,
% 0.30/0.50 ! [Z: int] :
% 0.30/0.50 ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( plus_plus_int @ Z @ Z ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_2_right
% 0.30/0.50 thf(fact_958_mult__2,axiom,
% 0.30/0.50 ! [Z: complex] :
% 0.30/0.50 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
% 0.30/0.50 = ( plus_plus_complex @ Z @ Z ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_2
% 0.30/0.50 thf(fact_959_mult__2,axiom,
% 0.30/0.50 ! [Z: real] :
% 0.30/0.50 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
% 0.30/0.50 = ( plus_plus_real @ Z @ Z ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_2
% 0.30/0.50 thf(fact_960_mult__2,axiom,
% 0.30/0.50 ! [Z: rat] :
% 0.30/0.50 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
% 0.30/0.50 = ( plus_plus_rat @ Z @ Z ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_2
% 0.30/0.50 thf(fact_961_mult__2,axiom,
% 0.30/0.50 ! [Z: nat] :
% 0.30/0.50 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
% 0.30/0.50 = ( plus_plus_nat @ Z @ Z ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_2
% 0.30/0.50 thf(fact_962_mult__2,axiom,
% 0.30/0.50 ! [Z: int] :
% 0.30/0.50 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
% 0.30/0.50 = ( plus_plus_int @ Z @ Z ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_2
% 0.30/0.50 thf(fact_963_power2__eq__square,axiom,
% 0.30/0.50 ! [A: complex] :
% 0.30/0.50 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( times_times_complex @ A @ A ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_eq_square
% 0.30/0.50 thf(fact_964_power2__eq__square,axiom,
% 0.30/0.50 ! [A: real] :
% 0.30/0.50 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( times_times_real @ A @ A ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_eq_square
% 0.30/0.50 thf(fact_965_power2__eq__square,axiom,
% 0.30/0.50 ! [A: rat] :
% 0.30/0.50 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( times_times_rat @ A @ A ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_eq_square
% 0.30/0.50 thf(fact_966_power2__eq__square,axiom,
% 0.30/0.50 ! [A: nat] :
% 0.30/0.50 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( times_times_nat @ A @ A ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_eq_square
% 0.30/0.50 thf(fact_967_power2__eq__square,axiom,
% 0.30/0.50 ! [A: int] :
% 0.30/0.50 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( times_times_int @ A @ A ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_eq_square
% 0.30/0.50 thf(fact_968_power4__eq__xxxx,axiom,
% 0.30/0.50 ! [X2: complex] :
% 0.30/0.50 ( ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % power4_eq_xxxx
% 0.30/0.50 thf(fact_969_power4__eq__xxxx,axiom,
% 0.30/0.50 ! [X2: real] :
% 0.30/0.50 ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = ( times_times_real @ ( times_times_real @ ( times_times_real @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % power4_eq_xxxx
% 0.30/0.50 thf(fact_970_power4__eq__xxxx,axiom,
% 0.30/0.50 ! [X2: rat] :
% 0.30/0.50 ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % power4_eq_xxxx
% 0.30/0.50 thf(fact_971_power4__eq__xxxx,axiom,
% 0.30/0.50 ! [X2: nat] :
% 0.30/0.50 ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % power4_eq_xxxx
% 0.30/0.50 thf(fact_972_power4__eq__xxxx,axiom,
% 0.30/0.50 ! [X2: int] :
% 0.30/0.50 ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = ( times_times_int @ ( times_times_int @ ( times_times_int @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % power4_eq_xxxx
% 0.30/0.50 thf(fact_973_power__even__eq,axiom,
% 0.30/0.50 ! [A: nat,N: nat] :
% 0.30/0.50 ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.50 = ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_even_eq
% 0.30/0.50 thf(fact_974_power__even__eq,axiom,
% 0.30/0.50 ! [A: real,N: nat] :
% 0.30/0.50 ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.50 = ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_even_eq
% 0.30/0.50 thf(fact_975_power__even__eq,axiom,
% 0.30/0.50 ! [A: int,N: nat] :
% 0.30/0.50 ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.50 = ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_even_eq
% 0.30/0.50 thf(fact_976_power__even__eq,axiom,
% 0.30/0.50 ! [A: complex,N: nat] :
% 0.30/0.50 ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.50 = ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_even_eq
% 0.30/0.50 thf(fact_977_div__nat__eqI,axiom,
% 0.30/0.50 ! [N: nat,Q3: nat,M: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
% 0.30/0.50 => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
% 0.30/0.50 => ( ( divide_divide_nat @ M @ N )
% 0.30/0.50 = Q3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_nat_eqI
% 0.30/0.50 thf(fact_978_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
% 0.30/0.50 ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 0.30/0.50 ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X2 )
% 0.30/0.50 = one_one_nat ) ).
% 0.30/0.50
% 0.30/0.50 % T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
% 0.30/0.50 thf(fact_979_diff__le__diff__pow,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N: 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 @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_le_diff_pow
% 0.30/0.50 thf(fact_980_power2__sum,axiom,
% 0.30/0.50 ! [X2: complex,Y3: complex] :
% 0.30/0.50 ( ( power_power_complex @ ( plus_plus_complex @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_sum
% 0.30/0.50 thf(fact_981_power2__sum,axiom,
% 0.30/0.50 ! [X2: real,Y3: real] :
% 0.30/0.50 ( ( power_power_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_sum
% 0.30/0.50 thf(fact_982_power2__sum,axiom,
% 0.30/0.50 ! [X2: rat,Y3: rat] :
% 0.30/0.50 ( ( power_power_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_sum
% 0.30/0.50 thf(fact_983_power2__sum,axiom,
% 0.30/0.50 ! [X2: nat,Y3: nat] :
% 0.30/0.50 ( ( power_power_nat @ ( plus_plus_nat @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_sum
% 0.30/0.50 thf(fact_984_power2__sum,axiom,
% 0.30/0.50 ! [X2: int,Y3: int] :
% 0.30/0.50 ( ( power_power_int @ ( plus_plus_int @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_sum
% 0.30/0.50 thf(fact_985_two__realpow__ge__one,axiom,
% 0.30/0.50 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.50
% 0.30/0.50 % two_realpow_ge_one
% 0.30/0.50 thf(fact_986_invar__vebt_Ointros_I2_J,axiom,
% 0.30/0.50 ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.30/0.50 ( ! [X5: vEBT_VEBT] :
% 0.30/0.50 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.50 => ( vEBT_invar_vebt @ X5 @ N ) )
% 0.30/0.50 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.30/0.50 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.30/0.50 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.50 => ( ( M = N )
% 0.30/0.50 => ( ( Deg
% 0.30/0.50 = ( plus_plus_nat @ N @ M ) )
% 0.30/0.50 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
% 0.30/0.50 => ( ! [X5: vEBT_VEBT] :
% 0.30/0.50 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.50 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
% 0.30/0.50 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % invar_vebt.intros(2)
% 0.30/0.50 thf(fact_987_invar__vebt_Ointros_I3_J,axiom,
% 0.30/0.50 ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.30/0.50 ( ! [X5: vEBT_VEBT] :
% 0.30/0.50 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.50 => ( vEBT_invar_vebt @ X5 @ N ) )
% 0.30/0.50 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.30/0.50 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.30/0.50 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.50 => ( ( M
% 0.30/0.50 = ( suc @ N ) )
% 0.30/0.50 => ( ( Deg
% 0.30/0.50 = ( plus_plus_nat @ N @ M ) )
% 0.30/0.50 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
% 0.30/0.50 => ( ! [X5: vEBT_VEBT] :
% 0.30/0.50 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.50 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
% 0.30/0.50 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % invar_vebt.intros(3)
% 0.30/0.50 thf(fact_988_field__sum__of__halves,axiom,
% 0.30/0.50 ! [X2: real] :
% 0.30/0.50 ( ( plus_plus_real @ ( divide_divide_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = X2 ) ).
% 0.30/0.50
% 0.30/0.50 % field_sum_of_halves
% 0.30/0.50 thf(fact_989_field__sum__of__halves,axiom,
% 0.30/0.50 ! [X2: rat] :
% 0.30/0.50 ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = X2 ) ).
% 0.30/0.50
% 0.30/0.50 % field_sum_of_halves
% 0.30/0.50 thf(fact_990_nested__mint,axiom,
% 0.30/0.50 ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
% 0.30/0.50 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 0.30/0.50 => ( ( N
% 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 @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nested_mint
% 0.30/0.50 thf(fact_991_summaxma,axiom,
% 0.30/0.50 ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.50 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
% 0.30/0.50 => ( ( Mi != Ma )
% 0.30/0.50 => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
% 0.30/0.50 = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % summaxma
% 0.30/0.50 thf(fact_992_sum__squares__bound,axiom,
% 0.30/0.50 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % sum_squares_bound
% 0.30/0.50 thf(fact_993_sum__squares__bound,axiom,
% 0.30/0.50 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % sum_squares_bound
% 0.30/0.51 thf(fact_994_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_995_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_996_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_997_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_998_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_999_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_1000_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_1001_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_1002_mul__def,axiom,
% 0.30/0.51 ( vEBT_VEBT_mul
% 0.30/0.51 = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).
% 0.30/0.51
% 0.30/0.51 % mul_def
% 0.30/0.51 thf(fact_1003_add__def,axiom,
% 0.30/0.51 ( vEBT_VEBT_add
% 0.30/0.51 = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_def
% 0.30/0.51 thf(fact_1004_vebt__maxt_Osimps_I3_J,axiom,
% 0.30/0.51 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.30/0.51 ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 0.30/0.51 = ( some_nat @ Ma ) ) ).
% 0.30/0.51
% 0.30/0.51 % vebt_maxt.simps(3)
% 0.30/0.51 thf(fact_1005_vebt__mint_Osimps_I3_J,axiom,
% 0.30/0.51 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.30/0.51 ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 0.30/0.51 = ( some_nat @ Mi ) ) ).
% 0.30/0.51
% 0.30/0.51 % vebt_mint.simps(3)
% 0.30/0.51 thf(fact_1006_vebt__insert_Osimps_I4_J,axiom,
% 0.30/0.51 ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 0.30/0.51 ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X2 )
% 0.30/0.51 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) ) ).
% 0.30/0.51
% 0.30/0.51 % vebt_insert.simps(4)
% 0.30/0.51 thf(fact_1007_enat__ord__number_I1_J,axiom,
% 0.30/0.51 ! [M: num,N: num] :
% 0.30/0.51 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.30/0.51 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % enat_ord_number(1)
% 0.30/0.51 thf(fact_1008_add__shift,axiom,
% 0.30/0.51 ! [X2: nat,Y3: nat,Z: nat] :
% 0.30/0.51 ( ( ( plus_plus_nat @ X2 @ Y3 )
% 0.30/0.51 = Z )
% 0.30/0.51 = ( ( vEBT_VEBT_add @ ( some_nat @ X2 ) @ ( some_nat @ Y3 ) )
% 0.30/0.51 = ( some_nat @ Z ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_shift
% 0.30/0.51 thf(fact_1009_mul__shift,axiom,
% 0.30/0.51 ! [X2: nat,Y3: nat,Z: nat] :
% 0.30/0.51 ( ( ( times_times_nat @ X2 @ Y3 )
% 0.30/0.51 = Z )
% 0.30/0.51 = ( ( vEBT_VEBT_mul @ ( some_nat @ X2 ) @ ( some_nat @ Y3 ) )
% 0.30/0.51 = ( some_nat @ Z ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mul_shift
% 0.30/0.51 thf(fact_1010_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_1011_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_1012_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_1013_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_1014_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_1015_semiring__norm_I13_J,axiom,
% 0.30/0.51 ! [M: num,N: num] :
% 0.30/0.51 ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.51 = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % semiring_norm(13)
% 0.30/0.51 thf(fact_1016_semiring__norm_I12_J,axiom,
% 0.30/0.51 ! [N: num] :
% 0.30/0.51 ( ( times_times_num @ one @ N )
% 0.30/0.51 = N ) ).
% 0.30/0.51
% 0.30/0.51 % semiring_norm(12)
% 0.30/0.51 thf(fact_1017_semiring__norm_I11_J,axiom,
% 0.30/0.51 ! [M: num] :
% 0.30/0.51 ( ( times_times_num @ M @ one )
% 0.30/0.51 = M ) ).
% 0.30/0.51
% 0.30/0.51 % semiring_norm(11)
% 0.30/0.51 thf(fact_1018_num__double,axiom,
% 0.30/0.51 ! [N: num] :
% 0.30/0.51 ( ( times_times_num @ ( bit0 @ one ) @ N )
% 0.30/0.51 = ( bit0 @ N ) ) ).
% 0.30/0.51
% 0.30/0.51 % num_double
% 0.30/0.51 thf(fact_1019_power__mult__numeral,axiom,
% 0.30/0.51 ! [A: nat,M: num,N: num] :
% 0.30/0.51 ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.51 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_mult_numeral
% 0.30/0.51 thf(fact_1020_power__mult__numeral,axiom,
% 0.30/0.51 ! [A: real,M: num,N: num] :
% 0.30/0.51 ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.51 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_mult_numeral
% 0.30/0.51 thf(fact_1021_power__mult__numeral,axiom,
% 0.30/0.51 ! [A: int,M: num,N: num] :
% 0.30/0.51 ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.51 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_mult_numeral
% 0.30/0.51 thf(fact_1022_power__mult__numeral,axiom,
% 0.30/0.51 ! [A: complex,M: num,N: num] :
% 0.30/0.51 ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.51 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_mult_numeral
% 0.30/0.51 thf(fact_1023_option_Ocollapse,axiom,
% 0.30/0.51 ! [Option: option4927543243414619207at_nat] :
% 0.30/0.51 ( ( Option != none_P5556105721700978146at_nat )
% 0.30/0.51 => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
% 0.30/0.51 = Option ) ) ).
% 0.30/0.51
% 0.30/0.51 % option.collapse
% 0.30/0.51 thf(fact_1024_option_Ocollapse,axiom,
% 0.30/0.51 ! [Option: option_nat] :
% 0.30/0.51 ( ( Option != none_nat )
% 0.30/0.51 => ( ( some_nat @ ( the_nat @ Option ) )
% 0.30/0.51 = Option ) ) ).
% 0.30/0.51
% 0.30/0.51 % option.collapse
% 0.30/0.51 thf(fact_1025_option_Ocollapse,axiom,
% 0.30/0.51 ! [Option: option_num] :
% 0.30/0.51 ( ( Option != none_num )
% 0.30/0.51 => ( ( some_num @ ( the_num @ Option ) )
% 0.30/0.51 = Option ) ) ).
% 0.30/0.51
% 0.30/0.51 % option.collapse
% 0.30/0.51 thf(fact_1026_enat__ord__number_I2_J,axiom,
% 0.30/0.51 ! [M: num,N: num] :
% 0.30/0.51 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.30/0.51 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % enat_ord_number(2)
% 0.30/0.51 thf(fact_1027_real__arch__pow,axiom,
% 0.30/0.51 ! [X2: real,Y3: real] :
% 0.30/0.51 ( ( ord_less_real @ one_one_real @ X2 )
% 0.30/0.51 => ? [N2: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X2 @ N2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % real_arch_pow
% 0.30/0.51 thf(fact_1028_add__diff__assoc__enat,axiom,
% 0.30/0.51 ! [Z: extended_enat,Y3: extended_enat,X2: extended_enat] :
% 0.30/0.51 ( ( ord_le2932123472753598470d_enat @ Z @ Y3 )
% 0.30/0.51 => ( ( plus_p3455044024723400733d_enat @ X2 @ ( minus_3235023915231533773d_enat @ Y3 @ Z ) )
% 0.30/0.51 = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X2 @ Y3 ) @ Z ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_assoc_enat
% 0.30/0.51 thf(fact_1029_less__eq__real__def,axiom,
% 0.30/0.51 ( ord_less_eq_real
% 0.30/0.51 = ( ^ [X: real,Y: real] :
% 0.30/0.51 ( ( ord_less_real @ X @ Y )
% 0.30/0.51 | ( X = Y ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_eq_real_def
% 0.30/0.51 thf(fact_1030_complete__real,axiom,
% 0.30/0.51 ! [S3: set_real] :
% 0.30/0.51 ( ? [X3: real] : ( member_real @ X3 @ S3 )
% 0.30/0.51 => ( ? [Z4: real] :
% 0.30/0.51 ! [X5: real] :
% 0.30/0.51 ( ( member_real @ X5 @ S3 )
% 0.30/0.51 => ( ord_less_eq_real @ X5 @ Z4 ) )
% 0.30/0.51 => ? [Y4: real] :
% 0.30/0.51 ( ! [X3: real] :
% 0.30/0.51 ( ( member_real @ X3 @ S3 )
% 0.30/0.51 => ( ord_less_eq_real @ X3 @ Y4 ) )
% 0.30/0.51 & ! [Z4: real] :
% 0.30/0.51 ( ! [X5: real] :
% 0.30/0.51 ( ( member_real @ X5 @ S3 )
% 0.30/0.51 => ( ord_less_eq_real @ X5 @ Z4 ) )
% 0.30/0.51 => ( ord_less_eq_real @ Y4 @ Z4 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % complete_real
% 0.30/0.51 thf(fact_1031_option_Osel,axiom,
% 0.30/0.51 ! [X22: product_prod_nat_nat] :
% 0.30/0.51 ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.30/0.51 = X22 ) ).
% 0.30/0.51
% 0.30/0.51 % option.sel
% 0.30/0.51 thf(fact_1032_option_Osel,axiom,
% 0.30/0.51 ! [X22: nat] :
% 0.30/0.51 ( ( the_nat @ ( some_nat @ X22 ) )
% 0.30/0.51 = X22 ) ).
% 0.30/0.51
% 0.30/0.51 % option.sel
% 0.30/0.51 thf(fact_1033_option_Osel,axiom,
% 0.30/0.51 ! [X22: num] :
% 0.30/0.51 ( ( the_num @ ( some_num @ X22 ) )
% 0.30/0.51 = X22 ) ).
% 0.30/0.51
% 0.30/0.51 % option.sel
% 0.30/0.51 thf(fact_1034_option_Oexpand,axiom,
% 0.30/0.51 ! [Option: option_nat,Option2: option_nat] :
% 0.30/0.51 ( ( ( Option = none_nat )
% 0.30/0.51 = ( Option2 = none_nat ) )
% 0.30/0.51 => ( ( ( Option != none_nat )
% 0.30/0.51 => ( ( Option2 != none_nat )
% 0.30/0.51 => ( ( the_nat @ Option )
% 0.30/0.51 = ( the_nat @ Option2 ) ) ) )
% 0.30/0.51 => ( Option = Option2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % option.expand
% 0.30/0.51 thf(fact_1035_option_Oexpand,axiom,
% 0.30/0.51 ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
% 0.30/0.51 ( ( ( Option = none_P5556105721700978146at_nat )
% 0.30/0.51 = ( Option2 = none_P5556105721700978146at_nat ) )
% 0.30/0.51 => ( ( ( Option != none_P5556105721700978146at_nat )
% 0.30/0.51 => ( ( Option2 != none_P5556105721700978146at_nat )
% 0.30/0.51 => ( ( the_Pr8591224930841456533at_nat @ Option )
% 0.30/0.51 = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
% 0.30/0.51 => ( Option = Option2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % option.expand
% 0.30/0.51 thf(fact_1036_option_Oexpand,axiom,
% 0.30/0.51 ! [Option: option_num,Option2: option_num] :
% 0.30/0.51 ( ( ( Option = none_num )
% 0.30/0.51 = ( Option2 = none_num ) )
% 0.30/0.51 => ( ( ( Option != none_num )
% 0.30/0.51 => ( ( Option2 != none_num )
% 0.30/0.51 => ( ( the_num @ Option )
% 0.30/0.51 = ( the_num @ Option2 ) ) ) )
% 0.30/0.51 => ( Option = Option2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % option.expand
% 0.30/0.51 thf(fact_1037_four__x__squared,axiom,
% 0.30/0.51 ! [X2: real] :
% 0.30/0.51 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.51 = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % four_x_squared
% 0.30/0.51 thf(fact_1038_L2__set__mult__ineq__lemma,axiom,
% 0.30/0.51 ! [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 ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % L2_set_mult_ineq_lemma
% 0.30/0.51 thf(fact_1039_div__mult2__numeral__eq,axiom,
% 0.30/0.51 ! [A: nat,K: num,L: num] :
% 0.30/0.51 ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
% 0.30/0.51 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % div_mult2_numeral_eq
% 0.30/0.51 thf(fact_1040_div__mult2__numeral__eq,axiom,
% 0.30/0.51 ! [A: int,K: num,L: num] :
% 0.30/0.51 ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
% 0.30/0.51 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % div_mult2_numeral_eq
% 0.30/0.51 thf(fact_1041_option_Oexhaust__sel,axiom,
% 0.30/0.51 ! [Option: option4927543243414619207at_nat] :
% 0.30/0.51 ( ( Option != none_P5556105721700978146at_nat )
% 0.30/0.51 => ( Option
% 0.30/0.51 = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % option.exhaust_sel
% 0.30/0.51 thf(fact_1042_option_Oexhaust__sel,axiom,
% 0.30/0.51 ! [Option: option_nat] :
% 0.30/0.51 ( ( Option != none_nat )
% 0.30/0.51 => ( Option
% 0.30/0.51 = ( some_nat @ ( the_nat @ Option ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % option.exhaust_sel
% 0.30/0.51 thf(fact_1043_option_Oexhaust__sel,axiom,
% 0.30/0.51 ! [Option: option_num] :
% 0.30/0.51 ( ( Option != none_num )
% 0.30/0.51 => ( Option
% 0.30/0.51 = ( some_num @ ( the_num @ Option ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % option.exhaust_sel
% 0.30/0.51 thf(fact_1044_linorder__neqE__linordered__idom,axiom,
% 0.30/0.51 ! [X2: real,Y3: real] :
% 0.30/0.51 ( ( X2 != Y3 )
% 0.30/0.51 => ( ~ ( ord_less_real @ X2 @ Y3 )
% 0.30/0.51 => ( ord_less_real @ Y3 @ X2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % linorder_neqE_linordered_idom
% 0.30/0.51 thf(fact_1045_linorder__neqE__linordered__idom,axiom,
% 0.30/0.51 ! [X2: rat,Y3: rat] :
% 0.30/0.51 ( ( X2 != Y3 )
% 0.30/0.51 => ( ~ ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.51 => ( ord_less_rat @ Y3 @ X2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % linorder_neqE_linordered_idom
% 0.30/0.51 thf(fact_1046_linorder__neqE__linordered__idom,axiom,
% 0.30/0.51 ! [X2: int,Y3: int] :
% 0.30/0.51 ( ( X2 != Y3 )
% 0.30/0.51 => ( ~ ( ord_less_int @ X2 @ Y3 )
% 0.30/0.51 => ( ord_less_int @ Y3 @ X2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % linorder_neqE_linordered_idom
% 0.30/0.51 thf(fact_1047_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_1048_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_1049_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_1050_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_1051_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_1052_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_1053_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_1054_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_1055_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_1056_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_1057_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_1058_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_1059_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_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_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_1067_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_1068_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_1069_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_1070_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_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_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_1078_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_1079_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_1080_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_1081_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_1082_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_1083_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_1084_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_1085_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_1086_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_1087_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_1088_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_1089_less__1__mult,axiom,
% 0.30/0.51 ! [M: real,N: real] :
% 0.30/0.51 ( ( ord_less_real @ one_one_real @ M )
% 0.30/0.51 => ( ( ord_less_real @ one_one_real @ N )
% 0.30/0.51 => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_1_mult
% 0.30/0.51 thf(fact_1090_less__1__mult,axiom,
% 0.30/0.51 ! [M: rat,N: rat] :
% 0.30/0.51 ( ( ord_less_rat @ one_one_rat @ M )
% 0.30/0.51 => ( ( ord_less_rat @ one_one_rat @ N )
% 0.30/0.51 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_1_mult
% 0.30/0.51 thf(fact_1091_less__1__mult,axiom,
% 0.30/0.51 ! [M: nat,N: nat] :
% 0.30/0.51 ( ( ord_less_nat @ one_one_nat @ M )
% 0.30/0.51 => ( ( ord_less_nat @ one_one_nat @ N )
% 0.30/0.51 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_1_mult
% 0.30/0.51 thf(fact_1092_less__1__mult,axiom,
% 0.30/0.51 ! [M: int,N: int] :
% 0.30/0.51 ( ( ord_less_int @ one_one_int @ M )
% 0.30/0.51 => ( ( ord_less_int @ one_one_int @ N )
% 0.30/0.51 => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_1_mult
% 0.30/0.51 thf(fact_1093_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_1094_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_1095_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_1096_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_1097_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_1098_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_1099_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_1100_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_1101_add__le__add__imp__diff__le,axiom,
% 0.30/0.51 ! [I2: real,K: real,N: real,J: real] :
% 0.30/0.51 ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
% 0.30/0.51 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.30/0.51 => ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
% 0.30/0.51 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.30/0.51 => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_le_add_imp_diff_le
% 0.30/0.51 thf(fact_1102_add__le__add__imp__diff__le,axiom,
% 0.30/0.51 ! [I2: rat,K: rat,N: rat,J: rat] :
% 0.30/0.51 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N )
% 0.30/0.51 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 0.30/0.51 => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N )
% 0.30/0.51 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 0.30/0.51 => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_le_add_imp_diff_le
% 0.30/0.51 thf(fact_1103_add__le__add__imp__diff__le,axiom,
% 0.30/0.51 ! [I2: nat,K: nat,N: nat,J: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
% 0.30/0.51 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.30/0.51 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
% 0.30/0.51 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.30/0.51 => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_le_add_imp_diff_le
% 0.30/0.51 thf(fact_1104_add__le__add__imp__diff__le,axiom,
% 0.30/0.51 ! [I2: int,K: int,N: int,J: int] :
% 0.30/0.51 ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
% 0.30/0.51 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.30/0.51 => ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
% 0.30/0.51 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.30/0.51 => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_le_add_imp_diff_le
% 0.30/0.51 thf(fact_1105_add__le__imp__le__diff,axiom,
% 0.30/0.51 ! [I2: real,K: real,N: real] :
% 0.30/0.51 ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
% 0.30/0.51 => ( ord_less_eq_real @ I2 @ ( minus_minus_real @ N @ K ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_le_imp_le_diff
% 0.30/0.51 thf(fact_1106_add__le__imp__le__diff,axiom,
% 0.30/0.51 ! [I2: rat,K: rat,N: rat] :
% 0.30/0.51 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N )
% 0.30/0.51 => ( ord_less_eq_rat @ I2 @ ( minus_minus_rat @ N @ K ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_le_imp_le_diff
% 0.30/0.51 thf(fact_1107_add__le__imp__le__diff,axiom,
% 0.30/0.51 ! [I2: nat,K: nat,N: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
% 0.30/0.51 => ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_le_imp_le_diff
% 0.30/0.51 thf(fact_1108_add__le__imp__le__diff,axiom,
% 0.30/0.51 ! [I2: int,K: int,N: int] :
% 0.30/0.51 ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
% 0.30/0.51 => ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N @ K ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_le_imp_le_diff
% 0.30/0.51 thf(fact_1109_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_1110_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_1111_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_1112_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_1113_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_1114_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_1115_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_1116_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_1117_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_1118_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_1119_square__diff__square__factored,axiom,
% 0.30/0.51 ! [X2: real,Y3: real] :
% 0.30/0.51 ( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) )
% 0.30/0.51 = ( times_times_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( minus_minus_real @ X2 @ Y3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % square_diff_square_factored
% 0.30/0.51 thf(fact_1120_square__diff__square__factored,axiom,
% 0.30/0.51 ! [X2: rat,Y3: rat] :
% 0.30/0.51 ( ( minus_minus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 0.30/0.51 = ( times_times_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ ( minus_minus_rat @ X2 @ Y3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % square_diff_square_factored
% 0.30/0.51 thf(fact_1121_square__diff__square__factored,axiom,
% 0.30/0.51 ! [X2: int,Y3: int] :
% 0.30/0.51 ( ( minus_minus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) )
% 0.30/0.51 = ( times_times_int @ ( plus_plus_int @ X2 @ Y3 ) @ ( minus_minus_int @ X2 @ Y3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % square_diff_square_factored
% 0.30/0.51 thf(fact_1122_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_1123_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_1124_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_1125_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_1126_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_1127_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.30/0.51 ! [X2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
% 0.30/0.51 ( ( ( vEBT_V1502963449132264192at_nat @ X2 @ Xa2 @ Xb )
% 0.30/0.51 = Y3 )
% 0.30/0.51 => ( ( ( Xa2 = none_P5556105721700978146at_nat )
% 0.30/0.51 => ( Y3 != 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 => ( ( Xb = none_P5556105721700978146at_nat )
% 0.30/0.51 => ( Y3 != none_P5556105721700978146at_nat ) ) )
% 0.30/0.51 => ~ ! [A4: product_prod_nat_nat] :
% 0.30/0.51 ( ( Xa2
% 0.30/0.51 = ( some_P7363390416028606310at_nat @ A4 ) )
% 0.30/0.51 => ! [B4: product_prod_nat_nat] :
% 0.30/0.51 ( ( Xb
% 0.30/0.51 = ( some_P7363390416028606310at_nat @ B4 ) )
% 0.30/0.51 => ( Y3
% 0.30/0.51 != ( some_P7363390416028606310at_nat @ ( X2 @ A4 @ B4 ) ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.option_shift.elims
% 0.30/0.51 thf(fact_1128_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.30/0.51 ! [X2: num > num > num,Xa2: option_num,Xb: option_num,Y3: option_num] :
% 0.30/0.51 ( ( ( vEBT_V819420779217536731ft_num @ X2 @ Xa2 @ Xb )
% 0.30/0.51 = Y3 )
% 0.30/0.51 => ( ( ( Xa2 = none_num )
% 0.30/0.51 => ( Y3 != none_num ) )
% 0.30/0.51 => ( ( ? [V2: num] :
% 0.30/0.51 ( Xa2
% 0.30/0.51 = ( some_num @ V2 ) )
% 0.30/0.51 => ( ( Xb = none_num )
% 0.30/0.51 => ( Y3 != none_num ) ) )
% 0.30/0.51 => ~ ! [A4: num] :
% 0.30/0.51 ( ( Xa2
% 0.30/0.51 = ( some_num @ A4 ) )
% 0.30/0.51 => ! [B4: num] :
% 0.30/0.51 ( ( Xb
% 0.30/0.51 = ( some_num @ B4 ) )
% 0.30/0.51 => ( Y3
% 0.30/0.51 != ( some_num @ ( X2 @ A4 @ B4 ) ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.option_shift.elims
% 0.30/0.51 thf(fact_1129_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.30/0.51 ! [X2: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y3: option_nat] :
% 0.30/0.51 ( ( ( vEBT_V4262088993061758097ft_nat @ X2 @ Xa2 @ Xb )
% 0.30/0.51 = Y3 )
% 0.30/0.51 => ( ( ( Xa2 = none_nat )
% 0.30/0.51 => ( Y3 != none_nat ) )
% 0.30/0.51 => ( ( ? [V2: nat] :
% 0.30/0.51 ( Xa2
% 0.30/0.51 = ( some_nat @ V2 ) )
% 0.30/0.51 => ( ( Xb = none_nat )
% 0.30/0.51 => ( Y3 != none_nat ) ) )
% 0.30/0.51 => ~ ! [A4: nat] :
% 0.30/0.51 ( ( Xa2
% 0.30/0.51 = ( some_nat @ A4 ) )
% 0.30/0.51 => ! [B4: nat] :
% 0.30/0.51 ( ( Xb
% 0.30/0.51 = ( some_nat @ B4 ) )
% 0.30/0.51 => ( Y3
% 0.30/0.51 != ( some_nat @ ( X2 @ A4 @ B4 ) ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.option_shift.elims
% 0.30/0.51 thf(fact_1130_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_1131_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_1132_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_1133_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_1134_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_1135_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_1136_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_1137_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_1138_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_1139_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_1140_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_1141_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_1142_square__diff__one__factored,axiom,
% 0.30/0.51 ! [X2: complex] :
% 0.30/0.51 ( ( minus_minus_complex @ ( times_times_complex @ X2 @ X2 ) @ one_one_complex )
% 0.30/0.51 = ( times_times_complex @ ( plus_plus_complex @ X2 @ one_one_complex ) @ ( minus_minus_complex @ X2 @ one_one_complex ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % square_diff_one_factored
% 0.30/0.51 thf(fact_1143_square__diff__one__factored,axiom,
% 0.30/0.51 ! [X2: real] :
% 0.30/0.51 ( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ one_one_real )
% 0.30/0.51 = ( times_times_real @ ( plus_plus_real @ X2 @ one_one_real ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % square_diff_one_factored
% 0.30/0.51 thf(fact_1144_square__diff__one__factored,axiom,
% 0.30/0.51 ! [X2: rat] :
% 0.30/0.51 ( ( minus_minus_rat @ ( times_times_rat @ X2 @ X2 ) @ one_one_rat )
% 0.30/0.51 = ( times_times_rat @ ( plus_plus_rat @ X2 @ one_one_rat ) @ ( minus_minus_rat @ X2 @ one_one_rat ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % square_diff_one_factored
% 0.30/0.51 thf(fact_1145_square__diff__one__factored,axiom,
% 0.30/0.51 ! [X2: int] :
% 0.30/0.51 ( ( minus_minus_int @ ( times_times_int @ X2 @ X2 ) @ one_one_int )
% 0.30/0.51 = ( times_times_int @ ( plus_plus_int @ X2 @ one_one_int ) @ ( minus_minus_int @ X2 @ one_one_int ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % square_diff_one_factored
% 0.30/0.51 thf(fact_1146_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_1147_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_1148_Suc__double__not__eq__double,axiom,
% 0.30/0.51 ! [M: nat,N: 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 ) ) @ N ) ) ).
% 0.30/0.51
% 0.30/0.51 % Suc_double_not_eq_double
% 0.30/0.51 thf(fact_1149_double__not__eq__Suc__double,axiom,
% 0.30/0.51 ! [M: nat,N: 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 ) ) @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % double_not_eq_Suc_double
% 0.30/0.51 thf(fact_1150_times__divide__eq__left,axiom,
% 0.30/0.51 ! [B: complex,C: complex,A: complex] :
% 0.30/0.51 ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
% 0.30/0.51 = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).
% 0.30/0.51
% 0.30/0.51 % times_divide_eq_left
% 0.30/0.51 thf(fact_1151_times__divide__eq__left,axiom,
% 0.30/0.51 ! [B: real,C: real,A: real] :
% 0.30/0.51 ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.30/0.51 = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% 0.30/0.51
% 0.30/0.51 % times_divide_eq_left
% 0.30/0.51 thf(fact_1152_times__divide__eq__left,axiom,
% 0.30/0.51 ! [B: rat,C: rat,A: rat] :
% 0.30/0.51 ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.30/0.51 = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).
% 0.30/0.51
% 0.30/0.51 % times_divide_eq_left
% 0.30/0.51 thf(fact_1153_divide__divide__eq__left,axiom,
% 0.30/0.51 ! [A: complex,B: complex,C: complex] :
% 0.30/0.51 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 0.30/0.51 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_divide_eq_left
% 0.30/0.51 thf(fact_1154_divide__divide__eq__left,axiom,
% 0.30/0.51 ! [A: real,B: real,C: real] :
% 0.30/0.51 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 0.30/0.51 = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_divide_eq_left
% 0.30/0.51 thf(fact_1155_divide__divide__eq__left,axiom,
% 0.30/0.51 ! [A: rat,B: rat,C: rat] :
% 0.30/0.51 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 0.30/0.51 = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_divide_eq_left
% 0.30/0.51 thf(fact_1156_divide__divide__eq__right,axiom,
% 0.30/0.51 ! [A: complex,B: complex,C: complex] :
% 0.30/0.51 ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.30/0.51 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_divide_eq_right
% 0.30/0.51 thf(fact_1157_divide__divide__eq__right,axiom,
% 0.30/0.51 ! [A: real,B: real,C: real] :
% 0.30/0.51 ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.30/0.51 = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_divide_eq_right
% 0.30/0.51 thf(fact_1158_divide__divide__eq__right,axiom,
% 0.30/0.51 ! [A: rat,B: rat,C: rat] :
% 0.30/0.51 ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.51 = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_divide_eq_right
% 0.30/0.51 thf(fact_1159_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_1160_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_1161_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_1162_pred__less__length__list,axiom,
% 0.30/0.51 ! [Deg: nat,X2: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
% 0.30/0.51 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.51 => ( ( ord_less_eq_nat @ X2 @ Ma )
% 0.30/0.51 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.30/0.51 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 0.30/0.51 = ( if_option_nat
% 0.30/0.51 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 != none_nat )
% 0.30/0.51 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( 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 @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 @ ( if_option_nat
% 0.30/0.51 @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.51 = none_nat )
% 0.30/0.51 @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( 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 @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pred_less_length_list
% 0.30/0.51 thf(fact_1163_pred__lesseq__max,axiom,
% 0.30/0.51 ! [Deg: nat,X2: nat,Ma: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.51 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.51 => ( ( ord_less_eq_nat @ X2 @ Ma )
% 0.30/0.51 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 0.30/0.51 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.30/0.51 @ ( if_option_nat
% 0.30/0.51 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 != none_nat )
% 0.30/0.51 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( 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 @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 @ ( if_option_nat
% 0.30/0.51 @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.51 = none_nat )
% 0.30/0.51 @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( 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 @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.51 @ none_nat ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pred_lesseq_max
% 0.30/0.51 thf(fact_1164_set__vebt_H__def,axiom,
% 0.30/0.51 ( vEBT_VEBT_set_vebt
% 0.30/0.51 = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % set_vebt'_def
% 0.30/0.51 thf(fact_1165_real__divide__square__eq,axiom,
% 0.30/0.51 ! [R2: real,A: real] :
% 0.30/0.51 ( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
% 0.30/0.51 = ( divide_divide_real @ A @ R2 ) ) ).
% 0.30/0.51
% 0.30/0.51 % real_divide_square_eq
% 0.30/0.51 thf(fact_1166_succ__less__length__list,axiom,
% 0.30/0.51 ! [Deg: nat,Mi: nat,X2: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 0.30/0.51 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.51 => ( ( ord_less_eq_nat @ Mi @ X2 )
% 0.30/0.51 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.30/0.51 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 0.30/0.51 = ( if_option_nat
% 0.30/0.51 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 != none_nat )
% 0.30/0.51 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( 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 @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 @ ( if_option_nat
% 0.30/0.51 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( 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 @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % succ_less_length_list
% 0.30/0.51 thf(fact_1167_succ__greatereq__min,axiom,
% 0.30/0.51 ! [Deg: nat,Mi: nat,X2: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.51 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.51 => ( ( ord_less_eq_nat @ Mi @ X2 )
% 0.30/0.51 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 0.30/0.51 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.30/0.51 @ ( if_option_nat
% 0.30/0.51 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 != none_nat )
% 0.30/0.51 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( 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 @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 @ ( if_option_nat
% 0.30/0.51 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( 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 @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.51 @ none_nat ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % succ_greatereq_min
% 0.30/0.51 thf(fact_1168_mult__commute__abs,axiom,
% 0.30/0.51 ! [C: real] :
% 0.30/0.51 ( ( ^ [X: real] : ( times_times_real @ X @ 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_1169_mult__commute__abs,axiom,
% 0.30/0.51 ! [C: rat] :
% 0.30/0.51 ( ( ^ [X: rat] : ( times_times_rat @ X @ 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_1170_mult__commute__abs,axiom,
% 0.30/0.51 ! [C: nat] :
% 0.30/0.51 ( ( ^ [X: nat] : ( times_times_nat @ X @ 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_1171_mult__commute__abs,axiom,
% 0.30/0.51 ! [C: int] :
% 0.30/0.51 ( ( ^ [X: int] : ( times_times_int @ X @ 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_1172_enat__less__induct,axiom,
% 0.30/0.51 ! [P: extended_enat > $o,N: extended_enat] :
% 0.30/0.51 ( ! [N2: extended_enat] :
% 0.30/0.51 ( ! [M2: extended_enat] :
% 0.30/0.51 ( ( ord_le72135733267957522d_enat @ M2 @ N2 )
% 0.30/0.51 => ( P @ M2 ) )
% 0.30/0.51 => ( P @ N2 ) )
% 0.30/0.51 => ( P @ N ) ) ).
% 0.30/0.51
% 0.30/0.51 % enat_less_induct
% 0.30/0.51 thf(fact_1173_lambda__one,axiom,
% 0.30/0.51 ( ( ^ [X: complex] : X )
% 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_1174_lambda__one,axiom,
% 0.30/0.51 ( ( ^ [X: real] : X )
% 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_1175_lambda__one,axiom,
% 0.30/0.51 ( ( ^ [X: rat] : X )
% 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_1176_lambda__one,axiom,
% 0.30/0.51 ( ( ^ [X: nat] : X )
% 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_1177_lambda__one,axiom,
% 0.30/0.51 ( ( ^ [X: int] : X )
% 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_1178_set__vebt__def,axiom,
% 0.30/0.51 ( vEBT_set_vebt
% 0.30/0.51 = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % set_vebt_def
% 0.30/0.51 thf(fact_1179_numeral__code_I2_J,axiom,
% 0.30/0.51 ! [N: num] :
% 0.30/0.51 ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 0.30/0.51 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % numeral_code(2)
% 0.30/0.51 thf(fact_1180_numeral__code_I2_J,axiom,
% 0.30/0.51 ! [N: num] :
% 0.30/0.51 ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 0.30/0.51 = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % numeral_code(2)
% 0.30/0.51 thf(fact_1181_numeral__code_I2_J,axiom,
% 0.30/0.51 ! [N: num] :
% 0.30/0.51 ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 0.30/0.51 = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % numeral_code(2)
% 0.30/0.51 thf(fact_1182_numeral__code_I2_J,axiom,
% 0.30/0.51 ! [N: num] :
% 0.30/0.51 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 0.30/0.51 = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % numeral_code(2)
% 0.30/0.51 thf(fact_1183_numeral__code_I2_J,axiom,
% 0.30/0.51 ! [N: num] :
% 0.30/0.51 ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 0.30/0.51 = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % numeral_code(2)
% 0.30/0.51 thf(fact_1184_power__numeral__even,axiom,
% 0.30/0.51 ! [Z: complex,W: num] :
% 0.30/0.51 ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.30/0.51 = ( times_times_complex @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_numeral_even
% 0.30/0.51 thf(fact_1185_power__numeral__even,axiom,
% 0.30/0.51 ! [Z: real,W: num] :
% 0.30/0.51 ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.30/0.51 = ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_numeral_even
% 0.30/0.51 thf(fact_1186_power__numeral__even,axiom,
% 0.30/0.51 ! [Z: rat,W: num] :
% 0.30/0.51 ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.30/0.51 = ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_numeral_even
% 0.30/0.51 thf(fact_1187_power__numeral__even,axiom,
% 0.30/0.51 ! [Z: nat,W: num] :
% 0.30/0.51 ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.30/0.51 = ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_numeral_even
% 0.30/0.51 thf(fact_1188_power__numeral__even,axiom,
% 0.30/0.51 ! [Z: int,W: num] :
% 0.30/0.51 ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.30/0.51 = ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % power_numeral_even
% 0.30/0.51 thf(fact_1189_linordered__field__no__ub,axiom,
% 0.30/0.51 ! [X3: real] :
% 0.30/0.51 ? [X_12: real] : ( ord_less_real @ X3 @ X_12 ) ).
% 0.30/0.51
% 0.30/0.51 % linordered_field_no_ub
% 0.30/0.51 thf(fact_1190_linordered__field__no__ub,axiom,
% 0.30/0.51 ! [X3: rat] :
% 0.30/0.51 ? [X_12: rat] : ( ord_less_rat @ X3 @ X_12 ) ).
% 0.30/0.51
% 0.30/0.51 % linordered_field_no_ub
% 0.30/0.51 thf(fact_1191_linordered__field__no__lb,axiom,
% 0.30/0.51 ! [X3: real] :
% 0.30/0.51 ? [Y4: real] : ( ord_less_real @ Y4 @ X3 ) ).
% 0.30/0.51
% 0.30/0.51 % linordered_field_no_lb
% 0.30/0.51 thf(fact_1192_linordered__field__no__lb,axiom,
% 0.30/0.51 ! [X3: rat] :
% 0.30/0.51 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X3 ) ).
% 0.30/0.51
% 0.30/0.51 % linordered_field_no_lb
% 0.30/0.51 thf(fact_1193_divide__divide__eq__left_H,axiom,
% 0.30/0.51 ! [A: complex,B: complex,C: complex] :
% 0.30/0.51 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 0.30/0.51 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_divide_eq_left'
% 0.30/0.51 thf(fact_1194_divide__divide__eq__left_H,axiom,
% 0.30/0.51 ! [A: real,B: real,C: real] :
% 0.30/0.51 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 0.30/0.51 = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_divide_eq_left'
% 0.30/0.51 thf(fact_1195_divide__divide__eq__left_H,axiom,
% 0.30/0.51 ! [A: rat,B: rat,C: rat] :
% 0.30/0.51 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 0.30/0.51 = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_divide_eq_left'
% 0.30/0.51 thf(fact_1196_divide__divide__times__eq,axiom,
% 0.30/0.51 ! [X2: complex,Y3: complex,Z: complex,W: complex] :
% 0.30/0.51 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X2 @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 0.30/0.51 = ( divide1717551699836669952omplex @ ( times_times_complex @ X2 @ W ) @ ( times_times_complex @ Y3 @ Z ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_divide_times_eq
% 0.30/0.51 thf(fact_1197_divide__divide__times__eq,axiom,
% 0.30/0.51 ! [X2: real,Y3: real,Z: real,W: real] :
% 0.30/0.51 ( ( divide_divide_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
% 0.30/0.51 = ( divide_divide_real @ ( times_times_real @ X2 @ W ) @ ( times_times_real @ Y3 @ Z ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_divide_times_eq
% 0.30/0.51 thf(fact_1198_divide__divide__times__eq,axiom,
% 0.30/0.51 ! [X2: rat,Y3: rat,Z: rat,W: rat] :
% 0.30/0.51 ( ( divide_divide_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
% 0.30/0.51 = ( divide_divide_rat @ ( times_times_rat @ X2 @ W ) @ ( times_times_rat @ Y3 @ Z ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_divide_times_eq
% 0.30/0.51 thf(fact_1199_times__divide__times__eq,axiom,
% 0.30/0.51 ! [X2: complex,Y3: complex,Z: complex,W: complex] :
% 0.30/0.51 ( ( times_times_complex @ ( divide1717551699836669952omplex @ X2 @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 0.30/0.51 = ( divide1717551699836669952omplex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ Y3 @ W ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % times_divide_times_eq
% 0.30/0.51 thf(fact_1200_times__divide__times__eq,axiom,
% 0.30/0.51 ! [X2: real,Y3: real,Z: real,W: real] :
% 0.30/0.51 ( ( times_times_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
% 0.30/0.51 = ( divide_divide_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y3 @ W ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % times_divide_times_eq
% 0.30/0.51 thf(fact_1201_times__divide__times__eq,axiom,
% 0.30/0.51 ! [X2: rat,Y3: rat,Z: rat,W: rat] :
% 0.30/0.51 ( ( times_times_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
% 0.30/0.51 = ( divide_divide_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y3 @ W ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % times_divide_times_eq
% 0.30/0.51 thf(fact_1202_add__divide__distrib,axiom,
% 0.30/0.51 ! [A: complex,B: complex,C: complex] :
% 0.30/0.51 ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
% 0.30/0.51 = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_divide_distrib
% 0.30/0.51 thf(fact_1203_add__divide__distrib,axiom,
% 0.30/0.51 ! [A: real,B: real,C: real] :
% 0.30/0.51 ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.30/0.51 = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_divide_distrib
% 0.30/0.51 thf(fact_1204_add__divide__distrib,axiom,
% 0.30/0.51 ! [A: rat,B: rat,C: rat] :
% 0.30/0.51 ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.30/0.51 = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_divide_distrib
% 0.30/0.51 thf(fact_1205_diff__divide__distrib,axiom,
% 0.30/0.51 ! [A: complex,B: complex,C: complex] :
% 0.30/0.51 ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
% 0.30/0.51 = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % diff_divide_distrib
% 0.30/0.51 thf(fact_1206_diff__divide__distrib,axiom,
% 0.30/0.51 ! [A: real,B: real,C: real] :
% 0.30/0.51 ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.30/0.51 = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % diff_divide_distrib
% 0.30/0.51 thf(fact_1207_diff__divide__distrib,axiom,
% 0.30/0.51 ! [A: rat,B: rat,C: rat] :
% 0.30/0.51 ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.30/0.51 = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % diff_divide_distrib
% 0.30/0.51 thf(fact_1208_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
% 0.30/0.51 ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 0.30/0.51 ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 0.30/0.51 = ( if_nat
% 0.30/0.51 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.30/0.51 & ~ ( ( X2 = Mi )
% 0.30/0.51 | ( X2 = Ma ) ) )
% 0.30/0.51 @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
% 0.30/0.51 @ one_one_nat ) ) ).
% 0.30/0.51
% 0.30/0.51 % T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
% 0.30/0.51 thf(fact_1209_gt__half__sum,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 @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % gt_half_sum
% 0.30/0.51 thf(fact_1210_gt__half__sum,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 @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % gt_half_sum
% 0.30/0.51 thf(fact_1211_less__half__sum,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 @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_half_sum
% 0.30/0.51 thf(fact_1212_less__half__sum,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 @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_half_sum
% 0.30/0.51 thf(fact_1213_vebt__succ_Osimps_I6_J,axiom,
% 0.30/0.51 ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.51 ( ( ( ord_less_nat @ X2 @ Mi )
% 0.30/0.51 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 0.30/0.51 = ( some_nat @ Mi ) ) )
% 0.30/0.51 & ( ~ ( ord_less_nat @ X2 @ Mi )
% 0.30/0.51 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 0.30/0.51 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.30/0.51 @ ( if_option_nat
% 0.30/0.51 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 @ ( if_option_nat
% 0.30/0.51 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( 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_1214_vebt__pred_Osimps_I7_J,axiom,
% 0.30/0.51 ! [Ma: nat,X2: nat,Mi: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.51 ( ( ( ord_less_nat @ Ma @ X2 )
% 0.30/0.51 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 0.30/0.51 = ( some_nat @ Ma ) ) )
% 0.30/0.51 & ( ~ ( ord_less_nat @ Ma @ X2 )
% 0.30/0.51 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 0.30/0.51 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.30/0.51 @ ( if_option_nat
% 0.30/0.51 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 @ ( if_option_nat
% 0.30/0.51 @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 ) @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( 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_1215_vebt__member_Osimps_I5_J,axiom,
% 0.30/0.51 ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 0.30/0.51 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 0.30/0.51 = ( ( X2 != Mi )
% 0.30/0.51 => ( ( X2 != Ma )
% 0.30/0.51 => ( ~ ( ord_less_nat @ X2 @ Mi )
% 0.30/0.51 & ( ~ ( ord_less_nat @ X2 @ Mi )
% 0.30/0.51 => ( ~ ( ord_less_nat @ Ma @ X2 )
% 0.30/0.51 & ( ~ ( ord_less_nat @ Ma @ X2 )
% 0.30/0.51 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.30/0.51 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % vebt_member.simps(5)
% 0.30/0.51 thf(fact_1216_pred__empty,axiom,
% 0.30/0.51 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.30/0.51 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.51 => ( ( ( vEBT_vebt_pred @ T @ X2 )
% 0.30/0.51 = none_nat )
% 0.30/0.51 = ( ( collect_nat
% 0.30/0.51 @ ^ [Y: nat] :
% 0.30/0.51 ( ( vEBT_vebt_member @ T @ Y )
% 0.30/0.51 & ( ord_less_nat @ Y @ X2 ) ) )
% 0.30/0.51 = bot_bot_set_nat ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pred_empty
% 0.30/0.51 thf(fact_1217_succ__empty,axiom,
% 0.30/0.51 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.30/0.51 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.51 => ( ( ( vEBT_vebt_succ @ T @ X2 )
% 0.30/0.51 = none_nat )
% 0.30/0.51 = ( ( collect_nat
% 0.30/0.51 @ ^ [Y: nat] :
% 0.30/0.51 ( ( vEBT_vebt_member @ T @ Y )
% 0.30/0.51 & ( ord_less_nat @ X2 @ Y ) ) )
% 0.30/0.51 = bot_bot_set_nat ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % succ_empty
% 0.30/0.51 thf(fact_1218_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 0.30/0.51 ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
% 0.30/0.51 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc ) @ X2 )
% 0.30/0.51 = ( ( X2 = Mi )
% 0.30/0.51 | ( X2 = Ma )
% 0.30/0.51 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.30/0.51 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.membermima.simps(4)
% 0.30/0.51 thf(fact_1219_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 0.30/0.51 ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
% 0.30/0.51 ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) @ X2 )
% 0.30/0.51 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.30/0.51 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.naive_member.simps(3)
% 0.30/0.51 thf(fact_1220_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 0.30/0.51 ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X2: nat] :
% 0.30/0.51 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) @ X2 )
% 0.30/0.51 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.30/0.51 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.membermima.simps(5)
% 0.30/0.51 thf(fact_1221_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_1222_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_1223_both__member__options__def,axiom,
% 0.30/0.51 ( vEBT_V8194947554948674370ptions
% 0.30/0.51 = ( ^ [T2: vEBT_VEBT,X: nat] :
% 0.30/0.51 ( ( vEBT_V5719532721284313246member @ T2 @ X )
% 0.30/0.51 | ( vEBT_VEBT_membermima @ T2 @ X ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % both_member_options_def
% 0.30/0.51 thf(fact_1224_member__valid__both__member__options,axiom,
% 0.30/0.51 ! [Tree: vEBT_VEBT,N: nat,X2: nat] :
% 0.30/0.51 ( ( vEBT_invar_vebt @ Tree @ N )
% 0.30/0.51 => ( ( vEBT_vebt_member @ Tree @ X2 )
% 0.30/0.51 => ( ( vEBT_V5719532721284313246member @ Tree @ X2 )
% 0.30/0.51 | ( vEBT_VEBT_membermima @ Tree @ X2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % member_valid_both_member_options
% 0.30/0.51 thf(fact_1225_mint__corr__help__empty,axiom,
% 0.30/0.51 ! [T: vEBT_VEBT,N: nat] :
% 0.30/0.51 ( ( vEBT_invar_vebt @ T @ N )
% 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_1226_maxt__corr__help__empty,axiom,
% 0.30/0.51 ! [T: vEBT_VEBT,N: nat] :
% 0.30/0.51 ( ( vEBT_invar_vebt @ T @ N )
% 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_1227_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_1228_vebt__member_Osimps_I2_J,axiom,
% 0.30/0.51 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
% 0.30/0.51 ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X2 ) ).
% 0.30/0.51
% 0.30/0.51 % vebt_member.simps(2)
% 0.30/0.51 thf(fact_1229_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_1230_is__succ__in__set__def,axiom,
% 0.30/0.51 ( vEBT_is_succ_in_set
% 0.30/0.51 = ( ^ [Xs: set_nat,X: nat,Y: nat] :
% 0.30/0.51 ( ( member_nat @ Y @ Xs )
% 0.30/0.51 & ( ord_less_nat @ X @ Y )
% 0.30/0.51 & ! [Z3: nat] :
% 0.30/0.51 ( ( member_nat @ Z3 @ Xs )
% 0.30/0.51 => ( ( ord_less_nat @ X @ Z3 )
% 0.30/0.51 => ( ord_less_eq_nat @ Y @ Z3 ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % is_succ_in_set_def
% 0.30/0.51 thf(fact_1231_is__pred__in__set__def,axiom,
% 0.30/0.51 ( vEBT_is_pred_in_set
% 0.30/0.51 = ( ^ [Xs: set_nat,X: nat,Y: nat] :
% 0.30/0.51 ( ( member_nat @ Y @ Xs )
% 0.30/0.51 & ( ord_less_nat @ Y @ X )
% 0.30/0.51 & ! [Z3: nat] :
% 0.30/0.51 ( ( member_nat @ Z3 @ Xs )
% 0.30/0.51 => ( ( ord_less_nat @ Z3 @ X )
% 0.30/0.51 => ( ord_less_eq_nat @ Z3 @ Y ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % is_pred_in_set_def
% 0.30/0.51 thf(fact_1232_buildup__gives__empty,axiom,
% 0.30/0.51 ! [N: nat] :
% 0.30/0.51 ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
% 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_1233_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 0.30/0.51 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.30/0.51 ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 0.30/0.51 => ( ! [Mi2: nat,Ma2: nat] :
% 0.30/0.51 ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.30/0.51 ( X2
% 0.30/0.51 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 0.30/0.51 => ~ ( ( Xa2 = Mi2 )
% 0.30/0.51 | ( Xa2 = Ma2 ) ) )
% 0.30/0.51 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
% 0.30/0.51 ( ? [Vc2: vEBT_VEBT] :
% 0.30/0.51 ( X2
% 0.30/0.51 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
% 0.30/0.51 => ~ ( ( Xa2 = Mi2 )
% 0.30/0.51 | ( Xa2 = Ma2 )
% 0.30/0.51 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.51 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.30/0.51 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
% 0.30/0.51 => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
% 0.30/0.51 ( ? [Vd2: vEBT_VEBT] :
% 0.30/0.51 ( X2
% 0.30/0.51 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
% 0.30/0.51 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.51 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.30/0.51 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.membermima.elims(2)
% 0.30/0.51 thf(fact_1234_divmod__step__eq,axiom,
% 0.30/0.51 ! [L: num,R2: nat,Q3: nat] :
% 0.30/0.51 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 0.30/0.51 => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
% 0.30/0.51 = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
% 0.30/0.51 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 0.30/0.51 => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
% 0.30/0.51 = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divmod_step_eq
% 0.30/0.51 thf(fact_1235_divmod__step__eq,axiom,
% 0.30/0.51 ! [L: num,R2: int,Q3: int] :
% 0.30/0.51 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 0.30/0.51 => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.30/0.51 = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
% 0.30/0.51 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 0.30/0.51 => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.30/0.51 = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divmod_step_eq
% 0.30/0.51 thf(fact_1236_divmod__step__eq,axiom,
% 0.30/0.51 ! [L: num,R2: code_integer,Q3: code_integer] :
% 0.30/0.51 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 0.30/0.51 => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
% 0.30/0.51 = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
% 0.30/0.51 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 0.30/0.51 => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
% 0.30/0.51 = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divmod_step_eq
% 0.30/0.51 thf(fact_1237_nth__mem,axiom,
% 0.30/0.51 ! [N: nat,Xs2: list_P6011104703257516679at_nat] :
% 0.30/0.51 ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
% 0.30/0.51 => ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ N ) @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nth_mem
% 0.30/0.51 thf(fact_1238_nth__mem,axiom,
% 0.30/0.51 ! [N: nat,Xs2: list_complex] :
% 0.30/0.51 ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 0.30/0.51 => ( member_complex @ ( nth_complex @ Xs2 @ N ) @ ( set_complex2 @ Xs2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nth_mem
% 0.30/0.51 thf(fact_1239_nth__mem,axiom,
% 0.30/0.51 ! [N: nat,Xs2: list_real] :
% 0.30/0.51 ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 0.30/0.51 => ( member_real @ ( nth_real @ Xs2 @ N ) @ ( set_real2 @ Xs2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nth_mem
% 0.30/0.51 thf(fact_1240_nth__mem,axiom,
% 0.30/0.51 ! [N: nat,Xs2: list_VEBT_VEBT] :
% 0.30/0.51 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.51 => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nth_mem
% 0.30/0.51 thf(fact_1241_nth__mem,axiom,
% 0.30/0.51 ! [N: nat,Xs2: list_o] :
% 0.30/0.51 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.51 => ( member_o @ ( nth_o @ Xs2 @ N ) @ ( set_o2 @ Xs2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nth_mem
% 0.30/0.51 thf(fact_1242_nth__mem,axiom,
% 0.30/0.51 ! [N: nat,Xs2: list_nat] :
% 0.30/0.51 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.51 => ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nth_mem
% 0.30/0.51 thf(fact_1243_nth__mem,axiom,
% 0.30/0.51 ! [N: nat,Xs2: list_int] :
% 0.30/0.51 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.51 => ( member_int @ ( nth_int @ Xs2 @ N ) @ ( set_int2 @ Xs2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nth_mem
% 0.30/0.51 thf(fact_1244_list__ball__nth,axiom,
% 0.30/0.51 ! [N: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.30/0.51 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.51 => ( ! [X5: vEBT_VEBT] :
% 0.30/0.51 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.30/0.51 => ( P @ X5 ) )
% 0.30/0.51 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % list_ball_nth
% 0.30/0.51 thf(fact_1245_list__ball__nth,axiom,
% 0.30/0.51 ! [N: nat,Xs2: list_o,P: $o > $o] :
% 0.30/0.51 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.51 => ( ! [X5: $o] :
% 0.30/0.51 ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
% 0.30/0.51 => ( P @ X5 ) )
% 0.30/0.51 => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % list_ball_nth
% 0.30/0.51 thf(fact_1246_list__ball__nth,axiom,
% 0.30/0.51 ! [N: nat,Xs2: list_nat,P: nat > $o] :
% 0.30/0.51 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.51 => ( ! [X5: nat] :
% 0.30/0.51 ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
% 0.30/0.51 => ( P @ X5 ) )
% 0.30/0.51 => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % list_ball_nth
% 0.30/0.51 thf(fact_1247_list__ball__nth,axiom,
% 0.30/0.51 ! [N: nat,Xs2: list_int,P: int > $o] :
% 0.30/0.51 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.51 => ( ! [X5: int] :
% 0.30/0.51 ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
% 0.30/0.51 => ( P @ X5 ) )
% 0.30/0.51 => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % list_ball_nth
% 0.30/0.51 thf(fact_1248_in__set__conv__nth,axiom,
% 0.30/0.51 ! [X2: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
% 0.30/0.51 ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
% 0.30/0.51 = ( ? [I4: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
% 0.30/0.51 & ( ( nth_Pr7617993195940197384at_nat @ Xs2 @ I4 )
% 0.30/0.51 = X2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % in_set_conv_nth
% 0.30/0.51 thf(fact_1249_in__set__conv__nth,axiom,
% 0.30/0.51 ! [X2: complex,Xs2: list_complex] :
% 0.30/0.51 ( ( member_complex @ X2 @ ( set_complex2 @ Xs2 ) )
% 0.30/0.51 = ( ? [I4: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 0.30/0.51 & ( ( nth_complex @ Xs2 @ I4 )
% 0.30/0.51 = X2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % in_set_conv_nth
% 0.30/0.51 thf(fact_1250_in__set__conv__nth,axiom,
% 0.30/0.51 ! [X2: real,Xs2: list_real] :
% 0.30/0.51 ( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
% 0.30/0.51 = ( ? [I4: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs2 ) )
% 0.30/0.51 & ( ( nth_real @ Xs2 @ I4 )
% 0.30/0.51 = X2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % in_set_conv_nth
% 0.30/0.51 thf(fact_1251_in__set__conv__nth,axiom,
% 0.30/0.51 ! [X2: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
% 0.30/0.51 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.30/0.51 = ( ? [I4: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.51 & ( ( nth_VEBT_VEBT @ Xs2 @ I4 )
% 0.30/0.51 = X2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % in_set_conv_nth
% 0.30/0.51 thf(fact_1252_in__set__conv__nth,axiom,
% 0.30/0.51 ! [X2: $o,Xs2: list_o] :
% 0.30/0.51 ( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
% 0.30/0.51 = ( ? [I4: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.51 & ( ( nth_o @ Xs2 @ I4 )
% 0.30/0.51 = X2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % in_set_conv_nth
% 0.30/0.51 thf(fact_1253_in__set__conv__nth,axiom,
% 0.30/0.51 ! [X2: nat,Xs2: list_nat] :
% 0.30/0.51 ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
% 0.30/0.51 = ( ? [I4: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.51 & ( ( nth_nat @ Xs2 @ I4 )
% 0.30/0.51 = X2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % in_set_conv_nth
% 0.30/0.51 thf(fact_1254_in__set__conv__nth,axiom,
% 0.30/0.51 ! [X2: int,Xs2: list_int] :
% 0.30/0.51 ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
% 0.30/0.51 = ( ? [I4: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.51 & ( ( nth_int @ Xs2 @ I4 )
% 0.30/0.51 = X2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % in_set_conv_nth
% 0.30/0.51 thf(fact_1255_all__nth__imp__all__set,axiom,
% 0.30/0.51 ! [Xs2: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,X2: product_prod_nat_nat] :
% 0.30/0.51 ( ! [I3: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
% 0.30/0.51 => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ I3 ) ) )
% 0.30/0.51 => ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
% 0.30/0.51 => ( P @ X2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % all_nth_imp_all_set
% 0.30/0.51 thf(fact_1256_all__nth__imp__all__set,axiom,
% 0.30/0.51 ! [Xs2: list_complex,P: complex > $o,X2: complex] :
% 0.30/0.51 ( ! [I3: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 0.30/0.51 => ( P @ ( nth_complex @ Xs2 @ I3 ) ) )
% 0.30/0.51 => ( ( member_complex @ X2 @ ( set_complex2 @ Xs2 ) )
% 0.30/0.51 => ( P @ X2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % all_nth_imp_all_set
% 0.30/0.51 thf(fact_1257_all__nth__imp__all__set,axiom,
% 0.30/0.51 ! [Xs2: list_real,P: real > $o,X2: real] :
% 0.30/0.51 ( ! [I3: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
% 0.30/0.51 => ( P @ ( nth_real @ Xs2 @ I3 ) ) )
% 0.30/0.51 => ( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
% 0.30/0.51 => ( P @ X2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % all_nth_imp_all_set
% 0.30/0.51 thf(fact_1258_all__nth__imp__all__set,axiom,
% 0.30/0.51 ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X2: vEBT_VEBT] :
% 0.30/0.51 ( ! [I3: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.51 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) )
% 0.30/0.51 => ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.30/0.51 => ( P @ X2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % all_nth_imp_all_set
% 0.30/0.51 thf(fact_1259_all__nth__imp__all__set,axiom,
% 0.30/0.51 ! [Xs2: list_o,P: $o > $o,X2: $o] :
% 0.30/0.51 ( ! [I3: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.51 => ( P @ ( nth_o @ Xs2 @ I3 ) ) )
% 0.30/0.51 => ( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
% 0.30/0.51 => ( P @ X2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % all_nth_imp_all_set
% 0.30/0.51 thf(fact_1260_all__nth__imp__all__set,axiom,
% 0.30/0.51 ! [Xs2: list_nat,P: nat > $o,X2: nat] :
% 0.30/0.51 ( ! [I3: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.51 => ( P @ ( nth_nat @ Xs2 @ I3 ) ) )
% 0.30/0.51 => ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
% 0.30/0.51 => ( P @ X2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % all_nth_imp_all_set
% 0.30/0.51 thf(fact_1261_all__nth__imp__all__set,axiom,
% 0.30/0.51 ! [Xs2: list_int,P: int > $o,X2: int] :
% 0.30/0.51 ( ! [I3: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.51 => ( P @ ( nth_int @ Xs2 @ I3 ) ) )
% 0.30/0.51 => ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
% 0.30/0.51 => ( P @ X2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % all_nth_imp_all_set
% 0.30/0.51 thf(fact_1262_all__set__conv__all__nth,axiom,
% 0.30/0.51 ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.30/0.51 ( ( ! [X: vEBT_VEBT] :
% 0.30/0.51 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.30/0.51 => ( P @ X ) ) )
% 0.30/0.51 = ( ! [I4: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.51 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I4 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % all_set_conv_all_nth
% 0.30/0.51 thf(fact_1263_all__set__conv__all__nth,axiom,
% 0.30/0.51 ! [Xs2: list_o,P: $o > $o] :
% 0.30/0.51 ( ( ! [X: $o] :
% 0.30/0.51 ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
% 0.30/0.51 => ( P @ X ) ) )
% 0.30/0.51 = ( ! [I4: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.51 => ( P @ ( nth_o @ Xs2 @ I4 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % all_set_conv_all_nth
% 0.30/0.51 thf(fact_1264_all__set__conv__all__nth,axiom,
% 0.30/0.51 ! [Xs2: list_nat,P: nat > $o] :
% 0.30/0.51 ( ( ! [X: nat] :
% 0.30/0.51 ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
% 0.30/0.51 => ( P @ X ) ) )
% 0.30/0.51 = ( ! [I4: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.51 => ( P @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % all_set_conv_all_nth
% 0.30/0.51 thf(fact_1265_all__set__conv__all__nth,axiom,
% 0.30/0.51 ! [Xs2: list_int,P: int > $o] :
% 0.30/0.51 ( ( ! [X: int] :
% 0.30/0.51 ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
% 0.30/0.51 => ( P @ X ) ) )
% 0.30/0.51 = ( ! [I4: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.51 => ( P @ ( nth_int @ Xs2 @ I4 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % all_set_conv_all_nth
% 0.30/0.51 thf(fact_1266_vebt__pred_Oelims,axiom,
% 0.30/0.51 ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 0.30/0.51 ( ( ( vEBT_vebt_pred @ X2 @ Xa2 )
% 0.30/0.51 = Y3 )
% 0.30/0.51 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.30/0.51 ( X2
% 0.30/0.51 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.51 => ( ( Xa2 = zero_zero_nat )
% 0.30/0.51 => ( Y3 != none_nat ) ) )
% 0.30/0.51 => ( ! [A4: $o] :
% 0.30/0.51 ( ? [Uw2: $o] :
% 0.30/0.51 ( X2
% 0.30/0.51 = ( vEBT_Leaf @ A4 @ Uw2 ) )
% 0.30/0.51 => ( ( Xa2
% 0.30/0.51 = ( suc @ zero_zero_nat ) )
% 0.30/0.51 => ~ ( ( A4
% 0.30/0.51 => ( Y3
% 0.30/0.51 = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.51 & ( ~ A4
% 0.30/0.51 => ( Y3 = none_nat ) ) ) ) )
% 0.30/0.51 => ( ! [A4: $o,B4: $o] :
% 0.30/0.51 ( ( X2
% 0.30/0.51 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.51 => ( ? [Va3: nat] :
% 0.30/0.51 ( Xa2
% 0.30/0.51 = ( suc @ ( suc @ Va3 ) ) )
% 0.30/0.51 => ~ ( ( B4
% 0.30/0.51 => ( Y3
% 0.30/0.51 = ( some_nat @ one_one_nat ) ) )
% 0.30/0.51 & ( ~ B4
% 0.30/0.51 => ( ( A4
% 0.30/0.51 => ( Y3
% 0.30/0.51 = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.51 & ( ~ A4
% 0.30/0.51 => ( Y3 = none_nat ) ) ) ) ) ) )
% 0.30/0.51 => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
% 0.30/0.51 ( X2
% 0.30/0.51 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
% 0.30/0.51 => ( Y3 != none_nat ) )
% 0.30/0.51 => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
% 0.30/0.51 ( X2
% 0.30/0.51 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) )
% 0.30/0.51 => ( Y3 != none_nat ) )
% 0.30/0.51 => ( ( ? [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
% 0.30/0.51 ( X2
% 0.30/0.51 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
% 0.30/0.51 => ( Y3 != none_nat ) )
% 0.30/0.51 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.51 ( ( X2
% 0.30/0.51 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.51 => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.51 => ( Y3
% 0.30/0.51 = ( some_nat @ Ma2 ) ) )
% 0.30/0.51 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.51 => ( Y3
% 0.30/0.51 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 != none_nat )
% 0.30/0.51 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 @ ( if_option_nat
% 0.30/0.51 @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.51 = none_nat )
% 0.30/0.51 @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.51 @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % vebt_pred.elims
% 0.30/0.51 thf(fact_1267_vebt__succ_Oelims,axiom,
% 0.30/0.51 ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 0.30/0.51 ( ( ( vEBT_vebt_succ @ X2 @ Xa2 )
% 0.30/0.51 = Y3 )
% 0.30/0.51 => ( ! [Uu2: $o,B4: $o] :
% 0.30/0.51 ( ( X2
% 0.30/0.51 = ( vEBT_Leaf @ Uu2 @ B4 ) )
% 0.30/0.51 => ( ( Xa2 = zero_zero_nat )
% 0.30/0.51 => ~ ( ( B4
% 0.30/0.51 => ( Y3
% 0.30/0.51 = ( some_nat @ one_one_nat ) ) )
% 0.30/0.51 & ( ~ B4
% 0.30/0.51 => ( Y3 = none_nat ) ) ) ) )
% 0.30/0.51 => ( ( ? [Uv2: $o,Uw2: $o] :
% 0.30/0.51 ( X2
% 0.30/0.51 = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 0.30/0.51 => ( ? [N2: nat] :
% 0.30/0.51 ( Xa2
% 0.30/0.51 = ( suc @ N2 ) )
% 0.30/0.51 => ( Y3 != none_nat ) ) )
% 0.30/0.51 => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.51 ( X2
% 0.30/0.51 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 0.30/0.51 => ( Y3 != none_nat ) )
% 0.30/0.51 => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.30/0.51 ( X2
% 0.30/0.51 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 0.30/0.51 => ( Y3 != none_nat ) )
% 0.30/0.51 => ( ( ? [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
% 0.30/0.51 ( X2
% 0.30/0.51 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
% 0.30/0.51 => ( Y3 != none_nat ) )
% 0.30/0.51 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.51 ( ( X2
% 0.30/0.51 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.51 => ~ ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.51 => ( Y3
% 0.30/0.51 = ( some_nat @ Mi2 ) ) )
% 0.30/0.51 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.51 => ( Y3
% 0.30/0.51 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 != none_nat )
% 0.30/0.51 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 @ ( if_option_nat
% 0.30/0.51 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.51 @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % vebt_succ.elims
% 0.30/0.51 thf(fact_1268_valid__tree__deg__neq__0,axiom,
% 0.30/0.51 ! [T: vEBT_VEBT] :
% 0.30/0.51 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 0.30/0.51
% 0.30/0.51 % valid_tree_deg_neq_0
% 0.30/0.51 thf(fact_1269_valid__0__not,axiom,
% 0.30/0.51 ! [T: vEBT_VEBT] :
% 0.30/0.51 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 0.30/0.51
% 0.30/0.51 % valid_0_not
% 0.30/0.51 thf(fact_1270_buildup__nothing__in__min__max,axiom,
% 0.30/0.51 ! [N: nat,X2: nat] :
% 0.30/0.51 ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X2 ) ).
% 0.30/0.51
% 0.30/0.51 % buildup_nothing_in_min_max
% 0.30/0.51 thf(fact_1271_buildup__nothing__in__leaf,axiom,
% 0.30/0.51 ! [N: nat,X2: nat] :
% 0.30/0.51 ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X2 ) ).
% 0.30/0.51
% 0.30/0.51 % buildup_nothing_in_leaf
% 0.30/0.51 thf(fact_1272_deg__not__0,axiom,
% 0.30/0.51 ! [T: vEBT_VEBT,N: nat] :
% 0.30/0.51 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.51 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.51
% 0.30/0.51 % deg_not_0
% 0.30/0.51 thf(fact_1273_Leaf__0__not,axiom,
% 0.30/0.51 ! [A: $o,B: $o] :
% 0.30/0.51 ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).
% 0.30/0.51
% 0.30/0.51 % Leaf_0_not
% 0.30/0.51 thf(fact_1274_deg1Leaf,axiom,
% 0.30/0.51 ! [T: vEBT_VEBT] :
% 0.30/0.51 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 0.30/0.51 = ( ? [A3: $o,B2: $o] :
% 0.30/0.51 ( T
% 0.30/0.51 = ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % deg1Leaf
% 0.30/0.51 thf(fact_1275_deg__1__Leaf,axiom,
% 0.30/0.51 ! [T: vEBT_VEBT] :
% 0.30/0.51 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 0.30/0.51 => ? [A4: $o,B4: $o] :
% 0.30/0.51 ( T
% 0.30/0.51 = ( vEBT_Leaf @ A4 @ B4 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % deg_1_Leaf
% 0.30/0.51 thf(fact_1276_deg__1__Leafy,axiom,
% 0.30/0.51 ! [T: vEBT_VEBT,N: nat] :
% 0.30/0.51 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.51 => ( ( N = one_one_nat )
% 0.30/0.51 => ? [A4: $o,B4: $o] :
% 0.30/0.51 ( T
% 0.30/0.51 = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % deg_1_Leafy
% 0.30/0.51 thf(fact_1277_buildup__gives__valid,axiom,
% 0.30/0.51 ! [N: nat] :
% 0.30/0.51 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.51 => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% 0.30/0.51
% 0.30/0.51 % buildup_gives_valid
% 0.30/0.51 thf(fact_1278_zdiv__numeral__Bit0,axiom,
% 0.30/0.51 ! [V: num,W: num] :
% 0.30/0.51 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.30/0.51 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zdiv_numeral_Bit0
% 0.30/0.51 thf(fact_1279_VEBT_Oinject_I2_J,axiom,
% 0.30/0.51 ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
% 0.30/0.51 ( ( ( vEBT_Leaf @ X21 @ X222 )
% 0.30/0.51 = ( vEBT_Leaf @ Y21 @ Y22 ) )
% 0.30/0.51 = ( ( X21 = Y21 )
% 0.30/0.51 & ( X222 = Y22 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT.inject(2)
% 0.30/0.51 thf(fact_1280_le__zero__eq,axiom,
% 0.30/0.51 ! [N: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.30/0.51 = ( N = zero_zero_nat ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_zero_eq
% 0.30/0.51 thf(fact_1281_not__gr__zero,axiom,
% 0.30/0.51 ! [N: nat] :
% 0.30/0.51 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.30/0.51 = ( N = zero_zero_nat ) ) ).
% 0.30/0.51
% 0.30/0.51 % not_gr_zero
% 0.30/0.51 thf(fact_1282_mult__cancel__right,axiom,
% 0.30/0.51 ! [A: complex,C: complex,B: complex] :
% 0.30/0.51 ( ( ( times_times_complex @ A @ C )
% 0.30/0.51 = ( times_times_complex @ B @ C ) )
% 0.30/0.51 = ( ( C = zero_zero_complex )
% 0.30/0.51 | ( A = B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_cancel_right
% 0.30/0.51 thf(fact_1283_mult__cancel__right,axiom,
% 0.30/0.51 ! [A: real,C: real,B: real] :
% 0.30/0.51 ( ( ( times_times_real @ A @ C )
% 0.30/0.51 = ( times_times_real @ B @ C ) )
% 0.30/0.51 = ( ( C = zero_zero_real )
% 0.30/0.51 | ( A = B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_cancel_right
% 0.30/0.51 thf(fact_1284_mult__cancel__right,axiom,
% 0.30/0.51 ! [A: rat,C: rat,B: rat] :
% 0.30/0.51 ( ( ( times_times_rat @ A @ C )
% 0.30/0.51 = ( times_times_rat @ B @ C ) )
% 0.30/0.51 = ( ( C = zero_zero_rat )
% 0.30/0.51 | ( A = B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_cancel_right
% 0.30/0.51 thf(fact_1285_mult__cancel__right,axiom,
% 0.30/0.51 ! [A: nat,C: nat,B: nat] :
% 0.30/0.51 ( ( ( times_times_nat @ A @ C )
% 0.30/0.51 = ( times_times_nat @ B @ C ) )
% 0.30/0.51 = ( ( C = zero_zero_nat )
% 0.30/0.51 | ( A = B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_cancel_right
% 0.30/0.51 thf(fact_1286_mult__cancel__right,axiom,
% 0.30/0.51 ! [A: int,C: int,B: int] :
% 0.30/0.51 ( ( ( times_times_int @ A @ C )
% 0.30/0.51 = ( times_times_int @ B @ C ) )
% 0.30/0.51 = ( ( C = zero_zero_int )
% 0.30/0.51 | ( A = B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_cancel_right
% 0.30/0.51 thf(fact_1287_mult__cancel__left,axiom,
% 0.30/0.51 ! [C: complex,A: complex,B: complex] :
% 0.30/0.51 ( ( ( times_times_complex @ C @ A )
% 0.30/0.51 = ( times_times_complex @ C @ B ) )
% 0.30/0.51 = ( ( C = zero_zero_complex )
% 0.30/0.51 | ( A = B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_cancel_left
% 0.30/0.51 thf(fact_1288_mult__cancel__left,axiom,
% 0.30/0.51 ! [C: real,A: real,B: real] :
% 0.30/0.51 ( ( ( times_times_real @ C @ A )
% 0.30/0.51 = ( times_times_real @ C @ B ) )
% 0.30/0.51 = ( ( C = zero_zero_real )
% 0.30/0.51 | ( A = B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_cancel_left
% 0.30/0.51 thf(fact_1289_mult__cancel__left,axiom,
% 0.30/0.51 ! [C: rat,A: rat,B: rat] :
% 0.30/0.51 ( ( ( times_times_rat @ C @ A )
% 0.30/0.51 = ( times_times_rat @ C @ B ) )
% 0.30/0.51 = ( ( C = zero_zero_rat )
% 0.30/0.51 | ( A = B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_cancel_left
% 0.30/0.51 thf(fact_1290_mult__cancel__left,axiom,
% 0.30/0.51 ! [C: nat,A: nat,B: nat] :
% 0.30/0.51 ( ( ( times_times_nat @ C @ A )
% 0.30/0.51 = ( times_times_nat @ C @ B ) )
% 0.30/0.51 = ( ( C = zero_zero_nat )
% 0.30/0.51 | ( A = B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_cancel_left
% 0.30/0.51 thf(fact_1291_mult__cancel__left,axiom,
% 0.30/0.51 ! [C: int,A: int,B: int] :
% 0.30/0.51 ( ( ( times_times_int @ C @ A )
% 0.30/0.51 = ( times_times_int @ C @ B ) )
% 0.30/0.51 = ( ( C = zero_zero_int )
% 0.30/0.51 | ( A = B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_cancel_left
% 0.30/0.51 thf(fact_1292_mult__eq__0__iff,axiom,
% 0.30/0.51 ! [A: complex,B: complex] :
% 0.30/0.51 ( ( ( times_times_complex @ A @ B )
% 0.30/0.51 = zero_zero_complex )
% 0.30/0.51 = ( ( A = zero_zero_complex )
% 0.30/0.51 | ( B = zero_zero_complex ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_eq_0_iff
% 0.30/0.51 thf(fact_1293_mult__eq__0__iff,axiom,
% 0.30/0.51 ! [A: real,B: real] :
% 0.30/0.51 ( ( ( times_times_real @ A @ B )
% 0.30/0.51 = zero_zero_real )
% 0.30/0.51 = ( ( A = zero_zero_real )
% 0.30/0.51 | ( B = zero_zero_real ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_eq_0_iff
% 0.30/0.51 thf(fact_1294_mult__eq__0__iff,axiom,
% 0.30/0.51 ! [A: rat,B: rat] :
% 0.30/0.51 ( ( ( times_times_rat @ A @ B )
% 0.30/0.51 = zero_zero_rat )
% 0.30/0.51 = ( ( A = zero_zero_rat )
% 0.30/0.51 | ( B = zero_zero_rat ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_eq_0_iff
% 0.30/0.51 thf(fact_1295_mult__eq__0__iff,axiom,
% 0.30/0.51 ! [A: nat,B: nat] :
% 0.30/0.51 ( ( ( times_times_nat @ A @ B )
% 0.30/0.51 = zero_zero_nat )
% 0.30/0.51 = ( ( A = zero_zero_nat )
% 0.30/0.51 | ( B = zero_zero_nat ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_eq_0_iff
% 0.30/0.51 thf(fact_1296_mult__eq__0__iff,axiom,
% 0.30/0.51 ! [A: int,B: int] :
% 0.30/0.51 ( ( ( times_times_int @ A @ B )
% 0.30/0.51 = zero_zero_int )
% 0.30/0.51 = ( ( A = zero_zero_int )
% 0.30/0.51 | ( B = zero_zero_int ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_eq_0_iff
% 0.30/0.51 thf(fact_1297_mult__zero__right,axiom,
% 0.30/0.51 ! [A: complex] :
% 0.30/0.51 ( ( times_times_complex @ A @ zero_zero_complex )
% 0.30/0.51 = zero_zero_complex ) ).
% 0.30/0.51
% 0.30/0.51 % mult_zero_right
% 0.30/0.51 thf(fact_1298_mult__zero__right,axiom,
% 0.30/0.51 ! [A: real] :
% 0.30/0.51 ( ( times_times_real @ A @ zero_zero_real )
% 0.30/0.51 = zero_zero_real ) ).
% 0.30/0.51
% 0.30/0.51 % mult_zero_right
% 0.30/0.51 thf(fact_1299_mult__zero__right,axiom,
% 0.30/0.51 ! [A: rat] :
% 0.30/0.51 ( ( times_times_rat @ A @ zero_zero_rat )
% 0.30/0.51 = zero_zero_rat ) ).
% 0.30/0.51
% 0.30/0.51 % mult_zero_right
% 0.30/0.51 thf(fact_1300_mult__zero__right,axiom,
% 0.30/0.51 ! [A: nat] :
% 0.30/0.51 ( ( times_times_nat @ A @ zero_zero_nat )
% 0.30/0.51 = zero_zero_nat ) ).
% 0.30/0.51
% 0.30/0.51 % mult_zero_right
% 0.30/0.51 thf(fact_1301_mult__zero__right,axiom,
% 0.30/0.51 ! [A: int] :
% 0.30/0.51 ( ( times_times_int @ A @ zero_zero_int )
% 0.30/0.51 = zero_zero_int ) ).
% 0.30/0.51
% 0.30/0.51 % mult_zero_right
% 0.30/0.51 thf(fact_1302_mult__zero__left,axiom,
% 0.30/0.51 ! [A: complex] :
% 0.30/0.51 ( ( times_times_complex @ zero_zero_complex @ A )
% 0.30/0.51 = zero_zero_complex ) ).
% 0.30/0.51
% 0.30/0.51 % mult_zero_left
% 0.30/0.51 thf(fact_1303_mult__zero__left,axiom,
% 0.30/0.51 ! [A: real] :
% 0.30/0.51 ( ( times_times_real @ zero_zero_real @ A )
% 0.30/0.51 = zero_zero_real ) ).
% 0.30/0.51
% 0.30/0.51 % mult_zero_left
% 0.30/0.51 thf(fact_1304_mult__zero__left,axiom,
% 0.30/0.51 ! [A: rat] :
% 0.30/0.51 ( ( times_times_rat @ zero_zero_rat @ A )
% 0.30/0.51 = zero_zero_rat ) ).
% 0.30/0.51
% 0.30/0.51 % mult_zero_left
% 0.30/0.51 thf(fact_1305_mult__zero__left,axiom,
% 0.30/0.51 ! [A: nat] :
% 0.30/0.51 ( ( times_times_nat @ zero_zero_nat @ A )
% 0.30/0.51 = zero_zero_nat ) ).
% 0.30/0.51
% 0.30/0.51 % mult_zero_left
% 0.30/0.51 thf(fact_1306_mult__zero__left,axiom,
% 0.30/0.51 ! [A: int] :
% 0.30/0.51 ( ( times_times_int @ zero_zero_int @ A )
% 0.30/0.51 = zero_zero_int ) ).
% 0.30/0.51
% 0.30/0.51 % mult_zero_left
% 0.30/0.51 thf(fact_1307_add_Oright__neutral,axiom,
% 0.30/0.51 ! [A: complex] :
% 0.30/0.51 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % add.right_neutral
% 0.30/0.51 thf(fact_1308_add_Oright__neutral,axiom,
% 0.30/0.51 ! [A: real] :
% 0.30/0.51 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % add.right_neutral
% 0.30/0.51 thf(fact_1309_add_Oright__neutral,axiom,
% 0.30/0.51 ! [A: rat] :
% 0.30/0.51 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % add.right_neutral
% 0.30/0.51 thf(fact_1310_add_Oright__neutral,axiom,
% 0.30/0.51 ! [A: nat] :
% 0.30/0.51 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % add.right_neutral
% 0.30/0.51 thf(fact_1311_add_Oright__neutral,axiom,
% 0.30/0.51 ! [A: int] :
% 0.30/0.51 ( ( plus_plus_int @ A @ zero_zero_int )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % add.right_neutral
% 0.30/0.51 thf(fact_1312_double__zero__sym,axiom,
% 0.30/0.51 ! [A: real] :
% 0.30/0.51 ( ( zero_zero_real
% 0.30/0.51 = ( plus_plus_real @ A @ A ) )
% 0.30/0.51 = ( A = zero_zero_real ) ) ).
% 0.30/0.51
% 0.30/0.51 % double_zero_sym
% 0.30/0.51 thf(fact_1313_double__zero__sym,axiom,
% 0.30/0.51 ! [A: rat] :
% 0.30/0.51 ( ( zero_zero_rat
% 0.30/0.51 = ( plus_plus_rat @ A @ A ) )
% 0.30/0.51 = ( A = zero_zero_rat ) ) ).
% 0.30/0.51
% 0.30/0.51 % double_zero_sym
% 0.30/0.51 thf(fact_1314_double__zero__sym,axiom,
% 0.30/0.51 ! [A: int] :
% 0.30/0.51 ( ( zero_zero_int
% 0.30/0.51 = ( plus_plus_int @ A @ A ) )
% 0.30/0.51 = ( A = zero_zero_int ) ) ).
% 0.30/0.51
% 0.30/0.51 % double_zero_sym
% 0.30/0.51 thf(fact_1315_add__cancel__left__left,axiom,
% 0.30/0.51 ! [B: complex,A: complex] :
% 0.30/0.51 ( ( ( plus_plus_complex @ B @ A )
% 0.30/0.51 = A )
% 0.30/0.51 = ( B = zero_zero_complex ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_left_left
% 0.30/0.51 thf(fact_1316_add__cancel__left__left,axiom,
% 0.30/0.51 ! [B: real,A: real] :
% 0.30/0.51 ( ( ( plus_plus_real @ B @ A )
% 0.30/0.51 = A )
% 0.30/0.51 = ( B = zero_zero_real ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_left_left
% 0.30/0.51 thf(fact_1317_add__cancel__left__left,axiom,
% 0.30/0.51 ! [B: rat,A: rat] :
% 0.30/0.51 ( ( ( plus_plus_rat @ B @ A )
% 0.30/0.51 = A )
% 0.30/0.51 = ( B = zero_zero_rat ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_left_left
% 0.30/0.51 thf(fact_1318_add__cancel__left__left,axiom,
% 0.30/0.51 ! [B: nat,A: nat] :
% 0.30/0.51 ( ( ( plus_plus_nat @ B @ A )
% 0.30/0.51 = A )
% 0.30/0.51 = ( B = zero_zero_nat ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_left_left
% 0.30/0.51 thf(fact_1319_add__cancel__left__left,axiom,
% 0.30/0.51 ! [B: int,A: int] :
% 0.30/0.51 ( ( ( plus_plus_int @ B @ A )
% 0.30/0.51 = A )
% 0.30/0.51 = ( B = zero_zero_int ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_left_left
% 0.30/0.51 thf(fact_1320_add__cancel__left__right,axiom,
% 0.30/0.51 ! [A: complex,B: complex] :
% 0.30/0.51 ( ( ( plus_plus_complex @ A @ B )
% 0.30/0.51 = A )
% 0.30/0.51 = ( B = zero_zero_complex ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_left_right
% 0.30/0.51 thf(fact_1321_add__cancel__left__right,axiom,
% 0.30/0.51 ! [A: real,B: real] :
% 0.30/0.51 ( ( ( plus_plus_real @ A @ B )
% 0.30/0.51 = A )
% 0.30/0.51 = ( B = zero_zero_real ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_left_right
% 0.30/0.51 thf(fact_1322_add__cancel__left__right,axiom,
% 0.30/0.51 ! [A: rat,B: rat] :
% 0.30/0.51 ( ( ( plus_plus_rat @ A @ B )
% 0.30/0.51 = A )
% 0.30/0.51 = ( B = zero_zero_rat ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_left_right
% 0.30/0.51 thf(fact_1323_add__cancel__left__right,axiom,
% 0.30/0.51 ! [A: nat,B: nat] :
% 0.30/0.51 ( ( ( plus_plus_nat @ A @ B )
% 0.30/0.51 = A )
% 0.30/0.51 = ( B = zero_zero_nat ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_left_right
% 0.30/0.51 thf(fact_1324_add__cancel__left__right,axiom,
% 0.30/0.51 ! [A: int,B: int] :
% 0.30/0.51 ( ( ( plus_plus_int @ A @ B )
% 0.30/0.51 = A )
% 0.30/0.51 = ( B = zero_zero_int ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_left_right
% 0.30/0.51 thf(fact_1325_add__cancel__right__left,axiom,
% 0.30/0.51 ! [A: complex,B: complex] :
% 0.30/0.51 ( ( A
% 0.30/0.51 = ( plus_plus_complex @ B @ A ) )
% 0.30/0.51 = ( B = zero_zero_complex ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_right_left
% 0.30/0.51 thf(fact_1326_add__cancel__right__left,axiom,
% 0.30/0.51 ! [A: real,B: real] :
% 0.30/0.51 ( ( A
% 0.30/0.51 = ( plus_plus_real @ B @ A ) )
% 0.30/0.51 = ( B = zero_zero_real ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_right_left
% 0.30/0.51 thf(fact_1327_add__cancel__right__left,axiom,
% 0.30/0.51 ! [A: rat,B: rat] :
% 0.30/0.51 ( ( A
% 0.30/0.51 = ( plus_plus_rat @ B @ A ) )
% 0.30/0.51 = ( B = zero_zero_rat ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_right_left
% 0.30/0.51 thf(fact_1328_add__cancel__right__left,axiom,
% 0.30/0.51 ! [A: nat,B: nat] :
% 0.30/0.51 ( ( A
% 0.30/0.51 = ( plus_plus_nat @ B @ A ) )
% 0.30/0.51 = ( B = zero_zero_nat ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_right_left
% 0.30/0.51 thf(fact_1329_add__cancel__right__left,axiom,
% 0.30/0.51 ! [A: int,B: int] :
% 0.30/0.51 ( ( A
% 0.30/0.51 = ( plus_plus_int @ B @ A ) )
% 0.30/0.51 = ( B = zero_zero_int ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_right_left
% 0.30/0.51 thf(fact_1330_add__cancel__right__right,axiom,
% 0.30/0.51 ! [A: complex,B: complex] :
% 0.30/0.51 ( ( A
% 0.30/0.51 = ( plus_plus_complex @ A @ B ) )
% 0.30/0.51 = ( B = zero_zero_complex ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_right_right
% 0.30/0.51 thf(fact_1331_add__cancel__right__right,axiom,
% 0.30/0.51 ! [A: real,B: real] :
% 0.30/0.51 ( ( A
% 0.30/0.51 = ( plus_plus_real @ A @ B ) )
% 0.30/0.51 = ( B = zero_zero_real ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_right_right
% 0.30/0.51 thf(fact_1332_add__cancel__right__right,axiom,
% 0.30/0.51 ! [A: rat,B: rat] :
% 0.30/0.51 ( ( A
% 0.30/0.51 = ( plus_plus_rat @ A @ B ) )
% 0.30/0.51 = ( B = zero_zero_rat ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_right_right
% 0.30/0.51 thf(fact_1333_add__cancel__right__right,axiom,
% 0.30/0.51 ! [A: nat,B: nat] :
% 0.30/0.51 ( ( A
% 0.30/0.51 = ( plus_plus_nat @ A @ B ) )
% 0.30/0.51 = ( B = zero_zero_nat ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_right_right
% 0.30/0.51 thf(fact_1334_add__cancel__right__right,axiom,
% 0.30/0.51 ! [A: int,B: int] :
% 0.30/0.51 ( ( A
% 0.30/0.51 = ( plus_plus_int @ A @ B ) )
% 0.30/0.51 = ( B = zero_zero_int ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_cancel_right_right
% 0.30/0.51 thf(fact_1335_add__eq__0__iff__both__eq__0,axiom,
% 0.30/0.51 ! [X2: nat,Y3: nat] :
% 0.30/0.51 ( ( ( plus_plus_nat @ X2 @ Y3 )
% 0.30/0.51 = zero_zero_nat )
% 0.30/0.51 = ( ( X2 = zero_zero_nat )
% 0.30/0.51 & ( Y3 = zero_zero_nat ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_eq_0_iff_both_eq_0
% 0.30/0.51 thf(fact_1336_zero__eq__add__iff__both__eq__0,axiom,
% 0.30/0.51 ! [X2: nat,Y3: nat] :
% 0.30/0.51 ( ( zero_zero_nat
% 0.30/0.51 = ( plus_plus_nat @ X2 @ Y3 ) )
% 0.30/0.51 = ( ( X2 = zero_zero_nat )
% 0.30/0.51 & ( Y3 = zero_zero_nat ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % zero_eq_add_iff_both_eq_0
% 0.30/0.51 thf(fact_1337_add__0,axiom,
% 0.30/0.51 ! [A: complex] :
% 0.30/0.51 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % add_0
% 0.30/0.51 thf(fact_1338_add__0,axiom,
% 0.30/0.51 ! [A: real] :
% 0.30/0.51 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % add_0
% 0.30/0.51 thf(fact_1339_add__0,axiom,
% 0.30/0.51 ! [A: rat] :
% 0.30/0.51 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % add_0
% 0.30/0.51 thf(fact_1340_add__0,axiom,
% 0.30/0.51 ! [A: nat] :
% 0.30/0.51 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % add_0
% 0.30/0.51 thf(fact_1341_add__0,axiom,
% 0.30/0.51 ! [A: int] :
% 0.30/0.51 ( ( plus_plus_int @ zero_zero_int @ A )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % add_0
% 0.30/0.51 thf(fact_1342_diff__self,axiom,
% 0.30/0.51 ! [A: complex] :
% 0.30/0.51 ( ( minus_minus_complex @ A @ A )
% 0.30/0.51 = zero_zero_complex ) ).
% 0.30/0.51
% 0.30/0.51 % diff_self
% 0.30/0.51 thf(fact_1343_diff__self,axiom,
% 0.30/0.51 ! [A: real] :
% 0.30/0.51 ( ( minus_minus_real @ A @ A )
% 0.30/0.51 = zero_zero_real ) ).
% 0.30/0.51
% 0.30/0.51 % diff_self
% 0.30/0.51 thf(fact_1344_diff__self,axiom,
% 0.30/0.51 ! [A: rat] :
% 0.30/0.51 ( ( minus_minus_rat @ A @ A )
% 0.30/0.51 = zero_zero_rat ) ).
% 0.30/0.51
% 0.30/0.51 % diff_self
% 0.30/0.51 thf(fact_1345_diff__self,axiom,
% 0.30/0.51 ! [A: int] :
% 0.30/0.51 ( ( minus_minus_int @ A @ A )
% 0.30/0.51 = zero_zero_int ) ).
% 0.30/0.51
% 0.30/0.51 % diff_self
% 0.30/0.51 thf(fact_1346_diff__0__right,axiom,
% 0.30/0.51 ! [A: complex] :
% 0.30/0.51 ( ( minus_minus_complex @ A @ zero_zero_complex )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % diff_0_right
% 0.30/0.51 thf(fact_1347_diff__0__right,axiom,
% 0.30/0.51 ! [A: real] :
% 0.30/0.51 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % diff_0_right
% 0.30/0.51 thf(fact_1348_diff__0__right,axiom,
% 0.30/0.51 ! [A: rat] :
% 0.30/0.51 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % diff_0_right
% 0.30/0.51 thf(fact_1349_diff__0__right,axiom,
% 0.30/0.51 ! [A: int] :
% 0.30/0.51 ( ( minus_minus_int @ A @ zero_zero_int )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % diff_0_right
% 0.30/0.51 thf(fact_1350_zero__diff,axiom,
% 0.30/0.51 ! [A: nat] :
% 0.30/0.51 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 0.30/0.51 = zero_zero_nat ) ).
% 0.30/0.51
% 0.30/0.51 % zero_diff
% 0.30/0.51 thf(fact_1351_diff__zero,axiom,
% 0.30/0.51 ! [A: complex] :
% 0.30/0.51 ( ( minus_minus_complex @ A @ zero_zero_complex )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % diff_zero
% 0.30/0.51 thf(fact_1352_diff__zero,axiom,
% 0.30/0.51 ! [A: real] :
% 0.30/0.51 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % diff_zero
% 0.30/0.51 thf(fact_1353_diff__zero,axiom,
% 0.30/0.51 ! [A: rat] :
% 0.30/0.51 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % diff_zero
% 0.30/0.51 thf(fact_1354_diff__zero,axiom,
% 0.30/0.51 ! [A: nat] :
% 0.30/0.51 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % diff_zero
% 0.30/0.51 thf(fact_1355_diff__zero,axiom,
% 0.30/0.51 ! [A: int] :
% 0.30/0.51 ( ( minus_minus_int @ A @ zero_zero_int )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % diff_zero
% 0.30/0.51 thf(fact_1356_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.51 ! [A: complex] :
% 0.30/0.51 ( ( minus_minus_complex @ A @ A )
% 0.30/0.51 = zero_zero_complex ) ).
% 0.30/0.51
% 0.30/0.51 % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.51 thf(fact_1357_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.51 ! [A: real] :
% 0.30/0.51 ( ( minus_minus_real @ A @ A )
% 0.30/0.51 = zero_zero_real ) ).
% 0.30/0.51
% 0.30/0.51 % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.51 thf(fact_1358_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.51 ! [A: rat] :
% 0.30/0.51 ( ( minus_minus_rat @ A @ A )
% 0.30/0.51 = zero_zero_rat ) ).
% 0.30/0.51
% 0.30/0.51 % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.51 thf(fact_1359_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.51 ! [A: nat] :
% 0.30/0.51 ( ( minus_minus_nat @ A @ A )
% 0.30/0.51 = zero_zero_nat ) ).
% 0.30/0.51
% 0.30/0.51 % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.51 thf(fact_1360_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.51 ! [A: int] :
% 0.30/0.51 ( ( minus_minus_int @ A @ A )
% 0.30/0.51 = zero_zero_int ) ).
% 0.30/0.51
% 0.30/0.51 % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.51 thf(fact_1361_division__ring__divide__zero,axiom,
% 0.30/0.51 ! [A: complex] :
% 0.30/0.51 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.30/0.51 = zero_zero_complex ) ).
% 0.30/0.51
% 0.30/0.51 % division_ring_divide_zero
% 0.30/0.51 thf(fact_1362_division__ring__divide__zero,axiom,
% 0.30/0.51 ! [A: real] :
% 0.30/0.51 ( ( divide_divide_real @ A @ zero_zero_real )
% 0.30/0.51 = zero_zero_real ) ).
% 0.30/0.51
% 0.30/0.51 % division_ring_divide_zero
% 0.30/0.51 thf(fact_1363_division__ring__divide__zero,axiom,
% 0.30/0.51 ! [A: rat] :
% 0.30/0.51 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.30/0.51 = zero_zero_rat ) ).
% 0.30/0.51
% 0.30/0.51 % division_ring_divide_zero
% 0.30/0.51 thf(fact_1364_divide__cancel__right,axiom,
% 0.30/0.51 ! [A: complex,C: complex,B: complex] :
% 0.30/0.51 ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.30/0.51 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.30/0.51 = ( ( C = zero_zero_complex )
% 0.30/0.51 | ( A = B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_cancel_right
% 0.30/0.51 thf(fact_1365_divide__cancel__right,axiom,
% 0.30/0.51 ! [A: real,C: real,B: real] :
% 0.30/0.51 ( ( ( divide_divide_real @ A @ C )
% 0.30/0.51 = ( divide_divide_real @ B @ C ) )
% 0.30/0.51 = ( ( C = zero_zero_real )
% 0.30/0.51 | ( A = B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_cancel_right
% 0.30/0.51 thf(fact_1366_divide__cancel__right,axiom,
% 0.30/0.51 ! [A: rat,C: rat,B: rat] :
% 0.30/0.51 ( ( ( divide_divide_rat @ A @ C )
% 0.30/0.51 = ( divide_divide_rat @ B @ C ) )
% 0.30/0.51 = ( ( C = zero_zero_rat )
% 0.30/0.51 | ( A = B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_cancel_right
% 0.30/0.51 thf(fact_1367_divide__cancel__left,axiom,
% 0.30/0.51 ! [C: complex,A: complex,B: complex] :
% 0.30/0.51 ( ( ( divide1717551699836669952omplex @ C @ A )
% 0.30/0.51 = ( divide1717551699836669952omplex @ C @ B ) )
% 0.30/0.51 = ( ( C = zero_zero_complex )
% 0.30/0.51 | ( A = B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_cancel_left
% 0.30/0.51 thf(fact_1368_divide__cancel__left,axiom,
% 0.30/0.51 ! [C: real,A: real,B: real] :
% 0.30/0.51 ( ( ( divide_divide_real @ C @ A )
% 0.30/0.51 = ( divide_divide_real @ C @ B ) )
% 0.30/0.51 = ( ( C = zero_zero_real )
% 0.30/0.51 | ( A = B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_cancel_left
% 0.30/0.51 thf(fact_1369_divide__cancel__left,axiom,
% 0.30/0.51 ! [C: rat,A: rat,B: rat] :
% 0.30/0.51 ( ( ( divide_divide_rat @ C @ A )
% 0.30/0.51 = ( divide_divide_rat @ C @ B ) )
% 0.30/0.51 = ( ( C = zero_zero_rat )
% 0.30/0.51 | ( A = B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_cancel_left
% 0.30/0.51 thf(fact_1370_divide__eq__0__iff,axiom,
% 0.30/0.51 ! [A: complex,B: complex] :
% 0.30/0.51 ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.30/0.51 = zero_zero_complex )
% 0.30/0.51 = ( ( A = zero_zero_complex )
% 0.30/0.51 | ( B = zero_zero_complex ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_eq_0_iff
% 0.30/0.51 thf(fact_1371_divide__eq__0__iff,axiom,
% 0.30/0.51 ! [A: real,B: real] :
% 0.30/0.51 ( ( ( divide_divide_real @ A @ B )
% 0.30/0.51 = zero_zero_real )
% 0.30/0.51 = ( ( A = zero_zero_real )
% 0.30/0.51 | ( B = zero_zero_real ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_eq_0_iff
% 0.30/0.51 thf(fact_1372_divide__eq__0__iff,axiom,
% 0.30/0.51 ! [A: rat,B: rat] :
% 0.30/0.51 ( ( ( divide_divide_rat @ A @ B )
% 0.30/0.51 = zero_zero_rat )
% 0.30/0.51 = ( ( A = zero_zero_rat )
% 0.30/0.51 | ( B = zero_zero_rat ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % divide_eq_0_iff
% 0.30/0.51 thf(fact_1373_bits__div__by__0,axiom,
% 0.30/0.51 ! [A: nat] :
% 0.30/0.51 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.30/0.51 = zero_zero_nat ) ).
% 0.30/0.51
% 0.30/0.51 % bits_div_by_0
% 0.30/0.51 thf(fact_1374_bits__div__by__0,axiom,
% 0.30/0.51 ! [A: int] :
% 0.30/0.51 ( ( divide_divide_int @ A @ zero_zero_int )
% 0.30/0.51 = zero_zero_int ) ).
% 0.30/0.51
% 0.30/0.51 % bits_div_by_0
% 0.30/0.51 thf(fact_1375_bits__div__0,axiom,
% 0.30/0.51 ! [A: nat] :
% 0.30/0.51 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.30/0.51 = zero_zero_nat ) ).
% 0.30/0.51
% 0.30/0.51 % bits_div_0
% 0.30/0.51 thf(fact_1376_bits__div__0,axiom,
% 0.30/0.51 ! [A: int] :
% 0.30/0.51 ( ( divide_divide_int @ zero_zero_int @ A )
% 0.30/0.51 = zero_zero_int ) ).
% 0.30/0.51
% 0.30/0.51 % bits_div_0
% 0.30/0.51 thf(fact_1377_div__by__0,axiom,
% 0.30/0.51 ! [A: complex] :
% 0.30/0.51 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.30/0.51 = zero_zero_complex ) ).
% 0.30/0.51
% 0.30/0.51 % div_by_0
% 0.30/0.51 thf(fact_1378_div__by__0,axiom,
% 0.30/0.51 ! [A: real] :
% 0.30/0.51 ( ( divide_divide_real @ A @ zero_zero_real )
% 0.30/0.51 = zero_zero_real ) ).
% 0.30/0.51
% 0.30/0.51 % div_by_0
% 0.30/0.51 thf(fact_1379_div__by__0,axiom,
% 0.30/0.51 ! [A: rat] :
% 0.30/0.51 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.30/0.51 = zero_zero_rat ) ).
% 0.30/0.51
% 0.30/0.51 % div_by_0
% 0.30/0.51 thf(fact_1380_div__by__0,axiom,
% 0.30/0.51 ! [A: nat] :
% 0.30/0.51 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.30/0.51 = zero_zero_nat ) ).
% 0.30/0.51
% 0.30/0.51 % div_by_0
% 0.30/0.51 thf(fact_1381_div__by__0,axiom,
% 0.30/0.51 ! [A: int] :
% 0.30/0.51 ( ( divide_divide_int @ A @ zero_zero_int )
% 0.30/0.51 = zero_zero_int ) ).
% 0.30/0.51
% 0.30/0.51 % div_by_0
% 0.30/0.51 thf(fact_1382_div__0,axiom,
% 0.30/0.51 ! [A: complex] :
% 0.30/0.51 ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 0.30/0.51 = zero_zero_complex ) ).
% 0.30/0.51
% 0.30/0.51 % div_0
% 0.30/0.51 thf(fact_1383_div__0,axiom,
% 0.30/0.51 ! [A: real] :
% 0.30/0.51 ( ( divide_divide_real @ zero_zero_real @ A )
% 0.30/0.51 = zero_zero_real ) ).
% 0.30/0.51
% 0.30/0.51 % div_0
% 0.30/0.51 thf(fact_1384_div__0,axiom,
% 0.30/0.51 ! [A: rat] :
% 0.30/0.51 ( ( divide_divide_rat @ zero_zero_rat @ A )
% 0.30/0.52 = zero_zero_rat ) ).
% 0.30/0.52
% 0.30/0.52 % div_0
% 0.30/0.52 thf(fact_1385_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 % div_0
% 0.30/0.52 thf(fact_1386_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 % div_0
% 0.30/0.52 thf(fact_1387_less__nat__zero__code,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % less_nat_zero_code
% 0.30/0.52 thf(fact_1388_neq0__conv,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( N != zero_zero_nat )
% 0.30/0.52 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.52
% 0.30/0.52 % neq0_conv
% 0.30/0.52 thf(fact_1389_bot__nat__0_Onot__eq__extremum,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( A != zero_zero_nat )
% 0.30/0.52 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.30/0.52
% 0.30/0.52 % bot_nat_0.not_eq_extremum
% 0.30/0.52 thf(fact_1390_bot__nat__0_Oextremum,axiom,
% 0.30/0.52 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.30/0.52
% 0.30/0.52 % bot_nat_0.extremum
% 0.30/0.52 thf(fact_1391_le0,axiom,
% 0.30/0.52 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.30/0.52
% 0.30/0.52 % le0
% 0.30/0.52 thf(fact_1392_add__is__0,axiom,
% 0.30/0.52 ! [M: nat,N: nat] :
% 0.30/0.52 ( ( ( plus_plus_nat @ M @ N )
% 0.30/0.52 = zero_zero_nat )
% 0.30/0.52 = ( ( M = zero_zero_nat )
% 0.30/0.52 & ( N = zero_zero_nat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_is_0
% 0.30/0.52 thf(fact_1393_Nat_Oadd__0__right,axiom,
% 0.30/0.52 ! [M: nat] :
% 0.30/0.52 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 0.30/0.52 = M ) ).
% 0.30/0.52
% 0.30/0.52 % Nat.add_0_right
% 0.30/0.52 thf(fact_1394_diff__0__eq__0,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 0.30/0.52 = zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % diff_0_eq_0
% 0.30/0.52 thf(fact_1395_diff__self__eq__0,axiom,
% 0.30/0.52 ! [M: nat] :
% 0.30/0.52 ( ( minus_minus_nat @ M @ M )
% 0.30/0.52 = zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % diff_self_eq_0
% 0.30/0.52 thf(fact_1396_mult__is__0,axiom,
% 0.30/0.52 ! [M: nat,N: nat] :
% 0.30/0.52 ( ( ( times_times_nat @ M @ N )
% 0.30/0.52 = zero_zero_nat )
% 0.30/0.52 = ( ( M = zero_zero_nat )
% 0.30/0.52 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_is_0
% 0.30/0.52 thf(fact_1397_mult__0__right,axiom,
% 0.30/0.52 ! [M: nat] :
% 0.30/0.52 ( ( times_times_nat @ M @ zero_zero_nat )
% 0.30/0.52 = zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % mult_0_right
% 0.30/0.52 thf(fact_1398_mult__cancel1,axiom,
% 0.30/0.52 ! [K: nat,M: nat,N: nat] :
% 0.30/0.52 ( ( ( times_times_nat @ K @ M )
% 0.30/0.52 = ( times_times_nat @ K @ N ) )
% 0.30/0.52 = ( ( M = N )
% 0.30/0.52 | ( K = zero_zero_nat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel1
% 0.30/0.52 thf(fact_1399_mult__cancel2,axiom,
% 0.30/0.52 ! [M: nat,K: nat,N: nat] :
% 0.30/0.52 ( ( ( times_times_nat @ M @ K )
% 0.30/0.52 = ( times_times_nat @ N @ K ) )
% 0.30/0.52 = ( ( M = N )
% 0.30/0.52 | ( K = zero_zero_nat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel2
% 0.30/0.52 thf(fact_1400_add__le__same__cancel1,axiom,
% 0.30/0.52 ! [B: real,A: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.30/0.52 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_same_cancel1
% 0.30/0.52 thf(fact_1401_add__le__same__cancel1,axiom,
% 0.30/0.52 ! [B: rat,A: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 0.30/0.52 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_same_cancel1
% 0.30/0.52 thf(fact_1402_add__le__same__cancel1,axiom,
% 0.30/0.52 ! [B: nat,A: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.30/0.52 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_same_cancel1
% 0.30/0.52 thf(fact_1403_add__le__same__cancel1,axiom,
% 0.30/0.52 ! [B: int,A: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.30/0.52 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_same_cancel1
% 0.30/0.52 thf(fact_1404_add__le__same__cancel2,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.30/0.52 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_same_cancel2
% 0.30/0.52 thf(fact_1405_add__le__same__cancel2,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.30/0.52 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_same_cancel2
% 0.30/0.52 thf(fact_1406_add__le__same__cancel2,axiom,
% 0.30/0.52 ! [A: nat,B: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.30/0.52 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_same_cancel2
% 0.30/0.52 thf(fact_1407_add__le__same__cancel2,axiom,
% 0.30/0.52 ! [A: int,B: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.30/0.52 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_same_cancel2
% 0.30/0.52 thf(fact_1408_le__add__same__cancel1,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.30/0.52 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_add_same_cancel1
% 0.30/0.52 thf(fact_1409_le__add__same__cancel1,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.30/0.52 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_add_same_cancel1
% 0.30/0.52 thf(fact_1410_le__add__same__cancel1,axiom,
% 0.30/0.52 ! [A: nat,B: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.30/0.52 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_add_same_cancel1
% 0.30/0.52 thf(fact_1411_le__add__same__cancel1,axiom,
% 0.30/0.52 ! [A: int,B: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.30/0.52 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_add_same_cancel1
% 0.30/0.52 thf(fact_1412_le__add__same__cancel2,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.30/0.52 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_add_same_cancel2
% 0.30/0.52 thf(fact_1413_le__add__same__cancel2,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.30/0.52 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_add_same_cancel2
% 0.30/0.52 thf(fact_1414_le__add__same__cancel2,axiom,
% 0.30/0.52 ! [A: nat,B: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.30/0.52 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_add_same_cancel2
% 0.30/0.52 thf(fact_1415_le__add__same__cancel2,axiom,
% 0.30/0.52 ! [A: int,B: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.30/0.52 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_add_same_cancel2
% 0.30/0.52 thf(fact_1416_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.30/0.52 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % double_add_le_zero_iff_single_add_le_zero
% 0.30/0.52 thf(fact_1417_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.30/0.52 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % double_add_le_zero_iff_single_add_le_zero
% 0.30/0.52 thf(fact_1418_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.30/0.52 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.30/0.52
% 0.30/0.52 % double_add_le_zero_iff_single_add_le_zero
% 0.30/0.52 thf(fact_1419_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.30/0.52 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_le_double_add_iff_zero_le_single_add
% 0.30/0.52 thf(fact_1420_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.30/0.52 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_le_double_add_iff_zero_le_single_add
% 0.30/0.52 thf(fact_1421_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.30/0.52 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_le_double_add_iff_zero_le_single_add
% 0.30/0.52 thf(fact_1422_add__less__same__cancel1,axiom,
% 0.30/0.52 ! [B: real,A: real] :
% 0.30/0.52 ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.30/0.52 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_same_cancel1
% 0.30/0.52 thf(fact_1423_add__less__same__cancel1,axiom,
% 0.30/0.52 ! [B: rat,A: rat] :
% 0.30/0.52 ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 0.30/0.52 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_same_cancel1
% 0.30/0.52 thf(fact_1424_add__less__same__cancel1,axiom,
% 0.30/0.52 ! [B: nat,A: nat] :
% 0.30/0.52 ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.30/0.52 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_same_cancel1
% 0.30/0.52 thf(fact_1425_add__less__same__cancel1,axiom,
% 0.30/0.52 ! [B: int,A: int] :
% 0.30/0.52 ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.30/0.52 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_same_cancel1
% 0.30/0.52 thf(fact_1426_add__less__same__cancel2,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.30/0.52 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_same_cancel2
% 0.30/0.52 thf(fact_1427_add__less__same__cancel2,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.30/0.52 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_same_cancel2
% 0.30/0.52 thf(fact_1428_add__less__same__cancel2,axiom,
% 0.30/0.52 ! [A: nat,B: nat] :
% 0.30/0.52 ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.30/0.52 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_same_cancel2
% 0.30/0.52 thf(fact_1429_add__less__same__cancel2,axiom,
% 0.30/0.52 ! [A: int,B: int] :
% 0.30/0.52 ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.30/0.52 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_same_cancel2
% 0.30/0.52 thf(fact_1430_less__add__same__cancel1,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.30/0.52 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_add_same_cancel1
% 0.30/0.52 thf(fact_1431_less__add__same__cancel1,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.30/0.52 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_add_same_cancel1
% 0.30/0.52 thf(fact_1432_less__add__same__cancel1,axiom,
% 0.30/0.52 ! [A: nat,B: nat] :
% 0.30/0.52 ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.30/0.52 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_add_same_cancel1
% 0.30/0.52 thf(fact_1433_less__add__same__cancel1,axiom,
% 0.30/0.52 ! [A: int,B: int] :
% 0.30/0.52 ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.30/0.52 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_add_same_cancel1
% 0.30/0.52 thf(fact_1434_less__add__same__cancel2,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.30/0.52 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_add_same_cancel2
% 0.30/0.52 thf(fact_1435_less__add__same__cancel2,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.30/0.52 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_add_same_cancel2
% 0.30/0.52 thf(fact_1436_less__add__same__cancel2,axiom,
% 0.30/0.52 ! [A: nat,B: nat] :
% 0.30/0.52 ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.30/0.52 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_add_same_cancel2
% 0.30/0.52 thf(fact_1437_less__add__same__cancel2,axiom,
% 0.30/0.52 ! [A: int,B: int] :
% 0.30/0.52 ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.30/0.52 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_add_same_cancel2
% 0.30/0.52 thf(fact_1438_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.30/0.52 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % double_add_less_zero_iff_single_add_less_zero
% 0.30/0.52 thf(fact_1439_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.30/0.52 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % double_add_less_zero_iff_single_add_less_zero
% 0.30/0.52 thf(fact_1440_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.30/0.52 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.30/0.52
% 0.30/0.52 % double_add_less_zero_iff_single_add_less_zero
% 0.30/0.52 thf(fact_1441_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.30/0.52 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_less_double_add_iff_zero_less_single_add
% 0.30/0.52 thf(fact_1442_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.30/0.52 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_less_double_add_iff_zero_less_single_add
% 0.30/0.52 thf(fact_1443_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.30/0.52 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_less_double_add_iff_zero_less_single_add
% 0.30/0.52 thf(fact_1444_diff__ge__0__iff__ge,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.30/0.52 = ( ord_less_eq_real @ B @ A ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_ge_0_iff_ge
% 0.30/0.52 thf(fact_1445_diff__ge__0__iff__ge,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 0.30/0.52 = ( ord_less_eq_rat @ B @ A ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_ge_0_iff_ge
% 0.30/0.52 thf(fact_1446_diff__ge__0__iff__ge,axiom,
% 0.30/0.52 ! [A: int,B: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.30/0.52 = ( ord_less_eq_int @ B @ A ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_ge_0_iff_ge
% 0.30/0.52 thf(fact_1447_diff__gt__0__iff__gt,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.30/0.52 = ( ord_less_real @ B @ A ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_gt_0_iff_gt
% 0.30/0.52 thf(fact_1448_diff__gt__0__iff__gt,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 0.30/0.52 = ( ord_less_rat @ B @ A ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_gt_0_iff_gt
% 0.30/0.52 thf(fact_1449_diff__gt__0__iff__gt,axiom,
% 0.30/0.52 ! [A: int,B: int] :
% 0.30/0.52 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.30/0.52 = ( ord_less_int @ B @ A ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_gt_0_iff_gt
% 0.30/0.52 thf(fact_1450_mult__cancel__left1,axiom,
% 0.30/0.52 ! [C: complex,B: complex] :
% 0.30/0.52 ( ( C
% 0.30/0.52 = ( times_times_complex @ C @ B ) )
% 0.30/0.52 = ( ( C = zero_zero_complex )
% 0.30/0.52 | ( B = one_one_complex ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_left1
% 0.30/0.52 thf(fact_1451_mult__cancel__left1,axiom,
% 0.30/0.52 ! [C: real,B: real] :
% 0.30/0.52 ( ( C
% 0.30/0.52 = ( times_times_real @ C @ B ) )
% 0.30/0.52 = ( ( C = zero_zero_real )
% 0.30/0.52 | ( B = one_one_real ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_left1
% 0.30/0.52 thf(fact_1452_mult__cancel__left1,axiom,
% 0.30/0.52 ! [C: rat,B: rat] :
% 0.30/0.52 ( ( C
% 0.30/0.52 = ( times_times_rat @ C @ B ) )
% 0.30/0.52 = ( ( C = zero_zero_rat )
% 0.30/0.52 | ( B = one_one_rat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_left1
% 0.30/0.52 thf(fact_1453_mult__cancel__left1,axiom,
% 0.30/0.52 ! [C: int,B: int] :
% 0.30/0.52 ( ( C
% 0.30/0.52 = ( times_times_int @ C @ B ) )
% 0.30/0.52 = ( ( C = zero_zero_int )
% 0.30/0.52 | ( B = one_one_int ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_left1
% 0.30/0.52 thf(fact_1454_mult__cancel__left2,axiom,
% 0.30/0.52 ! [C: complex,A: complex] :
% 0.30/0.52 ( ( ( times_times_complex @ C @ A )
% 0.30/0.52 = C )
% 0.30/0.52 = ( ( C = zero_zero_complex )
% 0.30/0.52 | ( A = one_one_complex ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_left2
% 0.30/0.52 thf(fact_1455_mult__cancel__left2,axiom,
% 0.30/0.52 ! [C: real,A: real] :
% 0.30/0.52 ( ( ( times_times_real @ C @ A )
% 0.30/0.52 = C )
% 0.30/0.52 = ( ( C = zero_zero_real )
% 0.30/0.52 | ( A = one_one_real ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_left2
% 0.30/0.52 thf(fact_1456_mult__cancel__left2,axiom,
% 0.30/0.52 ! [C: rat,A: rat] :
% 0.30/0.52 ( ( ( times_times_rat @ C @ A )
% 0.30/0.52 = C )
% 0.30/0.52 = ( ( C = zero_zero_rat )
% 0.30/0.52 | ( A = one_one_rat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_left2
% 0.30/0.52 thf(fact_1457_mult__cancel__left2,axiom,
% 0.30/0.52 ! [C: int,A: int] :
% 0.30/0.52 ( ( ( times_times_int @ C @ A )
% 0.30/0.52 = C )
% 0.30/0.52 = ( ( C = zero_zero_int )
% 0.30/0.52 | ( A = one_one_int ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_left2
% 0.30/0.52 thf(fact_1458_mult__cancel__right1,axiom,
% 0.30/0.52 ! [C: complex,B: complex] :
% 0.30/0.52 ( ( C
% 0.30/0.52 = ( times_times_complex @ B @ C ) )
% 0.30/0.52 = ( ( C = zero_zero_complex )
% 0.30/0.52 | ( B = one_one_complex ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_right1
% 0.30/0.52 thf(fact_1459_mult__cancel__right1,axiom,
% 0.30/0.52 ! [C: real,B: real] :
% 0.30/0.52 ( ( C
% 0.30/0.52 = ( times_times_real @ B @ C ) )
% 0.30/0.52 = ( ( C = zero_zero_real )
% 0.30/0.52 | ( B = one_one_real ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_right1
% 0.30/0.52 thf(fact_1460_mult__cancel__right1,axiom,
% 0.30/0.52 ! [C: rat,B: rat] :
% 0.30/0.52 ( ( C
% 0.30/0.52 = ( times_times_rat @ B @ C ) )
% 0.30/0.52 = ( ( C = zero_zero_rat )
% 0.30/0.52 | ( B = one_one_rat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_right1
% 0.30/0.52 thf(fact_1461_mult__cancel__right1,axiom,
% 0.30/0.52 ! [C: int,B: int] :
% 0.30/0.52 ( ( C
% 0.30/0.52 = ( times_times_int @ B @ C ) )
% 0.30/0.52 = ( ( C = zero_zero_int )
% 0.30/0.52 | ( B = one_one_int ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_right1
% 0.30/0.52 thf(fact_1462_mult__cancel__right2,axiom,
% 0.30/0.52 ! [A: complex,C: complex] :
% 0.30/0.52 ( ( ( times_times_complex @ A @ C )
% 0.30/0.52 = C )
% 0.30/0.52 = ( ( C = zero_zero_complex )
% 0.30/0.52 | ( A = one_one_complex ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_right2
% 0.30/0.52 thf(fact_1463_mult__cancel__right2,axiom,
% 0.30/0.52 ! [A: real,C: real] :
% 0.30/0.52 ( ( ( times_times_real @ A @ C )
% 0.30/0.52 = C )
% 0.30/0.52 = ( ( C = zero_zero_real )
% 0.30/0.52 | ( A = one_one_real ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_right2
% 0.30/0.52 thf(fact_1464_mult__cancel__right2,axiom,
% 0.30/0.52 ! [A: rat,C: rat] :
% 0.30/0.52 ( ( ( times_times_rat @ A @ C )
% 0.30/0.52 = C )
% 0.30/0.52 = ( ( C = zero_zero_rat )
% 0.30/0.52 | ( A = one_one_rat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_right2
% 0.30/0.52 thf(fact_1465_mult__cancel__right2,axiom,
% 0.30/0.52 ! [A: int,C: int] :
% 0.30/0.52 ( ( ( times_times_int @ A @ C )
% 0.30/0.52 = C )
% 0.30/0.52 = ( ( C = zero_zero_int )
% 0.30/0.52 | ( A = one_one_int ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_right2
% 0.30/0.52 thf(fact_1466_sum__squares__eq__zero__iff,axiom,
% 0.30/0.52 ! [X2: real,Y3: real] :
% 0.30/0.52 ( ( ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) )
% 0.30/0.52 = zero_zero_real )
% 0.30/0.52 = ( ( X2 = zero_zero_real )
% 0.30/0.52 & ( Y3 = zero_zero_real ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % sum_squares_eq_zero_iff
% 0.30/0.52 thf(fact_1467_sum__squares__eq__zero__iff,axiom,
% 0.30/0.52 ! [X2: rat,Y3: rat] :
% 0.30/0.52 ( ( ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 0.30/0.52 = zero_zero_rat )
% 0.30/0.52 = ( ( X2 = zero_zero_rat )
% 0.30/0.52 & ( Y3 = zero_zero_rat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % sum_squares_eq_zero_iff
% 0.30/0.52 thf(fact_1468_sum__squares__eq__zero__iff,axiom,
% 0.30/0.52 ! [X2: int,Y3: int] :
% 0.30/0.52 ( ( ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) )
% 0.30/0.52 = zero_zero_int )
% 0.30/0.52 = ( ( X2 = zero_zero_int )
% 0.30/0.52 & ( Y3 = zero_zero_int ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % sum_squares_eq_zero_iff
% 0.30/0.52 thf(fact_1469_diff__numeral__special_I9_J,axiom,
% 0.30/0.52 ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 0.30/0.52 = zero_zero_complex ) ).
% 0.30/0.52
% 0.30/0.52 % diff_numeral_special(9)
% 0.30/0.52 thf(fact_1470_diff__numeral__special_I9_J,axiom,
% 0.30/0.52 ( ( minus_minus_real @ one_one_real @ one_one_real )
% 0.30/0.52 = zero_zero_real ) ).
% 0.30/0.52
% 0.30/0.52 % diff_numeral_special(9)
% 0.30/0.52 thf(fact_1471_diff__numeral__special_I9_J,axiom,
% 0.30/0.52 ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 0.30/0.52 = zero_zero_rat ) ).
% 0.30/0.52
% 0.30/0.52 % diff_numeral_special(9)
% 0.30/0.52 thf(fact_1472_diff__numeral__special_I9_J,axiom,
% 0.30/0.52 ( ( minus_minus_int @ one_one_int @ one_one_int )
% 0.30/0.52 = zero_zero_int ) ).
% 0.30/0.52
% 0.30/0.52 % diff_numeral_special(9)
% 0.30/0.52 thf(fact_1473_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.30/0.52 ! [C: complex,A: complex,B: complex] :
% 0.30/0.52 ( ( C != zero_zero_complex )
% 0.30/0.52 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
% 0.30/0.52 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_divide_mult_cancel_right2
% 0.30/0.52 thf(fact_1474_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.30/0.52 ! [C: real,A: real,B: real] :
% 0.30/0.52 ( ( C != zero_zero_real )
% 0.30/0.52 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
% 0.30/0.52 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_divide_mult_cancel_right2
% 0.30/0.52 thf(fact_1475_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.30/0.52 ! [C: rat,A: rat,B: rat] :
% 0.30/0.52 ( ( C != zero_zero_rat )
% 0.30/0.52 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.52 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_divide_mult_cancel_right2
% 0.30/0.52 thf(fact_1476_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.30/0.52 ! [C: complex,A: complex,B: complex] :
% 0.30/0.52 ( ( C != zero_zero_complex )
% 0.30/0.52 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 0.30/0.52 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_divide_mult_cancel_right
% 0.30/0.52 thf(fact_1477_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.30/0.52 ! [C: real,A: real,B: real] :
% 0.30/0.52 ( ( C != zero_zero_real )
% 0.30/0.52 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.30/0.52 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_divide_mult_cancel_right
% 0.30/0.52 thf(fact_1478_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.30/0.52 ! [C: rat,A: rat,B: rat] :
% 0.30/0.52 ( ( C != zero_zero_rat )
% 0.30/0.52 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.30/0.52 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_divide_mult_cancel_right
% 0.30/0.52 thf(fact_1479_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.30/0.52 ! [C: complex,A: complex,B: complex] :
% 0.30/0.52 ( ( C != zero_zero_complex )
% 0.30/0.52 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
% 0.30/0.52 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_divide_mult_cancel_left2
% 0.30/0.52 thf(fact_1480_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.30/0.52 ! [C: real,A: real,B: real] :
% 0.30/0.52 ( ( C != zero_zero_real )
% 0.30/0.52 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
% 0.30/0.52 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_divide_mult_cancel_left2
% 0.30/0.52 thf(fact_1481_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.30/0.52 ! [C: rat,A: rat,B: rat] :
% 0.30/0.52 ( ( C != zero_zero_rat )
% 0.30/0.52 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
% 0.30/0.52 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_divide_mult_cancel_left2
% 0.30/0.52 thf(fact_1482_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.30/0.52 ! [C: complex,A: complex,B: complex] :
% 0.30/0.52 ( ( C != zero_zero_complex )
% 0.30/0.52 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.30/0.52 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_divide_mult_cancel_left
% 0.30/0.52 thf(fact_1483_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.30/0.52 ! [C: real,A: real,B: real] :
% 0.30/0.52 ( ( C != zero_zero_real )
% 0.30/0.52 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.52 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_divide_mult_cancel_left
% 0.30/0.52 thf(fact_1484_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.30/0.52 ! [C: rat,A: rat,B: rat] :
% 0.30/0.52 ( ( C != zero_zero_rat )
% 0.30/0.52 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.52 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_divide_mult_cancel_left
% 0.30/0.52 thf(fact_1485_mult__divide__mult__cancel__left__if,axiom,
% 0.30/0.52 ! [C: complex,A: complex,B: complex] :
% 0.30/0.52 ( ( ( C = zero_zero_complex )
% 0.30/0.52 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.30/0.52 = zero_zero_complex ) )
% 0.30/0.52 & ( ( C != zero_zero_complex )
% 0.30/0.52 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.30/0.52 = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_divide_mult_cancel_left_if
% 0.30/0.52 thf(fact_1486_mult__divide__mult__cancel__left__if,axiom,
% 0.30/0.52 ! [C: real,A: real,B: real] :
% 0.30/0.52 ( ( ( C = zero_zero_real )
% 0.30/0.52 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.52 = zero_zero_real ) )
% 0.30/0.52 & ( ( C != zero_zero_real )
% 0.30/0.52 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.52 = ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_divide_mult_cancel_left_if
% 0.30/0.52 thf(fact_1487_mult__divide__mult__cancel__left__if,axiom,
% 0.30/0.52 ! [C: rat,A: rat,B: rat] :
% 0.30/0.52 ( ( ( C = zero_zero_rat )
% 0.30/0.52 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.52 = zero_zero_rat ) )
% 0.30/0.52 & ( ( C != zero_zero_rat )
% 0.30/0.52 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.52 = ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_divide_mult_cancel_left_if
% 0.30/0.52 thf(fact_1488_div__mult__mult1,axiom,
% 0.30/0.52 ! [C: nat,A: nat,B: nat] :
% 0.30/0.52 ( ( C != zero_zero_nat )
% 0.30/0.52 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.30/0.52 = ( divide_divide_nat @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_mult_mult1
% 0.30/0.52 thf(fact_1489_div__mult__mult1,axiom,
% 0.30/0.52 ! [C: int,A: int,B: int] :
% 0.30/0.52 ( ( C != zero_zero_int )
% 0.30/0.52 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.52 = ( divide_divide_int @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_mult_mult1
% 0.30/0.52 thf(fact_1490_div__mult__mult2,axiom,
% 0.30/0.52 ! [C: nat,A: nat,B: nat] :
% 0.30/0.52 ( ( C != zero_zero_nat )
% 0.30/0.52 => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.30/0.52 = ( divide_divide_nat @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_mult_mult2
% 0.30/0.52 thf(fact_1491_div__mult__mult2,axiom,
% 0.30/0.52 ! [C: int,A: int,B: int] :
% 0.30/0.52 ( ( C != zero_zero_int )
% 0.30/0.52 => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.30/0.52 = ( divide_divide_int @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_mult_mult2
% 0.30/0.52 thf(fact_1492_div__mult__mult1__if,axiom,
% 0.30/0.52 ! [C: nat,A: nat,B: nat] :
% 0.30/0.52 ( ( ( C = zero_zero_nat )
% 0.30/0.52 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.30/0.52 = zero_zero_nat ) )
% 0.30/0.52 & ( ( C != zero_zero_nat )
% 0.30/0.52 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.30/0.52 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_mult_mult1_if
% 0.30/0.52 thf(fact_1493_div__mult__mult1__if,axiom,
% 0.30/0.52 ! [C: int,A: int,B: int] :
% 0.30/0.52 ( ( ( C = zero_zero_int )
% 0.30/0.52 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.52 = zero_zero_int ) )
% 0.30/0.52 & ( ( C != zero_zero_int )
% 0.30/0.52 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.52 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_mult_mult1_if
% 0.30/0.52 thf(fact_1494_nonzero__mult__div__cancel__right,axiom,
% 0.30/0.52 ! [B: complex,A: complex] :
% 0.30/0.52 ( ( B != zero_zero_complex )
% 0.30/0.52 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
% 0.30/0.52 = A ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_div_cancel_right
% 0.30/0.52 thf(fact_1495_nonzero__mult__div__cancel__right,axiom,
% 0.30/0.52 ! [B: real,A: real] :
% 0.30/0.52 ( ( B != zero_zero_real )
% 0.30/0.52 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
% 0.30/0.52 = A ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_div_cancel_right
% 0.30/0.52 thf(fact_1496_nonzero__mult__div__cancel__right,axiom,
% 0.30/0.52 ! [B: rat,A: rat] :
% 0.30/0.52 ( ( B != zero_zero_rat )
% 0.30/0.52 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
% 0.30/0.52 = A ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_div_cancel_right
% 0.30/0.52 thf(fact_1497_nonzero__mult__div__cancel__right,axiom,
% 0.30/0.52 ! [B: nat,A: nat] :
% 0.30/0.52 ( ( B != zero_zero_nat )
% 0.30/0.52 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
% 0.30/0.52 = A ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_div_cancel_right
% 0.30/0.52 thf(fact_1498_nonzero__mult__div__cancel__right,axiom,
% 0.30/0.52 ! [B: int,A: int] :
% 0.30/0.52 ( ( B != zero_zero_int )
% 0.30/0.52 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
% 0.30/0.52 = A ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_div_cancel_right
% 0.30/0.52 thf(fact_1499_nonzero__mult__div__cancel__left,axiom,
% 0.30/0.52 ! [A: complex,B: complex] :
% 0.30/0.52 ( ( A != zero_zero_complex )
% 0.30/0.52 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
% 0.30/0.52 = B ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_div_cancel_left
% 0.30/0.52 thf(fact_1500_nonzero__mult__div__cancel__left,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( A != zero_zero_real )
% 0.30/0.52 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
% 0.30/0.52 = B ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_div_cancel_left
% 0.30/0.52 thf(fact_1501_nonzero__mult__div__cancel__left,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( A != zero_zero_rat )
% 0.30/0.52 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
% 0.30/0.52 = B ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_div_cancel_left
% 0.30/0.52 thf(fact_1502_nonzero__mult__div__cancel__left,axiom,
% 0.30/0.52 ! [A: nat,B: nat] :
% 0.30/0.52 ( ( A != zero_zero_nat )
% 0.30/0.52 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
% 0.30/0.52 = B ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_div_cancel_left
% 0.30/0.52 thf(fact_1503_nonzero__mult__div__cancel__left,axiom,
% 0.30/0.52 ! [A: int,B: int] :
% 0.30/0.52 ( ( A != zero_zero_int )
% 0.30/0.52 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
% 0.30/0.52 = B ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_mult_div_cancel_left
% 0.30/0.52 thf(fact_1504_diff__add__zero,axiom,
% 0.30/0.52 ! [A: nat,B: nat] :
% 0.30/0.52 ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.30/0.52 = zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % diff_add_zero
% 0.30/0.52 thf(fact_1505_divide__eq__1__iff,axiom,
% 0.30/0.52 ! [A: complex,B: complex] :
% 0.30/0.52 ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.30/0.52 = one_one_complex )
% 0.30/0.52 = ( ( B != zero_zero_complex )
% 0.30/0.52 & ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_eq_1_iff
% 0.30/0.52 thf(fact_1506_divide__eq__1__iff,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ( divide_divide_real @ A @ B )
% 0.30/0.52 = one_one_real )
% 0.30/0.52 = ( ( B != zero_zero_real )
% 0.30/0.52 & ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_eq_1_iff
% 0.30/0.52 thf(fact_1507_divide__eq__1__iff,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ( divide_divide_rat @ A @ B )
% 0.30/0.52 = one_one_rat )
% 0.30/0.52 = ( ( B != zero_zero_rat )
% 0.30/0.52 & ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_eq_1_iff
% 0.30/0.52 thf(fact_1508_one__eq__divide__iff,axiom,
% 0.30/0.52 ! [A: complex,B: complex] :
% 0.30/0.52 ( ( one_one_complex
% 0.30/0.52 = ( divide1717551699836669952omplex @ A @ B ) )
% 0.30/0.52 = ( ( B != zero_zero_complex )
% 0.30/0.52 & ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_eq_divide_iff
% 0.30/0.52 thf(fact_1509_one__eq__divide__iff,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( one_one_real
% 0.30/0.52 = ( divide_divide_real @ A @ B ) )
% 0.30/0.52 = ( ( B != zero_zero_real )
% 0.30/0.52 & ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_eq_divide_iff
% 0.30/0.52 thf(fact_1510_one__eq__divide__iff,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( one_one_rat
% 0.30/0.52 = ( divide_divide_rat @ A @ B ) )
% 0.30/0.52 = ( ( B != zero_zero_rat )
% 0.30/0.52 & ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_eq_divide_iff
% 0.30/0.52 thf(fact_1511_divide__self,axiom,
% 0.30/0.52 ! [A: complex] :
% 0.30/0.52 ( ( A != zero_zero_complex )
% 0.30/0.52 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.30/0.52 = one_one_complex ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_self
% 0.30/0.52 thf(fact_1512_divide__self,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( A != zero_zero_real )
% 0.30/0.52 => ( ( divide_divide_real @ A @ A )
% 0.30/0.52 = one_one_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_self
% 0.30/0.52 thf(fact_1513_divide__self,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( A != zero_zero_rat )
% 0.30/0.52 => ( ( divide_divide_rat @ A @ A )
% 0.30/0.52 = one_one_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_self
% 0.30/0.52 thf(fact_1514_divide__self__if,axiom,
% 0.30/0.52 ! [A: complex] :
% 0.30/0.52 ( ( ( A = zero_zero_complex )
% 0.30/0.52 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.30/0.52 = zero_zero_complex ) )
% 0.30/0.52 & ( ( A != zero_zero_complex )
% 0.30/0.52 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.30/0.52 = one_one_complex ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_self_if
% 0.30/0.52 thf(fact_1515_divide__self__if,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( ( A = zero_zero_real )
% 0.30/0.52 => ( ( divide_divide_real @ A @ A )
% 0.30/0.52 = zero_zero_real ) )
% 0.30/0.52 & ( ( A != zero_zero_real )
% 0.30/0.52 => ( ( divide_divide_real @ A @ A )
% 0.30/0.52 = one_one_real ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_self_if
% 0.30/0.52 thf(fact_1516_divide__self__if,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( ( A = zero_zero_rat )
% 0.30/0.52 => ( ( divide_divide_rat @ A @ A )
% 0.30/0.52 = zero_zero_rat ) )
% 0.30/0.52 & ( ( A != zero_zero_rat )
% 0.30/0.52 => ( ( divide_divide_rat @ A @ A )
% 0.30/0.52 = one_one_rat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_self_if
% 0.30/0.52 thf(fact_1517_divide__eq__eq__1,axiom,
% 0.30/0.52 ! [B: real,A: real] :
% 0.30/0.52 ( ( ( divide_divide_real @ B @ A )
% 0.30/0.52 = one_one_real )
% 0.30/0.52 = ( ( A != zero_zero_real )
% 0.30/0.52 & ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_eq_eq_1
% 0.30/0.52 thf(fact_1518_divide__eq__eq__1,axiom,
% 0.30/0.52 ! [B: rat,A: rat] :
% 0.30/0.52 ( ( ( divide_divide_rat @ B @ A )
% 0.30/0.52 = one_one_rat )
% 0.30/0.52 = ( ( A != zero_zero_rat )
% 0.30/0.52 & ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_eq_eq_1
% 0.30/0.52 thf(fact_1519_eq__divide__eq__1,axiom,
% 0.30/0.52 ! [B: real,A: real] :
% 0.30/0.52 ( ( one_one_real
% 0.30/0.52 = ( divide_divide_real @ B @ A ) )
% 0.30/0.52 = ( ( A != zero_zero_real )
% 0.30/0.52 & ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % eq_divide_eq_1
% 0.30/0.52 thf(fact_1520_eq__divide__eq__1,axiom,
% 0.30/0.52 ! [B: rat,A: rat] :
% 0.30/0.52 ( ( one_one_rat
% 0.30/0.52 = ( divide_divide_rat @ B @ A ) )
% 0.30/0.52 = ( ( A != zero_zero_rat )
% 0.30/0.52 & ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % eq_divide_eq_1
% 0.30/0.52 thf(fact_1521_one__divide__eq__0__iff,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( ( divide_divide_real @ one_one_real @ A )
% 0.30/0.52 = zero_zero_real )
% 0.30/0.52 = ( A = zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_divide_eq_0_iff
% 0.30/0.52 thf(fact_1522_one__divide__eq__0__iff,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( ( divide_divide_rat @ one_one_rat @ A )
% 0.30/0.52 = zero_zero_rat )
% 0.30/0.52 = ( A = zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_divide_eq_0_iff
% 0.30/0.52 thf(fact_1523_zero__eq__1__divide__iff,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( zero_zero_real
% 0.30/0.52 = ( divide_divide_real @ one_one_real @ A ) )
% 0.30/0.52 = ( A = zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_eq_1_divide_iff
% 0.30/0.52 thf(fact_1524_zero__eq__1__divide__iff,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( zero_zero_rat
% 0.30/0.52 = ( divide_divide_rat @ one_one_rat @ A ) )
% 0.30/0.52 = ( A = zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_eq_1_divide_iff
% 0.30/0.52 thf(fact_1525_div__self,axiom,
% 0.30/0.52 ! [A: complex] :
% 0.30/0.52 ( ( A != zero_zero_complex )
% 0.30/0.52 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.30/0.52 = one_one_complex ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_self
% 0.30/0.52 thf(fact_1526_div__self,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( A != zero_zero_real )
% 0.30/0.52 => ( ( divide_divide_real @ A @ A )
% 0.30/0.52 = one_one_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_self
% 0.30/0.52 thf(fact_1527_div__self,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( A != zero_zero_rat )
% 0.30/0.52 => ( ( divide_divide_rat @ A @ A )
% 0.30/0.52 = one_one_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_self
% 0.30/0.52 thf(fact_1528_div__self,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( A != zero_zero_nat )
% 0.30/0.52 => ( ( divide_divide_nat @ A @ A )
% 0.30/0.52 = one_one_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_self
% 0.30/0.52 thf(fact_1529_div__self,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( A != zero_zero_int )
% 0.30/0.52 => ( ( divide_divide_int @ A @ A )
% 0.30/0.52 = one_one_int ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_self
% 0.30/0.52 thf(fact_1530_power__0__Suc,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
% 0.30/0.52 = zero_zero_rat ) ).
% 0.30/0.52
% 0.30/0.52 % power_0_Suc
% 0.30/0.52 thf(fact_1531_power__0__Suc,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.30/0.52 = zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % power_0_Suc
% 0.30/0.52 thf(fact_1532_power__0__Suc,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 0.30/0.52 = zero_zero_real ) ).
% 0.30/0.52
% 0.30/0.52 % power_0_Suc
% 0.30/0.52 thf(fact_1533_power__0__Suc,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 0.30/0.52 = zero_zero_int ) ).
% 0.30/0.52
% 0.30/0.52 % power_0_Suc
% 0.30/0.52 thf(fact_1534_power__0__Suc,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
% 0.30/0.52 = zero_zero_complex ) ).
% 0.30/0.52
% 0.30/0.52 % power_0_Suc
% 0.30/0.52 thf(fact_1535_power__zero__numeral,axiom,
% 0.30/0.52 ! [K: num] :
% 0.30/0.52 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 0.30/0.52 = zero_zero_rat ) ).
% 0.30/0.52
% 0.30/0.52 % power_zero_numeral
% 0.30/0.52 thf(fact_1536_power__zero__numeral,axiom,
% 0.30/0.52 ! [K: num] :
% 0.30/0.52 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 0.30/0.52 = zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % power_zero_numeral
% 0.30/0.52 thf(fact_1537_power__zero__numeral,axiom,
% 0.30/0.52 ! [K: num] :
% 0.30/0.52 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 0.30/0.52 = zero_zero_real ) ).
% 0.30/0.52
% 0.30/0.52 % power_zero_numeral
% 0.30/0.52 thf(fact_1538_power__zero__numeral,axiom,
% 0.30/0.52 ! [K: num] :
% 0.30/0.52 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 0.30/0.52 = zero_zero_int ) ).
% 0.30/0.52
% 0.30/0.52 % power_zero_numeral
% 0.30/0.52 thf(fact_1539_power__zero__numeral,axiom,
% 0.30/0.52 ! [K: num] :
% 0.30/0.52 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 0.30/0.52 = zero_zero_complex ) ).
% 0.30/0.52
% 0.30/0.52 % power_zero_numeral
% 0.30/0.52 thf(fact_1540_power__Suc0__right,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % power_Suc0_right
% 0.30/0.52 thf(fact_1541_power__Suc0__right,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % power_Suc0_right
% 0.30/0.52 thf(fact_1542_power__Suc0__right,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % power_Suc0_right
% 0.30/0.52 thf(fact_1543_power__Suc0__right,axiom,
% 0.30/0.52 ! [A: complex] :
% 0.30/0.52 ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % power_Suc0_right
% 0.30/0.52 thf(fact_1544_zero__less__Suc,axiom,
% 0.30/0.52 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_less_Suc
% 0.30/0.52 thf(fact_1545_less__Suc0,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.30/0.52 = ( N = zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_Suc0
% 0.30/0.52 thf(fact_1546_add__gr__0,axiom,
% 0.30/0.52 ! [M: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.52 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.52 | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_gr_0
% 0.30/0.52 thf(fact_1547_mult__eq__1__iff,axiom,
% 0.30/0.52 ! [M: nat,N: nat] :
% 0.30/0.52 ( ( ( times_times_nat @ M @ N )
% 0.30/0.52 = ( suc @ zero_zero_nat ) )
% 0.30/0.52 = ( ( M
% 0.30/0.52 = ( suc @ zero_zero_nat ) )
% 0.30/0.52 & ( N
% 0.30/0.52 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_eq_1_iff
% 0.30/0.52 thf(fact_1548_one__eq__mult__iff,axiom,
% 0.30/0.52 ! [M: nat,N: nat] :
% 0.30/0.52 ( ( ( suc @ zero_zero_nat )
% 0.30/0.52 = ( times_times_nat @ M @ N ) )
% 0.30/0.52 = ( ( M
% 0.30/0.52 = ( suc @ zero_zero_nat ) )
% 0.30/0.52 & ( N
% 0.30/0.52 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_eq_mult_iff
% 0.30/0.52 thf(fact_1549_div__by__Suc__0,axiom,
% 0.30/0.52 ! [M: nat] :
% 0.30/0.52 ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.30/0.52 = M ) ).
% 0.30/0.52
% 0.30/0.52 % div_by_Suc_0
% 0.30/0.52 thf(fact_1550_less__one,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ N @ one_one_nat )
% 0.30/0.52 = ( N = zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_one
% 0.30/0.52 thf(fact_1551_zero__less__diff,axiom,
% 0.30/0.52 ! [N: nat,M: nat] :
% 0.30/0.52 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 0.30/0.52 = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_less_diff
% 0.30/0.52 thf(fact_1552_nat__0__less__mult__iff,axiom,
% 0.30/0.52 ! [M: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 0.30/0.52 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.52 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nat_0_less_mult_iff
% 0.30/0.52 thf(fact_1553_mult__less__cancel2,axiom,
% 0.30/0.52 ! [M: nat,K: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.30/0.52 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.52 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_less_cancel2
% 0.30/0.52 thf(fact_1554_nat__mult__less__cancel__disj,axiom,
% 0.30/0.52 ! [K: nat,M: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.52 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.52 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nat_mult_less_cancel_disj
% 0.30/0.52 thf(fact_1555_div__less,axiom,
% 0.30/0.52 ! [M: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ M @ N )
% 0.30/0.52 => ( ( divide_divide_nat @ M @ N )
% 0.30/0.52 = zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_less
% 0.30/0.52 thf(fact_1556_diff__is__0__eq_H,axiom,
% 0.30/0.52 ! [M: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.52 => ( ( minus_minus_nat @ M @ N )
% 0.30/0.52 = zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_is_0_eq'
% 0.30/0.52 thf(fact_1557_diff__is__0__eq,axiom,
% 0.30/0.52 ! [M: nat,N: nat] :
% 0.30/0.52 ( ( ( minus_minus_nat @ M @ N )
% 0.30/0.52 = zero_zero_nat )
% 0.30/0.52 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_is_0_eq
% 0.30/0.52 thf(fact_1558_nat__power__eq__Suc__0__iff,axiom,
% 0.30/0.52 ! [X2: nat,M: nat] :
% 0.30/0.52 ( ( ( power_power_nat @ X2 @ M )
% 0.30/0.52 = ( suc @ zero_zero_nat ) )
% 0.30/0.52 = ( ( M = zero_zero_nat )
% 0.30/0.52 | ( X2
% 0.30/0.52 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nat_power_eq_Suc_0_iff
% 0.30/0.52 thf(fact_1559_power__Suc__0,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.52 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_Suc_0
% 0.30/0.52 thf(fact_1560_nat__zero__less__power__iff,axiom,
% 0.30/0.52 ! [X2: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N ) )
% 0.30/0.52 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 0.30/0.52 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nat_zero_less_power_iff
% 0.30/0.52 thf(fact_1561_nat__mult__div__cancel__disj,axiom,
% 0.30/0.52 ! [K: nat,M: nat,N: nat] :
% 0.30/0.52 ( ( ( K = zero_zero_nat )
% 0.30/0.52 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.52 = zero_zero_nat ) )
% 0.30/0.52 & ( ( K != zero_zero_nat )
% 0.30/0.52 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.52 = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nat_mult_div_cancel_disj
% 0.30/0.52 thf(fact_1562_divide__le__0__1__iff,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.30/0.52 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_le_0_1_iff
% 0.30/0.52 thf(fact_1563_divide__le__0__1__iff,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.30/0.52 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_le_0_1_iff
% 0.30/0.52 thf(fact_1564_zero__le__divide__1__iff,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.30/0.52 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_le_divide_1_iff
% 0.30/0.52 thf(fact_1565_zero__le__divide__1__iff,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.30/0.52 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_le_divide_1_iff
% 0.30/0.52 thf(fact_1566_divide__less__0__1__iff,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.30/0.52 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_less_0_1_iff
% 0.30/0.52 thf(fact_1567_divide__less__0__1__iff,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.30/0.52 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_less_0_1_iff
% 0.30/0.52 thf(fact_1568_divide__less__eq__1__neg,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.52 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.30/0.52 = ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_less_eq_1_neg
% 0.30/0.52 thf(fact_1569_divide__less__eq__1__neg,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.52 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.30/0.52 = ( ord_less_rat @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_less_eq_1_neg
% 0.30/0.52 thf(fact_1570_divide__less__eq__1__pos,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.52 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.30/0.52 = ( ord_less_real @ B @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_less_eq_1_pos
% 0.30/0.52 thf(fact_1571_divide__less__eq__1__pos,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.52 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.30/0.52 = ( ord_less_rat @ B @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_less_eq_1_pos
% 0.30/0.52 thf(fact_1572_less__divide__eq__1__neg,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.52 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.30/0.52 = ( ord_less_real @ B @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_divide_eq_1_neg
% 0.30/0.52 thf(fact_1573_less__divide__eq__1__neg,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.52 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.30/0.52 = ( ord_less_rat @ B @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_divide_eq_1_neg
% 0.30/0.52 thf(fact_1574_less__divide__eq__1__pos,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.52 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.30/0.52 = ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_divide_eq_1_pos
% 0.30/0.52 thf(fact_1575_less__divide__eq__1__pos,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.52 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.30/0.52 = ( ord_less_rat @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_divide_eq_1_pos
% 0.30/0.52 thf(fact_1576_zero__less__divide__1__iff,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.30/0.52 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_less_divide_1_iff
% 0.30/0.52 thf(fact_1577_zero__less__divide__1__iff,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.30/0.52 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_less_divide_1_iff
% 0.30/0.52 thf(fact_1578_eq__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.52 ! [A: complex,B: complex,W: num] :
% 0.30/0.52 ( ( A
% 0.30/0.52 = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.52 = ( ( ( ( numera6690914467698888265omplex @ W )
% 0.30/0.52 != zero_zero_complex )
% 0.30/0.52 => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.52 = B ) )
% 0.30/0.52 & ( ( ( numera6690914467698888265omplex @ W )
% 0.30/0.52 = zero_zero_complex )
% 0.30/0.52 => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % eq_divide_eq_numeral1(1)
% 0.30/0.52 thf(fact_1579_eq__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.52 ! [A: real,B: real,W: num] :
% 0.30/0.52 ( ( A
% 0.30/0.52 = ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.52 = ( ( ( ( numeral_numeral_real @ W )
% 0.30/0.52 != zero_zero_real )
% 0.30/0.52 => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
% 0.30/0.52 = B ) )
% 0.30/0.52 & ( ( ( numeral_numeral_real @ W )
% 0.30/0.52 = zero_zero_real )
% 0.30/0.52 => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % eq_divide_eq_numeral1(1)
% 0.30/0.52 thf(fact_1580_eq__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.52 ! [A: rat,B: rat,W: num] :
% 0.30/0.52 ( ( A
% 0.30/0.52 = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 0.30/0.52 = ( ( ( ( numeral_numeral_rat @ W )
% 0.30/0.52 != zero_zero_rat )
% 0.30/0.52 => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
% 0.30/0.52 = B ) )
% 0.30/0.52 & ( ( ( numeral_numeral_rat @ W )
% 0.30/0.52 = zero_zero_rat )
% 0.30/0.52 => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % eq_divide_eq_numeral1(1)
% 0.30/0.52 thf(fact_1581_divide__eq__eq__numeral1_I1_J,axiom,
% 0.30/0.52 ! [B: complex,W: num,A: complex] :
% 0.30/0.52 ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.52 = A )
% 0.30/0.52 = ( ( ( ( numera6690914467698888265omplex @ W )
% 0.30/0.52 != zero_zero_complex )
% 0.30/0.52 => ( B
% 0.30/0.52 = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
% 0.30/0.52 & ( ( ( numera6690914467698888265omplex @ W )
% 0.30/0.52 = zero_zero_complex )
% 0.30/0.52 => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_eq_eq_numeral1(1)
% 0.30/0.52 thf(fact_1582_divide__eq__eq__numeral1_I1_J,axiom,
% 0.30/0.52 ! [B: real,W: num,A: real] :
% 0.30/0.52 ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
% 0.30/0.52 = A )
% 0.30/0.52 = ( ( ( ( numeral_numeral_real @ W )
% 0.30/0.52 != zero_zero_real )
% 0.30/0.52 => ( B
% 0.30/0.52 = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
% 0.30/0.52 & ( ( ( numeral_numeral_real @ W )
% 0.30/0.52 = zero_zero_real )
% 0.30/0.52 => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_eq_eq_numeral1(1)
% 0.30/0.52 thf(fact_1583_divide__eq__eq__numeral1_I1_J,axiom,
% 0.30/0.52 ! [B: rat,W: num,A: rat] :
% 0.30/0.52 ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
% 0.30/0.52 = A )
% 0.30/0.52 = ( ( ( ( numeral_numeral_rat @ W )
% 0.30/0.52 != zero_zero_rat )
% 0.30/0.52 => ( B
% 0.30/0.52 = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
% 0.30/0.52 & ( ( ( numeral_numeral_rat @ W )
% 0.30/0.52 = zero_zero_rat )
% 0.30/0.52 => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_eq_eq_numeral1(1)
% 0.30/0.52 thf(fact_1584_nonzero__divide__mult__cancel__right,axiom,
% 0.30/0.52 ! [B: complex,A: complex] :
% 0.30/0.52 ( ( B != zero_zero_complex )
% 0.30/0.52 => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
% 0.30/0.52 = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_divide_mult_cancel_right
% 0.30/0.52 thf(fact_1585_nonzero__divide__mult__cancel__right,axiom,
% 0.30/0.52 ! [B: real,A: real] :
% 0.30/0.52 ( ( B != zero_zero_real )
% 0.30/0.52 => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
% 0.30/0.52 = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_divide_mult_cancel_right
% 0.30/0.52 thf(fact_1586_nonzero__divide__mult__cancel__right,axiom,
% 0.30/0.52 ! [B: rat,A: rat] :
% 0.30/0.52 ( ( B != zero_zero_rat )
% 0.30/0.52 => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
% 0.30/0.52 = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_divide_mult_cancel_right
% 0.30/0.52 thf(fact_1587_nonzero__divide__mult__cancel__left,axiom,
% 0.30/0.52 ! [A: complex,B: complex] :
% 0.30/0.52 ( ( A != zero_zero_complex )
% 0.30/0.52 => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
% 0.30/0.52 = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_divide_mult_cancel_left
% 0.30/0.52 thf(fact_1588_nonzero__divide__mult__cancel__left,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( A != zero_zero_real )
% 0.30/0.52 => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
% 0.30/0.52 = ( divide_divide_real @ one_one_real @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_divide_mult_cancel_left
% 0.30/0.52 thf(fact_1589_nonzero__divide__mult__cancel__left,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( A != zero_zero_rat )
% 0.30/0.52 => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
% 0.30/0.52 = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nonzero_divide_mult_cancel_left
% 0.30/0.52 thf(fact_1590_div__mult__self4,axiom,
% 0.30/0.52 ! [B: nat,C: nat,A: nat] :
% 0.30/0.52 ( ( B != zero_zero_nat )
% 0.30/0.52 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 0.30/0.52 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_mult_self4
% 0.30/0.52 thf(fact_1591_div__mult__self4,axiom,
% 0.30/0.52 ! [B: int,C: int,A: int] :
% 0.30/0.52 ( ( B != zero_zero_int )
% 0.30/0.52 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 0.30/0.52 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_mult_self4
% 0.30/0.52 thf(fact_1592_div__mult__self3,axiom,
% 0.30/0.52 ! [B: nat,C: nat,A: nat] :
% 0.30/0.52 ( ( B != zero_zero_nat )
% 0.30/0.52 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 0.30/0.52 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_mult_self3
% 0.30/0.52 thf(fact_1593_div__mult__self3,axiom,
% 0.30/0.52 ! [B: int,C: int,A: int] :
% 0.30/0.52 ( ( B != zero_zero_int )
% 0.30/0.52 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 0.30/0.52 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_mult_self3
% 0.30/0.52 thf(fact_1594_div__mult__self2,axiom,
% 0.30/0.52 ! [B: nat,A: nat,C: nat] :
% 0.30/0.52 ( ( B != zero_zero_nat )
% 0.30/0.52 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 0.30/0.52 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_mult_self2
% 0.30/0.52 thf(fact_1595_div__mult__self2,axiom,
% 0.30/0.52 ! [B: int,A: int,C: int] :
% 0.30/0.52 ( ( B != zero_zero_int )
% 0.30/0.52 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 0.30/0.52 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_mult_self2
% 0.30/0.52 thf(fact_1596_div__mult__self1,axiom,
% 0.30/0.52 ! [B: nat,A: nat,C: nat] :
% 0.30/0.52 ( ( B != zero_zero_nat )
% 0.30/0.52 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 0.30/0.52 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_mult_self1
% 0.30/0.52 thf(fact_1597_div__mult__self1,axiom,
% 0.30/0.52 ! [B: int,A: int,C: int] :
% 0.30/0.52 ( ( B != zero_zero_int )
% 0.30/0.52 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 0.30/0.52 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_mult_self1
% 0.30/0.52 thf(fact_1598_power__eq__0__iff,axiom,
% 0.30/0.52 ! [A: rat,N: nat] :
% 0.30/0.52 ( ( ( power_power_rat @ A @ N )
% 0.30/0.52 = zero_zero_rat )
% 0.30/0.52 = ( ( A = zero_zero_rat )
% 0.30/0.52 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_eq_0_iff
% 0.30/0.52 thf(fact_1599_power__eq__0__iff,axiom,
% 0.30/0.52 ! [A: nat,N: nat] :
% 0.30/0.52 ( ( ( power_power_nat @ A @ N )
% 0.30/0.52 = zero_zero_nat )
% 0.30/0.52 = ( ( A = zero_zero_nat )
% 0.30/0.52 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_eq_0_iff
% 0.30/0.52 thf(fact_1600_power__eq__0__iff,axiom,
% 0.30/0.52 ! [A: real,N: nat] :
% 0.30/0.52 ( ( ( power_power_real @ A @ N )
% 0.30/0.52 = zero_zero_real )
% 0.30/0.52 = ( ( A = zero_zero_real )
% 0.30/0.52 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_eq_0_iff
% 0.30/0.52 thf(fact_1601_power__eq__0__iff,axiom,
% 0.30/0.52 ! [A: int,N: nat] :
% 0.30/0.52 ( ( ( power_power_int @ A @ N )
% 0.30/0.52 = zero_zero_int )
% 0.30/0.52 = ( ( A = zero_zero_int )
% 0.30/0.52 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_eq_0_iff
% 0.30/0.52 thf(fact_1602_power__eq__0__iff,axiom,
% 0.30/0.52 ! [A: complex,N: nat] :
% 0.30/0.52 ( ( ( power_power_complex @ A @ N )
% 0.30/0.52 = zero_zero_complex )
% 0.30/0.52 = ( ( A = zero_zero_complex )
% 0.30/0.52 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_eq_0_iff
% 0.30/0.52 thf(fact_1603_Suc__pred,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.30/0.52 = N ) ) ).
% 0.30/0.52
% 0.30/0.52 % Suc_pred
% 0.30/0.52 thf(fact_1604_one__le__mult__iff,axiom,
% 0.30/0.52 ! [M: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 0.30/0.52 = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.30/0.52 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_le_mult_iff
% 0.30/0.52 thf(fact_1605_mult__le__cancel2,axiom,
% 0.30/0.52 ! [M: nat,K: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.30/0.52 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.52 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_le_cancel2
% 0.30/0.52 thf(fact_1606_nat__mult__le__cancel__disj,axiom,
% 0.30/0.52 ! [K: nat,M: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.52 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.52 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nat_mult_le_cancel_disj
% 0.30/0.52 thf(fact_1607_div__mult__self1__is__m,axiom,
% 0.30/0.52 ! [N: nat,M: nat] :
% 0.30/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52 => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 0.30/0.52 = M ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_mult_self1_is_m
% 0.30/0.52 thf(fact_1608_div__mult__self__is__m,axiom,
% 0.30/0.52 ! [N: nat,M: nat] :
% 0.30/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52 => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 0.30/0.52 = M ) ) ).
% 0.30/0.52
% 0.30/0.52 % div_mult_self_is_m
% 0.30/0.52 thf(fact_1609_divide__le__eq__1__neg,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.52 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.30/0.52 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_le_eq_1_neg
% 0.30/0.52 thf(fact_1610_divide__le__eq__1__neg,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.52 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.30/0.52 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_le_eq_1_neg
% 0.30/0.52 thf(fact_1611_divide__le__eq__1__pos,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.52 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.30/0.52 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_le_eq_1_pos
% 0.30/0.52 thf(fact_1612_divide__le__eq__1__pos,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.52 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.30/0.52 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_le_eq_1_pos
% 0.30/0.52 thf(fact_1613_le__divide__eq__1__neg,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.52 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.30/0.52 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_divide_eq_1_neg
% 0.30/0.52 thf(fact_1614_le__divide__eq__1__neg,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.52 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.30/0.52 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_divide_eq_1_neg
% 0.30/0.52 thf(fact_1615_le__divide__eq__1__pos,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.52 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.30/0.52 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_divide_eq_1_pos
% 0.30/0.52 thf(fact_1616_le__divide__eq__1__pos,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.52 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.30/0.52 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_divide_eq_1_pos
% 0.30/0.52 thf(fact_1617_power__strict__decreasing__iff,axiom,
% 0.30/0.52 ! [B: real,M: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.52 => ( ( ord_less_real @ B @ one_one_real )
% 0.30/0.52 => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 0.30/0.52 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_strict_decreasing_iff
% 0.30/0.52 thf(fact_1618_power__strict__decreasing__iff,axiom,
% 0.30/0.52 ! [B: rat,M: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.30/0.52 => ( ( ord_less_rat @ B @ one_one_rat )
% 0.30/0.52 => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 0.30/0.52 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_strict_decreasing_iff
% 0.30/0.52 thf(fact_1619_power__strict__decreasing__iff,axiom,
% 0.30/0.52 ! [B: nat,M: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.52 => ( ( ord_less_nat @ B @ one_one_nat )
% 0.30/0.52 => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 0.30/0.52 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_strict_decreasing_iff
% 0.30/0.52 thf(fact_1620_power__strict__decreasing__iff,axiom,
% 0.30/0.52 ! [B: int,M: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.52 => ( ( ord_less_int @ B @ one_one_int )
% 0.30/0.52 => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 0.30/0.52 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_strict_decreasing_iff
% 0.30/0.52 thf(fact_1621_power__mono__iff,axiom,
% 0.30/0.52 ! [A: real,B: real,N: nat] :
% 0.30/0.52 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52 => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 0.30/0.52 = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_mono_iff
% 0.30/0.52 thf(fact_1622_power__mono__iff,axiom,
% 0.30/0.52 ! [A: rat,B: rat,N: nat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 0.30/0.52 = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_mono_iff
% 0.30/0.52 thf(fact_1623_power__mono__iff,axiom,
% 0.30/0.52 ! [A: nat,B: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.30/0.52 = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_mono_iff
% 0.30/0.52 thf(fact_1624_power__mono__iff,axiom,
% 0.30/0.52 ! [A: int,B: int,N: nat] :
% 0.30/0.52 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52 => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.30/0.52 = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_mono_iff
% 0.30/0.52 thf(fact_1625_zero__eq__power2,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52 = zero_zero_rat )
% 0.30/0.52 = ( A = zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_eq_power2
% 0.30/0.52 thf(fact_1626_zero__eq__power2,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52 = zero_zero_nat )
% 0.30/0.52 = ( A = zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_eq_power2
% 0.30/0.52 thf(fact_1627_zero__eq__power2,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52 = zero_zero_real )
% 0.30/0.52 = ( A = zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_eq_power2
% 0.30/0.52 thf(fact_1628_zero__eq__power2,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52 = zero_zero_int )
% 0.30/0.52 = ( A = zero_zero_int ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_eq_power2
% 0.30/0.52 thf(fact_1629_zero__eq__power2,axiom,
% 0.30/0.52 ! [A: complex] :
% 0.30/0.52 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52 = zero_zero_complex )
% 0.30/0.52 = ( A = zero_zero_complex ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_eq_power2
% 0.30/0.52 thf(fact_1630_Suc__diff__1,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52 => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.30/0.52 = N ) ) ).
% 0.30/0.52
% 0.30/0.52 % Suc_diff_1
% 0.30/0.52 thf(fact_1631_bits__1__div__2,axiom,
% 0.30/0.52 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52 = zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % bits_1_div_2
% 0.30/0.52 thf(fact_1632_bits__1__div__2,axiom,
% 0.30/0.52 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.52 = zero_zero_int ) ).
% 0.30/0.52
% 0.30/0.52 % bits_1_div_2
% 0.30/0.52 thf(fact_1633_one__div__two__eq__zero,axiom,
% 0.30/0.52 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52 = zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % one_div_two_eq_zero
% 0.30/0.52 thf(fact_1634_one__div__two__eq__zero,axiom,
% 0.30/0.52 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.52 = zero_zero_int ) ).
% 0.30/0.52
% 0.30/0.52 % one_div_two_eq_zero
% 0.30/0.52 thf(fact_1635_power__decreasing__iff,axiom,
% 0.30/0.52 ! [B: real,M: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.52 => ( ( ord_less_real @ B @ one_one_real )
% 0.30/0.52 => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 0.30/0.52 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_decreasing_iff
% 0.30/0.52 thf(fact_1636_power__decreasing__iff,axiom,
% 0.30/0.52 ! [B: rat,M: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.30/0.52 => ( ( ord_less_rat @ B @ one_one_rat )
% 0.30/0.52 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 0.30/0.52 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_decreasing_iff
% 0.30/0.52 thf(fact_1637_power__decreasing__iff,axiom,
% 0.30/0.52 ! [B: nat,M: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.52 => ( ( ord_less_nat @ B @ one_one_nat )
% 0.30/0.52 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 0.30/0.52 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_decreasing_iff
% 0.30/0.52 thf(fact_1638_power__decreasing__iff,axiom,
% 0.30/0.52 ! [B: int,M: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.52 => ( ( ord_less_int @ B @ one_one_int )
% 0.30/0.52 => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 0.30/0.52 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_decreasing_iff
% 0.30/0.52 thf(fact_1639_power2__eq__iff__nonneg,axiom,
% 0.30/0.52 ! [X2: real,Y3: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.52 => ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52 = ( X2 = Y3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power2_eq_iff_nonneg
% 0.30/0.52 thf(fact_1640_power2__eq__iff__nonneg,axiom,
% 0.30/0.52 ! [X2: rat,Y3: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.30/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.52 => ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52 = ( X2 = Y3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power2_eq_iff_nonneg
% 0.30/0.52 thf(fact_1641_power2__eq__iff__nonneg,axiom,
% 0.30/0.52 ! [X2: nat,Y3: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
% 0.30/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.30/0.52 => ( ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52 = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52 = ( X2 = Y3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power2_eq_iff_nonneg
% 0.30/0.52 thf(fact_1642_power2__eq__iff__nonneg,axiom,
% 0.30/0.52 ! [X2: int,Y3: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.52 => ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52 = ( X2 = Y3 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power2_eq_iff_nonneg
% 0.30/0.52 thf(fact_1643_power2__less__eq__zero__iff,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 0.30/0.52 = ( A = zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % power2_less_eq_zero_iff
% 0.30/0.52 thf(fact_1644_power2__less__eq__zero__iff,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 0.30/0.52 = ( A = zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % power2_less_eq_zero_iff
% 0.30/0.52 thf(fact_1645_power2__less__eq__zero__iff,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.30/0.52 = ( A = zero_zero_int ) ) ).
% 0.30/0.52
% 0.30/0.52 % power2_less_eq_zero_iff
% 0.30/0.52 thf(fact_1646_zero__less__power2,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52 = ( A != zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_less_power2
% 0.30/0.52 thf(fact_1647_zero__less__power2,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52 = ( A != zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_less_power2
% 0.30/0.52 thf(fact_1648_zero__less__power2,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52 = ( A != zero_zero_int ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_less_power2
% 0.30/0.52 thf(fact_1649_sum__power2__eq__zero__iff,axiom,
% 0.30/0.52 ! [X2: rat,Y3: rat] :
% 0.30/0.52 ( ( ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52 = zero_zero_rat )
% 0.30/0.52 = ( ( X2 = zero_zero_rat )
% 0.30/0.52 & ( Y3 = zero_zero_rat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % sum_power2_eq_zero_iff
% 0.30/0.52 thf(fact_1650_sum__power2__eq__zero__iff,axiom,
% 0.30/0.52 ! [X2: real,Y3: real] :
% 0.30/0.52 ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52 = zero_zero_real )
% 0.30/0.52 = ( ( X2 = zero_zero_real )
% 0.30/0.52 & ( Y3 = zero_zero_real ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % sum_power2_eq_zero_iff
% 0.30/0.52 thf(fact_1651_sum__power2__eq__zero__iff,axiom,
% 0.30/0.52 ! [X2: int,Y3: int] :
% 0.30/0.52 ( ( ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52 = zero_zero_int )
% 0.30/0.52 = ( ( X2 = zero_zero_int )
% 0.30/0.52 & ( Y3 = zero_zero_int ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % sum_power2_eq_zero_iff
% 0.30/0.52 thf(fact_1652_VEBT__internal_Ovalid_H_Ocases,axiom,
% 0.30/0.52 ! [X2: produc9072475918466114483BT_nat] :
% 0.30/0.52 ( ! [Uu2: $o,Uv2: $o,D3: nat] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D3 ) )
% 0.30/0.52 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Deg3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT_internal.valid'.cases
% 0.30/0.52 thf(fact_1653_VEBT_Osize_I4_J,axiom,
% 0.30/0.52 ! [X21: $o,X222: $o] :
% 0.30/0.52 ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 0.30/0.52 = zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT.size(4)
% 0.30/0.52 thf(fact_1654_vebt__buildup_Osimps_I1_J,axiom,
% 0.30/0.52 ( ( vEBT_vebt_buildup @ zero_zero_nat )
% 0.30/0.52 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 0.30/0.52
% 0.30/0.52 % vebt_buildup.simps(1)
% 0.30/0.52 thf(fact_1655_zero__reorient,axiom,
% 0.30/0.52 ! [X2: complex] :
% 0.30/0.52 ( ( zero_zero_complex = X2 )
% 0.30/0.52 = ( X2 = zero_zero_complex ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_reorient
% 0.30/0.52 thf(fact_1656_zero__reorient,axiom,
% 0.30/0.52 ! [X2: real] :
% 0.30/0.52 ( ( zero_zero_real = X2 )
% 0.30/0.52 = ( X2 = zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_reorient
% 0.30/0.52 thf(fact_1657_zero__reorient,axiom,
% 0.30/0.52 ! [X2: rat] :
% 0.30/0.52 ( ( zero_zero_rat = X2 )
% 0.30/0.52 = ( X2 = zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_reorient
% 0.30/0.52 thf(fact_1658_zero__reorient,axiom,
% 0.30/0.52 ! [X2: nat] :
% 0.30/0.52 ( ( zero_zero_nat = X2 )
% 0.30/0.52 = ( X2 = zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_reorient
% 0.30/0.52 thf(fact_1659_zero__reorient,axiom,
% 0.30/0.52 ! [X2: int] :
% 0.30/0.52 ( ( zero_zero_int = X2 )
% 0.30/0.52 = ( X2 = zero_zero_int ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_reorient
% 0.30/0.52 thf(fact_1660_vebt__buildup_Osimps_I2_J,axiom,
% 0.30/0.52 ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
% 0.30/0.52 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 0.30/0.52
% 0.30/0.52 % vebt_buildup.simps(2)
% 0.30/0.52 thf(fact_1661_VEBT__internal_Onaive__member_Ocases,axiom,
% 0.30/0.52 ! [X2: produc9072475918466114483BT_nat] :
% 0.30/0.52 ( ! [A4: $o,B4: $o,X5: nat] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X5 ) )
% 0.30/0.52 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
% 0.30/0.52 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT,X5: nat] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) @ X5 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT_internal.naive_member.cases
% 0.30/0.52 thf(fact_1662_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.30/0.52 ! [X2: produc5542196010084753463at_nat] :
% 0.30/0.52 ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 0.30/0.52 => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 0.30/0.52 => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A4 ) @ ( some_P7363390416028606310at_nat @ B4 ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT_internal.option_shift.cases
% 0.30/0.52 thf(fact_1663_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.30/0.52 ! [X2: produc8306885398267862888on_nat] :
% 0.30/0.52 ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 0.30/0.52 => ( ! [Uw2: nat > nat > nat,V2: nat] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 0.30/0.52 => ~ ! [F2: nat > nat > nat,A4: nat,B4: nat] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A4 ) @ ( some_nat @ B4 ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT_internal.option_shift.cases
% 0.30/0.52 thf(fact_1664_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.30/0.52 ! [X2: produc1193250871479095198on_num] :
% 0.30/0.52 ( ! [Uu2: num > num > num,Uv2: option_num] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 0.30/0.52 => ( ! [Uw2: num > num > num,V2: num] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 0.30/0.52 => ~ ! [F2: num > num > num,A4: num,B4: num] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A4 ) @ ( some_num @ B4 ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT_internal.option_shift.cases
% 0.30/0.52 thf(fact_1665_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.30/0.52 ! [X2: produc5491161045314408544at_nat] :
% 0.30/0.52 ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 0.30/0.52 => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 0.30/0.52 => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat,Y4: product_prod_nat_nat] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X5 ) @ ( some_P7363390416028606310at_nat @ Y4 ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT_internal.option_comp_shift.cases
% 0.30/0.52 thf(fact_1666_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.30/0.52 ! [X2: produc2233624965454879586on_nat] :
% 0.30/0.52 ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 0.30/0.52 => ( ! [Uw2: nat > nat > $o,V2: nat] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 0.30/0.52 => ~ ! [F2: nat > nat > $o,X5: nat,Y4: nat] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X5 ) @ ( some_nat @ Y4 ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT_internal.option_comp_shift.cases
% 0.30/0.52 thf(fact_1667_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.30/0.52 ! [X2: produc7036089656553540234on_num] :
% 0.30/0.52 ( ! [Uu2: num > num > $o,Uv2: option_num] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 0.30/0.52 => ( ! [Uw2: num > num > $o,V2: num] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 0.30/0.52 => ~ ! [F2: num > num > $o,X5: num,Y4: num] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X5 ) @ ( some_num @ Y4 ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT_internal.option_comp_shift.cases
% 0.30/0.52 thf(fact_1668_invar__vebt_Ointros_I1_J,axiom,
% 0.30/0.52 ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % invar_vebt.intros(1)
% 0.30/0.52 thf(fact_1669_power__0__left,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( ( N = zero_zero_nat )
% 0.30/0.52 => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.30/0.52 = one_one_rat ) )
% 0.30/0.52 & ( ( N != zero_zero_nat )
% 0.30/0.52 => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.30/0.52 = zero_zero_rat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_0_left
% 0.30/0.52 thf(fact_1670_power__0__left,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( ( N = zero_zero_nat )
% 0.30/0.52 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.30/0.52 = one_one_nat ) )
% 0.30/0.52 & ( ( N != zero_zero_nat )
% 0.30/0.52 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.30/0.52 = zero_zero_nat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_0_left
% 0.30/0.52 thf(fact_1671_power__0__left,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( ( N = zero_zero_nat )
% 0.30/0.52 => ( ( power_power_real @ zero_zero_real @ N )
% 0.30/0.52 = one_one_real ) )
% 0.30/0.52 & ( ( N != zero_zero_nat )
% 0.30/0.52 => ( ( power_power_real @ zero_zero_real @ N )
% 0.30/0.52 = zero_zero_real ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_0_left
% 0.30/0.52 thf(fact_1672_power__0__left,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( ( N = zero_zero_nat )
% 0.30/0.52 => ( ( power_power_int @ zero_zero_int @ N )
% 0.30/0.52 = one_one_int ) )
% 0.30/0.52 & ( ( N != zero_zero_nat )
% 0.30/0.52 => ( ( power_power_int @ zero_zero_int @ N )
% 0.30/0.52 = zero_zero_int ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_0_left
% 0.30/0.52 thf(fact_1673_power__0__left,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( ( N = zero_zero_nat )
% 0.30/0.52 => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.30/0.52 = one_one_complex ) )
% 0.30/0.52 & ( ( N != zero_zero_nat )
% 0.30/0.52 => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.30/0.52 = zero_zero_complex ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_0_left
% 0.30/0.52 thf(fact_1674_zero__power,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52 => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.30/0.52 = zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_power
% 0.30/0.52 thf(fact_1675_zero__power,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.30/0.52 = zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_power
% 0.30/0.52 thf(fact_1676_zero__power,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52 => ( ( power_power_real @ zero_zero_real @ N )
% 0.30/0.52 = zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_power
% 0.30/0.52 thf(fact_1677_zero__power,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52 => ( ( power_power_int @ zero_zero_int @ N )
% 0.30/0.52 = zero_zero_int ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_power
% 0.30/0.52 thf(fact_1678_zero__power,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52 => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.30/0.52 = zero_zero_complex ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_power
% 0.30/0.52 thf(fact_1679_vebt__member_Osimps_I1_J,axiom,
% 0.30/0.52 ! [A: $o,B: $o,X2: nat] :
% 0.30/0.52 ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X2 )
% 0.30/0.52 = ( ( ( X2 = zero_zero_nat )
% 0.30/0.52 => A )
% 0.30/0.52 & ( ( X2 != zero_zero_nat )
% 0.30/0.52 => ( ( ( X2 = one_one_nat )
% 0.30/0.52 => B )
% 0.30/0.52 & ( X2 = one_one_nat ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % vebt_member.simps(1)
% 0.30/0.52 thf(fact_1680_VEBT_Odistinct_I1_J,axiom,
% 0.30/0.52 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
% 0.30/0.52 ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 0.30/0.52 != ( vEBT_Leaf @ X21 @ X222 ) ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT.distinct(1)
% 0.30/0.52 thf(fact_1681_VEBT_Oexhaust,axiom,
% 0.30/0.52 ! [Y3: vEBT_VEBT] :
% 0.30/0.52 ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
% 0.30/0.52 ( Y3
% 0.30/0.52 != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
% 0.30/0.52 => ~ ! [X212: $o,X223: $o] :
% 0.30/0.52 ( Y3
% 0.30/0.52 != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT.exhaust
% 0.30/0.52 thf(fact_1682_vebt__insert_Osimps_I1_J,axiom,
% 0.30/0.52 ! [X2: nat,A: $o,B: $o] :
% 0.30/0.52 ( ( ( X2 = zero_zero_nat )
% 0.30/0.52 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X2 )
% 0.30/0.52 = ( vEBT_Leaf @ $true @ B ) ) )
% 0.30/0.52 & ( ( X2 != zero_zero_nat )
% 0.30/0.52 => ( ( ( X2 = one_one_nat )
% 0.30/0.52 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X2 )
% 0.30/0.52 = ( vEBT_Leaf @ A @ $true ) ) )
% 0.30/0.52 & ( ( X2 != one_one_nat )
% 0.30/0.52 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X2 )
% 0.30/0.52 = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % vebt_insert.simps(1)
% 0.30/0.52 thf(fact_1683_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 0.30/0.52 ! [A: $o,B: $o,X2: nat] :
% 0.30/0.52 ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X2 )
% 0.30/0.52 = ( ( ( X2 = zero_zero_nat )
% 0.30/0.52 => A )
% 0.30/0.52 & ( ( X2 != zero_zero_nat )
% 0.30/0.52 => ( ( ( X2 = one_one_nat )
% 0.30/0.52 => B )
% 0.30/0.52 & ( X2 = one_one_nat ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT_internal.naive_member.simps(1)
% 0.30/0.52 thf(fact_1684_vebt__pred_Osimps_I1_J,axiom,
% 0.30/0.52 ! [Uu: $o,Uv: $o] :
% 0.30/0.52 ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
% 0.30/0.52 = none_nat ) ).
% 0.30/0.52
% 0.30/0.52 % vebt_pred.simps(1)
% 0.30/0.52 thf(fact_1685_le__numeral__extra_I3_J,axiom,
% 0.30/0.52 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.30/0.52
% 0.30/0.52 % le_numeral_extra(3)
% 0.30/0.52 thf(fact_1686_le__numeral__extra_I3_J,axiom,
% 0.30/0.52 ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 0.30/0.52
% 0.30/0.52 % le_numeral_extra(3)
% 0.30/0.52 thf(fact_1687_le__numeral__extra_I3_J,axiom,
% 0.30/0.52 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.30/0.52
% 0.30/0.52 % le_numeral_extra(3)
% 0.30/0.52 thf(fact_1688_le__numeral__extra_I3_J,axiom,
% 0.30/0.52 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.30/0.52
% 0.30/0.52 % le_numeral_extra(3)
% 0.30/0.52 thf(fact_1689_zero__le,axiom,
% 0.30/0.52 ! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% 0.30/0.52
% 0.30/0.52 % zero_le
% 0.30/0.52 thf(fact_1690_field__lbound__gt__zero,axiom,
% 0.30/0.52 ! [D1: real,D22: real] :
% 0.30/0.52 ( ( ord_less_real @ zero_zero_real @ D1 )
% 0.30/0.52 => ( ( ord_less_real @ zero_zero_real @ D22 )
% 0.30/0.52 => ? [E2: real] :
% 0.30/0.52 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.30/0.52 & ( ord_less_real @ E2 @ D1 )
% 0.30/0.52 & ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % field_lbound_gt_zero
% 0.30/0.52 thf(fact_1691_field__lbound__gt__zero,axiom,
% 0.30/0.52 ! [D1: rat,D22: rat] :
% 0.30/0.52 ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 0.30/0.52 => ( ( ord_less_rat @ zero_zero_rat @ D22 )
% 0.30/0.52 => ? [E2: rat] :
% 0.30/0.52 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.30/0.52 & ( ord_less_rat @ E2 @ D1 )
% 0.30/0.52 & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % field_lbound_gt_zero
% 0.30/0.52 thf(fact_1692_less__numeral__extra_I3_J,axiom,
% 0.30/0.52 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 0.30/0.52
% 0.30/0.52 % less_numeral_extra(3)
% 0.30/0.52 thf(fact_1693_less__numeral__extra_I3_J,axiom,
% 0.30/0.52 ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 0.30/0.52
% 0.30/0.52 % less_numeral_extra(3)
% 0.30/0.52 thf(fact_1694_less__numeral__extra_I3_J,axiom,
% 0.30/0.52 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % less_numeral_extra(3)
% 0.30/0.52 thf(fact_1695_less__numeral__extra_I3_J,axiom,
% 0.30/0.52 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.30/0.52
% 0.30/0.52 % less_numeral_extra(3)
% 0.30/0.52 thf(fact_1696_zero__less__iff__neq__zero,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52 = ( N != zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_less_iff_neq_zero
% 0.30/0.52 thf(fact_1697_gr__implies__not__zero,axiom,
% 0.30/0.52 ! [M: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ M @ N )
% 0.30/0.52 => ( N != zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % gr_implies_not_zero
% 0.30/0.52 thf(fact_1698_not__less__zero,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % not_less_zero
% 0.30/0.52 thf(fact_1699_gr__zeroI,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( N != zero_zero_nat )
% 0.30/0.52 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.52
% 0.30/0.52 % gr_zeroI
% 0.30/0.52 thf(fact_1700_zero__neq__numeral,axiom,
% 0.30/0.52 ! [N: num] :
% 0.30/0.52 ( zero_zero_complex
% 0.30/0.52 != ( numera6690914467698888265omplex @ N ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_neq_numeral
% 0.30/0.52 thf(fact_1701_zero__neq__numeral,axiom,
% 0.30/0.52 ! [N: num] :
% 0.30/0.52 ( zero_zero_real
% 0.30/0.52 != ( numeral_numeral_real @ N ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_neq_numeral
% 0.30/0.52 thf(fact_1702_zero__neq__numeral,axiom,
% 0.30/0.52 ! [N: num] :
% 0.30/0.52 ( zero_zero_rat
% 0.30/0.52 != ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_neq_numeral
% 0.30/0.52 thf(fact_1703_zero__neq__numeral,axiom,
% 0.30/0.52 ! [N: num] :
% 0.30/0.52 ( zero_zero_nat
% 0.30/0.52 != ( numeral_numeral_nat @ N ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_neq_numeral
% 0.30/0.52 thf(fact_1704_zero__neq__numeral,axiom,
% 0.30/0.52 ! [N: num] :
% 0.30/0.52 ( zero_zero_int
% 0.30/0.52 != ( numeral_numeral_int @ N ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_neq_numeral
% 0.30/0.52 thf(fact_1705_mult__right__cancel,axiom,
% 0.30/0.52 ! [C: complex,A: complex,B: complex] :
% 0.30/0.52 ( ( C != zero_zero_complex )
% 0.30/0.52 => ( ( ( times_times_complex @ A @ C )
% 0.30/0.52 = ( times_times_complex @ B @ C ) )
% 0.30/0.52 = ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_right_cancel
% 0.30/0.52 thf(fact_1706_mult__right__cancel,axiom,
% 0.30/0.52 ! [C: real,A: real,B: real] :
% 0.30/0.52 ( ( C != zero_zero_real )
% 0.30/0.52 => ( ( ( times_times_real @ A @ C )
% 0.30/0.52 = ( times_times_real @ B @ C ) )
% 0.30/0.52 = ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_right_cancel
% 0.30/0.52 thf(fact_1707_mult__right__cancel,axiom,
% 0.30/0.52 ! [C: rat,A: rat,B: rat] :
% 0.30/0.52 ( ( C != zero_zero_rat )
% 0.30/0.52 => ( ( ( times_times_rat @ A @ C )
% 0.30/0.52 = ( times_times_rat @ B @ C ) )
% 0.30/0.52 = ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_right_cancel
% 0.30/0.52 thf(fact_1708_mult__right__cancel,axiom,
% 0.30/0.52 ! [C: nat,A: nat,B: nat] :
% 0.30/0.52 ( ( C != zero_zero_nat )
% 0.30/0.52 => ( ( ( times_times_nat @ A @ C )
% 0.30/0.52 = ( times_times_nat @ B @ C ) )
% 0.30/0.52 = ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_right_cancel
% 0.30/0.52 thf(fact_1709_mult__right__cancel,axiom,
% 0.30/0.52 ! [C: int,A: int,B: int] :
% 0.30/0.52 ( ( C != zero_zero_int )
% 0.30/0.52 => ( ( ( times_times_int @ A @ C )
% 0.30/0.52 = ( times_times_int @ B @ C ) )
% 0.30/0.52 = ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_right_cancel
% 0.30/0.52 thf(fact_1710_mult__left__cancel,axiom,
% 0.30/0.52 ! [C: complex,A: complex,B: complex] :
% 0.30/0.52 ( ( C != zero_zero_complex )
% 0.30/0.52 => ( ( ( times_times_complex @ C @ A )
% 0.30/0.52 = ( times_times_complex @ C @ B ) )
% 0.30/0.52 = ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_left_cancel
% 0.30/0.52 thf(fact_1711_mult__left__cancel,axiom,
% 0.30/0.52 ! [C: real,A: real,B: real] :
% 0.30/0.52 ( ( C != zero_zero_real )
% 0.30/0.52 => ( ( ( times_times_real @ C @ A )
% 0.30/0.52 = ( times_times_real @ C @ B ) )
% 0.30/0.52 = ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_left_cancel
% 0.30/0.52 thf(fact_1712_mult__left__cancel,axiom,
% 0.30/0.52 ! [C: rat,A: rat,B: rat] :
% 0.30/0.52 ( ( C != zero_zero_rat )
% 0.30/0.52 => ( ( ( times_times_rat @ C @ A )
% 0.30/0.52 = ( times_times_rat @ C @ B ) )
% 0.30/0.52 = ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_left_cancel
% 0.30/0.52 thf(fact_1713_mult__left__cancel,axiom,
% 0.30/0.52 ! [C: nat,A: nat,B: nat] :
% 0.30/0.52 ( ( C != zero_zero_nat )
% 0.30/0.52 => ( ( ( times_times_nat @ C @ A )
% 0.30/0.52 = ( times_times_nat @ C @ B ) )
% 0.30/0.52 = ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_left_cancel
% 0.30/0.52 thf(fact_1714_mult__left__cancel,axiom,
% 0.30/0.52 ! [C: int,A: int,B: int] :
% 0.30/0.52 ( ( C != zero_zero_int )
% 0.30/0.52 => ( ( ( times_times_int @ C @ A )
% 0.30/0.52 = ( times_times_int @ C @ B ) )
% 0.30/0.52 = ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_left_cancel
% 0.30/0.52 thf(fact_1715_no__zero__divisors,axiom,
% 0.30/0.52 ! [A: complex,B: complex] :
% 0.30/0.52 ( ( A != zero_zero_complex )
% 0.30/0.52 => ( ( B != zero_zero_complex )
% 0.30/0.52 => ( ( times_times_complex @ A @ B )
% 0.30/0.52 != zero_zero_complex ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % no_zero_divisors
% 0.30/0.52 thf(fact_1716_no__zero__divisors,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( A != zero_zero_real )
% 0.30/0.52 => ( ( B != zero_zero_real )
% 0.30/0.52 => ( ( times_times_real @ A @ B )
% 0.30/0.52 != zero_zero_real ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % no_zero_divisors
% 0.30/0.52 thf(fact_1717_no__zero__divisors,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( A != zero_zero_rat )
% 0.30/0.52 => ( ( B != zero_zero_rat )
% 0.30/0.52 => ( ( times_times_rat @ A @ B )
% 0.30/0.52 != zero_zero_rat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % no_zero_divisors
% 0.30/0.52 thf(fact_1718_no__zero__divisors,axiom,
% 0.30/0.52 ! [A: nat,B: nat] :
% 0.30/0.52 ( ( A != zero_zero_nat )
% 0.30/0.52 => ( ( B != zero_zero_nat )
% 0.30/0.52 => ( ( times_times_nat @ A @ B )
% 0.30/0.52 != zero_zero_nat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % no_zero_divisors
% 0.30/0.52 thf(fact_1719_no__zero__divisors,axiom,
% 0.30/0.52 ! [A: int,B: int] :
% 0.30/0.52 ( ( A != zero_zero_int )
% 0.30/0.52 => ( ( B != zero_zero_int )
% 0.30/0.52 => ( ( times_times_int @ A @ B )
% 0.30/0.52 != zero_zero_int ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % no_zero_divisors
% 0.30/0.52 thf(fact_1720_divisors__zero,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 % divisors_zero
% 0.30/0.52 thf(fact_1721_divisors__zero,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 % divisors_zero
% 0.30/0.52 thf(fact_1722_divisors__zero,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 % divisors_zero
% 0.30/0.52 thf(fact_1723_divisors__zero,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 % divisors_zero
% 0.30/0.52 thf(fact_1724_divisors__zero,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 % divisors_zero
% 0.30/0.52 thf(fact_1725_mult__not__zero,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_not_zero
% 0.30/0.52 thf(fact_1726_mult__not__zero,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_not_zero
% 0.30/0.52 thf(fact_1727_mult__not__zero,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_not_zero
% 0.30/0.52 thf(fact_1728_mult__not__zero,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_not_zero
% 0.30/0.52 thf(fact_1729_mult__not__zero,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_not_zero
% 0.30/0.52 thf(fact_1730_zero__neq__one,axiom,
% 0.30/0.52 zero_zero_complex != one_one_complex ).
% 0.30/0.52
% 0.30/0.52 % zero_neq_one
% 0.30/0.52 thf(fact_1731_zero__neq__one,axiom,
% 0.30/0.52 zero_zero_real != one_one_real ).
% 0.30/0.52
% 0.30/0.52 % zero_neq_one
% 0.30/0.52 thf(fact_1732_zero__neq__one,axiom,
% 0.30/0.52 zero_zero_rat != one_one_rat ).
% 0.30/0.52
% 0.30/0.52 % zero_neq_one
% 0.30/0.52 thf(fact_1733_zero__neq__one,axiom,
% 0.30/0.52 zero_zero_nat != one_one_nat ).
% 0.30/0.52
% 0.30/0.52 % zero_neq_one
% 0.30/0.52 thf(fact_1734_zero__neq__one,axiom,
% 0.30/0.52 zero_zero_int != one_one_int ).
% 0.30/0.52
% 0.30/0.52 % zero_neq_one
% 0.30/0.52 thf(fact_1735_comm__monoid__add__class_Oadd__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 % comm_monoid_add_class.add_0
% 0.30/0.52 thf(fact_1736_comm__monoid__add__class_Oadd__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 % comm_monoid_add_class.add_0
% 0.30/0.52 thf(fact_1737_comm__monoid__add__class_Oadd__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 % comm_monoid_add_class.add_0
% 0.30/0.52 thf(fact_1738_comm__monoid__add__class_Oadd__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 % comm_monoid_add_class.add_0
% 0.30/0.52 thf(fact_1739_comm__monoid__add__class_Oadd__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 % comm_monoid_add_class.add_0
% 0.30/0.52 thf(fact_1740_add_Ocomm__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.comm_neutral
% 0.30/0.52 thf(fact_1741_add_Ocomm__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.comm_neutral
% 0.30/0.52 thf(fact_1742_add_Ocomm__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.comm_neutral
% 0.30/0.52 thf(fact_1743_add_Ocomm__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.comm_neutral
% 0.30/0.52 thf(fact_1744_add_Ocomm__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.comm_neutral
% 0.30/0.52 thf(fact_1745_add_Ogroup__left__neutral,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.group_left_neutral
% 0.30/0.52 thf(fact_1746_add_Ogroup__left__neutral,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.group_left_neutral
% 0.30/0.52 thf(fact_1747_add_Ogroup__left__neutral,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.group_left_neutral
% 0.30/0.52 thf(fact_1748_add_Ogroup__left__neutral,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.group_left_neutral
% 0.30/0.52 thf(fact_1749_eq__iff__diff__eq__0,axiom,
% 0.30/0.52 ( ( ^ [Y6: complex,Z5: complex] : ( Y6 = Z5 ) )
% 0.30/0.52 = ( ^ [A3: complex,B2: complex] :
% 0.30/0.52 ( ( minus_minus_complex @ A3 @ B2 )
% 0.30/0.52 = zero_zero_complex ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % eq_iff_diff_eq_0
% 0.30/0.52 thf(fact_1750_eq__iff__diff__eq__0,axiom,
% 0.30/0.52 ( ( ^ [Y6: real,Z5: real] : ( Y6 = Z5 ) )
% 0.30/0.52 = ( ^ [A3: real,B2: real] :
% 0.30/0.52 ( ( minus_minus_real @ A3 @ B2 )
% 0.30/0.52 = zero_zero_real ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % eq_iff_diff_eq_0
% 0.30/0.52 thf(fact_1751_eq__iff__diff__eq__0,axiom,
% 0.30/0.52 ( ( ^ [Y6: rat,Z5: rat] : ( Y6 = Z5 ) )
% 0.30/0.52 = ( ^ [A3: rat,B2: rat] :
% 0.30/0.52 ( ( minus_minus_rat @ A3 @ B2 )
% 0.30/0.52 = zero_zero_rat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % eq_iff_diff_eq_0
% 0.30/0.52 thf(fact_1752_eq__iff__diff__eq__0,axiom,
% 0.30/0.52 ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 0.30/0.52 = ( ^ [A3: int,B2: int] :
% 0.30/0.52 ( ( minus_minus_int @ A3 @ B2 )
% 0.30/0.52 = zero_zero_int ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % eq_iff_diff_eq_0
% 0.30/0.52 thf(fact_1753_power__not__zero,axiom,
% 0.30/0.52 ! [A: rat,N: nat] :
% 0.30/0.52 ( ( A != zero_zero_rat )
% 0.30/0.52 => ( ( power_power_rat @ A @ N )
% 0.30/0.52 != zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_not_zero
% 0.30/0.52 thf(fact_1754_power__not__zero,axiom,
% 0.30/0.52 ! [A: nat,N: nat] :
% 0.30/0.52 ( ( A != zero_zero_nat )
% 0.30/0.52 => ( ( power_power_nat @ A @ N )
% 0.30/0.52 != zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_not_zero
% 0.30/0.52 thf(fact_1755_power__not__zero,axiom,
% 0.30/0.52 ! [A: real,N: nat] :
% 0.30/0.52 ( ( A != zero_zero_real )
% 0.30/0.52 => ( ( power_power_real @ A @ N )
% 0.30/0.52 != zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_not_zero
% 0.30/0.52 thf(fact_1756_power__not__zero,axiom,
% 0.30/0.52 ! [A: int,N: nat] :
% 0.30/0.52 ( ( A != zero_zero_int )
% 0.30/0.52 => ( ( power_power_int @ A @ N )
% 0.30/0.52 != zero_zero_int ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_not_zero
% 0.30/0.52 thf(fact_1757_power__not__zero,axiom,
% 0.30/0.52 ! [A: complex,N: nat] :
% 0.30/0.52 ( ( A != zero_zero_complex )
% 0.30/0.52 => ( ( power_power_complex @ A @ N )
% 0.30/0.52 != zero_zero_complex ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_not_zero
% 0.30/0.52 thf(fact_1758_num_Osize_I4_J,axiom,
% 0.30/0.52 ( ( size_size_num @ one )
% 0.30/0.52 = zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % num.size(4)
% 0.30/0.52 thf(fact_1759_nat_Odistinct_I1_J,axiom,
% 0.30/0.52 ! [X22: nat] :
% 0.30/0.52 ( zero_zero_nat
% 0.30/0.52 != ( suc @ X22 ) ) ).
% 0.30/0.52
% 0.30/0.52 % nat.distinct(1)
% 0.30/0.52 thf(fact_1760_old_Onat_Odistinct_I2_J,axiom,
% 0.30/0.52 ! [Nat2: nat] :
% 0.30/0.52 ( ( suc @ Nat2 )
% 0.30/0.52 != zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % old.nat.distinct(2)
% 0.30/0.52 thf(fact_1761_old_Onat_Odistinct_I1_J,axiom,
% 0.30/0.52 ! [Nat2: nat] :
% 0.30/0.52 ( zero_zero_nat
% 0.30/0.52 != ( suc @ Nat2 ) ) ).
% 0.30/0.52
% 0.30/0.52 % old.nat.distinct(1)
% 0.30/0.52 thf(fact_1762_nat_OdiscI,axiom,
% 0.30/0.52 ! [Nat: nat,X22: nat] :
% 0.30/0.52 ( ( Nat
% 0.30/0.52 = ( suc @ X22 ) )
% 0.30/0.52 => ( Nat != zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % nat.discI
% 0.30/0.52 thf(fact_1763_old_Onat_Oexhaust,axiom,
% 0.30/0.52 ! [Y3: nat] :
% 0.30/0.52 ( ( Y3 != zero_zero_nat )
% 0.30/0.52 => ~ ! [Nat3: nat] :
% 0.30/0.52 ( Y3
% 0.30/0.52 != ( suc @ Nat3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % old.nat.exhaust
% 0.30/0.52 thf(fact_1764_vebt__buildup_Ocases,axiom,
% 0.30/0.52 ! [X2: nat] :
% 0.30/0.52 ( ( X2 != zero_zero_nat )
% 0.30/0.52 => ( ( X2
% 0.30/0.52 != ( suc @ zero_zero_nat ) )
% 0.30/0.52 => ~ ! [Va3: nat] :
% 0.30/0.52 ( X2
% 0.30/0.52 != ( suc @ ( suc @ Va3 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % vebt_buildup.cases
% 0.30/0.52 thf(fact_1765_nat__induct,axiom,
% 0.30/0.52 ! [P: nat > $o,N: nat] :
% 0.30/0.52 ( ( P @ zero_zero_nat )
% 0.30/0.52 => ( ! [N2: nat] :
% 0.30/0.52 ( ( P @ N2 )
% 0.30/0.52 => ( P @ ( suc @ N2 ) ) )
% 0.30/0.52 => ( P @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nat_induct
% 0.30/0.52 thf(fact_1766_diff__induct,axiom,
% 0.30/0.52 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.30/0.52 ( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
% 0.30/0.52 => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
% 0.30/0.52 => ( ! [X5: nat,Y4: nat] :
% 0.30/0.52 ( ( P @ X5 @ Y4 )
% 0.30/0.52 => ( P @ ( suc @ X5 ) @ ( suc @ Y4 ) ) )
% 0.30/0.52 => ( P @ M @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_induct
% 0.30/0.52 thf(fact_1767_zero__induct,axiom,
% 0.30/0.52 ! [P: nat > $o,K: nat] :
% 0.30/0.52 ( ( P @ K )
% 0.30/0.52 => ( ! [N2: nat] :
% 0.30/0.52 ( ( P @ ( suc @ N2 ) )
% 0.30/0.52 => ( P @ N2 ) )
% 0.30/0.52 => ( P @ zero_zero_nat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_induct
% 0.30/0.52 thf(fact_1768_Suc__neq__Zero,axiom,
% 0.30/0.52 ! [M: nat] :
% 0.30/0.52 ( ( suc @ M )
% 0.30/0.52 != zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % Suc_neq_Zero
% 0.30/0.52 thf(fact_1769_Zero__neq__Suc,axiom,
% 0.30/0.52 ! [M: nat] :
% 0.30/0.52 ( zero_zero_nat
% 0.30/0.52 != ( suc @ M ) ) ).
% 0.30/0.52
% 0.30/0.52 % Zero_neq_Suc
% 0.30/0.52 thf(fact_1770_Zero__not__Suc,axiom,
% 0.30/0.52 ! [M: nat] :
% 0.30/0.52 ( zero_zero_nat
% 0.30/0.52 != ( suc @ M ) ) ).
% 0.30/0.52
% 0.30/0.52 % Zero_not_Suc
% 0.30/0.52 thf(fact_1771_not0__implies__Suc,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( N != zero_zero_nat )
% 0.30/0.52 => ? [M4: nat] :
% 0.30/0.52 ( N
% 0.30/0.52 = ( suc @ M4 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % not0_implies_Suc
% 0.30/0.52 thf(fact_1772_infinite__descent0,axiom,
% 0.30/0.52 ! [P: nat > $o,N: nat] :
% 0.30/0.52 ( ( P @ zero_zero_nat )
% 0.30/0.52 => ( ! [N2: nat] :
% 0.30/0.52 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.52 => ( ~ ( P @ N2 )
% 0.30/0.52 => ? [M2: nat] :
% 0.30/0.52 ( ( ord_less_nat @ M2 @ N2 )
% 0.30/0.52 & ~ ( P @ M2 ) ) ) )
% 0.30/0.52 => ( P @ N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % infinite_descent0
% 0.30/0.52 thf(fact_1773_gr__implies__not0,axiom,
% 0.30/0.52 ! [M: nat,N: nat] :
% 0.30/0.52 ( ( ord_less_nat @ M @ N )
% 0.30/0.52 => ( N != zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % gr_implies_not0
% 0.30/0.52 thf(fact_1774_less__zeroE,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % less_zeroE
% 0.30/0.52 thf(fact_1775_not__less0,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % not_less0
% 0.30/0.52 thf(fact_1776_not__gr0,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.30/0.52 = ( N = zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % not_gr0
% 0.30/0.52 thf(fact_1777_gr0I,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( N != zero_zero_nat )
% 0.30/0.52 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.52
% 0.30/0.52 % gr0I
% 0.30/0.52 thf(fact_1778_bot__nat__0_Oextremum__strict,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % bot_nat_0.extremum_strict
% 0.30/0.52 thf(fact_1779_VEBT__internal_OminNull_Osimps_I1_J,axiom,
% 0.30/0.52 vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT_internal.minNull.simps(1)
% 0.30/0.52 thf(fact_1780_VEBT__internal_OminNull_Osimps_I2_J,axiom,
% 0.30/0.52 ! [Uv: $o] :
% 0.30/0.52 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT_internal.minNull.simps(2)
% 0.30/0.52 thf(fact_1781_VEBT__internal_OminNull_Osimps_I3_J,axiom,
% 0.30/0.52 ! [Uu: $o] :
% 0.30/0.52 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT_internal.minNull.simps(3)
% 0.30/0.52 thf(fact_1782_less__eq__nat_Osimps_I1_J,axiom,
% 0.30/0.52 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.30/0.52
% 0.30/0.52 % less_eq_nat.simps(1)
% 0.30/0.52 thf(fact_1783_bot__nat__0_Oextremum__unique,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.52 = ( A = zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % bot_nat_0.extremum_unique
% 0.30/0.52 thf(fact_1784_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.52 => ( A = zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % bot_nat_0.extremum_uniqueI
% 0.30/0.52 thf(fact_1785_le__0__eq,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.30/0.52 = ( N = zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_0_eq
% 0.30/0.52 thf(fact_1786_plus__nat_Oadd__0,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 0.30/0.52 = N ) ).
% 0.30/0.52
% 0.30/0.52 % plus_nat.add_0
% 0.30/0.52 thf(fact_1787_add__eq__self__zero,axiom,
% 0.30/0.52 ! [M: nat,N: nat] :
% 0.30/0.52 ( ( ( plus_plus_nat @ M @ N )
% 0.30/0.52 = M )
% 0.30/0.52 => ( N = zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_eq_self_zero
% 0.30/0.52 thf(fact_1788_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
% 0.30/0.52 ! [Uu: $o,Uv: $o,Uw: nat] :
% 0.30/0.52 ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT_internal.membermima.simps(1)
% 0.30/0.52 thf(fact_1789_minus__nat_Odiff__0,axiom,
% 0.30/0.52 ! [M: nat] :
% 0.30/0.52 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 0.30/0.52 = M ) ).
% 0.30/0.52
% 0.30/0.52 % minus_nat.diff_0
% 0.30/0.52 thf(fact_1790_diffs0__imp__equal,axiom,
% 0.30/0.52 ! [M: nat,N: nat] :
% 0.30/0.52 ( ( ( minus_minus_nat @ M @ N )
% 0.30/0.52 = zero_zero_nat )
% 0.30/0.52 => ( ( ( minus_minus_nat @ N @ M )
% 0.30/0.52 = zero_zero_nat )
% 0.30/0.52 => ( M = N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diffs0_imp_equal
% 0.30/0.52 thf(fact_1791_nat__mult__eq__cancel__disj,axiom,
% 0.30/0.52 ! [K: nat,M: nat,N: nat] :
% 0.30/0.52 ( ( ( times_times_nat @ K @ M )
% 0.30/0.52 = ( times_times_nat @ K @ N ) )
% 0.30/0.52 = ( ( K = zero_zero_nat )
% 0.30/0.52 | ( M = N ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % nat_mult_eq_cancel_disj
% 0.30/0.52 thf(fact_1792_mult__0,axiom,
% 0.30/0.52 ! [N: nat] :
% 0.30/0.52 ( ( times_times_nat @ zero_zero_nat @ N )
% 0.30/0.52 = zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % mult_0
% 0.30/0.52 thf(fact_1793_power__eq__imp__eq__base,axiom,
% 0.30/0.52 ! [A: real,N: nat,B: real] :
% 0.30/0.52 ( ( ( power_power_real @ A @ N )
% 0.30/0.52 = ( power_power_real @ B @ N ) )
% 0.30/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52 => ( A = B ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_eq_imp_eq_base
% 0.30/0.52 thf(fact_1794_power__eq__imp__eq__base,axiom,
% 0.30/0.52 ! [A: rat,N: nat,B: rat] :
% 0.30/0.52 ( ( ( power_power_rat @ A @ N )
% 0.30/0.52 = ( power_power_rat @ B @ N ) )
% 0.30/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52 => ( A = B ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_eq_imp_eq_base
% 0.30/0.52 thf(fact_1795_power__eq__imp__eq__base,axiom,
% 0.30/0.52 ! [A: nat,N: nat,B: nat] :
% 0.30/0.52 ( ( ( power_power_nat @ A @ N )
% 0.30/0.52 = ( power_power_nat @ B @ N ) )
% 0.30/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52 => ( A = B ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_eq_imp_eq_base
% 0.30/0.52 thf(fact_1796_power__eq__imp__eq__base,axiom,
% 0.30/0.52 ! [A: int,N: nat,B: int] :
% 0.30/0.52 ( ( ( power_power_int @ A @ N )
% 0.30/0.52 = ( power_power_int @ B @ N ) )
% 0.30/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52 => ( A = B ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % power_eq_imp_eq_base
% 0.30/0.52 thf(fact_1797_power__eq__iff__eq__base,axiom,
% 0.30/0.53 ! [N: nat,A: real,B: real] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 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 => ( ( ( power_power_real @ A @ N )
% 0.30/0.53 = ( power_power_real @ B @ N ) )
% 0.30/0.53 = ( A = B ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_eq_iff_eq_base
% 0.30/0.53 thf(fact_1798_power__eq__iff__eq__base,axiom,
% 0.30/0.53 ! [N: nat,A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 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 => ( ( ( power_power_rat @ A @ N )
% 0.30/0.53 = ( power_power_rat @ B @ N ) )
% 0.30/0.53 = ( A = B ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_eq_iff_eq_base
% 0.30/0.53 thf(fact_1799_power__eq__iff__eq__base,axiom,
% 0.30/0.53 ! [N: nat,A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 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 => ( ( ( power_power_nat @ A @ N )
% 0.30/0.53 = ( power_power_nat @ B @ N ) )
% 0.30/0.53 = ( A = B ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_eq_iff_eq_base
% 0.30/0.53 thf(fact_1800_power__eq__iff__eq__base,axiom,
% 0.30/0.53 ! [N: nat,A: int,B: int] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 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 => ( ( ( power_power_int @ A @ N )
% 0.30/0.53 = ( power_power_int @ B @ N ) )
% 0.30/0.53 = ( A = B ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_eq_iff_eq_base
% 0.30/0.53 thf(fact_1801_lambda__zero,axiom,
% 0.30/0.53 ( ( ^ [H: complex] : zero_zero_complex )
% 0.30/0.53 = ( times_times_complex @ zero_zero_complex ) ) ).
% 0.30/0.53
% 0.30/0.53 % lambda_zero
% 0.30/0.53 thf(fact_1802_lambda__zero,axiom,
% 0.30/0.53 ( ( ^ [H: real] : zero_zero_real )
% 0.30/0.53 = ( times_times_real @ zero_zero_real ) ) ).
% 0.30/0.53
% 0.30/0.53 % lambda_zero
% 0.30/0.53 thf(fact_1803_lambda__zero,axiom,
% 0.30/0.53 ( ( ^ [H: rat] : zero_zero_rat )
% 0.30/0.53 = ( times_times_rat @ zero_zero_rat ) ) ).
% 0.30/0.53
% 0.30/0.53 % lambda_zero
% 0.30/0.53 thf(fact_1804_lambda__zero,axiom,
% 0.30/0.53 ( ( ^ [H: nat] : zero_zero_nat )
% 0.30/0.53 = ( times_times_nat @ zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % lambda_zero
% 0.30/0.53 thf(fact_1805_lambda__zero,axiom,
% 0.30/0.53 ( ( ^ [H: int] : zero_zero_int )
% 0.30/0.53 = ( times_times_int @ zero_zero_int ) ) ).
% 0.30/0.53
% 0.30/0.53 % lambda_zero
% 0.30/0.53 thf(fact_1806_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
% 0.30/0.53 ! [X2: produc9072475918466114483BT_nat] :
% 0.30/0.53 ( ! [A4: $o,B4: $o,X5: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X5 ) )
% 0.30/0.53 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X5: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) @ X5 ) )
% 0.30/0.53 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X5: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) @ X5 ) )
% 0.30/0.53 => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ X5 ) )
% 0.30/0.53 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X5 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
% 0.30/0.53 thf(fact_1807_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
% 0.30/0.53 ! [X2: produc9072475918466114483BT_nat] :
% 0.30/0.53 ( ! [A4: $o,B4: $o,X5: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X5 ) )
% 0.30/0.53 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X5: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X5 ) )
% 0.30/0.53 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X5: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X5 ) )
% 0.30/0.53 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X5 ) )
% 0.30/0.53 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X5 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
% 0.30/0.53 thf(fact_1808_VEBT__internal_Omembermima_Ocases,axiom,
% 0.30/0.53 ! [X2: produc9072475918466114483BT_nat] :
% 0.30/0.53 ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
% 0.30/0.53 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
% 0.30/0.53 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT,X5: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ X5 ) )
% 0.30/0.53 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ X5 ) )
% 0.30/0.53 => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X5: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ X5 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % VEBT_internal.membermima.cases
% 0.30/0.53 thf(fact_1809_vebt__pred_Ocases,axiom,
% 0.30/0.53 ! [X2: produc9072475918466114483BT_nat] :
% 0.30/0.53 ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
% 0.30/0.53 => ( ! [A4: $o,Uw2: $o] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
% 0.30/0.53 => ( ! [A4: $o,B4: $o,Va3: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ Va3 ) ) ) )
% 0.30/0.53 => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb2: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Vb2 ) )
% 0.30/0.53 => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) @ Vf ) )
% 0.30/0.53 => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj ) )
% 0.30/0.53 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X5 ) ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % vebt_pred.cases
% 0.30/0.53 thf(fact_1810_vebt__succ_Ocases,axiom,
% 0.30/0.53 ! [X2: produc9072475918466114483BT_nat] :
% 0.30/0.53 ( ! [Uu2: $o,B4: $o] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B4 ) @ zero_zero_nat ) )
% 0.30/0.53 => ( ! [Uv2: $o,Uw2: $o,N2: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N2 ) ) )
% 0.30/0.53 => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va2 ) )
% 0.30/0.53 => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve ) )
% 0.30/0.53 => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi ) )
% 0.30/0.53 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X5 ) ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % vebt_succ.cases
% 0.30/0.53 thf(fact_1811_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
% 0.30/0.53 ! [A: $o,B: $o] :
% 0.30/0.53 ( ( vEBT_T_m_i_n_t @ ( vEBT_Leaf @ A @ B ) )
% 0.30/0.53 = ( plus_plus_nat @ one_one_nat @ ( if_nat @ A @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
% 0.30/0.53 thf(fact_1812_power__strict__mono,axiom,
% 0.30/0.53 ! [A: real,B: real,N: nat] :
% 0.30/0.53 ( ( ord_less_real @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_strict_mono
% 0.30/0.53 thf(fact_1813_power__strict__mono,axiom,
% 0.30/0.53 ! [A: rat,B: rat,N: nat] :
% 0.30/0.53 ( ( ord_less_rat @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_strict_mono
% 0.30/0.53 thf(fact_1814_power__strict__mono,axiom,
% 0.30/0.53 ! [A: nat,B: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_nat @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_strict_mono
% 0.30/0.53 thf(fact_1815_power__strict__mono,axiom,
% 0.30/0.53 ! [A: int,B: int,N: nat] :
% 0.30/0.53 ( ( ord_less_int @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_strict_mono
% 0.30/0.53 thf(fact_1816_div__positive,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 @ B @ A )
% 0.30/0.53 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_positive
% 0.30/0.53 thf(fact_1817_div__positive,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 @ B @ A )
% 0.30/0.53 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_positive
% 0.30/0.53 thf(fact_1818_unique__euclidean__semiring__numeral__class_Odiv__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 => ( ( divide_divide_nat @ A @ B )
% 0.30/0.53 = zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % unique_euclidean_semiring_numeral_class.div_less
% 0.30/0.53 thf(fact_1819_unique__euclidean__semiring__numeral__class_Odiv__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 => ( ( divide_divide_int @ A @ B )
% 0.30/0.53 = zero_zero_int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % unique_euclidean_semiring_numeral_class.div_less
% 0.30/0.53 thf(fact_1820_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.30/0.53 ! [C: nat,A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.53 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.30/0.53 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.30/0.53 thf(fact_1821_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.30/0.53 ! [C: int,A: int,B: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.53 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.30/0.53 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.30/0.53 thf(fact_1822_vebt__pred_Osimps_I2_J,axiom,
% 0.30/0.53 ! [A: $o,Uw: $o] :
% 0.30/0.53 ( ( A
% 0.30/0.53 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.53 = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.53 & ( ~ A
% 0.30/0.53 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.53 = none_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % vebt_pred.simps(2)
% 0.30/0.53 thf(fact_1823_vebt__mint_Osimps_I1_J,axiom,
% 0.30/0.53 ! [A: $o,B: $o] :
% 0.30/0.53 ( ( A
% 0.30/0.53 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 0.30/0.53 = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.53 & ( ~ A
% 0.30/0.53 => ( ( B
% 0.30/0.53 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 0.30/0.53 = ( some_nat @ one_one_nat ) ) )
% 0.30/0.53 & ( ~ B
% 0.30/0.53 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 0.30/0.53 = none_nat ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % vebt_mint.simps(1)
% 0.30/0.53 thf(fact_1824_vebt__succ_Osimps_I1_J,axiom,
% 0.30/0.53 ! [B: $o,Uu: $o] :
% 0.30/0.53 ( ( B
% 0.30/0.53 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
% 0.30/0.53 = ( some_nat @ one_one_nat ) ) )
% 0.30/0.53 & ( ~ B
% 0.30/0.53 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
% 0.30/0.53 = none_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % vebt_succ.simps(1)
% 0.30/0.53 thf(fact_1825_vebt__maxt_Osimps_I1_J,axiom,
% 0.30/0.53 ! [B: $o,A: $o] :
% 0.30/0.53 ( ( B
% 0.30/0.53 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 0.30/0.53 = ( some_nat @ one_one_nat ) ) )
% 0.30/0.53 & ( ~ B
% 0.30/0.53 => ( ( A
% 0.30/0.53 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 0.30/0.53 = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.53 & ( ~ A
% 0.30/0.53 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 0.30/0.53 = none_nat ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % vebt_maxt.simps(1)
% 0.30/0.53 thf(fact_1826_zero__le__numeral,axiom,
% 0.30/0.53 ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_numeral
% 0.30/0.53 thf(fact_1827_zero__le__numeral,axiom,
% 0.30/0.53 ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_numeral
% 0.30/0.53 thf(fact_1828_zero__le__numeral,axiom,
% 0.30/0.53 ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_numeral
% 0.30/0.53 thf(fact_1829_zero__le__numeral,axiom,
% 0.30/0.53 ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_numeral
% 0.30/0.53 thf(fact_1830_not__numeral__le__zero,axiom,
% 0.30/0.53 ! [N: num] :
% 0.30/0.53 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 0.30/0.53
% 0.30/0.53 % not_numeral_le_zero
% 0.30/0.53 thf(fact_1831_not__numeral__le__zero,axiom,
% 0.30/0.53 ! [N: num] :
% 0.30/0.53 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 0.30/0.53
% 0.30/0.53 % not_numeral_le_zero
% 0.30/0.53 thf(fact_1832_not__numeral__le__zero,axiom,
% 0.30/0.53 ! [N: num] :
% 0.30/0.53 ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % not_numeral_le_zero
% 0.30/0.53 thf(fact_1833_not__numeral__le__zero,axiom,
% 0.30/0.53 ! [N: num] :
% 0.30/0.53 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 0.30/0.53
% 0.30/0.53 % not_numeral_le_zero
% 0.30/0.53 thf(fact_1834_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.30/0.53 ! [A: real,B: real,C: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.53 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.30/0.53 thf(fact_1835_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.30/0.53 ! [A: rat,B: rat,C: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.30/0.53 thf(fact_1836_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.30/0.53 ! [A: nat,B: nat,C: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.53 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.30/0.53 thf(fact_1837_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.30/0.53 ! [A: int,B: int,C: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.53 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.30/0.53 thf(fact_1838_zero__le__mult__iff,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 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_eq_real @ A @ zero_zero_real )
% 0.30/0.53 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_mult_iff
% 0.30/0.53 thf(fact_1839_zero__le__mult__iff,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 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_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_mult_iff
% 0.30/0.53 thf(fact_1840_zero__le__mult__iff,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 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_eq_int @ A @ zero_zero_int )
% 0.30/0.53 & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_mult_iff
% 0.30/0.53 thf(fact_1841_mult__nonneg__nonpos2,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.30/0.53 => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_nonneg_nonpos2
% 0.30/0.53 thf(fact_1842_mult__nonneg__nonpos2,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.30/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_nonneg_nonpos2
% 0.30/0.53 thf(fact_1843_mult__nonneg__nonpos2,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 @ B @ zero_zero_nat )
% 0.30/0.53 => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_nonneg_nonpos2
% 0.30/0.53 thf(fact_1844_mult__nonneg__nonpos2,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 @ B @ zero_zero_int )
% 0.30/0.53 => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_nonneg_nonpos2
% 0.30/0.53 thf(fact_1845_mult__nonpos__nonneg,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.53 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_nonpos_nonneg
% 0.30/0.53 thf(fact_1846_mult__nonpos__nonneg,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_nonpos_nonneg
% 0.30/0.53 thf(fact_1847_mult__nonpos__nonneg,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.53 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_nonpos_nonneg
% 0.30/0.53 thf(fact_1848_mult__nonpos__nonneg,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.53 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_nonpos_nonneg
% 0.30/0.53 thf(fact_1849_mult__nonneg__nonpos,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.30/0.53 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_nonneg_nonpos
% 0.30/0.53 thf(fact_1850_mult__nonneg__nonpos,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.30/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_nonneg_nonpos
% 0.30/0.53 thf(fact_1851_mult__nonneg__nonpos,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 @ B @ zero_zero_nat )
% 0.30/0.53 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_nonneg_nonpos
% 0.30/0.53 thf(fact_1852_mult__nonneg__nonpos,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 @ B @ zero_zero_int )
% 0.30/0.53 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_nonneg_nonpos
% 0.30/0.53 thf(fact_1853_mult__nonneg__nonneg,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 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_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_nonneg_nonneg
% 0.30/0.53 thf(fact_1854_mult__nonneg__nonneg,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 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_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_nonneg_nonneg
% 0.30/0.53 thf(fact_1855_mult__nonneg__nonneg,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 @ zero_zero_nat @ B )
% 0.30/0.53 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_nonneg_nonneg
% 0.30/0.53 thf(fact_1856_mult__nonneg__nonneg,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 @ zero_zero_int @ B )
% 0.30/0.53 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_nonneg_nonneg
% 0.30/0.53 thf(fact_1857_split__mult__neg__le,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.30/0.53 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.53 & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
% 0.30/0.53 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% 0.30/0.53
% 0.30/0.53 % split_mult_neg_le
% 0.30/0.53 thf(fact_1858_split__mult__neg__le,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.30/0.53 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
% 0.30/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% 0.30/0.53
% 0.30/0.53 % split_mult_neg_le
% 0.30/0.53 thf(fact_1859_split__mult__neg__le,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 @ B @ zero_zero_nat ) )
% 0.30/0.53 | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.53 & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 0.30/0.53 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % split_mult_neg_le
% 0.30/0.53 thf(fact_1860_split__mult__neg__le,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 @ B @ zero_zero_int ) )
% 0.30/0.53 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.53 & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
% 0.30/0.53 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% 0.30/0.53
% 0.30/0.53 % split_mult_neg_le
% 0.30/0.53 thf(fact_1861_mult__le__0__iff,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.30/0.53 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.30/0.53 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.53 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_le_0_iff
% 0.30/0.53 thf(fact_1862_mult__le__0__iff,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.30/0.53 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.30/0.53 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_le_0_iff
% 0.30/0.53 thf(fact_1863_mult__le__0__iff,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 0.30/0.53 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 0.30/0.53 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.53 & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_le_0_iff
% 0.30/0.53 thf(fact_1864_mult__right__mono,axiom,
% 0.30/0.53 ! [A: real,B: real,C: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.53 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_right_mono
% 0.30/0.53 thf(fact_1865_mult__right__mono,axiom,
% 0.30/0.53 ! [A: rat,B: rat,C: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_right_mono
% 0.30/0.53 thf(fact_1866_mult__right__mono,axiom,
% 0.30/0.53 ! [A: nat,B: nat,C: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.53 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_right_mono
% 0.30/0.53 thf(fact_1867_mult__right__mono,axiom,
% 0.30/0.53 ! [A: int,B: int,C: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.53 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_right_mono
% 0.30/0.53 thf(fact_1868_mult__right__mono__neg,axiom,
% 0.30/0.53 ! [B: real,A: real,C: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ B @ A )
% 0.30/0.53 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.53 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_right_mono_neg
% 0.30/0.53 thf(fact_1869_mult__right__mono__neg,axiom,
% 0.30/0.53 ! [B: rat,A: rat,C: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.53 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_right_mono_neg
% 0.30/0.53 thf(fact_1870_mult__right__mono__neg,axiom,
% 0.30/0.53 ! [B: int,A: int,C: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ B @ A )
% 0.30/0.53 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.53 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_right_mono_neg
% 0.30/0.53 thf(fact_1871_mult__left__mono,axiom,
% 0.30/0.53 ! [A: real,B: real,C: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.53 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_left_mono
% 0.30/0.53 thf(fact_1872_mult__left__mono,axiom,
% 0.30/0.53 ! [A: rat,B: rat,C: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_left_mono
% 0.30/0.53 thf(fact_1873_mult__left__mono,axiom,
% 0.30/0.53 ! [A: nat,B: nat,C: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.53 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_left_mono
% 0.30/0.53 thf(fact_1874_mult__left__mono,axiom,
% 0.30/0.53 ! [A: int,B: int,C: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.53 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_left_mono
% 0.30/0.53 thf(fact_1875_mult__nonpos__nonpos,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.53 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.30/0.53 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_nonpos_nonpos
% 0.30/0.53 thf(fact_1876_mult__nonpos__nonpos,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.30/0.53 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_nonpos_nonpos
% 0.30/0.53 thf(fact_1877_mult__nonpos__nonpos,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.53 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.30/0.53 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_nonpos_nonpos
% 0.30/0.53 thf(fact_1878_mult__left__mono__neg,axiom,
% 0.30/0.53 ! [B: real,A: real,C: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ B @ A )
% 0.30/0.53 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.53 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_left_mono_neg
% 0.30/0.53 thf(fact_1879_mult__left__mono__neg,axiom,
% 0.30/0.53 ! [B: rat,A: rat,C: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.53 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_left_mono_neg
% 0.30/0.53 thf(fact_1880_mult__left__mono__neg,axiom,
% 0.30/0.53 ! [B: int,A: int,C: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ B @ A )
% 0.30/0.53 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.53 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_left_mono_neg
% 0.30/0.53 thf(fact_1881_split__mult__pos__le,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 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_eq_real @ A @ zero_zero_real )
% 0.30/0.53 & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.30/0.53 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % split_mult_pos_le
% 0.30/0.53 thf(fact_1882_split__mult__pos__le,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 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_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 0.30/0.53 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % split_mult_pos_le
% 0.30/0.53 thf(fact_1883_split__mult__pos__le,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 @ zero_zero_int @ B ) )
% 0.30/0.53 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.53 & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 0.30/0.53 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % split_mult_pos_le
% 0.30/0.53 thf(fact_1884_zero__le__square,axiom,
% 0.30/0.53 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_square
% 0.30/0.53 thf(fact_1885_zero__le__square,axiom,
% 0.30/0.53 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_square
% 0.30/0.53 thf(fact_1886_zero__le__square,axiom,
% 0.30/0.53 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_square
% 0.30/0.53 thf(fact_1887_mult__mono_H,axiom,
% 0.30/0.53 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_real @ C @ D )
% 0.30/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.53 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_mono'
% 0.30/0.53 thf(fact_1888_mult__mono_H,axiom,
% 0.30/0.53 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_rat @ C @ D )
% 0.30/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_mono'
% 0.30/0.53 thf(fact_1889_mult__mono_H,axiom,
% 0.30/0.53 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_nat @ C @ D )
% 0.30/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.53 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_mono'
% 0.30/0.53 thf(fact_1890_mult__mono_H,axiom,
% 0.30/0.53 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_int @ C @ D )
% 0.30/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.53 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_mono'
% 0.30/0.53 thf(fact_1891_mult__mono,axiom,
% 0.30/0.53 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_real @ C @ D )
% 0.30/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.53 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_mono
% 0.30/0.53 thf(fact_1892_mult__mono,axiom,
% 0.30/0.53 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_rat @ C @ D )
% 0.30/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_mono
% 0.30/0.53 thf(fact_1893_mult__mono,axiom,
% 0.30/0.53 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_nat @ C @ D )
% 0.30/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.53 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_mono
% 0.30/0.53 thf(fact_1894_mult__mono,axiom,
% 0.30/0.53 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_int @ C @ D )
% 0.30/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.53 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_mono
% 0.30/0.53 thf(fact_1895_not__numeral__less__zero,axiom,
% 0.30/0.53 ! [N: num] :
% 0.30/0.53 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 0.30/0.53
% 0.30/0.53 % not_numeral_less_zero
% 0.30/0.53 thf(fact_1896_not__numeral__less__zero,axiom,
% 0.30/0.53 ! [N: num] :
% 0.30/0.53 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 0.30/0.53
% 0.30/0.53 % not_numeral_less_zero
% 0.30/0.53 thf(fact_1897_not__numeral__less__zero,axiom,
% 0.30/0.53 ! [N: num] :
% 0.30/0.53 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % not_numeral_less_zero
% 0.30/0.53 thf(fact_1898_not__numeral__less__zero,axiom,
% 0.30/0.53 ! [N: num] :
% 0.30/0.53 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 0.30/0.53
% 0.30/0.53 % not_numeral_less_zero
% 0.30/0.53 thf(fact_1899_zero__less__numeral,axiom,
% 0.30/0.53 ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_numeral
% 0.30/0.53 thf(fact_1900_zero__less__numeral,axiom,
% 0.30/0.53 ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_numeral
% 0.30/0.53 thf(fact_1901_zero__less__numeral,axiom,
% 0.30/0.53 ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_numeral
% 0.30/0.53 thf(fact_1902_zero__less__numeral,axiom,
% 0.30/0.53 ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_numeral
% 0.30/0.53 thf(fact_1903_zero__less__one__class_Ozero__le__one,axiom,
% 0.30/0.53 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.30/0.53
% 0.30/0.53 % zero_less_one_class.zero_le_one
% 0.30/0.53 thf(fact_1904_zero__less__one__class_Ozero__le__one,axiom,
% 0.30/0.53 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.30/0.53
% 0.30/0.53 % zero_less_one_class.zero_le_one
% 0.30/0.53 thf(fact_1905_zero__less__one__class_Ozero__le__one,axiom,
% 0.30/0.53 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.30/0.53
% 0.30/0.53 % zero_less_one_class.zero_le_one
% 0.30/0.53 thf(fact_1906_zero__less__one__class_Ozero__le__one,axiom,
% 0.30/0.53 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.30/0.53
% 0.30/0.53 % zero_less_one_class.zero_le_one
% 0.30/0.53 thf(fact_1907_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.30/0.53 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.30/0.53
% 0.30/0.53 % linordered_nonzero_semiring_class.zero_le_one
% 0.30/0.53 thf(fact_1908_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.30/0.53 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.30/0.53
% 0.30/0.53 % linordered_nonzero_semiring_class.zero_le_one
% 0.30/0.53 thf(fact_1909_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.30/0.53 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.30/0.53
% 0.30/0.53 % linordered_nonzero_semiring_class.zero_le_one
% 0.30/0.53 thf(fact_1910_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.30/0.53 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.30/0.53
% 0.30/0.53 % linordered_nonzero_semiring_class.zero_le_one
% 0.30/0.53 thf(fact_1911_not__one__le__zero,axiom,
% 0.30/0.53 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 0.30/0.53
% 0.30/0.53 % not_one_le_zero
% 0.30/0.53 thf(fact_1912_not__one__le__zero,axiom,
% 0.30/0.53 ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.30/0.53
% 0.30/0.53 % not_one_le_zero
% 0.30/0.53 thf(fact_1913_not__one__le__zero,axiom,
% 0.30/0.53 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % not_one_le_zero
% 0.30/0.53 thf(fact_1914_not__one__le__zero,axiom,
% 0.30/0.53 ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 0.30/0.53
% 0.30/0.53 % not_one_le_zero
% 0.30/0.53 thf(fact_1915_add__decreasing,axiom,
% 0.30/0.53 ! [A: real,C: real,B: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.53 => ( ( ord_less_eq_real @ C @ B )
% 0.30/0.53 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_decreasing
% 0.30/0.53 thf(fact_1916_add__decreasing,axiom,
% 0.30/0.53 ! [A: rat,C: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53 => ( ( ord_less_eq_rat @ C @ B )
% 0.30/0.53 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_decreasing
% 0.30/0.53 thf(fact_1917_add__decreasing,axiom,
% 0.30/0.53 ! [A: nat,C: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.53 => ( ( ord_less_eq_nat @ C @ B )
% 0.30/0.53 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_decreasing
% 0.30/0.53 thf(fact_1918_add__decreasing,axiom,
% 0.30/0.53 ! [A: int,C: int,B: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.53 => ( ( ord_less_eq_int @ C @ B )
% 0.30/0.53 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_decreasing
% 0.30/0.53 thf(fact_1919_add__increasing,axiom,
% 0.30/0.53 ! [A: real,B: real,C: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53 => ( ( ord_less_eq_real @ B @ C )
% 0.30/0.53 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_increasing
% 0.30/0.53 thf(fact_1920_add__increasing,axiom,
% 0.30/0.53 ! [A: rat,B: rat,C: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53 => ( ( ord_less_eq_rat @ B @ C )
% 0.30/0.53 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_increasing
% 0.30/0.53 thf(fact_1921_add__increasing,axiom,
% 0.30/0.53 ! [A: nat,B: nat,C: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53 => ( ( ord_less_eq_nat @ B @ C )
% 0.30/0.53 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_increasing
% 0.30/0.53 thf(fact_1922_add__increasing,axiom,
% 0.30/0.53 ! [A: int,B: int,C: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53 => ( ( ord_less_eq_int @ B @ C )
% 0.30/0.53 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_increasing
% 0.30/0.53 thf(fact_1923_add__decreasing2,axiom,
% 0.30/0.53 ! [C: real,A: real,B: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.53 => ( ( ord_less_eq_real @ A @ B )
% 0.30/0.53 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_decreasing2
% 0.30/0.53 thf(fact_1924_add__decreasing2,axiom,
% 0.30/0.53 ! [C: rat,A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.53 => ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.53 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_decreasing2
% 0.30/0.53 thf(fact_1925_add__decreasing2,axiom,
% 0.30/0.53 ! [C: nat,A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 0.30/0.53 => ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.53 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_decreasing2
% 0.30/0.53 thf(fact_1926_add__decreasing2,axiom,
% 0.30/0.53 ! [C: int,A: int,B: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.53 => ( ( ord_less_eq_int @ A @ B )
% 0.30/0.53 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_decreasing2
% 0.30/0.53 thf(fact_1927_add__increasing2,axiom,
% 0.30/0.53 ! [C: real,B: real,A: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.53 => ( ( ord_less_eq_real @ B @ A )
% 0.30/0.53 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_increasing2
% 0.30/0.53 thf(fact_1928_add__increasing2,axiom,
% 0.30/0.53 ! [C: rat,B: rat,A: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.53 => ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.53 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_increasing2
% 0.30/0.53 thf(fact_1929_add__increasing2,axiom,
% 0.30/0.53 ! [C: nat,B: nat,A: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.53 => ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.53 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_increasing2
% 0.30/0.53 thf(fact_1930_add__increasing2,axiom,
% 0.30/0.53 ! [C: int,B: int,A: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.53 => ( ( ord_less_eq_int @ B @ A )
% 0.30/0.53 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_increasing2
% 0.30/0.53 thf(fact_1931_add__nonneg__nonneg,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 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_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_nonneg_nonneg
% 0.30/0.53 thf(fact_1932_add__nonneg__nonneg,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 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_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_nonneg_nonneg
% 0.30/0.53 thf(fact_1933_add__nonneg__nonneg,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 @ zero_zero_nat @ B )
% 0.30/0.53 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_nonneg_nonneg
% 0.30/0.53 thf(fact_1934_add__nonneg__nonneg,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 @ zero_zero_int @ B )
% 0.30/0.53 => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_nonneg_nonneg
% 0.30/0.53 thf(fact_1935_add__nonpos__nonpos,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.53 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.30/0.53 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_nonpos_nonpos
% 0.30/0.53 thf(fact_1936_add__nonpos__nonpos,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.30/0.53 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_nonpos_nonpos
% 0.30/0.53 thf(fact_1937_add__nonpos__nonpos,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.53 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.30/0.53 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_nonpos_nonpos
% 0.30/0.53 thf(fact_1938_add__nonpos__nonpos,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.53 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.30/0.53 => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_nonpos_nonpos
% 0.30/0.53 thf(fact_1939_add__nonneg__eq__0__iff,axiom,
% 0.30/0.53 ! [X2: real,Y3: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.53 => ( ( ( plus_plus_real @ X2 @ Y3 )
% 0.30/0.53 = zero_zero_real )
% 0.30/0.53 = ( ( X2 = zero_zero_real )
% 0.30/0.53 & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_nonneg_eq_0_iff
% 0.30/0.53 thf(fact_1940_add__nonneg__eq__0__iff,axiom,
% 0.30/0.53 ! [X2: rat,Y3: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.30/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.53 => ( ( ( plus_plus_rat @ X2 @ Y3 )
% 0.30/0.53 = zero_zero_rat )
% 0.30/0.53 = ( ( X2 = zero_zero_rat )
% 0.30/0.53 & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_nonneg_eq_0_iff
% 0.30/0.53 thf(fact_1941_add__nonneg__eq__0__iff,axiom,
% 0.30/0.53 ! [X2: nat,Y3: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
% 0.30/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.30/0.53 => ( ( ( plus_plus_nat @ X2 @ Y3 )
% 0.30/0.53 = zero_zero_nat )
% 0.30/0.53 = ( ( X2 = zero_zero_nat )
% 0.30/0.53 & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_nonneg_eq_0_iff
% 0.30/0.53 thf(fact_1942_add__nonneg__eq__0__iff,axiom,
% 0.30/0.53 ! [X2: int,Y3: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.53 => ( ( ( plus_plus_int @ X2 @ Y3 )
% 0.30/0.53 = zero_zero_int )
% 0.30/0.53 = ( ( X2 = zero_zero_int )
% 0.30/0.53 & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_nonneg_eq_0_iff
% 0.30/0.53 thf(fact_1943_add__nonpos__eq__0__iff,axiom,
% 0.30/0.53 ! [X2: real,Y3: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.30/0.53 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.30/0.53 => ( ( ( plus_plus_real @ X2 @ Y3 )
% 0.30/0.53 = zero_zero_real )
% 0.30/0.53 = ( ( X2 = zero_zero_real )
% 0.30/0.53 & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_nonpos_eq_0_iff
% 0.30/0.53 thf(fact_1944_add__nonpos__eq__0__iff,axiom,
% 0.30/0.53 ! [X2: rat,Y3: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 0.30/0.53 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 0.30/0.53 => ( ( ( plus_plus_rat @ X2 @ Y3 )
% 0.30/0.53 = zero_zero_rat )
% 0.30/0.53 = ( ( X2 = zero_zero_rat )
% 0.30/0.53 & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_nonpos_eq_0_iff
% 0.30/0.53 thf(fact_1945_add__nonpos__eq__0__iff,axiom,
% 0.30/0.53 ! [X2: nat,Y3: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
% 0.30/0.53 => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
% 0.30/0.53 => ( ( ( plus_plus_nat @ X2 @ Y3 )
% 0.30/0.53 = zero_zero_nat )
% 0.30/0.53 = ( ( X2 = zero_zero_nat )
% 0.30/0.53 & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_nonpos_eq_0_iff
% 0.30/0.53 thf(fact_1946_add__nonpos__eq__0__iff,axiom,
% 0.30/0.53 ! [X2: int,Y3: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ X2 @ zero_zero_int )
% 0.30/0.53 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 0.30/0.53 => ( ( ( plus_plus_int @ X2 @ Y3 )
% 0.30/0.53 = zero_zero_int )
% 0.30/0.53 = ( ( X2 = zero_zero_int )
% 0.30/0.53 & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_nonpos_eq_0_iff
% 0.30/0.53 thf(fact_1947_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.30/0.53 ! [A: real,B: real,C: real] :
% 0.30/0.53 ( ( ord_less_real @ A @ B )
% 0.30/0.53 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.53 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.30/0.53 thf(fact_1948_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.30/0.53 ! [A: rat,B: rat,C: rat] :
% 0.30/0.53 ( ( ord_less_rat @ A @ B )
% 0.30/0.53 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.53 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.30/0.53 thf(fact_1949_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.30/0.53 ! [A: nat,B: nat,C: nat] :
% 0.30/0.53 ( ( ord_less_nat @ A @ B )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.53 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.30/0.53 thf(fact_1950_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.30/0.53 ! [A: int,B: int,C: int] :
% 0.30/0.53 ( ( ord_less_int @ A @ B )
% 0.30/0.53 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.53 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.30/0.53 thf(fact_1951_mult__less__cancel__right__disj,axiom,
% 0.30/0.53 ! [A: real,C: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.30/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.53 & ( ord_less_real @ A @ B ) )
% 0.30/0.53 | ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.53 & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_cancel_right_disj
% 0.30/0.53 thf(fact_1952_mult__less__cancel__right__disj,axiom,
% 0.30/0.53 ! [A: rat,C: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.30/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.53 & ( ord_less_rat @ A @ B ) )
% 0.30/0.53 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.53 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_cancel_right_disj
% 0.30/0.53 thf(fact_1953_mult__less__cancel__right__disj,axiom,
% 0.30/0.53 ! [A: int,C: int,B: int] :
% 0.30/0.53 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.30/0.53 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.53 & ( ord_less_int @ A @ B ) )
% 0.30/0.53 | ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.53 & ( ord_less_int @ B @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_cancel_right_disj
% 0.30/0.53 thf(fact_1954_mult__strict__right__mono,axiom,
% 0.30/0.53 ! [A: real,B: real,C: real] :
% 0.30/0.53 ( ( ord_less_real @ A @ B )
% 0.30/0.53 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.53 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_strict_right_mono
% 0.30/0.53 thf(fact_1955_mult__strict__right__mono,axiom,
% 0.30/0.53 ! [A: rat,B: rat,C: rat] :
% 0.30/0.53 ( ( ord_less_rat @ A @ B )
% 0.30/0.53 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.53 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_strict_right_mono
% 0.30/0.53 thf(fact_1956_mult__strict__right__mono,axiom,
% 0.30/0.53 ! [A: nat,B: nat,C: nat] :
% 0.30/0.53 ( ( ord_less_nat @ A @ B )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.53 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_strict_right_mono
% 0.30/0.53 thf(fact_1957_mult__strict__right__mono,axiom,
% 0.30/0.53 ! [A: int,B: int,C: int] :
% 0.30/0.53 ( ( ord_less_int @ A @ B )
% 0.30/0.53 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.53 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_strict_right_mono
% 0.30/0.53 thf(fact_1958_mult__strict__right__mono__neg,axiom,
% 0.30/0.53 ! [B: real,A: real,C: real] :
% 0.30/0.53 ( ( ord_less_real @ B @ A )
% 0.30/0.53 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.53 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_strict_right_mono_neg
% 0.30/0.53 thf(fact_1959_mult__strict__right__mono__neg,axiom,
% 0.30/0.53 ! [B: rat,A: rat,C: rat] :
% 0.30/0.53 ( ( ord_less_rat @ B @ A )
% 0.30/0.53 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.53 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_strict_right_mono_neg
% 0.30/0.53 thf(fact_1960_mult__strict__right__mono__neg,axiom,
% 0.30/0.53 ! [B: int,A: int,C: int] :
% 0.30/0.53 ( ( ord_less_int @ B @ A )
% 0.30/0.53 => ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.53 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_strict_right_mono_neg
% 0.30/0.53 thf(fact_1961_mult__less__cancel__left__disj,axiom,
% 0.30/0.53 ! [C: real,A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.53 & ( ord_less_real @ A @ B ) )
% 0.30/0.53 | ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.53 & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_cancel_left_disj
% 0.30/0.53 thf(fact_1962_mult__less__cancel__left__disj,axiom,
% 0.30/0.53 ! [C: rat,A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.53 & ( ord_less_rat @ A @ B ) )
% 0.30/0.53 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.53 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_cancel_left_disj
% 0.30/0.53 thf(fact_1963_mult__less__cancel__left__disj,axiom,
% 0.30/0.53 ! [C: int,A: int,B: int] :
% 0.30/0.53 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.53 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.53 & ( ord_less_int @ A @ B ) )
% 0.30/0.53 | ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.53 & ( ord_less_int @ B @ A ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_cancel_left_disj
% 0.30/0.53 thf(fact_1964_mult__strict__left__mono,axiom,
% 0.30/0.53 ! [A: real,B: real,C: real] :
% 0.30/0.53 ( ( ord_less_real @ A @ B )
% 0.30/0.53 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.53 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_strict_left_mono
% 0.30/0.53 thf(fact_1965_mult__strict__left__mono,axiom,
% 0.30/0.53 ! [A: rat,B: rat,C: rat] :
% 0.30/0.53 ( ( ord_less_rat @ A @ B )
% 0.30/0.53 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.53 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_strict_left_mono
% 0.30/0.53 thf(fact_1966_mult__strict__left__mono,axiom,
% 0.30/0.53 ! [A: nat,B: nat,C: nat] :
% 0.30/0.53 ( ( ord_less_nat @ A @ B )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.53 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_strict_left_mono
% 0.30/0.53 thf(fact_1967_mult__strict__left__mono,axiom,
% 0.30/0.53 ! [A: int,B: int,C: int] :
% 0.30/0.53 ( ( ord_less_int @ A @ B )
% 0.30/0.53 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.53 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_strict_left_mono
% 0.30/0.53 thf(fact_1968_mult__strict__left__mono__neg,axiom,
% 0.30/0.53 ! [B: real,A: real,C: real] :
% 0.30/0.53 ( ( ord_less_real @ B @ A )
% 0.30/0.53 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.53 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_strict_left_mono_neg
% 0.30/0.53 thf(fact_1969_mult__strict__left__mono__neg,axiom,
% 0.30/0.53 ! [B: rat,A: rat,C: rat] :
% 0.30/0.53 ( ( ord_less_rat @ B @ A )
% 0.30/0.53 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.53 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_strict_left_mono_neg
% 0.30/0.53 thf(fact_1970_mult__strict__left__mono__neg,axiom,
% 0.30/0.53 ! [B: int,A: int,C: int] :
% 0.30/0.53 ( ( ord_less_int @ B @ A )
% 0.30/0.53 => ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.53 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_strict_left_mono_neg
% 0.30/0.53 thf(fact_1971_mult__less__cancel__left__pos,axiom,
% 0.30/0.53 ! [C: real,A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.53 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.53 = ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_cancel_left_pos
% 0.30/0.53 thf(fact_1972_mult__less__cancel__left__pos,axiom,
% 0.30/0.53 ! [C: rat,A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.53 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.53 = ( ord_less_rat @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_cancel_left_pos
% 0.30/0.53 thf(fact_1973_mult__less__cancel__left__pos,axiom,
% 0.30/0.53 ! [C: int,A: int,B: int] :
% 0.30/0.53 ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.53 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.53 = ( ord_less_int @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_cancel_left_pos
% 0.30/0.53 thf(fact_1974_mult__less__cancel__left__neg,axiom,
% 0.30/0.53 ! [C: real,A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.53 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.53 = ( ord_less_real @ B @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_cancel_left_neg
% 0.30/0.53 thf(fact_1975_mult__less__cancel__left__neg,axiom,
% 0.30/0.53 ! [C: rat,A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.53 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.53 = ( ord_less_rat @ B @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_cancel_left_neg
% 0.30/0.53 thf(fact_1976_mult__less__cancel__left__neg,axiom,
% 0.30/0.53 ! [C: int,A: int,B: int] :
% 0.30/0.53 ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.53 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.53 = ( ord_less_int @ B @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_cancel_left_neg
% 0.30/0.53 thf(fact_1977_zero__less__mult__pos2,axiom,
% 0.30/0.53 ! [B: real,A: real] :
% 0.30/0.53 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
% 0.30/0.53 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_mult_pos2
% 0.30/0.53 thf(fact_1978_zero__less__mult__pos2,axiom,
% 0.30/0.53 ! [B: rat,A: rat] :
% 0.30/0.53 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
% 0.30/0.53 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_mult_pos2
% 0.30/0.53 thf(fact_1979_zero__less__mult__pos2,axiom,
% 0.30/0.53 ! [B: nat,A: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.53 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_mult_pos2
% 0.30/0.53 thf(fact_1980_zero__less__mult__pos2,axiom,
% 0.30/0.53 ! [B: int,A: int] :
% 0.30/0.53 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
% 0.30/0.53 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_mult_pos2
% 0.30/0.53 thf(fact_1981_zero__less__mult__pos,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.30/0.53 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_mult_pos
% 0.30/0.53 thf(fact_1982_zero__less__mult__pos,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.30/0.53 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_mult_pos
% 0.30/0.53 thf(fact_1983_zero__less__mult__pos,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.53 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_mult_pos
% 0.30/0.53 thf(fact_1984_zero__less__mult__pos,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.30/0.53 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_mult_pos
% 0.30/0.53 thf(fact_1985_zero__less__mult__iff,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.30/0.53 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53 & ( ord_less_real @ zero_zero_real @ B ) )
% 0.30/0.53 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.53 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_mult_iff
% 0.30/0.53 thf(fact_1986_zero__less__mult__iff,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.30/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 0.30/0.53 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.53 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_mult_iff
% 0.30/0.53 thf(fact_1987_zero__less__mult__iff,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.30/0.53 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53 & ( ord_less_int @ zero_zero_int @ B ) )
% 0.30/0.53 | ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.53 & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_mult_iff
% 0.30/0.53 thf(fact_1988_mult__pos__neg2,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 @ B @ zero_zero_real )
% 0.30/0.53 => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_pos_neg2
% 0.30/0.53 thf(fact_1989_mult__pos__neg2,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 @ B @ zero_zero_rat )
% 0.30/0.53 => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_pos_neg2
% 0.30/0.53 thf(fact_1990_mult__pos__neg2,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.53 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.30/0.53 => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_pos_neg2
% 0.30/0.53 thf(fact_1991_mult__pos__neg2,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.30/0.53 => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_pos_neg2
% 0.30/0.53 thf(fact_1992_mult__pos__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 @ zero_zero_real @ B )
% 0.30/0.53 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_pos_pos
% 0.30/0.53 thf(fact_1993_mult__pos__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 @ zero_zero_rat @ B )
% 0.30/0.53 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_pos_pos
% 0.30/0.53 thf(fact_1994_mult__pos__pos,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.53 => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_pos_pos
% 0.30/0.53 thf(fact_1995_mult__pos__pos,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.53 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_pos_pos
% 0.30/0.53 thf(fact_1996_mult__pos__neg,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 @ B @ zero_zero_real )
% 0.30/0.53 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_pos_neg
% 0.30/0.53 thf(fact_1997_mult__pos__neg,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 @ B @ zero_zero_rat )
% 0.30/0.53 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_pos_neg
% 0.30/0.53 thf(fact_1998_mult__pos__neg,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.53 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.30/0.53 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_pos_neg
% 0.30/0.53 thf(fact_1999_mult__pos__neg,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.30/0.53 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_pos_neg
% 0.30/0.53 thf(fact_2000_mult__neg__pos,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 @ zero_zero_real @ B )
% 0.30/0.53 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_neg_pos
% 0.30/0.53 thf(fact_2001_mult__neg__pos,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 @ zero_zero_rat @ B )
% 0.30/0.53 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_neg_pos
% 0.30/0.53 thf(fact_2002_mult__neg__pos,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.53 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_neg_pos
% 0.30/0.53 thf(fact_2003_mult__neg__pos,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.53 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.53 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_neg_pos
% 0.30/0.53 thf(fact_2004_mult__less__0__iff,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.30/0.53 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53 & ( ord_less_real @ B @ zero_zero_real ) )
% 0.30/0.53 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.53 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_0_iff
% 0.30/0.53 thf(fact_2005_mult__less__0__iff,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.30/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 0.30/0.53 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.53 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_0_iff
% 0.30/0.53 thf(fact_2006_mult__less__0__iff,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 0.30/0.53 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53 & ( ord_less_int @ B @ zero_zero_int ) )
% 0.30/0.53 | ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.53 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_0_iff
% 0.30/0.53 thf(fact_2007_not__square__less__zero,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 0.30/0.53
% 0.30/0.53 % not_square_less_zero
% 0.30/0.53 thf(fact_2008_not__square__less__zero,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 0.30/0.53
% 0.30/0.53 % not_square_less_zero
% 0.30/0.53 thf(fact_2009_not__square__less__zero,axiom,
% 0.30/0.53 ! [A: int] :
% 0.30/0.53 ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 0.30/0.53
% 0.30/0.53 % not_square_less_zero
% 0.30/0.53 thf(fact_2010_mult__neg__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 @ B @ zero_zero_real )
% 0.30/0.53 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_neg_neg
% 0.30/0.53 thf(fact_2011_mult__neg__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 @ B @ zero_zero_rat )
% 0.30/0.53 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_neg_neg
% 0.30/0.53 thf(fact_2012_mult__neg__neg,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.53 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.30/0.53 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_neg_neg
% 0.30/0.53 thf(fact_2013_less__numeral__extra_I1_J,axiom,
% 0.30/0.53 ord_less_real @ zero_zero_real @ one_one_real ).
% 0.30/0.53
% 0.30/0.53 % less_numeral_extra(1)
% 0.30/0.53 thf(fact_2014_less__numeral__extra_I1_J,axiom,
% 0.30/0.53 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.30/0.53
% 0.30/0.53 % less_numeral_extra(1)
% 0.30/0.53 thf(fact_2015_less__numeral__extra_I1_J,axiom,
% 0.30/0.53 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.30/0.53
% 0.30/0.53 % less_numeral_extra(1)
% 0.30/0.53 thf(fact_2016_less__numeral__extra_I1_J,axiom,
% 0.30/0.53 ord_less_int @ zero_zero_int @ one_one_int ).
% 0.30/0.53
% 0.30/0.53 % less_numeral_extra(1)
% 0.30/0.53 thf(fact_2017_zero__less__one,axiom,
% 0.30/0.53 ord_less_real @ zero_zero_real @ one_one_real ).
% 0.30/0.53
% 0.30/0.53 % zero_less_one
% 0.30/0.53 thf(fact_2018_zero__less__one,axiom,
% 0.30/0.53 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.30/0.53
% 0.30/0.53 % zero_less_one
% 0.30/0.53 thf(fact_2019_zero__less__one,axiom,
% 0.30/0.53 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.30/0.53
% 0.30/0.53 % zero_less_one
% 0.30/0.53 thf(fact_2020_zero__less__one,axiom,
% 0.30/0.53 ord_less_int @ zero_zero_int @ one_one_int ).
% 0.30/0.53
% 0.30/0.53 % zero_less_one
% 0.30/0.53 thf(fact_2021_not__one__less__zero,axiom,
% 0.30/0.53 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 0.30/0.53
% 0.30/0.53 % not_one_less_zero
% 0.30/0.53 thf(fact_2022_not__one__less__zero,axiom,
% 0.30/0.53 ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.30/0.53
% 0.30/0.53 % not_one_less_zero
% 0.30/0.53 thf(fact_2023_not__one__less__zero,axiom,
% 0.30/0.53 ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % not_one_less_zero
% 0.30/0.53 thf(fact_2024_not__one__less__zero,axiom,
% 0.30/0.53 ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 0.30/0.53
% 0.30/0.53 % not_one_less_zero
% 0.30/0.53 thf(fact_2025_add__neg__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 @ B @ zero_zero_real )
% 0.30/0.53 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_neg_neg
% 0.30/0.53 thf(fact_2026_add__neg__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 @ B @ zero_zero_rat )
% 0.30/0.53 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_neg_neg
% 0.30/0.53 thf(fact_2027_add__neg__neg,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.30/0.53 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.30/0.53 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_neg_neg
% 0.30/0.53 thf(fact_2028_add__neg__neg,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.53 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.30/0.53 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_neg_neg
% 0.30/0.53 thf(fact_2029_add__pos__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 @ zero_zero_real @ B )
% 0.30/0.53 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_pos_pos
% 0.30/0.53 thf(fact_2030_add__pos__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 @ zero_zero_rat @ B )
% 0.30/0.53 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_pos_pos
% 0.30/0.53 thf(fact_2031_add__pos__pos,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.53 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_pos_pos
% 0.30/0.53 thf(fact_2032_add__pos__pos,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.53 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_pos_pos
% 0.30/0.53 thf(fact_2033_canonically__ordered__monoid__add__class_OlessE,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_nat @ A @ B )
% 0.30/0.53 => ~ ! [C3: nat] :
% 0.30/0.53 ( ( B
% 0.30/0.53 = ( plus_plus_nat @ A @ C3 ) )
% 0.30/0.53 => ( C3 = zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % canonically_ordered_monoid_add_class.lessE
% 0.30/0.53 thf(fact_2034_pos__add__strict,axiom,
% 0.30/0.53 ! [A: real,B: real,C: real] :
% 0.30/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53 => ( ( ord_less_real @ B @ C )
% 0.30/0.53 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % pos_add_strict
% 0.30/0.53 thf(fact_2035_pos__add__strict,axiom,
% 0.30/0.53 ! [A: rat,B: rat,C: rat] :
% 0.30/0.53 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53 => ( ( ord_less_rat @ B @ C )
% 0.30/0.53 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % pos_add_strict
% 0.30/0.53 thf(fact_2036_pos__add__strict,axiom,
% 0.30/0.53 ! [A: nat,B: nat,C: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.53 => ( ( ord_less_nat @ B @ C )
% 0.30/0.53 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % pos_add_strict
% 0.30/0.53 thf(fact_2037_pos__add__strict,axiom,
% 0.30/0.53 ! [A: int,B: int,C: int] :
% 0.30/0.53 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53 => ( ( ord_less_int @ B @ C )
% 0.30/0.53 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % pos_add_strict
% 0.30/0.53 thf(fact_2038_add__less__zeroD,axiom,
% 0.30/0.53 ! [X2: real,Y3: real] :
% 0.30/0.53 ( ( ord_less_real @ ( plus_plus_real @ X2 @ Y3 ) @ zero_zero_real )
% 0.30/0.53 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.30/0.53 | ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_less_zeroD
% 0.30/0.53 thf(fact_2039_add__less__zeroD,axiom,
% 0.30/0.53 ! [X2: rat,Y3: rat] :
% 0.30/0.53 ( ( ord_less_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ zero_zero_rat )
% 0.30/0.53 => ( ( ord_less_rat @ X2 @ zero_zero_rat )
% 0.30/0.53 | ( ord_less_rat @ Y3 @ zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_less_zeroD
% 0.30/0.53 thf(fact_2040_add__less__zeroD,axiom,
% 0.30/0.53 ! [X2: int,Y3: int] :
% 0.30/0.53 ( ( ord_less_int @ ( plus_plus_int @ X2 @ Y3 ) @ zero_zero_int )
% 0.30/0.53 => ( ( ord_less_int @ X2 @ zero_zero_int )
% 0.30/0.53 | ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_less_zeroD
% 0.30/0.53 thf(fact_2041_le__iff__diff__le__0,axiom,
% 0.30/0.53 ( ord_less_eq_real
% 0.30/0.53 = ( ^ [A3: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % le_iff_diff_le_0
% 0.30/0.53 thf(fact_2042_le__iff__diff__le__0,axiom,
% 0.30/0.53 ( ord_less_eq_rat
% 0.30/0.53 = ( ^ [A3: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % le_iff_diff_le_0
% 0.30/0.53 thf(fact_2043_le__iff__diff__le__0,axiom,
% 0.30/0.53 ( ord_less_eq_int
% 0.30/0.53 = ( ^ [A3: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % le_iff_diff_le_0
% 0.30/0.53 thf(fact_2044_divide__le__0__iff,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.30/0.53 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.30/0.53 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.53 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_le_0_iff
% 0.30/0.53 thf(fact_2045_divide__le__0__iff,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 0.30/0.53 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.30/0.53 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_le_0_iff
% 0.30/0.53 thf(fact_2046_divide__right__mono,axiom,
% 0.30/0.53 ! [A: real,B: real,C: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.53 => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_right_mono
% 0.30/0.53 thf(fact_2047_divide__right__mono,axiom,
% 0.30/0.53 ! [A: rat,B: rat,C: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.53 => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_right_mono
% 0.30/0.53 thf(fact_2048_zero__le__divide__iff,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 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_eq_real @ A @ zero_zero_real )
% 0.30/0.53 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_divide_iff
% 0.30/0.53 thf(fact_2049_zero__le__divide__iff,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 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_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_divide_iff
% 0.30/0.53 thf(fact_2050_divide__nonneg__nonneg,axiom,
% 0.30/0.53 ! [X2: real,Y3: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.53 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_nonneg_nonneg
% 0.30/0.53 thf(fact_2051_divide__nonneg__nonneg,axiom,
% 0.30/0.53 ! [X2: rat,Y3: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.30/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.53 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_nonneg_nonneg
% 0.30/0.53 thf(fact_2052_divide__nonneg__nonpos,axiom,
% 0.30/0.53 ! [X2: real,Y3: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.53 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.30/0.53 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_nonneg_nonpos
% 0.30/0.53 thf(fact_2053_divide__nonneg__nonpos,axiom,
% 0.30/0.53 ! [X2: rat,Y3: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.30/0.53 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 0.30/0.53 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_nonneg_nonpos
% 0.30/0.53 thf(fact_2054_divide__nonpos__nonneg,axiom,
% 0.30/0.53 ! [X2: real,Y3: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.30/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.53 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_nonpos_nonneg
% 0.30/0.53 thf(fact_2055_divide__nonpos__nonneg,axiom,
% 0.30/0.53 ! [X2: rat,Y3: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 0.30/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.53 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_nonpos_nonneg
% 0.30/0.53 thf(fact_2056_divide__nonpos__nonpos,axiom,
% 0.30/0.53 ! [X2: real,Y3: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.30/0.53 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.30/0.53 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_nonpos_nonpos
% 0.30/0.53 thf(fact_2057_divide__nonpos__nonpos,axiom,
% 0.30/0.53 ! [X2: rat,Y3: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 0.30/0.53 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 0.30/0.53 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_nonpos_nonpos
% 0.30/0.53 thf(fact_2058_divide__right__mono__neg,axiom,
% 0.30/0.53 ! [A: real,B: real,C: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.53 => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_right_mono_neg
% 0.30/0.53 thf(fact_2059_divide__right__mono__neg,axiom,
% 0.30/0.53 ! [A: rat,B: rat,C: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.53 => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_right_mono_neg
% 0.30/0.53 thf(fact_2060_less__iff__diff__less__0,axiom,
% 0.30/0.53 ( ord_less_real
% 0.30/0.53 = ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_iff_diff_less_0
% 0.30/0.53 thf(fact_2061_less__iff__diff__less__0,axiom,
% 0.30/0.53 ( ord_less_rat
% 0.30/0.53 = ( ^ [A3: rat,B2: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_iff_diff_less_0
% 0.30/0.53 thf(fact_2062_less__iff__diff__less__0,axiom,
% 0.30/0.53 ( ord_less_int
% 0.30/0.53 = ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_iff_diff_less_0
% 0.30/0.53 thf(fact_2063_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I1_J,axiom,
% 0.30/0.53 ! [A: $o,B: $o,X2: nat] :
% 0.30/0.53 ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Leaf @ A @ B ) @ X2 )
% 0.30/0.53 = one_one_nat ) ).
% 0.30/0.53
% 0.30/0.53 % T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
% 0.30/0.53 thf(fact_2064_divide__strict__right__mono__neg,axiom,
% 0.30/0.53 ! [B: real,A: real,C: real] :
% 0.30/0.53 ( ( ord_less_real @ B @ A )
% 0.30/0.53 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.53 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_strict_right_mono_neg
% 0.30/0.53 thf(fact_2065_divide__strict__right__mono__neg,axiom,
% 0.30/0.53 ! [B: rat,A: rat,C: rat] :
% 0.30/0.53 ( ( ord_less_rat @ B @ A )
% 0.30/0.53 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.53 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_strict_right_mono_neg
% 0.30/0.53 thf(fact_2066_divide__strict__right__mono,axiom,
% 0.30/0.53 ! [A: real,B: real,C: real] :
% 0.30/0.53 ( ( ord_less_real @ A @ B )
% 0.30/0.53 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.53 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_strict_right_mono
% 0.30/0.53 thf(fact_2067_divide__strict__right__mono,axiom,
% 0.30/0.53 ! [A: rat,B: rat,C: rat] :
% 0.30/0.53 ( ( ord_less_rat @ A @ B )
% 0.30/0.53 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.53 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_strict_right_mono
% 0.30/0.53 thf(fact_2068_zero__less__divide__iff,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.53 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53 & ( ord_less_real @ zero_zero_real @ B ) )
% 0.30/0.53 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.53 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_divide_iff
% 0.30/0.53 thf(fact_2069_zero__less__divide__iff,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 0.30/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 0.30/0.53 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.53 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_divide_iff
% 0.30/0.53 thf(fact_2070_divide__less__cancel,axiom,
% 0.30/0.53 ! [A: real,C: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.30/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.53 => ( ord_less_real @ A @ B ) )
% 0.30/0.53 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.53 => ( ord_less_real @ B @ A ) )
% 0.30/0.53 & ( C != zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_less_cancel
% 0.30/0.53 thf(fact_2071_divide__less__cancel,axiom,
% 0.30/0.53 ! [A: rat,C: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.53 => ( ord_less_rat @ A @ B ) )
% 0.30/0.53 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.53 => ( ord_less_rat @ B @ A ) )
% 0.30/0.53 & ( C != zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_less_cancel
% 0.30/0.53 thf(fact_2072_divide__less__0__iff,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.30/0.53 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53 & ( ord_less_real @ B @ zero_zero_real ) )
% 0.30/0.53 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.53 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_less_0_iff
% 0.30/0.53 thf(fact_2073_divide__less__0__iff,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 0.30/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 0.30/0.53 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.53 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_less_0_iff
% 0.30/0.53 thf(fact_2074_divide__pos__pos,axiom,
% 0.30/0.53 ! [X2: real,Y3: real] :
% 0.30/0.53 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.53 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.53 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_pos_pos
% 0.30/0.53 thf(fact_2075_divide__pos__pos,axiom,
% 0.30/0.53 ! [X2: rat,Y3: rat] :
% 0.30/0.53 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 0.30/0.53 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.30/0.53 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_pos_pos
% 0.30/0.53 thf(fact_2076_divide__pos__neg,axiom,
% 0.30/0.53 ! [X2: real,Y3: real] :
% 0.30/0.53 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.53 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.30/0.53 => ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_pos_neg
% 0.30/0.53 thf(fact_2077_divide__pos__neg,axiom,
% 0.30/0.53 ! [X2: rat,Y3: rat] :
% 0.30/0.53 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 0.30/0.53 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.30/0.53 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_pos_neg
% 0.30/0.53 thf(fact_2078_divide__neg__pos,axiom,
% 0.30/0.53 ! [X2: real,Y3: real] :
% 0.30/0.53 ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.30/0.53 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.53 => ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_neg_pos
% 0.30/0.53 thf(fact_2079_divide__neg__pos,axiom,
% 0.30/0.53 ! [X2: rat,Y3: rat] :
% 0.30/0.53 ( ( ord_less_rat @ X2 @ zero_zero_rat )
% 0.30/0.53 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.30/0.53 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_neg_pos
% 0.30/0.53 thf(fact_2080_divide__neg__neg,axiom,
% 0.30/0.53 ! [X2: real,Y3: real] :
% 0.30/0.53 ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.30/0.53 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.30/0.53 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_neg_neg
% 0.30/0.53 thf(fact_2081_divide__neg__neg,axiom,
% 0.30/0.53 ! [X2: rat,Y3: rat] :
% 0.30/0.53 ( ( ord_less_rat @ X2 @ zero_zero_rat )
% 0.30/0.53 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.30/0.53 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_neg_neg
% 0.30/0.53 thf(fact_2082_power__mono,axiom,
% 0.30/0.53 ! [A: real,B: real,N: nat] :
% 0.30/0.53 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_mono
% 0.30/0.53 thf(fact_2083_power__mono,axiom,
% 0.30/0.53 ! [A: rat,B: rat,N: nat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_mono
% 0.30/0.53 thf(fact_2084_power__mono,axiom,
% 0.30/0.53 ! [A: nat,B: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_mono
% 0.30/0.53 thf(fact_2085_power__mono,axiom,
% 0.30/0.53 ! [A: int,B: int,N: nat] :
% 0.30/0.53 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_mono
% 0.30/0.53 thf(fact_2086_zero__le__power,axiom,
% 0.30/0.53 ! [A: real,N: nat] :
% 0.30/0.53 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_power
% 0.30/0.53 thf(fact_2087_zero__le__power,axiom,
% 0.30/0.53 ! [A: rat,N: nat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_power
% 0.30/0.53 thf(fact_2088_zero__le__power,axiom,
% 0.30/0.53 ! [A: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_power
% 0.30/0.53 thf(fact_2089_zero__le__power,axiom,
% 0.30/0.53 ! [A: int,N: nat] :
% 0.30/0.53 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_power
% 0.30/0.53 thf(fact_2090_zero__less__power,axiom,
% 0.30/0.53 ! [A: real,N: nat] :
% 0.30/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53 => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_power
% 0.30/0.53 thf(fact_2091_zero__less__power,axiom,
% 0.30/0.53 ! [A: rat,N: nat] :
% 0.30/0.53 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53 => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_power
% 0.30/0.53 thf(fact_2092_zero__less__power,axiom,
% 0.30/0.53 ! [A: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.53 => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_power
% 0.30/0.53 thf(fact_2093_zero__less__power,axiom,
% 0.30/0.53 ! [A: int,N: nat] :
% 0.30/0.53 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53 => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_power
% 0.30/0.53 thf(fact_2094_nonzero__eq__divide__eq,axiom,
% 0.30/0.53 ! [C: complex,A: complex,B: complex] :
% 0.30/0.53 ( ( C != zero_zero_complex )
% 0.30/0.53 => ( ( A
% 0.30/0.53 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.30/0.53 = ( ( times_times_complex @ A @ C )
% 0.30/0.53 = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_eq_divide_eq
% 0.30/0.53 thf(fact_2095_nonzero__eq__divide__eq,axiom,
% 0.30/0.53 ! [C: real,A: real,B: real] :
% 0.30/0.53 ( ( C != zero_zero_real )
% 0.30/0.53 => ( ( A
% 0.30/0.53 = ( divide_divide_real @ B @ C ) )
% 0.30/0.53 = ( ( times_times_real @ A @ C )
% 0.30/0.53 = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_eq_divide_eq
% 0.30/0.53 thf(fact_2096_nonzero__eq__divide__eq,axiom,
% 0.30/0.53 ! [C: rat,A: rat,B: rat] :
% 0.30/0.53 ( ( C != zero_zero_rat )
% 0.30/0.53 => ( ( A
% 0.30/0.53 = ( divide_divide_rat @ B @ C ) )
% 0.30/0.53 = ( ( times_times_rat @ A @ C )
% 0.30/0.53 = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_eq_divide_eq
% 0.30/0.53 thf(fact_2097_nonzero__divide__eq__eq,axiom,
% 0.30/0.53 ! [C: complex,B: complex,A: complex] :
% 0.30/0.53 ( ( C != zero_zero_complex )
% 0.30/0.53 => ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.30/0.53 = A )
% 0.30/0.53 = ( B
% 0.30/0.53 = ( times_times_complex @ A @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_divide_eq_eq
% 0.30/0.53 thf(fact_2098_nonzero__divide__eq__eq,axiom,
% 0.30/0.53 ! [C: real,B: real,A: real] :
% 0.30/0.53 ( ( C != zero_zero_real )
% 0.30/0.53 => ( ( ( divide_divide_real @ B @ C )
% 0.30/0.53 = A )
% 0.30/0.53 = ( B
% 0.30/0.53 = ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_divide_eq_eq
% 0.30/0.53 thf(fact_2099_nonzero__divide__eq__eq,axiom,
% 0.30/0.53 ! [C: rat,B: rat,A: rat] :
% 0.30/0.53 ( ( C != zero_zero_rat )
% 0.30/0.53 => ( ( ( divide_divide_rat @ B @ C )
% 0.30/0.53 = A )
% 0.30/0.53 = ( B
% 0.30/0.53 = ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_divide_eq_eq
% 0.30/0.53 thf(fact_2100_eq__divide__imp,axiom,
% 0.30/0.53 ! [C: complex,A: complex,B: complex] :
% 0.30/0.53 ( ( C != zero_zero_complex )
% 0.30/0.53 => ( ( ( times_times_complex @ A @ C )
% 0.30/0.53 = B )
% 0.30/0.53 => ( A
% 0.30/0.53 = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % eq_divide_imp
% 0.30/0.53 thf(fact_2101_eq__divide__imp,axiom,
% 0.30/0.53 ! [C: real,A: real,B: real] :
% 0.30/0.53 ( ( C != zero_zero_real )
% 0.30/0.53 => ( ( ( times_times_real @ A @ C )
% 0.30/0.53 = B )
% 0.30/0.53 => ( A
% 0.30/0.53 = ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % eq_divide_imp
% 0.30/0.53 thf(fact_2102_eq__divide__imp,axiom,
% 0.30/0.53 ! [C: rat,A: rat,B: rat] :
% 0.30/0.53 ( ( C != zero_zero_rat )
% 0.30/0.53 => ( ( ( times_times_rat @ A @ C )
% 0.30/0.53 = B )
% 0.30/0.53 => ( A
% 0.30/0.53 = ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % eq_divide_imp
% 0.30/0.53 thf(fact_2103_divide__eq__imp,axiom,
% 0.30/0.53 ! [C: complex,B: complex,A: complex] :
% 0.30/0.53 ( ( C != zero_zero_complex )
% 0.30/0.53 => ( ( B
% 0.30/0.53 = ( times_times_complex @ A @ C ) )
% 0.30/0.53 => ( ( divide1717551699836669952omplex @ B @ C )
% 0.30/0.53 = A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_eq_imp
% 0.30/0.53 thf(fact_2104_divide__eq__imp,axiom,
% 0.30/0.53 ! [C: real,B: real,A: real] :
% 0.30/0.53 ( ( C != zero_zero_real )
% 0.30/0.53 => ( ( B
% 0.30/0.53 = ( times_times_real @ A @ C ) )
% 0.30/0.53 => ( ( divide_divide_real @ B @ C )
% 0.30/0.53 = A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_eq_imp
% 0.30/0.53 thf(fact_2105_divide__eq__imp,axiom,
% 0.30/0.53 ! [C: rat,B: rat,A: rat] :
% 0.30/0.53 ( ( C != zero_zero_rat )
% 0.30/0.53 => ( ( B
% 0.30/0.53 = ( times_times_rat @ A @ C ) )
% 0.30/0.53 => ( ( divide_divide_rat @ B @ C )
% 0.30/0.53 = A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_eq_imp
% 0.30/0.53 thf(fact_2106_eq__divide__eq,axiom,
% 0.30/0.53 ! [A: complex,B: complex,C: complex] :
% 0.30/0.53 ( ( A
% 0.30/0.53 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.30/0.53 = ( ( ( C != zero_zero_complex )
% 0.30/0.53 => ( ( times_times_complex @ A @ C )
% 0.30/0.53 = B ) )
% 0.30/0.53 & ( ( C = zero_zero_complex )
% 0.30/0.53 => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % eq_divide_eq
% 0.30/0.53 thf(fact_2107_eq__divide__eq,axiom,
% 0.30/0.53 ! [A: real,B: real,C: real] :
% 0.30/0.53 ( ( A
% 0.30/0.53 = ( divide_divide_real @ B @ C ) )
% 0.30/0.53 = ( ( ( C != zero_zero_real )
% 0.30/0.53 => ( ( times_times_real @ A @ C )
% 0.30/0.53 = B ) )
% 0.30/0.53 & ( ( C = zero_zero_real )
% 0.30/0.53 => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % eq_divide_eq
% 0.30/0.53 thf(fact_2108_eq__divide__eq,axiom,
% 0.30/0.53 ! [A: rat,B: rat,C: rat] :
% 0.30/0.53 ( ( A
% 0.30/0.53 = ( divide_divide_rat @ B @ C ) )
% 0.30/0.53 = ( ( ( C != zero_zero_rat )
% 0.30/0.53 => ( ( times_times_rat @ A @ C )
% 0.30/0.53 = B ) )
% 0.30/0.53 & ( ( C = zero_zero_rat )
% 0.30/0.53 => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % eq_divide_eq
% 0.30/0.53 thf(fact_2109_divide__eq__eq,axiom,
% 0.30/0.53 ! [B: complex,C: complex,A: complex] :
% 0.30/0.53 ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.30/0.53 = A )
% 0.30/0.53 = ( ( ( C != zero_zero_complex )
% 0.30/0.53 => ( B
% 0.30/0.53 = ( times_times_complex @ A @ C ) ) )
% 0.30/0.53 & ( ( C = zero_zero_complex )
% 0.30/0.53 => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_eq_eq
% 0.30/0.53 thf(fact_2110_divide__eq__eq,axiom,
% 0.30/0.53 ! [B: real,C: real,A: real] :
% 0.30/0.53 ( ( ( divide_divide_real @ B @ C )
% 0.30/0.53 = A )
% 0.30/0.53 = ( ( ( C != zero_zero_real )
% 0.30/0.53 => ( B
% 0.30/0.53 = ( times_times_real @ A @ C ) ) )
% 0.30/0.53 & ( ( C = zero_zero_real )
% 0.30/0.53 => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_eq_eq
% 0.30/0.53 thf(fact_2111_divide__eq__eq,axiom,
% 0.30/0.53 ! [B: rat,C: rat,A: rat] :
% 0.30/0.53 ( ( ( divide_divide_rat @ B @ C )
% 0.30/0.53 = A )
% 0.30/0.53 = ( ( ( C != zero_zero_rat )
% 0.30/0.53 => ( B
% 0.30/0.53 = ( times_times_rat @ A @ C ) ) )
% 0.30/0.53 & ( ( C = zero_zero_rat )
% 0.30/0.53 => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_eq_eq
% 0.30/0.53 thf(fact_2112_frac__eq__eq,axiom,
% 0.30/0.53 ! [Y3: complex,Z: complex,X2: complex,W: complex] :
% 0.30/0.53 ( ( Y3 != zero_zero_complex )
% 0.30/0.53 => ( ( Z != zero_zero_complex )
% 0.30/0.53 => ( ( ( divide1717551699836669952omplex @ X2 @ Y3 )
% 0.30/0.53 = ( divide1717551699836669952omplex @ W @ Z ) )
% 0.30/0.53 = ( ( times_times_complex @ X2 @ Z )
% 0.30/0.53 = ( times_times_complex @ W @ Y3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % frac_eq_eq
% 0.30/0.53 thf(fact_2113_frac__eq__eq,axiom,
% 0.30/0.53 ! [Y3: real,Z: real,X2: real,W: real] :
% 0.30/0.53 ( ( Y3 != zero_zero_real )
% 0.30/0.53 => ( ( Z != zero_zero_real )
% 0.30/0.53 => ( ( ( divide_divide_real @ X2 @ Y3 )
% 0.30/0.53 = ( divide_divide_real @ W @ Z ) )
% 0.30/0.53 = ( ( times_times_real @ X2 @ Z )
% 0.30/0.53 = ( times_times_real @ W @ Y3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % frac_eq_eq
% 0.30/0.53 thf(fact_2114_frac__eq__eq,axiom,
% 0.30/0.53 ! [Y3: rat,Z: rat,X2: rat,W: rat] :
% 0.30/0.53 ( ( Y3 != zero_zero_rat )
% 0.30/0.53 => ( ( Z != zero_zero_rat )
% 0.30/0.53 => ( ( ( divide_divide_rat @ X2 @ Y3 )
% 0.30/0.53 = ( divide_divide_rat @ W @ Z ) )
% 0.30/0.53 = ( ( times_times_rat @ X2 @ Z )
% 0.30/0.53 = ( times_times_rat @ W @ Y3 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % frac_eq_eq
% 0.30/0.53 thf(fact_2115_right__inverse__eq,axiom,
% 0.30/0.53 ! [B: complex,A: complex] :
% 0.30/0.53 ( ( B != zero_zero_complex )
% 0.30/0.53 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.30/0.53 = one_one_complex )
% 0.30/0.53 = ( A = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % right_inverse_eq
% 0.30/0.53 thf(fact_2116_right__inverse__eq,axiom,
% 0.30/0.53 ! [B: real,A: real] :
% 0.30/0.53 ( ( B != zero_zero_real )
% 0.30/0.53 => ( ( ( divide_divide_real @ A @ B )
% 0.30/0.53 = one_one_real )
% 0.30/0.53 = ( A = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % right_inverse_eq
% 0.30/0.53 thf(fact_2117_right__inverse__eq,axiom,
% 0.30/0.53 ! [B: rat,A: rat] :
% 0.30/0.53 ( ( B != zero_zero_rat )
% 0.30/0.53 => ( ( ( divide_divide_rat @ A @ B )
% 0.30/0.53 = one_one_rat )
% 0.30/0.53 = ( A = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % right_inverse_eq
% 0.30/0.53 thf(fact_2118_length__pos__if__in__set,axiom,
% 0.30/0.53 ! [X2: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
% 0.30/0.53 ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
% 0.30/0.53 => ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % length_pos_if_in_set
% 0.30/0.53 thf(fact_2119_length__pos__if__in__set,axiom,
% 0.30/0.53 ! [X2: complex,Xs2: list_complex] :
% 0.30/0.53 ( ( member_complex @ X2 @ ( set_complex2 @ Xs2 ) )
% 0.30/0.53 => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % length_pos_if_in_set
% 0.30/0.53 thf(fact_2120_length__pos__if__in__set,axiom,
% 0.30/0.53 ! [X2: real,Xs2: list_real] :
% 0.30/0.53 ( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
% 0.30/0.53 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % length_pos_if_in_set
% 0.30/0.53 thf(fact_2121_length__pos__if__in__set,axiom,
% 0.30/0.53 ! [X2: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
% 0.30/0.53 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.30/0.53 => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % length_pos_if_in_set
% 0.30/0.53 thf(fact_2122_length__pos__if__in__set,axiom,
% 0.30/0.53 ! [X2: $o,Xs2: list_o] :
% 0.30/0.53 ( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
% 0.30/0.53 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % length_pos_if_in_set
% 0.30/0.53 thf(fact_2123_length__pos__if__in__set,axiom,
% 0.30/0.53 ! [X2: nat,Xs2: list_nat] :
% 0.30/0.53 ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
% 0.30/0.53 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % length_pos_if_in_set
% 0.30/0.53 thf(fact_2124_length__pos__if__in__set,axiom,
% 0.30/0.53 ! [X2: int,Xs2: list_int] :
% 0.30/0.53 ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
% 0.30/0.53 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % length_pos_if_in_set
% 0.30/0.53 thf(fact_2125_power__0,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ( ( power_power_rat @ A @ zero_zero_nat )
% 0.30/0.53 = one_one_rat ) ).
% 0.30/0.53
% 0.30/0.53 % power_0
% 0.30/0.53 thf(fact_2126_power__0,axiom,
% 0.30/0.53 ! [A: nat] :
% 0.30/0.53 ( ( power_power_nat @ A @ zero_zero_nat )
% 0.30/0.53 = one_one_nat ) ).
% 0.30/0.53
% 0.30/0.53 % power_0
% 0.30/0.53 thf(fact_2127_power__0,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ( ( power_power_real @ A @ zero_zero_nat )
% 0.30/0.53 = one_one_real ) ).
% 0.30/0.53
% 0.30/0.53 % power_0
% 0.30/0.53 thf(fact_2128_power__0,axiom,
% 0.30/0.53 ! [A: int] :
% 0.30/0.53 ( ( power_power_int @ A @ zero_zero_nat )
% 0.30/0.53 = one_one_int ) ).
% 0.30/0.53
% 0.30/0.53 % power_0
% 0.30/0.53 thf(fact_2129_power__0,axiom,
% 0.30/0.53 ! [A: complex] :
% 0.30/0.53 ( ( power_power_complex @ A @ zero_zero_nat )
% 0.30/0.53 = one_one_complex ) ).
% 0.30/0.53
% 0.30/0.53 % power_0
% 0.30/0.53 thf(fact_2130_less__Suc__eq__0__disj,axiom,
% 0.30/0.53 ! [M: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.30/0.53 = ( ( M = zero_zero_nat )
% 0.30/0.53 | ? [J3: nat] :
% 0.30/0.53 ( ( M
% 0.30/0.53 = ( suc @ J3 ) )
% 0.30/0.53 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_Suc_eq_0_disj
% 0.30/0.53 thf(fact_2131_gr0__implies__Suc,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53 => ? [M4: nat] :
% 0.30/0.53 ( N
% 0.30/0.53 = ( suc @ M4 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % gr0_implies_Suc
% 0.30/0.53 thf(fact_2132_All__less__Suc2,axiom,
% 0.30/0.53 ! [N: nat,P: nat > $o] :
% 0.30/0.53 ( ( ! [I4: nat] :
% 0.30/0.53 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.30/0.53 => ( P @ I4 ) ) )
% 0.30/0.53 = ( ( P @ zero_zero_nat )
% 0.30/0.53 & ! [I4: nat] :
% 0.30/0.53 ( ( ord_less_nat @ I4 @ N )
% 0.30/0.53 => ( P @ ( suc @ I4 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % All_less_Suc2
% 0.30/0.53 thf(fact_2133_gr0__conv__Suc,axiom,
% 0.30/0.53 ! [N: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53 = ( ? [M3: nat] :
% 0.30/0.53 ( N
% 0.30/0.53 = ( suc @ M3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % gr0_conv_Suc
% 0.30/0.53 thf(fact_2134_Ex__less__Suc2,axiom,
% 0.30/0.53 ! [N: nat,P: nat > $o] :
% 0.30/0.53 ( ( ? [I4: nat] :
% 0.30/0.53 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.30/0.53 & ( P @ I4 ) ) )
% 0.30/0.53 = ( ( P @ zero_zero_nat )
% 0.30/0.53 | ? [I4: nat] :
% 0.30/0.53 ( ( ord_less_nat @ I4 @ N )
% 0.30/0.53 & ( P @ ( suc @ I4 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % Ex_less_Suc2
% 0.30/0.53 thf(fact_2135_one__is__add,axiom,
% 0.30/0.53 ! [M: nat,N: nat] :
% 0.30/0.53 ( ( ( suc @ zero_zero_nat )
% 0.30/0.53 = ( plus_plus_nat @ M @ N ) )
% 0.30/0.53 = ( ( ( M
% 0.30/0.53 = ( suc @ zero_zero_nat ) )
% 0.30/0.53 & ( N = zero_zero_nat ) )
% 0.30/0.53 | ( ( M = zero_zero_nat )
% 0.30/0.53 & ( N
% 0.30/0.53 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % one_is_add
% 0.30/0.53 thf(fact_2136_add__is__1,axiom,
% 0.30/0.53 ! [M: nat,N: nat] :
% 0.30/0.53 ( ( ( plus_plus_nat @ M @ N )
% 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 & ( N = zero_zero_nat ) )
% 0.30/0.53 | ( ( M = zero_zero_nat )
% 0.30/0.53 & ( N
% 0.30/0.53 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_is_1
% 0.30/0.53 thf(fact_2137_option_Osize_I4_J,axiom,
% 0.30/0.53 ! [X22: product_prod_nat_nat] :
% 0.30/0.53 ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.30/0.53 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % option.size(4)
% 0.30/0.53 thf(fact_2138_option_Osize_I4_J,axiom,
% 0.30/0.53 ! [X22: nat] :
% 0.30/0.53 ( ( size_size_option_nat @ ( some_nat @ X22 ) )
% 0.30/0.53 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % option.size(4)
% 0.30/0.53 thf(fact_2139_option_Osize_I4_J,axiom,
% 0.30/0.53 ! [X22: num] :
% 0.30/0.53 ( ( size_size_option_num @ ( some_num @ X22 ) )
% 0.30/0.53 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % option.size(4)
% 0.30/0.53 thf(fact_2140_ex__least__nat__le,axiom,
% 0.30/0.53 ! [P: nat > $o,N: nat] :
% 0.30/0.53 ( ( P @ N )
% 0.30/0.53 => ( ~ ( P @ zero_zero_nat )
% 0.30/0.53 => ? [K3: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ K3 @ N )
% 0.30/0.53 & ! [I: nat] :
% 0.30/0.53 ( ( ord_less_nat @ I @ K3 )
% 0.30/0.53 => ~ ( P @ I ) )
% 0.30/0.53 & ( P @ K3 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % ex_least_nat_le
% 0.30/0.53 thf(fact_2141_less__imp__add__positive,axiom,
% 0.30/0.53 ! [I2: nat,J: nat] :
% 0.30/0.53 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.53 => ? [K3: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.30/0.53 & ( ( plus_plus_nat @ I2 @ K3 )
% 0.30/0.53 = J ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_imp_add_positive
% 0.30/0.53 thf(fact_2142_One__nat__def,axiom,
% 0.30/0.53 ( one_one_nat
% 0.30/0.53 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % One_nat_def
% 0.30/0.53 thf(fact_2143_option_Osize_I3_J,axiom,
% 0.30/0.53 ( ( size_size_option_nat @ none_nat )
% 0.30/0.53 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % option.size(3)
% 0.30/0.53 thf(fact_2144_option_Osize_I3_J,axiom,
% 0.30/0.53 ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
% 0.30/0.53 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % option.size(3)
% 0.30/0.53 thf(fact_2145_option_Osize_I3_J,axiom,
% 0.30/0.53 ( ( size_size_option_num @ none_num )
% 0.30/0.53 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % option.size(3)
% 0.30/0.53 thf(fact_2146_diff__less,axiom,
% 0.30/0.53 ! [N: nat,M: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.53 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % diff_less
% 0.30/0.53 thf(fact_2147_mult__less__mono2,axiom,
% 0.30/0.53 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.53 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.53 => ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_mono2
% 0.30/0.53 thf(fact_2148_mult__less__mono1,axiom,
% 0.30/0.53 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.53 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.53 => ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_mono1
% 0.30/0.53 thf(fact_2149_nat__mult__less__cancel1,axiom,
% 0.30/0.53 ! [K: nat,M: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.53 => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.53 = ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nat_mult_less_cancel1
% 0.30/0.53 thf(fact_2150_nat__mult__eq__cancel1,axiom,
% 0.30/0.53 ! [K: nat,M: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.53 => ( ( ( times_times_nat @ K @ M )
% 0.30/0.53 = ( times_times_nat @ K @ N ) )
% 0.30/0.53 = ( M = N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nat_mult_eq_cancel1
% 0.30/0.53 thf(fact_2151_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 0.30/0.53 ! [M: nat,N: nat] :
% 0.30/0.53 ( ( ( divide_divide_nat @ M @ N )
% 0.30/0.53 = zero_zero_nat )
% 0.30/0.53 = ( ( ord_less_nat @ M @ N )
% 0.30/0.53 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % Euclidean_Division.div_eq_0_iff
% 0.30/0.53 thf(fact_2152_diff__add__0,axiom,
% 0.30/0.53 ! [N: nat,M: nat] :
% 0.30/0.53 ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % diff_add_0
% 0.30/0.53 thf(fact_2153_nat__power__less__imp__less,axiom,
% 0.30/0.53 ! [I2: nat,M: nat,N: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ I2 )
% 0.30/0.53 => ( ( ord_less_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N ) )
% 0.30/0.53 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nat_power_less_imp_less
% 0.30/0.53 thf(fact_2154_mult__eq__self__implies__10,axiom,
% 0.30/0.53 ! [M: nat,N: nat] :
% 0.30/0.53 ( ( M
% 0.30/0.53 = ( times_times_nat @ M @ N ) )
% 0.30/0.53 => ( ( N = one_one_nat )
% 0.30/0.53 | ( M = zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_eq_self_implies_10
% 0.30/0.53 thf(fact_2155_vebt__insert_Osimps_I2_J,axiom,
% 0.30/0.53 ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
% 0.30/0.53 ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S ) @ X2 )
% 0.30/0.53 = ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S ) ) ).
% 0.30/0.53
% 0.30/0.53 % vebt_insert.simps(2)
% 0.30/0.53 thf(fact_2156_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_2157_vebt__pred_Osimps_I3_J,axiom,
% 0.30/0.53 ! [B: $o,A: $o,Va: nat] :
% 0.30/0.53 ( ( B
% 0.30/0.53 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.53 = ( some_nat @ one_one_nat ) ) )
% 0.30/0.53 & ( ~ B
% 0.30/0.53 => ( ( A
% 0.30/0.53 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.53 = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.53 & ( ~ A
% 0.30/0.53 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.53 = none_nat ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % vebt_pred.simps(3)
% 0.30/0.53 thf(fact_2158_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
% 0.30/0.53 ! [X2: vEBT_VEBT] :
% 0.30/0.53 ( ( X2
% 0.30/0.53 != ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.53 => ( ! [Uv2: $o] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.30/0.53 => ( ! [Uu2: $o] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.30/0.53 => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.30/0.53 => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
% 0.30/0.53 thf(fact_2159_div__geq,axiom,
% 0.30/0.53 ! [N: nat,M: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53 => ( ~ ( ord_less_nat @ M @ N )
% 0.30/0.53 => ( ( divide_divide_nat @ M @ N )
% 0.30/0.53 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_geq
% 0.30/0.53 thf(fact_2160_VEBT__internal_OminNull_Oelims_I3_J,axiom,
% 0.30/0.53 ! [X2: vEBT_VEBT] :
% 0.30/0.53 ( ~ ( vEBT_VEBT_minNull @ X2 )
% 0.30/0.53 => ( ! [Uv2: $o] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.30/0.53 => ( ! [Uu2: $o] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.30/0.53 => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.53 ( X2
% 0.30/0.53 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % VEBT_internal.minNull.elims(3)
% 0.30/0.53 thf(fact_2161_vebt__succ_Osimps_I2_J,axiom,
% 0.30/0.53 ! [Uv: $o,Uw: $o,N: nat] :
% 0.30/0.53 ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
% 0.30/0.53 = none_nat ) ).
% 0.30/0.53
% 0.30/0.53 % vebt_succ.simps(2)
% 0.30/0.53 thf(fact_2162_mult__less__le__imp__less,axiom,
% 0.30/0.53 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.53 ( ( ord_less_real @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_real @ C @ D )
% 0.30/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.53 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_le_imp_less
% 0.30/0.53 thf(fact_2163_mult__less__le__imp__less,axiom,
% 0.30/0.53 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.53 ( ( ord_less_rat @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_rat @ C @ D )
% 0.30/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.53 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_le_imp_less
% 0.30/0.53 thf(fact_2164_mult__less__le__imp__less,axiom,
% 0.30/0.53 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.53 ( ( ord_less_nat @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_nat @ C @ D )
% 0.30/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.53 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_le_imp_less
% 0.30/0.53 thf(fact_2165_mult__less__le__imp__less,axiom,
% 0.30/0.53 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.53 ( ( ord_less_int @ A @ B )
% 0.30/0.53 => ( ( ord_less_eq_int @ C @ D )
% 0.30/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.53 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_le_imp_less
% 0.30/0.53 thf(fact_2166_mult__le__less__imp__less,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_real @ C @ D )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_less_imp_less
% 0.30/0.54 thf(fact_2167_mult__le__less__imp__less,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_rat @ C @ D )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_less_imp_less
% 0.30/0.54 thf(fact_2168_mult__le__less__imp__less,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_nat @ C @ D )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.54 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_less_imp_less
% 0.30/0.54 thf(fact_2169_mult__le__less__imp__less,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_int @ C @ D )
% 0.30/0.54 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_less_imp_less
% 0.30/0.54 thf(fact_2170_mult__right__le__imp__le,axiom,
% 0.30/0.54 ! [A: real,C: real,B: real] :
% 0.30/0.54 ( ( ord_less_eq_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_eq_real @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_le_imp_le
% 0.30/0.54 thf(fact_2171_mult__right__le__imp__le,axiom,
% 0.30/0.54 ! [A: rat,C: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_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_eq_rat @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_le_imp_le
% 0.30/0.54 thf(fact_2172_mult__right__le__imp__le,axiom,
% 0.30/0.54 ! [A: nat,C: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.54 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_le_imp_le
% 0.30/0.54 thf(fact_2173_mult__right__le__imp__le,axiom,
% 0.30/0.54 ! [A: int,C: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_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_eq_int @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_le_imp_le
% 0.30/0.54 thf(fact_2174_mult__left__le__imp__le,axiom,
% 0.30/0.54 ! [C: real,A: real,B: real] :
% 0.30/0.54 ( ( ord_less_eq_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_eq_real @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_le_imp_le
% 0.30/0.54 thf(fact_2175_mult__left__le__imp__le,axiom,
% 0.30/0.54 ! [C: rat,A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_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_eq_rat @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_le_imp_le
% 0.30/0.54 thf(fact_2176_mult__left__le__imp__le,axiom,
% 0.30/0.54 ! [C: nat,A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.54 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_le_imp_le
% 0.30/0.54 thf(fact_2177_mult__left__le__imp__le,axiom,
% 0.30/0.54 ! [C: int,A: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_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_eq_int @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_le_imp_le
% 0.30/0.54 thf(fact_2178_mult__le__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_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.54 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_left_pos
% 0.30/0.54 thf(fact_2179_mult__le__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_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.54 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_left_pos
% 0.30/0.54 thf(fact_2180_mult__le__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_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.54 = ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_left_pos
% 0.30/0.54 thf(fact_2181_mult__le__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_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.54 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_left_neg
% 0.30/0.54 thf(fact_2182_mult__le__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_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.54 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_left_neg
% 0.30/0.54 thf(fact_2183_mult__le__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_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.54 = ( ord_less_eq_int @ B @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_left_neg
% 0.30/0.54 thf(fact_2184_mult__less__cancel__right,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_eq_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_real @ A @ B ) )
% 0.30/0.54 & ( ( ord_less_eq_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
% 0.30/0.54 thf(fact_2185_mult__less__cancel__right,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_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_rat @ A @ B ) )
% 0.30/0.54 & ( ( ord_less_eq_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
% 0.30/0.54 thf(fact_2186_mult__less__cancel__right,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_eq_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_int @ A @ B ) )
% 0.30/0.54 & ( ( ord_less_eq_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
% 0.30/0.54 thf(fact_2187_mult__strict__mono_H,axiom,
% 0.30/0.54 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.54 ( ( ord_less_real @ A @ B )
% 0.30/0.54 => ( ( ord_less_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_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_mono'
% 0.30/0.54 thf(fact_2188_mult__strict__mono_H,axiom,
% 0.30/0.54 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.54 ( ( ord_less_rat @ A @ B )
% 0.30/0.54 => ( ( ord_less_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_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_mono'
% 0.30/0.54 thf(fact_2189_mult__strict__mono_H,axiom,
% 0.30/0.54 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.54 ( ( ord_less_nat @ A @ B )
% 0.30/0.54 => ( ( ord_less_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_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_mono'
% 0.30/0.54 thf(fact_2190_mult__strict__mono_H,axiom,
% 0.30/0.54 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.54 ( ( ord_less_int @ A @ B )
% 0.30/0.54 => ( ( ord_less_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_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_mono'
% 0.30/0.54 thf(fact_2191_mult__right__less__imp__less,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_eq_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_less_imp_less
% 0.30/0.54 thf(fact_2192_mult__right__less__imp__less,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_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_less_imp_less
% 0.30/0.54 thf(fact_2193_mult__right__less__imp__less,axiom,
% 0.30/0.54 ! [A: nat,C: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.54 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_less_imp_less
% 0.30/0.54 thf(fact_2194_mult__right__less__imp__less,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_eq_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_int @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_less_imp_less
% 0.30/0.54 thf(fact_2195_mult__less__cancel__left,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_eq_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_real @ A @ B ) )
% 0.30/0.54 & ( ( ord_less_eq_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
% 0.30/0.54 thf(fact_2196_mult__less__cancel__left,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_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_rat @ A @ B ) )
% 0.30/0.54 & ( ( ord_less_eq_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
% 0.30/0.54 thf(fact_2197_mult__less__cancel__left,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_eq_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_int @ A @ B ) )
% 0.30/0.54 & ( ( ord_less_eq_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
% 0.30/0.54 thf(fact_2198_mult__strict__mono,axiom,
% 0.30/0.54 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.54 ( ( ord_less_real @ A @ B )
% 0.30/0.54 => ( ( ord_less_real @ C @ D )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_mono
% 0.30/0.54 thf(fact_2199_mult__strict__mono,axiom,
% 0.30/0.54 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.54 ( ( ord_less_rat @ A @ B )
% 0.30/0.54 => ( ( ord_less_rat @ C @ D )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_mono
% 0.30/0.54 thf(fact_2200_mult__strict__mono,axiom,
% 0.30/0.54 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.54 ( ( ord_less_nat @ A @ B )
% 0.30/0.54 => ( ( ord_less_nat @ C @ D )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.54 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_mono
% 0.30/0.54 thf(fact_2201_mult__strict__mono,axiom,
% 0.30/0.54 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.54 ( ( ord_less_int @ A @ B )
% 0.30/0.54 => ( ( ord_less_int @ C @ D )
% 0.30/0.54 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_mono
% 0.30/0.54 thf(fact_2202_mult__left__less__imp__less,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_eq_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_less_imp_less
% 0.30/0.54 thf(fact_2203_mult__left__less__imp__less,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_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_less_imp_less
% 0.30/0.54 thf(fact_2204_mult__left__less__imp__less,axiom,
% 0.30/0.54 ! [C: nat,A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.54 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_less_imp_less
% 0.30/0.54 thf(fact_2205_mult__left__less__imp__less,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_eq_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_int @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_less_imp_less
% 0.30/0.54 thf(fact_2206_mult__le__cancel__right,axiom,
% 0.30/0.54 ! [A: real,C: real,B: real] :
% 0.30/0.54 ( ( ord_less_eq_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_eq_real @ A @ B ) )
% 0.30/0.54 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_right
% 0.30/0.54 thf(fact_2207_mult__le__cancel__right,axiom,
% 0.30/0.54 ! [A: rat,C: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_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_eq_rat @ A @ B ) )
% 0.30/0.54 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_right
% 0.30/0.54 thf(fact_2208_mult__le__cancel__right,axiom,
% 0.30/0.54 ! [A: int,C: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_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_eq_int @ A @ B ) )
% 0.30/0.54 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.54 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_right
% 0.30/0.54 thf(fact_2209_mult__le__cancel__left,axiom,
% 0.30/0.54 ! [C: real,A: real,B: real] :
% 0.30/0.54 ( ( ord_less_eq_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_eq_real @ A @ B ) )
% 0.30/0.54 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_left
% 0.30/0.54 thf(fact_2210_mult__le__cancel__left,axiom,
% 0.30/0.54 ! [C: rat,A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_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_eq_rat @ A @ B ) )
% 0.30/0.54 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_left
% 0.30/0.54 thf(fact_2211_mult__le__cancel__left,axiom,
% 0.30/0.54 ! [C: int,A: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_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_eq_int @ A @ B ) )
% 0.30/0.54 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.54 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_left
% 0.30/0.54 thf(fact_2212_VEBT__internal_OminNull_Oelims_I2_J,axiom,
% 0.30/0.54 ! [X2: vEBT_VEBT] :
% 0.30/0.54 ( ( vEBT_VEBT_minNull @ X2 )
% 0.30/0.54 => ( ( X2
% 0.30/0.54 != ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.54 => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.54 ( X2
% 0.30/0.54 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % VEBT_internal.minNull.elims(2)
% 0.30/0.54 thf(fact_2213_field__le__epsilon,axiom,
% 0.30/0.54 ! [X2: real,Y3: real] :
% 0.30/0.54 ( ! [E2: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.30/0.54 => ( ord_less_eq_real @ X2 @ ( plus_plus_real @ Y3 @ E2 ) ) )
% 0.30/0.54 => ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 0.30/0.54
% 0.30/0.54 % field_le_epsilon
% 0.30/0.54 thf(fact_2214_field__le__epsilon,axiom,
% 0.30/0.54 ! [X2: rat,Y3: rat] :
% 0.30/0.54 ( ! [E2: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.30/0.54 => ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ Y3 @ E2 ) ) )
% 0.30/0.54 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 0.30/0.54
% 0.30/0.54 % field_le_epsilon
% 0.30/0.54 thf(fact_2215_add__neg__nonpos,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_eq_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_nonpos
% 0.30/0.54 thf(fact_2216_add__neg__nonpos,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_eq_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_nonpos
% 0.30/0.54 thf(fact_2217_add__neg__nonpos,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_eq_nat @ B @ zero_zero_nat )
% 0.30/0.54 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_neg_nonpos
% 0.30/0.54 thf(fact_2218_add__neg__nonpos,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_eq_int @ B @ zero_zero_int )
% 0.30/0.54 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_neg_nonpos
% 0.30/0.54 thf(fact_2219_add__nonneg__pos,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_real @ zero_zero_real @ B )
% 0.30/0.54 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonneg_pos
% 0.30/0.54 thf(fact_2220_add__nonneg__pos,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_rat @ zero_zero_rat @ B )
% 0.30/0.54 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonneg_pos
% 0.30/0.54 thf(fact_2221_add__nonneg__pos,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 @ zero_zero_nat @ B )
% 0.30/0.54 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonneg_pos
% 0.30/0.54 thf(fact_2222_add__nonneg__pos,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 @ zero_zero_int @ B )
% 0.30/0.54 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonneg_pos
% 0.30/0.54 thf(fact_2223_add__nonpos__neg,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_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_nonpos_neg
% 0.30/0.54 thf(fact_2224_add__nonpos__neg,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_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_nonpos_neg
% 0.30/0.54 thf(fact_2225_add__nonpos__neg,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_nat @ B @ zero_zero_nat )
% 0.30/0.54 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonpos_neg
% 0.30/0.54 thf(fact_2226_add__nonpos__neg,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_int @ B @ zero_zero_int )
% 0.30/0.54 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonpos_neg
% 0.30/0.54 thf(fact_2227_add__pos__nonneg,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_eq_real @ zero_zero_real @ B )
% 0.30/0.54 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_pos_nonneg
% 0.30/0.54 thf(fact_2228_add__pos__nonneg,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_eq_rat @ zero_zero_rat @ B )
% 0.30/0.54 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_pos_nonneg
% 0.30/0.54 thf(fact_2229_add__pos__nonneg,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_eq_nat @ zero_zero_nat @ B )
% 0.30/0.54 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_pos_nonneg
% 0.30/0.54 thf(fact_2230_add__pos__nonneg,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_eq_int @ zero_zero_int @ B )
% 0.30/0.54 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_pos_nonneg
% 0.30/0.54 thf(fact_2231_add__strict__increasing,axiom,
% 0.30/0.54 ! [A: real,B: real,C: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_eq_real @ B @ C )
% 0.30/0.54 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_strict_increasing
% 0.30/0.54 thf(fact_2232_add__strict__increasing,axiom,
% 0.30/0.54 ! [A: rat,B: rat,C: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_eq_rat @ B @ C )
% 0.30/0.54 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_strict_increasing
% 0.30/0.54 thf(fact_2233_add__strict__increasing,axiom,
% 0.30/0.54 ! [A: nat,B: nat,C: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_eq_nat @ B @ C )
% 0.30/0.54 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_strict_increasing
% 0.30/0.54 thf(fact_2234_add__strict__increasing,axiom,
% 0.30/0.54 ! [A: int,B: int,C: int] :
% 0.30/0.54 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_eq_int @ B @ C )
% 0.30/0.54 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_strict_increasing
% 0.30/0.54 thf(fact_2235_add__strict__increasing2,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_real @ B @ C )
% 0.30/0.54 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_strict_increasing2
% 0.30/0.54 thf(fact_2236_add__strict__increasing2,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_rat @ B @ C )
% 0.30/0.54 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_strict_increasing2
% 0.30/0.54 thf(fact_2237_add__strict__increasing2,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_nat @ B @ C )
% 0.30/0.54 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_strict_increasing2
% 0.30/0.54 thf(fact_2238_add__strict__increasing2,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_int @ B @ C )
% 0.30/0.54 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_strict_increasing2
% 0.30/0.54 thf(fact_2239_mult__left__le__one__le,axiom,
% 0.30/0.54 ! [X2: real,Y3: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.54 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.54 => ( ord_less_eq_real @ ( times_times_real @ Y3 @ X2 ) @ X2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_le_one_le
% 0.30/0.54 thf(fact_2240_mult__left__le__one__le,axiom,
% 0.30/0.54 ! [X2: rat,Y3: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.54 => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ Y3 @ X2 ) @ X2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_le_one_le
% 0.30/0.54 thf(fact_2241_mult__left__le__one__le,axiom,
% 0.30/0.54 ! [X2: int,Y3: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.54 => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 0.30/0.54 => ( ord_less_eq_int @ ( times_times_int @ Y3 @ X2 ) @ X2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_le_one_le
% 0.30/0.54 thf(fact_2242_mult__right__le__one__le,axiom,
% 0.30/0.54 ! [X2: real,Y3: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.54 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.54 => ( ord_less_eq_real @ ( times_times_real @ X2 @ Y3 ) @ X2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_le_one_le
% 0.30/0.54 thf(fact_2243_mult__right__le__one__le,axiom,
% 0.30/0.54 ! [X2: rat,Y3: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.54 => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ X2 @ Y3 ) @ X2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_le_one_le
% 0.30/0.54 thf(fact_2244_mult__right__le__one__le,axiom,
% 0.30/0.54 ! [X2: int,Y3: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.54 => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 0.30/0.54 => ( ord_less_eq_int @ ( times_times_int @ X2 @ Y3 ) @ X2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_le_one_le
% 0.30/0.54 thf(fact_2245_mult__le__one,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ A @ one_one_real )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.54 => ( ( ord_less_eq_real @ B @ one_one_real )
% 0.30/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_one
% 0.30/0.54 thf(fact_2246_mult__le__one,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.54 => ( ( ord_less_eq_rat @ B @ one_one_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_one
% 0.30/0.54 thf(fact_2247_mult__le__one,axiom,
% 0.30/0.54 ! [A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.54 => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 0.30/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_one
% 0.30/0.54 thf(fact_2248_mult__le__one,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ A @ one_one_int )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.54 => ( ( ord_less_eq_int @ B @ one_one_int )
% 0.30/0.54 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_one
% 0.30/0.54 thf(fact_2249_mult__left__le,axiom,
% 0.30/0.54 ! [C: real,A: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ C @ one_one_real )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_le
% 0.30/0.54 thf(fact_2250_mult__left__le,axiom,
% 0.30/0.54 ! [C: rat,A: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ C @ one_one_rat )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_le
% 0.30/0.54 thf(fact_2251_mult__left__le,axiom,
% 0.30/0.54 ! [C: nat,A: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_le
% 0.30/0.54 thf(fact_2252_mult__left__le,axiom,
% 0.30/0.54 ! [C: int,A: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ C @ one_one_int )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_le
% 0.30/0.54 thf(fact_2253_divide__nonpos__pos,axiom,
% 0.30/0.54 ! [X2: real,Y3: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.54 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_nonpos_pos
% 0.30/0.54 thf(fact_2254_divide__nonpos__pos,axiom,
% 0.30/0.54 ! [X2: rat,Y3: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.30/0.54 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_nonpos_pos
% 0.30/0.54 thf(fact_2255_divide__nonpos__neg,axiom,
% 0.30/0.54 ! [X2: real,Y3: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.30/0.54 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.30/0.54 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_nonpos_neg
% 0.30/0.54 thf(fact_2256_divide__nonpos__neg,axiom,
% 0.30/0.54 ! [X2: rat,Y3: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 0.30/0.54 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_nonpos_neg
% 0.30/0.54 thf(fact_2257_divide__nonneg__pos,axiom,
% 0.30/0.54 ! [X2: real,Y3: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.54 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_nonneg_pos
% 0.30/0.54 thf(fact_2258_divide__nonneg__pos,axiom,
% 0.30/0.54 ! [X2: rat,Y3: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.30/0.54 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_nonneg_pos
% 0.30/0.54 thf(fact_2259_divide__nonneg__neg,axiom,
% 0.30/0.54 ! [X2: real,Y3: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.54 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.30/0.54 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_nonneg_neg
% 0.30/0.54 thf(fact_2260_divide__nonneg__neg,axiom,
% 0.30/0.54 ! [X2: rat,Y3: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.30/0.54 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_nonneg_neg
% 0.30/0.54 thf(fact_2261_divide__le__cancel,axiom,
% 0.30/0.54 ! [A: real,C: real,B: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.30/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_eq_real @ A @ B ) )
% 0.30/0.54 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_le_cancel
% 0.30/0.54 thf(fact_2262_divide__le__cancel,axiom,
% 0.30/0.54 ! [A: rat,C: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_eq_rat @ A @ B ) )
% 0.30/0.54 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_le_cancel
% 0.30/0.54 thf(fact_2263_frac__less2,axiom,
% 0.30/0.54 ! [X2: real,Y3: real,W: real,Z: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.54 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.30/0.54 => ( ( ord_less_real @ W @ Z )
% 0.30/0.54 => ( ord_less_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % frac_less2
% 0.30/0.54 thf(fact_2264_frac__less2,axiom,
% 0.30/0.54 ! [X2: rat,Y3: rat,W: rat,Z: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 0.30/0.54 => ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.30/0.54 => ( ( ord_less_rat @ W @ Z )
% 0.30/0.54 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % frac_less2
% 0.30/0.54 thf(fact_2265_frac__less,axiom,
% 0.30/0.54 ! [X2: real,Y3: real,W: real,Z: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.54 => ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.30/0.54 => ( ( ord_less_eq_real @ W @ Z )
% 0.30/0.54 => ( ord_less_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % frac_less
% 0.30/0.54 thf(fact_2266_frac__less,axiom,
% 0.30/0.54 ! [X2: rat,Y3: rat,W: rat,Z: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.30/0.54 => ( ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.30/0.54 => ( ( ord_less_eq_rat @ W @ Z )
% 0.30/0.54 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % frac_less
% 0.30/0.54 thf(fact_2267_frac__le,axiom,
% 0.30/0.54 ! [Y3: real,X2: real,W: real,Z: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.54 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.30/0.54 => ( ( ord_less_eq_real @ W @ Z )
% 0.30/0.54 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % frac_le
% 0.30/0.54 thf(fact_2268_frac__le,axiom,
% 0.30/0.54 ! [Y3: rat,X2: rat,W: rat,Z: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.54 => ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.30/0.54 => ( ( ord_less_eq_rat @ W @ Z )
% 0.30/0.54 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % frac_le
% 0.30/0.54 thf(fact_2269_sum__squares__le__zero__iff,axiom,
% 0.30/0.54 ! [X2: real,Y3: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real )
% 0.30/0.54 = ( ( X2 = zero_zero_real )
% 0.30/0.54 & ( Y3 = zero_zero_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sum_squares_le_zero_iff
% 0.30/0.54 thf(fact_2270_sum__squares__le__zero__iff,axiom,
% 0.30/0.54 ! [X2: rat,Y3: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat )
% 0.30/0.54 = ( ( X2 = zero_zero_rat )
% 0.30/0.54 & ( Y3 = zero_zero_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sum_squares_le_zero_iff
% 0.30/0.54 thf(fact_2271_sum__squares__le__zero__iff,axiom,
% 0.30/0.54 ! [X2: int,Y3: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int )
% 0.30/0.54 = ( ( X2 = zero_zero_int )
% 0.30/0.54 & ( Y3 = zero_zero_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sum_squares_le_zero_iff
% 0.30/0.54 thf(fact_2272_sum__squares__ge__zero,axiom,
% 0.30/0.54 ! [X2: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sum_squares_ge_zero
% 0.30/0.54 thf(fact_2273_sum__squares__ge__zero,axiom,
% 0.30/0.54 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sum_squares_ge_zero
% 0.30/0.54 thf(fact_2274_sum__squares__ge__zero,axiom,
% 0.30/0.54 ! [X2: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sum_squares_ge_zero
% 0.30/0.54 thf(fact_2275_sum__squares__gt__zero__iff,axiom,
% 0.30/0.54 ! [X2: real,Y3: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) )
% 0.30/0.54 = ( ( X2 != zero_zero_real )
% 0.30/0.54 | ( Y3 != zero_zero_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sum_squares_gt_zero_iff
% 0.30/0.54 thf(fact_2276_sum__squares__gt__zero__iff,axiom,
% 0.30/0.54 ! [X2: rat,Y3: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) )
% 0.30/0.54 = ( ( X2 != zero_zero_rat )
% 0.30/0.54 | ( Y3 != zero_zero_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sum_squares_gt_zero_iff
% 0.30/0.54 thf(fact_2277_sum__squares__gt__zero__iff,axiom,
% 0.30/0.54 ! [X2: int,Y3: int] :
% 0.30/0.54 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) )
% 0.30/0.54 = ( ( X2 != zero_zero_int )
% 0.30/0.54 | ( Y3 != zero_zero_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % sum_squares_gt_zero_iff
% 0.30/0.54 thf(fact_2278_not__sum__squares__lt__zero,axiom,
% 0.30/0.54 ! [X2: real,Y3: real] :
% 0.30/0.54 ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real ) ).
% 0.30/0.54
% 0.30/0.54 % not_sum_squares_lt_zero
% 0.30/0.54 thf(fact_2279_not__sum__squares__lt__zero,axiom,
% 0.30/0.54 ! [X2: rat,Y3: rat] :
% 0.30/0.54 ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat ) ).
% 0.30/0.54
% 0.30/0.54 % not_sum_squares_lt_zero
% 0.30/0.54 thf(fact_2280_not__sum__squares__lt__zero,axiom,
% 0.30/0.54 ! [X2: int,Y3: int] :
% 0.30/0.54 ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int ) ).
% 0.30/0.54
% 0.30/0.54 % not_sum_squares_lt_zero
% 0.30/0.54 thf(fact_2281_power__less__imp__less__base,axiom,
% 0.30/0.54 ! [A: real,N: nat,B: real] :
% 0.30/0.54 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.54 => ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_less_imp_less_base
% 0.30/0.54 thf(fact_2282_power__less__imp__less__base,axiom,
% 0.30/0.54 ! [A: rat,N: nat,B: rat] :
% 0.30/0.54 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.54 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_less_imp_less_base
% 0.30/0.54 thf(fact_2283_power__less__imp__less__base,axiom,
% 0.30/0.54 ! [A: nat,N: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.54 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_less_imp_less_base
% 0.30/0.54 thf(fact_2284_power__less__imp__less__base,axiom,
% 0.30/0.54 ! [A: int,N: nat,B: int] :
% 0.30/0.54 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.54 => ( ord_less_int @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_less_imp_less_base
% 0.30/0.54 thf(fact_2285_zero__less__two,axiom,
% 0.30/0.54 ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_two
% 0.30/0.54 thf(fact_2286_zero__less__two,axiom,
% 0.30/0.54 ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_two
% 0.30/0.54 thf(fact_2287_zero__less__two,axiom,
% 0.30/0.54 ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_two
% 0.30/0.54 thf(fact_2288_zero__less__two,axiom,
% 0.30/0.54 ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_two
% 0.30/0.54 thf(fact_2289_divide__strict__left__mono__neg,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 @ C @ zero_zero_real )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.30/0.54 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_strict_left_mono_neg
% 0.30/0.54 thf(fact_2290_divide__strict__left__mono__neg,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 @ C @ zero_zero_rat )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.30/0.54 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_strict_left_mono_neg
% 0.30/0.54 thf(fact_2291_divide__strict__left__mono,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 @ zero_zero_real @ C )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.30/0.54 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_strict_left_mono
% 0.30/0.54 thf(fact_2292_divide__strict__left__mono,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 @ zero_zero_rat @ C )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.30/0.54 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_strict_left_mono
% 0.30/0.54 thf(fact_2293_mult__imp__less__div__pos,axiom,
% 0.30/0.54 ! [Y3: real,Z: real,X2: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.54 => ( ( ord_less_real @ ( times_times_real @ Z @ Y3 ) @ X2 )
% 0.30/0.54 => ( ord_less_real @ Z @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_imp_less_div_pos
% 0.30/0.54 thf(fact_2294_mult__imp__less__div__pos,axiom,
% 0.30/0.54 ! [Y3: rat,Z: rat,X2: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.30/0.54 => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y3 ) @ X2 )
% 0.30/0.54 => ( ord_less_rat @ Z @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_imp_less_div_pos
% 0.30/0.54 thf(fact_2295_mult__imp__div__pos__less,axiom,
% 0.30/0.54 ! [Y3: real,X2: real,Z: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.54 => ( ( ord_less_real @ X2 @ ( times_times_real @ Z @ Y3 ) )
% 0.30/0.54 => ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ Z ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_imp_div_pos_less
% 0.30/0.54 thf(fact_2296_mult__imp__div__pos__less,axiom,
% 0.30/0.54 ! [Y3: rat,X2: rat,Z: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.30/0.54 => ( ( ord_less_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) )
% 0.30/0.54 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ Z ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_imp_div_pos_less
% 0.30/0.54 thf(fact_2297_pos__less__divide__eq,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 @ A @ ( divide_divide_real @ B @ C ) )
% 0.30/0.54 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % pos_less_divide_eq
% 0.30/0.54 thf(fact_2298_pos__less__divide__eq,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 @ A @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.54 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % pos_less_divide_eq
% 0.30/0.54 thf(fact_2299_pos__divide__less__eq,axiom,
% 0.30/0.54 ! [C: real,B: real,A: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.30/0.54 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % pos_divide_less_eq
% 0.30/0.54 thf(fact_2300_pos__divide__less__eq,axiom,
% 0.30/0.54 ! [C: rat,B: rat,A: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.30/0.54 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % pos_divide_less_eq
% 0.30/0.54 thf(fact_2301_neg__less__divide__eq,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 @ A @ ( divide_divide_real @ B @ C ) )
% 0.30/0.54 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % neg_less_divide_eq
% 0.30/0.54 thf(fact_2302_neg__less__divide__eq,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 @ A @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.54 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % neg_less_divide_eq
% 0.30/0.54 thf(fact_2303_neg__divide__less__eq,axiom,
% 0.30/0.54 ! [C: real,B: real,A: real] :
% 0.30/0.54 ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.30/0.54 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % neg_divide_less_eq
% 0.30/0.54 thf(fact_2304_neg__divide__less__eq,axiom,
% 0.30/0.54 ! [C: rat,B: rat,A: rat] :
% 0.30/0.54 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.30/0.54 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % neg_divide_less_eq
% 0.30/0.54 thf(fact_2305_less__divide__eq,axiom,
% 0.30/0.54 ! [A: real,B: real,C: real] :
% 0.30/0.54 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.30/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.30/0.54 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.30/0.54 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % less_divide_eq
% 0.30/0.54 thf(fact_2306_less__divide__eq,axiom,
% 0.30/0.54 ! [A: rat,B: rat,C: rat] :
% 0.30/0.54 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.30/0.54 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.54 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % less_divide_eq
% 0.30/0.54 thf(fact_2307_divide__less__eq,axiom,
% 0.30/0.54 ! [B: real,C: real,A: real] :
% 0.30/0.54 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.30/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.30/0.54 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.30/0.54 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_less_eq
% 0.30/0.54 thf(fact_2308_divide__less__eq,axiom,
% 0.30/0.54 ! [B: rat,C: rat,A: rat] :
% 0.30/0.54 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.30/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.54 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.30/0.54 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_less_eq
% 0.30/0.54 thf(fact_2309_less__divide__eq__1,axiom,
% 0.30/0.54 ! [B: real,A: real] :
% 0.30/0.54 ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.30/0.54 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54 & ( ord_less_real @ A @ B ) )
% 0.30/0.54 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.54 & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % less_divide_eq_1
% 0.30/0.54 thf(fact_2310_less__divide__eq__1,axiom,
% 0.30/0.54 ! [B: rat,A: rat] :
% 0.30/0.54 ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.30/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54 & ( ord_less_rat @ A @ B ) )
% 0.30/0.54 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.54 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % less_divide_eq_1
% 0.30/0.54 thf(fact_2311_divide__less__eq__1,axiom,
% 0.30/0.54 ! [B: real,A: real] :
% 0.30/0.54 ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.30/0.54 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54 & ( ord_less_real @ B @ A ) )
% 0.30/0.54 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.54 & ( ord_less_real @ A @ B ) )
% 0.30/0.54 | ( A = zero_zero_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_less_eq_1
% 0.30/0.54 thf(fact_2312_divide__less__eq__1,axiom,
% 0.30/0.54 ! [B: rat,A: rat] :
% 0.30/0.54 ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.30/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54 & ( ord_less_rat @ B @ A ) )
% 0.30/0.54 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.54 & ( ord_less_rat @ A @ B ) )
% 0.30/0.54 | ( A = zero_zero_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_less_eq_1
% 0.30/0.54 thf(fact_2313_power__le__one,axiom,
% 0.30/0.54 ! [A: real,N: nat] :
% 0.30/0.54 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.30/0.54 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_le_one
% 0.30/0.54 thf(fact_2314_power__le__one,axiom,
% 0.30/0.54 ! [A: rat,N: nat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_le_one
% 0.30/0.54 thf(fact_2315_power__le__one,axiom,
% 0.30/0.54 ! [A: nat,N: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.30/0.54 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_le_one
% 0.30/0.54 thf(fact_2316_power__le__one,axiom,
% 0.30/0.54 ! [A: int,N: nat] :
% 0.30/0.54 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.30/0.54 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_le_one
% 0.30/0.54 thf(fact_2317_eq__divide__eq__numeral_I1_J,axiom,
% 0.30/0.54 ! [W: num,B: complex,C: complex] :
% 0.30/0.54 ( ( ( numera6690914467698888265omplex @ W )
% 0.30/0.54 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.30/0.54 = ( ( ( C != zero_zero_complex )
% 0.30/0.54 => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
% 0.30/0.54 = B ) )
% 0.30/0.54 & ( ( C = zero_zero_complex )
% 0.30/0.54 => ( ( numera6690914467698888265omplex @ W )
% 0.30/0.54 = zero_zero_complex ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_divide_eq_numeral(1)
% 0.30/0.54 thf(fact_2318_eq__divide__eq__numeral_I1_J,axiom,
% 0.30/0.54 ! [W: num,B: real,C: real] :
% 0.30/0.54 ( ( ( numeral_numeral_real @ W )
% 0.30/0.54 = ( divide_divide_real @ B @ C ) )
% 0.30/0.54 = ( ( ( C != zero_zero_real )
% 0.30/0.54 => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
% 0.30/0.54 = B ) )
% 0.30/0.54 & ( ( C = zero_zero_real )
% 0.30/0.54 => ( ( numeral_numeral_real @ W )
% 0.30/0.54 = zero_zero_real ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_divide_eq_numeral(1)
% 0.30/0.54 thf(fact_2319_eq__divide__eq__numeral_I1_J,axiom,
% 0.30/0.54 ! [W: num,B: rat,C: rat] :
% 0.30/0.54 ( ( ( numeral_numeral_rat @ W )
% 0.30/0.54 = ( divide_divide_rat @ B @ C ) )
% 0.30/0.54 = ( ( ( C != zero_zero_rat )
% 0.30/0.54 => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
% 0.30/0.54 = B ) )
% 0.30/0.54 & ( ( C = zero_zero_rat )
% 0.30/0.54 => ( ( numeral_numeral_rat @ W )
% 0.30/0.54 = zero_zero_rat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_divide_eq_numeral(1)
% 0.30/0.54 thf(fact_2320_divide__eq__eq__numeral_I1_J,axiom,
% 0.30/0.54 ! [B: complex,C: complex,W: num] :
% 0.30/0.54 ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.30/0.54 = ( numera6690914467698888265omplex @ W ) )
% 0.30/0.54 = ( ( ( C != zero_zero_complex )
% 0.30/0.54 => ( B
% 0.30/0.54 = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
% 0.30/0.54 & ( ( C = zero_zero_complex )
% 0.30/0.54 => ( ( numera6690914467698888265omplex @ W )
% 0.30/0.54 = zero_zero_complex ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_eq_eq_numeral(1)
% 0.30/0.54 thf(fact_2321_divide__eq__eq__numeral_I1_J,axiom,
% 0.30/0.54 ! [B: real,C: real,W: num] :
% 0.30/0.54 ( ( ( divide_divide_real @ B @ C )
% 0.30/0.54 = ( numeral_numeral_real @ W ) )
% 0.30/0.54 = ( ( ( C != zero_zero_real )
% 0.30/0.54 => ( B
% 0.30/0.54 = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.30/0.54 & ( ( C = zero_zero_real )
% 0.30/0.54 => ( ( numeral_numeral_real @ W )
% 0.30/0.54 = zero_zero_real ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_eq_eq_numeral(1)
% 0.30/0.54 thf(fact_2322_divide__eq__eq__numeral_I1_J,axiom,
% 0.30/0.54 ! [B: rat,C: rat,W: num] :
% 0.30/0.54 ( ( ( divide_divide_rat @ B @ C )
% 0.30/0.54 = ( numeral_numeral_rat @ W ) )
% 0.30/0.54 = ( ( ( C != zero_zero_rat )
% 0.30/0.54 => ( B
% 0.30/0.54 = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.30/0.54 & ( ( C = zero_zero_rat )
% 0.30/0.54 => ( ( numeral_numeral_rat @ W )
% 0.30/0.54 = zero_zero_rat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_eq_eq_numeral(1)
% 0.30/0.54 thf(fact_2323_divide__add__eq__iff,axiom,
% 0.30/0.54 ! [Z: complex,X2: complex,Y3: complex] :
% 0.30/0.54 ( ( Z != zero_zero_complex )
% 0.30/0.54 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Z ) @ Y3 )
% 0.30/0.54 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_add_eq_iff
% 0.30/0.54 thf(fact_2324_divide__add__eq__iff,axiom,
% 0.30/0.54 ! [Z: real,X2: real,Y3: real] :
% 0.30/0.54 ( ( Z != zero_zero_real )
% 0.30/0.54 => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Z ) @ Y3 )
% 0.30/0.54 = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_add_eq_iff
% 0.30/0.54 thf(fact_2325_divide__add__eq__iff,axiom,
% 0.30/0.54 ! [Z: rat,X2: rat,Y3: rat] :
% 0.30/0.54 ( ( Z != zero_zero_rat )
% 0.30/0.54 => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Z ) @ Y3 )
% 0.30/0.54 = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_add_eq_iff
% 0.30/0.54 thf(fact_2326_add__divide__eq__iff,axiom,
% 0.30/0.54 ! [Z: complex,X2: complex,Y3: complex] :
% 0.30/0.54 ( ( Z != zero_zero_complex )
% 0.30/0.54 => ( ( plus_plus_complex @ X2 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 0.30/0.54 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_divide_eq_iff
% 0.30/0.54 thf(fact_2327_add__divide__eq__iff,axiom,
% 0.30/0.54 ! [Z: real,X2: real,Y3: real] :
% 0.30/0.54 ( ( Z != zero_zero_real )
% 0.30/0.54 => ( ( plus_plus_real @ X2 @ ( divide_divide_real @ Y3 @ Z ) )
% 0.30/0.54 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_divide_eq_iff
% 0.30/0.54 thf(fact_2328_add__divide__eq__iff,axiom,
% 0.30/0.54 ! [Z: rat,X2: rat,Y3: rat] :
% 0.30/0.54 ( ( Z != zero_zero_rat )
% 0.30/0.54 => ( ( plus_plus_rat @ X2 @ ( divide_divide_rat @ Y3 @ Z ) )
% 0.30/0.54 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_divide_eq_iff
% 0.30/0.54 thf(fact_2329_add__num__frac,axiom,
% 0.30/0.54 ! [Y3: complex,Z: complex,X2: complex] :
% 0.30/0.54 ( ( Y3 != zero_zero_complex )
% 0.30/0.54 => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X2 @ Y3 ) )
% 0.30/0.54 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_num_frac
% 0.30/0.54 thf(fact_2330_add__num__frac,axiom,
% 0.30/0.54 ! [Y3: real,Z: real,X2: real] :
% 0.30/0.54 ( ( Y3 != zero_zero_real )
% 0.30/0.54 => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X2 @ Y3 ) )
% 0.30/0.54 = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_num_frac
% 0.30/0.54 thf(fact_2331_add__num__frac,axiom,
% 0.30/0.54 ! [Y3: rat,Z: rat,X2: rat] :
% 0.30/0.54 ( ( Y3 != zero_zero_rat )
% 0.30/0.54 => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X2 @ Y3 ) )
% 0.30/0.54 = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_num_frac
% 0.30/0.54 thf(fact_2332_add__frac__num,axiom,
% 0.30/0.54 ! [Y3: complex,X2: complex,Z: complex] :
% 0.30/0.54 ( ( Y3 != zero_zero_complex )
% 0.30/0.54 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Y3 ) @ Z )
% 0.30/0.54 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_frac_num
% 0.30/0.54 thf(fact_2333_add__frac__num,axiom,
% 0.30/0.54 ! [Y3: real,X2: real,Z: real] :
% 0.30/0.54 ( ( Y3 != zero_zero_real )
% 0.30/0.54 => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y3 ) @ Z )
% 0.30/0.54 = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_frac_num
% 0.30/0.54 thf(fact_2334_add__frac__num,axiom,
% 0.30/0.54 ! [Y3: rat,X2: rat,Z: rat] :
% 0.30/0.54 ( ( Y3 != zero_zero_rat )
% 0.30/0.54 => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ Z )
% 0.30/0.54 = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_frac_num
% 0.30/0.54 thf(fact_2335_add__frac__eq,axiom,
% 0.30/0.54 ! [Y3: complex,Z: complex,X2: complex,W: complex] :
% 0.30/0.54 ( ( Y3 != zero_zero_complex )
% 0.30/0.54 => ( ( Z != zero_zero_complex )
% 0.30/0.54 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 0.30/0.54 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_frac_eq
% 0.30/0.54 thf(fact_2336_add__frac__eq,axiom,
% 0.30/0.54 ! [Y3: real,Z: real,X2: real,W: real] :
% 0.30/0.54 ( ( Y3 != zero_zero_real )
% 0.30/0.54 => ( ( Z != zero_zero_real )
% 0.30/0.54 => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 0.30/0.54 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_frac_eq
% 0.30/0.54 thf(fact_2337_add__frac__eq,axiom,
% 0.30/0.54 ! [Y3: rat,Z: rat,X2: rat,W: rat] :
% 0.30/0.54 ( ( Y3 != zero_zero_rat )
% 0.30/0.54 => ( ( Z != zero_zero_rat )
% 0.30/0.54 => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.30/0.54 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_frac_eq
% 0.30/0.54 thf(fact_2338_add__divide__eq__if__simps_I1_J,axiom,
% 0.30/0.54 ! [Z: complex,A: complex,B: complex] :
% 0.30/0.54 ( ( ( Z = zero_zero_complex )
% 0.30/0.54 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.30/0.54 = A ) )
% 0.30/0.54 & ( ( Z != zero_zero_complex )
% 0.30/0.54 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.30/0.54 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_divide_eq_if_simps(1)
% 0.30/0.54 thf(fact_2339_add__divide__eq__if__simps_I1_J,axiom,
% 0.30/0.54 ! [Z: real,A: real,B: real] :
% 0.30/0.54 ( ( ( Z = zero_zero_real )
% 0.30/0.54 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.30/0.54 = A ) )
% 0.30/0.54 & ( ( Z != zero_zero_real )
% 0.30/0.54 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.30/0.54 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_divide_eq_if_simps(1)
% 0.30/0.54 thf(fact_2340_add__divide__eq__if__simps_I1_J,axiom,
% 0.30/0.54 ! [Z: rat,A: rat,B: rat] :
% 0.30/0.54 ( ( ( Z = zero_zero_rat )
% 0.30/0.54 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.30/0.54 = A ) )
% 0.30/0.54 & ( ( Z != zero_zero_rat )
% 0.30/0.54 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.30/0.54 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_divide_eq_if_simps(1)
% 0.30/0.54 thf(fact_2341_add__divide__eq__if__simps_I2_J,axiom,
% 0.30/0.54 ! [Z: complex,A: complex,B: complex] :
% 0.30/0.54 ( ( ( Z = zero_zero_complex )
% 0.30/0.54 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.30/0.54 = B ) )
% 0.30/0.54 & ( ( Z != zero_zero_complex )
% 0.30/0.54 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.30/0.54 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_divide_eq_if_simps(2)
% 0.30/0.54 thf(fact_2342_add__divide__eq__if__simps_I2_J,axiom,
% 0.30/0.54 ! [Z: real,A: real,B: real] :
% 0.30/0.54 ( ( ( Z = zero_zero_real )
% 0.30/0.54 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.30/0.54 = B ) )
% 0.30/0.54 & ( ( Z != zero_zero_real )
% 0.30/0.54 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.30/0.54 = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_divide_eq_if_simps(2)
% 0.30/0.54 thf(fact_2343_add__divide__eq__if__simps_I2_J,axiom,
% 0.30/0.54 ! [Z: rat,A: rat,B: rat] :
% 0.30/0.54 ( ( ( Z = zero_zero_rat )
% 0.30/0.54 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.30/0.54 = B ) )
% 0.30/0.54 & ( ( Z != zero_zero_rat )
% 0.30/0.54 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.30/0.54 = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_divide_eq_if_simps(2)
% 0.30/0.54 thf(fact_2344_power__inject__base,axiom,
% 0.30/0.54 ! [A: real,N: nat,B: real] :
% 0.30/0.54 ( ( ( power_power_real @ A @ ( suc @ N ) )
% 0.30/0.54 = ( power_power_real @ B @ ( suc @ N ) ) )
% 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 => ( A = B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_inject_base
% 0.30/0.54 thf(fact_2345_power__inject__base,axiom,
% 0.30/0.54 ! [A: rat,N: nat,B: rat] :
% 0.30/0.54 ( ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.30/0.54 = ( power_power_rat @ B @ ( suc @ N ) ) )
% 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 => ( A = B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_inject_base
% 0.30/0.54 thf(fact_2346_power__inject__base,axiom,
% 0.30/0.54 ! [A: nat,N: nat,B: nat] :
% 0.30/0.54 ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.30/0.54 = ( power_power_nat @ B @ ( suc @ N ) ) )
% 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 => ( A = B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_inject_base
% 0.30/0.54 thf(fact_2347_power__inject__base,axiom,
% 0.30/0.54 ! [A: int,N: nat,B: int] :
% 0.30/0.54 ( ( ( power_power_int @ A @ ( suc @ N ) )
% 0.30/0.54 = ( power_power_int @ B @ ( suc @ N ) ) )
% 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 => ( A = B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_inject_base
% 0.30/0.54 thf(fact_2348_power__le__imp__le__base,axiom,
% 0.30/0.54 ! [A: real,N: nat,B: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.54 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_le_imp_le_base
% 0.30/0.54 thf(fact_2349_power__le__imp__le__base,axiom,
% 0.30/0.54 ! [A: rat,N: nat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.54 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_le_imp_le_base
% 0.30/0.54 thf(fact_2350_power__le__imp__le__base,axiom,
% 0.30/0.54 ! [A: nat,N: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.54 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_le_imp_le_base
% 0.30/0.54 thf(fact_2351_power__le__imp__le__base,axiom,
% 0.30/0.54 ! [A: int,N: nat,B: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.54 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_le_imp_le_base
% 0.30/0.54 thf(fact_2352_div__add__self1,axiom,
% 0.30/0.54 ! [B: nat,A: nat] :
% 0.30/0.54 ( ( B != zero_zero_nat )
% 0.30/0.54 => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.30/0.54 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % div_add_self1
% 0.30/0.54 thf(fact_2353_div__add__self1,axiom,
% 0.30/0.54 ! [B: int,A: int] :
% 0.30/0.54 ( ( B != zero_zero_int )
% 0.30/0.54 => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.30/0.54 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % div_add_self1
% 0.30/0.54 thf(fact_2354_div__add__self2,axiom,
% 0.30/0.54 ! [B: nat,A: nat] :
% 0.30/0.54 ( ( B != zero_zero_nat )
% 0.30/0.54 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.30/0.54 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % div_add_self2
% 0.30/0.54 thf(fact_2355_div__add__self2,axiom,
% 0.30/0.54 ! [B: int,A: int] :
% 0.30/0.54 ( ( B != zero_zero_int )
% 0.30/0.54 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.30/0.54 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % div_add_self2
% 0.30/0.54 thf(fact_2356_divide__diff__eq__iff,axiom,
% 0.30/0.54 ! [Z: complex,X2: complex,Y3: complex] :
% 0.30/0.54 ( ( Z != zero_zero_complex )
% 0.30/0.54 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X2 @ Z ) @ Y3 )
% 0.30/0.54 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_diff_eq_iff
% 0.30/0.54 thf(fact_2357_divide__diff__eq__iff,axiom,
% 0.30/0.54 ! [Z: real,X2: real,Y3: real] :
% 0.30/0.54 ( ( Z != zero_zero_real )
% 0.30/0.54 => ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Z ) @ Y3 )
% 0.30/0.54 = ( divide_divide_real @ ( minus_minus_real @ X2 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_diff_eq_iff
% 0.30/0.54 thf(fact_2358_divide__diff__eq__iff,axiom,
% 0.30/0.54 ! [Z: rat,X2: rat,Y3: rat] :
% 0.30/0.54 ( ( Z != zero_zero_rat )
% 0.30/0.54 => ( ( minus_minus_rat @ ( divide_divide_rat @ X2 @ Z ) @ Y3 )
% 0.30/0.54 = ( divide_divide_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_diff_eq_iff
% 0.30/0.54 thf(fact_2359_diff__divide__eq__iff,axiom,
% 0.30/0.54 ! [Z: complex,X2: complex,Y3: complex] :
% 0.30/0.54 ( ( Z != zero_zero_complex )
% 0.30/0.54 => ( ( minus_minus_complex @ X2 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 0.30/0.54 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % diff_divide_eq_iff
% 0.30/0.54 thf(fact_2360_diff__divide__eq__iff,axiom,
% 0.30/0.54 ! [Z: real,X2: real,Y3: real] :
% 0.30/0.54 ( ( Z != zero_zero_real )
% 0.30/0.54 => ( ( minus_minus_real @ X2 @ ( divide_divide_real @ Y3 @ Z ) )
% 0.30/0.54 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % diff_divide_eq_iff
% 0.30/0.54 thf(fact_2361_diff__divide__eq__iff,axiom,
% 0.30/0.54 ! [Z: rat,X2: rat,Y3: rat] :
% 0.30/0.54 ( ( Z != zero_zero_rat )
% 0.30/0.54 => ( ( minus_minus_rat @ X2 @ ( divide_divide_rat @ Y3 @ Z ) )
% 0.30/0.54 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % diff_divide_eq_iff
% 0.30/0.54 thf(fact_2362_diff__frac__eq,axiom,
% 0.30/0.54 ! [Y3: complex,Z: complex,X2: complex,W: complex] :
% 0.30/0.54 ( ( Y3 != zero_zero_complex )
% 0.30/0.54 => ( ( Z != zero_zero_complex )
% 0.30/0.54 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X2 @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 0.30/0.54 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % diff_frac_eq
% 0.30/0.54 thf(fact_2363_diff__frac__eq,axiom,
% 0.30/0.54 ! [Y3: real,Z: real,X2: real,W: real] :
% 0.30/0.54 ( ( Y3 != zero_zero_real )
% 0.30/0.54 => ( ( Z != zero_zero_real )
% 0.30/0.54 => ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 0.30/0.54 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % diff_frac_eq
% 0.30/0.54 thf(fact_2364_diff__frac__eq,axiom,
% 0.30/0.54 ! [Y3: rat,Z: rat,X2: rat,W: rat] :
% 0.30/0.54 ( ( Y3 != zero_zero_rat )
% 0.30/0.54 => ( ( Z != zero_zero_rat )
% 0.30/0.54 => ( ( minus_minus_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.30/0.54 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % diff_frac_eq
% 0.30/0.54 thf(fact_2365_add__divide__eq__if__simps_I4_J,axiom,
% 0.30/0.54 ! [Z: complex,A: complex,B: complex] :
% 0.30/0.54 ( ( ( Z = zero_zero_complex )
% 0.30/0.54 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.30/0.54 = A ) )
% 0.30/0.54 & ( ( Z != zero_zero_complex )
% 0.30/0.54 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.30/0.54 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_divide_eq_if_simps(4)
% 0.30/0.54 thf(fact_2366_add__divide__eq__if__simps_I4_J,axiom,
% 0.30/0.54 ! [Z: real,A: real,B: real] :
% 0.30/0.54 ( ( ( Z = zero_zero_real )
% 0.30/0.54 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.30/0.54 = A ) )
% 0.30/0.54 & ( ( Z != zero_zero_real )
% 0.30/0.54 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.30/0.54 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_divide_eq_if_simps(4)
% 0.30/0.54 thf(fact_2367_add__divide__eq__if__simps_I4_J,axiom,
% 0.30/0.54 ! [Z: rat,A: rat,B: rat] :
% 0.30/0.54 ( ( ( Z = zero_zero_rat )
% 0.30/0.54 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.30/0.54 = A ) )
% 0.30/0.54 & ( ( Z != zero_zero_rat )
% 0.30/0.54 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.30/0.54 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_divide_eq_if_simps(4)
% 0.30/0.54 thf(fact_2368_numeral__1__eq__Suc__0,axiom,
% 0.30/0.54 ( ( numeral_numeral_nat @ one )
% 0.30/0.54 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.54
% 0.30/0.54 % numeral_1_eq_Suc_0
% 0.30/0.54 thf(fact_2369_subset__code_I1_J,axiom,
% 0.30/0.54 ! [Xs2: list_P6011104703257516679at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.30/0.54 ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ B3 )
% 0.30/0.54 = ( ! [X: product_prod_nat_nat] :
% 0.30/0.54 ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
% 0.30/0.54 => ( member8440522571783428010at_nat @ X @ B3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % subset_code(1)
% 0.30/0.54 thf(fact_2370_subset__code_I1_J,axiom,
% 0.30/0.54 ! [Xs2: list_complex,B3: set_complex] :
% 0.30/0.54 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ B3 )
% 0.30/0.54 = ( ! [X: complex] :
% 0.30/0.54 ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
% 0.30/0.54 => ( member_complex @ X @ B3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % subset_code(1)
% 0.30/0.54 thf(fact_2371_subset__code_I1_J,axiom,
% 0.30/0.54 ! [Xs2: list_real,B3: set_real] :
% 0.30/0.54 ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B3 )
% 0.30/0.54 = ( ! [X: real] :
% 0.30/0.54 ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
% 0.30/0.54 => ( member_real @ X @ B3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % subset_code(1)
% 0.30/0.54 thf(fact_2372_subset__code_I1_J,axiom,
% 0.30/0.54 ! [Xs2: list_VEBT_VEBT,B3: set_VEBT_VEBT] :
% 0.30/0.54 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ B3 )
% 0.30/0.54 = ( ! [X: vEBT_VEBT] :
% 0.30/0.54 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.30/0.54 => ( member_VEBT_VEBT @ X @ B3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % subset_code(1)
% 0.30/0.54 thf(fact_2373_subset__code_I1_J,axiom,
% 0.30/0.54 ! [Xs2: list_nat,B3: set_nat] :
% 0.30/0.54 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B3 )
% 0.30/0.54 = ( ! [X: nat] :
% 0.30/0.54 ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
% 0.30/0.54 => ( member_nat @ X @ B3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % subset_code(1)
% 0.30/0.54 thf(fact_2374_subset__code_I1_J,axiom,
% 0.30/0.54 ! [Xs2: list_int,B3: set_int] :
% 0.30/0.54 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B3 )
% 0.30/0.54 = ( ! [X: int] :
% 0.30/0.54 ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
% 0.30/0.54 => ( member_int @ X @ B3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % subset_code(1)
% 0.30/0.54 thf(fact_2375_num_Osize_I5_J,axiom,
% 0.30/0.54 ! [X22: num] :
% 0.30/0.54 ( ( size_size_num @ ( bit0 @ X22 ) )
% 0.30/0.54 = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % num.size(5)
% 0.30/0.54 thf(fact_2376_neq__if__length__neq,axiom,
% 0.30/0.54 ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.30/0.54 ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.30/0.54 != ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.30/0.54 => ( Xs2 != Ys ) ) ).
% 0.30/0.54
% 0.30/0.54 % neq_if_length_neq
% 0.30/0.54 thf(fact_2377_neq__if__length__neq,axiom,
% 0.30/0.54 ! [Xs2: list_o,Ys: list_o] :
% 0.30/0.54 ( ( ( size_size_list_o @ Xs2 )
% 0.30/0.54 != ( size_size_list_o @ Ys ) )
% 0.30/0.54 => ( Xs2 != Ys ) ) ).
% 0.30/0.54
% 0.30/0.54 % neq_if_length_neq
% 0.30/0.54 thf(fact_2378_neq__if__length__neq,axiom,
% 0.30/0.54 ! [Xs2: list_nat,Ys: list_nat] :
% 0.30/0.54 ( ( ( size_size_list_nat @ Xs2 )
% 0.30/0.54 != ( size_size_list_nat @ Ys ) )
% 0.30/0.54 => ( Xs2 != Ys ) ) ).
% 0.30/0.54
% 0.30/0.54 % neq_if_length_neq
% 0.30/0.54 thf(fact_2379_neq__if__length__neq,axiom,
% 0.30/0.54 ! [Xs2: list_int,Ys: list_int] :
% 0.30/0.54 ( ( ( size_size_list_int @ Xs2 )
% 0.30/0.54 != ( size_size_list_int @ Ys ) )
% 0.30/0.54 => ( Xs2 != Ys ) ) ).
% 0.30/0.54
% 0.30/0.54 % neq_if_length_neq
% 0.30/0.54 thf(fact_2380_Ex__list__of__length,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ? [Xs3: list_VEBT_VEBT] :
% 0.30/0.54 ( ( size_s6755466524823107622T_VEBT @ Xs3 )
% 0.30/0.54 = N ) ).
% 0.30/0.54
% 0.30/0.54 % Ex_list_of_length
% 0.30/0.54 thf(fact_2381_Ex__list__of__length,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ? [Xs3: list_o] :
% 0.30/0.54 ( ( size_size_list_o @ Xs3 )
% 0.30/0.54 = N ) ).
% 0.30/0.54
% 0.30/0.54 % Ex_list_of_length
% 0.30/0.54 thf(fact_2382_Ex__list__of__length,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ? [Xs3: list_nat] :
% 0.30/0.54 ( ( size_size_list_nat @ Xs3 )
% 0.30/0.54 = N ) ).
% 0.30/0.54
% 0.30/0.54 % Ex_list_of_length
% 0.30/0.54 thf(fact_2383_Ex__list__of__length,axiom,
% 0.30/0.54 ! [N: nat] :
% 0.30/0.54 ? [Xs3: list_int] :
% 0.30/0.54 ( ( size_size_list_int @ Xs3 )
% 0.30/0.54 = N ) ).
% 0.30/0.54
% 0.30/0.54 % Ex_list_of_length
% 0.30/0.54 thf(fact_2384_ex__least__nat__less,axiom,
% 0.30/0.54 ! [P: nat > $o,N: nat] :
% 0.30/0.54 ( ( P @ N )
% 0.30/0.54 => ( ~ ( P @ zero_zero_nat )
% 0.30/0.54 => ? [K3: nat] :
% 0.30/0.54 ( ( ord_less_nat @ K3 @ N )
% 0.30/0.54 & ! [I: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ I @ K3 )
% 0.30/0.54 => ~ ( P @ I ) )
% 0.30/0.54 & ( P @ ( suc @ K3 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % ex_least_nat_less
% 0.30/0.54 thf(fact_2385_nat__induct__non__zero,axiom,
% 0.30/0.54 ! [N: nat,P: nat > $o] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54 => ( ( P @ one_one_nat )
% 0.30/0.54 => ( ! [N2: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.54 => ( ( P @ N2 )
% 0.30/0.54 => ( P @ ( suc @ N2 ) ) ) )
% 0.30/0.54 => ( P @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nat_induct_non_zero
% 0.30/0.54 thf(fact_2386_diff__Suc__less,axiom,
% 0.30/0.54 ! [N: nat,I2: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54 => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% 0.30/0.54
% 0.30/0.54 % diff_Suc_less
% 0.30/0.54 thf(fact_2387_one__less__mult,axiom,
% 0.30/0.54 ! [N: nat,M: nat] :
% 0.30/0.54 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.54 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.30/0.54 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % one_less_mult
% 0.30/0.54 thf(fact_2388_n__less__m__mult__n,axiom,
% 0.30/0.54 ! [N: nat,M: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.30/0.54 => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % n_less_m_mult_n
% 0.30/0.54 thf(fact_2389_n__less__n__mult__m,axiom,
% 0.30/0.54 ! [N: nat,M: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.30/0.54 => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % n_less_n_mult_m
% 0.30/0.54 thf(fact_2390_nat__mult__le__cancel1,axiom,
% 0.30/0.54 ! [K: nat,M: nat,N: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.54 => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.54 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nat_mult_le_cancel1
% 0.30/0.54 thf(fact_2391_power__gt__expt,axiom,
% 0.30/0.54 ! [N: nat,K: nat] :
% 0.30/0.54 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.54 => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_gt_expt
% 0.30/0.54 thf(fact_2392_nat__diff__split__asm,axiom,
% 0.30/0.54 ! [P: nat > $o,A: nat,B: nat] :
% 0.30/0.54 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 0.30/0.54 = ( ~ ( ( ( ord_less_nat @ A @ B )
% 0.30/0.54 & ~ ( P @ zero_zero_nat ) )
% 0.30/0.54 | ? [D2: nat] :
% 0.30/0.54 ( ( A
% 0.30/0.54 = ( plus_plus_nat @ B @ D2 ) )
% 0.30/0.54 & ~ ( P @ D2 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nat_diff_split_asm
% 0.30/0.54 thf(fact_2393_nat__diff__split,axiom,
% 0.30/0.54 ! [P: nat > $o,A: nat,B: nat] :
% 0.30/0.54 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 0.30/0.54 = ( ( ( ord_less_nat @ A @ B )
% 0.30/0.54 => ( P @ zero_zero_nat ) )
% 0.30/0.54 & ! [D2: nat] :
% 0.30/0.54 ( ( A
% 0.30/0.54 = ( plus_plus_nat @ B @ D2 ) )
% 0.30/0.54 => ( P @ D2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nat_diff_split
% 0.30/0.54 thf(fact_2394_div__le__mono2,axiom,
% 0.30/0.54 ! [M: nat,N: nat,K: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.54 => ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.54 => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % div_le_mono2
% 0.30/0.54 thf(fact_2395_div__greater__zero__iff,axiom,
% 0.30/0.54 ! [M: nat,N: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 0.30/0.54 = ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.54 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % div_greater_zero_iff
% 0.30/0.54 thf(fact_2396_nat__one__le__power,axiom,
% 0.30/0.54 ! [I2: nat,N: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I2 )
% 0.30/0.54 => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I2 @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nat_one_le_power
% 0.30/0.54 thf(fact_2397_div__less__dividend,axiom,
% 0.30/0.54 ! [N: nat,M: nat] :
% 0.30/0.54 ( ( ord_less_nat @ one_one_nat @ N )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.54 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % div_less_dividend
% 0.30/0.54 thf(fact_2398_div__eq__dividend__iff,axiom,
% 0.30/0.54 ! [M: nat,N: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.54 => ( ( ( divide_divide_nat @ M @ N )
% 0.30/0.54 = M )
% 0.30/0.54 = ( N = one_one_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % div_eq_dividend_iff
% 0.30/0.54 thf(fact_2399_div__less__iff__less__mult,axiom,
% 0.30/0.54 ! [Q3: nat,M: nat,N: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ Q3 )
% 0.30/0.54 => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N )
% 0.30/0.54 = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % div_less_iff_less_mult
% 0.30/0.54 thf(fact_2400_nat__mult__div__cancel1,axiom,
% 0.30/0.54 ! [K: nat,M: nat,N: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.54 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.54 = ( divide_divide_nat @ M @ N ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nat_mult_div_cancel1
% 0.30/0.54 thf(fact_2401_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
% 0.30/0.54 ! [A: $o,B: $o] :
% 0.30/0.54 ( ( vEBT_T_m_a_x_t @ ( vEBT_Leaf @ A @ B ) )
% 0.30/0.54 = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
% 0.30/0.54 thf(fact_2402_vebt__insert_Osimps_I3_J,axiom,
% 0.30/0.54 ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
% 0.30/0.54 ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ X2 )
% 0.30/0.54 = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) ) ).
% 0.30/0.54
% 0.30/0.54 % vebt_insert.simps(3)
% 0.30/0.54 thf(fact_2403_vebt__member_Osimps_I3_J,axiom,
% 0.30/0.54 ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
% 0.30/0.54 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X2 ) ).
% 0.30/0.54
% 0.30/0.54 % vebt_member.simps(3)
% 0.30/0.54 thf(fact_2404_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
% 0.30/0.54 ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
% 0.30/0.54 ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S ) @ X2 )
% 0.30/0.54 = one_one_nat ) ).
% 0.30/0.54
% 0.30/0.54 % T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
% 0.30/0.54 thf(fact_2405_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_2406_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
% 0.30/0.54 ! [X2: vEBT_VEBT,Y3: nat] :
% 0.30/0.54 ( ( ( vEBT_T_m_i_n_t @ X2 )
% 0.30/0.54 = Y3 )
% 0.30/0.54 => ( ! [A4: $o] :
% 0.30/0.54 ( ? [B4: $o] :
% 0.30/0.54 ( X2
% 0.30/0.54 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.54 => ( Y3
% 0.30/0.54 != ( plus_plus_nat @ one_one_nat @ ( if_nat @ A4 @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
% 0.30/0.54 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.54 ( X2
% 0.30/0.54 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.54 => ( Y3 != one_one_nat ) )
% 0.30/0.54 => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.54 ( X2
% 0.30/0.54 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.30/0.54 => ( Y3 != one_one_nat ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
% 0.30/0.54 thf(fact_2407_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
% 0.30/0.54 ! [X2: vEBT_VEBT] :
% 0.30/0.54 ( ! [A4: $o,B4: $o] :
% 0.30/0.54 ( X2
% 0.30/0.54 != ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.54 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.54 ( X2
% 0.30/0.54 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.54 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.54 ( X2
% 0.30/0.54 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
% 0.30/0.54 thf(fact_2408_VEBT__internal_OminNull_Oelims_I1_J,axiom,
% 0.30/0.54 ! [X2: vEBT_VEBT,Y3: $o] :
% 0.30/0.54 ( ( ( vEBT_VEBT_minNull @ X2 )
% 0.30/0.54 = Y3 )
% 0.30/0.54 => ( ( ( X2
% 0.30/0.54 = ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.54 => ~ Y3 )
% 0.30/0.54 => ( ( ? [Uv2: $o] :
% 0.30/0.54 ( X2
% 0.30/0.54 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.30/0.54 => Y3 )
% 0.30/0.54 => ( ( ? [Uu2: $o] :
% 0.30/0.54 ( X2
% 0.30/0.54 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.30/0.54 => Y3 )
% 0.30/0.54 => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.54 ( X2
% 0.30/0.54 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.30/0.54 => ~ Y3 )
% 0.30/0.54 => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.54 ( X2
% 0.30/0.54 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
% 0.30/0.54 => Y3 ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % VEBT_internal.minNull.elims(1)
% 0.30/0.54 thf(fact_2409_field__le__mult__one__interval,axiom,
% 0.30/0.54 ! [X2: real,Y3: real] :
% 0.30/0.54 ( ! [Z2: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ Z2 )
% 0.30/0.54 => ( ( ord_less_real @ Z2 @ one_one_real )
% 0.30/0.54 => ( ord_less_eq_real @ ( times_times_real @ Z2 @ X2 ) @ Y3 ) ) )
% 0.30/0.54 => ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 0.30/0.54
% 0.30/0.54 % field_le_mult_one_interval
% 0.30/0.54 thf(fact_2410_field__le__mult__one__interval,axiom,
% 0.30/0.54 ! [X2: rat,Y3: rat] :
% 0.30/0.54 ( ! [Z2: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 0.30/0.54 => ( ( ord_less_rat @ Z2 @ one_one_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X2 ) @ Y3 ) ) )
% 0.30/0.54 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 0.30/0.54
% 0.30/0.54 % field_le_mult_one_interval
% 0.30/0.54 thf(fact_2411_mult__le__cancel__left1,axiom,
% 0.30/0.54 ! [C: real,B: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
% 0.30/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_eq_real @ one_one_real @ B ) )
% 0.30/0.54 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_left1
% 0.30/0.54 thf(fact_2412_mult__le__cancel__left1,axiom,
% 0.30/0.54 ! [C: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
% 0.30/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 0.30/0.54 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_left1
% 0.30/0.54 thf(fact_2413_mult__le__cancel__left1,axiom,
% 0.30/0.54 ! [C: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
% 0.30/0.54 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_eq_int @ one_one_int @ B ) )
% 0.30/0.54 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.54 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_left1
% 0.30/0.54 thf(fact_2414_mult__le__cancel__left2,axiom,
% 0.30/0.54 ! [C: real,A: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 0.30/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.30/0.54 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_left2
% 0.30/0.54 thf(fact_2415_mult__le__cancel__left2,axiom,
% 0.30/0.54 ! [C: rat,A: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.30/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.30/0.54 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_left2
% 0.30/0.54 thf(fact_2416_mult__le__cancel__left2,axiom,
% 0.30/0.54 ! [C: int,A: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 0.30/0.54 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.30/0.54 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.54 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_left2
% 0.30/0.54 thf(fact_2417_mult__le__cancel__right1,axiom,
% 0.30/0.54 ! [C: real,B: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
% 0.30/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_eq_real @ one_one_real @ B ) )
% 0.30/0.54 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_right1
% 0.30/0.54 thf(fact_2418_mult__le__cancel__right1,axiom,
% 0.30/0.54 ! [C: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
% 0.30/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 0.30/0.54 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_right1
% 0.30/0.54 thf(fact_2419_mult__le__cancel__right1,axiom,
% 0.30/0.54 ! [C: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
% 0.30/0.54 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_eq_int @ one_one_int @ B ) )
% 0.30/0.54 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.54 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_right1
% 0.30/0.54 thf(fact_2420_mult__le__cancel__right2,axiom,
% 0.30/0.54 ! [A: real,C: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 0.30/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.30/0.54 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_right2
% 0.30/0.54 thf(fact_2421_mult__le__cancel__right2,axiom,
% 0.30/0.54 ! [A: rat,C: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.30/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.30/0.54 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_right2
% 0.30/0.54 thf(fact_2422_mult__le__cancel__right2,axiom,
% 0.30/0.54 ! [A: int,C: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 0.30/0.54 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.30/0.54 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.54 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_cancel_right2
% 0.30/0.54 thf(fact_2423_mult__less__cancel__left1,axiom,
% 0.30/0.54 ! [C: real,B: real] :
% 0.30/0.54 ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
% 0.30/0.54 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_real @ one_one_real @ B ) )
% 0.30/0.54 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_left1
% 0.30/0.54 thf(fact_2424_mult__less__cancel__left1,axiom,
% 0.30/0.54 ! [C: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
% 0.30/0.54 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_rat @ one_one_rat @ B ) )
% 0.30/0.54 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_left1
% 0.30/0.54 thf(fact_2425_mult__less__cancel__left1,axiom,
% 0.30/0.54 ! [C: int,B: int] :
% 0.30/0.54 ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
% 0.30/0.54 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_int @ one_one_int @ B ) )
% 0.30/0.54 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.54 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_left1
% 0.30/0.54 thf(fact_2426_mult__less__cancel__left2,axiom,
% 0.30/0.54 ! [C: real,A: real] :
% 0.30/0.54 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 0.30/0.54 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_real @ A @ one_one_real ) )
% 0.30/0.54 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_left2
% 0.30/0.54 thf(fact_2427_mult__less__cancel__left2,axiom,
% 0.30/0.54 ! [C: rat,A: rat] :
% 0.30/0.54 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.30/0.54 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_rat @ A @ one_one_rat ) )
% 0.30/0.54 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_left2
% 0.30/0.54 thf(fact_2428_mult__less__cancel__left2,axiom,
% 0.30/0.54 ! [C: int,A: int] :
% 0.30/0.54 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 0.30/0.54 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_int @ A @ one_one_int ) )
% 0.30/0.54 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.54 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_left2
% 0.30/0.54 thf(fact_2429_mult__less__cancel__right1,axiom,
% 0.30/0.54 ! [C: real,B: real] :
% 0.30/0.54 ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
% 0.30/0.54 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_real @ one_one_real @ B ) )
% 0.30/0.54 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_right1
% 0.30/0.54 thf(fact_2430_mult__less__cancel__right1,axiom,
% 0.30/0.54 ! [C: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
% 0.30/0.54 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_rat @ one_one_rat @ B ) )
% 0.30/0.54 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_right1
% 0.30/0.54 thf(fact_2431_mult__less__cancel__right1,axiom,
% 0.30/0.54 ! [C: int,B: int] :
% 0.30/0.54 ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
% 0.30/0.54 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_int @ one_one_int @ B ) )
% 0.30/0.54 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.54 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_right1
% 0.30/0.54 thf(fact_2432_mult__less__cancel__right2,axiom,
% 0.30/0.54 ! [A: real,C: real] :
% 0.30/0.54 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 0.30/0.54 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_real @ A @ one_one_real ) )
% 0.30/0.54 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_right2
% 0.30/0.54 thf(fact_2433_mult__less__cancel__right2,axiom,
% 0.30/0.54 ! [A: rat,C: rat] :
% 0.30/0.54 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.30/0.54 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_rat @ A @ one_one_rat ) )
% 0.30/0.54 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_right2
% 0.30/0.54 thf(fact_2434_mult__less__cancel__right2,axiom,
% 0.30/0.54 ! [A: int,C: int] :
% 0.30/0.54 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 0.30/0.54 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_int @ A @ one_one_int ) )
% 0.30/0.54 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.54 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_right2
% 0.30/0.54 thf(fact_2435_divide__left__mono__neg,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 @ C @ zero_zero_real )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.30/0.54 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_left_mono_neg
% 0.30/0.54 thf(fact_2436_divide__left__mono__neg,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 @ C @ zero_zero_rat )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.30/0.54 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_left_mono_neg
% 0.30/0.54 thf(fact_2437_mult__imp__le__div__pos,axiom,
% 0.30/0.54 ! [Y3: real,Z: real,X2: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.54 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y3 ) @ X2 )
% 0.30/0.54 => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_imp_le_div_pos
% 0.30/0.54 thf(fact_2438_mult__imp__le__div__pos,axiom,
% 0.30/0.54 ! [Y3: rat,Z: rat,X2: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.30/0.54 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y3 ) @ X2 )
% 0.30/0.54 => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_imp_le_div_pos
% 0.30/0.54 thf(fact_2439_mult__imp__div__pos__le,axiom,
% 0.30/0.54 ! [Y3: real,X2: real,Z: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.54 => ( ( ord_less_eq_real @ X2 @ ( times_times_real @ Z @ Y3 ) )
% 0.30/0.54 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ Z ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_imp_div_pos_le
% 0.30/0.54 thf(fact_2440_mult__imp__div__pos__le,axiom,
% 0.30/0.54 ! [Y3: rat,X2: rat,Z: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.30/0.54 => ( ( ord_less_eq_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) )
% 0.30/0.54 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ Z ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_imp_div_pos_le
% 0.30/0.54 thf(fact_2441_pos__le__divide__eq,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_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.30/0.54 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % pos_le_divide_eq
% 0.30/0.54 thf(fact_2442_pos__le__divide__eq,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_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.54 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % pos_le_divide_eq
% 0.30/0.54 thf(fact_2443_pos__divide__le__eq,axiom,
% 0.30/0.54 ! [C: real,B: real,A: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.30/0.54 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % pos_divide_le_eq
% 0.30/0.54 thf(fact_2444_pos__divide__le__eq,axiom,
% 0.30/0.54 ! [C: rat,B: rat,A: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.30/0.54 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % pos_divide_le_eq
% 0.30/0.54 thf(fact_2445_neg__le__divide__eq,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_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.30/0.54 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % neg_le_divide_eq
% 0.30/0.54 thf(fact_2446_neg__le__divide__eq,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_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.54 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % neg_le_divide_eq
% 0.30/0.54 thf(fact_2447_neg__divide__le__eq,axiom,
% 0.30/0.54 ! [C: real,B: real,A: real] :
% 0.30/0.54 ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.30/0.54 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % neg_divide_le_eq
% 0.30/0.54 thf(fact_2448_neg__divide__le__eq,axiom,
% 0.30/0.54 ! [C: rat,B: rat,A: rat] :
% 0.30/0.54 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.30/0.54 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % neg_divide_le_eq
% 0.30/0.54 thf(fact_2449_divide__left__mono,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 @ zero_zero_real @ C )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.30/0.54 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_left_mono
% 0.30/0.54 thf(fact_2450_divide__left__mono,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 @ zero_zero_rat @ C )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.30/0.54 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_left_mono
% 0.30/0.54 thf(fact_2451_le__divide__eq,axiom,
% 0.30/0.54 ! [A: real,B: real,C: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.30/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.30/0.54 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.30/0.54 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % le_divide_eq
% 0.30/0.54 thf(fact_2452_le__divide__eq,axiom,
% 0.30/0.54 ! [A: rat,B: rat,C: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.30/0.54 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.54 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % le_divide_eq
% 0.30/0.54 thf(fact_2453_divide__le__eq,axiom,
% 0.30/0.54 ! [B: real,C: real,A: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.30/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.30/0.54 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.30/0.54 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_le_eq
% 0.30/0.54 thf(fact_2454_divide__le__eq,axiom,
% 0.30/0.54 ! [B: rat,C: rat,A: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.30/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.54 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.30/0.54 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_le_eq
% 0.30/0.54 thf(fact_2455_divide__le__eq__1,axiom,
% 0.30/0.54 ! [B: real,A: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.30/0.54 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54 & ( ord_less_eq_real @ B @ A ) )
% 0.30/0.54 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.54 & ( ord_less_eq_real @ A @ B ) )
% 0.30/0.54 | ( A = zero_zero_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_le_eq_1
% 0.30/0.54 thf(fact_2456_divide__le__eq__1,axiom,
% 0.30/0.54 ! [B: rat,A: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.30/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54 & ( ord_less_eq_rat @ B @ A ) )
% 0.30/0.54 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.54 & ( ord_less_eq_rat @ A @ B ) )
% 0.30/0.54 | ( A = zero_zero_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_le_eq_1
% 0.30/0.54 thf(fact_2457_le__divide__eq__1,axiom,
% 0.30/0.54 ! [B: real,A: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.30/0.54 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54 & ( ord_less_eq_real @ A @ B ) )
% 0.30/0.54 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.54 & ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % le_divide_eq_1
% 0.30/0.54 thf(fact_2458_le__divide__eq__1,axiom,
% 0.30/0.54 ! [B: rat,A: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.30/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54 & ( ord_less_eq_rat @ A @ B ) )
% 0.30/0.54 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.54 & ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % le_divide_eq_1
% 0.30/0.54 thf(fact_2459_convex__bound__le,axiom,
% 0.30/0.54 ! [X2: real,A: real,Y3: real,U: real,V: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ X2 @ A )
% 0.30/0.54 => ( ( ord_less_eq_real @ Y3 @ A )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.30/0.54 => ( ( ( plus_plus_real @ U @ V )
% 0.30/0.54 = one_one_real )
% 0.30/0.54 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X2 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % convex_bound_le
% 0.30/0.54 thf(fact_2460_convex__bound__le,axiom,
% 0.30/0.54 ! [X2: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ X2 @ A )
% 0.30/0.54 => ( ( ord_less_eq_rat @ Y3 @ A )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.30/0.54 => ( ( ( plus_plus_rat @ U @ V )
% 0.30/0.54 = one_one_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X2 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % convex_bound_le
% 0.30/0.54 thf(fact_2461_convex__bound__le,axiom,
% 0.30/0.54 ! [X2: int,A: int,Y3: int,U: int,V: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ X2 @ A )
% 0.30/0.54 => ( ( ord_less_eq_int @ Y3 @ A )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.30/0.54 => ( ( ( plus_plus_int @ U @ V )
% 0.30/0.54 = one_one_int )
% 0.30/0.54 => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X2 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % convex_bound_le
% 0.30/0.54 thf(fact_2462_less__divide__eq__numeral_I1_J,axiom,
% 0.30/0.54 ! [W: num,B: real,C: real] :
% 0.30/0.54 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 0.30/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.30/0.54 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.30/0.54 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % less_divide_eq_numeral(1)
% 0.30/0.54 thf(fact_2463_less__divide__eq__numeral_I1_J,axiom,
% 0.30/0.54 ! [W: num,B: rat,C: rat] :
% 0.30/0.54 ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.30/0.54 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.30/0.54 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % less_divide_eq_numeral(1)
% 0.30/0.54 thf(fact_2464_divide__less__eq__numeral_I1_J,axiom,
% 0.30/0.54 ! [B: real,C: real,W: num] :
% 0.30/0.54 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 0.30/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.30/0.54 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.30/0.54 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_less_eq_numeral(1)
% 0.30/0.54 thf(fact_2465_divide__less__eq__numeral_I1_J,axiom,
% 0.30/0.54 ! [B: rat,C: rat,W: num] :
% 0.30/0.54 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 0.30/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.30/0.54 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.30/0.54 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divide_less_eq_numeral(1)
% 0.30/0.54 thf(fact_2466_frac__le__eq,axiom,
% 0.30/0.54 ! [Y3: real,Z: real,X2: real,W: real] :
% 0.30/0.54 ( ( Y3 != zero_zero_real )
% 0.30/0.54 => ( ( Z != zero_zero_real )
% 0.30/0.54 => ( ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 0.30/0.54 = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % frac_le_eq
% 0.30/0.54 thf(fact_2467_frac__le__eq,axiom,
% 0.30/0.54 ! [Y3: rat,Z: rat,X2: rat,W: rat] :
% 0.30/0.54 ( ( Y3 != zero_zero_rat )
% 0.30/0.54 => ( ( Z != zero_zero_rat )
% 0.30/0.54 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.30/0.54 = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % frac_le_eq
% 0.30/0.54 thf(fact_2468_power__Suc__less,axiom,
% 0.30/0.54 ! [A: real,N: nat] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_real @ A @ one_one_real )
% 0.30/0.54 => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_Suc_less
% 0.30/0.54 thf(fact_2469_power__Suc__less,axiom,
% 0.30/0.54 ! [A: rat,N: nat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.30/0.54 => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_Suc_less
% 0.30/0.54 thf(fact_2470_power__Suc__less,axiom,
% 0.30/0.54 ! [A: nat,N: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_nat @ A @ one_one_nat )
% 0.30/0.54 => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_Suc_less
% 0.30/0.54 thf(fact_2471_power__Suc__less,axiom,
% 0.30/0.54 ! [A: int,N: nat] :
% 0.30/0.54 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_int @ A @ one_one_int )
% 0.30/0.54 => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_Suc_less
% 0.30/0.54 thf(fact_2472_frac__less__eq,axiom,
% 0.30/0.54 ! [Y3: real,Z: real,X2: real,W: real] :
% 0.30/0.54 ( ( Y3 != zero_zero_real )
% 0.30/0.54 => ( ( Z != zero_zero_real )
% 0.30/0.54 => ( ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 0.30/0.54 = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % frac_less_eq
% 0.30/0.54 thf(fact_2473_frac__less__eq,axiom,
% 0.30/0.54 ! [Y3: rat,Z: rat,X2: rat,W: rat] :
% 0.30/0.54 ( ( Y3 != zero_zero_rat )
% 0.30/0.54 => ( ( Z != zero_zero_rat )
% 0.30/0.54 => ( ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.30/0.54 = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % frac_less_eq
% 0.30/0.54 thf(fact_2474_vebt__mint_Oelims,axiom,
% 0.30/0.54 ! [X2: vEBT_VEBT,Y3: option_nat] :
% 0.30/0.54 ( ( ( vEBT_vebt_mint @ X2 )
% 0.30/0.54 = Y3 )
% 0.30/0.54 => ( ! [A4: $o,B4: $o] :
% 0.30/0.54 ( ( X2
% 0.30/0.54 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.54 => ~ ( ( A4
% 0.30/0.54 => ( Y3
% 0.30/0.54 = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.54 & ( ~ A4
% 0.30/0.54 => ( ( B4
% 0.30/0.54 => ( Y3
% 0.30/0.54 = ( some_nat @ one_one_nat ) ) )
% 0.30/0.54 & ( ~ B4
% 0.30/0.54 => ( Y3 = none_nat ) ) ) ) ) )
% 0.30/0.54 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.54 ( X2
% 0.30/0.54 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.54 => ( Y3 != none_nat ) )
% 0.30/0.54 => ~ ! [Mi2: nat] :
% 0.30/0.54 ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.54 ( X2
% 0.30/0.54 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.30/0.54 => ( Y3
% 0.30/0.54 != ( some_nat @ Mi2 ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % vebt_mint.elims
% 0.30/0.54 thf(fact_2475_power__Suc__le__self,axiom,
% 0.30/0.54 ! [A: real,N: nat] :
% 0.30/0.54 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.30/0.54 => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_Suc_le_self
% 0.30/0.54 thf(fact_2476_power__Suc__le__self,axiom,
% 0.30/0.54 ! [A: rat,N: nat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_Suc_le_self
% 0.30/0.54 thf(fact_2477_power__Suc__le__self,axiom,
% 0.30/0.54 ! [A: nat,N: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.30/0.54 => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_Suc_le_self
% 0.30/0.54 thf(fact_2478_power__Suc__le__self,axiom,
% 0.30/0.54 ! [A: int,N: nat] :
% 0.30/0.54 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.30/0.54 => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_Suc_le_self
% 0.30/0.54 thf(fact_2479_power__Suc__less__one,axiom,
% 0.30/0.54 ! [A: real,N: nat] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_real @ A @ one_one_real )
% 0.30/0.54 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_Suc_less_one
% 0.30/0.54 thf(fact_2480_power__Suc__less__one,axiom,
% 0.30/0.54 ! [A: rat,N: nat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.30/0.54 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_Suc_less_one
% 0.30/0.54 thf(fact_2481_power__Suc__less__one,axiom,
% 0.30/0.54 ! [A: nat,N: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_nat @ A @ one_one_nat )
% 0.30/0.54 => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_Suc_less_one
% 0.30/0.54 thf(fact_2482_power__Suc__less__one,axiom,
% 0.30/0.54 ! [A: int,N: nat] :
% 0.30/0.54 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_int @ A @ one_one_int )
% 0.30/0.54 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_Suc_less_one
% 0.30/0.54 thf(fact_2483_vebt__maxt_Oelims,axiom,
% 0.30/0.54 ! [X2: vEBT_VEBT,Y3: option_nat] :
% 0.30/0.54 ( ( ( vEBT_vebt_maxt @ X2 )
% 0.30/0.54 = Y3 )
% 0.30/0.54 => ( ! [A4: $o,B4: $o] :
% 0.30/0.54 ( ( X2
% 0.30/0.54 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.54 => ~ ( ( B4
% 0.30/0.54 => ( Y3
% 0.30/0.54 = ( some_nat @ one_one_nat ) ) )
% 0.30/0.54 & ( ~ B4
% 0.30/0.54 => ( ( A4
% 0.30/0.54 => ( Y3
% 0.30/0.54 = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.54 & ( ~ A4
% 0.30/0.54 => ( Y3 = none_nat ) ) ) ) ) )
% 0.30/0.54 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.54 ( X2
% 0.30/0.54 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.54 => ( Y3 != none_nat ) )
% 0.30/0.54 => ~ ! [Mi2: nat,Ma2: nat] :
% 0.30/0.54 ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.54 ( X2
% 0.30/0.54 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.30/0.54 => ( Y3
% 0.30/0.54 != ( some_nat @ Ma2 ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % vebt_maxt.elims
% 0.30/0.54 thf(fact_2484_power__strict__decreasing,axiom,
% 0.30/0.54 ! [N: nat,N5: nat,A: real] :
% 0.30/0.54 ( ( ord_less_nat @ N @ N5 )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_real @ A @ one_one_real )
% 0.30/0.54 => ( ord_less_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_strict_decreasing
% 0.30/0.54 thf(fact_2485_power__strict__decreasing,axiom,
% 0.30/0.54 ! [N: nat,N5: nat,A: rat] :
% 0.30/0.54 ( ( ord_less_nat @ N @ N5 )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.30/0.54 => ( ord_less_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_strict_decreasing
% 0.30/0.54 thf(fact_2486_power__strict__decreasing,axiom,
% 0.30/0.54 ! [N: nat,N5: nat,A: nat] :
% 0.30/0.54 ( ( ord_less_nat @ N @ N5 )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_nat @ A @ one_one_nat )
% 0.30/0.54 => ( ord_less_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_strict_decreasing
% 0.30/0.54 thf(fact_2487_power__strict__decreasing,axiom,
% 0.30/0.54 ! [N: nat,N5: nat,A: int] :
% 0.30/0.54 ( ( ord_less_nat @ N @ N5 )
% 0.30/0.54 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_int @ A @ one_one_int )
% 0.30/0.54 => ( ord_less_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_strict_decreasing
% 0.30/0.54 thf(fact_2488_power__decreasing,axiom,
% 0.30/0.54 ! [N: nat,N5: nat,A: real] :
% 0.30/0.54 ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.30/0.54 => ( ord_less_eq_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_decreasing
% 0.30/0.54 thf(fact_2489_power__decreasing,axiom,
% 0.30/0.54 ! [N: nat,N5: nat,A: rat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_decreasing
% 0.30/0.54 thf(fact_2490_power__decreasing,axiom,
% 0.30/0.54 ! [N: nat,N5: nat,A: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.30/0.54 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_decreasing
% 0.30/0.54 thf(fact_2491_power__decreasing,axiom,
% 0.30/0.54 ! [N: nat,N5: nat,A: int] :
% 0.30/0.54 ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.30/0.54 => ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_decreasing
% 0.30/0.54 thf(fact_2492_zero__power2,axiom,
% 0.30/0.54 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.54 = zero_zero_rat ) ).
% 0.30/0.54
% 0.30/0.54 % zero_power2
% 0.30/0.54 thf(fact_2493_zero__power2,axiom,
% 0.30/0.54 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.54 = zero_zero_nat ) ).
% 0.30/0.54
% 0.30/0.54 % zero_power2
% 0.30/0.54 thf(fact_2494_zero__power2,axiom,
% 0.30/0.54 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.54 = zero_zero_real ) ).
% 0.30/0.54
% 0.30/0.54 % zero_power2
% 0.30/0.54 thf(fact_2495_zero__power2,axiom,
% 0.30/0.54 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.54 = zero_zero_int ) ).
% 0.30/0.54
% 0.30/0.54 % zero_power2
% 0.30/0.54 thf(fact_2496_zero__power2,axiom,
% 0.30/0.54 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.54 = zero_zero_complex ) ).
% 0.30/0.54
% 0.30/0.54 % zero_power2
% 0.30/0.54 thf(fact_2497_self__le__power,axiom,
% 0.30/0.54 ! [A: real,N: nat] :
% 0.30/0.54 ( ( ord_less_eq_real @ one_one_real @ A )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54 => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % self_le_power
% 0.30/0.54 thf(fact_2498_self__le__power,axiom,
% 0.30/0.54 ! [A: rat,N: nat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54 => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % self_le_power
% 0.30/0.54 thf(fact_2499_self__le__power,axiom,
% 0.30/0.54 ! [A: nat,N: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54 => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % self_le_power
% 0.30/0.54 thf(fact_2500_self__le__power,axiom,
% 0.30/0.54 ! [A: int,N: nat] :
% 0.30/0.54 ( ( ord_less_eq_int @ one_one_int @ A )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54 => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % self_le_power
% 0.30/0.54 thf(fact_2501_one__less__power,axiom,
% 0.30/0.54 ! [A: real,N: nat] :
% 0.30/0.54 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % one_less_power
% 0.30/0.54 thf(fact_2502_one__less__power,axiom,
% 0.30/0.54 ! [A: rat,N: nat] :
% 0.30/0.54 ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % one_less_power
% 0.30/0.54 thf(fact_2503_one__less__power,axiom,
% 0.30/0.54 ! [A: nat,N: nat] :
% 0.30/0.54 ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % one_less_power
% 0.30/0.54 thf(fact_2504_one__less__power,axiom,
% 0.30/0.54 ! [A: int,N: nat] :
% 0.30/0.54 ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % one_less_power
% 0.30/0.54 thf(fact_2505_numeral__2__eq__2,axiom,
% 0.30/0.54 ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 0.30/0.54 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % numeral_2_eq_2
% 0.30/0.54 thf(fact_2506_pos2,axiom,
% 0.30/0.54 ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 0.30/0.54
% 0.30/0.54 % pos2
% 0.30/0.54 thf(fact_2507_power__diff,axiom,
% 0.30/0.54 ! [A: complex,N: nat,M: nat] :
% 0.30/0.54 ( ( A != zero_zero_complex )
% 0.30/0.54 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.54 => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.54 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_diff
% 0.30/0.54 thf(fact_2508_power__diff,axiom,
% 0.30/0.54 ! [A: real,N: nat,M: nat] :
% 0.30/0.55 ( ( A != zero_zero_real )
% 0.30/0.55 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.55 => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.55 = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_diff
% 0.30/0.55 thf(fact_2509_power__diff,axiom,
% 0.30/0.55 ! [A: rat,N: nat,M: nat] :
% 0.30/0.55 ( ( A != zero_zero_rat )
% 0.30/0.55 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.55 => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.55 = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_diff
% 0.30/0.55 thf(fact_2510_power__diff,axiom,
% 0.30/0.55 ! [A: nat,N: nat,M: nat] :
% 0.30/0.55 ( ( A != zero_zero_nat )
% 0.30/0.55 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.55 => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.55 = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_diff
% 0.30/0.55 thf(fact_2511_power__diff,axiom,
% 0.30/0.55 ! [A: int,N: nat,M: nat] :
% 0.30/0.55 ( ( A != zero_zero_int )
% 0.30/0.55 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.55 => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.55 = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_diff
% 0.30/0.55 thf(fact_2512_Suc__diff__eq__diff__pred,axiom,
% 0.30/0.55 ! [N: nat,M: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.30/0.55 = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Suc_diff_eq_diff_pred
% 0.30/0.55 thf(fact_2513_Suc__pred_H,axiom,
% 0.30/0.55 ! [N: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55 => ( N
% 0.30/0.55 = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Suc_pred'
% 0.30/0.55 thf(fact_2514_div__if,axiom,
% 0.30/0.55 ( divide_divide_nat
% 0.30/0.55 = ( ^ [M3: nat,N3: nat] :
% 0.30/0.55 ( if_nat
% 0.30/0.55 @ ( ( ord_less_nat @ M3 @ N3 )
% 0.30/0.55 | ( N3 = zero_zero_nat ) )
% 0.30/0.55 @ zero_zero_nat
% 0.30/0.55 @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N3 ) @ N3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % div_if
% 0.30/0.55 thf(fact_2515_add__eq__if,axiom,
% 0.30/0.55 ( plus_plus_nat
% 0.30/0.55 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_eq_if
% 0.30/0.55 thf(fact_2516_less__eq__div__iff__mult__less__eq,axiom,
% 0.30/0.55 ! [Q3: nat,M: nat,N: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ Q3 )
% 0.30/0.55 => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q3 ) )
% 0.30/0.55 = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q3 ) @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % less_eq_div_iff_mult_less_eq
% 0.30/0.55 thf(fact_2517_dividend__less__times__div,axiom,
% 0.30/0.55 ! [N: nat,M: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % dividend_less_times_div
% 0.30/0.55 thf(fact_2518_dividend__less__div__times,axiom,
% 0.30/0.55 ! [N: nat,M: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % dividend_less_div_times
% 0.30/0.55 thf(fact_2519_split__div,axiom,
% 0.30/0.55 ! [P: nat > $o,M: nat,N: nat] :
% 0.30/0.55 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 0.30/0.55 = ( ( ( N = zero_zero_nat )
% 0.30/0.55 => ( P @ zero_zero_nat ) )
% 0.30/0.55 & ( ( N != zero_zero_nat )
% 0.30/0.55 => ! [I4: nat,J3: nat] :
% 0.30/0.55 ( ( ord_less_nat @ J3 @ N )
% 0.30/0.55 => ( ( M
% 0.30/0.55 = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 0.30/0.55 => ( P @ I4 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % split_div
% 0.30/0.55 thf(fact_2520_mult__eq__if,axiom,
% 0.30/0.55 ( times_times_nat
% 0.30/0.55 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_eq_if
% 0.30/0.55 thf(fact_2521_vebt__member_Osimps_I4_J,axiom,
% 0.30/0.55 ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
% 0.30/0.55 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X2 ) ).
% 0.30/0.55
% 0.30/0.55 % vebt_member.simps(4)
% 0.30/0.55 thf(fact_2522_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
% 0.30/0.55 ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
% 0.30/0.55 ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ X2 )
% 0.30/0.55 = one_one_nat ) ).
% 0.30/0.55
% 0.30/0.55 % T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
% 0.30/0.55 thf(fact_2523_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
% 0.30/0.55 ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X2: nat] :
% 0.30/0.55 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X2 )
% 0.30/0.55 = ( ( X2 = Mi )
% 0.30/0.55 | ( X2 = Ma ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % VEBT_internal.membermima.simps(3)
% 0.30/0.55 thf(fact_2524_vebt__pred_Osimps_I5_J,axiom,
% 0.30/0.55 ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
% 0.30/0.55 ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve2 ) @ Vf2 )
% 0.30/0.55 = none_nat ) ).
% 0.30/0.55
% 0.30/0.55 % vebt_pred.simps(5)
% 0.30/0.55 thf(fact_2525_vebt__succ_Osimps_I4_J,axiom,
% 0.30/0.55 ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve2: nat] :
% 0.30/0.55 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve2 )
% 0.30/0.55 = none_nat ) ).
% 0.30/0.55
% 0.30/0.55 % vebt_succ.simps(4)
% 0.30/0.55 thf(fact_2526_convex__bound__lt,axiom,
% 0.30/0.55 ! [X2: real,A: real,Y3: real,U: real,V: real] :
% 0.30/0.55 ( ( ord_less_real @ X2 @ A )
% 0.30/0.55 => ( ( ord_less_real @ Y3 @ A )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.30/0.55 => ( ( ( plus_plus_real @ U @ V )
% 0.30/0.55 = one_one_real )
% 0.30/0.55 => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X2 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % convex_bound_lt
% 0.30/0.55 thf(fact_2527_convex__bound__lt,axiom,
% 0.30/0.55 ! [X2: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 0.30/0.55 ( ( ord_less_rat @ X2 @ A )
% 0.30/0.55 => ( ( ord_less_rat @ Y3 @ A )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.30/0.55 => ( ( ( plus_plus_rat @ U @ V )
% 0.30/0.55 = one_one_rat )
% 0.30/0.55 => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X2 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % convex_bound_lt
% 0.30/0.55 thf(fact_2528_convex__bound__lt,axiom,
% 0.30/0.55 ! [X2: int,A: int,Y3: int,U: int,V: int] :
% 0.30/0.55 ( ( ord_less_int @ X2 @ A )
% 0.30/0.55 => ( ( ord_less_int @ Y3 @ A )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.30/0.55 => ( ( ( plus_plus_int @ U @ V )
% 0.30/0.55 = one_one_int )
% 0.30/0.55 => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X2 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % convex_bound_lt
% 0.30/0.55 thf(fact_2529_le__divide__eq__numeral_I1_J,axiom,
% 0.30/0.55 ! [W: num,B: real,C: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 0.30/0.55 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.30/0.55 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.30/0.55 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55 => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % le_divide_eq_numeral(1)
% 0.30/0.55 thf(fact_2530_le__divide__eq__numeral_I1_J,axiom,
% 0.30/0.55 ! [W: num,B: rat,C: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.30/0.55 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.30/0.55 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % le_divide_eq_numeral(1)
% 0.30/0.55 thf(fact_2531_divide__le__eq__numeral_I1_J,axiom,
% 0.30/0.55 ! [B: real,C: real,W: num] :
% 0.30/0.55 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 0.30/0.55 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.30/0.55 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.30/0.55 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55 => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_le_eq_numeral(1)
% 0.30/0.55 thf(fact_2532_divide__le__eq__numeral_I1_J,axiom,
% 0.30/0.55 ! [B: rat,C: rat,W: num] :
% 0.30/0.55 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 0.30/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.30/0.55 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.30/0.55 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_le_eq_numeral(1)
% 0.30/0.55 thf(fact_2533_half__gt__zero__iff,axiom,
% 0.30/0.55 ! [A: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.55 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/0.55
% 0.30/0.55 % half_gt_zero_iff
% 0.30/0.55 thf(fact_2534_half__gt__zero__iff,axiom,
% 0.30/0.55 ! [A: rat] :
% 0.30/0.55 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.30/0.55 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.55
% 0.30/0.55 % half_gt_zero_iff
% 0.30/0.55 thf(fact_2535_half__gt__zero,axiom,
% 0.30/0.55 ! [A: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.55 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % half_gt_zero
% 0.30/0.55 thf(fact_2536_half__gt__zero,axiom,
% 0.30/0.55 ! [A: rat] :
% 0.30/0.55 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.55 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % half_gt_zero
% 0.30/0.55 thf(fact_2537_scaling__mono,axiom,
% 0.30/0.55 ! [U: real,V: real,R2: real,S: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ U @ V )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 0.30/0.55 => ( ( ord_less_eq_real @ R2 @ S )
% 0.30/0.55 => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % scaling_mono
% 0.30/0.55 thf(fact_2538_scaling__mono,axiom,
% 0.30/0.55 ! [U: rat,V: rat,R2: rat,S: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ U @ V )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 0.30/0.55 => ( ( ord_less_eq_rat @ R2 @ S )
% 0.30/0.55 => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % scaling_mono
% 0.30/0.55 thf(fact_2539_power2__le__imp__le,axiom,
% 0.30/0.55 ! [X2: real,Y3: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.55 => ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power2_le_imp_le
% 0.30/0.55 thf(fact_2540_power2__le__imp__le,axiom,
% 0.30/0.55 ! [X2: rat,Y3: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.55 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power2_le_imp_le
% 0.30/0.55 thf(fact_2541_power2__le__imp__le,axiom,
% 0.30/0.55 ! [X2: nat,Y3: nat] :
% 0.30/0.55 ( ( ord_less_eq_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.30/0.55 => ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power2_le_imp_le
% 0.30/0.55 thf(fact_2542_power2__le__imp__le,axiom,
% 0.30/0.55 ! [X2: int,Y3: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.55 => ( ord_less_eq_int @ X2 @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power2_le_imp_le
% 0.30/0.55 thf(fact_2543_power2__eq__imp__eq,axiom,
% 0.30/0.55 ! [X2: real,Y3: real] :
% 0.30/0.55 ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.55 => ( X2 = Y3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power2_eq_imp_eq
% 0.30/0.55 thf(fact_2544_power2__eq__imp__eq,axiom,
% 0.30/0.55 ! [X2: rat,Y3: rat] :
% 0.30/0.55 ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.55 => ( X2 = Y3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power2_eq_imp_eq
% 0.30/0.55 thf(fact_2545_power2__eq__imp__eq,axiom,
% 0.30/0.55 ! [X2: nat,Y3: nat] :
% 0.30/0.55 ( ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55 = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
% 0.30/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.30/0.55 => ( X2 = Y3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power2_eq_imp_eq
% 0.30/0.55 thf(fact_2546_power2__eq__imp__eq,axiom,
% 0.30/0.55 ! [X2: int,Y3: int] :
% 0.30/0.55 ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.55 => ( X2 = Y3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power2_eq_imp_eq
% 0.30/0.55 thf(fact_2547_zero__le__power2,axiom,
% 0.30/0.55 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_le_power2
% 0.30/0.55 thf(fact_2548_zero__le__power2,axiom,
% 0.30/0.55 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_le_power2
% 0.30/0.55 thf(fact_2549_zero__le__power2,axiom,
% 0.30/0.55 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_le_power2
% 0.30/0.55 thf(fact_2550_power2__less__0,axiom,
% 0.30/0.55 ! [A: real] :
% 0.30/0.55 ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 0.30/0.55
% 0.30/0.55 % power2_less_0
% 0.30/0.55 thf(fact_2551_power2__less__0,axiom,
% 0.30/0.55 ! [A: rat] :
% 0.30/0.55 ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 0.30/0.55
% 0.30/0.55 % power2_less_0
% 0.30/0.55 thf(fact_2552_power2__less__0,axiom,
% 0.30/0.55 ! [A: int] :
% 0.30/0.55 ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 0.30/0.55
% 0.30/0.55 % power2_less_0
% 0.30/0.55 thf(fact_2553_exp__add__not__zero__imp__right,axiom,
% 0.30/0.55 ! [M: nat,N: nat] :
% 0.30/0.55 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.55 != zero_zero_nat )
% 0.30/0.55 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.55 != zero_zero_nat ) ) ).
% 0.30/0.55
% 0.30/0.55 % exp_add_not_zero_imp_right
% 0.30/0.55 thf(fact_2554_exp__add__not__zero__imp__right,axiom,
% 0.30/0.55 ! [M: nat,N: nat] :
% 0.30/0.55 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.55 != zero_zero_int )
% 0.30/0.55 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.30/0.55 != zero_zero_int ) ) ).
% 0.30/0.55
% 0.30/0.55 % exp_add_not_zero_imp_right
% 0.30/0.55 thf(fact_2555_exp__add__not__zero__imp__left,axiom,
% 0.30/0.55 ! [M: nat,N: nat] :
% 0.30/0.55 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.55 != zero_zero_nat )
% 0.30/0.55 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.30/0.55 != zero_zero_nat ) ) ).
% 0.30/0.55
% 0.30/0.55 % exp_add_not_zero_imp_left
% 0.30/0.55 thf(fact_2556_exp__add__not__zero__imp__left,axiom,
% 0.30/0.55 ! [M: nat,N: nat] :
% 0.30/0.55 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.55 != zero_zero_int )
% 0.30/0.55 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.30/0.55 != zero_zero_int ) ) ).
% 0.30/0.55
% 0.30/0.55 % exp_add_not_zero_imp_left
% 0.30/0.55 thf(fact_2557_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.30/0.55 ! [N: nat,M: nat] :
% 0.30/0.55 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.55 != zero_zero_nat )
% 0.30/0.55 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 0.30/0.55 != zero_zero_nat ) ) ).
% 0.30/0.55
% 0.30/0.55 % exp_not_zero_imp_exp_diff_not_zero
% 0.30/0.55 thf(fact_2558_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.30/0.55 ! [N: nat,M: nat] :
% 0.30/0.55 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.30/0.55 != zero_zero_int )
% 0.30/0.55 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 0.30/0.55 != zero_zero_int ) ) ).
% 0.30/0.55
% 0.30/0.55 % exp_not_zero_imp_exp_diff_not_zero
% 0.30/0.55 thf(fact_2559_power__diff__power__eq,axiom,
% 0.30/0.55 ! [A: nat,N: nat,M: nat] :
% 0.30/0.55 ( ( A != zero_zero_nat )
% 0.30/0.55 => ( ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.55 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.30/0.55 = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.30/0.55 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.30/0.55 = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_diff_power_eq
% 0.30/0.55 thf(fact_2560_power__diff__power__eq,axiom,
% 0.30/0.55 ! [A: int,N: nat,M: nat] :
% 0.30/0.55 ( ( A != zero_zero_int )
% 0.30/0.55 => ( ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.55 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.30/0.55 = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 0.30/0.55 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.30/0.55 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.30/0.55 = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_diff_power_eq
% 0.30/0.55 thf(fact_2561_less__2__cases__iff,axiom,
% 0.30/0.55 ! [N: nat] :
% 0.30/0.55 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55 = ( ( N = zero_zero_nat )
% 0.30/0.55 | ( N
% 0.30/0.55 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % less_2_cases_iff
% 0.30/0.55 thf(fact_2562_less__2__cases,axiom,
% 0.30/0.55 ! [N: nat] :
% 0.30/0.55 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55 => ( ( N = zero_zero_nat )
% 0.30/0.55 | ( N
% 0.30/0.55 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % less_2_cases
% 0.30/0.55 thf(fact_2563_nat__induct2,axiom,
% 0.30/0.55 ! [P: nat > $o,N: nat] :
% 0.30/0.55 ( ( P @ zero_zero_nat )
% 0.30/0.55 => ( ( P @ one_one_nat )
% 0.30/0.55 => ( ! [N2: nat] :
% 0.30/0.55 ( ( P @ N2 )
% 0.30/0.55 => ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.55 => ( P @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nat_induct2
% 0.30/0.55 thf(fact_2564_power__eq__if,axiom,
% 0.30/0.55 ( power_power_complex
% 0.30/0.55 = ( ^ [P4: complex,M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P4 @ ( power_power_complex @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_eq_if
% 0.30/0.55 thf(fact_2565_power__eq__if,axiom,
% 0.30/0.55 ( power_power_real
% 0.30/0.55 = ( ^ [P4: real,M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P4 @ ( power_power_real @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_eq_if
% 0.30/0.55 thf(fact_2566_power__eq__if,axiom,
% 0.30/0.55 ( power_power_rat
% 0.30/0.55 = ( ^ [P4: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P4 @ ( power_power_rat @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_eq_if
% 0.30/0.55 thf(fact_2567_power__eq__if,axiom,
% 0.30/0.55 ( power_power_nat
% 0.30/0.55 = ( ^ [P4: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P4 @ ( power_power_nat @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_eq_if
% 0.30/0.55 thf(fact_2568_power__eq__if,axiom,
% 0.30/0.55 ( power_power_int
% 0.30/0.55 = ( ^ [P4: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P4 @ ( power_power_int @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_eq_if
% 0.30/0.55 thf(fact_2569_power__minus__mult,axiom,
% 0.30/0.55 ! [N: nat,A: complex] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55 => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.30/0.55 = ( power_power_complex @ A @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_minus_mult
% 0.30/0.55 thf(fact_2570_power__minus__mult,axiom,
% 0.30/0.55 ! [N: nat,A: real] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55 => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.30/0.55 = ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_minus_mult
% 0.30/0.55 thf(fact_2571_power__minus__mult,axiom,
% 0.30/0.55 ! [N: nat,A: rat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55 => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.30/0.55 = ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_minus_mult
% 0.30/0.55 thf(fact_2572_power__minus__mult,axiom,
% 0.30/0.55 ! [N: nat,A: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55 => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.30/0.55 = ( power_power_nat @ A @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_minus_mult
% 0.30/0.55 thf(fact_2573_power__minus__mult,axiom,
% 0.30/0.55 ! [N: nat,A: int] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55 => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.30/0.55 = ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_minus_mult
% 0.30/0.55 thf(fact_2574_le__div__geq,axiom,
% 0.30/0.55 ! [N: nat,M: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.55 => ( ( divide_divide_nat @ M @ N )
% 0.30/0.55 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % le_div_geq
% 0.30/0.55 thf(fact_2575_split__div_H,axiom,
% 0.30/0.55 ! [P: nat > $o,M: nat,N: nat] :
% 0.30/0.55 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 0.30/0.55 = ( ( ( N = zero_zero_nat )
% 0.30/0.55 & ( P @ zero_zero_nat ) )
% 0.30/0.55 | ? [Q4: nat] :
% 0.30/0.55 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
% 0.30/0.55 & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
% 0.30/0.55 & ( P @ Q4 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % split_div'
% 0.30/0.55 thf(fact_2576_vebt__pred_Osimps_I6_J,axiom,
% 0.30/0.55 ! [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
% 0.30/0.55 ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 )
% 0.30/0.55 = none_nat ) ).
% 0.30/0.55
% 0.30/0.55 % vebt_pred.simps(6)
% 0.30/0.55 thf(fact_2577_vebt__succ_Osimps_I5_J,axiom,
% 0.30/0.55 ! [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
% 0.30/0.55 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 )
% 0.30/0.55 = none_nat ) ).
% 0.30/0.55
% 0.30/0.55 % vebt_succ.simps(5)
% 0.30/0.55 thf(fact_2578_power2__less__imp__less,axiom,
% 0.30/0.55 ! [X2: real,Y3: real] :
% 0.30/0.55 ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.55 => ( ord_less_real @ X2 @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power2_less_imp_less
% 0.30/0.55 thf(fact_2579_power2__less__imp__less,axiom,
% 0.30/0.55 ! [X2: rat,Y3: rat] :
% 0.30/0.55 ( ( ord_less_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.55 => ( ord_less_rat @ X2 @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power2_less_imp_less
% 0.30/0.55 thf(fact_2580_power2__less__imp__less,axiom,
% 0.30/0.55 ! [X2: nat,Y3: nat] :
% 0.30/0.55 ( ( ord_less_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.30/0.55 => ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power2_less_imp_less
% 0.30/0.55 thf(fact_2581_power2__less__imp__less,axiom,
% 0.30/0.55 ! [X2: int,Y3: int] :
% 0.30/0.55 ( ( ord_less_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.55 => ( ord_less_int @ X2 @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power2_less_imp_less
% 0.30/0.55 thf(fact_2582_sum__power2__ge__zero,axiom,
% 0.30/0.55 ! [X2: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sum_power2_ge_zero
% 0.30/0.55 thf(fact_2583_sum__power2__ge__zero,axiom,
% 0.30/0.55 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sum_power2_ge_zero
% 0.30/0.55 thf(fact_2584_sum__power2__ge__zero,axiom,
% 0.30/0.55 ! [X2: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sum_power2_ge_zero
% 0.30/0.55 thf(fact_2585_sum__power2__le__zero__iff,axiom,
% 0.30/0.55 ! [X2: real,Y3: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 0.30/0.55 = ( ( X2 = zero_zero_real )
% 0.30/0.55 & ( Y3 = zero_zero_real ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sum_power2_le_zero_iff
% 0.30/0.55 thf(fact_2586_sum__power2__le__zero__iff,axiom,
% 0.30/0.55 ! [X2: rat,Y3: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 0.30/0.55 = ( ( X2 = zero_zero_rat )
% 0.30/0.55 & ( Y3 = zero_zero_rat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sum_power2_le_zero_iff
% 0.30/0.55 thf(fact_2587_sum__power2__le__zero__iff,axiom,
% 0.30/0.55 ! [X2: int,Y3: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 0.30/0.55 = ( ( X2 = zero_zero_int )
% 0.30/0.55 & ( Y3 = zero_zero_int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sum_power2_le_zero_iff
% 0.30/0.55 thf(fact_2588_sum__power2__gt__zero__iff,axiom,
% 0.30/0.55 ! [X2: real,Y3: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.55 = ( ( X2 != zero_zero_real )
% 0.30/0.55 | ( Y3 != zero_zero_real ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sum_power2_gt_zero_iff
% 0.30/0.55 thf(fact_2589_sum__power2__gt__zero__iff,axiom,
% 0.30/0.55 ! [X2: rat,Y3: rat] :
% 0.30/0.55 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.55 = ( ( X2 != zero_zero_rat )
% 0.30/0.55 | ( Y3 != zero_zero_rat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sum_power2_gt_zero_iff
% 0.30/0.55 thf(fact_2590_sum__power2__gt__zero__iff,axiom,
% 0.30/0.55 ! [X2: int,Y3: int] :
% 0.30/0.55 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.55 = ( ( X2 != zero_zero_int )
% 0.30/0.55 | ( Y3 != zero_zero_int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sum_power2_gt_zero_iff
% 0.30/0.55 thf(fact_2591_not__sum__power2__lt__zero,axiom,
% 0.30/0.55 ! [X2: real,Y3: real] :
% 0.30/0.55 ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 0.30/0.55
% 0.30/0.55 % not_sum_power2_lt_zero
% 0.30/0.55 thf(fact_2592_not__sum__power2__lt__zero,axiom,
% 0.30/0.55 ! [X2: rat,Y3: rat] :
% 0.30/0.55 ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 0.30/0.55
% 0.30/0.55 % not_sum_power2_lt_zero
% 0.30/0.55 thf(fact_2593_not__sum__power2__lt__zero,axiom,
% 0.30/0.55 ! [X2: int,Y3: int] :
% 0.30/0.55 ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 0.30/0.55
% 0.30/0.55 % not_sum_power2_lt_zero
% 0.30/0.55 thf(fact_2594_zero__le__even__power_H,axiom,
% 0.30/0.55 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_le_even_power'
% 0.30/0.55 thf(fact_2595_zero__le__even__power_H,axiom,
% 0.30/0.55 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_le_even_power'
% 0.30/0.55 thf(fact_2596_zero__le__even__power_H,axiom,
% 0.30/0.55 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_le_even_power'
% 0.30/0.55 thf(fact_2597_nat__bit__induct,axiom,
% 0.30/0.55 ! [P: nat > $o,N: nat] :
% 0.30/0.55 ( ( P @ zero_zero_nat )
% 0.30/0.55 => ( ! [N2: nat] :
% 0.30/0.55 ( ( P @ N2 )
% 0.30/0.55 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.55 => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.30/0.55 => ( ! [N2: nat] :
% 0.30/0.55 ( ( P @ N2 )
% 0.30/0.55 => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.30/0.55 => ( P @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nat_bit_induct
% 0.30/0.55 thf(fact_2598_Suc__n__div__2__gt__zero,axiom,
% 0.30/0.55 ! [N: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Suc_n_div_2_gt_zero
% 0.30/0.55 thf(fact_2599_div__2__gt__zero,axiom,
% 0.30/0.55 ! [N: nat] :
% 0.30/0.55 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.55 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % div_2_gt_zero
% 0.30/0.55 thf(fact_2600_length__induct,axiom,
% 0.30/0.55 ! [P: list_VEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
% 0.30/0.55 ( ! [Xs3: list_VEBT_VEBT] :
% 0.30/0.55 ( ! [Ys2: list_VEBT_VEBT] :
% 0.30/0.55 ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
% 0.30/0.55 => ( P @ Ys2 ) )
% 0.30/0.55 => ( P @ Xs3 ) )
% 0.30/0.55 => ( P @ Xs2 ) ) ).
% 0.30/0.55
% 0.30/0.55 % length_induct
% 0.30/0.55 thf(fact_2601_length__induct,axiom,
% 0.30/0.55 ! [P: list_o > $o,Xs2: list_o] :
% 0.30/0.55 ( ! [Xs3: list_o] :
% 0.30/0.55 ( ! [Ys2: list_o] :
% 0.30/0.55 ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
% 0.30/0.55 => ( P @ Ys2 ) )
% 0.30/0.55 => ( P @ Xs3 ) )
% 0.30/0.55 => ( P @ Xs2 ) ) ).
% 0.30/0.55
% 0.30/0.55 % length_induct
% 0.30/0.55 thf(fact_2602_length__induct,axiom,
% 0.30/0.55 ! [P: list_nat > $o,Xs2: list_nat] :
% 0.30/0.55 ( ! [Xs3: list_nat] :
% 0.30/0.55 ( ! [Ys2: list_nat] :
% 0.30/0.55 ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
% 0.30/0.55 => ( P @ Ys2 ) )
% 0.30/0.55 => ( P @ Xs3 ) )
% 0.30/0.55 => ( P @ Xs2 ) ) ).
% 0.30/0.55
% 0.30/0.55 % length_induct
% 0.30/0.55 thf(fact_2603_length__induct,axiom,
% 0.30/0.55 ! [P: list_int > $o,Xs2: list_int] :
% 0.30/0.55 ( ! [Xs3: list_int] :
% 0.30/0.55 ( ! [Ys2: list_int] :
% 0.30/0.55 ( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs3 ) )
% 0.30/0.55 => ( P @ Ys2 ) )
% 0.30/0.55 => ( P @ Xs3 ) )
% 0.30/0.55 => ( P @ Xs2 ) ) ).
% 0.30/0.55
% 0.30/0.55 % length_induct
% 0.30/0.55 thf(fact_2604_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.55 ( ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ! [A4: $o,B4: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.55 => A4 )
% 0.30/0.55 & ( ( Xa2 != zero_zero_nat )
% 0.30/0.55 => ( ( ( Xa2 = one_one_nat )
% 0.30/0.55 => B4 )
% 0.30/0.55 & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 0.30/0.55 => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.30/0.55 => Y3 )
% 0.30/0.55 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
% 0.30/0.55 ( ? [S2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % VEBT_internal.naive_member.elims(1)
% 0.30/0.55 thf(fact_2605_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.30/0.55 ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 0.30/0.55 => ( ! [A4: $o,B4: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.55 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.55 => A4 )
% 0.30/0.55 & ( ( Xa2 != zero_zero_nat )
% 0.30/0.55 => ( ( ( Xa2 = one_one_nat )
% 0.30/0.55 => B4 )
% 0.30/0.55 & ( Xa2 = one_one_nat ) ) ) ) )
% 0.30/0.55 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
% 0.30/0.55 ( ? [S2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) )
% 0.30/0.55 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % VEBT_internal.naive_member.elims(2)
% 0.30/0.55 thf(fact_2606_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.30/0.55 ( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 0.30/0.55 => ( ! [A4: $o,B4: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.55 => ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.55 => A4 )
% 0.30/0.55 & ( ( Xa2 != zero_zero_nat )
% 0.30/0.55 => ( ( ( Xa2 = one_one_nat )
% 0.30/0.55 => B4 )
% 0.30/0.55 & ( Xa2 = one_one_nat ) ) ) ) )
% 0.30/0.55 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.30/0.55 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
% 0.30/0.55 ( ? [S2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) )
% 0.30/0.55 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % VEBT_internal.naive_member.elims(3)
% 0.30/0.55 thf(fact_2607_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Y3: nat] :
% 0.30/0.55 ( ( ( vEBT_T_m_a_x_t @ X2 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ! [A4: $o,B4: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 != ( plus_plus_nat @ one_one_nat @ ( if_nat @ B4 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
% 0.30/0.55 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.55 => ( Y3 != one_one_nat ) )
% 0.30/0.55 => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.30/0.55 => ( Y3 != one_one_nat ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
% 0.30/0.55 thf(fact_2608_odd__0__le__power__imp__0__le,axiom,
% 0.30/0.55 ! [A: real,N: nat] :
% 0.30/0.55 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.55 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.55
% 0.30/0.55 % odd_0_le_power_imp_0_le
% 0.30/0.55 thf(fact_2609_odd__0__le__power__imp__0__le,axiom,
% 0.30/0.55 ! [A: rat,N: nat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.55 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.55
% 0.30/0.55 % odd_0_le_power_imp_0_le
% 0.30/0.55 thf(fact_2610_odd__0__le__power__imp__0__le,axiom,
% 0.30/0.55 ! [A: int,N: nat] :
% 0.30/0.55 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.55 => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.30/0.55
% 0.30/0.55 % odd_0_le_power_imp_0_le
% 0.30/0.55 thf(fact_2611_odd__power__less__zero,axiom,
% 0.30/0.55 ! [A: real,N: nat] :
% 0.30/0.55 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.55 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% 0.30/0.55
% 0.30/0.55 % odd_power_less_zero
% 0.30/0.55 thf(fact_2612_odd__power__less__zero,axiom,
% 0.30/0.55 ! [A: rat,N: nat] :
% 0.30/0.55 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% 0.30/0.55
% 0.30/0.55 % odd_power_less_zero
% 0.30/0.55 thf(fact_2613_odd__power__less__zero,axiom,
% 0.30/0.55 ! [A: int,N: nat] :
% 0.30/0.55 ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.55 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% 0.30/0.55
% 0.30/0.55 % odd_power_less_zero
% 0.30/0.55 thf(fact_2614_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 0.30/0.55 ! [X2: nat,N: nat,M: nat] :
% 0.30/0.55 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.30/0.55 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.55 => ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % VEBT_internal.exp_split_high_low(1)
% 0.30/0.55 thf(fact_2615_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 0.30/0.55 ! [X2: nat,N: nat,M: nat] :
% 0.30/0.55 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.30/0.55 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.55 => ( ord_less_nat @ ( vEBT_VEBT_low @ X2 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % VEBT_internal.exp_split_high_low(2)
% 0.30/0.55 thf(fact_2616_vebt__member_Oelims_I2_J,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.30/0.55 ( ( vEBT_vebt_member @ X2 @ Xa2 )
% 0.30/0.55 => ( ! [A4: $o,B4: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.55 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.55 => A4 )
% 0.30/0.55 & ( ( Xa2 != zero_zero_nat )
% 0.30/0.55 => ( ( ( Xa2 = one_one_nat )
% 0.30/0.55 => B4 )
% 0.30/0.55 & ( Xa2 = one_one_nat ) ) ) ) )
% 0.30/0.55 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
% 0.30/0.55 ( ? [Summary2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.55 => ~ ( ( Xa2 != Mi2 )
% 0.30/0.55 => ( ( Xa2 != Ma2 )
% 0.30/0.55 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % vebt_member.elims(2)
% 0.30/0.55 thf(fact_2617_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.55 ( ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.55 => Y3 )
% 0.30/0.55 => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.30/0.55 => Y3 )
% 0.30/0.55 => ( ! [Mi2: nat,Ma2: nat] :
% 0.30/0.55 ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( ~ ( ( Xa2 = Mi2 )
% 0.30/0.55 | ( Xa2 = Ma2 ) ) ) ) )
% 0.30/0.55 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
% 0.30/0.55 ( ? [Vc2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( ~ ( ( Xa2 = Mi2 )
% 0.30/0.55 | ( Xa2 = Ma2 )
% 0.30/0.55 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) )
% 0.30/0.55 => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
% 0.30/0.55 ( ? [Vd2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % VEBT_internal.membermima.elims(1)
% 0.30/0.55 thf(fact_2618_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.30/0.55 ( ~ ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 0.30/0.55 => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.55 ( X2
% 0.30/0.55 != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.55 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.30/0.55 => ( ! [Mi2: nat,Ma2: nat] :
% 0.30/0.55 ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 0.30/0.55 => ( ( Xa2 = Mi2 )
% 0.30/0.55 | ( Xa2 = Ma2 ) ) )
% 0.30/0.55 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
% 0.30/0.55 ( ? [Vc2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
% 0.30/0.55 => ( ( Xa2 = Mi2 )
% 0.30/0.55 | ( Xa2 = Ma2 )
% 0.30/0.55 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
% 0.30/0.55 => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
% 0.30/0.55 ( ? [Vd2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
% 0.30/0.55 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % VEBT_internal.membermima.elims(3)
% 0.30/0.55 thf(fact_2619_vebt__member_Oelims_I1_J,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.55 ( ( ( vEBT_vebt_member @ X2 @ Xa2 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ! [A4: $o,B4: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.55 => A4 )
% 0.30/0.55 & ( ( Xa2 != zero_zero_nat )
% 0.30/0.55 => ( ( ( Xa2 = one_one_nat )
% 0.30/0.55 => B4 )
% 0.30/0.55 & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 0.30/0.55 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.55 => Y3 )
% 0.30/0.55 => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.30/0.55 => Y3 )
% 0.30/0.55 => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.30/0.55 => Y3 )
% 0.30/0.55 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
% 0.30/0.55 ( ? [Summary2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( ~ ( ( Xa2 != Mi2 )
% 0.30/0.55 => ( ( Xa2 != Ma2 )
% 0.30/0.55 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % vebt_member.elims(1)
% 0.30/0.55 thf(fact_2620_vebt__member_Oelims_I3_J,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.30/0.55 ( ~ ( vEBT_vebt_member @ X2 @ Xa2 )
% 0.30/0.55 => ( ! [A4: $o,B4: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.55 => ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.55 => A4 )
% 0.30/0.55 & ( ( Xa2 != zero_zero_nat )
% 0.30/0.55 => ( ( ( Xa2 = one_one_nat )
% 0.30/0.55 => B4 )
% 0.30/0.55 & ( Xa2 = one_one_nat ) ) ) ) )
% 0.30/0.55 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.55 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.30/0.55 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.30/0.55 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
% 0.30/0.55 ( ? [Summary2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.55 => ( ( Xa2 != Mi2 )
% 0.30/0.55 => ( ( Xa2 != Ma2 )
% 0.30/0.55 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % vebt_member.elims(3)
% 0.30/0.55 thf(fact_2621_arith__geo__mean,axiom,
% 0.30/0.55 ! [U: real,X2: real,Y3: real] :
% 0.30/0.55 ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55 = ( times_times_real @ X2 @ Y3 ) )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.55 => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % arith_geo_mean
% 0.30/0.55 thf(fact_2622_arith__geo__mean,axiom,
% 0.30/0.55 ! [U: rat,X2: rat,Y3: rat] :
% 0.30/0.55 ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55 = ( times_times_rat @ X2 @ Y3 ) )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.55 => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % arith_geo_mean
% 0.30/0.55 thf(fact_2623_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat,Y3: nat] :
% 0.30/0.55 ( ( ( vEBT_T_i_n_s_e_r_t2 @ X2 @ Xa2 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( ? [A4: $o,B4: $o] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.55 => ( Y3 != one_one_nat ) )
% 0.30/0.55 => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 0.30/0.55 => ( Y3 != one_one_nat ) )
% 0.30/0.55 => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 0.30/0.55 => ( Y3 != one_one_nat ) )
% 0.30/0.55 => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.55 => ( Y3 != one_one_nat ) )
% 0.30/0.55 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 != ( if_nat
% 0.30/0.55 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 & ~ ( ( Xa2 = Mi2 )
% 0.30/0.55 | ( Xa2 = Ma2 ) ) )
% 0.30/0.55 @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
% 0.30/0.55 @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
% 0.30/0.55 thf(fact_2624_invar__vebt_Osimps,axiom,
% 0.30/0.55 ( vEBT_invar_vebt
% 0.30/0.55 = ( ^ [A1: vEBT_VEBT,A22: nat] :
% 0.30/0.55 ( ( ? [A3: $o,B2: $o] :
% 0.30/0.55 ( A1
% 0.30/0.55 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.30/0.55 & ( A22
% 0.30/0.55 = ( suc @ zero_zero_nat ) ) )
% 0.30/0.55 | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
% 0.30/0.55 ( ( A1
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList3 @ Summary3 ) )
% 0.30/0.55 & ! [X: vEBT_VEBT] :
% 0.30/0.55 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.55 => ( vEBT_invar_vebt @ X @ N3 ) )
% 0.30/0.55 & ( vEBT_invar_vebt @ Summary3 @ N3 )
% 0.30/0.55 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 0.30/0.55 & ( A22
% 0.30/0.55 = ( plus_plus_nat @ N3 @ N3 ) )
% 0.30/0.55 & ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X4 )
% 0.30/0.55 & ! [X: vEBT_VEBT] :
% 0.30/0.55 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.55 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 0.30/0.55 | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
% 0.30/0.55 ( ( A1
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList3 @ Summary3 ) )
% 0.30/0.55 & ! [X: vEBT_VEBT] :
% 0.30/0.55 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.55 => ( vEBT_invar_vebt @ X @ N3 ) )
% 0.30/0.55 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
% 0.30/0.55 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 0.30/0.55 & ( A22
% 0.30/0.55 = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
% 0.30/0.55 & ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X4 )
% 0.30/0.55 & ! [X: vEBT_VEBT] :
% 0.30/0.55 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.55 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 0.30/0.55 | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.30/0.55 ( ( A1
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList3 @ Summary3 ) )
% 0.30/0.55 & ! [X: vEBT_VEBT] :
% 0.30/0.55 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.55 => ( vEBT_invar_vebt @ X @ N3 ) )
% 0.30/0.55 & ( vEBT_invar_vebt @ Summary3 @ N3 )
% 0.30/0.55 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 0.30/0.55 & ( A22
% 0.30/0.55 = ( plus_plus_nat @ N3 @ N3 ) )
% 0.30/0.55 & ! [I4: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 0.30/0.55 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X4 ) )
% 0.30/0.55 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 0.30/0.55 & ( ( Mi3 = Ma3 )
% 0.30/0.55 => ! [X: vEBT_VEBT] :
% 0.30/0.55 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.55 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 0.30/0.55 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.55 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 0.30/0.55 & ( ( Mi3 != Ma3 )
% 0.30/0.55 => ! [I4: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 0.30/0.55 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
% 0.30/0.55 = I4 )
% 0.30/0.55 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
% 0.30/0.55 & ! [X: nat] :
% 0.30/0.55 ( ( ( ( vEBT_VEBT_high @ X @ N3 )
% 0.30/0.55 = I4 )
% 0.30/0.55 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X @ N3 ) ) )
% 0.30/0.55 => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.55 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) )
% 0.30/0.55 | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.30/0.55 ( ( A1
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList3 @ Summary3 ) )
% 0.30/0.55 & ! [X: vEBT_VEBT] :
% 0.30/0.55 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.55 => ( vEBT_invar_vebt @ X @ N3 ) )
% 0.30/0.55 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
% 0.30/0.55 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 0.30/0.55 & ( A22
% 0.30/0.55 = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
% 0.30/0.55 & ! [I4: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 0.30/0.55 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X4 ) )
% 0.30/0.55 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 0.30/0.55 & ( ( Mi3 = Ma3 )
% 0.30/0.55 => ! [X: vEBT_VEBT] :
% 0.30/0.55 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.55 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 0.30/0.55 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.55 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 0.30/0.55 & ( ( Mi3 != Ma3 )
% 0.30/0.55 => ! [I4: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 0.30/0.55 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
% 0.30/0.55 = I4 )
% 0.30/0.55 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
% 0.30/0.55 & ! [X: nat] :
% 0.30/0.55 ( ( ( ( vEBT_VEBT_high @ X @ N3 )
% 0.30/0.55 = I4 )
% 0.30/0.55 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X @ N3 ) ) )
% 0.30/0.55 => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.55 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % invar_vebt.simps
% 0.30/0.55 thf(fact_2625_invar__vebt_Ocases,axiom,
% 0.30/0.55 ! [A12: vEBT_VEBT,A23: nat] :
% 0.30/0.55 ( ( vEBT_invar_vebt @ A12 @ A23 )
% 0.30/0.55 => ( ( ? [A4: $o,B4: $o] :
% 0.30/0.55 ( A12
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.55 => ( A23
% 0.30/0.55 != ( suc @ zero_zero_nat ) ) )
% 0.30/0.55 => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 0.30/0.55 ( ( A12
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.30/0.55 => ( ( A23 = Deg2 )
% 0.30/0.55 => ( ! [X3: vEBT_VEBT] :
% 0.30/0.55 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_invar_vebt @ X3 @ N2 ) )
% 0.30/0.55 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.30/0.55 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.30/0.55 => ( ( M4 = N2 )
% 0.30/0.55 => ( ( Deg2
% 0.30/0.55 = ( plus_plus_nat @ N2 @ M4 ) )
% 0.30/0.55 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
% 0.30/0.55 => ~ ! [X3: vEBT_VEBT] :
% 0.30/0.55 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.55 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ) ) ) ) ) ) )
% 0.30/0.55 => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 0.30/0.55 ( ( A12
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.30/0.55 => ( ( A23 = Deg2 )
% 0.30/0.55 => ( ! [X3: vEBT_VEBT] :
% 0.30/0.55 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_invar_vebt @ X3 @ N2 ) )
% 0.30/0.55 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.30/0.55 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.30/0.55 => ( ( M4
% 0.30/0.55 = ( suc @ N2 ) )
% 0.30/0.55 => ( ( Deg2
% 0.30/0.55 = ( plus_plus_nat @ N2 @ M4 ) )
% 0.30/0.55 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
% 0.30/0.55 => ~ ! [X3: vEBT_VEBT] :
% 0.30/0.55 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.55 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ) ) ) ) ) ) )
% 0.30/0.55 => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 0.30/0.55 ( ( A12
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.30/0.55 => ( ( A23 = Deg2 )
% 0.30/0.55 => ( ! [X3: vEBT_VEBT] :
% 0.30/0.55 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_invar_vebt @ X3 @ N2 ) )
% 0.30/0.55 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.30/0.55 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.30/0.55 => ( ( M4 = N2 )
% 0.30/0.55 => ( ( Deg2
% 0.30/0.55 = ( plus_plus_nat @ N2 @ M4 ) )
% 0.30/0.55 => ( ! [I: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.30/0.55 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ X4 ) )
% 0.30/0.55 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
% 0.30/0.55 => ( ( ( Mi2 = Ma2 )
% 0.30/0.55 => ! [X3: vEBT_VEBT] :
% 0.30/0.55 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.55 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
% 0.30/0.55 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.30/0.55 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.55 => ~ ( ( Mi2 != Ma2 )
% 0.30/0.55 => ! [I: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.30/0.55 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 0.30/0.55 = I )
% 0.30/0.55 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 0.30/0.55 & ! [X3: nat] :
% 0.30/0.55 ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
% 0.30/0.55 = I )
% 0.30/0.55 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
% 0.30/0.55 => ( ( ord_less_nat @ Mi2 @ X3 )
% 0.30/0.55 & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.55 => ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 0.30/0.55 ( ( A12
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.30/0.55 => ( ( A23 = Deg2 )
% 0.30/0.55 => ( ! [X3: vEBT_VEBT] :
% 0.30/0.55 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_invar_vebt @ X3 @ N2 ) )
% 0.30/0.55 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.30/0.55 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.30/0.55 => ( ( M4
% 0.30/0.55 = ( suc @ N2 ) )
% 0.30/0.55 => ( ( Deg2
% 0.30/0.55 = ( plus_plus_nat @ N2 @ M4 ) )
% 0.30/0.55 => ( ! [I: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.30/0.55 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ X4 ) )
% 0.30/0.55 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
% 0.30/0.55 => ( ( ( Mi2 = Ma2 )
% 0.30/0.55 => ! [X3: vEBT_VEBT] :
% 0.30/0.55 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.55 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
% 0.30/0.55 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.30/0.55 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.55 => ~ ( ( Mi2 != Ma2 )
% 0.30/0.55 => ! [I: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.30/0.55 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 0.30/0.55 = I )
% 0.30/0.55 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 0.30/0.55 & ! [X3: nat] :
% 0.30/0.55 ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
% 0.30/0.55 = I )
% 0.30/0.55 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
% 0.30/0.55 => ( ( ord_less_nat @ Mi2 @ X3 )
% 0.30/0.55 & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % invar_vebt.cases
% 0.30/0.55 thf(fact_2626_list__eq__iff__nth__eq,axiom,
% 0.30/0.55 ( ( ^ [Y6: list_VEBT_VEBT,Z5: list_VEBT_VEBT] : ( Y6 = Z5 ) )
% 0.30/0.55 = ( ^ [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 0.30/0.55 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.30/0.55 = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
% 0.30/0.55 & ! [I4: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.30/0.55 => ( ( nth_VEBT_VEBT @ Xs @ I4 )
% 0.30/0.55 = ( nth_VEBT_VEBT @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % list_eq_iff_nth_eq
% 0.30/0.55 thf(fact_2627_list__eq__iff__nth__eq,axiom,
% 0.30/0.55 ( ( ^ [Y6: list_o,Z5: list_o] : ( Y6 = Z5 ) )
% 0.30/0.55 = ( ^ [Xs: list_o,Ys3: list_o] :
% 0.30/0.55 ( ( ( size_size_list_o @ Xs )
% 0.30/0.55 = ( size_size_list_o @ Ys3 ) )
% 0.30/0.55 & ! [I4: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 0.30/0.55 => ( ( nth_o @ Xs @ I4 )
% 0.30/0.55 = ( nth_o @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % list_eq_iff_nth_eq
% 0.30/0.55 thf(fact_2628_list__eq__iff__nth__eq,axiom,
% 0.30/0.55 ( ( ^ [Y6: list_nat,Z5: list_nat] : ( Y6 = Z5 ) )
% 0.30/0.55 = ( ^ [Xs: list_nat,Ys3: list_nat] :
% 0.30/0.55 ( ( ( size_size_list_nat @ Xs )
% 0.30/0.55 = ( size_size_list_nat @ Ys3 ) )
% 0.30/0.55 & ! [I4: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 0.30/0.55 => ( ( nth_nat @ Xs @ I4 )
% 0.30/0.55 = ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % list_eq_iff_nth_eq
% 0.30/0.55 thf(fact_2629_list__eq__iff__nth__eq,axiom,
% 0.30/0.55 ( ( ^ [Y6: list_int,Z5: list_int] : ( Y6 = Z5 ) )
% 0.30/0.55 = ( ^ [Xs: list_int,Ys3: list_int] :
% 0.30/0.55 ( ( ( size_size_list_int @ Xs )
% 0.30/0.55 = ( size_size_list_int @ Ys3 ) )
% 0.30/0.55 & ! [I4: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 0.30/0.55 => ( ( nth_int @ Xs @ I4 )
% 0.30/0.55 = ( nth_int @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % list_eq_iff_nth_eq
% 0.30/0.55 thf(fact_2630_Skolem__list__nth,axiom,
% 0.30/0.55 ! [K: nat,P: nat > vEBT_VEBT > $o] :
% 0.30/0.55 ( ( ! [I4: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I4 @ K )
% 0.30/0.55 => ? [X4: vEBT_VEBT] : ( P @ I4 @ X4 ) ) )
% 0.30/0.55 = ( ? [Xs: list_VEBT_VEBT] :
% 0.30/0.55 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.30/0.55 = K )
% 0.30/0.55 & ! [I4: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I4 @ K )
% 0.30/0.55 => ( P @ I4 @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Skolem_list_nth
% 0.30/0.55 thf(fact_2631_Skolem__list__nth,axiom,
% 0.30/0.55 ! [K: nat,P: nat > $o > $o] :
% 0.30/0.55 ( ( ! [I4: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I4 @ K )
% 0.30/0.55 => ? [X4: $o] : ( P @ I4 @ X4 ) ) )
% 0.30/0.55 = ( ? [Xs: list_o] :
% 0.30/0.55 ( ( ( size_size_list_o @ Xs )
% 0.30/0.55 = K )
% 0.30/0.55 & ! [I4: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I4 @ K )
% 0.30/0.55 => ( P @ I4 @ ( nth_o @ Xs @ I4 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Skolem_list_nth
% 0.30/0.55 thf(fact_2632_Skolem__list__nth,axiom,
% 0.30/0.55 ! [K: nat,P: nat > nat > $o] :
% 0.30/0.55 ( ( ! [I4: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I4 @ K )
% 0.30/0.55 => ? [X4: nat] : ( P @ I4 @ X4 ) ) )
% 0.30/0.55 = ( ? [Xs: list_nat] :
% 0.30/0.55 ( ( ( size_size_list_nat @ Xs )
% 0.30/0.55 = K )
% 0.30/0.55 & ! [I4: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I4 @ K )
% 0.30/0.55 => ( P @ I4 @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Skolem_list_nth
% 0.30/0.55 thf(fact_2633_Skolem__list__nth,axiom,
% 0.30/0.55 ! [K: nat,P: nat > int > $o] :
% 0.30/0.55 ( ( ! [I4: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I4 @ K )
% 0.30/0.55 => ? [X4: int] : ( P @ I4 @ X4 ) ) )
% 0.30/0.55 = ( ? [Xs: list_int] :
% 0.30/0.55 ( ( ( size_size_list_int @ Xs )
% 0.30/0.55 = K )
% 0.30/0.55 & ! [I4: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I4 @ K )
% 0.30/0.55 => ( P @ I4 @ ( nth_int @ Xs @ I4 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Skolem_list_nth
% 0.30/0.55 thf(fact_2634_nth__equalityI,axiom,
% 0.30/0.55 ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.30/0.55 ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.30/0.55 = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.30/0.55 => ( ! [I3: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.55 => ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
% 0.30/0.55 = ( nth_VEBT_VEBT @ Ys @ I3 ) ) )
% 0.30/0.55 => ( Xs2 = Ys ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nth_equalityI
% 0.30/0.55 thf(fact_2635_nth__equalityI,axiom,
% 0.30/0.55 ! [Xs2: list_o,Ys: list_o] :
% 0.30/0.55 ( ( ( size_size_list_o @ Xs2 )
% 0.30/0.55 = ( size_size_list_o @ Ys ) )
% 0.30/0.55 => ( ! [I3: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.55 => ( ( nth_o @ Xs2 @ I3 )
% 0.30/0.55 = ( nth_o @ Ys @ I3 ) ) )
% 0.30/0.55 => ( Xs2 = Ys ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nth_equalityI
% 0.30/0.55 thf(fact_2636_nth__equalityI,axiom,
% 0.30/0.55 ! [Xs2: list_nat,Ys: list_nat] :
% 0.30/0.55 ( ( ( size_size_list_nat @ Xs2 )
% 0.30/0.55 = ( size_size_list_nat @ Ys ) )
% 0.30/0.55 => ( ! [I3: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.55 => ( ( nth_nat @ Xs2 @ I3 )
% 0.30/0.55 = ( nth_nat @ Ys @ I3 ) ) )
% 0.30/0.55 => ( Xs2 = Ys ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nth_equalityI
% 0.30/0.55 thf(fact_2637_nth__equalityI,axiom,
% 0.30/0.55 ! [Xs2: list_int,Ys: list_int] :
% 0.30/0.55 ( ( ( size_size_list_int @ Xs2 )
% 0.30/0.55 = ( size_size_list_int @ Ys ) )
% 0.30/0.55 => ( ! [I3: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.55 => ( ( nth_int @ Xs2 @ I3 )
% 0.30/0.55 = ( nth_int @ Ys @ I3 ) ) )
% 0.30/0.55 => ( Xs2 = Ys ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nth_equalityI
% 0.30/0.55 thf(fact_2638_discrete,axiom,
% 0.30/0.55 ( ord_less_nat
% 0.30/0.55 = ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % discrete
% 0.30/0.55 thf(fact_2639_discrete,axiom,
% 0.30/0.55 ( ord_less_int
% 0.30/0.55 = ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % discrete
% 0.30/0.55 thf(fact_2640_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat,Y3: nat] :
% 0.30/0.55 ( ( ( vEBT_T_m_e_m_b_e_r2 @ X2 @ Xa2 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( ? [A4: $o,B4: $o] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.55 => ( Y3 != one_one_nat ) )
% 0.30/0.55 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.55 => ( Y3 != one_one_nat ) )
% 0.30/0.55 => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.30/0.55 => ( Y3 != one_one_nat ) )
% 0.30/0.55 => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.30/0.55 => ( Y3 != one_one_nat ) )
% 0.30/0.55 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
% 0.30/0.55 ( ? [Summary2: vEBT_VEBT] :
% 0.30/0.55 ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.55 => ( Y3
% 0.30/0.55 != ( plus_plus_nat @ one_one_nat
% 0.30/0.55 @ ( if_nat @ ( Xa2 = Mi2 ) @ zero_zero_nat
% 0.30/0.55 @ ( if_nat @ ( Xa2 = Ma2 ) @ zero_zero_nat
% 0.30/0.55 @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ zero_zero_nat
% 0.30/0.55 @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa2 ) @ zero_zero_nat
% 0.30/0.55 @ ( if_nat
% 0.30/0.55 @ ( ( ord_less_nat @ Mi2 @ Xa2 )
% 0.30/0.55 & ( ord_less_nat @ Xa2 @ Ma2 ) )
% 0.30/0.55 @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
% 0.30/0.55 @ zero_zero_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
% 0.30/0.55 thf(fact_2641_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
% 0.30/0.55 ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 0.30/0.55 ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 0.30/0.55 = ( plus_plus_nat @ one_one_nat
% 0.30/0.55 @ ( if_nat @ ( X2 = Mi ) @ zero_zero_nat
% 0.30/0.55 @ ( if_nat @ ( X2 = Ma ) @ zero_zero_nat
% 0.30/0.55 @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ zero_zero_nat
% 0.30/0.55 @ ( if_nat @ ( ord_less_nat @ Ma @ X2 ) @ zero_zero_nat
% 0.30/0.55 @ ( if_nat
% 0.30/0.55 @ ( ( ord_less_nat @ Mi @ X2 )
% 0.30/0.55 & ( ord_less_nat @ X2 @ Ma ) )
% 0.30/0.55 @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
% 0.30/0.55 @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
% 0.30/0.55 thf(fact_2642_inrange,axiom,
% 0.30/0.55 ! [T: vEBT_VEBT,N: nat] :
% 0.30/0.55 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.55 => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % inrange
% 0.30/0.55 thf(fact_2643_set__bit__0,axiom,
% 0.30/0.55 ! [A: int] :
% 0.30/0.55 ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
% 0.30/0.55 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % set_bit_0
% 0.30/0.55 thf(fact_2644_set__bit__0,axiom,
% 0.30/0.55 ! [A: nat] :
% 0.30/0.55 ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
% 0.30/0.55 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % set_bit_0
% 0.30/0.55 thf(fact_2645_vebt__succ_Opelims,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 0.30/0.55 ( ( ( vEBT_vebt_succ @ X2 @ Xa2 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.30/0.55 => ( ! [Uu2: $o,B4: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ Uu2 @ B4 ) )
% 0.30/0.55 => ( ( Xa2 = zero_zero_nat )
% 0.30/0.55 => ( ( ( B4
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( some_nat @ one_one_nat ) ) )
% 0.30/0.55 & ( ~ B4
% 0.30/0.55 => ( Y3 = none_nat ) ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B4 ) @ zero_zero_nat ) ) ) ) )
% 0.30/0.55 => ( ! [Uv2: $o,Uw2: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 0.30/0.55 => ! [N2: nat] :
% 0.30/0.55 ( ( Xa2
% 0.30/0.55 = ( suc @ N2 ) )
% 0.30/0.55 => ( ( Y3 = none_nat )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N2 ) ) ) ) ) )
% 0.30/0.55 => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 0.30/0.55 => ( ( Y3 = none_nat )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 0.30/0.55 => ( ( Y3 = none_nat )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
% 0.30/0.55 => ( ( Y3 = none_nat )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Xa2 ) ) ) )
% 0.30/0.55 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.55 => ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( some_nat @ Mi2 ) ) )
% 0.30/0.55 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 @ ( if_option_nat
% 0.30/0.55 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.55 != none_nat )
% 0.30/0.55 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.55 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.55 @ ( if_option_nat
% 0.30/0.55 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.55 = none_nat )
% 0.30/0.55 @ none_nat
% 0.30/0.55 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.55 @ none_nat ) ) ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % vebt_succ.pelims
% 0.30/0.55 thf(fact_2646_vebt__pred_Opelims,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 0.30/0.55 ( ( ( vEBT_vebt_pred @ X2 @ Xa2 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.30/0.55 => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.55 => ( ( Xa2 = zero_zero_nat )
% 0.30/0.55 => ( ( Y3 = none_nat )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
% 0.30/0.55 => ( ! [A4: $o,Uw2: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ Uw2 ) )
% 0.30/0.55 => ( ( Xa2
% 0.30/0.55 = ( suc @ zero_zero_nat ) )
% 0.30/0.55 => ( ( ( A4
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.55 & ( ~ A4
% 0.30/0.55 => ( Y3 = none_nat ) ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
% 0.30/0.55 => ( ! [A4: $o,B4: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.55 => ! [Va3: nat] :
% 0.30/0.55 ( ( Xa2
% 0.30/0.55 = ( suc @ ( suc @ Va3 ) ) )
% 0.30/0.55 => ( ( ( B4
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( some_nat @ one_one_nat ) ) )
% 0.30/0.55 & ( ~ B4
% 0.30/0.55 => ( ( A4
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.55 & ( ~ A4
% 0.30/0.55 => ( Y3 = none_nat ) ) ) ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
% 0.30/0.55 => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
% 0.30/0.55 => ( ( Y3 = none_nat )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) )
% 0.30/0.55 => ( ( Y3 = none_nat )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) @ Xa2 ) ) ) )
% 0.30/0.55 => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
% 0.30/0.55 => ( ( Y3 = none_nat )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Xa2 ) ) ) )
% 0.30/0.55 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.55 => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( some_nat @ Ma2 ) ) )
% 0.30/0.55 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55 => ( Y3
% 0.30/0.55 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 @ ( if_option_nat
% 0.30/0.55 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.55 != none_nat )
% 0.30/0.55 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.55 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.55 @ ( if_option_nat
% 0.30/0.55 @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.55 = none_nat )
% 0.30/0.55 @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
% 0.30/0.55 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.55 @ none_nat ) ) ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % vebt_pred.pelims
% 0.30/0.55 thf(fact_2647_double__eq__0__iff,axiom,
% 0.30/0.55 ! [A: real] :
% 0.30/0.55 ( ( ( plus_plus_real @ A @ A )
% 0.30/0.55 = zero_zero_real )
% 0.30/0.55 = ( A = zero_zero_real ) ) ).
% 0.30/0.55
% 0.30/0.55 % double_eq_0_iff
% 0.30/0.55 thf(fact_2648_double__eq__0__iff,axiom,
% 0.30/0.55 ! [A: rat] :
% 0.30/0.55 ( ( ( plus_plus_rat @ A @ A )
% 0.30/0.55 = zero_zero_rat )
% 0.30/0.55 = ( A = zero_zero_rat ) ) ).
% 0.30/0.55
% 0.30/0.55 % double_eq_0_iff
% 0.30/0.55 thf(fact_2649_double__eq__0__iff,axiom,
% 0.30/0.55 ! [A: int] :
% 0.30/0.55 ( ( ( plus_plus_int @ A @ A )
% 0.30/0.55 = zero_zero_int )
% 0.30/0.55 = ( A = zero_zero_int ) ) ).
% 0.30/0.55
% 0.30/0.55 % double_eq_0_iff
% 0.30/0.55 thf(fact_2650_unset__bit__0,axiom,
% 0.30/0.55 ! [A: int] :
% 0.30/0.55 ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
% 0.30/0.55 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % unset_bit_0
% 0.30/0.55 thf(fact_2651_unset__bit__0,axiom,
% 0.30/0.55 ! [A: nat] :
% 0.30/0.55 ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
% 0.30/0.55 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % unset_bit_0
% 0.30/0.55 thf(fact_2652_mult__le__cancel__iff2,axiom,
% 0.30/0.55 ! [Z: real,X2: real,Y3: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.30/0.55 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X2 ) @ ( times_times_real @ Z @ Y3 ) )
% 0.30/0.55 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_cancel_iff2
% 0.30/0.55 thf(fact_2653_mult__le__cancel__iff2,axiom,
% 0.30/0.55 ! [Z: rat,X2: rat,Y3: rat] :
% 0.30/0.55 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.30/0.55 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X2 ) @ ( times_times_rat @ Z @ Y3 ) )
% 0.30/0.55 = ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_cancel_iff2
% 0.30/0.55 thf(fact_2654_mult__le__cancel__iff2,axiom,
% 0.30/0.55 ! [Z: int,X2: int,Y3: int] :
% 0.30/0.55 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.30/0.55 => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X2 ) @ ( times_times_int @ Z @ Y3 ) )
% 0.30/0.55 = ( ord_less_eq_int @ X2 @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_cancel_iff2
% 0.30/0.55 thf(fact_2655_div__neg__neg__trivial,axiom,
% 0.30/0.55 ! [K: int,L: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.30/0.55 => ( ( ord_less_int @ L @ K )
% 0.30/0.55 => ( ( divide_divide_int @ K @ L )
% 0.30/0.55 = zero_zero_int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % div_neg_neg_trivial
% 0.30/0.55 thf(fact_2656_div__pos__pos__trivial,axiom,
% 0.30/0.55 ! [K: int,L: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.55 => ( ( ord_less_int @ K @ L )
% 0.30/0.55 => ( ( divide_divide_int @ K @ L )
% 0.30/0.55 = zero_zero_int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % div_pos_pos_trivial
% 0.30/0.55 thf(fact_2657_i0__less,axiom,
% 0.30/0.55 ! [N: extended_enat] :
% 0.30/0.55 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.30/0.55 = ( N != zero_z5237406670263579293d_enat ) ) ).
% 0.30/0.55
% 0.30/0.55 % i0_less
% 0.30/0.55 thf(fact_2658_idiff__0,axiom,
% 0.30/0.55 ! [N: extended_enat] :
% 0.30/0.55 ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.30/0.55 = zero_z5237406670263579293d_enat ) ).
% 0.30/0.55
% 0.30/0.55 % idiff_0
% 0.30/0.55 thf(fact_2659_idiff__0__right,axiom,
% 0.30/0.55 ! [N: extended_enat] :
% 0.30/0.55 ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
% 0.30/0.55 = N ) ).
% 0.30/0.55
% 0.30/0.55 % idiff_0_right
% 0.30/0.55 thf(fact_2660_not__real__square__gt__zero,axiom,
% 0.30/0.55 ! [X2: real] :
% 0.30/0.55 ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X2 @ X2 ) ) )
% 0.30/0.55 = ( X2 = zero_zero_real ) ) ).
% 0.30/0.55
% 0.30/0.55 % not_real_square_gt_zero
% 0.30/0.55 thf(fact_2661_half__nonnegative__int__iff,axiom,
% 0.30/0.55 ! [K: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.30/0.55 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.30/0.55
% 0.30/0.55 % half_nonnegative_int_iff
% 0.30/0.55 thf(fact_2662_half__negative__int__iff,axiom,
% 0.30/0.55 ! [K: int] :
% 0.30/0.55 ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.30/0.55 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.55
% 0.30/0.55 % half_negative_int_iff
% 0.30/0.55 thf(fact_2663_zdiv__mono1,axiom,
% 0.30/0.55 ! [A: int,A5: int,B: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ A @ A5 )
% 0.30/0.55 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.55 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A5 @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zdiv_mono1
% 0.30/0.55 thf(fact_2664_zdiv__mono2,axiom,
% 0.30/0.55 ! [A: int,B5: 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 @ B5 )
% 0.30/0.55 => ( ( ord_less_eq_int @ B5 @ B )
% 0.30/0.55 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B5 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zdiv_mono2
% 0.30/0.55 thf(fact_2665_zdiv__eq__0__iff,axiom,
% 0.30/0.55 ! [I2: int,K: int] :
% 0.30/0.55 ( ( ( divide_divide_int @ I2 @ K )
% 0.30/0.55 = zero_zero_int )
% 0.30/0.55 = ( ( K = zero_zero_int )
% 0.30/0.55 | ( ( ord_less_eq_int @ zero_zero_int @ I2 )
% 0.30/0.55 & ( ord_less_int @ I2 @ K ) )
% 0.30/0.55 | ( ( ord_less_eq_int @ I2 @ zero_zero_int )
% 0.30/0.55 & ( ord_less_int @ K @ I2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zdiv_eq_0_iff
% 0.30/0.55 thf(fact_2666_zdiv__mono1__neg,axiom,
% 0.30/0.55 ! [A: int,A5: int,B: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ A @ A5 )
% 0.30/0.55 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.30/0.55 => ( ord_less_eq_int @ ( divide_divide_int @ A5 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zdiv_mono1_neg
% 0.30/0.55 thf(fact_2667_zdiv__mono2__neg,axiom,
% 0.30/0.55 ! [A: int,B5: int,B: int] :
% 0.30/0.55 ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.55 => ( ( ord_less_int @ zero_zero_int @ B5 )
% 0.30/0.55 => ( ( ord_less_eq_int @ B5 @ B )
% 0.30/0.55 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B5 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zdiv_mono2_neg
% 0.30/0.55 thf(fact_2668_div__int__pos__iff,axiom,
% 0.30/0.55 ! [K: int,L: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
% 0.30/0.55 = ( ( K = zero_zero_int )
% 0.30/0.55 | ( L = zero_zero_int )
% 0.30/0.55 | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.55 & ( ord_less_eq_int @ zero_zero_int @ L ) )
% 0.30/0.55 | ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.55 & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % div_int_pos_iff
% 0.30/0.55 thf(fact_2669_div__positive__int,axiom,
% 0.30/0.55 ! [L: int,K: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ L @ K )
% 0.30/0.55 => ( ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.55 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % div_positive_int
% 0.30/0.55 thf(fact_2670_int__div__less__self,axiom,
% 0.30/0.55 ! [X2: int,K: int] :
% 0.30/0.55 ( ( ord_less_int @ zero_zero_int @ X2 )
% 0.30/0.55 => ( ( ord_less_int @ one_one_int @ K )
% 0.30/0.55 => ( ord_less_int @ ( divide_divide_int @ X2 @ K ) @ X2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % int_div_less_self
% 0.30/0.55 thf(fact_2671_div__nonneg__neg__le0,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 @ B @ zero_zero_int )
% 0.30/0.55 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % div_nonneg_neg_le0
% 0.30/0.55 thf(fact_2672_div__nonpos__pos__le0,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 @ zero_zero_int @ B )
% 0.30/0.55 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % div_nonpos_pos_le0
% 0.30/0.55 thf(fact_2673_pos__imp__zdiv__pos__iff,axiom,
% 0.30/0.55 ! [K: int,I2: int] :
% 0.30/0.55 ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.55 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I2 @ K ) )
% 0.30/0.55 = ( ord_less_eq_int @ K @ I2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % pos_imp_zdiv_pos_iff
% 0.30/0.55 thf(fact_2674_neg__imp__zdiv__nonneg__iff,axiom,
% 0.30/0.55 ! [B: int,A: int] :
% 0.30/0.55 ( ( ord_less_int @ B @ zero_zero_int )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.30/0.55 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % neg_imp_zdiv_nonneg_iff
% 0.30/0.55 thf(fact_2675_pos__imp__zdiv__nonneg__iff,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_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.30/0.55 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % pos_imp_zdiv_nonneg_iff
% 0.30/0.55 thf(fact_2676_nonneg1__imp__zdiv__pos__iff,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 @ ( divide_divide_int @ A @ B ) )
% 0.30/0.55 = ( ( ord_less_eq_int @ B @ A )
% 0.30/0.55 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nonneg1_imp_zdiv_pos_iff
% 0.30/0.55 thf(fact_2677_zdiv__zmult2__eq,axiom,
% 0.30/0.55 ! [C: int,A: int,B: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.55 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.30/0.55 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zdiv_zmult2_eq
% 0.30/0.55 thf(fact_2678_int__div__pos__eq,axiom,
% 0.30/0.55 ! [A: int,B: int,Q3: int,R2: int] :
% 0.30/0.55 ( ( A
% 0.30/0.55 = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.30/0.55 => ( ( ord_less_int @ R2 @ B )
% 0.30/0.55 => ( ( divide_divide_int @ A @ B )
% 0.30/0.55 = Q3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % int_div_pos_eq
% 0.30/0.55 thf(fact_2679_int__div__neg__eq,axiom,
% 0.30/0.55 ! [A: int,B: int,Q3: int,R2: int] :
% 0.30/0.55 ( ( A
% 0.30/0.55 = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 0.30/0.55 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.30/0.55 => ( ( ord_less_int @ B @ R2 )
% 0.30/0.55 => ( ( divide_divide_int @ A @ B )
% 0.30/0.55 = Q3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % int_div_neg_eq
% 0.30/0.55 thf(fact_2680_div__pos__geq,axiom,
% 0.30/0.55 ! [L: int,K: int] :
% 0.30/0.55 ( ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.55 => ( ( ord_less_eq_int @ L @ K )
% 0.30/0.55 => ( ( divide_divide_int @ K @ L )
% 0.30/0.55 = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % div_pos_geq
% 0.30/0.55 thf(fact_2681_split__zdiv,axiom,
% 0.30/0.55 ! [P: int > $o,N: int,K: int] :
% 0.30/0.55 ( ( P @ ( divide_divide_int @ N @ K ) )
% 0.30/0.55 = ( ( ( K = zero_zero_int )
% 0.30/0.55 => ( P @ zero_zero_int ) )
% 0.30/0.55 & ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.55 => ! [I4: int,J3: int] :
% 0.30/0.55 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.30/0.55 & ( ord_less_int @ J3 @ K )
% 0.30/0.55 & ( N
% 0.30/0.55 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.30/0.55 => ( P @ I4 ) ) )
% 0.30/0.55 & ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.55 => ! [I4: int,J3: int] :
% 0.30/0.55 ( ( ( ord_less_int @ K @ J3 )
% 0.30/0.55 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.30/0.55 & ( N
% 0.30/0.55 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.30/0.55 => ( P @ I4 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % split_zdiv
% 0.30/0.55 thf(fact_2682_not__iless0,axiom,
% 0.30/0.55 ! [N: extended_enat] :
% 0.30/0.55 ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% 0.30/0.55
% 0.30/0.55 % not_iless0
% 0.30/0.55 thf(fact_2683_enat__0__less__mult__iff,axiom,
% 0.30/0.55 ! [M: extended_enat,N: extended_enat] :
% 0.30/0.55 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
% 0.30/0.55 = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 0.30/0.55 & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % enat_0_less_mult_iff
% 0.30/0.55 thf(fact_2684_iadd__is__0,axiom,
% 0.30/0.55 ! [M: extended_enat,N: extended_enat] :
% 0.30/0.55 ( ( ( plus_p3455044024723400733d_enat @ M @ N )
% 0.30/0.55 = zero_z5237406670263579293d_enat )
% 0.30/0.55 = ( ( M = zero_z5237406670263579293d_enat )
% 0.30/0.55 & ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % iadd_is_0
% 0.30/0.55 thf(fact_2685_i0__lb,axiom,
% 0.30/0.55 ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% 0.30/0.55
% 0.30/0.55 % i0_lb
% 0.30/0.55 thf(fact_2686_ile0__eq,axiom,
% 0.30/0.55 ! [N: extended_enat] :
% 0.30/0.55 ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
% 0.30/0.55 = ( N = zero_z5237406670263579293d_enat ) ) ).
% 0.30/0.55
% 0.30/0.55 % ile0_eq
% 0.30/0.55 thf(fact_2687_all__nat__less,axiom,
% 0.30/0.55 ! [N: nat,P: nat > $o] :
% 0.30/0.55 ( ( ! [M3: nat] :
% 0.30/0.55 ( ( ord_less_eq_nat @ M3 @ N )
% 0.30/0.55 => ( P @ M3 ) ) )
% 0.30/0.55 = ( ! [X: nat] :
% 0.30/0.55 ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.55 => ( P @ X ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % all_nat_less
% 0.30/0.55 thf(fact_2688_ex__nat__less,axiom,
% 0.30/0.55 ! [N: nat,P: nat > $o] :
% 0.30/0.55 ( ( ? [M3: nat] :
% 0.30/0.55 ( ( ord_less_eq_nat @ M3 @ N )
% 0.30/0.55 & ( P @ M3 ) ) )
% 0.30/0.55 = ( ? [X: nat] :
% 0.30/0.55 ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.55 & ( P @ X ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % ex_nat_less
% 0.30/0.55 thf(fact_2689_not__exp__less__eq__0__int,axiom,
% 0.30/0.55 ! [N: nat] :
% 0.30/0.55 ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% 0.30/0.55
% 0.30/0.55 % not_exp_less_eq_0_int
% 0.30/0.55 thf(fact_2690_pos__zdiv__mult__2,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 => ( ( 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 ) )
% 0.30/0.55 = ( divide_divide_int @ B @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % pos_zdiv_mult_2
% 0.30/0.55 thf(fact_2691_neg__zdiv__mult__2,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 => ( ( 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 ) )
% 0.30/0.55 = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % neg_zdiv_mult_2
% 0.30/0.55 thf(fact_2692_realpow__pos__nth2,axiom,
% 0.30/0.55 ! [A: real,N: nat] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.55 => ? [R3: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.30/0.55 & ( ( power_power_real @ R3 @ ( suc @ N ) )
% 0.30/0.55 = A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % realpow_pos_nth2
% 0.30/0.55 thf(fact_2693_real__arch__pow__inv,axiom,
% 0.30/0.55 ! [Y3: real,X2: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.55 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.30/0.55 => ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X2 @ N2 ) @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % real_arch_pow_inv
% 0.30/0.55 thf(fact_2694_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I1_J,axiom,
% 0.30/0.55 ! [A: $o,B: $o,X2: nat] :
% 0.30/0.55 ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Leaf @ A @ B ) @ X2 )
% 0.30/0.55 = one_one_nat ) ).
% 0.30/0.55
% 0.30/0.55 % T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
% 0.30/0.55 thf(fact_2695_int__power__div__base,axiom,
% 0.30/0.55 ! [M: nat,K: int] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.55 => ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.55 => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 0.30/0.55 = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % int_power_div_base
% 0.30/0.55 thf(fact_2696_realpow__pos__nth__unique,axiom,
% 0.30/0.55 ! [N: nat,A: real] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.55 => ? [X5: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ X5 )
% 0.30/0.55 & ( ( power_power_real @ X5 @ N )
% 0.30/0.55 = A )
% 0.30/0.55 & ! [Y5: real] :
% 0.30/0.55 ( ( ( ord_less_real @ zero_zero_real @ Y5 )
% 0.30/0.55 & ( ( power_power_real @ Y5 @ N )
% 0.30/0.55 = A ) )
% 0.30/0.55 => ( Y5 = X5 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % realpow_pos_nth_unique
% 0.30/0.55 thf(fact_2697_realpow__pos__nth,axiom,
% 0.30/0.55 ! [N: nat,A: real] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.55 => ? [R3: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.30/0.55 & ( ( power_power_real @ R3 @ N )
% 0.30/0.55 = A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % realpow_pos_nth
% 0.30/0.55 thf(fact_2698_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
% 0.30/0.55 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
% 0.30/0.55 ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X2 )
% 0.30/0.55 = one_one_nat ) ).
% 0.30/0.55
% 0.30/0.55 % T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
% 0.30/0.55 thf(fact_2699_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
% 0.30/0.55 ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
% 0.30/0.55 ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X2 )
% 0.30/0.55 = one_one_nat ) ).
% 0.30/0.55
% 0.30/0.55 % T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
% 0.30/0.55 thf(fact_2700_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
% 0.30/0.55 ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
% 0.30/0.55 ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X2 )
% 0.30/0.55 = one_one_nat ) ).
% 0.30/0.55
% 0.30/0.55 % T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
% 0.30/0.55 thf(fact_2701_member__bound__height_H,axiom,
% 0.30/0.55 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.30/0.55 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.55 => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r2 @ T @ X2 ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % member_bound_height'
% 0.30/0.55 thf(fact_2702_mult__less__iff1,axiom,
% 0.30/0.55 ! [Z: real,X2: real,Y3: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.30/0.55 => ( ( ord_less_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 0.30/0.55 = ( ord_less_real @ X2 @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_less_iff1
% 0.30/0.55 thf(fact_2703_mult__less__iff1,axiom,
% 0.30/0.55 ! [Z: rat,X2: rat,Y3: rat] :
% 0.30/0.55 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.30/0.55 => ( ( ord_less_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 0.30/0.55 = ( ord_less_rat @ X2 @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_less_iff1
% 0.30/0.55 thf(fact_2704_mult__less__iff1,axiom,
% 0.30/0.55 ! [Z: int,X2: int,Y3: int] :
% 0.30/0.55 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.30/0.55 => ( ( ord_less_int @ ( times_times_int @ X2 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 0.30/0.55 = ( ord_less_int @ X2 @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_less_iff1
% 0.30/0.55 thf(fact_2705_mult__le__cancel__iff1,axiom,
% 0.30/0.55 ! [Z: real,X2: real,Y3: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.30/0.55 => ( ( ord_less_eq_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 0.30/0.55 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_cancel_iff1
% 0.30/0.55 thf(fact_2706_mult__le__cancel__iff1,axiom,
% 0.30/0.55 ! [Z: rat,X2: rat,Y3: rat] :
% 0.30/0.55 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.30/0.55 => ( ( ord_less_eq_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 0.30/0.55 = ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_cancel_iff1
% 0.30/0.55 thf(fact_2707_mult__le__cancel__iff1,axiom,
% 0.30/0.55 ! [Z: int,X2: int,Y3: int] :
% 0.30/0.55 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.30/0.55 => ( ( ord_less_eq_int @ ( times_times_int @ X2 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 0.30/0.55 = ( ord_less_eq_int @ X2 @ Y3 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_cancel_iff1
% 0.30/0.55 thf(fact_2708_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat,Y3: nat] :
% 0.30/0.55 ( ( ( vEBT_T_i_n_s_e_r_t2 @ X2 @ Xa2 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.30/0.55 => ( ! [A4: $o,B4: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.55 => ( ( Y3 = one_one_nat )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 0.30/0.55 => ( ( Y3 = one_one_nat )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 0.30/0.55 => ( ( Y3 = one_one_nat )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.55 => ( ( Y3 = one_one_nat )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.55 => ( ( Y3
% 0.30/0.55 = ( if_nat
% 0.30/0.55 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 & ~ ( ( Xa2 = Mi2 )
% 0.30/0.55 | ( Xa2 = Ma2 ) ) )
% 0.30/0.55 @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
% 0.30/0.55 @ one_one_nat ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
% 0.30/0.55 thf(fact_2709_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat,Y3: nat] :
% 0.30/0.55 ( ( ( vEBT_T_m_e_m_b_e_r2 @ X2 @ Xa2 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.30/0.55 => ( ! [A4: $o,B4: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.55 => ( ( Y3 = one_one_nat )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.55 => ( ( Y3 = one_one_nat )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.30/0.55 => ( ( Y3 = one_one_nat )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.30/0.55 => ( ( Y3 = one_one_nat )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.55 => ( ( Y3
% 0.30/0.55 = ( plus_plus_nat @ one_one_nat
% 0.30/0.55 @ ( if_nat @ ( Xa2 = Mi2 ) @ zero_zero_nat
% 0.30/0.55 @ ( if_nat @ ( Xa2 = Ma2 ) @ zero_zero_nat
% 0.30/0.55 @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ zero_zero_nat
% 0.30/0.55 @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa2 ) @ zero_zero_nat
% 0.30/0.55 @ ( if_nat
% 0.30/0.55 @ ( ( ord_less_nat @ Mi2 @ Xa2 )
% 0.30/0.55 & ( ord_less_nat @ Xa2 @ Ma2 ) )
% 0.30/0.55 @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
% 0.30/0.55 @ zero_zero_nat ) ) ) ) ) ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
% 0.30/0.55 thf(fact_2710_vebt__member_Opelims_I3_J,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.30/0.55 ( ~ ( vEBT_vebt_member @ X2 @ Xa2 )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.30/0.55 => ( ! [A4: $o,B4: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
% 0.30/0.55 => ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.55 => A4 )
% 0.30/0.55 & ( ( Xa2 != zero_zero_nat )
% 0.30/0.55 => ( ( ( Xa2 = one_one_nat )
% 0.30/0.55 => B4 )
% 0.30/0.55 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.30/0.55 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 0.30/0.55 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
% 0.30/0.55 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
% 0.30/0.55 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) )
% 0.30/0.55 => ( ( Xa2 != Mi2 )
% 0.30/0.55 => ( ( Xa2 != Ma2 )
% 0.30/0.55 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % vebt_member.pelims(3)
% 0.30/0.55 thf(fact_2711_vebt__member_Opelims_I1_J,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.55 ( ( ( vEBT_vebt_member @ X2 @ Xa2 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.30/0.55 => ( ! [A4: $o,B4: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.55 => ( ( Y3
% 0.30/0.55 = ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.55 => A4 )
% 0.30/0.55 & ( ( Xa2 != zero_zero_nat )
% 0.30/0.55 => ( ( ( Xa2 = one_one_nat )
% 0.30/0.55 => B4 )
% 0.30/0.55 & ( Xa2 = one_one_nat ) ) ) ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.55 => ( ~ Y3
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.30/0.55 => ( ~ Y3
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.30/0.55 => ( ~ Y3
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.55 => ( ( Y3
% 0.30/0.55 = ( ( Xa2 != Mi2 )
% 0.30/0.55 => ( ( Xa2 != Ma2 )
% 0.30/0.55 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % vebt_member.pelims(1)
% 0.30/0.55 thf(fact_2712_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.30/0.55 ( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.30/0.55 => ( ! [A4: $o,B4: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
% 0.30/0.55 => ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.55 => A4 )
% 0.30/0.55 & ( ( Xa2 != zero_zero_nat )
% 0.30/0.55 => ( ( ( Xa2 = one_one_nat )
% 0.30/0.55 => B4 )
% 0.30/0.55 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.30/0.55 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 0.30/0.55 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) @ Xa2 ) )
% 0.30/0.55 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % VEBT_internal.naive_member.pelims(3)
% 0.30/0.55 thf(fact_2713_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.30/0.55 ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.30/0.55 => ( ! [A4: $o,B4: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
% 0.30/0.55 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.55 => A4 )
% 0.30/0.55 & ( ( Xa2 != zero_zero_nat )
% 0.30/0.55 => ( ( ( Xa2 = one_one_nat )
% 0.30/0.55 => B4 )
% 0.30/0.55 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.30/0.55 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) @ Xa2 ) )
% 0.30/0.55 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % VEBT_internal.naive_member.pelims(2)
% 0.30/0.55 thf(fact_2714_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.55 ( ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.30/0.55 => ( ! [A4: $o,B4: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.55 => ( ( Y3
% 0.30/0.55 = ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.55 => A4 )
% 0.30/0.55 & ( ( Xa2 != zero_zero_nat )
% 0.30/0.55 => ( ( ( Xa2 = one_one_nat )
% 0.30/0.55 => B4 )
% 0.30/0.55 & ( Xa2 = one_one_nat ) ) ) ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.30/0.55 => ( ~ Y3
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) )
% 0.30/0.55 => ( ( Y3
% 0.30/0.55 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) @ Xa2 ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % VEBT_internal.naive_member.pelims(1)
% 0.30/0.55 thf(fact_2715_vebt__member_Opelims_I2_J,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.30/0.55 ( ( vEBT_vebt_member @ X2 @ Xa2 )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.30/0.55 => ( ! [A4: $o,B4: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
% 0.30/0.55 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.55 => A4 )
% 0.30/0.55 & ( ( Xa2 != zero_zero_nat )
% 0.30/0.55 => ( ( ( Xa2 = one_one_nat )
% 0.30/0.55 => B4 )
% 0.30/0.55 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.30/0.55 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) )
% 0.30/0.55 => ~ ( ( Xa2 != Mi2 )
% 0.30/0.55 => ( ( Xa2 != Ma2 )
% 0.30/0.55 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % vebt_member.pelims(2)
% 0.30/0.55 thf(fact_2716_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.30/0.55 ( ~ ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.30/0.55 => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
% 0.30/0.55 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
% 0.30/0.55 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) )
% 0.30/0.55 => ( ( Xa2 = Mi2 )
% 0.30/0.55 | ( Xa2 = Ma2 ) ) ) )
% 0.30/0.55 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
% 0.30/0.55 => ( ( Xa2 = Mi2 )
% 0.30/0.55 | ( Xa2 = Ma2 )
% 0.30/0.55 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
% 0.30/0.55 => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa2 ) )
% 0.30/0.55 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % VEBT_internal.membermima.pelims(3)
% 0.30/0.55 thf(fact_2717_unset__bit__nonnegative__int__iff,axiom,
% 0.30/0.55 ! [N: nat,K: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
% 0.30/0.55 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.30/0.55
% 0.30/0.55 % unset_bit_nonnegative_int_iff
% 0.30/0.55 thf(fact_2718_set__bit__nonnegative__int__iff,axiom,
% 0.30/0.55 ! [N: nat,K: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
% 0.30/0.55 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.30/0.55
% 0.30/0.55 % set_bit_nonnegative_int_iff
% 0.30/0.55 thf(fact_2719_unset__bit__negative__int__iff,axiom,
% 0.30/0.55 ! [N: nat,K: int] :
% 0.30/0.55 ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
% 0.30/0.55 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.55
% 0.30/0.55 % unset_bit_negative_int_iff
% 0.30/0.55 thf(fact_2720_set__bit__negative__int__iff,axiom,
% 0.30/0.55 ! [N: nat,K: int] :
% 0.30/0.55 ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
% 0.30/0.55 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.55
% 0.30/0.55 % set_bit_negative_int_iff
% 0.30/0.55 thf(fact_2721_zle__add1__eq__le,axiom,
% 0.30/0.55 ! [W: int,Z: int] :
% 0.30/0.55 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 0.30/0.55 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.30/0.55
% 0.30/0.55 % zle_add1_eq_le
% 0.30/0.55 thf(fact_2722_zle__diff1__eq,axiom,
% 0.30/0.55 ! [W: int,Z: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
% 0.30/0.55 = ( ord_less_int @ W @ Z ) ) ).
% 0.30/0.55
% 0.30/0.55 % zle_diff1_eq
% 0.30/0.55 thf(fact_2723_less__eq__int__code_I1_J,axiom,
% 0.30/0.55 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.30/0.55
% 0.30/0.55 % less_eq_int_code(1)
% 0.30/0.55 thf(fact_2724_zero__one__enat__neq_I1_J,axiom,
% 0.30/0.55 zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% 0.30/0.55
% 0.30/0.55 % zero_one_enat_neq(1)
% 0.30/0.55 thf(fact_2725_int__one__le__iff__zero__less,axiom,
% 0.30/0.55 ! [Z: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ one_one_int @ Z )
% 0.30/0.55 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.30/0.55
% 0.30/0.55 % int_one_le_iff_zero_less
% 0.30/0.55 thf(fact_2726_odd__less__0__iff,axiom,
% 0.30/0.55 ! [Z: int] :
% 0.30/0.55 ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
% 0.30/0.55 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.30/0.55
% 0.30/0.55 % odd_less_0_iff
% 0.30/0.55 thf(fact_2727_le__imp__0__less,axiom,
% 0.30/0.55 ! [Z: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.55 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % le_imp_0_less
% 0.30/0.55 thf(fact_2728_odd__nonzero,axiom,
% 0.30/0.55 ! [Z: int] :
% 0.30/0.55 ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
% 0.30/0.55 != zero_zero_int ) ).
% 0.30/0.55
% 0.30/0.55 % odd_nonzero
% 0.30/0.55 thf(fact_2729_q__pos__lemma,axiom,
% 0.30/0.55 ! [B5: int,Q5: int,R4: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) )
% 0.30/0.55 => ( ( ord_less_int @ R4 @ B5 )
% 0.30/0.55 => ( ( ord_less_int @ zero_zero_int @ B5 )
% 0.30/0.55 => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % q_pos_lemma
% 0.30/0.55 thf(fact_2730_zmult__zless__mono2,axiom,
% 0.30/0.55 ! [I2: int,J: int,K: int] :
% 0.30/0.55 ( ( ord_less_int @ I2 @ J )
% 0.30/0.55 => ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.55 => ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zmult_zless_mono2
% 0.30/0.55 thf(fact_2731_pos__zmult__eq__1__iff,axiom,
% 0.30/0.55 ! [M: int,N: int] :
% 0.30/0.55 ( ( ord_less_int @ zero_zero_int @ M )
% 0.30/0.55 => ( ( ( times_times_int @ M @ N )
% 0.30/0.55 = one_one_int )
% 0.30/0.55 = ( ( M = one_one_int )
% 0.30/0.55 & ( N = one_one_int ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % pos_zmult_eq_1_iff
% 0.30/0.55 thf(fact_2732_zdiv__mono2__lemma,axiom,
% 0.30/0.55 ! [B: int,Q3: int,R2: int,B5: int,Q5: int,R4: int] :
% 0.30/0.55 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 )
% 0.30/0.55 = ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) )
% 0.30/0.55 => ( ( ord_less_int @ R4 @ B5 )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.30/0.55 => ( ( ord_less_int @ zero_zero_int @ B5 )
% 0.30/0.55 => ( ( ord_less_eq_int @ B5 @ B )
% 0.30/0.55 => ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zdiv_mono2_lemma
% 0.30/0.55 thf(fact_2733_zdiv__mono2__neg__lemma,axiom,
% 0.30/0.55 ! [B: int,Q3: int,R2: int,B5: int,Q5: int,R4: int] :
% 0.30/0.55 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 )
% 0.30/0.55 = ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) )
% 0.30/0.55 => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) @ zero_zero_int )
% 0.30/0.55 => ( ( ord_less_int @ R2 @ B )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 0.30/0.55 => ( ( ord_less_int @ zero_zero_int @ B5 )
% 0.30/0.55 => ( ( ord_less_eq_int @ B5 @ B )
% 0.30/0.55 => ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zdiv_mono2_neg_lemma
% 0.30/0.55 thf(fact_2734_unique__quotient__lemma,axiom,
% 0.30/0.55 ! [B: int,Q5: int,R4: int,Q3: int,R2: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 0.30/0.55 => ( ( ord_less_int @ R4 @ B )
% 0.30/0.55 => ( ( ord_less_int @ R2 @ B )
% 0.30/0.55 => ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % unique_quotient_lemma
% 0.30/0.55 thf(fact_2735_unique__quotient__lemma__neg,axiom,
% 0.30/0.55 ! [B: int,Q5: int,R4: int,Q3: int,R2: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 0.30/0.55 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.30/0.55 => ( ( ord_less_int @ B @ R2 )
% 0.30/0.55 => ( ( ord_less_int @ B @ R4 )
% 0.30/0.55 => ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % unique_quotient_lemma_neg
% 0.30/0.55 thf(fact_2736_minus__int__code_I1_J,axiom,
% 0.30/0.55 ! [K: int] :
% 0.30/0.55 ( ( minus_minus_int @ K @ zero_zero_int )
% 0.30/0.55 = K ) ).
% 0.30/0.55
% 0.30/0.55 % minus_int_code(1)
% 0.30/0.55 thf(fact_2737_times__int__code_I2_J,axiom,
% 0.30/0.55 ! [L: int] :
% 0.30/0.55 ( ( times_times_int @ zero_zero_int @ L )
% 0.30/0.55 = zero_zero_int ) ).
% 0.30/0.55
% 0.30/0.55 % times_int_code(2)
% 0.30/0.55 thf(fact_2738_times__int__code_I1_J,axiom,
% 0.30/0.55 ! [K: int] :
% 0.30/0.55 ( ( times_times_int @ K @ zero_zero_int )
% 0.30/0.55 = zero_zero_int ) ).
% 0.30/0.55
% 0.30/0.55 % times_int_code(1)
% 0.30/0.55 thf(fact_2739_imult__is__0,axiom,
% 0.30/0.55 ! [M: extended_enat,N: extended_enat] :
% 0.30/0.55 ( ( ( times_7803423173614009249d_enat @ M @ N )
% 0.30/0.55 = zero_z5237406670263579293d_enat )
% 0.30/0.55 = ( ( M = zero_z5237406670263579293d_enat )
% 0.30/0.55 | ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % imult_is_0
% 0.30/0.55 thf(fact_2740_int__distrib_I3_J,axiom,
% 0.30/0.55 ! [Z1: int,Z22: int,W: int] :
% 0.30/0.55 ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
% 0.30/0.55 = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % int_distrib(3)
% 0.30/0.55 thf(fact_2741_int__distrib_I4_J,axiom,
% 0.30/0.55 ! [W: int,Z1: int,Z22: int] :
% 0.30/0.55 ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
% 0.30/0.55 = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % int_distrib(4)
% 0.30/0.55 thf(fact_2742_int__less__induct,axiom,
% 0.30/0.55 ! [I2: int,K: int,P: int > $o] :
% 0.30/0.55 ( ( ord_less_int @ I2 @ K )
% 0.30/0.55 => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 0.30/0.55 => ( ! [I3: int] :
% 0.30/0.55 ( ( ord_less_int @ I3 @ K )
% 0.30/0.55 => ( ( P @ I3 )
% 0.30/0.55 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 0.30/0.55 => ( P @ I2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % int_less_induct
% 0.30/0.55 thf(fact_2743_int__le__induct,axiom,
% 0.30/0.55 ! [I2: int,K: int,P: int > $o] :
% 0.30/0.55 ( ( ord_less_eq_int @ I2 @ K )
% 0.30/0.55 => ( ( P @ K )
% 0.30/0.55 => ( ! [I3: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ I3 @ K )
% 0.30/0.55 => ( ( P @ I3 )
% 0.30/0.55 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 0.30/0.55 => ( P @ I2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % int_le_induct
% 0.30/0.55 thf(fact_2744_int__induct,axiom,
% 0.30/0.55 ! [P: int > $o,K: int,I2: int] :
% 0.30/0.55 ( ( P @ K )
% 0.30/0.55 => ( ! [I3: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ K @ I3 )
% 0.30/0.55 => ( ( P @ I3 )
% 0.30/0.55 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 0.30/0.55 => ( ! [I3: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ I3 @ K )
% 0.30/0.55 => ( ( P @ I3 )
% 0.30/0.55 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 0.30/0.55 => ( P @ I2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % int_induct
% 0.30/0.55 thf(fact_2745_int__distrib_I1_J,axiom,
% 0.30/0.55 ! [Z1: int,Z22: int,W: int] :
% 0.30/0.55 ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
% 0.30/0.55 = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % int_distrib(1)
% 0.30/0.55 thf(fact_2746_int__distrib_I2_J,axiom,
% 0.30/0.55 ! [W: int,Z1: int,Z22: int] :
% 0.30/0.55 ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
% 0.30/0.55 = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % int_distrib(2)
% 0.30/0.55 thf(fact_2747_add1__zle__eq,axiom,
% 0.30/0.55 ! [W: int,Z: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
% 0.30/0.55 = ( ord_less_int @ W @ Z ) ) ).
% 0.30/0.55
% 0.30/0.55 % add1_zle_eq
% 0.30/0.55 thf(fact_2748_int__ge__induct,axiom,
% 0.30/0.55 ! [K: int,I2: int,P: int > $o] :
% 0.30/0.55 ( ( ord_less_eq_int @ K @ I2 )
% 0.30/0.55 => ( ( P @ K )
% 0.30/0.55 => ( ! [I3: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ K @ I3 )
% 0.30/0.55 => ( ( P @ I3 )
% 0.30/0.55 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 0.30/0.55 => ( P @ I2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % int_ge_induct
% 0.30/0.55 thf(fact_2749_int__gr__induct,axiom,
% 0.30/0.55 ! [K: int,I2: int,P: int > $o] :
% 0.30/0.55 ( ( ord_less_int @ K @ I2 )
% 0.30/0.55 => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 0.30/0.55 => ( ! [I3: int] :
% 0.30/0.55 ( ( ord_less_int @ K @ I3 )
% 0.30/0.55 => ( ( P @ I3 )
% 0.30/0.55 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 0.30/0.55 => ( P @ I2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % int_gr_induct
% 0.30/0.55 thf(fact_2750_zless__add1__eq,axiom,
% 0.30/0.55 ! [W: int,Z: int] :
% 0.30/0.55 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 0.30/0.55 = ( ( ord_less_int @ W @ Z )
% 0.30/0.55 | ( W = Z ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zless_add1_eq
% 0.30/0.55 thf(fact_2751_zless__imp__add1__zle,axiom,
% 0.30/0.55 ! [W: int,Z: int] :
% 0.30/0.55 ( ( ord_less_int @ W @ Z )
% 0.30/0.55 => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% 0.30/0.55
% 0.30/0.55 % zless_imp_add1_zle
% 0.30/0.55 thf(fact_2752_unset__bit__less__eq,axiom,
% 0.30/0.55 ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% 0.30/0.55
% 0.30/0.55 % unset_bit_less_eq
% 0.30/0.55 thf(fact_2753_set__bit__greater__eq,axiom,
% 0.30/0.55 ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% 0.30/0.55
% 0.30/0.55 % set_bit_greater_eq
% 0.30/0.55 thf(fact_2754_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.30/0.55 ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.30/0.55 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) )
% 0.30/0.55 => ~ ( ( Xa2 = Mi2 )
% 0.30/0.55 | ( Xa2 = Ma2 ) ) ) )
% 0.30/0.55 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
% 0.30/0.55 => ~ ( ( Xa2 = Mi2 )
% 0.30/0.55 | ( Xa2 = Ma2 )
% 0.30/0.55 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
% 0.30/0.55 => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa2 ) )
% 0.30/0.55 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % VEBT_internal.membermima.pelims(2)
% 0.30/0.55 thf(fact_2755_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 0.30/0.55 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.55 ( ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 0.30/0.55 = Y3 )
% 0.30/0.55 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.30/0.55 => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.55 => ( ~ Y3
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.30/0.55 => ( ~ Y3
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 0.30/0.55 => ( ( Y3
% 0.30/0.55 = ( ( Xa2 = Mi2 )
% 0.30/0.55 | ( Xa2 = Ma2 ) ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
% 0.30/0.55 => ( ( Y3
% 0.30/0.55 = ( ( Xa2 = Mi2 )
% 0.30/0.55 | ( Xa2 = Ma2 )
% 0.30/0.55 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa2 ) ) ) )
% 0.30/0.55 => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.30/0.55 ( ( X2
% 0.30/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
% 0.30/0.55 => ( ( Y3
% 0.30/0.55 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.55 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.30/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
% 0.30/0.55 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % VEBT_internal.membermima.pelims(1)
% 0.30/0.55 thf(fact_2756_atLeastatMost__empty,axiom,
% 0.30/0.55 ! [B: rat,A: rat] :
% 0.30/0.55 ( ( ord_less_rat @ B @ A )
% 0.30/0.55 => ( ( set_or633870826150836451st_rat @ A @ B )
% 0.30/0.55 = bot_bot_set_rat ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty
% 0.30/0.55 thf(fact_2757_atLeastatMost__empty,axiom,
% 0.30/0.55 ! [B: num,A: num] :
% 0.30/0.55 ( ( ord_less_num @ B @ A )
% 0.30/0.55 => ( ( set_or7049704709247886629st_num @ A @ B )
% 0.30/0.55 = bot_bot_set_num ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty
% 0.30/0.55 thf(fact_2758_atLeastatMost__empty,axiom,
% 0.30/0.55 ! [B: nat,A: nat] :
% 0.30/0.55 ( ( ord_less_nat @ B @ A )
% 0.30/0.55 => ( ( set_or1269000886237332187st_nat @ A @ B )
% 0.30/0.55 = bot_bot_set_nat ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty
% 0.30/0.55 thf(fact_2759_atLeastatMost__empty,axiom,
% 0.30/0.55 ! [B: int,A: int] :
% 0.30/0.55 ( ( ord_less_int @ B @ A )
% 0.30/0.55 => ( ( set_or1266510415728281911st_int @ A @ B )
% 0.30/0.55 = bot_bot_set_int ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty
% 0.30/0.55 thf(fact_2760_atLeastatMost__empty,axiom,
% 0.30/0.55 ! [B: real,A: real] :
% 0.30/0.55 ( ( ord_less_real @ B @ A )
% 0.30/0.55 => ( ( set_or1222579329274155063t_real @ A @ B )
% 0.30/0.55 = bot_bot_set_real ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty
% 0.30/0.55 thf(fact_2761_atLeastatMost__subset__iff,axiom,
% 0.30/0.55 ! [A: set_int,B: set_int,C: set_int,D: set_int] :
% 0.30/0.55 ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 0.30/0.55 = ( ~ ( ord_less_eq_set_int @ A @ B )
% 0.30/0.55 | ( ( ord_less_eq_set_int @ C @ A )
% 0.30/0.55 & ( ord_less_eq_set_int @ B @ D ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_subset_iff
% 0.30/0.55 thf(fact_2762_atLeastatMost__subset__iff,axiom,
% 0.30/0.55 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.55 ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 0.30/0.55 = ( ~ ( ord_less_eq_rat @ A @ B )
% 0.30/0.55 | ( ( ord_less_eq_rat @ C @ A )
% 0.30/0.55 & ( ord_less_eq_rat @ B @ D ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_subset_iff
% 0.30/0.55 thf(fact_2763_atLeastatMost__subset__iff,axiom,
% 0.30/0.55 ! [A: num,B: num,C: num,D: num] :
% 0.30/0.55 ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 0.30/0.55 = ( ~ ( ord_less_eq_num @ A @ B )
% 0.30/0.55 | ( ( ord_less_eq_num @ C @ A )
% 0.30/0.55 & ( ord_less_eq_num @ B @ D ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_subset_iff
% 0.30/0.55 thf(fact_2764_atLeastatMost__subset__iff,axiom,
% 0.30/0.55 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.55 ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 0.30/0.55 = ( ~ ( ord_less_eq_nat @ A @ B )
% 0.30/0.55 | ( ( ord_less_eq_nat @ C @ A )
% 0.30/0.55 & ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_subset_iff
% 0.30/0.55 thf(fact_2765_atLeastatMost__subset__iff,axiom,
% 0.30/0.55 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.55 ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 0.30/0.55 = ( ~ ( ord_less_eq_int @ A @ B )
% 0.30/0.55 | ( ( ord_less_eq_int @ C @ A )
% 0.30/0.55 & ( ord_less_eq_int @ B @ D ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_subset_iff
% 0.30/0.55 thf(fact_2766_atLeastatMost__subset__iff,axiom,
% 0.30/0.55 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.55 ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 0.30/0.55 = ( ~ ( ord_less_eq_real @ A @ B )
% 0.30/0.55 | ( ( ord_less_eq_real @ C @ A )
% 0.30/0.55 & ( ord_less_eq_real @ B @ D ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_subset_iff
% 0.30/0.55 thf(fact_2767_atLeastatMost__empty__iff,axiom,
% 0.30/0.55 ! [A: set_int,B: set_int] :
% 0.30/0.55 ( ( ( set_or370866239135849197et_int @ A @ B )
% 0.30/0.55 = bot_bot_set_set_int )
% 0.30/0.55 = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty_iff
% 0.30/0.55 thf(fact_2768_atLeastatMost__empty__iff,axiom,
% 0.30/0.55 ! [A: rat,B: rat] :
% 0.30/0.55 ( ( ( set_or633870826150836451st_rat @ A @ B )
% 0.30/0.55 = bot_bot_set_rat )
% 0.30/0.55 = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty_iff
% 0.30/0.55 thf(fact_2769_atLeastatMost__empty__iff,axiom,
% 0.30/0.55 ! [A: num,B: num] :
% 0.30/0.55 ( ( ( set_or7049704709247886629st_num @ A @ B )
% 0.30/0.55 = bot_bot_set_num )
% 0.30/0.55 = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty_iff
% 0.30/0.55 thf(fact_2770_atLeastatMost__empty__iff,axiom,
% 0.30/0.55 ! [A: nat,B: nat] :
% 0.30/0.55 ( ( ( set_or1269000886237332187st_nat @ A @ B )
% 0.30/0.55 = bot_bot_set_nat )
% 0.30/0.55 = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty_iff
% 0.30/0.55 thf(fact_2771_atLeastatMost__empty__iff,axiom,
% 0.30/0.55 ! [A: int,B: int] :
% 0.30/0.55 ( ( ( set_or1266510415728281911st_int @ A @ B )
% 0.30/0.55 = bot_bot_set_int )
% 0.30/0.55 = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty_iff
% 0.30/0.55 thf(fact_2772_atLeastatMost__empty__iff,axiom,
% 0.30/0.55 ! [A: real,B: real] :
% 0.30/0.55 ( ( ( set_or1222579329274155063t_real @ A @ B )
% 0.30/0.55 = bot_bot_set_real )
% 0.30/0.55 = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty_iff
% 0.30/0.55 thf(fact_2773_atLeastatMost__empty__iff2,axiom,
% 0.30/0.55 ! [A: set_int,B: set_int] :
% 0.30/0.55 ( ( bot_bot_set_set_int
% 0.30/0.55 = ( set_or370866239135849197et_int @ A @ B ) )
% 0.30/0.55 = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty_iff2
% 0.30/0.55 thf(fact_2774_atLeastatMost__empty__iff2,axiom,
% 0.30/0.55 ! [A: rat,B: rat] :
% 0.30/0.55 ( ( bot_bot_set_rat
% 0.30/0.55 = ( set_or633870826150836451st_rat @ A @ B ) )
% 0.30/0.55 = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty_iff2
% 0.30/0.55 thf(fact_2775_atLeastatMost__empty__iff2,axiom,
% 0.30/0.55 ! [A: num,B: num] :
% 0.30/0.55 ( ( bot_bot_set_num
% 0.30/0.55 = ( set_or7049704709247886629st_num @ A @ B ) )
% 0.30/0.55 = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty_iff2
% 0.30/0.55 thf(fact_2776_atLeastatMost__empty__iff2,axiom,
% 0.30/0.55 ! [A: nat,B: nat] :
% 0.30/0.55 ( ( bot_bot_set_nat
% 0.30/0.55 = ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.30/0.55 = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty_iff2
% 0.30/0.55 thf(fact_2777_atLeastatMost__empty__iff2,axiom,
% 0.30/0.55 ! [A: int,B: int] :
% 0.30/0.55 ( ( bot_bot_set_int
% 0.30/0.55 = ( set_or1266510415728281911st_int @ A @ B ) )
% 0.30/0.55 = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty_iff2
% 0.30/0.55 thf(fact_2778_atLeastatMost__empty__iff2,axiom,
% 0.30/0.55 ! [A: real,B: real] :
% 0.30/0.55 ( ( bot_bot_set_real
% 0.30/0.55 = ( set_or1222579329274155063t_real @ A @ B ) )
% 0.30/0.55 = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastatMost_empty_iff2
% 0.30/0.55 thf(fact_2779_decr__mult__lemma,axiom,
% 0.30/0.55 ! [D: int,P: int > $o,K: int] :
% 0.30/0.55 ( ( ord_less_int @ zero_zero_int @ D )
% 0.30/0.55 => ( ! [X5: int] :
% 0.30/0.55 ( ( P @ X5 )
% 0.30/0.55 => ( P @ ( minus_minus_int @ X5 @ D ) ) )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.55 => ! [X3: int] :
% 0.30/0.55 ( ( P @ X3 )
% 0.30/0.55 => ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % decr_mult_lemma
% 0.30/0.55 thf(fact_2780_incr__mult__lemma,axiom,
% 0.30/0.55 ! [D: int,P: int > $o,K: int] :
% 0.30/0.55 ( ( ord_less_int @ zero_zero_int @ D )
% 0.30/0.55 => ( ! [X5: int] :
% 0.30/0.55 ( ( P @ X5 )
% 0.30/0.55 => ( P @ ( plus_plus_int @ X5 @ D ) ) )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.55 => ! [X3: int] :
% 0.30/0.55 ( ( P @ X3 )
% 0.30/0.55 => ( P @ ( plus_plus_int @ X3 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % incr_mult_lemma
% 0.30/0.55 thf(fact_2781_Icc__eq__Icc,axiom,
% 0.30/0.55 ! [L: set_int,H2: set_int,L3: set_int,H3: set_int] :
% 0.30/0.55 ( ( ( set_or370866239135849197et_int @ L @ H2 )
% 0.30/0.55 = ( set_or370866239135849197et_int @ L3 @ H3 ) )
% 0.30/0.55 = ( ( ( L = L3 )
% 0.30/0.55 & ( H2 = H3 ) )
% 0.30/0.55 | ( ~ ( ord_less_eq_set_int @ L @ H2 )
% 0.30/0.55 & ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Icc_eq_Icc
% 0.30/0.55 thf(fact_2782_Icc__eq__Icc,axiom,
% 0.30/0.55 ! [L: rat,H2: rat,L3: rat,H3: rat] :
% 0.30/0.55 ( ( ( set_or633870826150836451st_rat @ L @ H2 )
% 0.30/0.55 = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
% 0.30/0.55 = ( ( ( L = L3 )
% 0.30/0.55 & ( H2 = H3 ) )
% 0.30/0.55 | ( ~ ( ord_less_eq_rat @ L @ H2 )
% 0.30/0.55 & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Icc_eq_Icc
% 0.30/0.55 thf(fact_2783_Icc__eq__Icc,axiom,
% 0.30/0.55 ! [L: num,H2: num,L3: num,H3: num] :
% 0.30/0.55 ( ( ( set_or7049704709247886629st_num @ L @ H2 )
% 0.30/0.55 = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
% 0.30/0.55 = ( ( ( L = L3 )
% 0.30/0.55 & ( H2 = H3 ) )
% 0.30/0.55 | ( ~ ( ord_less_eq_num @ L @ H2 )
% 0.30/0.55 & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Icc_eq_Icc
% 0.30/0.55 thf(fact_2784_Icc__eq__Icc,axiom,
% 0.30/0.55 ! [L: nat,H2: nat,L3: nat,H3: nat] :
% 0.30/0.55 ( ( ( set_or1269000886237332187st_nat @ L @ H2 )
% 0.30/0.55 = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
% 0.30/0.55 = ( ( ( L = L3 )
% 0.30/0.55 & ( H2 = H3 ) )
% 0.30/0.55 | ( ~ ( ord_less_eq_nat @ L @ H2 )
% 0.30/0.55 & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Icc_eq_Icc
% 0.30/0.55 thf(fact_2785_Icc__eq__Icc,axiom,
% 0.30/0.55 ! [L: int,H2: int,L3: int,H3: int] :
% 0.30/0.55 ( ( ( set_or1266510415728281911st_int @ L @ H2 )
% 0.30/0.55 = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
% 0.30/0.55 = ( ( ( L = L3 )
% 0.30/0.55 & ( H2 = H3 ) )
% 0.30/0.55 | ( ~ ( ord_less_eq_int @ L @ H2 )
% 0.30/0.55 & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Icc_eq_Icc
% 0.30/0.55 thf(fact_2786_Icc__eq__Icc,axiom,
% 0.30/0.55 ! [L: real,H2: real,L3: real,H3: real] :
% 0.30/0.55 ( ( ( set_or1222579329274155063t_real @ L @ H2 )
% 0.30/0.55 = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
% 0.30/0.55 = ( ( ( L = L3 )
% 0.30/0.55 & ( H2 = H3 ) )
% 0.30/0.55 | ( ~ ( ord_less_eq_real @ L @ H2 )
% 0.30/0.55 & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Icc_eq_Icc
% 0.30/0.55 thf(fact_2787_atLeastAtMost__iff,axiom,
% 0.30/0.55 ! [I2: set_int,L: set_int,U: set_int] :
% 0.30/0.55 ( ( member_set_int @ I2 @ ( set_or370866239135849197et_int @ L @ U ) )
% 0.30/0.55 = ( ( ord_less_eq_set_int @ L @ I2 )
% 0.30/0.55 & ( ord_less_eq_set_int @ I2 @ U ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastAtMost_iff
% 0.30/0.55 thf(fact_2788_atLeastAtMost__iff,axiom,
% 0.30/0.55 ! [I2: rat,L: rat,U: rat] :
% 0.30/0.55 ( ( member_rat @ I2 @ ( set_or633870826150836451st_rat @ L @ U ) )
% 0.30/0.55 = ( ( ord_less_eq_rat @ L @ I2 )
% 0.30/0.55 & ( ord_less_eq_rat @ I2 @ U ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastAtMost_iff
% 0.30/0.55 thf(fact_2789_atLeastAtMost__iff,axiom,
% 0.30/0.55 ! [I2: num,L: num,U: num] :
% 0.30/0.55 ( ( member_num @ I2 @ ( set_or7049704709247886629st_num @ L @ U ) )
% 0.30/0.55 = ( ( ord_less_eq_num @ L @ I2 )
% 0.30/0.55 & ( ord_less_eq_num @ I2 @ U ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastAtMost_iff
% 0.30/0.55 thf(fact_2790_atLeastAtMost__iff,axiom,
% 0.30/0.55 ! [I2: nat,L: nat,U: nat] :
% 0.30/0.55 ( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 0.30/0.55 = ( ( ord_less_eq_nat @ L @ I2 )
% 0.30/0.55 & ( ord_less_eq_nat @ I2 @ U ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastAtMost_iff
% 0.30/0.55 thf(fact_2791_atLeastAtMost__iff,axiom,
% 0.30/0.55 ! [I2: int,L: int,U: int] :
% 0.30/0.55 ( ( member_int @ I2 @ ( set_or1266510415728281911st_int @ L @ U ) )
% 0.30/0.55 = ( ( ord_less_eq_int @ L @ I2 )
% 0.30/0.55 & ( ord_less_eq_int @ I2 @ U ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastAtMost_iff
% 0.30/0.55 thf(fact_2792_atLeastAtMost__iff,axiom,
% 0.30/0.55 ! [I2: real,L: real,U: real] :
% 0.30/0.55 ( ( member_real @ I2 @ ( set_or1222579329274155063t_real @ L @ U ) )
% 0.30/0.55 = ( ( ord_less_eq_real @ L @ I2 )
% 0.30/0.55 & ( ord_less_eq_real @ I2 @ U ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % atLeastAtMost_iff
% 0.30/0.55 thf(fact_2793_aset_I2_J,axiom,
% 0.30/0.55 ! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
% 0.30/0.55 ( ! [X5: int] :
% 0.30/0.55 ( ! [Xa: int] :
% 0.30/0.55 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.55 => ! [Xb2: int] :
% 0.30/0.55 ( ( member_int @ Xb2 @ A2 )
% 0.30/0.55 => ( X5
% 0.30/0.55 != ( minus_minus_int @ Xb2 @ Xa ) ) ) )
% 0.30/0.55 => ( ( P @ X5 )
% 0.30/0.55 => ( P @ ( plus_plus_int @ X5 @ D4 ) ) ) )
% 0.30/0.55 => ( ! [X5: int] :
% 0.30/0.55 ( ! [Xa: int] :
% 0.30/0.55 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.55 => ! [Xb2: int] :
% 0.30/0.55 ( ( member_int @ Xb2 @ A2 )
% 0.30/0.55 => ( X5
% 0.30/0.55 != ( minus_minus_int @ Xb2 @ Xa ) ) ) )
% 0.30/0.55 => ( ( Q @ X5 )
% 0.30/0.55 => ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) )
% 0.30/0.55 => ! [X3: int] :
% 0.30/0.55 ( ! [Xa3: int] :
% 0.30/0.55 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.55 => ! [Xb3: int] :
% 0.30/0.55 ( ( member_int @ Xb3 @ A2 )
% 0.30/0.55 => ( X3
% 0.30/0.55 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.55 => ( ( ( P @ X3 )
% 0.30/0.55 | ( Q @ X3 ) )
% 0.30/0.55 => ( ( P @ ( plus_plus_int @ X3 @ D4 ) )
% 0.30/0.55 | ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % aset(2)
% 0.30/0.55 thf(fact_2794_aset_I1_J,axiom,
% 0.30/0.55 ! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
% 0.30/0.55 ( ! [X5: int] :
% 0.30/0.55 ( ! [Xa: int] :
% 0.30/0.55 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.55 => ! [Xb2: int] :
% 0.30/0.55 ( ( member_int @ Xb2 @ A2 )
% 0.30/0.55 => ( X5
% 0.30/0.55 != ( minus_minus_int @ Xb2 @ Xa ) ) ) )
% 0.30/0.55 => ( ( P @ X5 )
% 0.30/0.55 => ( P @ ( plus_plus_int @ X5 @ D4 ) ) ) )
% 0.30/0.55 => ( ! [X5: int] :
% 0.30/0.55 ( ! [Xa: int] :
% 0.30/0.55 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.55 => ! [Xb2: int] :
% 0.30/0.55 ( ( member_int @ Xb2 @ A2 )
% 0.30/0.55 => ( X5
% 0.30/0.55 != ( minus_minus_int @ Xb2 @ Xa ) ) ) )
% 0.30/0.55 => ( ( Q @ X5 )
% 0.30/0.55 => ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) )
% 0.30/0.55 => ! [X3: int] :
% 0.30/0.55 ( ! [Xa3: int] :
% 0.30/0.55 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.55 => ! [Xb3: int] :
% 0.30/0.55 ( ( member_int @ Xb3 @ A2 )
% 0.30/0.55 => ( X3
% 0.30/0.55 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.55 => ( ( ( P @ X3 )
% 0.30/0.55 & ( Q @ X3 ) )
% 0.30/0.55 => ( ( P @ ( plus_plus_int @ X3 @ D4 ) )
% 0.30/0.55 & ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % aset(1)
% 0.30/0.55 thf(fact_2795_bset_I2_J,axiom,
% 0.30/0.55 ! [D4: int,B3: set_int,P: int > $o,Q: int > $o] :
% 0.30/0.55 ( ! [X5: int] :
% 0.30/0.55 ( ! [Xa: int] :
% 0.30/0.55 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.55 => ! [Xb2: int] :
% 0.30/0.55 ( ( member_int @ Xb2 @ B3 )
% 0.30/0.55 => ( X5
% 0.30/0.55 != ( plus_plus_int @ Xb2 @ Xa ) ) ) )
% 0.30/0.55 => ( ( P @ X5 )
% 0.30/0.55 => ( P @ ( minus_minus_int @ X5 @ D4 ) ) ) )
% 0.30/0.55 => ( ! [X5: int] :
% 0.30/0.55 ( ! [Xa: int] :
% 0.30/0.55 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.55 => ! [Xb2: int] :
% 0.30/0.55 ( ( member_int @ Xb2 @ B3 )
% 0.30/0.55 => ( X5
% 0.30/0.55 != ( plus_plus_int @ Xb2 @ Xa ) ) ) )
% 0.30/0.55 => ( ( Q @ X5 )
% 0.30/0.55 => ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) )
% 0.30/0.55 => ! [X3: int] :
% 0.30/0.55 ( ! [Xa3: int] :
% 0.30/0.55 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.55 => ! [Xb3: int] :
% 0.30/0.55 ( ( member_int @ Xb3 @ B3 )
% 0.30/0.55 => ( X3
% 0.30/0.55 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.55 => ( ( ( P @ X3 )
% 0.30/0.55 | ( Q @ X3 ) )
% 0.30/0.55 => ( ( P @ ( minus_minus_int @ X3 @ D4 ) )
% 0.30/0.55 | ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bset(2)
% 0.30/0.55 thf(fact_2796_bset_I1_J,axiom,
% 0.30/0.55 ! [D4: int,B3: set_int,P: int > $o,Q: int > $o] :
% 0.30/0.55 ( ! [X5: int] :
% 0.30/0.55 ( ! [Xa: int] :
% 0.30/0.55 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.55 => ! [Xb2: int] :
% 0.30/0.55 ( ( member_int @ Xb2 @ B3 )
% 0.30/0.55 => ( X5
% 0.30/0.55 != ( plus_plus_int @ Xb2 @ Xa ) ) ) )
% 0.30/0.55 => ( ( P @ X5 )
% 0.30/0.55 => ( P @ ( minus_minus_int @ X5 @ D4 ) ) ) )
% 0.30/0.55 => ( ! [X5: int] :
% 0.30/0.55 ( ! [Xa: int] :
% 0.30/0.55 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.55 => ! [Xb2: int] :
% 0.30/0.55 ( ( member_int @ Xb2 @ B3 )
% 0.30/0.55 => ( X5
% 0.30/0.55 != ( plus_plus_int @ Xb2 @ Xa ) ) ) )
% 0.30/0.55 => ( ( Q @ X5 )
% 0.30/0.55 => ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) )
% 0.30/0.55 => ! [X3: int] :
% 0.30/0.55 ( ! [Xa3: int] :
% 0.30/0.55 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.55 => ! [Xb3: int] :
% 0.30/0.55 ( ( member_int @ Xb3 @ B3 )
% 0.30/0.55 => ( X3
% 0.30/0.55 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.55 => ( ( ( P @ X3 )
% 0.30/0.55 & ( Q @ X3 ) )
% 0.30/0.55 => ( ( P @ ( minus_minus_int @ X3 @ D4 ) )
% 0.30/0.55 & ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % bset(1)
% 0.30/0.55 thf(fact_2797_minf_I7_J,axiom,
% 0.30/0.55 ! [T: real] :
% 0.30/0.55 ? [Z2: real] :
% 0.30/0.55 ! [X3: real] :
% 0.30/0.55 ( ( ord_less_real @ X3 @ Z2 )
% 0.30/0.55 => ~ ( ord_less_real @ T @ X3 ) ) ).
% 0.30/0.55
% 0.30/0.55 % minf(7)
% 0.30/0.55 thf(fact_2798_minf_I7_J,axiom,
% 0.30/0.55 ! [T: rat] :
% 0.30/0.55 ? [Z2: rat] :
% 0.30/0.55 ! [X3: rat] :
% 0.30/0.55 ( ( ord_less_rat @ X3 @ Z2 )
% 0.30/0.55 => ~ ( ord_less_rat @ T @ X3 ) ) ).
% 0.30/0.55
% 0.30/0.55 % minf(7)
% 0.30/0.55 thf(fact_2799_minf_I7_J,axiom,
% 0.30/0.55 ! [T: num] :
% 0.30/0.55 ? [Z2: num] :
% 0.30/0.55 ! [X3: num] :
% 0.30/0.55 ( ( ord_less_num @ X3 @ Z2 )
% 0.30/0.55 => ~ ( ord_less_num @ T @ X3 ) ) ).
% 0.30/0.55
% 0.30/0.55 % minf(7)
% 0.30/0.55 thf(fact_2800_minf_I7_J,axiom,
% 0.30/0.55 ! [T: nat] :
% 0.30/0.55 ? [Z2: nat] :
% 0.30/0.55 ! [X3: nat] :
% 0.30/0.55 ( ( ord_less_nat @ X3 @ Z2 )
% 0.30/0.55 => ~ ( ord_less_nat @ T @ X3 ) ) ).
% 0.30/0.55
% 0.30/0.55 % minf(7)
% 0.30/0.55 thf(fact_2801_minf_I7_J,axiom,
% 0.30/0.55 ! [T: int] :
% 0.30/0.55 ? [Z2: int] :
% 0.30/0.55 ! [X3: int] :
% 0.30/0.55 ( ( ord_less_int @ X3 @ Z2 )
% 0.30/0.55 => ~ ( ord_less_int @ T @ X3 ) ) ).
% 0.30/0.55
% 0.30/0.55 % minf(7)
% 0.30/0.55 thf(fact_2802_minf_I5_J,axiom,
% 0.30/0.55 ! [T: real] :
% 0.30/0.55 ? [Z2: real] :
% 0.30/0.55 ! [X3: real] :
% 0.30/0.55 ( ( ord_less_real @ X3 @ Z2 )
% 0.30/0.55 => ( ord_less_real @ X3 @ T ) ) ).
% 0.30/0.55
% 0.30/0.55 % minf(5)
% 0.30/0.55 thf(fact_2803_minf_I5_J,axiom,
% 0.30/0.55 ! [T: rat] :
% 0.30/0.55 ? [Z2: rat] :
% 0.30/0.55 ! [X3: rat] :
% 0.30/0.55 ( ( ord_less_rat @ X3 @ Z2 )
% 0.30/0.55 => ( ord_less_rat @ X3 @ T ) ) ).
% 0.30/0.55
% 0.30/0.55 % minf(5)
% 0.30/0.55 thf(fact_2804_minf_I5_J,axiom,
% 0.30/0.55 ! [T: num] :
% 0.30/0.55 ? [Z2: num] :
% 0.30/0.55 ! [X3: num] :
% 0.30/0.55 ( ( ord_less_num @ X3 @ Z2 )
% 0.30/0.55 => ( ord_less_num @ X3 @ T ) ) ).
% 0.30/0.55
% 0.30/0.55 % minf(5)
% 0.30/0.55 thf(fact_2805_minf_I5_J,axiom,
% 0.30/0.55 ! [T: nat] :
% 0.30/0.55 ? [Z2: nat] :
% 0.30/0.55 ! [X3: nat] :
% 0.30/0.55 ( ( ord_less_nat @ X3 @ Z2 )
% 0.30/0.55 => ( ord_less_nat @ X3 @ T ) ) ).
% 0.30/0.55
% 0.30/0.55 % minf(5)
% 0.30/0.55 thf(fact_2806_minf_I5_J,axiom,
% 0.30/0.55 ! [T: int] :
% 0.30/0.55 ? [Z2: int] :
% 0.30/0.55 ! [X3: int] :
% 0.30/0.55 ( ( ord_less_int @ X3 @ Z2 )
% 0.30/0.55 => ( ord_less_int @ X3 @ T ) ) ).
% 0.30/0.55
% 0.30/0.55 % minf(5)
% 0.30/0.55 thf(fact_2807_minf_I4_J,axiom,
% 0.30/0.55 ! [T: real] :
% 0.30/0.55 ? [Z2: real] :
% 0.30/0.55 ! [X3: real] :
% 0.30/0.55 ( ( ord_less_real @ X3 @ Z2 )
% 0.30/0.55 => ( X3 != T ) ) ).
% 0.30/0.55
% 0.30/0.55 % minf(4)
% 0.30/0.55 thf(fact_2808_minf_I4_J,axiom,
% 0.30/0.55 ! [T: rat] :
% 0.30/0.55 ? [Z2: rat] :
% 0.30/0.55 ! [X3: rat] :
% 0.30/0.55 ( ( ord_less_rat @ X3 @ Z2 )
% 0.30/0.56 => ( X3 != T ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(4)
% 0.30/0.56 thf(fact_2809_minf_I4_J,axiom,
% 0.30/0.56 ! [T: num] :
% 0.30/0.56 ? [Z2: num] :
% 0.30/0.56 ! [X3: num] :
% 0.30/0.56 ( ( ord_less_num @ X3 @ Z2 )
% 0.30/0.56 => ( X3 != T ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(4)
% 0.30/0.56 thf(fact_2810_minf_I4_J,axiom,
% 0.30/0.56 ! [T: nat] :
% 0.30/0.56 ? [Z2: nat] :
% 0.30/0.56 ! [X3: nat] :
% 0.30/0.56 ( ( ord_less_nat @ X3 @ Z2 )
% 0.30/0.56 => ( X3 != T ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(4)
% 0.30/0.56 thf(fact_2811_minf_I4_J,axiom,
% 0.30/0.56 ! [T: int] :
% 0.30/0.56 ? [Z2: int] :
% 0.30/0.56 ! [X3: int] :
% 0.30/0.56 ( ( ord_less_int @ X3 @ Z2 )
% 0.30/0.56 => ( X3 != T ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(4)
% 0.30/0.56 thf(fact_2812_minf_I3_J,axiom,
% 0.30/0.56 ! [T: real] :
% 0.30/0.56 ? [Z2: real] :
% 0.30/0.56 ! [X3: real] :
% 0.30/0.56 ( ( ord_less_real @ X3 @ Z2 )
% 0.30/0.56 => ( X3 != T ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(3)
% 0.30/0.56 thf(fact_2813_minf_I3_J,axiom,
% 0.30/0.56 ! [T: rat] :
% 0.30/0.56 ? [Z2: rat] :
% 0.30/0.56 ! [X3: rat] :
% 0.30/0.56 ( ( ord_less_rat @ X3 @ Z2 )
% 0.30/0.56 => ( X3 != T ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(3)
% 0.30/0.56 thf(fact_2814_minf_I3_J,axiom,
% 0.30/0.56 ! [T: num] :
% 0.30/0.56 ? [Z2: num] :
% 0.30/0.56 ! [X3: num] :
% 0.30/0.56 ( ( ord_less_num @ X3 @ Z2 )
% 0.30/0.56 => ( X3 != T ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(3)
% 0.30/0.56 thf(fact_2815_minf_I3_J,axiom,
% 0.30/0.56 ! [T: nat] :
% 0.30/0.56 ? [Z2: nat] :
% 0.30/0.56 ! [X3: nat] :
% 0.30/0.56 ( ( ord_less_nat @ X3 @ Z2 )
% 0.30/0.56 => ( X3 != T ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(3)
% 0.30/0.56 thf(fact_2816_minf_I3_J,axiom,
% 0.30/0.56 ! [T: int] :
% 0.30/0.56 ? [Z2: int] :
% 0.30/0.56 ! [X3: int] :
% 0.30/0.56 ( ( ord_less_int @ X3 @ Z2 )
% 0.30/0.56 => ( X3 != T ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(3)
% 0.30/0.56 thf(fact_2817_minf_I2_J,axiom,
% 0.30/0.56 ! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
% 0.30/0.56 ( ? [Z4: real] :
% 0.30/0.56 ! [X5: real] :
% 0.30/0.56 ( ( ord_less_real @ X5 @ Z4 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: real] :
% 0.30/0.56 ! [X5: real] :
% 0.30/0.56 ( ( ord_less_real @ X5 @ Z4 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: real] :
% 0.30/0.56 ! [X3: real] :
% 0.30/0.56 ( ( ord_less_real @ X3 @ Z2 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 | ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(2)
% 0.30/0.56 thf(fact_2818_minf_I2_J,axiom,
% 0.30/0.56 ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.30/0.56 ( ? [Z4: rat] :
% 0.30/0.56 ! [X5: rat] :
% 0.30/0.56 ( ( ord_less_rat @ X5 @ Z4 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: rat] :
% 0.30/0.56 ! [X5: rat] :
% 0.30/0.56 ( ( ord_less_rat @ X5 @ Z4 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: rat] :
% 0.30/0.56 ! [X3: rat] :
% 0.30/0.56 ( ( ord_less_rat @ X3 @ Z2 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 | ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(2)
% 0.30/0.56 thf(fact_2819_minf_I2_J,axiom,
% 0.30/0.56 ! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
% 0.30/0.56 ( ? [Z4: num] :
% 0.30/0.56 ! [X5: num] :
% 0.30/0.56 ( ( ord_less_num @ X5 @ Z4 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: num] :
% 0.30/0.56 ! [X5: num] :
% 0.30/0.56 ( ( ord_less_num @ X5 @ Z4 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: num] :
% 0.30/0.56 ! [X3: num] :
% 0.30/0.56 ( ( ord_less_num @ X3 @ Z2 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 | ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(2)
% 0.30/0.56 thf(fact_2820_minf_I2_J,axiom,
% 0.30/0.56 ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.30/0.56 ( ? [Z4: nat] :
% 0.30/0.56 ! [X5: nat] :
% 0.30/0.56 ( ( ord_less_nat @ X5 @ Z4 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: nat] :
% 0.30/0.56 ! [X5: nat] :
% 0.30/0.56 ( ( ord_less_nat @ X5 @ Z4 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: nat] :
% 0.30/0.56 ! [X3: nat] :
% 0.30/0.56 ( ( ord_less_nat @ X3 @ Z2 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 | ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(2)
% 0.30/0.56 thf(fact_2821_minf_I2_J,axiom,
% 0.30/0.56 ! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
% 0.30/0.56 ( ? [Z4: int] :
% 0.30/0.56 ! [X5: int] :
% 0.30/0.56 ( ( ord_less_int @ X5 @ Z4 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: int] :
% 0.30/0.56 ! [X5: int] :
% 0.30/0.56 ( ( ord_less_int @ X5 @ Z4 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: int] :
% 0.30/0.56 ! [X3: int] :
% 0.30/0.56 ( ( ord_less_int @ X3 @ Z2 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 | ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(2)
% 0.30/0.56 thf(fact_2822_minf_I1_J,axiom,
% 0.30/0.56 ! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
% 0.30/0.56 ( ? [Z4: real] :
% 0.30/0.56 ! [X5: real] :
% 0.30/0.56 ( ( ord_less_real @ X5 @ Z4 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: real] :
% 0.30/0.56 ! [X5: real] :
% 0.30/0.56 ( ( ord_less_real @ X5 @ Z4 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: real] :
% 0.30/0.56 ! [X3: real] :
% 0.30/0.56 ( ( ord_less_real @ X3 @ Z2 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 & ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(1)
% 0.30/0.56 thf(fact_2823_minf_I1_J,axiom,
% 0.30/0.56 ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.30/0.56 ( ? [Z4: rat] :
% 0.30/0.56 ! [X5: rat] :
% 0.30/0.56 ( ( ord_less_rat @ X5 @ Z4 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: rat] :
% 0.30/0.56 ! [X5: rat] :
% 0.30/0.56 ( ( ord_less_rat @ X5 @ Z4 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: rat] :
% 0.30/0.56 ! [X3: rat] :
% 0.30/0.56 ( ( ord_less_rat @ X3 @ Z2 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 & ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(1)
% 0.30/0.56 thf(fact_2824_minf_I1_J,axiom,
% 0.30/0.56 ! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
% 0.30/0.56 ( ? [Z4: num] :
% 0.30/0.56 ! [X5: num] :
% 0.30/0.56 ( ( ord_less_num @ X5 @ Z4 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: num] :
% 0.30/0.56 ! [X5: num] :
% 0.30/0.56 ( ( ord_less_num @ X5 @ Z4 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: num] :
% 0.30/0.56 ! [X3: num] :
% 0.30/0.56 ( ( ord_less_num @ X3 @ Z2 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 & ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(1)
% 0.30/0.56 thf(fact_2825_minf_I1_J,axiom,
% 0.30/0.56 ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.30/0.56 ( ? [Z4: nat] :
% 0.30/0.56 ! [X5: nat] :
% 0.30/0.56 ( ( ord_less_nat @ X5 @ Z4 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: nat] :
% 0.30/0.56 ! [X5: nat] :
% 0.30/0.56 ( ( ord_less_nat @ X5 @ Z4 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: nat] :
% 0.30/0.56 ! [X3: nat] :
% 0.30/0.56 ( ( ord_less_nat @ X3 @ Z2 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 & ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(1)
% 0.30/0.56 thf(fact_2826_minf_I1_J,axiom,
% 0.30/0.56 ! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
% 0.30/0.56 ( ? [Z4: int] :
% 0.30/0.56 ! [X5: int] :
% 0.30/0.56 ( ( ord_less_int @ X5 @ Z4 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: int] :
% 0.30/0.56 ! [X5: int] :
% 0.30/0.56 ( ( ord_less_int @ X5 @ Z4 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: int] :
% 0.30/0.56 ! [X3: int] :
% 0.30/0.56 ( ( ord_less_int @ X3 @ Z2 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 & ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(1)
% 0.30/0.56 thf(fact_2827_pinf_I7_J,axiom,
% 0.30/0.56 ! [T: real] :
% 0.30/0.56 ? [Z2: real] :
% 0.30/0.56 ! [X3: real] :
% 0.30/0.56 ( ( ord_less_real @ Z2 @ X3 )
% 0.30/0.56 => ( ord_less_real @ T @ X3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(7)
% 0.30/0.56 thf(fact_2828_pinf_I7_J,axiom,
% 0.30/0.56 ! [T: rat] :
% 0.30/0.56 ? [Z2: rat] :
% 0.30/0.56 ! [X3: rat] :
% 0.30/0.56 ( ( ord_less_rat @ Z2 @ X3 )
% 0.30/0.56 => ( ord_less_rat @ T @ X3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(7)
% 0.30/0.56 thf(fact_2829_pinf_I7_J,axiom,
% 0.30/0.56 ! [T: num] :
% 0.30/0.56 ? [Z2: num] :
% 0.30/0.56 ! [X3: num] :
% 0.30/0.56 ( ( ord_less_num @ Z2 @ X3 )
% 0.30/0.56 => ( ord_less_num @ T @ X3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(7)
% 0.30/0.56 thf(fact_2830_pinf_I7_J,axiom,
% 0.30/0.56 ! [T: nat] :
% 0.30/0.56 ? [Z2: nat] :
% 0.30/0.56 ! [X3: nat] :
% 0.30/0.56 ( ( ord_less_nat @ Z2 @ X3 )
% 0.30/0.56 => ( ord_less_nat @ T @ X3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(7)
% 0.30/0.56 thf(fact_2831_pinf_I7_J,axiom,
% 0.30/0.56 ! [T: int] :
% 0.30/0.56 ? [Z2: int] :
% 0.30/0.56 ! [X3: int] :
% 0.30/0.56 ( ( ord_less_int @ Z2 @ X3 )
% 0.30/0.56 => ( ord_less_int @ T @ X3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(7)
% 0.30/0.56 thf(fact_2832_pinf_I5_J,axiom,
% 0.30/0.56 ! [T: real] :
% 0.30/0.56 ? [Z2: real] :
% 0.30/0.56 ! [X3: real] :
% 0.30/0.56 ( ( ord_less_real @ Z2 @ X3 )
% 0.30/0.56 => ~ ( ord_less_real @ X3 @ T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(5)
% 0.30/0.56 thf(fact_2833_pinf_I5_J,axiom,
% 0.30/0.56 ! [T: rat] :
% 0.30/0.56 ? [Z2: rat] :
% 0.30/0.56 ! [X3: rat] :
% 0.30/0.56 ( ( ord_less_rat @ Z2 @ X3 )
% 0.30/0.56 => ~ ( ord_less_rat @ X3 @ T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(5)
% 0.30/0.56 thf(fact_2834_pinf_I5_J,axiom,
% 0.30/0.56 ! [T: num] :
% 0.30/0.56 ? [Z2: num] :
% 0.30/0.56 ! [X3: num] :
% 0.30/0.56 ( ( ord_less_num @ Z2 @ X3 )
% 0.30/0.56 => ~ ( ord_less_num @ X3 @ T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(5)
% 0.30/0.56 thf(fact_2835_pinf_I5_J,axiom,
% 0.30/0.56 ! [T: nat] :
% 0.30/0.56 ? [Z2: nat] :
% 0.30/0.56 ! [X3: nat] :
% 0.30/0.56 ( ( ord_less_nat @ Z2 @ X3 )
% 0.30/0.56 => ~ ( ord_less_nat @ X3 @ T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(5)
% 0.30/0.56 thf(fact_2836_pinf_I5_J,axiom,
% 0.30/0.56 ! [T: int] :
% 0.30/0.56 ? [Z2: int] :
% 0.30/0.56 ! [X3: int] :
% 0.30/0.56 ( ( ord_less_int @ Z2 @ X3 )
% 0.30/0.56 => ~ ( ord_less_int @ X3 @ T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(5)
% 0.30/0.56 thf(fact_2837_pinf_I4_J,axiom,
% 0.30/0.56 ! [T: real] :
% 0.30/0.56 ? [Z2: real] :
% 0.30/0.56 ! [X3: real] :
% 0.30/0.56 ( ( ord_less_real @ Z2 @ X3 )
% 0.30/0.56 => ( X3 != T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(4)
% 0.30/0.56 thf(fact_2838_pinf_I4_J,axiom,
% 0.30/0.56 ! [T: rat] :
% 0.30/0.56 ? [Z2: rat] :
% 0.30/0.56 ! [X3: rat] :
% 0.30/0.56 ( ( ord_less_rat @ Z2 @ X3 )
% 0.30/0.56 => ( X3 != T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(4)
% 0.30/0.56 thf(fact_2839_pinf_I4_J,axiom,
% 0.30/0.56 ! [T: num] :
% 0.30/0.56 ? [Z2: num] :
% 0.30/0.56 ! [X3: num] :
% 0.30/0.56 ( ( ord_less_num @ Z2 @ X3 )
% 0.30/0.56 => ( X3 != T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(4)
% 0.30/0.56 thf(fact_2840_pinf_I4_J,axiom,
% 0.30/0.56 ! [T: nat] :
% 0.30/0.56 ? [Z2: nat] :
% 0.30/0.56 ! [X3: nat] :
% 0.30/0.56 ( ( ord_less_nat @ Z2 @ X3 )
% 0.30/0.56 => ( X3 != T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(4)
% 0.30/0.56 thf(fact_2841_pinf_I4_J,axiom,
% 0.30/0.56 ! [T: int] :
% 0.30/0.56 ? [Z2: int] :
% 0.30/0.56 ! [X3: int] :
% 0.30/0.56 ( ( ord_less_int @ Z2 @ X3 )
% 0.30/0.56 => ( X3 != T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(4)
% 0.30/0.56 thf(fact_2842_pinf_I3_J,axiom,
% 0.30/0.56 ! [T: real] :
% 0.30/0.56 ? [Z2: real] :
% 0.30/0.56 ! [X3: real] :
% 0.30/0.56 ( ( ord_less_real @ Z2 @ X3 )
% 0.30/0.56 => ( X3 != T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(3)
% 0.30/0.56 thf(fact_2843_pinf_I3_J,axiom,
% 0.30/0.56 ! [T: rat] :
% 0.30/0.56 ? [Z2: rat] :
% 0.30/0.56 ! [X3: rat] :
% 0.30/0.56 ( ( ord_less_rat @ Z2 @ X3 )
% 0.30/0.56 => ( X3 != T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(3)
% 0.30/0.56 thf(fact_2844_pinf_I3_J,axiom,
% 0.30/0.56 ! [T: num] :
% 0.30/0.56 ? [Z2: num] :
% 0.30/0.56 ! [X3: num] :
% 0.30/0.56 ( ( ord_less_num @ Z2 @ X3 )
% 0.30/0.56 => ( X3 != T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(3)
% 0.30/0.56 thf(fact_2845_pinf_I3_J,axiom,
% 0.30/0.56 ! [T: nat] :
% 0.30/0.56 ? [Z2: nat] :
% 0.30/0.56 ! [X3: nat] :
% 0.30/0.56 ( ( ord_less_nat @ Z2 @ X3 )
% 0.30/0.56 => ( X3 != T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(3)
% 0.30/0.56 thf(fact_2846_pinf_I3_J,axiom,
% 0.30/0.56 ! [T: int] :
% 0.30/0.56 ? [Z2: int] :
% 0.30/0.56 ! [X3: int] :
% 0.30/0.56 ( ( ord_less_int @ Z2 @ X3 )
% 0.30/0.56 => ( X3 != T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(3)
% 0.30/0.56 thf(fact_2847_pinf_I2_J,axiom,
% 0.30/0.56 ! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
% 0.30/0.56 ( ? [Z4: real] :
% 0.30/0.56 ! [X5: real] :
% 0.30/0.56 ( ( ord_less_real @ Z4 @ X5 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: real] :
% 0.30/0.56 ! [X5: real] :
% 0.30/0.56 ( ( ord_less_real @ Z4 @ X5 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: real] :
% 0.30/0.56 ! [X3: real] :
% 0.30/0.56 ( ( ord_less_real @ Z2 @ X3 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 | ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(2)
% 0.30/0.56 thf(fact_2848_pinf_I2_J,axiom,
% 0.30/0.56 ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.30/0.56 ( ? [Z4: rat] :
% 0.30/0.56 ! [X5: rat] :
% 0.30/0.56 ( ( ord_less_rat @ Z4 @ X5 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: rat] :
% 0.30/0.56 ! [X5: rat] :
% 0.30/0.56 ( ( ord_less_rat @ Z4 @ X5 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: rat] :
% 0.30/0.56 ! [X3: rat] :
% 0.30/0.56 ( ( ord_less_rat @ Z2 @ X3 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 | ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(2)
% 0.30/0.56 thf(fact_2849_pinf_I2_J,axiom,
% 0.30/0.56 ! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
% 0.30/0.56 ( ? [Z4: num] :
% 0.30/0.56 ! [X5: num] :
% 0.30/0.56 ( ( ord_less_num @ Z4 @ X5 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: num] :
% 0.30/0.56 ! [X5: num] :
% 0.30/0.56 ( ( ord_less_num @ Z4 @ X5 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: num] :
% 0.30/0.56 ! [X3: num] :
% 0.30/0.56 ( ( ord_less_num @ Z2 @ X3 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 | ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(2)
% 0.30/0.56 thf(fact_2850_pinf_I2_J,axiom,
% 0.30/0.56 ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.30/0.56 ( ? [Z4: nat] :
% 0.30/0.56 ! [X5: nat] :
% 0.30/0.56 ( ( ord_less_nat @ Z4 @ X5 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: nat] :
% 0.30/0.56 ! [X5: nat] :
% 0.30/0.56 ( ( ord_less_nat @ Z4 @ X5 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: nat] :
% 0.30/0.56 ! [X3: nat] :
% 0.30/0.56 ( ( ord_less_nat @ Z2 @ X3 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 | ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(2)
% 0.30/0.56 thf(fact_2851_pinf_I2_J,axiom,
% 0.30/0.56 ! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
% 0.30/0.56 ( ? [Z4: int] :
% 0.30/0.56 ! [X5: int] :
% 0.30/0.56 ( ( ord_less_int @ Z4 @ X5 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: int] :
% 0.30/0.56 ! [X5: int] :
% 0.30/0.56 ( ( ord_less_int @ Z4 @ X5 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: int] :
% 0.30/0.56 ! [X3: int] :
% 0.30/0.56 ( ( ord_less_int @ Z2 @ X3 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 | ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(2)
% 0.30/0.56 thf(fact_2852_pinf_I1_J,axiom,
% 0.30/0.56 ! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
% 0.30/0.56 ( ? [Z4: real] :
% 0.30/0.56 ! [X5: real] :
% 0.30/0.56 ( ( ord_less_real @ Z4 @ X5 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: real] :
% 0.30/0.56 ! [X5: real] :
% 0.30/0.56 ( ( ord_less_real @ Z4 @ X5 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: real] :
% 0.30/0.56 ! [X3: real] :
% 0.30/0.56 ( ( ord_less_real @ Z2 @ X3 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 & ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(1)
% 0.30/0.56 thf(fact_2853_pinf_I1_J,axiom,
% 0.30/0.56 ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.30/0.56 ( ? [Z4: rat] :
% 0.30/0.56 ! [X5: rat] :
% 0.30/0.56 ( ( ord_less_rat @ Z4 @ X5 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: rat] :
% 0.30/0.56 ! [X5: rat] :
% 0.30/0.56 ( ( ord_less_rat @ Z4 @ X5 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: rat] :
% 0.30/0.56 ! [X3: rat] :
% 0.30/0.56 ( ( ord_less_rat @ Z2 @ X3 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 & ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(1)
% 0.30/0.56 thf(fact_2854_pinf_I1_J,axiom,
% 0.30/0.56 ! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
% 0.30/0.56 ( ? [Z4: num] :
% 0.30/0.56 ! [X5: num] :
% 0.30/0.56 ( ( ord_less_num @ Z4 @ X5 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: num] :
% 0.30/0.56 ! [X5: num] :
% 0.30/0.56 ( ( ord_less_num @ Z4 @ X5 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: num] :
% 0.30/0.56 ! [X3: num] :
% 0.30/0.56 ( ( ord_less_num @ Z2 @ X3 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 & ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(1)
% 0.30/0.56 thf(fact_2855_pinf_I1_J,axiom,
% 0.30/0.56 ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.30/0.56 ( ? [Z4: nat] :
% 0.30/0.56 ! [X5: nat] :
% 0.30/0.56 ( ( ord_less_nat @ Z4 @ X5 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: nat] :
% 0.30/0.56 ! [X5: nat] :
% 0.30/0.56 ( ( ord_less_nat @ Z4 @ X5 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: nat] :
% 0.30/0.56 ! [X3: nat] :
% 0.30/0.56 ( ( ord_less_nat @ Z2 @ X3 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 & ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(1)
% 0.30/0.56 thf(fact_2856_pinf_I1_J,axiom,
% 0.30/0.56 ! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
% 0.30/0.56 ( ? [Z4: int] :
% 0.30/0.56 ! [X5: int] :
% 0.30/0.56 ( ( ord_less_int @ Z4 @ X5 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [Z4: int] :
% 0.30/0.56 ! [X5: int] :
% 0.30/0.56 ( ( ord_less_int @ Z4 @ X5 )
% 0.30/0.56 => ( ( Q @ X5 )
% 0.30/0.56 = ( Q6 @ X5 ) ) )
% 0.30/0.56 => ? [Z2: int] :
% 0.30/0.56 ! [X3: int] :
% 0.30/0.56 ( ( ord_less_int @ Z2 @ X3 )
% 0.30/0.56 => ( ( ( P @ X3 )
% 0.30/0.56 & ( Q @ X3 ) )
% 0.30/0.56 = ( ( P5 @ X3 )
% 0.30/0.56 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(1)
% 0.30/0.56 thf(fact_2857_bounded__Max__nat,axiom,
% 0.30/0.56 ! [P: nat > $o,X2: nat,M7: nat] :
% 0.30/0.56 ( ( P @ X2 )
% 0.30/0.56 => ( ! [X5: nat] :
% 0.30/0.56 ( ( P @ X5 )
% 0.30/0.56 => ( ord_less_eq_nat @ X5 @ M7 ) )
% 0.30/0.56 => ~ ! [M4: nat] :
% 0.30/0.56 ( ( P @ M4 )
% 0.30/0.56 => ~ ! [X3: nat] :
% 0.30/0.56 ( ( P @ X3 )
% 0.30/0.56 => ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % bounded_Max_nat
% 0.30/0.56 thf(fact_2858_fold__atLeastAtMost__nat_Ocases,axiom,
% 0.30/0.56 ! [X2: produc3368934014287244435at_num] :
% 0.30/0.56 ~ ! [F2: nat > num > num,A4: nat,B4: nat,Acc: num] :
% 0.30/0.56 ( X2
% 0.30/0.56 != ( produc851828971589881931at_num @ F2 @ ( produc1195630363706982562at_num @ A4 @ ( product_Pair_nat_num @ B4 @ Acc ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % fold_atLeastAtMost_nat.cases
% 0.30/0.56 thf(fact_2859_fold__atLeastAtMost__nat_Ocases,axiom,
% 0.30/0.56 ! [X2: produc4471711990508489141at_nat] :
% 0.30/0.56 ~ ! [F2: nat > nat > nat,A4: nat,B4: nat,Acc: nat] :
% 0.30/0.56 ( X2
% 0.30/0.56 != ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A4 @ ( product_Pair_nat_nat @ B4 @ Acc ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % fold_atLeastAtMost_nat.cases
% 0.30/0.56 thf(fact_2860_periodic__finite__ex,axiom,
% 0.30/0.56 ! [D: int,P: int > $o] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ D )
% 0.30/0.56 => ( ! [X5: int,K3: int] :
% 0.30/0.56 ( ( P @ X5 )
% 0.30/0.56 = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) )
% 0.30/0.56 => ( ( ? [X4: int] : ( P @ X4 ) )
% 0.30/0.56 = ( ? [X: int] :
% 0.30/0.56 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 0.30/0.56 & ( P @ X ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % periodic_finite_ex
% 0.30/0.56 thf(fact_2861_bset_I3_J,axiom,
% 0.30/0.56 ! [D4: int,T: int,B3: set_int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B3 )
% 0.30/0.56 => ! [X3: int] :
% 0.30/0.56 ( ! [Xa3: int] :
% 0.30/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56 => ! [Xb3: int] :
% 0.30/0.56 ( ( member_int @ Xb3 @ B3 )
% 0.30/0.56 => ( X3
% 0.30/0.56 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56 => ( ( X3 = T )
% 0.30/0.56 => ( ( minus_minus_int @ X3 @ D4 )
% 0.30/0.56 = T ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % bset(3)
% 0.30/0.56 thf(fact_2862_bset_I4_J,axiom,
% 0.30/0.56 ! [D4: int,T: int,B3: set_int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56 => ( ( member_int @ T @ B3 )
% 0.30/0.56 => ! [X3: int] :
% 0.30/0.56 ( ! [Xa3: int] :
% 0.30/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56 => ! [Xb3: int] :
% 0.30/0.56 ( ( member_int @ Xb3 @ B3 )
% 0.30/0.56 => ( X3
% 0.30/0.56 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56 => ( ( X3 != T )
% 0.30/0.56 => ( ( minus_minus_int @ X3 @ D4 )
% 0.30/0.56 != T ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % bset(4)
% 0.30/0.56 thf(fact_2863_bset_I5_J,axiom,
% 0.30/0.56 ! [D4: int,B3: set_int,T: int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56 => ! [X3: int] :
% 0.30/0.56 ( ! [Xa3: int] :
% 0.30/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56 => ! [Xb3: int] :
% 0.30/0.56 ( ( member_int @ Xb3 @ B3 )
% 0.30/0.56 => ( X3
% 0.30/0.56 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56 => ( ( ord_less_int @ X3 @ T )
% 0.30/0.56 => ( ord_less_int @ ( minus_minus_int @ X3 @ D4 ) @ T ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % bset(5)
% 0.30/0.56 thf(fact_2864_bset_I7_J,axiom,
% 0.30/0.56 ! [D4: int,T: int,B3: set_int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56 => ( ( member_int @ T @ B3 )
% 0.30/0.56 => ! [X3: int] :
% 0.30/0.56 ( ! [Xa3: int] :
% 0.30/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56 => ! [Xb3: int] :
% 0.30/0.56 ( ( member_int @ Xb3 @ B3 )
% 0.30/0.56 => ( X3
% 0.30/0.56 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56 => ( ( ord_less_int @ T @ X3 )
% 0.30/0.56 => ( ord_less_int @ T @ ( minus_minus_int @ X3 @ D4 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % bset(7)
% 0.30/0.56 thf(fact_2865_aset_I3_J,axiom,
% 0.30/0.56 ! [D4: int,T: int,A2: set_int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 0.30/0.56 => ! [X3: int] :
% 0.30/0.56 ( ! [Xa3: int] :
% 0.30/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56 => ! [Xb3: int] :
% 0.30/0.56 ( ( member_int @ Xb3 @ A2 )
% 0.30/0.56 => ( X3
% 0.30/0.56 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56 => ( ( X3 = T )
% 0.30/0.56 => ( ( plus_plus_int @ X3 @ D4 )
% 0.30/0.56 = T ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % aset(3)
% 0.30/0.56 thf(fact_2866_aset_I4_J,axiom,
% 0.30/0.56 ! [D4: int,T: int,A2: set_int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56 => ( ( member_int @ T @ A2 )
% 0.30/0.56 => ! [X3: int] :
% 0.30/0.56 ( ! [Xa3: int] :
% 0.30/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56 => ! [Xb3: int] :
% 0.30/0.56 ( ( member_int @ Xb3 @ A2 )
% 0.30/0.56 => ( X3
% 0.30/0.56 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56 => ( ( X3 != T )
% 0.30/0.56 => ( ( plus_plus_int @ X3 @ D4 )
% 0.30/0.56 != T ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % aset(4)
% 0.30/0.56 thf(fact_2867_aset_I5_J,axiom,
% 0.30/0.56 ! [D4: int,T: int,A2: set_int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56 => ( ( member_int @ T @ A2 )
% 0.30/0.56 => ! [X3: int] :
% 0.30/0.56 ( ! [Xa3: int] :
% 0.30/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56 => ! [Xb3: int] :
% 0.30/0.56 ( ( member_int @ Xb3 @ A2 )
% 0.30/0.56 => ( X3
% 0.30/0.56 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56 => ( ( ord_less_int @ X3 @ T )
% 0.30/0.56 => ( ord_less_int @ ( plus_plus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % aset(5)
% 0.30/0.56 thf(fact_2868_aset_I7_J,axiom,
% 0.30/0.56 ! [D4: int,A2: set_int,T: int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56 => ! [X3: int] :
% 0.30/0.56 ( ! [Xa3: int] :
% 0.30/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56 => ! [Xb3: int] :
% 0.30/0.56 ( ( member_int @ Xb3 @ A2 )
% 0.30/0.56 => ( X3
% 0.30/0.56 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56 => ( ( ord_less_int @ T @ X3 )
% 0.30/0.56 => ( ord_less_int @ T @ ( plus_plus_int @ X3 @ D4 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % aset(7)
% 0.30/0.56 thf(fact_2869_bset_I6_J,axiom,
% 0.30/0.56 ! [D4: int,B3: set_int,T: int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56 => ! [X3: int] :
% 0.30/0.56 ( ! [Xa3: int] :
% 0.30/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56 => ! [Xb3: int] :
% 0.30/0.56 ( ( member_int @ Xb3 @ B3 )
% 0.30/0.56 => ( X3
% 0.30/0.56 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56 => ( ( ord_less_eq_int @ X3 @ T )
% 0.30/0.56 => ( ord_less_eq_int @ ( minus_minus_int @ X3 @ D4 ) @ T ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % bset(6)
% 0.30/0.56 thf(fact_2870_bset_I8_J,axiom,
% 0.30/0.56 ! [D4: int,T: int,B3: set_int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B3 )
% 0.30/0.56 => ! [X3: int] :
% 0.30/0.56 ( ! [Xa3: int] :
% 0.30/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56 => ! [Xb3: int] :
% 0.30/0.56 ( ( member_int @ Xb3 @ B3 )
% 0.30/0.56 => ( X3
% 0.30/0.56 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56 => ( ( ord_less_eq_int @ T @ X3 )
% 0.30/0.56 => ( ord_less_eq_int @ T @ ( minus_minus_int @ X3 @ D4 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % bset(8)
% 0.30/0.56 thf(fact_2871_aset_I6_J,axiom,
% 0.30/0.56 ! [D4: int,T: int,A2: set_int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 0.30/0.56 => ! [X3: int] :
% 0.30/0.56 ( ! [Xa3: int] :
% 0.30/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56 => ! [Xb3: int] :
% 0.30/0.56 ( ( member_int @ Xb3 @ A2 )
% 0.30/0.56 => ( X3
% 0.30/0.56 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56 => ( ( ord_less_eq_int @ X3 @ T )
% 0.30/0.56 => ( ord_less_eq_int @ ( plus_plus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % aset(6)
% 0.30/0.56 thf(fact_2872_aset_I8_J,axiom,
% 0.30/0.56 ! [D4: int,A2: set_int,T: int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56 => ! [X3: int] :
% 0.30/0.56 ( ! [Xa3: int] :
% 0.30/0.56 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56 => ! [Xb3: int] :
% 0.30/0.56 ( ( member_int @ Xb3 @ A2 )
% 0.30/0.56 => ( X3
% 0.30/0.56 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56 => ( ( ord_less_eq_int @ T @ X3 )
% 0.30/0.56 => ( ord_less_eq_int @ T @ ( plus_plus_int @ X3 @ D4 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % aset(8)
% 0.30/0.56 thf(fact_2873_cppi,axiom,
% 0.30/0.56 ! [D4: int,P: int > $o,P5: int > $o,A2: set_int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56 => ( ? [Z4: int] :
% 0.30/0.56 ! [X5: int] :
% 0.30/0.56 ( ( ord_less_int @ Z4 @ X5 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ! [X5: int] :
% 0.30/0.56 ( ! [Xa: int] :
% 0.30/0.56 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56 => ! [Xb2: int] :
% 0.30/0.56 ( ( member_int @ Xb2 @ A2 )
% 0.30/0.56 => ( X5
% 0.30/0.56 != ( minus_minus_int @ Xb2 @ Xa ) ) ) )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 => ( P @ ( plus_plus_int @ X5 @ D4 ) ) ) )
% 0.30/0.56 => ( ! [X5: int,K3: int] :
% 0.30/0.56 ( ( P5 @ X5 )
% 0.30/0.56 = ( P5 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 0.30/0.56 => ( ( ? [X4: int] : ( P @ X4 ) )
% 0.30/0.56 = ( ? [X: int] :
% 0.30/0.56 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56 & ( P5 @ X ) )
% 0.30/0.56 | ? [X: int] :
% 0.30/0.56 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56 & ? [Y: int] :
% 0.30/0.56 ( ( member_int @ Y @ A2 )
% 0.30/0.56 & ( P @ ( minus_minus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % cppi
% 0.30/0.56 thf(fact_2874_cpmi,axiom,
% 0.30/0.56 ! [D4: int,P: int > $o,P5: int > $o,B3: set_int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56 => ( ? [Z4: int] :
% 0.30/0.56 ! [X5: int] :
% 0.30/0.56 ( ( ord_less_int @ X5 @ Z4 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ! [X5: int] :
% 0.30/0.56 ( ! [Xa: int] :
% 0.30/0.56 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56 => ! [Xb2: int] :
% 0.30/0.56 ( ( member_int @ Xb2 @ B3 )
% 0.30/0.56 => ( X5
% 0.30/0.56 != ( plus_plus_int @ Xb2 @ Xa ) ) ) )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 => ( P @ ( minus_minus_int @ X5 @ D4 ) ) ) )
% 0.30/0.56 => ( ! [X5: int,K3: int] :
% 0.30/0.56 ( ( P5 @ X5 )
% 0.30/0.56 = ( P5 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 0.30/0.56 => ( ( ? [X4: int] : ( P @ X4 ) )
% 0.30/0.56 = ( ? [X: int] :
% 0.30/0.56 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56 & ( P5 @ X ) )
% 0.30/0.56 | ? [X: int] :
% 0.30/0.56 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56 & ? [Y: int] :
% 0.30/0.56 ( ( member_int @ Y @ B3 )
% 0.30/0.56 & ( P @ ( plus_plus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % cpmi
% 0.30/0.56 thf(fact_2875_pinf_I6_J,axiom,
% 0.30/0.56 ! [T: real] :
% 0.30/0.56 ? [Z2: real] :
% 0.30/0.56 ! [X3: real] :
% 0.30/0.56 ( ( ord_less_real @ Z2 @ X3 )
% 0.30/0.56 => ~ ( ord_less_eq_real @ X3 @ T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(6)
% 0.30/0.56 thf(fact_2876_pinf_I6_J,axiom,
% 0.30/0.56 ! [T: rat] :
% 0.30/0.56 ? [Z2: rat] :
% 0.30/0.56 ! [X3: rat] :
% 0.30/0.56 ( ( ord_less_rat @ Z2 @ X3 )
% 0.30/0.56 => ~ ( ord_less_eq_rat @ X3 @ T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(6)
% 0.30/0.56 thf(fact_2877_pinf_I6_J,axiom,
% 0.30/0.56 ! [T: num] :
% 0.30/0.56 ? [Z2: num] :
% 0.30/0.56 ! [X3: num] :
% 0.30/0.56 ( ( ord_less_num @ Z2 @ X3 )
% 0.30/0.56 => ~ ( ord_less_eq_num @ X3 @ T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(6)
% 0.30/0.56 thf(fact_2878_pinf_I6_J,axiom,
% 0.30/0.56 ! [T: nat] :
% 0.30/0.56 ? [Z2: nat] :
% 0.30/0.56 ! [X3: nat] :
% 0.30/0.56 ( ( ord_less_nat @ Z2 @ X3 )
% 0.30/0.56 => ~ ( ord_less_eq_nat @ X3 @ T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(6)
% 0.30/0.56 thf(fact_2879_pinf_I6_J,axiom,
% 0.30/0.56 ! [T: int] :
% 0.30/0.56 ? [Z2: int] :
% 0.30/0.56 ! [X3: int] :
% 0.30/0.56 ( ( ord_less_int @ Z2 @ X3 )
% 0.30/0.56 => ~ ( ord_less_eq_int @ X3 @ T ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(6)
% 0.30/0.56 thf(fact_2880_pinf_I8_J,axiom,
% 0.30/0.56 ! [T: real] :
% 0.30/0.56 ? [Z2: real] :
% 0.30/0.56 ! [X3: real] :
% 0.30/0.56 ( ( ord_less_real @ Z2 @ X3 )
% 0.30/0.56 => ( ord_less_eq_real @ T @ X3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(8)
% 0.30/0.56 thf(fact_2881_pinf_I8_J,axiom,
% 0.30/0.56 ! [T: rat] :
% 0.30/0.56 ? [Z2: rat] :
% 0.30/0.56 ! [X3: rat] :
% 0.30/0.56 ( ( ord_less_rat @ Z2 @ X3 )
% 0.30/0.56 => ( ord_less_eq_rat @ T @ X3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(8)
% 0.30/0.56 thf(fact_2882_pinf_I8_J,axiom,
% 0.30/0.56 ! [T: num] :
% 0.30/0.56 ? [Z2: num] :
% 0.30/0.56 ! [X3: num] :
% 0.30/0.56 ( ( ord_less_num @ Z2 @ X3 )
% 0.30/0.56 => ( ord_less_eq_num @ T @ X3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(8)
% 0.30/0.56 thf(fact_2883_pinf_I8_J,axiom,
% 0.30/0.56 ! [T: nat] :
% 0.30/0.56 ? [Z2: nat] :
% 0.30/0.56 ! [X3: nat] :
% 0.30/0.56 ( ( ord_less_nat @ Z2 @ X3 )
% 0.30/0.56 => ( ord_less_eq_nat @ T @ X3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(8)
% 0.30/0.56 thf(fact_2884_pinf_I8_J,axiom,
% 0.30/0.56 ! [T: int] :
% 0.30/0.56 ? [Z2: int] :
% 0.30/0.56 ! [X3: int] :
% 0.30/0.56 ( ( ord_less_int @ Z2 @ X3 )
% 0.30/0.56 => ( ord_less_eq_int @ T @ X3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % pinf(8)
% 0.30/0.56 thf(fact_2885_minf_I6_J,axiom,
% 0.30/0.56 ! [T: real] :
% 0.30/0.56 ? [Z2: real] :
% 0.30/0.56 ! [X3: real] :
% 0.30/0.56 ( ( ord_less_real @ X3 @ Z2 )
% 0.30/0.56 => ( ord_less_eq_real @ X3 @ T ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(6)
% 0.30/0.56 thf(fact_2886_minf_I6_J,axiom,
% 0.30/0.56 ! [T: rat] :
% 0.30/0.56 ? [Z2: rat] :
% 0.30/0.56 ! [X3: rat] :
% 0.30/0.56 ( ( ord_less_rat @ X3 @ Z2 )
% 0.30/0.56 => ( ord_less_eq_rat @ X3 @ T ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(6)
% 0.30/0.56 thf(fact_2887_minf_I6_J,axiom,
% 0.30/0.56 ! [T: num] :
% 0.30/0.56 ? [Z2: num] :
% 0.30/0.56 ! [X3: num] :
% 0.30/0.56 ( ( ord_less_num @ X3 @ Z2 )
% 0.30/0.56 => ( ord_less_eq_num @ X3 @ T ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(6)
% 0.30/0.56 thf(fact_2888_minf_I6_J,axiom,
% 0.30/0.56 ! [T: nat] :
% 0.30/0.56 ? [Z2: nat] :
% 0.30/0.56 ! [X3: nat] :
% 0.30/0.56 ( ( ord_less_nat @ X3 @ Z2 )
% 0.30/0.56 => ( ord_less_eq_nat @ X3 @ T ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(6)
% 0.30/0.56 thf(fact_2889_minf_I6_J,axiom,
% 0.30/0.56 ! [T: int] :
% 0.30/0.56 ? [Z2: int] :
% 0.30/0.56 ! [X3: int] :
% 0.30/0.56 ( ( ord_less_int @ X3 @ Z2 )
% 0.30/0.56 => ( ord_less_eq_int @ X3 @ T ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(6)
% 0.30/0.56 thf(fact_2890_minf_I8_J,axiom,
% 0.30/0.56 ! [T: real] :
% 0.30/0.56 ? [Z2: real] :
% 0.30/0.56 ! [X3: real] :
% 0.30/0.56 ( ( ord_less_real @ X3 @ Z2 )
% 0.30/0.56 => ~ ( ord_less_eq_real @ T @ X3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(8)
% 0.30/0.56 thf(fact_2891_minf_I8_J,axiom,
% 0.30/0.56 ! [T: rat] :
% 0.30/0.56 ? [Z2: rat] :
% 0.30/0.56 ! [X3: rat] :
% 0.30/0.56 ( ( ord_less_rat @ X3 @ Z2 )
% 0.30/0.56 => ~ ( ord_less_eq_rat @ T @ X3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(8)
% 0.30/0.56 thf(fact_2892_minf_I8_J,axiom,
% 0.30/0.56 ! [T: num] :
% 0.30/0.56 ? [Z2: num] :
% 0.30/0.56 ! [X3: num] :
% 0.30/0.56 ( ( ord_less_num @ X3 @ Z2 )
% 0.30/0.56 => ~ ( ord_less_eq_num @ T @ X3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(8)
% 0.30/0.56 thf(fact_2893_minf_I8_J,axiom,
% 0.30/0.56 ! [T: nat] :
% 0.30/0.56 ? [Z2: nat] :
% 0.30/0.56 ! [X3: nat] :
% 0.30/0.56 ( ( ord_less_nat @ X3 @ Z2 )
% 0.30/0.56 => ~ ( ord_less_eq_nat @ T @ X3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(8)
% 0.30/0.56 thf(fact_2894_minf_I8_J,axiom,
% 0.30/0.56 ! [T: int] :
% 0.30/0.56 ? [Z2: int] :
% 0.30/0.56 ! [X3: int] :
% 0.30/0.56 ( ( ord_less_int @ X3 @ Z2 )
% 0.30/0.56 => ~ ( ord_less_eq_int @ T @ X3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % minf(8)
% 0.30/0.56 thf(fact_2895_inf__period_I2_J,axiom,
% 0.30/0.56 ! [P: real > $o,D4: real,Q: real > $o] :
% 0.30/0.56 ( ! [X5: real,K3: real] :
% 0.30/0.56 ( ( P @ X5 )
% 0.30/0.56 = ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 0.30/0.56 => ( ! [X5: real,K3: real] :
% 0.30/0.56 ( ( Q @ X5 )
% 0.30/0.56 = ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 0.30/0.56 => ! [X3: real,K4: real] :
% 0.30/0.56 ( ( ( P @ X3 )
% 0.30/0.56 | ( Q @ X3 ) )
% 0.30/0.56 = ( ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) )
% 0.30/0.56 | ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_period(2)
% 0.30/0.56 thf(fact_2896_inf__period_I2_J,axiom,
% 0.30/0.56 ! [P: rat > $o,D4: rat,Q: rat > $o] :
% 0.30/0.56 ( ! [X5: rat,K3: rat] :
% 0.30/0.56 ( ( P @ X5 )
% 0.30/0.56 = ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 0.30/0.56 => ( ! [X5: rat,K3: rat] :
% 0.30/0.56 ( ( Q @ X5 )
% 0.30/0.56 = ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 0.30/0.56 => ! [X3: rat,K4: rat] :
% 0.30/0.56 ( ( ( P @ X3 )
% 0.30/0.56 | ( Q @ X3 ) )
% 0.30/0.56 = ( ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) )
% 0.30/0.56 | ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_period(2)
% 0.30/0.56 thf(fact_2897_inf__period_I2_J,axiom,
% 0.30/0.56 ! [P: int > $o,D4: int,Q: int > $o] :
% 0.30/0.56 ( ! [X5: int,K3: int] :
% 0.30/0.56 ( ( P @ X5 )
% 0.30/0.56 = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 0.30/0.56 => ( ! [X5: int,K3: int] :
% 0.30/0.56 ( ( Q @ X5 )
% 0.30/0.56 = ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 0.30/0.56 => ! [X3: int,K4: int] :
% 0.30/0.56 ( ( ( P @ X3 )
% 0.30/0.56 | ( Q @ X3 ) )
% 0.30/0.56 = ( ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) )
% 0.30/0.56 | ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_period(2)
% 0.30/0.56 thf(fact_2898_inf__period_I1_J,axiom,
% 0.30/0.56 ! [P: real > $o,D4: real,Q: real > $o] :
% 0.30/0.56 ( ! [X5: real,K3: real] :
% 0.30/0.56 ( ( P @ X5 )
% 0.30/0.56 = ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 0.30/0.56 => ( ! [X5: real,K3: real] :
% 0.30/0.56 ( ( Q @ X5 )
% 0.30/0.56 = ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 0.30/0.56 => ! [X3: real,K4: real] :
% 0.30/0.56 ( ( ( P @ X3 )
% 0.30/0.56 & ( Q @ X3 ) )
% 0.30/0.56 = ( ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) )
% 0.30/0.56 & ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_period(1)
% 0.30/0.56 thf(fact_2899_inf__period_I1_J,axiom,
% 0.30/0.56 ! [P: rat > $o,D4: rat,Q: rat > $o] :
% 0.30/0.56 ( ! [X5: rat,K3: rat] :
% 0.30/0.56 ( ( P @ X5 )
% 0.30/0.56 = ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 0.30/0.56 => ( ! [X5: rat,K3: rat] :
% 0.30/0.56 ( ( Q @ X5 )
% 0.30/0.56 = ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 0.30/0.56 => ! [X3: rat,K4: rat] :
% 0.30/0.56 ( ( ( P @ X3 )
% 0.30/0.56 & ( Q @ X3 ) )
% 0.30/0.56 = ( ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) )
% 0.30/0.56 & ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_period(1)
% 0.30/0.56 thf(fact_2900_inf__period_I1_J,axiom,
% 0.30/0.56 ! [P: int > $o,D4: int,Q: int > $o] :
% 0.30/0.56 ( ! [X5: int,K3: int] :
% 0.30/0.56 ( ( P @ X5 )
% 0.30/0.56 = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 0.30/0.56 => ( ! [X5: int,K3: int] :
% 0.30/0.56 ( ( Q @ X5 )
% 0.30/0.56 = ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 0.30/0.56 => ! [X3: int,K4: int] :
% 0.30/0.56 ( ( ( P @ X3 )
% 0.30/0.56 & ( Q @ X3 ) )
% 0.30/0.56 = ( ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) )
% 0.30/0.56 & ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % inf_period(1)
% 0.30/0.56 thf(fact_2901_conj__le__cong,axiom,
% 0.30/0.56 ! [X2: int,X7: int,P: $o,P5: $o] :
% 0.30/0.56 ( ( X2 = X7 )
% 0.30/0.56 => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 0.30/0.56 => ( P = P5 ) )
% 0.30/0.56 => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.56 & P )
% 0.30/0.56 = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 0.30/0.56 & P5 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % conj_le_cong
% 0.30/0.56 thf(fact_2902_imp__le__cong,axiom,
% 0.30/0.56 ! [X2: int,X7: int,P: $o,P5: $o] :
% 0.30/0.56 ( ( X2 = X7 )
% 0.30/0.56 => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 0.30/0.56 => ( P = P5 ) )
% 0.30/0.56 => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.56 => P )
% 0.30/0.56 = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 0.30/0.56 => P5 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % imp_le_cong
% 0.30/0.56 thf(fact_2903_atLeastatMost__psubset__iff,axiom,
% 0.30/0.56 ! [A: set_int,B: set_int,C: set_int,D: set_int] :
% 0.30/0.56 ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 0.30/0.56 = ( ( ~ ( ord_less_eq_set_int @ A @ B )
% 0.30/0.56 | ( ( ord_less_eq_set_int @ C @ A )
% 0.30/0.56 & ( ord_less_eq_set_int @ B @ D )
% 0.30/0.56 & ( ( ord_less_set_int @ C @ A )
% 0.30/0.56 | ( ord_less_set_int @ B @ D ) ) ) )
% 0.30/0.56 & ( ord_less_eq_set_int @ C @ D ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastatMost_psubset_iff
% 0.30/0.56 thf(fact_2904_atLeastatMost__psubset__iff,axiom,
% 0.30/0.56 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.56 ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 0.30/0.56 = ( ( ~ ( ord_less_eq_rat @ A @ B )
% 0.30/0.56 | ( ( ord_less_eq_rat @ C @ A )
% 0.30/0.56 & ( ord_less_eq_rat @ B @ D )
% 0.30/0.56 & ( ( ord_less_rat @ C @ A )
% 0.30/0.56 | ( ord_less_rat @ B @ D ) ) ) )
% 0.30/0.56 & ( ord_less_eq_rat @ C @ D ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastatMost_psubset_iff
% 0.30/0.56 thf(fact_2905_atLeastatMost__psubset__iff,axiom,
% 0.30/0.56 ! [A: num,B: num,C: num,D: num] :
% 0.30/0.56 ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 0.30/0.56 = ( ( ~ ( ord_less_eq_num @ A @ B )
% 0.30/0.56 | ( ( ord_less_eq_num @ C @ A )
% 0.30/0.56 & ( ord_less_eq_num @ B @ D )
% 0.30/0.56 & ( ( ord_less_num @ C @ A )
% 0.30/0.56 | ( ord_less_num @ B @ D ) ) ) )
% 0.30/0.56 & ( ord_less_eq_num @ C @ D ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastatMost_psubset_iff
% 0.30/0.56 thf(fact_2906_atLeastatMost__psubset__iff,axiom,
% 0.30/0.56 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.56 ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 0.30/0.56 = ( ( ~ ( ord_less_eq_nat @ A @ B )
% 0.30/0.56 | ( ( ord_less_eq_nat @ C @ A )
% 0.30/0.56 & ( ord_less_eq_nat @ B @ D )
% 0.30/0.56 & ( ( ord_less_nat @ C @ A )
% 0.30/0.56 | ( ord_less_nat @ B @ D ) ) ) )
% 0.30/0.56 & ( ord_less_eq_nat @ C @ D ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastatMost_psubset_iff
% 0.30/0.56 thf(fact_2907_atLeastatMost__psubset__iff,axiom,
% 0.30/0.56 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.56 ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 0.30/0.56 = ( ( ~ ( ord_less_eq_int @ A @ B )
% 0.30/0.56 | ( ( ord_less_eq_int @ C @ A )
% 0.30/0.56 & ( ord_less_eq_int @ B @ D )
% 0.30/0.56 & ( ( ord_less_int @ C @ A )
% 0.30/0.56 | ( ord_less_int @ B @ D ) ) ) )
% 0.30/0.56 & ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastatMost_psubset_iff
% 0.30/0.56 thf(fact_2908_atLeastatMost__psubset__iff,axiom,
% 0.30/0.56 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.56 ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 0.30/0.56 = ( ( ~ ( ord_less_eq_real @ A @ B )
% 0.30/0.56 | ( ( ord_less_eq_real @ C @ A )
% 0.30/0.56 & ( ord_less_eq_real @ B @ D )
% 0.30/0.56 & ( ( ord_less_real @ C @ A )
% 0.30/0.56 | ( ord_less_real @ B @ D ) ) ) )
% 0.30/0.56 & ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % atLeastatMost_psubset_iff
% 0.30/0.56 thf(fact_2909_minusinfinity,axiom,
% 0.30/0.56 ! [D: int,P1: int > $o,P: int > $o] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ D )
% 0.30/0.56 => ( ! [X5: int,K3: int] :
% 0.30/0.56 ( ( P1 @ X5 )
% 0.30/0.56 = ( P1 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) )
% 0.30/0.56 => ( ? [Z4: int] :
% 0.30/0.56 ! [X5: int] :
% 0.30/0.56 ( ( ord_less_int @ X5 @ Z4 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P1 @ X5 ) ) )
% 0.30/0.56 => ( ? [X_1: int] : ( P1 @ X_1 )
% 0.30/0.56 => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minusinfinity
% 0.30/0.56 thf(fact_2910_plusinfinity,axiom,
% 0.30/0.56 ! [D: int,P5: int > $o,P: int > $o] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ D )
% 0.30/0.56 => ( ! [X5: int,K3: int] :
% 0.30/0.56 ( ( P5 @ X5 )
% 0.30/0.56 = ( P5 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) )
% 0.30/0.56 => ( ? [Z4: int] :
% 0.30/0.56 ! [X5: int] :
% 0.30/0.56 ( ( ord_less_int @ Z4 @ X5 )
% 0.30/0.56 => ( ( P @ X5 )
% 0.30/0.56 = ( P5 @ X5 ) ) )
% 0.30/0.56 => ( ? [X_1: int] : ( P5 @ X_1 )
% 0.30/0.56 => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % plusinfinity
% 0.30/0.56 thf(fact_2911_Diff__eq__empty__iff,axiom,
% 0.30/0.56 ! [A2: set_real,B3: set_real] :
% 0.30/0.56 ( ( ( minus_minus_set_real @ A2 @ B3 )
% 0.30/0.56 = bot_bot_set_real )
% 0.30/0.56 = ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_eq_empty_iff
% 0.30/0.56 thf(fact_2912_Diff__eq__empty__iff,axiom,
% 0.30/0.56 ! [A2: set_nat,B3: set_nat] :
% 0.30/0.56 ( ( ( minus_minus_set_nat @ A2 @ B3 )
% 0.30/0.56 = bot_bot_set_nat )
% 0.30/0.56 = ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_eq_empty_iff
% 0.30/0.56 thf(fact_2913_Diff__eq__empty__iff,axiom,
% 0.30/0.56 ! [A2: set_int,B3: set_int] :
% 0.30/0.56 ( ( ( minus_minus_set_int @ A2 @ B3 )
% 0.30/0.56 = bot_bot_set_int )
% 0.30/0.56 = ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_eq_empty_iff
% 0.30/0.56 thf(fact_2914_empty__subsetI,axiom,
% 0.30/0.56 ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% 0.30/0.56
% 0.30/0.56 % empty_subsetI
% 0.30/0.56 thf(fact_2915_empty__subsetI,axiom,
% 0.30/0.56 ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% 0.30/0.56
% 0.30/0.56 % empty_subsetI
% 0.30/0.56 thf(fact_2916_empty__subsetI,axiom,
% 0.30/0.56 ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% 0.30/0.56
% 0.30/0.56 % empty_subsetI
% 0.30/0.56 thf(fact_2917_subset__empty,axiom,
% 0.30/0.56 ! [A2: set_nat] :
% 0.30/0.56 ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 0.30/0.56 = ( A2 = bot_bot_set_nat ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_empty
% 0.30/0.56 thf(fact_2918_subset__empty,axiom,
% 0.30/0.56 ! [A2: set_real] :
% 0.30/0.56 ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
% 0.30/0.56 = ( A2 = bot_bot_set_real ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_empty
% 0.30/0.56 thf(fact_2919_subset__empty,axiom,
% 0.30/0.56 ! [A2: set_int] :
% 0.30/0.56 ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 0.30/0.56 = ( A2 = bot_bot_set_int ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_empty
% 0.30/0.56 thf(fact_2920_Bolzano,axiom,
% 0.30/0.56 ! [A: real,B: real,P: real > real > $o] :
% 0.30/0.56 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.56 => ( ! [A4: real,B4: real,C3: real] :
% 0.30/0.56 ( ( P @ A4 @ B4 )
% 0.30/0.56 => ( ( P @ B4 @ C3 )
% 0.30/0.56 => ( ( ord_less_eq_real @ A4 @ B4 )
% 0.30/0.56 => ( ( ord_less_eq_real @ B4 @ C3 )
% 0.30/0.56 => ( P @ A4 @ C3 ) ) ) ) )
% 0.30/0.56 => ( ! [X5: real] :
% 0.30/0.56 ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.56 => ( ( ord_less_eq_real @ X5 @ B )
% 0.30/0.56 => ? [D5: real] :
% 0.30/0.56 ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.30/0.56 & ! [A4: real,B4: real] :
% 0.30/0.56 ( ( ( ord_less_eq_real @ A4 @ X5 )
% 0.30/0.56 & ( ord_less_eq_real @ X5 @ B4 )
% 0.30/0.56 & ( ord_less_real @ ( minus_minus_real @ B4 @ A4 ) @ D5 ) )
% 0.30/0.56 => ( P @ A4 @ B4 ) ) ) ) )
% 0.30/0.56 => ( P @ A @ B ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Bolzano
% 0.30/0.56 thf(fact_2921_divides__aux__eq,axiom,
% 0.30/0.56 ! [Q3: nat,R2: nat] :
% 0.30/0.56 ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
% 0.30/0.56 = ( R2 = zero_zero_nat ) ) ).
% 0.30/0.56
% 0.30/0.56 % divides_aux_eq
% 0.30/0.56 thf(fact_2922_divides__aux__eq,axiom,
% 0.30/0.56 ! [Q3: int,R2: int] :
% 0.30/0.56 ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.30/0.56 = ( R2 = zero_zero_int ) ) ).
% 0.30/0.56
% 0.30/0.56 % divides_aux_eq
% 0.30/0.56 thf(fact_2923_neg__eucl__rel__int__mult__2,axiom,
% 0.30/0.56 ! [B: int,A: int,Q3: int,R2: int] :
% 0.30/0.56 ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.30/0.56 => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.30/0.56 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q3 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % neg_eucl_rel_int_mult_2
% 0.30/0.56 thf(fact_2924_low__def,axiom,
% 0.30/0.56 ( vEBT_VEBT_low
% 0.30/0.56 = ( ^ [X: nat,N3: nat] : ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % low_def
% 0.30/0.56 thf(fact_2925_subsetI,axiom,
% 0.30/0.56 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.30/0.56 ( ! [X5: product_prod_nat_nat] :
% 0.30/0.56 ( ( member8440522571783428010at_nat @ X5 @ A2 )
% 0.30/0.56 => ( member8440522571783428010at_nat @ X5 @ B3 ) )
% 0.30/0.56 => ( ord_le3146513528884898305at_nat @ A2 @ B3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % subsetI
% 0.30/0.56 thf(fact_2926_subsetI,axiom,
% 0.30/0.56 ! [A2: set_complex,B3: set_complex] :
% 0.30/0.56 ( ! [X5: complex] :
% 0.30/0.56 ( ( member_complex @ X5 @ A2 )
% 0.30/0.56 => ( member_complex @ X5 @ B3 ) )
% 0.30/0.56 => ( ord_le211207098394363844omplex @ A2 @ B3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % subsetI
% 0.30/0.56 thf(fact_2927_subsetI,axiom,
% 0.30/0.56 ! [A2: set_real,B3: set_real] :
% 0.30/0.56 ( ! [X5: real] :
% 0.30/0.56 ( ( member_real @ X5 @ A2 )
% 0.30/0.56 => ( member_real @ X5 @ B3 ) )
% 0.30/0.56 => ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % subsetI
% 0.30/0.56 thf(fact_2928_subsetI,axiom,
% 0.30/0.56 ! [A2: set_nat,B3: set_nat] :
% 0.30/0.56 ( ! [X5: nat] :
% 0.30/0.56 ( ( member_nat @ X5 @ A2 )
% 0.30/0.56 => ( member_nat @ X5 @ B3 ) )
% 0.30/0.56 => ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % subsetI
% 0.30/0.56 thf(fact_2929_subsetI,axiom,
% 0.30/0.56 ! [A2: set_int,B3: set_int] :
% 0.30/0.56 ( ! [X5: int] :
% 0.30/0.56 ( ( member_int @ X5 @ A2 )
% 0.30/0.56 => ( member_int @ X5 @ B3 ) )
% 0.30/0.56 => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % subsetI
% 0.30/0.56 thf(fact_2930_subset__antisym,axiom,
% 0.30/0.56 ! [A2: set_int,B3: set_int] :
% 0.30/0.56 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.30/0.56 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.30/0.56 => ( A2 = B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_antisym
% 0.30/0.56 thf(fact_2931_mod__mod__trivial,axiom,
% 0.30/0.56 ! [A: nat,B: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.30/0.56 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mod_trivial
% 0.30/0.56 thf(fact_2932_mod__mod__trivial,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.30/0.56 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mod_trivial
% 0.30/0.56 thf(fact_2933_mod__mod__trivial,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.30/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mod_trivial
% 0.30/0.56 thf(fact_2934_Diff__idemp,axiom,
% 0.30/0.56 ! [A2: set_nat,B3: set_nat] :
% 0.30/0.56 ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ B3 )
% 0.30/0.56 = ( minus_minus_set_nat @ A2 @ B3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_idemp
% 0.30/0.56 thf(fact_2935_Diff__iff,axiom,
% 0.30/0.56 ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.30/0.56 ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B3 ) )
% 0.30/0.56 = ( ( member8440522571783428010at_nat @ C @ A2 )
% 0.30/0.56 & ~ ( member8440522571783428010at_nat @ C @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_iff
% 0.30/0.56 thf(fact_2936_Diff__iff,axiom,
% 0.30/0.56 ! [C: complex,A2: set_complex,B3: set_complex] :
% 0.30/0.56 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.30/0.56 = ( ( member_complex @ C @ A2 )
% 0.30/0.56 & ~ ( member_complex @ C @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_iff
% 0.30/0.56 thf(fact_2937_Diff__iff,axiom,
% 0.30/0.56 ! [C: real,A2: set_real,B3: set_real] :
% 0.30/0.56 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 0.30/0.56 = ( ( member_real @ C @ A2 )
% 0.30/0.56 & ~ ( member_real @ C @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_iff
% 0.30/0.56 thf(fact_2938_Diff__iff,axiom,
% 0.30/0.56 ! [C: int,A2: set_int,B3: set_int] :
% 0.30/0.56 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
% 0.30/0.56 = ( ( member_int @ C @ A2 )
% 0.30/0.56 & ~ ( member_int @ C @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_iff
% 0.30/0.56 thf(fact_2939_Diff__iff,axiom,
% 0.30/0.56 ! [C: nat,A2: set_nat,B3: set_nat] :
% 0.30/0.56 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 0.30/0.56 = ( ( member_nat @ C @ A2 )
% 0.30/0.56 & ~ ( member_nat @ C @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_iff
% 0.30/0.56 thf(fact_2940_DiffI,axiom,
% 0.30/0.56 ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.30/0.56 ( ( member8440522571783428010at_nat @ C @ A2 )
% 0.30/0.56 => ( ~ ( member8440522571783428010at_nat @ C @ B3 )
% 0.30/0.56 => ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffI
% 0.30/0.56 thf(fact_2941_DiffI,axiom,
% 0.30/0.56 ! [C: complex,A2: set_complex,B3: set_complex] :
% 0.30/0.56 ( ( member_complex @ C @ A2 )
% 0.30/0.56 => ( ~ ( member_complex @ C @ B3 )
% 0.30/0.56 => ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffI
% 0.30/0.56 thf(fact_2942_DiffI,axiom,
% 0.30/0.56 ! [C: real,A2: set_real,B3: set_real] :
% 0.30/0.56 ( ( member_real @ C @ A2 )
% 0.30/0.56 => ( ~ ( member_real @ C @ B3 )
% 0.30/0.56 => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffI
% 0.30/0.56 thf(fact_2943_DiffI,axiom,
% 0.30/0.56 ! [C: int,A2: set_int,B3: set_int] :
% 0.30/0.56 ( ( member_int @ C @ A2 )
% 0.30/0.56 => ( ~ ( member_int @ C @ B3 )
% 0.30/0.56 => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffI
% 0.30/0.56 thf(fact_2944_DiffI,axiom,
% 0.30/0.56 ! [C: nat,A2: set_nat,B3: set_nat] :
% 0.30/0.56 ( ( member_nat @ C @ A2 )
% 0.30/0.56 => ( ~ ( member_nat @ C @ B3 )
% 0.30/0.56 => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffI
% 0.30/0.56 thf(fact_2945_mod__0,axiom,
% 0.30/0.56 ! [A: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 0.30/0.56 = zero_zero_nat ) ).
% 0.30/0.56
% 0.30/0.56 % mod_0
% 0.30/0.56 thf(fact_2946_mod__0,axiom,
% 0.30/0.56 ! [A: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 0.30/0.56 = zero_zero_int ) ).
% 0.30/0.56
% 0.30/0.56 % mod_0
% 0.30/0.56 thf(fact_2947_mod__0,axiom,
% 0.30/0.56 ! [A: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.56 = zero_z3403309356797280102nteger ) ).
% 0.30/0.56
% 0.30/0.56 % mod_0
% 0.30/0.56 thf(fact_2948_mod__by__0,axiom,
% 0.30/0.56 ! [A: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ A @ zero_zero_nat )
% 0.30/0.56 = A ) ).
% 0.30/0.56
% 0.30/0.56 % mod_by_0
% 0.30/0.56 thf(fact_2949_mod__by__0,axiom,
% 0.30/0.56 ! [A: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ A @ zero_zero_int )
% 0.30/0.56 = A ) ).
% 0.30/0.56
% 0.30/0.56 % mod_by_0
% 0.30/0.56 thf(fact_2950_mod__by__0,axiom,
% 0.30/0.56 ! [A: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
% 0.30/0.56 = A ) ).
% 0.30/0.56
% 0.30/0.56 % mod_by_0
% 0.30/0.56 thf(fact_2951_mod__self,axiom,
% 0.30/0.56 ! [A: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ A @ A )
% 0.30/0.56 = zero_zero_nat ) ).
% 0.30/0.56
% 0.30/0.56 % mod_self
% 0.30/0.56 thf(fact_2952_mod__self,axiom,
% 0.30/0.56 ! [A: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ A @ A )
% 0.30/0.56 = zero_zero_int ) ).
% 0.30/0.56
% 0.30/0.56 % mod_self
% 0.30/0.56 thf(fact_2953_mod__self,axiom,
% 0.30/0.56 ! [A: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ A @ A )
% 0.30/0.56 = zero_z3403309356797280102nteger ) ).
% 0.30/0.56
% 0.30/0.56 % mod_self
% 0.30/0.56 thf(fact_2954_bits__mod__0,axiom,
% 0.30/0.56 ! [A: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 0.30/0.56 = zero_zero_nat ) ).
% 0.30/0.56
% 0.30/0.56 % bits_mod_0
% 0.30/0.56 thf(fact_2955_bits__mod__0,axiom,
% 0.30/0.56 ! [A: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 0.30/0.56 = zero_zero_int ) ).
% 0.30/0.56
% 0.30/0.56 % bits_mod_0
% 0.30/0.56 thf(fact_2956_bits__mod__0,axiom,
% 0.30/0.56 ! [A: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.56 = zero_z3403309356797280102nteger ) ).
% 0.30/0.56
% 0.30/0.56 % bits_mod_0
% 0.30/0.56 thf(fact_2957_mod__add__self1,axiom,
% 0.30/0.56 ! [B: nat,A: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.30/0.56 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_add_self1
% 0.30/0.56 thf(fact_2958_mod__add__self1,axiom,
% 0.30/0.56 ! [B: int,A: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.30/0.56 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_add_self1
% 0.30/0.56 thf(fact_2959_mod__add__self1,axiom,
% 0.30/0.56 ! [B: code_integer,A: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
% 0.30/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_add_self1
% 0.30/0.56 thf(fact_2960_mod__add__self2,axiom,
% 0.30/0.56 ! [A: nat,B: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.30/0.56 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_add_self2
% 0.30/0.56 thf(fact_2961_mod__add__self2,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.30/0.56 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_add_self2
% 0.30/0.56 thf(fact_2962_mod__add__self2,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
% 0.30/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_add_self2
% 0.30/0.56 thf(fact_2963_minus__mod__self2,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.30/0.56 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_mod_self2
% 0.30/0.56 thf(fact_2964_minus__mod__self2,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
% 0.30/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_mod_self2
% 0.30/0.56 thf(fact_2965_Diff__empty,axiom,
% 0.30/0.56 ! [A2: set_int] :
% 0.30/0.56 ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
% 0.30/0.56 = A2 ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_empty
% 0.30/0.56 thf(fact_2966_Diff__empty,axiom,
% 0.30/0.56 ! [A2: set_real] :
% 0.30/0.56 ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
% 0.30/0.56 = A2 ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_empty
% 0.30/0.56 thf(fact_2967_Diff__empty,axiom,
% 0.30/0.56 ! [A2: set_nat] :
% 0.30/0.56 ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
% 0.30/0.56 = A2 ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_empty
% 0.30/0.56 thf(fact_2968_empty__Diff,axiom,
% 0.30/0.56 ! [A2: set_int] :
% 0.30/0.56 ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
% 0.30/0.56 = bot_bot_set_int ) ).
% 0.30/0.56
% 0.30/0.56 % empty_Diff
% 0.30/0.56 thf(fact_2969_empty__Diff,axiom,
% 0.30/0.56 ! [A2: set_real] :
% 0.30/0.56 ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
% 0.30/0.56 = bot_bot_set_real ) ).
% 0.30/0.56
% 0.30/0.56 % empty_Diff
% 0.30/0.56 thf(fact_2970_empty__Diff,axiom,
% 0.30/0.56 ! [A2: set_nat] :
% 0.30/0.56 ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
% 0.30/0.56 = bot_bot_set_nat ) ).
% 0.30/0.56
% 0.30/0.56 % empty_Diff
% 0.30/0.56 thf(fact_2971_Diff__cancel,axiom,
% 0.30/0.56 ! [A2: set_int] :
% 0.30/0.56 ( ( minus_minus_set_int @ A2 @ A2 )
% 0.30/0.56 = bot_bot_set_int ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_cancel
% 0.30/0.56 thf(fact_2972_Diff__cancel,axiom,
% 0.30/0.56 ! [A2: set_real] :
% 0.30/0.56 ( ( minus_minus_set_real @ A2 @ A2 )
% 0.30/0.56 = bot_bot_set_real ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_cancel
% 0.30/0.56 thf(fact_2973_Diff__cancel,axiom,
% 0.30/0.56 ! [A2: set_nat] :
% 0.30/0.56 ( ( minus_minus_set_nat @ A2 @ A2 )
% 0.30/0.56 = bot_bot_set_nat ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_cancel
% 0.30/0.56 thf(fact_2974_psubsetI,axiom,
% 0.30/0.56 ! [A2: set_int,B3: set_int] :
% 0.30/0.56 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.30/0.56 => ( ( A2 != B3 )
% 0.30/0.56 => ( ord_less_set_int @ A2 @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % psubsetI
% 0.30/0.56 thf(fact_2975_mod__less,axiom,
% 0.30/0.56 ! [M: nat,N: nat] :
% 0.30/0.56 ( ( ord_less_nat @ M @ N )
% 0.30/0.56 => ( ( modulo_modulo_nat @ M @ N )
% 0.30/0.56 = M ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_less
% 0.30/0.56 thf(fact_2976_mod__mult__self1__is__0,axiom,
% 0.30/0.56 ! [B: nat,A: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
% 0.30/0.56 = zero_zero_nat ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_self1_is_0
% 0.30/0.56 thf(fact_2977_mod__mult__self1__is__0,axiom,
% 0.30/0.56 ! [B: int,A: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
% 0.30/0.56 = zero_zero_int ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_self1_is_0
% 0.30/0.56 thf(fact_2978_mod__mult__self1__is__0,axiom,
% 0.30/0.56 ! [B: code_integer,A: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
% 0.30/0.56 = zero_z3403309356797280102nteger ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_self1_is_0
% 0.30/0.56 thf(fact_2979_mod__mult__self2__is__0,axiom,
% 0.30/0.56 ! [A: nat,B: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
% 0.30/0.56 = zero_zero_nat ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_self2_is_0
% 0.30/0.56 thf(fact_2980_mod__mult__self2__is__0,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
% 0.30/0.56 = zero_zero_int ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_self2_is_0
% 0.30/0.56 thf(fact_2981_mod__mult__self2__is__0,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 0.30/0.56 = zero_z3403309356797280102nteger ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_self2_is_0
% 0.30/0.56 thf(fact_2982_bits__mod__by__1,axiom,
% 0.30/0.56 ! [A: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.30/0.56 = zero_zero_nat ) ).
% 0.30/0.56
% 0.30/0.56 % bits_mod_by_1
% 0.30/0.56 thf(fact_2983_bits__mod__by__1,axiom,
% 0.30/0.56 ! [A: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ A @ one_one_int )
% 0.30/0.56 = zero_zero_int ) ).
% 0.30/0.56
% 0.30/0.56 % bits_mod_by_1
% 0.30/0.56 thf(fact_2984_bits__mod__by__1,axiom,
% 0.30/0.56 ! [A: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.30/0.56 = zero_z3403309356797280102nteger ) ).
% 0.30/0.56
% 0.30/0.56 % bits_mod_by_1
% 0.30/0.56 thf(fact_2985_mod__by__1,axiom,
% 0.30/0.56 ! [A: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.30/0.56 = zero_zero_nat ) ).
% 0.30/0.56
% 0.30/0.56 % mod_by_1
% 0.30/0.56 thf(fact_2986_mod__by__1,axiom,
% 0.30/0.56 ! [A: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ A @ one_one_int )
% 0.30/0.56 = zero_zero_int ) ).
% 0.30/0.56
% 0.30/0.56 % mod_by_1
% 0.30/0.56 thf(fact_2987_mod__by__1,axiom,
% 0.30/0.56 ! [A: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.30/0.56 = zero_z3403309356797280102nteger ) ).
% 0.30/0.56
% 0.30/0.56 % mod_by_1
% 0.30/0.56 thf(fact_2988_bits__mod__div__trivial,axiom,
% 0.30/0.56 ! [A: nat,B: nat] :
% 0.30/0.56 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.30/0.56 = zero_zero_nat ) ).
% 0.30/0.56
% 0.30/0.56 % bits_mod_div_trivial
% 0.30/0.56 thf(fact_2989_bits__mod__div__trivial,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.30/0.56 = zero_zero_int ) ).
% 0.30/0.56
% 0.30/0.56 % bits_mod_div_trivial
% 0.30/0.56 thf(fact_2990_bits__mod__div__trivial,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.30/0.56 = zero_z3403309356797280102nteger ) ).
% 0.30/0.56
% 0.30/0.56 % bits_mod_div_trivial
% 0.30/0.56 thf(fact_2991_mod__div__trivial,axiom,
% 0.30/0.56 ! [A: nat,B: nat] :
% 0.30/0.56 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.30/0.56 = zero_zero_nat ) ).
% 0.30/0.56
% 0.30/0.56 % mod_div_trivial
% 0.30/0.56 thf(fact_2992_mod__div__trivial,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.30/0.56 = zero_zero_int ) ).
% 0.30/0.56
% 0.30/0.56 % mod_div_trivial
% 0.30/0.56 thf(fact_2993_mod__div__trivial,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.30/0.56 = zero_z3403309356797280102nteger ) ).
% 0.30/0.56
% 0.30/0.56 % mod_div_trivial
% 0.30/0.56 thf(fact_2994_mod__mult__self1,axiom,
% 0.30/0.56 ! [A: nat,C: nat,B: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 0.30/0.56 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_self1
% 0.30/0.56 thf(fact_2995_mod__mult__self1,axiom,
% 0.30/0.56 ! [A: int,C: int,B: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 0.30/0.56 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_self1
% 0.30/0.56 thf(fact_2996_mod__mult__self1,axiom,
% 0.30/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
% 0.30/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_self1
% 0.30/0.56 thf(fact_2997_mod__mult__self2,axiom,
% 0.30/0.56 ! [A: nat,B: nat,C: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 0.30/0.56 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_self2
% 0.30/0.56 thf(fact_2998_mod__mult__self2,axiom,
% 0.30/0.56 ! [A: int,B: int,C: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 0.30/0.56 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_self2
% 0.30/0.56 thf(fact_2999_mod__mult__self2,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
% 0.30/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_self2
% 0.30/0.56 thf(fact_3000_mod__mult__self3,axiom,
% 0.30/0.56 ! [C: nat,B: nat,A: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 0.30/0.56 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_self3
% 0.30/0.56 thf(fact_3001_mod__mult__self3,axiom,
% 0.30/0.56 ! [C: int,B: int,A: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 0.30/0.56 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_self3
% 0.30/0.56 thf(fact_3002_mod__mult__self3,axiom,
% 0.30/0.56 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
% 0.30/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_self3
% 0.30/0.56 thf(fact_3003_mod__mult__self4,axiom,
% 0.30/0.56 ! [B: nat,C: nat,A: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 0.30/0.56 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_self4
% 0.30/0.56 thf(fact_3004_mod__mult__self4,axiom,
% 0.30/0.56 ! [B: int,C: int,A: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 0.30/0.56 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_self4
% 0.30/0.56 thf(fact_3005_mod__mult__self4,axiom,
% 0.30/0.56 ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
% 0.30/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_self4
% 0.30/0.56 thf(fact_3006_mod__by__Suc__0,axiom,
% 0.30/0.56 ! [M: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.30/0.56 = zero_zero_nat ) ).
% 0.30/0.56
% 0.30/0.56 % mod_by_Suc_0
% 0.30/0.56 thf(fact_3007_Suc__mod__mult__self1,axiom,
% 0.30/0.56 ! [M: nat,K: nat,N: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
% 0.30/0.56 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.56
% 0.30/0.56 % Suc_mod_mult_self1
% 0.30/0.56 thf(fact_3008_Suc__mod__mult__self2,axiom,
% 0.30/0.56 ! [M: nat,N: nat,K: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
% 0.30/0.56 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.56
% 0.30/0.56 % Suc_mod_mult_self2
% 0.30/0.56 thf(fact_3009_Suc__mod__mult__self3,axiom,
% 0.30/0.56 ! [K: nat,N: nat,M: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
% 0.30/0.56 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.56
% 0.30/0.56 % Suc_mod_mult_self3
% 0.30/0.56 thf(fact_3010_Suc__mod__mult__self4,axiom,
% 0.30/0.56 ! [N: nat,K: nat,M: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
% 0.30/0.56 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.56
% 0.30/0.56 % Suc_mod_mult_self4
% 0.30/0.56 thf(fact_3011_bits__one__mod__two__eq__one,axiom,
% 0.30/0.56 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56 = one_one_nat ) ).
% 0.30/0.56
% 0.30/0.56 % bits_one_mod_two_eq_one
% 0.30/0.56 thf(fact_3012_bits__one__mod__two__eq__one,axiom,
% 0.30/0.56 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.56 = one_one_int ) ).
% 0.30/0.56
% 0.30/0.56 % bits_one_mod_two_eq_one
% 0.30/0.56 thf(fact_3013_bits__one__mod__two__eq__one,axiom,
% 0.30/0.56 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.56 = one_one_Code_integer ) ).
% 0.30/0.56
% 0.30/0.56 % bits_one_mod_two_eq_one
% 0.30/0.56 thf(fact_3014_one__mod__two__eq__one,axiom,
% 0.30/0.56 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56 = one_one_nat ) ).
% 0.30/0.56
% 0.30/0.56 % one_mod_two_eq_one
% 0.30/0.56 thf(fact_3015_one__mod__two__eq__one,axiom,
% 0.30/0.56 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.56 = one_one_int ) ).
% 0.30/0.56
% 0.30/0.56 % one_mod_two_eq_one
% 0.30/0.56 thf(fact_3016_one__mod__two__eq__one,axiom,
% 0.30/0.56 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.56 = one_one_Code_integer ) ).
% 0.30/0.56
% 0.30/0.56 % one_mod_two_eq_one
% 0.30/0.56 thf(fact_3017_mod2__Suc__Suc,axiom,
% 0.30/0.56 ! [M: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56 = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod2_Suc_Suc
% 0.30/0.56 thf(fact_3018_Suc__times__numeral__mod__eq,axiom,
% 0.30/0.56 ! [K: num,N: nat] :
% 0.30/0.56 ( ( ( numeral_numeral_nat @ K )
% 0.30/0.56 != one_one_nat )
% 0.30/0.56 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.56 = one_one_nat ) ) ).
% 0.30/0.56
% 0.30/0.56 % Suc_times_numeral_mod_eq
% 0.30/0.56 thf(fact_3019_not__mod__2__eq__1__eq__0,axiom,
% 0.30/0.56 ! [A: nat] :
% 0.30/0.56 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56 != one_one_nat )
% 0.30/0.56 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56 = zero_zero_nat ) ) ).
% 0.30/0.56
% 0.30/0.56 % not_mod_2_eq_1_eq_0
% 0.30/0.56 thf(fact_3020_not__mod__2__eq__1__eq__0,axiom,
% 0.30/0.56 ! [A: int] :
% 0.30/0.56 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.56 != one_one_int )
% 0.30/0.56 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.56 = zero_zero_int ) ) ).
% 0.30/0.56
% 0.30/0.56 % not_mod_2_eq_1_eq_0
% 0.30/0.56 thf(fact_3021_not__mod__2__eq__1__eq__0,axiom,
% 0.30/0.56 ! [A: code_integer] :
% 0.30/0.56 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.56 != one_one_Code_integer )
% 0.30/0.56 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.56 = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.56
% 0.30/0.56 % not_mod_2_eq_1_eq_0
% 0.30/0.56 thf(fact_3022_not__mod__2__eq__0__eq__1,axiom,
% 0.30/0.56 ! [A: nat] :
% 0.30/0.56 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56 != zero_zero_nat )
% 0.30/0.56 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56 = one_one_nat ) ) ).
% 0.30/0.56
% 0.30/0.56 % not_mod_2_eq_0_eq_1
% 0.30/0.56 thf(fact_3023_not__mod__2__eq__0__eq__1,axiom,
% 0.30/0.56 ! [A: int] :
% 0.30/0.56 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.56 != zero_zero_int )
% 0.30/0.56 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.56 = one_one_int ) ) ).
% 0.30/0.56
% 0.30/0.56 % not_mod_2_eq_0_eq_1
% 0.30/0.56 thf(fact_3024_not__mod__2__eq__0__eq__1,axiom,
% 0.30/0.56 ! [A: code_integer] :
% 0.30/0.56 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.56 != zero_z3403309356797280102nteger )
% 0.30/0.56 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.56 = one_one_Code_integer ) ) ).
% 0.30/0.56
% 0.30/0.56 % not_mod_2_eq_0_eq_1
% 0.30/0.56 thf(fact_3025_not__mod2__eq__Suc__0__eq__0,axiom,
% 0.30/0.56 ! [N: nat] :
% 0.30/0.56 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56 != ( suc @ zero_zero_nat ) )
% 0.30/0.56 = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56 = zero_zero_nat ) ) ).
% 0.30/0.56
% 0.30/0.56 % not_mod2_eq_Suc_0_eq_0
% 0.30/0.56 thf(fact_3026_add__self__mod__2,axiom,
% 0.30/0.56 ! [M: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56 = zero_zero_nat ) ).
% 0.30/0.56
% 0.30/0.56 % add_self_mod_2
% 0.30/0.56 thf(fact_3027_mod2__gr__0,axiom,
% 0.30/0.56 ! [M: nat] :
% 0.30/0.56 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.56 = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56 = one_one_nat ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod2_gr_0
% 0.30/0.56 thf(fact_3028_minus__set__def,axiom,
% 0.30/0.56 ( minus_1356011639430497352at_nat
% 0.30/0.56 = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 0.30/0.56 ( collec3392354462482085612at_nat
% 0.30/0.56 @ ( minus_2270307095948843157_nat_o
% 0.30/0.56 @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A6 )
% 0.30/0.56 @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_set_def
% 0.30/0.56 thf(fact_3029_minus__set__def,axiom,
% 0.30/0.56 ( minus_811609699411566653omplex
% 0.30/0.56 = ( ^ [A6: set_complex,B6: set_complex] :
% 0.30/0.56 ( collect_complex
% 0.30/0.56 @ ( minus_8727706125548526216plex_o
% 0.30/0.56 @ ^ [X: complex] : ( member_complex @ X @ A6 )
% 0.30/0.56 @ ^ [X: complex] : ( member_complex @ X @ B6 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_set_def
% 0.30/0.56 thf(fact_3030_minus__set__def,axiom,
% 0.30/0.56 ( minus_minus_set_real
% 0.30/0.56 = ( ^ [A6: set_real,B6: set_real] :
% 0.30/0.56 ( collect_real
% 0.30/0.56 @ ( minus_minus_real_o
% 0.30/0.56 @ ^ [X: real] : ( member_real @ X @ A6 )
% 0.30/0.56 @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_set_def
% 0.30/0.56 thf(fact_3031_minus__set__def,axiom,
% 0.30/0.56 ( minus_7954133019191499631st_nat
% 0.30/0.56 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 0.30/0.56 ( collect_list_nat
% 0.30/0.56 @ ( minus_1139252259498527702_nat_o
% 0.30/0.56 @ ^ [X: list_nat] : ( member_list_nat @ X @ A6 )
% 0.30/0.56 @ ^ [X: list_nat] : ( member_list_nat @ X @ B6 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_set_def
% 0.30/0.56 thf(fact_3032_minus__set__def,axiom,
% 0.30/0.56 ( minus_minus_set_int
% 0.30/0.56 = ( ^ [A6: set_int,B6: set_int] :
% 0.30/0.56 ( collect_int
% 0.30/0.56 @ ( minus_minus_int_o
% 0.30/0.56 @ ^ [X: int] : ( member_int @ X @ A6 )
% 0.30/0.56 @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_set_def
% 0.30/0.56 thf(fact_3033_minus__set__def,axiom,
% 0.30/0.56 ( minus_minus_set_nat
% 0.30/0.56 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.30/0.56 ( collect_nat
% 0.30/0.56 @ ( minus_minus_nat_o
% 0.30/0.56 @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 0.30/0.56 @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_set_def
% 0.30/0.56 thf(fact_3034_set__diff__eq,axiom,
% 0.30/0.56 ( minus_1356011639430497352at_nat
% 0.30/0.56 = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 0.30/0.56 ( collec3392354462482085612at_nat
% 0.30/0.56 @ ^ [X: product_prod_nat_nat] :
% 0.30/0.56 ( ( member8440522571783428010at_nat @ X @ A6 )
% 0.30/0.56 & ~ ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_diff_eq
% 0.30/0.56 thf(fact_3035_set__diff__eq,axiom,
% 0.30/0.56 ( minus_811609699411566653omplex
% 0.30/0.56 = ( ^ [A6: set_complex,B6: set_complex] :
% 0.30/0.56 ( collect_complex
% 0.30/0.56 @ ^ [X: complex] :
% 0.30/0.56 ( ( member_complex @ X @ A6 )
% 0.30/0.56 & ~ ( member_complex @ X @ B6 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_diff_eq
% 0.30/0.56 thf(fact_3036_set__diff__eq,axiom,
% 0.30/0.56 ( minus_minus_set_real
% 0.30/0.56 = ( ^ [A6: set_real,B6: set_real] :
% 0.30/0.56 ( collect_real
% 0.30/0.56 @ ^ [X: real] :
% 0.30/0.56 ( ( member_real @ X @ A6 )
% 0.30/0.56 & ~ ( member_real @ X @ B6 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_diff_eq
% 0.30/0.56 thf(fact_3037_set__diff__eq,axiom,
% 0.30/0.56 ( minus_7954133019191499631st_nat
% 0.30/0.56 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 0.30/0.56 ( collect_list_nat
% 0.30/0.56 @ ^ [X: list_nat] :
% 0.30/0.56 ( ( member_list_nat @ X @ A6 )
% 0.30/0.56 & ~ ( member_list_nat @ X @ B6 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_diff_eq
% 0.30/0.56 thf(fact_3038_set__diff__eq,axiom,
% 0.30/0.56 ( minus_minus_set_int
% 0.30/0.56 = ( ^ [A6: set_int,B6: set_int] :
% 0.30/0.56 ( collect_int
% 0.30/0.56 @ ^ [X: int] :
% 0.30/0.56 ( ( member_int @ X @ A6 )
% 0.30/0.56 & ~ ( member_int @ X @ B6 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_diff_eq
% 0.30/0.56 thf(fact_3039_set__diff__eq,axiom,
% 0.30/0.56 ( minus_minus_set_nat
% 0.30/0.56 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.30/0.56 ( collect_nat
% 0.30/0.56 @ ^ [X: nat] :
% 0.30/0.56 ( ( member_nat @ X @ A6 )
% 0.30/0.56 & ~ ( member_nat @ X @ B6 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_diff_eq
% 0.30/0.56 thf(fact_3040_DiffD2,axiom,
% 0.30/0.56 ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.30/0.56 ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B3 ) )
% 0.30/0.56 => ~ ( member8440522571783428010at_nat @ C @ B3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffD2
% 0.30/0.56 thf(fact_3041_DiffD2,axiom,
% 0.30/0.56 ! [C: complex,A2: set_complex,B3: set_complex] :
% 0.30/0.56 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.30/0.56 => ~ ( member_complex @ C @ B3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffD2
% 0.30/0.56 thf(fact_3042_DiffD2,axiom,
% 0.30/0.56 ! [C: real,A2: set_real,B3: set_real] :
% 0.30/0.56 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 0.30/0.56 => ~ ( member_real @ C @ B3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffD2
% 0.30/0.56 thf(fact_3043_DiffD2,axiom,
% 0.30/0.56 ! [C: int,A2: set_int,B3: set_int] :
% 0.30/0.56 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
% 0.30/0.56 => ~ ( member_int @ C @ B3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffD2
% 0.30/0.56 thf(fact_3044_DiffD2,axiom,
% 0.30/0.56 ! [C: nat,A2: set_nat,B3: set_nat] :
% 0.30/0.56 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 0.30/0.56 => ~ ( member_nat @ C @ B3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffD2
% 0.30/0.56 thf(fact_3045_DiffD1,axiom,
% 0.30/0.56 ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.30/0.56 ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B3 ) )
% 0.30/0.56 => ( member8440522571783428010at_nat @ C @ A2 ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffD1
% 0.30/0.56 thf(fact_3046_DiffD1,axiom,
% 0.30/0.56 ! [C: complex,A2: set_complex,B3: set_complex] :
% 0.30/0.56 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.30/0.56 => ( member_complex @ C @ A2 ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffD1
% 0.30/0.56 thf(fact_3047_DiffD1,axiom,
% 0.30/0.56 ! [C: real,A2: set_real,B3: set_real] :
% 0.30/0.56 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 0.30/0.56 => ( member_real @ C @ A2 ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffD1
% 0.30/0.56 thf(fact_3048_DiffD1,axiom,
% 0.30/0.56 ! [C: int,A2: set_int,B3: set_int] :
% 0.30/0.56 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
% 0.30/0.56 => ( member_int @ C @ A2 ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffD1
% 0.30/0.56 thf(fact_3049_DiffD1,axiom,
% 0.30/0.56 ! [C: nat,A2: set_nat,B3: set_nat] :
% 0.30/0.56 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 0.30/0.56 => ( member_nat @ C @ A2 ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffD1
% 0.30/0.56 thf(fact_3050_DiffE,axiom,
% 0.30/0.56 ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.30/0.56 ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B3 ) )
% 0.30/0.56 => ~ ( ( member8440522571783428010at_nat @ C @ A2 )
% 0.30/0.56 => ( member8440522571783428010at_nat @ C @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffE
% 0.30/0.56 thf(fact_3051_DiffE,axiom,
% 0.30/0.56 ! [C: complex,A2: set_complex,B3: set_complex] :
% 0.30/0.56 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.30/0.56 => ~ ( ( member_complex @ C @ A2 )
% 0.30/0.56 => ( member_complex @ C @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffE
% 0.30/0.56 thf(fact_3052_DiffE,axiom,
% 0.30/0.56 ! [C: real,A2: set_real,B3: set_real] :
% 0.30/0.56 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 0.30/0.56 => ~ ( ( member_real @ C @ A2 )
% 0.30/0.56 => ( member_real @ C @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffE
% 0.30/0.56 thf(fact_3053_DiffE,axiom,
% 0.30/0.56 ! [C: int,A2: set_int,B3: set_int] :
% 0.30/0.56 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
% 0.30/0.56 => ~ ( ( member_int @ C @ A2 )
% 0.30/0.56 => ( member_int @ C @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffE
% 0.30/0.56 thf(fact_3054_DiffE,axiom,
% 0.30/0.56 ! [C: nat,A2: set_nat,B3: set_nat] :
% 0.30/0.56 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 0.30/0.56 => ~ ( ( member_nat @ C @ A2 )
% 0.30/0.56 => ( member_nat @ C @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % DiffE
% 0.30/0.56 thf(fact_3055_psubset__imp__ex__mem,axiom,
% 0.30/0.56 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.30/0.56 ( ( ord_le7866589430770878221at_nat @ A2 @ B3 )
% 0.30/0.56 => ? [B4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B4 @ ( minus_1356011639430497352at_nat @ B3 @ A2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % psubset_imp_ex_mem
% 0.30/0.56 thf(fact_3056_psubset__imp__ex__mem,axiom,
% 0.30/0.56 ! [A2: set_complex,B3: set_complex] :
% 0.30/0.56 ( ( ord_less_set_complex @ A2 @ B3 )
% 0.30/0.56 => ? [B4: complex] : ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B3 @ A2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % psubset_imp_ex_mem
% 0.30/0.56 thf(fact_3057_psubset__imp__ex__mem,axiom,
% 0.30/0.56 ! [A2: set_real,B3: set_real] :
% 0.30/0.56 ( ( ord_less_set_real @ A2 @ B3 )
% 0.30/0.56 => ? [B4: real] : ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % psubset_imp_ex_mem
% 0.30/0.56 thf(fact_3058_psubset__imp__ex__mem,axiom,
% 0.30/0.56 ! [A2: set_int,B3: set_int] :
% 0.30/0.56 ( ( ord_less_set_int @ A2 @ B3 )
% 0.30/0.56 => ? [B4: int] : ( member_int @ B4 @ ( minus_minus_set_int @ B3 @ A2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % psubset_imp_ex_mem
% 0.30/0.56 thf(fact_3059_psubset__imp__ex__mem,axiom,
% 0.30/0.56 ! [A2: set_nat,B3: set_nat] :
% 0.30/0.56 ( ( ord_less_set_nat @ A2 @ B3 )
% 0.30/0.56 => ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B3 @ A2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % psubset_imp_ex_mem
% 0.30/0.56 thf(fact_3060_mod__mult__eq,axiom,
% 0.30/0.56 ! [A: nat,C: nat,B: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.30/0.56 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_eq
% 0.30/0.56 thf(fact_3061_mod__mult__eq,axiom,
% 0.30/0.56 ! [A: int,C: int,B: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.30/0.56 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_eq
% 0.30/0.56 thf(fact_3062_mod__mult__eq,axiom,
% 0.30/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.30/0.56 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_eq
% 0.30/0.56 thf(fact_3063_mod__mult__cong,axiom,
% 0.30/0.56 ! [A: nat,C: nat,A5: nat,B: nat,B5: nat] :
% 0.30/0.56 ( ( ( modulo_modulo_nat @ A @ C )
% 0.30/0.56 = ( modulo_modulo_nat @ A5 @ C ) )
% 0.30/0.56 => ( ( ( modulo_modulo_nat @ B @ C )
% 0.30/0.56 = ( modulo_modulo_nat @ B5 @ C ) )
% 0.30/0.56 => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.30/0.56 = ( modulo_modulo_nat @ ( times_times_nat @ A5 @ B5 ) @ C ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_cong
% 0.30/0.56 thf(fact_3064_mod__mult__cong,axiom,
% 0.30/0.56 ! [A: int,C: int,A5: int,B: int,B5: int] :
% 0.30/0.56 ( ( ( modulo_modulo_int @ A @ C )
% 0.30/0.56 = ( modulo_modulo_int @ A5 @ C ) )
% 0.30/0.56 => ( ( ( modulo_modulo_int @ B @ C )
% 0.30/0.56 = ( modulo_modulo_int @ B5 @ C ) )
% 0.30/0.56 => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
% 0.30/0.56 = ( modulo_modulo_int @ ( times_times_int @ A5 @ B5 ) @ C ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_cong
% 0.30/0.56 thf(fact_3065_mod__mult__cong,axiom,
% 0.30/0.56 ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B5: code_integer] :
% 0.30/0.56 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.30/0.56 = ( modulo364778990260209775nteger @ A5 @ C ) )
% 0.30/0.56 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 0.30/0.56 = ( modulo364778990260209775nteger @ B5 @ C ) )
% 0.30/0.56 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.30/0.56 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A5 @ B5 ) @ C ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_cong
% 0.30/0.56 thf(fact_3066_mod__mult__mult2,axiom,
% 0.30/0.56 ! [A: nat,C: nat,B: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.30/0.56 = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_mult2
% 0.30/0.56 thf(fact_3067_mod__mult__mult2,axiom,
% 0.30/0.56 ! [A: int,C: int,B: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.30/0.56 = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_mult2
% 0.30/0.56 thf(fact_3068_mod__mult__mult2,axiom,
% 0.30/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.30/0.56 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_mult2
% 0.30/0.56 thf(fact_3069_mult__mod__right,axiom,
% 0.30/0.56 ! [C: nat,A: nat,B: nat] :
% 0.30/0.56 ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 0.30/0.56 = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mult_mod_right
% 0.30/0.56 thf(fact_3070_mult__mod__right,axiom,
% 0.30/0.56 ! [C: int,A: int,B: int] :
% 0.30/0.56 ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.56 = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mult_mod_right
% 0.30/0.56 thf(fact_3071_mult__mod__right,axiom,
% 0.30/0.56 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.30/0.56 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mult_mod_right
% 0.30/0.56 thf(fact_3072_mod__mult__left__eq,axiom,
% 0.30/0.56 ! [A: nat,C: nat,B: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 0.30/0.56 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_left_eq
% 0.30/0.56 thf(fact_3073_mod__mult__left__eq,axiom,
% 0.30/0.56 ! [A: int,C: int,B: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 0.30/0.56 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_left_eq
% 0.30/0.56 thf(fact_3074_mod__mult__left__eq,axiom,
% 0.30/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 0.30/0.56 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_left_eq
% 0.30/0.56 thf(fact_3075_mod__mult__right__eq,axiom,
% 0.30/0.56 ! [A: nat,B: nat,C: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.30/0.56 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_right_eq
% 0.30/0.56 thf(fact_3076_mod__mult__right__eq,axiom,
% 0.30/0.56 ! [A: int,B: int,C: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.30/0.56 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_right_eq
% 0.30/0.56 thf(fact_3077_mod__mult__right__eq,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.30/0.56 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_right_eq
% 0.30/0.56 thf(fact_3078_mod__add__eq,axiom,
% 0.30/0.56 ! [A: nat,C: nat,B: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.30/0.56 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_add_eq
% 0.30/0.56 thf(fact_3079_mod__add__eq,axiom,
% 0.30/0.56 ! [A: int,C: int,B: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.30/0.56 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_add_eq
% 0.30/0.56 thf(fact_3080_mod__add__eq,axiom,
% 0.30/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.30/0.56 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_add_eq
% 0.30/0.56 thf(fact_3081_mod__add__cong,axiom,
% 0.30/0.56 ! [A: nat,C: nat,A5: nat,B: nat,B5: nat] :
% 0.30/0.56 ( ( ( modulo_modulo_nat @ A @ C )
% 0.30/0.56 = ( modulo_modulo_nat @ A5 @ C ) )
% 0.30/0.56 => ( ( ( modulo_modulo_nat @ B @ C )
% 0.30/0.56 = ( modulo_modulo_nat @ B5 @ C ) )
% 0.30/0.56 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.56 = ( modulo_modulo_nat @ ( plus_plus_nat @ A5 @ B5 ) @ C ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_add_cong
% 0.30/0.56 thf(fact_3082_mod__add__cong,axiom,
% 0.30/0.56 ! [A: int,C: int,A5: int,B: int,B5: int] :
% 0.30/0.56 ( ( ( modulo_modulo_int @ A @ C )
% 0.30/0.56 = ( modulo_modulo_int @ A5 @ C ) )
% 0.30/0.56 => ( ( ( modulo_modulo_int @ B @ C )
% 0.30/0.56 = ( modulo_modulo_int @ B5 @ C ) )
% 0.30/0.56 => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.56 = ( modulo_modulo_int @ ( plus_plus_int @ A5 @ B5 ) @ C ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_add_cong
% 0.30/0.56 thf(fact_3083_mod__add__cong,axiom,
% 0.30/0.56 ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B5: code_integer] :
% 0.30/0.56 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.30/0.56 = ( modulo364778990260209775nteger @ A5 @ C ) )
% 0.30/0.56 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 0.30/0.56 = ( modulo364778990260209775nteger @ B5 @ C ) )
% 0.30/0.56 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.30/0.56 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A5 @ B5 ) @ C ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_add_cong
% 0.30/0.56 thf(fact_3084_mod__add__left__eq,axiom,
% 0.30/0.56 ! [A: nat,C: nat,B: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 0.30/0.56 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_add_left_eq
% 0.30/0.56 thf(fact_3085_mod__add__left__eq,axiom,
% 0.30/0.56 ! [A: int,C: int,B: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 0.30/0.56 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_add_left_eq
% 0.30/0.56 thf(fact_3086_mod__add__left__eq,axiom,
% 0.30/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 0.30/0.56 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_add_left_eq
% 0.30/0.56 thf(fact_3087_mod__add__right__eq,axiom,
% 0.30/0.56 ! [A: nat,B: nat,C: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.30/0.56 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_add_right_eq
% 0.30/0.56 thf(fact_3088_mod__add__right__eq,axiom,
% 0.30/0.56 ! [A: int,B: int,C: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.30/0.56 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_add_right_eq
% 0.30/0.56 thf(fact_3089_mod__add__right__eq,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.30/0.56 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_add_right_eq
% 0.30/0.56 thf(fact_3090_mod__diff__eq,axiom,
% 0.30/0.56 ! [A: int,C: int,B: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.30/0.56 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_diff_eq
% 0.30/0.56 thf(fact_3091_mod__diff__eq,axiom,
% 0.30/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.30/0.56 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_diff_eq
% 0.30/0.56 thf(fact_3092_mod__diff__cong,axiom,
% 0.30/0.56 ! [A: int,C: int,A5: int,B: int,B5: int] :
% 0.30/0.56 ( ( ( modulo_modulo_int @ A @ C )
% 0.30/0.56 = ( modulo_modulo_int @ A5 @ C ) )
% 0.30/0.56 => ( ( ( modulo_modulo_int @ B @ C )
% 0.30/0.56 = ( modulo_modulo_int @ B5 @ C ) )
% 0.30/0.56 => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.56 = ( modulo_modulo_int @ ( minus_minus_int @ A5 @ B5 ) @ C ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_diff_cong
% 0.30/0.56 thf(fact_3093_mod__diff__cong,axiom,
% 0.30/0.56 ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B5: code_integer] :
% 0.30/0.56 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.30/0.56 = ( modulo364778990260209775nteger @ A5 @ C ) )
% 0.30/0.56 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 0.30/0.56 = ( modulo364778990260209775nteger @ B5 @ C ) )
% 0.30/0.56 => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 0.30/0.56 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A5 @ B5 ) @ C ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_diff_cong
% 0.30/0.56 thf(fact_3094_mod__diff__left__eq,axiom,
% 0.30/0.56 ! [A: int,C: int,B: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 0.30/0.56 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_diff_left_eq
% 0.30/0.56 thf(fact_3095_mod__diff__left__eq,axiom,
% 0.30/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 0.30/0.56 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_diff_left_eq
% 0.30/0.56 thf(fact_3096_mod__diff__right__eq,axiom,
% 0.30/0.56 ! [A: int,B: int,C: int] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.30/0.56 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_diff_right_eq
% 0.30/0.56 thf(fact_3097_mod__diff__right__eq,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.30/0.56 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_diff_right_eq
% 0.30/0.56 thf(fact_3098_power__mod,axiom,
% 0.30/0.56 ! [A: nat,B: nat,N: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N ) @ B )
% 0.30/0.56 = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % power_mod
% 0.30/0.56 thf(fact_3099_power__mod,axiom,
% 0.30/0.56 ! [A: int,B: int,N: nat] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N ) @ B )
% 0.30/0.56 = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % power_mod
% 0.30/0.56 thf(fact_3100_power__mod,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer,N: nat] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N ) @ B )
% 0.30/0.56 = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % power_mod
% 0.30/0.56 thf(fact_3101_mod__Suc__Suc__eq,axiom,
% 0.30/0.56 ! [M: nat,N: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
% 0.30/0.56 = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_Suc_Suc_eq
% 0.30/0.56 thf(fact_3102_mod__Suc__eq,axiom,
% 0.30/0.56 ! [M: nat,N: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
% 0.30/0.56 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_Suc_eq
% 0.30/0.56 thf(fact_3103_mod__less__eq__dividend,axiom,
% 0.30/0.56 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% 0.30/0.56
% 0.30/0.56 % mod_less_eq_dividend
% 0.30/0.56 thf(fact_3104_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.56 => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).
% 0.30/0.56
% 0.30/0.56 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.30/0.56 thf(fact_3105_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.30/0.56 ! [A: nat,B: nat] :
% 0.30/0.56 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.56 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% 0.30/0.56
% 0.30/0.56 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.30/0.56 thf(fact_3106_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.56 => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% 0.30/0.56
% 0.30/0.56 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.30/0.56 thf(fact_3107_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.30/0.56 ! [B: nat,A: nat] :
% 0.30/0.56 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.56 => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.30/0.56 thf(fact_3108_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.30/0.56 ! [B: int,A: int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.56 => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.30/0.56 thf(fact_3109_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.30/0.56 ! [B: code_integer,A: code_integer] :
% 0.30/0.56 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.56 => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.30/0.56 thf(fact_3110_mod__eq__self__iff__div__eq__0,axiom,
% 0.30/0.56 ! [A: nat,B: nat] :
% 0.30/0.56 ( ( ( modulo_modulo_nat @ A @ B )
% 0.30/0.56 = A )
% 0.30/0.56 = ( ( divide_divide_nat @ A @ B )
% 0.30/0.56 = zero_zero_nat ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_eq_self_iff_div_eq_0
% 0.30/0.56 thf(fact_3111_mod__eq__self__iff__div__eq__0,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( ( ( modulo_modulo_int @ A @ B )
% 0.30/0.56 = A )
% 0.30/0.56 = ( ( divide_divide_int @ A @ B )
% 0.30/0.56 = zero_zero_int ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_eq_self_iff_div_eq_0
% 0.30/0.56 thf(fact_3112_mod__eq__self__iff__div__eq__0,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.30/0.56 = A )
% 0.30/0.56 = ( ( divide6298287555418463151nteger @ A @ B )
% 0.30/0.56 = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_eq_self_iff_div_eq_0
% 0.30/0.56 thf(fact_3113_cong__exp__iff__simps_I9_J,axiom,
% 0.30/0.56 ! [M: num,Q3: num,N: num] :
% 0.30/0.56 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.30/0.56 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 0.30/0.56 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
% 0.30/0.56 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % cong_exp_iff_simps(9)
% 0.30/0.56 thf(fact_3114_cong__exp__iff__simps_I9_J,axiom,
% 0.30/0.56 ! [M: num,Q3: num,N: num] :
% 0.30/0.56 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.30/0.56 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 0.30/0.56 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
% 0.30/0.56 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % cong_exp_iff_simps(9)
% 0.30/0.56 thf(fact_3115_cong__exp__iff__simps_I9_J,axiom,
% 0.30/0.56 ! [M: num,Q3: num,N: num] :
% 0.30/0.56 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.30/0.56 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 0.30/0.56 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 0.30/0.56 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % cong_exp_iff_simps(9)
% 0.30/0.56 thf(fact_3116_cong__exp__iff__simps_I4_J,axiom,
% 0.30/0.56 ! [M: num,N: num] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
% 0.30/0.56 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % cong_exp_iff_simps(4)
% 0.30/0.56 thf(fact_3117_cong__exp__iff__simps_I4_J,axiom,
% 0.30/0.56 ! [M: num,N: num] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
% 0.30/0.56 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % cong_exp_iff_simps(4)
% 0.30/0.56 thf(fact_3118_cong__exp__iff__simps_I4_J,axiom,
% 0.30/0.56 ! [M: num,N: num] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
% 0.30/0.56 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % cong_exp_iff_simps(4)
% 0.30/0.56 thf(fact_3119_mod__eqE,axiom,
% 0.30/0.56 ! [A: int,C: int,B: int] :
% 0.30/0.56 ( ( ( modulo_modulo_int @ A @ C )
% 0.30/0.56 = ( modulo_modulo_int @ B @ C ) )
% 0.30/0.56 => ~ ! [D3: int] :
% 0.30/0.56 ( B
% 0.30/0.56 != ( plus_plus_int @ A @ ( times_times_int @ C @ D3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_eqE
% 0.30/0.56 thf(fact_3120_mod__eqE,axiom,
% 0.30/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.56 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.30/0.56 = ( modulo364778990260209775nteger @ B @ C ) )
% 0.30/0.56 => ~ ! [D3: code_integer] :
% 0.30/0.56 ( B
% 0.30/0.56 != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_eqE
% 0.30/0.56 thf(fact_3121_div__add1__eq,axiom,
% 0.30/0.56 ! [A: nat,B: nat,C: nat] :
% 0.30/0.56 ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.56 = ( 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 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % div_add1_eq
% 0.30/0.56 thf(fact_3122_div__add1__eq,axiom,
% 0.30/0.56 ! [A: int,B: int,C: int] :
% 0.30/0.56 ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.56 = ( 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 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % div_add1_eq
% 0.30/0.56 thf(fact_3123_div__add1__eq,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56 ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.30/0.56 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % div_add1_eq
% 0.30/0.56 thf(fact_3124_mod__Suc,axiom,
% 0.30/0.56 ! [M: nat,N: nat] :
% 0.30/0.56 ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 0.30/0.56 = N )
% 0.30/0.56 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 0.30/0.56 = zero_zero_nat ) )
% 0.30/0.56 & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 0.30/0.56 != N )
% 0.30/0.56 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 0.30/0.56 = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_Suc
% 0.30/0.56 thf(fact_3125_mod__induct,axiom,
% 0.30/0.56 ! [P: nat > $o,N: nat,P6: nat,M: nat] :
% 0.30/0.56 ( ( P @ N )
% 0.30/0.56 => ( ( ord_less_nat @ N @ P6 )
% 0.30/0.56 => ( ( ord_less_nat @ M @ P6 )
% 0.30/0.56 => ( ! [N2: nat] :
% 0.30/0.56 ( ( ord_less_nat @ N2 @ P6 )
% 0.30/0.56 => ( ( P @ N2 )
% 0.30/0.56 => ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P6 ) ) ) )
% 0.30/0.56 => ( P @ M ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_induct
% 0.30/0.56 thf(fact_3126_mod__less__divisor,axiom,
% 0.30/0.56 ! [N: nat,M: nat] :
% 0.30/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.56 => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_less_divisor
% 0.30/0.56 thf(fact_3127_mod__Suc__le__divisor,axiom,
% 0.30/0.56 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% 0.30/0.56
% 0.30/0.56 % mod_Suc_le_divisor
% 0.30/0.56 thf(fact_3128_mod__eq__0D,axiom,
% 0.30/0.56 ! [M: nat,D: nat] :
% 0.30/0.56 ( ( ( modulo_modulo_nat @ M @ D )
% 0.30/0.56 = zero_zero_nat )
% 0.30/0.56 => ? [Q2: nat] :
% 0.30/0.56 ( M
% 0.30/0.56 = ( times_times_nat @ D @ Q2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_eq_0D
% 0.30/0.56 thf(fact_3129_mod__geq,axiom,
% 0.30/0.56 ! [M: nat,N: nat] :
% 0.30/0.56 ( ~ ( ord_less_nat @ M @ N )
% 0.30/0.56 => ( ( modulo_modulo_nat @ M @ N )
% 0.30/0.56 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_geq
% 0.30/0.56 thf(fact_3130_mod__if,axiom,
% 0.30/0.56 ( modulo_modulo_nat
% 0.30/0.56 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N3 ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N3 ) @ N3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_if
% 0.30/0.56 thf(fact_3131_le__mod__geq,axiom,
% 0.30/0.56 ! [N: nat,M: nat] :
% 0.30/0.56 ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.56 => ( ( modulo_modulo_nat @ M @ N )
% 0.30/0.56 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % le_mod_geq
% 0.30/0.56 thf(fact_3132_nat__mod__eq__iff,axiom,
% 0.30/0.56 ! [X2: nat,N: nat,Y3: nat] :
% 0.30/0.56 ( ( ( modulo_modulo_nat @ X2 @ N )
% 0.30/0.56 = ( modulo_modulo_nat @ Y3 @ N ) )
% 0.30/0.56 = ( ? [Q1: nat,Q22: nat] :
% 0.30/0.56 ( ( plus_plus_nat @ X2 @ ( times_times_nat @ N @ Q1 ) )
% 0.30/0.56 = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % nat_mod_eq_iff
% 0.30/0.56 thf(fact_3133_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.30/0.56 ! [B: code_integer,A: code_integer] :
% 0.30/0.56 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.56 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.30/0.56 thf(fact_3134_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.30/0.56 ! [B: nat,A: nat] :
% 0.30/0.56 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.56 => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.30/0.56 thf(fact_3135_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.30/0.56 ! [B: int,A: int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.56 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.30/0.56 thf(fact_3136_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.56 => ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.30/0.56 => ( ( modulo364778990260209775nteger @ A @ B )
% 0.30/0.56 = A ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % unique_euclidean_semiring_numeral_class.mod_less
% 0.30/0.56 thf(fact_3137_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.30/0.56 ! [A: nat,B: nat] :
% 0.30/0.56 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.56 => ( ( ord_less_nat @ A @ B )
% 0.30/0.56 => ( ( modulo_modulo_nat @ A @ B )
% 0.30/0.56 = A ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % unique_euclidean_semiring_numeral_class.mod_less
% 0.30/0.56 thf(fact_3138_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.56 => ( ( ord_less_int @ A @ B )
% 0.30/0.56 => ( ( modulo_modulo_int @ A @ B )
% 0.30/0.56 = A ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % unique_euclidean_semiring_numeral_class.mod_less
% 0.30/0.56 thf(fact_3139_cong__exp__iff__simps_I2_J,axiom,
% 0.30/0.56 ! [N: num,Q3: num] :
% 0.30/0.56 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.30/0.56 = zero_zero_nat )
% 0.30/0.56 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
% 0.30/0.56 = zero_zero_nat ) ) ).
% 0.30/0.56
% 0.30/0.56 % cong_exp_iff_simps(2)
% 0.30/0.56 thf(fact_3140_cong__exp__iff__simps_I2_J,axiom,
% 0.30/0.56 ! [N: num,Q3: num] :
% 0.30/0.56 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.30/0.56 = zero_zero_int )
% 0.30/0.56 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
% 0.30/0.56 = zero_zero_int ) ) ).
% 0.30/0.56
% 0.30/0.56 % cong_exp_iff_simps(2)
% 0.30/0.56 thf(fact_3141_cong__exp__iff__simps_I2_J,axiom,
% 0.30/0.56 ! [N: num,Q3: num] :
% 0.30/0.56 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.30/0.56 = zero_z3403309356797280102nteger )
% 0.30/0.56 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 0.30/0.56 = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.56
% 0.30/0.56 % cong_exp_iff_simps(2)
% 0.30/0.56 thf(fact_3142_cong__exp__iff__simps_I1_J,axiom,
% 0.30/0.56 ! [N: num] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
% 0.30/0.56 = zero_zero_nat ) ).
% 0.30/0.56
% 0.30/0.56 % cong_exp_iff_simps(1)
% 0.30/0.56 thf(fact_3143_cong__exp__iff__simps_I1_J,axiom,
% 0.30/0.56 ! [N: num] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
% 0.30/0.56 = zero_zero_int ) ).
% 0.30/0.56
% 0.30/0.56 % cong_exp_iff_simps(1)
% 0.30/0.56 thf(fact_3144_cong__exp__iff__simps_I1_J,axiom,
% 0.30/0.56 ! [N: num] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
% 0.30/0.56 = zero_z3403309356797280102nteger ) ).
% 0.30/0.56
% 0.30/0.56 % cong_exp_iff_simps(1)
% 0.30/0.56 thf(fact_3145_cong__exp__iff__simps_I6_J,axiom,
% 0.30/0.56 ! [Q3: num,N: num] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.30/0.56 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % cong_exp_iff_simps(6)
% 0.30/0.56 thf(fact_3146_cong__exp__iff__simps_I6_J,axiom,
% 0.30/0.56 ! [Q3: num,N: num] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.30/0.56 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % cong_exp_iff_simps(6)
% 0.30/0.56 thf(fact_3147_cong__exp__iff__simps_I6_J,axiom,
% 0.30/0.56 ! [Q3: num,N: num] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.30/0.56 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % cong_exp_iff_simps(6)
% 0.30/0.56 thf(fact_3148_cong__exp__iff__simps_I8_J,axiom,
% 0.30/0.56 ! [M: num,Q3: num] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.30/0.56 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % cong_exp_iff_simps(8)
% 0.30/0.56 thf(fact_3149_cong__exp__iff__simps_I8_J,axiom,
% 0.30/0.56 ! [M: num,Q3: num] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.30/0.56 != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % cong_exp_iff_simps(8)
% 0.30/0.56 thf(fact_3150_cong__exp__iff__simps_I8_J,axiom,
% 0.30/0.56 ! [M: num,Q3: num] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.30/0.56 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % cong_exp_iff_simps(8)
% 0.30/0.56 thf(fact_3151_div__mult1__eq,axiom,
% 0.30/0.56 ! [A: nat,B: nat,C: nat] :
% 0.30/0.56 ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.30/0.56 = ( 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 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % div_mult1_eq
% 0.30/0.56 thf(fact_3152_div__mult1__eq,axiom,
% 0.30/0.56 ! [A: int,B: int,C: int] :
% 0.30/0.56 ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
% 0.30/0.56 = ( 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 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % div_mult1_eq
% 0.30/0.56 thf(fact_3153_div__mult1__eq,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56 ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.30/0.56 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % div_mult1_eq
% 0.30/0.56 thf(fact_3154_mult__div__mod__eq,axiom,
% 0.30/0.56 ! [B: nat,A: nat] :
% 0.30/0.56 ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
% 0.30/0.56 = A ) ).
% 0.30/0.56
% 0.30/0.56 % mult_div_mod_eq
% 0.30/0.56 thf(fact_3155_mult__div__mod__eq,axiom,
% 0.30/0.56 ! [B: int,A: int] :
% 0.30/0.56 ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.56 = A ) ).
% 0.30/0.56
% 0.30/0.56 % mult_div_mod_eq
% 0.30/0.56 thf(fact_3156_mult__div__mod__eq,axiom,
% 0.30/0.56 ! [B: code_integer,A: code_integer] :
% 0.30/0.56 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.30/0.56 = A ) ).
% 0.30/0.56
% 0.30/0.56 % mult_div_mod_eq
% 0.30/0.56 thf(fact_3157_mod__mult__div__eq,axiom,
% 0.30/0.56 ! [A: nat,B: nat] :
% 0.30/0.56 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 0.30/0.56 = A ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_div_eq
% 0.30/0.56 thf(fact_3158_mod__mult__div__eq,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 0.30/0.56 = A ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_div_eq
% 0.30/0.56 thf(fact_3159_mod__mult__div__eq,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 0.30/0.56 = A ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult_div_eq
% 0.30/0.56 thf(fact_3160_mod__div__mult__eq,axiom,
% 0.30/0.56 ! [A: nat,B: nat] :
% 0.30/0.56 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 0.30/0.56 = A ) ).
% 0.30/0.56
% 0.30/0.56 % mod_div_mult_eq
% 0.30/0.56 thf(fact_3161_mod__div__mult__eq,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 0.30/0.56 = A ) ).
% 0.30/0.56
% 0.30/0.56 % mod_div_mult_eq
% 0.30/0.56 thf(fact_3162_mod__div__mult__eq,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 0.30/0.56 = A ) ).
% 0.30/0.56
% 0.30/0.56 % mod_div_mult_eq
% 0.30/0.56 thf(fact_3163_div__mult__mod__eq,axiom,
% 0.30/0.56 ! [A: nat,B: nat] :
% 0.30/0.56 ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
% 0.30/0.56 = A ) ).
% 0.30/0.56
% 0.30/0.56 % div_mult_mod_eq
% 0.30/0.56 thf(fact_3164_div__mult__mod__eq,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.56 = A ) ).
% 0.30/0.56
% 0.30/0.56 % div_mult_mod_eq
% 0.30/0.56 thf(fact_3165_div__mult__mod__eq,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.30/0.56 = A ) ).
% 0.30/0.56
% 0.30/0.56 % div_mult_mod_eq
% 0.30/0.56 thf(fact_3166_mod__div__decomp,axiom,
% 0.30/0.56 ! [A: nat,B: nat] :
% 0.30/0.56 ( A
% 0.30/0.56 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_div_decomp
% 0.30/0.56 thf(fact_3167_mod__div__decomp,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( A
% 0.30/0.56 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_div_decomp
% 0.30/0.56 thf(fact_3168_mod__div__decomp,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer] :
% 0.30/0.56 ( A
% 0.30/0.56 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_div_decomp
% 0.30/0.56 thf(fact_3169_cancel__div__mod__rules_I1_J,axiom,
% 0.30/0.56 ! [A: nat,B: nat,C: nat] :
% 0.30/0.56 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 0.30/0.56 = ( plus_plus_nat @ A @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % cancel_div_mod_rules(1)
% 0.30/0.56 thf(fact_3170_cancel__div__mod__rules_I1_J,axiom,
% 0.30/0.56 ! [A: int,B: int,C: int] :
% 0.30/0.56 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 0.30/0.56 = ( plus_plus_int @ A @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % cancel_div_mod_rules(1)
% 0.30/0.56 thf(fact_3171_cancel__div__mod__rules_I1_J,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 0.30/0.56 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % cancel_div_mod_rules(1)
% 0.30/0.56 thf(fact_3172_cancel__div__mod__rules_I2_J,axiom,
% 0.30/0.56 ! [B: nat,A: nat,C: nat] :
% 0.30/0.56 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 0.30/0.56 = ( plus_plus_nat @ A @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % cancel_div_mod_rules(2)
% 0.30/0.56 thf(fact_3173_cancel__div__mod__rules_I2_J,axiom,
% 0.30/0.56 ! [B: int,A: int,C: int] :
% 0.30/0.56 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 0.30/0.56 = ( plus_plus_int @ A @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % cancel_div_mod_rules(2)
% 0.30/0.56 thf(fact_3174_cancel__div__mod__rules_I2_J,axiom,
% 0.30/0.56 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.56 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 0.30/0.56 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 0.30/0.56
% 0.30/0.56 % cancel_div_mod_rules(2)
% 0.30/0.56 thf(fact_3175_minus__mult__div__eq__mod,axiom,
% 0.30/0.56 ! [A: nat,B: nat] :
% 0.30/0.56 ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 0.30/0.56 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_mult_div_eq_mod
% 0.30/0.56 thf(fact_3176_minus__mult__div__eq__mod,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 0.30/0.56 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_mult_div_eq_mod
% 0.30/0.56 thf(fact_3177_minus__mult__div__eq__mod,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 0.30/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_mult_div_eq_mod
% 0.30/0.56 thf(fact_3178_minus__mod__eq__mult__div,axiom,
% 0.30/0.56 ! [A: nat,B: nat] :
% 0.30/0.56 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 0.30/0.56 = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_mod_eq_mult_div
% 0.30/0.56 thf(fact_3179_minus__mod__eq__mult__div,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.56 = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_mod_eq_mult_div
% 0.30/0.56 thf(fact_3180_minus__mod__eq__mult__div,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.30/0.56 = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_mod_eq_mult_div
% 0.30/0.56 thf(fact_3181_minus__mod__eq__div__mult,axiom,
% 0.30/0.56 ! [A: nat,B: nat] :
% 0.30/0.56 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 0.30/0.56 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_mod_eq_div_mult
% 0.30/0.56 thf(fact_3182_minus__mod__eq__div__mult,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.56 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_mod_eq_div_mult
% 0.30/0.56 thf(fact_3183_minus__mod__eq__div__mult,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.30/0.56 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_mod_eq_div_mult
% 0.30/0.56 thf(fact_3184_minus__div__mult__eq__mod,axiom,
% 0.30/0.56 ! [A: nat,B: nat] :
% 0.30/0.56 ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 0.30/0.56 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_div_mult_eq_mod
% 0.30/0.56 thf(fact_3185_minus__div__mult__eq__mod,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 0.30/0.56 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_div_mult_eq_mod
% 0.30/0.56 thf(fact_3186_minus__div__mult__eq__mod,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 0.30/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % minus_div_mult_eq_mod
% 0.30/0.56 thf(fact_3187_mod__le__divisor,axiom,
% 0.30/0.56 ! [N: nat,M: nat] :
% 0.30/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.56 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_le_divisor
% 0.30/0.56 thf(fact_3188_mod__eq__nat1E,axiom,
% 0.30/0.56 ! [M: nat,Q3: nat,N: nat] :
% 0.30/0.56 ( ( ( modulo_modulo_nat @ M @ Q3 )
% 0.30/0.56 = ( modulo_modulo_nat @ N @ Q3 ) )
% 0.30/0.56 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.56 => ~ ! [S2: nat] :
% 0.30/0.56 ( M
% 0.30/0.56 != ( plus_plus_nat @ N @ ( times_times_nat @ Q3 @ S2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_eq_nat1E
% 0.30/0.56 thf(fact_3189_mod__eq__nat2E,axiom,
% 0.30/0.56 ! [M: nat,Q3: nat,N: nat] :
% 0.30/0.56 ( ( ( modulo_modulo_nat @ M @ Q3 )
% 0.30/0.56 = ( modulo_modulo_nat @ N @ Q3 ) )
% 0.30/0.56 => ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.56 => ~ ! [S2: nat] :
% 0.30/0.56 ( N
% 0.30/0.56 != ( plus_plus_nat @ M @ ( times_times_nat @ Q3 @ S2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_eq_nat2E
% 0.30/0.56 thf(fact_3190_nat__mod__eq__lemma,axiom,
% 0.30/0.56 ! [X2: nat,N: nat,Y3: nat] :
% 0.30/0.56 ( ( ( modulo_modulo_nat @ X2 @ N )
% 0.30/0.56 = ( modulo_modulo_nat @ Y3 @ N ) )
% 0.30/0.56 => ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.30/0.56 => ? [Q2: nat] :
% 0.30/0.56 ( X2
% 0.30/0.56 = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q2 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % nat_mod_eq_lemma
% 0.30/0.56 thf(fact_3191_eucl__rel__int__dividesI,axiom,
% 0.30/0.56 ! [L: int,K: int,Q3: int] :
% 0.30/0.56 ( ( L != zero_zero_int )
% 0.30/0.56 => ( ( K
% 0.30/0.56 = ( times_times_int @ Q3 @ L ) )
% 0.30/0.56 => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ zero_zero_int ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % eucl_rel_int_dividesI
% 0.30/0.56 thf(fact_3192_mod__mult2__eq,axiom,
% 0.30/0.56 ! [M: nat,N: nat,Q3: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q3 ) )
% 0.30/0.56 = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q3 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_mult2_eq
% 0.30/0.56 thf(fact_3193_modulo__nat__def,axiom,
% 0.30/0.56 ( modulo_modulo_nat
% 0.30/0.56 = ( ^ [M3: nat,N3: nat] : ( minus_minus_nat @ M3 @ ( times_times_nat @ ( divide_divide_nat @ M3 @ N3 ) @ N3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % modulo_nat_def
% 0.30/0.56 thf(fact_3194_in__mono,axiom,
% 0.30/0.56 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 0.30/0.56 ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
% 0.30/0.56 => ( ( member8440522571783428010at_nat @ X2 @ A2 )
% 0.30/0.56 => ( member8440522571783428010at_nat @ X2 @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % in_mono
% 0.30/0.56 thf(fact_3195_in__mono,axiom,
% 0.30/0.56 ! [A2: set_complex,B3: set_complex,X2: complex] :
% 0.30/0.56 ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.30/0.56 => ( ( member_complex @ X2 @ A2 )
% 0.30/0.56 => ( member_complex @ X2 @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % in_mono
% 0.30/0.56 thf(fact_3196_in__mono,axiom,
% 0.30/0.56 ! [A2: set_real,B3: set_real,X2: real] :
% 0.30/0.56 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.30/0.56 => ( ( member_real @ X2 @ A2 )
% 0.30/0.56 => ( member_real @ X2 @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % in_mono
% 0.30/0.56 thf(fact_3197_in__mono,axiom,
% 0.30/0.56 ! [A2: set_nat,B3: set_nat,X2: nat] :
% 0.30/0.56 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.30/0.56 => ( ( member_nat @ X2 @ A2 )
% 0.30/0.56 => ( member_nat @ X2 @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % in_mono
% 0.30/0.56 thf(fact_3198_in__mono,axiom,
% 0.30/0.56 ! [A2: set_int,B3: set_int,X2: int] :
% 0.30/0.56 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.30/0.56 => ( ( member_int @ X2 @ A2 )
% 0.30/0.56 => ( member_int @ X2 @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % in_mono
% 0.30/0.56 thf(fact_3199_subsetD,axiom,
% 0.30/0.56 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
% 0.30/0.56 ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
% 0.30/0.56 => ( ( member8440522571783428010at_nat @ C @ A2 )
% 0.30/0.56 => ( member8440522571783428010at_nat @ C @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subsetD
% 0.30/0.56 thf(fact_3200_subsetD,axiom,
% 0.30/0.56 ! [A2: set_complex,B3: set_complex,C: complex] :
% 0.30/0.56 ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.30/0.56 => ( ( member_complex @ C @ A2 )
% 0.30/0.56 => ( member_complex @ C @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subsetD
% 0.30/0.56 thf(fact_3201_subsetD,axiom,
% 0.30/0.56 ! [A2: set_real,B3: set_real,C: real] :
% 0.30/0.56 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.30/0.56 => ( ( member_real @ C @ A2 )
% 0.30/0.56 => ( member_real @ C @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subsetD
% 0.30/0.56 thf(fact_3202_subsetD,axiom,
% 0.30/0.56 ! [A2: set_nat,B3: set_nat,C: nat] :
% 0.30/0.56 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.30/0.56 => ( ( member_nat @ C @ A2 )
% 0.30/0.56 => ( member_nat @ C @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subsetD
% 0.30/0.56 thf(fact_3203_subsetD,axiom,
% 0.30/0.56 ! [A2: set_int,B3: set_int,C: int] :
% 0.30/0.56 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.30/0.56 => ( ( member_int @ C @ A2 )
% 0.30/0.56 => ( member_int @ C @ B3 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subsetD
% 0.30/0.56 thf(fact_3204_equalityE,axiom,
% 0.30/0.56 ! [A2: set_int,B3: set_int] :
% 0.30/0.56 ( ( A2 = B3 )
% 0.30/0.56 => ~ ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.30/0.56 => ~ ( ord_less_eq_set_int @ B3 @ A2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % equalityE
% 0.30/0.56 thf(fact_3205_subset__eq,axiom,
% 0.30/0.56 ( ord_le3146513528884898305at_nat
% 0.30/0.56 = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 0.30/0.56 ! [X: product_prod_nat_nat] :
% 0.30/0.56 ( ( member8440522571783428010at_nat @ X @ A6 )
% 0.30/0.56 => ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_eq
% 0.30/0.56 thf(fact_3206_subset__eq,axiom,
% 0.30/0.56 ( ord_le211207098394363844omplex
% 0.30/0.56 = ( ^ [A6: set_complex,B6: set_complex] :
% 0.30/0.56 ! [X: complex] :
% 0.30/0.56 ( ( member_complex @ X @ A6 )
% 0.30/0.56 => ( member_complex @ X @ B6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_eq
% 0.30/0.56 thf(fact_3207_subset__eq,axiom,
% 0.30/0.56 ( ord_less_eq_set_real
% 0.30/0.56 = ( ^ [A6: set_real,B6: set_real] :
% 0.30/0.56 ! [X: real] :
% 0.30/0.56 ( ( member_real @ X @ A6 )
% 0.30/0.56 => ( member_real @ X @ B6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_eq
% 0.30/0.56 thf(fact_3208_subset__eq,axiom,
% 0.30/0.56 ( ord_less_eq_set_nat
% 0.30/0.56 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.30/0.56 ! [X: nat] :
% 0.30/0.56 ( ( member_nat @ X @ A6 )
% 0.30/0.56 => ( member_nat @ X @ B6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_eq
% 0.30/0.56 thf(fact_3209_subset__eq,axiom,
% 0.30/0.56 ( ord_less_eq_set_int
% 0.30/0.56 = ( ^ [A6: set_int,B6: set_int] :
% 0.30/0.56 ! [X: int] :
% 0.30/0.56 ( ( member_int @ X @ A6 )
% 0.30/0.56 => ( member_int @ X @ B6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_eq
% 0.30/0.56 thf(fact_3210_equalityD1,axiom,
% 0.30/0.56 ! [A2: set_int,B3: set_int] :
% 0.30/0.56 ( ( A2 = B3 )
% 0.30/0.56 => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % equalityD1
% 0.30/0.56 thf(fact_3211_equalityD2,axiom,
% 0.30/0.56 ! [A2: set_int,B3: set_int] :
% 0.30/0.56 ( ( A2 = B3 )
% 0.30/0.56 => ( ord_less_eq_set_int @ B3 @ A2 ) ) ).
% 0.30/0.56
% 0.30/0.56 % equalityD2
% 0.30/0.56 thf(fact_3212_subset__iff,axiom,
% 0.30/0.56 ( ord_le3146513528884898305at_nat
% 0.30/0.56 = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 0.30/0.56 ! [T2: product_prod_nat_nat] :
% 0.30/0.56 ( ( member8440522571783428010at_nat @ T2 @ A6 )
% 0.30/0.56 => ( member8440522571783428010at_nat @ T2 @ B6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_iff
% 0.30/0.56 thf(fact_3213_subset__iff,axiom,
% 0.30/0.56 ( ord_le211207098394363844omplex
% 0.30/0.56 = ( ^ [A6: set_complex,B6: set_complex] :
% 0.30/0.56 ! [T2: complex] :
% 0.30/0.56 ( ( member_complex @ T2 @ A6 )
% 0.30/0.56 => ( member_complex @ T2 @ B6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_iff
% 0.30/0.56 thf(fact_3214_subset__iff,axiom,
% 0.30/0.56 ( ord_less_eq_set_real
% 0.30/0.56 = ( ^ [A6: set_real,B6: set_real] :
% 0.30/0.56 ! [T2: real] :
% 0.30/0.56 ( ( member_real @ T2 @ A6 )
% 0.30/0.56 => ( member_real @ T2 @ B6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_iff
% 0.30/0.56 thf(fact_3215_subset__iff,axiom,
% 0.30/0.56 ( ord_less_eq_set_nat
% 0.30/0.56 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.30/0.56 ! [T2: nat] :
% 0.30/0.56 ( ( member_nat @ T2 @ A6 )
% 0.30/0.56 => ( member_nat @ T2 @ B6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_iff
% 0.30/0.56 thf(fact_3216_subset__iff,axiom,
% 0.30/0.56 ( ord_less_eq_set_int
% 0.30/0.56 = ( ^ [A6: set_int,B6: set_int] :
% 0.30/0.56 ! [T2: int] :
% 0.30/0.56 ( ( member_int @ T2 @ A6 )
% 0.30/0.56 => ( member_int @ T2 @ B6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_iff
% 0.30/0.56 thf(fact_3217_subset__refl,axiom,
% 0.30/0.56 ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% 0.30/0.56
% 0.30/0.56 % subset_refl
% 0.30/0.56 thf(fact_3218_Collect__mono,axiom,
% 0.30/0.56 ! [P: complex > $o,Q: complex > $o] :
% 0.30/0.56 ( ! [X5: complex] :
% 0.30/0.56 ( ( P @ X5 )
% 0.30/0.56 => ( Q @ X5 ) )
% 0.30/0.56 => ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Collect_mono
% 0.30/0.56 thf(fact_3219_Collect__mono,axiom,
% 0.30/0.56 ! [P: real > $o,Q: real > $o] :
% 0.30/0.56 ( ! [X5: real] :
% 0.30/0.56 ( ( P @ X5 )
% 0.30/0.56 => ( Q @ X5 ) )
% 0.30/0.56 => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Collect_mono
% 0.30/0.56 thf(fact_3220_Collect__mono,axiom,
% 0.30/0.56 ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.30/0.56 ( ! [X5: list_nat] :
% 0.30/0.56 ( ( P @ X5 )
% 0.30/0.56 => ( Q @ X5 ) )
% 0.30/0.56 => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Collect_mono
% 0.30/0.56 thf(fact_3221_Collect__mono,axiom,
% 0.30/0.56 ! [P: nat > $o,Q: nat > $o] :
% 0.30/0.56 ( ! [X5: nat] :
% 0.30/0.56 ( ( P @ X5 )
% 0.30/0.56 => ( Q @ X5 ) )
% 0.30/0.56 => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Collect_mono
% 0.30/0.56 thf(fact_3222_Collect__mono,axiom,
% 0.30/0.56 ! [P: int > $o,Q: int > $o] :
% 0.30/0.56 ( ! [X5: int] :
% 0.30/0.56 ( ( P @ X5 )
% 0.30/0.56 => ( Q @ X5 ) )
% 0.30/0.56 => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Collect_mono
% 0.30/0.56 thf(fact_3223_subset__trans,axiom,
% 0.30/0.56 ! [A2: set_int,B3: set_int,C4: set_int] :
% 0.30/0.56 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.30/0.56 => ( ( ord_less_eq_set_int @ B3 @ C4 )
% 0.30/0.56 => ( ord_less_eq_set_int @ A2 @ C4 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_trans
% 0.30/0.56 thf(fact_3224_set__eq__subset,axiom,
% 0.30/0.56 ( ( ^ [Y6: set_int,Z5: set_int] : ( Y6 = Z5 ) )
% 0.30/0.56 = ( ^ [A6: set_int,B6: set_int] :
% 0.30/0.56 ( ( ord_less_eq_set_int @ A6 @ B6 )
% 0.30/0.56 & ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_eq_subset
% 0.30/0.56 thf(fact_3225_Collect__mono__iff,axiom,
% 0.30/0.56 ! [P: complex > $o,Q: complex > $o] :
% 0.30/0.56 ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
% 0.30/0.56 = ( ! [X: complex] :
% 0.30/0.56 ( ( P @ X )
% 0.30/0.56 => ( Q @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Collect_mono_iff
% 0.30/0.56 thf(fact_3226_Collect__mono__iff,axiom,
% 0.30/0.56 ! [P: real > $o,Q: real > $o] :
% 0.30/0.56 ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
% 0.30/0.56 = ( ! [X: real] :
% 0.30/0.56 ( ( P @ X )
% 0.30/0.56 => ( Q @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Collect_mono_iff
% 0.30/0.56 thf(fact_3227_Collect__mono__iff,axiom,
% 0.30/0.56 ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.30/0.56 ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
% 0.30/0.56 = ( ! [X: list_nat] :
% 0.30/0.56 ( ( P @ X )
% 0.30/0.56 => ( Q @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Collect_mono_iff
% 0.30/0.56 thf(fact_3228_Collect__mono__iff,axiom,
% 0.30/0.56 ! [P: nat > $o,Q: nat > $o] :
% 0.30/0.56 ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 0.30/0.56 = ( ! [X: nat] :
% 0.30/0.56 ( ( P @ X )
% 0.30/0.56 => ( Q @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Collect_mono_iff
% 0.30/0.56 thf(fact_3229_Collect__mono__iff,axiom,
% 0.30/0.56 ! [P: int > $o,Q: int > $o] :
% 0.30/0.56 ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
% 0.30/0.56 = ( ! [X: int] :
% 0.30/0.56 ( ( P @ X )
% 0.30/0.56 => ( Q @ X ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Collect_mono_iff
% 0.30/0.56 thf(fact_3230_subset__iff__psubset__eq,axiom,
% 0.30/0.56 ( ord_less_eq_set_int
% 0.30/0.56 = ( ^ [A6: set_int,B6: set_int] :
% 0.30/0.56 ( ( ord_less_set_int @ A6 @ B6 )
% 0.30/0.56 | ( A6 = B6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_iff_psubset_eq
% 0.30/0.56 thf(fact_3231_subset__psubset__trans,axiom,
% 0.30/0.56 ! [A2: set_int,B3: set_int,C4: set_int] :
% 0.30/0.56 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.30/0.56 => ( ( ord_less_set_int @ B3 @ C4 )
% 0.30/0.56 => ( ord_less_set_int @ A2 @ C4 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_psubset_trans
% 0.30/0.56 thf(fact_3232_subset__not__subset__eq,axiom,
% 0.30/0.56 ( ord_less_set_int
% 0.30/0.56 = ( ^ [A6: set_int,B6: set_int] :
% 0.30/0.56 ( ( ord_less_eq_set_int @ A6 @ B6 )
% 0.30/0.56 & ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_not_subset_eq
% 0.30/0.56 thf(fact_3233_psubset__subset__trans,axiom,
% 0.30/0.56 ! [A2: set_int,B3: set_int,C4: set_int] :
% 0.30/0.56 ( ( ord_less_set_int @ A2 @ B3 )
% 0.30/0.56 => ( ( ord_less_eq_set_int @ B3 @ C4 )
% 0.30/0.56 => ( ord_less_set_int @ A2 @ C4 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % psubset_subset_trans
% 0.30/0.56 thf(fact_3234_psubset__imp__subset,axiom,
% 0.30/0.56 ! [A2: set_int,B3: set_int] :
% 0.30/0.56 ( ( ord_less_set_int @ A2 @ B3 )
% 0.30/0.56 => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% 0.30/0.56
% 0.30/0.56 % psubset_imp_subset
% 0.30/0.56 thf(fact_3235_psubset__eq,axiom,
% 0.30/0.56 ( ord_less_set_int
% 0.30/0.56 = ( ^ [A6: set_int,B6: set_int] :
% 0.30/0.56 ( ( ord_less_eq_set_int @ A6 @ B6 )
% 0.30/0.56 & ( A6 != B6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % psubset_eq
% 0.30/0.56 thf(fact_3236_psubsetE,axiom,
% 0.30/0.56 ! [A2: set_int,B3: set_int] :
% 0.30/0.56 ( ( ord_less_set_int @ A2 @ B3 )
% 0.30/0.56 => ~ ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.30/0.56 => ( ord_less_eq_set_int @ B3 @ A2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % psubsetE
% 0.30/0.56 thf(fact_3237_Diff__mono,axiom,
% 0.30/0.56 ! [A2: set_nat,C4: set_nat,D4: set_nat,B3: set_nat] :
% 0.30/0.56 ( ( ord_less_eq_set_nat @ A2 @ C4 )
% 0.30/0.56 => ( ( ord_less_eq_set_nat @ D4 @ B3 )
% 0.30/0.56 => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ ( minus_minus_set_nat @ C4 @ D4 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_mono
% 0.30/0.56 thf(fact_3238_Diff__mono,axiom,
% 0.30/0.56 ! [A2: set_int,C4: set_int,D4: set_int,B3: set_int] :
% 0.30/0.56 ( ( ord_less_eq_set_int @ A2 @ C4 )
% 0.30/0.56 => ( ( ord_less_eq_set_int @ D4 @ B3 )
% 0.30/0.56 => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B3 ) @ ( minus_minus_set_int @ C4 @ D4 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_mono
% 0.30/0.56 thf(fact_3239_Diff__subset,axiom,
% 0.30/0.56 ! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ A2 ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_subset
% 0.30/0.56 thf(fact_3240_Diff__subset,axiom,
% 0.30/0.56 ! [A2: set_int,B3: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B3 ) @ A2 ) ).
% 0.30/0.56
% 0.30/0.56 % Diff_subset
% 0.30/0.56 thf(fact_3241_double__diff,axiom,
% 0.30/0.56 ! [A2: set_nat,B3: set_nat,C4: set_nat] :
% 0.30/0.56 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.30/0.56 => ( ( ord_less_eq_set_nat @ B3 @ C4 )
% 0.30/0.56 => ( ( minus_minus_set_nat @ B3 @ ( minus_minus_set_nat @ C4 @ A2 ) )
% 0.30/0.56 = A2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % double_diff
% 0.30/0.56 thf(fact_3242_double__diff,axiom,
% 0.30/0.56 ! [A2: set_int,B3: set_int,C4: set_int] :
% 0.30/0.56 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.30/0.56 => ( ( ord_less_eq_set_int @ B3 @ C4 )
% 0.30/0.56 => ( ( minus_minus_set_int @ B3 @ ( minus_minus_set_int @ C4 @ A2 ) )
% 0.30/0.56 = A2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % double_diff
% 0.30/0.56 thf(fact_3243_split__mod,axiom,
% 0.30/0.56 ! [P: nat > $o,M: nat,N: nat] :
% 0.30/0.56 ( ( P @ ( modulo_modulo_nat @ M @ N ) )
% 0.30/0.56 = ( ( ( N = zero_zero_nat )
% 0.30/0.56 => ( P @ M ) )
% 0.30/0.56 & ( ( N != zero_zero_nat )
% 0.30/0.56 => ! [I4: nat,J3: nat] :
% 0.30/0.56 ( ( ord_less_nat @ J3 @ N )
% 0.30/0.56 => ( ( M
% 0.30/0.56 = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 0.30/0.56 => ( P @ J3 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % split_mod
% 0.30/0.56 thf(fact_3244_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.30/0.56 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.30/0.56 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.30/0.56 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.30/0.56 thf(fact_3245_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.30/0.56 ! [C: nat,A: nat,B: nat] :
% 0.30/0.56 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.56 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.30/0.56 = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.30/0.56 thf(fact_3246_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.30/0.56 ! [C: int,A: int,B: int] :
% 0.30/0.56 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.56 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 0.30/0.56 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.30/0.56 thf(fact_3247_less__eq__set__def,axiom,
% 0.30/0.56 ( ord_le3146513528884898305at_nat
% 0.30/0.56 = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 0.30/0.56 ( ord_le704812498762024988_nat_o
% 0.30/0.56 @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A6 )
% 0.30/0.56 @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % less_eq_set_def
% 0.30/0.56 thf(fact_3248_less__eq__set__def,axiom,
% 0.30/0.56 ( ord_le211207098394363844omplex
% 0.30/0.56 = ( ^ [A6: set_complex,B6: set_complex] :
% 0.30/0.56 ( ord_le4573692005234683329plex_o
% 0.30/0.56 @ ^ [X: complex] : ( member_complex @ X @ A6 )
% 0.30/0.56 @ ^ [X: complex] : ( member_complex @ X @ B6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % less_eq_set_def
% 0.30/0.56 thf(fact_3249_less__eq__set__def,axiom,
% 0.30/0.56 ( ord_less_eq_set_real
% 0.30/0.56 = ( ^ [A6: set_real,B6: set_real] :
% 0.30/0.56 ( ord_less_eq_real_o
% 0.30/0.56 @ ^ [X: real] : ( member_real @ X @ A6 )
% 0.30/0.56 @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % less_eq_set_def
% 0.30/0.56 thf(fact_3250_less__eq__set__def,axiom,
% 0.30/0.56 ( ord_less_eq_set_nat
% 0.30/0.56 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.30/0.56 ( ord_less_eq_nat_o
% 0.30/0.56 @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 0.30/0.56 @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % less_eq_set_def
% 0.30/0.56 thf(fact_3251_less__eq__set__def,axiom,
% 0.30/0.56 ( ord_less_eq_set_int
% 0.30/0.56 = ( ^ [A6: set_int,B6: set_int] :
% 0.30/0.56 ( ord_less_eq_int_o
% 0.30/0.56 @ ^ [X: int] : ( member_int @ X @ A6 )
% 0.30/0.56 @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % less_eq_set_def
% 0.30/0.56 thf(fact_3252_Collect__subset,axiom,
% 0.30/0.56 ! [A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
% 0.30/0.56 ( ord_le3146513528884898305at_nat
% 0.30/0.56 @ ( collec3392354462482085612at_nat
% 0.30/0.56 @ ^ [X: product_prod_nat_nat] :
% 0.30/0.56 ( ( member8440522571783428010at_nat @ X @ A2 )
% 0.30/0.56 & ( P @ X ) ) )
% 0.30/0.56 @ A2 ) ).
% 0.30/0.56
% 0.30/0.56 % Collect_subset
% 0.30/0.56 thf(fact_3253_Collect__subset,axiom,
% 0.30/0.56 ! [A2: set_complex,P: complex > $o] :
% 0.30/0.56 ( ord_le211207098394363844omplex
% 0.30/0.56 @ ( collect_complex
% 0.30/0.56 @ ^ [X: complex] :
% 0.30/0.56 ( ( member_complex @ X @ A2 )
% 0.30/0.56 & ( P @ X ) ) )
% 0.30/0.56 @ A2 ) ).
% 0.30/0.56
% 0.30/0.56 % Collect_subset
% 0.30/0.56 thf(fact_3254_Collect__subset,axiom,
% 0.30/0.56 ! [A2: set_real,P: real > $o] :
% 0.30/0.56 ( ord_less_eq_set_real
% 0.30/0.56 @ ( collect_real
% 0.30/0.56 @ ^ [X: real] :
% 0.30/0.56 ( ( member_real @ X @ A2 )
% 0.30/0.56 & ( P @ X ) ) )
% 0.30/0.56 @ A2 ) ).
% 0.30/0.56
% 0.30/0.56 % Collect_subset
% 0.30/0.56 thf(fact_3255_Collect__subset,axiom,
% 0.30/0.56 ! [A2: set_list_nat,P: list_nat > $o] :
% 0.30/0.56 ( ord_le6045566169113846134st_nat
% 0.30/0.56 @ ( collect_list_nat
% 0.30/0.56 @ ^ [X: list_nat] :
% 0.30/0.56 ( ( member_list_nat @ X @ A2 )
% 0.30/0.56 & ( P @ X ) ) )
% 0.30/0.56 @ A2 ) ).
% 0.30/0.56
% 0.30/0.56 % Collect_subset
% 0.30/0.56 thf(fact_3256_Collect__subset,axiom,
% 0.30/0.56 ! [A2: set_nat,P: nat > $o] :
% 0.30/0.56 ( ord_less_eq_set_nat
% 0.30/0.56 @ ( collect_nat
% 0.30/0.56 @ ^ [X: nat] :
% 0.30/0.56 ( ( member_nat @ X @ A2 )
% 0.30/0.56 & ( P @ X ) ) )
% 0.30/0.56 @ A2 ) ).
% 0.30/0.56
% 0.30/0.56 % Collect_subset
% 0.30/0.56 thf(fact_3257_Collect__subset,axiom,
% 0.30/0.56 ! [A2: set_int,P: int > $o] :
% 0.30/0.56 ( ord_less_eq_set_int
% 0.30/0.56 @ ( collect_int
% 0.30/0.56 @ ^ [X: int] :
% 0.30/0.56 ( ( member_int @ X @ A2 )
% 0.30/0.56 & ( P @ X ) ) )
% 0.30/0.56 @ A2 ) ).
% 0.30/0.56
% 0.30/0.56 % Collect_subset
% 0.30/0.56 thf(fact_3258_Suc__times__mod__eq,axiom,
% 0.30/0.56 ! [M: nat,N: nat] :
% 0.30/0.56 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.30/0.56 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
% 0.30/0.56 = one_one_nat ) ) ).
% 0.30/0.56
% 0.30/0.56 % Suc_times_mod_eq
% 0.30/0.56 thf(fact_3259_divmod__digit__0_I2_J,axiom,
% 0.30/0.56 ! [B: nat,A: nat] :
% 0.30/0.56 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.56 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.56 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
% 0.30/0.56 = ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % divmod_digit_0(2)
% 0.30/0.56 thf(fact_3260_divmod__digit__0_I2_J,axiom,
% 0.30/0.56 ! [B: int,A: int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.56 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.56 => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
% 0.30/0.56 = ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % divmod_digit_0(2)
% 0.30/0.56 thf(fact_3261_divmod__digit__0_I2_J,axiom,
% 0.30/0.56 ! [B: code_integer,A: code_integer] :
% 0.30/0.56 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.56 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.56 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
% 0.30/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % divmod_digit_0(2)
% 0.30/0.56 thf(fact_3262_bits__stable__imp__add__self,axiom,
% 0.30/0.56 ! [A: nat] :
% 0.30/0.56 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56 = A )
% 0.30/0.56 => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.56 = zero_zero_nat ) ) ).
% 0.30/0.56
% 0.30/0.56 % bits_stable_imp_add_self
% 0.30/0.56 thf(fact_3263_bits__stable__imp__add__self,axiom,
% 0.30/0.56 ! [A: int] :
% 0.30/0.56 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.56 = A )
% 0.30/0.56 => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.30/0.56 = zero_zero_int ) ) ).
% 0.30/0.56
% 0.30/0.56 % bits_stable_imp_add_self
% 0.30/0.56 thf(fact_3264_bits__stable__imp__add__self,axiom,
% 0.30/0.56 ! [A: code_integer] :
% 0.30/0.56 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.56 = A )
% 0.30/0.56 => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.30/0.56 = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.56
% 0.30/0.56 % bits_stable_imp_add_self
% 0.30/0.56 thf(fact_3265_div__exp__mod__exp__eq,axiom,
% 0.30/0.56 ! [A: nat,N: nat,M: nat] :
% 0.30/0.56 ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.56 = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % div_exp_mod_exp_eq
% 0.30/0.56 thf(fact_3266_div__exp__mod__exp__eq,axiom,
% 0.30/0.56 ! [A: int,N: nat,M: nat] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.56 = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % div_exp_mod_exp_eq
% 0.30/0.56 thf(fact_3267_div__exp__mod__exp__eq,axiom,
% 0.30/0.56 ! [A: code_integer,N: nat,M: nat] :
% 0.30/0.56 ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.56 = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % div_exp_mod_exp_eq
% 0.30/0.56 thf(fact_3268_divmod__digit__0_I1_J,axiom,
% 0.30/0.56 ! [B: nat,A: nat] :
% 0.30/0.56 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.56 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.56 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.56 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % divmod_digit_0(1)
% 0.30/0.56 thf(fact_3269_divmod__digit__0_I1_J,axiom,
% 0.30/0.56 ! [B: int,A: int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.56 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.56 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.56 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % divmod_digit_0(1)
% 0.30/0.56 thf(fact_3270_divmod__digit__0_I1_J,axiom,
% 0.30/0.56 ! [B: code_integer,A: code_integer] :
% 0.30/0.56 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.56 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.56 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.56 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % divmod_digit_0(1)
% 0.30/0.56 thf(fact_3271_mult__exp__mod__exp__eq,axiom,
% 0.30/0.56 ! [M: nat,N: nat,A: nat] :
% 0.30/0.56 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.56 => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.56 = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mult_exp_mod_exp_eq
% 0.30/0.56 thf(fact_3272_mult__exp__mod__exp__eq,axiom,
% 0.30/0.56 ! [M: nat,N: nat,A: int] :
% 0.30/0.56 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.56 => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.56 = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mult_exp_mod_exp_eq
% 0.30/0.56 thf(fact_3273_mult__exp__mod__exp__eq,axiom,
% 0.30/0.56 ! [M: nat,N: nat,A: code_integer] :
% 0.30/0.56 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.56 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.56 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mult_exp_mod_exp_eq
% 0.30/0.56 thf(fact_3274_eucl__rel__int__iff,axiom,
% 0.30/0.56 ! [K: int,L: int,Q3: int,R2: int] :
% 0.30/0.56 ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.30/0.56 = ( ( K
% 0.30/0.56 = ( plus_plus_int @ ( times_times_int @ L @ Q3 ) @ R2 ) )
% 0.30/0.56 & ( ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.30/0.56 & ( ord_less_int @ R2 @ L ) ) )
% 0.30/0.56 & ( ~ ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.56 => ( ( ( ord_less_int @ L @ zero_zero_int )
% 0.30/0.56 => ( ( ord_less_int @ L @ R2 )
% 0.30/0.56 & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
% 0.30/0.56 & ( ~ ( ord_less_int @ L @ zero_zero_int )
% 0.30/0.56 => ( Q3 = zero_zero_int ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % eucl_rel_int_iff
% 0.30/0.56 thf(fact_3275_mod__double__modulus,axiom,
% 0.30/0.56 ! [M: code_integer,X2: code_integer] :
% 0.30/0.56 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 0.30/0.56 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X2 )
% 0.30/0.56 => ( ( ( modulo364778990260209775nteger @ X2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.56 = ( modulo364778990260209775nteger @ X2 @ M ) )
% 0.30/0.56 | ( ( modulo364778990260209775nteger @ X2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.56 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X2 @ M ) @ M ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_double_modulus
% 0.30/0.56 thf(fact_3276_mod__double__modulus,axiom,
% 0.30/0.56 ! [M: nat,X2: nat] :
% 0.30/0.56 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.56 => ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
% 0.30/0.56 => ( ( ( modulo_modulo_nat @ X2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.56 = ( modulo_modulo_nat @ X2 @ M ) )
% 0.30/0.56 | ( ( modulo_modulo_nat @ X2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.56 = ( plus_plus_nat @ ( modulo_modulo_nat @ X2 @ M ) @ M ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_double_modulus
% 0.30/0.56 thf(fact_3277_mod__double__modulus,axiom,
% 0.30/0.56 ! [M: int,X2: int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ M )
% 0.30/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.56 => ( ( ( modulo_modulo_int @ X2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.56 = ( modulo_modulo_int @ X2 @ M ) )
% 0.30/0.56 | ( ( modulo_modulo_int @ X2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.56 = ( plus_plus_int @ ( modulo_modulo_int @ X2 @ M ) @ M ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_double_modulus
% 0.30/0.56 thf(fact_3278_divmod__digit__1_I2_J,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.56 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.56 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.56 => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.56 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % divmod_digit_1(2)
% 0.30/0.56 thf(fact_3279_divmod__digit__1_I2_J,axiom,
% 0.30/0.56 ! [A: nat,B: nat] :
% 0.30/0.56 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.56 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.56 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.56 => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.56 = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % divmod_digit_1(2)
% 0.30/0.56 thf(fact_3280_divmod__digit__1_I2_J,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.56 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.56 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.56 => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.56 = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % divmod_digit_1(2)
% 0.30/0.56 thf(fact_3281_unset__bit__Suc,axiom,
% 0.30/0.56 ! [N: nat,A: code_integer] :
% 0.30/0.56 ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
% 0.30/0.56 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % unset_bit_Suc
% 0.30/0.56 thf(fact_3282_unset__bit__Suc,axiom,
% 0.30/0.56 ! [N: nat,A: int] :
% 0.30/0.56 ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
% 0.30/0.56 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % unset_bit_Suc
% 0.30/0.56 thf(fact_3283_unset__bit__Suc,axiom,
% 0.30/0.56 ! [N: nat,A: nat] :
% 0.30/0.56 ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
% 0.30/0.56 = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % unset_bit_Suc
% 0.30/0.56 thf(fact_3284_set__bit__Suc,axiom,
% 0.30/0.56 ! [N: nat,A: code_integer] :
% 0.30/0.56 ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
% 0.30/0.56 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_bit_Suc
% 0.30/0.56 thf(fact_3285_set__bit__Suc,axiom,
% 0.30/0.56 ! [N: nat,A: int] :
% 0.30/0.56 ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
% 0.30/0.56 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_bit_Suc
% 0.30/0.56 thf(fact_3286_set__bit__Suc,axiom,
% 0.30/0.56 ! [N: nat,A: nat] :
% 0.30/0.56 ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
% 0.30/0.56 = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_bit_Suc
% 0.30/0.56 thf(fact_3287_divmod__digit__1_I1_J,axiom,
% 0.30/0.56 ! [A: code_integer,B: code_integer] :
% 0.30/0.56 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.56 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.56 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.56 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
% 0.30/0.56 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % divmod_digit_1(1)
% 0.30/0.56 thf(fact_3288_divmod__digit__1_I1_J,axiom,
% 0.30/0.56 ! [A: nat,B: nat] :
% 0.30/0.56 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.56 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.56 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.56 => ( ( 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 )
% 0.30/0.56 = ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % divmod_digit_1(1)
% 0.30/0.56 thf(fact_3289_divmod__digit__1_I1_J,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.56 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.56 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.56 => ( ( 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 )
% 0.30/0.56 = ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % divmod_digit_1(1)
% 0.30/0.56 thf(fact_3290_pos__eucl__rel__int__mult__2,axiom,
% 0.30/0.56 ! [B: int,A: int,Q3: int,R2: int] :
% 0.30/0.56 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.56 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.30/0.56 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q3 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % pos_eucl_rel_int_mult_2
% 0.30/0.56 thf(fact_3291_verit__le__mono__div,axiom,
% 0.30/0.56 ! [A2: nat,B3: nat,N: nat] :
% 0.30/0.56 ( ( ord_less_nat @ A2 @ B3 )
% 0.30/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.56 => ( ord_less_eq_nat
% 0.30/0.56 @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N )
% 0.30/0.56 @ ( if_nat
% 0.30/0.56 @ ( ( modulo_modulo_nat @ B3 @ N )
% 0.30/0.56 = zero_zero_nat )
% 0.30/0.56 @ one_one_nat
% 0.30/0.56 @ zero_zero_nat ) )
% 0.30/0.56 @ ( divide_divide_nat @ B3 @ N ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % verit_le_mono_div
% 0.30/0.56 thf(fact_3292_div__mod__decomp,axiom,
% 0.30/0.56 ! [A2: nat,N: nat] :
% 0.30/0.56 ( A2
% 0.30/0.56 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N ) @ N ) @ ( modulo_modulo_nat @ A2 @ N ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % div_mod_decomp
% 0.30/0.56 thf(fact_3293_flip__bit__Suc,axiom,
% 0.30/0.56 ! [N: nat,A: code_integer] :
% 0.30/0.56 ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
% 0.30/0.56 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % flip_bit_Suc
% 0.30/0.56 thf(fact_3294_flip__bit__Suc,axiom,
% 0.30/0.56 ! [N: nat,A: int] :
% 0.30/0.56 ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
% 0.30/0.56 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % flip_bit_Suc
% 0.30/0.56 thf(fact_3295_flip__bit__Suc,axiom,
% 0.30/0.56 ! [N: nat,A: nat] :
% 0.30/0.56 ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
% 0.30/0.56 = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % flip_bit_Suc
% 0.30/0.56 thf(fact_3296_div__less__mono,axiom,
% 0.30/0.56 ! [A2: nat,B3: nat,N: nat] :
% 0.30/0.56 ( ( ord_less_nat @ A2 @ B3 )
% 0.30/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.56 => ( ( ( modulo_modulo_nat @ A2 @ N )
% 0.30/0.56 = zero_zero_nat )
% 0.30/0.56 => ( ( ( modulo_modulo_nat @ B3 @ N )
% 0.30/0.56 = zero_zero_nat )
% 0.30/0.56 => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B3 @ N ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % div_less_mono
% 0.30/0.56 thf(fact_3297_product__nth,axiom,
% 0.30/0.56 ! [N: nat,Xs2: list_num,Ys: list_num] :
% 0.30/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_num @ Xs2 ) @ ( size_size_list_num @ Ys ) ) )
% 0.30/0.56 => ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs2 @ Ys ) @ N )
% 0.30/0.56 = ( product_Pair_num_num @ ( nth_num @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys ) ) ) @ ( nth_num @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % product_nth
% 0.30/0.56 thf(fact_3298_product__nth,axiom,
% 0.30/0.56 ! [N: nat,Xs2: list_Code_integer,Ys: list_o] :
% 0.30/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s3445333598471063425nteger @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
% 0.30/0.56 => ( ( nth_Pr8522763379788166057eger_o @ ( produc3607205314601156340eger_o @ Xs2 @ Ys ) @ N )
% 0.30/0.56 = ( produc6677183202524767010eger_o @ ( nth_Code_integer @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % product_nth
% 0.30/0.56 thf(fact_3299_product__nth,axiom,
% 0.30/0.56 ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.30/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 0.30/0.56 => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) @ N )
% 0.30/0.56 = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % product_nth
% 0.30/0.56 thf(fact_3300_product__nth,axiom,
% 0.30/0.56 ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_o] :
% 0.30/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
% 0.30/0.56 => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) @ N )
% 0.30/0.56 = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % product_nth
% 0.30/0.56 thf(fact_3301_product__nth,axiom,
% 0.30/0.56 ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_nat] :
% 0.30/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
% 0.30/0.56 => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) @ N )
% 0.30/0.56 = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % product_nth
% 0.30/0.56 thf(fact_3302_product__nth,axiom,
% 0.30/0.56 ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_int] :
% 0.30/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
% 0.30/0.56 => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys ) @ N )
% 0.30/0.56 = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % product_nth
% 0.30/0.56 thf(fact_3303_product__nth,axiom,
% 0.30/0.56 ! [N: nat,Xs2: list_o,Ys: list_VEBT_VEBT] :
% 0.30/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 0.30/0.56 => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys ) @ N )
% 0.30/0.56 = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % product_nth
% 0.30/0.56 thf(fact_3304_product__nth,axiom,
% 0.30/0.56 ! [N: nat,Xs2: list_o,Ys: list_o] :
% 0.30/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
% 0.30/0.56 => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs2 @ Ys ) @ N )
% 0.30/0.56 = ( product_Pair_o_o @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % product_nth
% 0.30/0.56 thf(fact_3305_product__nth,axiom,
% 0.30/0.56 ! [N: nat,Xs2: list_o,Ys: list_nat] :
% 0.30/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
% 0.30/0.56 => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs2 @ Ys ) @ N )
% 0.30/0.56 = ( product_Pair_o_nat @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % product_nth
% 0.30/0.56 thf(fact_3306_product__nth,axiom,
% 0.30/0.56 ! [N: nat,Xs2: list_o,Ys: list_int] :
% 0.30/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
% 0.30/0.56 => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs2 @ Ys ) @ N )
% 0.30/0.56 = ( product_Pair_o_int @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % product_nth
% 0.30/0.56 thf(fact_3307_obtain__set__succ,axiom,
% 0.30/0.56 ! [X2: nat,Z: nat,A2: set_nat,B3: set_nat] :
% 0.30/0.56 ( ( ord_less_nat @ X2 @ Z )
% 0.30/0.56 => ( ( vEBT_VEBT_max_in_set @ A2 @ Z )
% 0.30/0.56 => ( ( finite_finite_nat @ B3 )
% 0.30/0.56 => ( ( A2 = B3 )
% 0.30/0.56 => ? [X_12: nat] : ( vEBT_is_succ_in_set @ A2 @ X2 @ X_12 ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % obtain_set_succ
% 0.30/0.56 thf(fact_3308_obtain__set__pred,axiom,
% 0.30/0.56 ! [Z: nat,X2: nat,A2: set_nat] :
% 0.30/0.56 ( ( ord_less_nat @ Z @ X2 )
% 0.30/0.56 => ( ( vEBT_VEBT_min_in_set @ A2 @ Z )
% 0.30/0.56 => ( ( finite_finite_nat @ A2 )
% 0.30/0.56 => ? [X_12: nat] : ( vEBT_is_pred_in_set @ A2 @ X2 @ X_12 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % obtain_set_pred
% 0.30/0.56 thf(fact_3309_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
% 0.30/0.56 ! [A: $o,B: $o,X2: nat] :
% 0.30/0.56 ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Leaf @ A @ B ) @ X2 )
% 0.30/0.56 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( X2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
% 0.30/0.56 thf(fact_3310_set__vebt__finite,axiom,
% 0.30/0.56 ! [T: vEBT_VEBT,N: nat] :
% 0.30/0.56 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.56 => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % set_vebt_finite
% 0.30/0.56 thf(fact_3311_pred__none__empty,axiom,
% 0.30/0.56 ! [Xs2: set_nat,A: nat] :
% 0.30/0.56 ( ~ ? [X_12: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A @ X_12 )
% 0.30/0.56 => ( ( finite_finite_nat @ Xs2 )
% 0.30/0.56 => ~ ? [X3: nat] :
% 0.30/0.56 ( ( member_nat @ X3 @ Xs2 )
% 0.30/0.56 & ( ord_less_nat @ X3 @ A ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % pred_none_empty
% 0.30/0.56 thf(fact_3312_succ__none__empty,axiom,
% 0.30/0.56 ! [Xs2: set_nat,A: nat] :
% 0.30/0.56 ( ~ ? [X_12: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A @ X_12 )
% 0.30/0.56 => ( ( finite_finite_nat @ Xs2 )
% 0.30/0.56 => ~ ? [X3: nat] :
% 0.30/0.56 ( ( member_nat @ X3 @ Xs2 )
% 0.30/0.56 & ( ord_less_nat @ A @ X3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % succ_none_empty
% 0.30/0.56 thf(fact_3313_verit__eq__simplify_I8_J,axiom,
% 0.30/0.56 ! [X22: num,Y2: num] :
% 0.30/0.56 ( ( ( bit0 @ X22 )
% 0.30/0.56 = ( bit0 @ Y2 ) )
% 0.30/0.56 = ( X22 = Y2 ) ) ).
% 0.30/0.56
% 0.30/0.56 % verit_eq_simplify(8)
% 0.30/0.56 thf(fact_3314_List_Ofinite__set,axiom,
% 0.30/0.56 ! [Xs2: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) ) ).
% 0.30/0.56
% 0.30/0.56 % List.finite_set
% 0.30/0.56 thf(fact_3315_List_Ofinite__set,axiom,
% 0.30/0.56 ! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).
% 0.30/0.56
% 0.30/0.56 % List.finite_set
% 0.30/0.56 thf(fact_3316_List_Ofinite__set,axiom,
% 0.30/0.56 ! [Xs2: list_int] : ( finite_finite_int @ ( set_int2 @ Xs2 ) ) ).
% 0.30/0.56
% 0.30/0.56 % List.finite_set
% 0.30/0.56 thf(fact_3317_List_Ofinite__set,axiom,
% 0.30/0.56 ! [Xs2: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs2 ) ) ).
% 0.30/0.56
% 0.30/0.56 % List.finite_set
% 0.30/0.56 thf(fact_3318_flip__bit__nonnegative__int__iff,axiom,
% 0.30/0.56 ! [N: nat,K: int] :
% 0.30/0.56 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
% 0.30/0.56 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.30/0.56
% 0.30/0.56 % flip_bit_nonnegative_int_iff
% 0.30/0.56 thf(fact_3319_flip__bit__negative__int__iff,axiom,
% 0.30/0.56 ! [N: nat,K: int] :
% 0.30/0.56 ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
% 0.30/0.56 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.56
% 0.30/0.56 % flip_bit_negative_int_iff
% 0.30/0.56 thf(fact_3320_infinite__Icc__iff,axiom,
% 0.30/0.56 ! [A: rat,B: rat] :
% 0.30/0.56 ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
% 0.30/0.56 = ( ord_less_rat @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % infinite_Icc_iff
% 0.30/0.56 thf(fact_3321_infinite__Icc__iff,axiom,
% 0.30/0.56 ! [A: real,B: real] :
% 0.30/0.56 ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
% 0.30/0.56 = ( ord_less_real @ A @ B ) ) ).
% 0.30/0.56
% 0.30/0.56 % infinite_Icc_iff
% 0.30/0.56 thf(fact_3322_mod__neg__neg__trivial,axiom,
% 0.30/0.56 ! [K: int,L: int] :
% 0.30/0.56 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.30/0.56 => ( ( ord_less_int @ L @ K )
% 0.30/0.56 => ( ( modulo_modulo_int @ K @ L )
% 0.30/0.56 = K ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_neg_neg_trivial
% 0.30/0.56 thf(fact_3323_mod__pos__pos__trivial,axiom,
% 0.30/0.56 ! [K: int,L: int] :
% 0.30/0.56 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.56 => ( ( ord_less_int @ K @ L )
% 0.30/0.56 => ( ( modulo_modulo_int @ K @ L )
% 0.30/0.56 = K ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % mod_pos_pos_trivial
% 0.30/0.56 thf(fact_3324_length__product,axiom,
% 0.30/0.56 ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.30/0.56 ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) )
% 0.30/0.56 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % length_product
% 0.30/0.56 thf(fact_3325_length__product,axiom,
% 0.30/0.56 ! [Xs2: list_VEBT_VEBT,Ys: list_o] :
% 0.30/0.56 ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) )
% 0.30/0.56 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % length_product
% 0.30/0.56 thf(fact_3326_length__product,axiom,
% 0.30/0.56 ! [Xs2: list_VEBT_VEBT,Ys: list_nat] :
% 0.30/0.56 ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) )
% 0.30/0.56 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % length_product
% 0.30/0.56 thf(fact_3327_length__product,axiom,
% 0.30/0.56 ! [Xs2: list_VEBT_VEBT,Ys: list_int] :
% 0.30/0.56 ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys ) )
% 0.30/0.56 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % length_product
% 0.30/0.56 thf(fact_3328_length__product,axiom,
% 0.30/0.56 ! [Xs2: list_o,Ys: list_VEBT_VEBT] :
% 0.30/0.56 ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys ) )
% 0.30/0.56 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % length_product
% 0.30/0.56 thf(fact_3329_length__product,axiom,
% 0.30/0.56 ! [Xs2: list_o,Ys: list_o] :
% 0.30/0.56 ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs2 @ Ys ) )
% 0.30/0.56 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % length_product
% 0.30/0.56 thf(fact_3330_length__product,axiom,
% 0.30/0.56 ! [Xs2: list_o,Ys: list_nat] :
% 0.30/0.56 ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs2 @ Ys ) )
% 0.30/0.56 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % length_product
% 0.30/0.56 thf(fact_3331_length__product,axiom,
% 0.30/0.56 ! [Xs2: list_o,Ys: list_int] :
% 0.30/0.56 ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs2 @ Ys ) )
% 0.30/0.56 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % length_product
% 0.30/0.56 thf(fact_3332_length__product,axiom,
% 0.30/0.56 ! [Xs2: list_nat,Ys: list_VEBT_VEBT] :
% 0.30/0.56 ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys ) )
% 0.30/0.56 = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % length_product
% 0.30/0.56 thf(fact_3333_length__product,axiom,
% 0.30/0.56 ! [Xs2: list_nat,Ys: list_o] :
% 0.30/0.56 ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs2 @ Ys ) )
% 0.30/0.56 = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % length_product
% 0.30/0.56 thf(fact_3334_zmod__numeral__Bit0,axiom,
% 0.30/0.56 ! [V: num,W: num] :
% 0.30/0.56 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.30/0.56 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % zmod_numeral_Bit0
% 0.30/0.56 thf(fact_3335_bounded__nat__set__is__finite,axiom,
% 0.30/0.56 ! [N5: set_nat,N: nat] :
% 0.30/0.56 ( ! [X5: nat] :
% 0.30/0.56 ( ( member_nat @ X5 @ N5 )
% 0.30/0.56 => ( ord_less_nat @ X5 @ N ) )
% 0.30/0.56 => ( finite_finite_nat @ N5 ) ) ).
% 0.30/0.56
% 0.30/0.56 % bounded_nat_set_is_finite
% 0.30/0.56 thf(fact_3336_finite__nat__set__iff__bounded,axiom,
% 0.30/0.56 ( finite_finite_nat
% 0.30/0.56 = ( ^ [N6: set_nat] :
% 0.30/0.56 ? [M3: nat] :
% 0.30/0.56 ! [X: nat] :
% 0.30/0.56 ( ( member_nat @ X @ N6 )
% 0.30/0.56 => ( ord_less_nat @ X @ M3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_nat_set_iff_bounded
% 0.30/0.56 thf(fact_3337_finite__nat__set__iff__bounded__le,axiom,
% 0.30/0.56 ( finite_finite_nat
% 0.30/0.56 = ( ^ [N6: set_nat] :
% 0.30/0.56 ? [M3: nat] :
% 0.30/0.56 ! [X: nat] :
% 0.30/0.56 ( ( member_nat @ X @ N6 )
% 0.30/0.56 => ( ord_less_eq_nat @ X @ M3 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_nat_set_iff_bounded_le
% 0.30/0.56 thf(fact_3338_finite__list,axiom,
% 0.30/0.56 ! [A2: set_VEBT_VEBT] :
% 0.30/0.56 ( ( finite5795047828879050333T_VEBT @ A2 )
% 0.30/0.56 => ? [Xs3: list_VEBT_VEBT] :
% 0.30/0.56 ( ( set_VEBT_VEBT2 @ Xs3 )
% 0.30/0.56 = A2 ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_list
% 0.30/0.56 thf(fact_3339_finite__list,axiom,
% 0.30/0.56 ! [A2: set_nat] :
% 0.30/0.56 ( ( finite_finite_nat @ A2 )
% 0.30/0.56 => ? [Xs3: list_nat] :
% 0.30/0.56 ( ( set_nat2 @ Xs3 )
% 0.30/0.56 = A2 ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_list
% 0.30/0.56 thf(fact_3340_finite__list,axiom,
% 0.30/0.56 ! [A2: set_int] :
% 0.30/0.56 ( ( finite_finite_int @ A2 )
% 0.30/0.56 => ? [Xs3: list_int] :
% 0.30/0.56 ( ( set_int2 @ Xs3 )
% 0.30/0.56 = A2 ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_list
% 0.30/0.56 thf(fact_3341_finite__list,axiom,
% 0.30/0.56 ! [A2: set_complex] :
% 0.30/0.56 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.56 => ? [Xs3: list_complex] :
% 0.30/0.56 ( ( set_complex2 @ Xs3 )
% 0.30/0.56 = A2 ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_list
% 0.30/0.56 thf(fact_3342_finite__M__bounded__by__nat,axiom,
% 0.30/0.56 ! [P: nat > $o,I2: nat] :
% 0.30/0.56 ( finite_finite_nat
% 0.30/0.56 @ ( collect_nat
% 0.30/0.56 @ ^ [K2: nat] :
% 0.30/0.56 ( ( P @ K2 )
% 0.30/0.56 & ( ord_less_nat @ K2 @ I2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_M_bounded_by_nat
% 0.30/0.56 thf(fact_3343_finite__less__ub,axiom,
% 0.30/0.56 ! [F: nat > nat,U: nat] :
% 0.30/0.56 ( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
% 0.30/0.56 => ( finite_finite_nat
% 0.30/0.56 @ ( collect_nat
% 0.30/0.56 @ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_less_ub
% 0.30/0.56 thf(fact_3344_finite__lists__length__eq,axiom,
% 0.30/0.56 ! [A2: set_complex,N: nat] :
% 0.30/0.56 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.56 => ( finite8712137658972009173omplex
% 0.30/0.56 @ ( collect_list_complex
% 0.30/0.56 @ ^ [Xs: list_complex] :
% 0.30/0.56 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
% 0.30/0.56 & ( ( size_s3451745648224563538omplex @ Xs )
% 0.30/0.56 = N ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_lists_length_eq
% 0.30/0.56 thf(fact_3345_finite__lists__length__eq,axiom,
% 0.30/0.56 ! [A2: set_VEBT_VEBT,N: nat] :
% 0.30/0.56 ( ( finite5795047828879050333T_VEBT @ A2 )
% 0.30/0.56 => ( finite3004134309566078307T_VEBT
% 0.30/0.56 @ ( collec5608196760682091941T_VEBT
% 0.30/0.56 @ ^ [Xs: list_VEBT_VEBT] :
% 0.30/0.56 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
% 0.30/0.56 & ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.30/0.56 = N ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_lists_length_eq
% 0.30/0.56 thf(fact_3346_finite__lists__length__eq,axiom,
% 0.30/0.56 ! [A2: set_o,N: nat] :
% 0.30/0.56 ( ( finite_finite_o @ A2 )
% 0.30/0.56 => ( finite_finite_list_o
% 0.30/0.56 @ ( collect_list_o
% 0.30/0.56 @ ^ [Xs: list_o] :
% 0.30/0.56 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
% 0.30/0.56 & ( ( size_size_list_o @ Xs )
% 0.30/0.56 = N ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_lists_length_eq
% 0.30/0.56 thf(fact_3347_finite__lists__length__eq,axiom,
% 0.30/0.56 ! [A2: set_nat,N: nat] :
% 0.30/0.56 ( ( finite_finite_nat @ A2 )
% 0.30/0.56 => ( finite8100373058378681591st_nat
% 0.30/0.56 @ ( collect_list_nat
% 0.30/0.56 @ ^ [Xs: list_nat] :
% 0.30/0.56 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
% 0.30/0.56 & ( ( size_size_list_nat @ Xs )
% 0.30/0.56 = N ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_lists_length_eq
% 0.30/0.56 thf(fact_3348_finite__lists__length__eq,axiom,
% 0.30/0.56 ! [A2: set_int,N: nat] :
% 0.30/0.56 ( ( finite_finite_int @ A2 )
% 0.30/0.56 => ( finite3922522038869484883st_int
% 0.30/0.56 @ ( collect_list_int
% 0.30/0.56 @ ^ [Xs: list_int] :
% 0.30/0.56 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
% 0.30/0.56 & ( ( size_size_list_int @ Xs )
% 0.30/0.56 = N ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_lists_length_eq
% 0.30/0.56 thf(fact_3349_zmod__le__nonneg__dividend,axiom,
% 0.30/0.56 ! [M: int,K: int] :
% 0.30/0.56 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 0.30/0.56 => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% 0.30/0.56
% 0.30/0.56 % zmod_le_nonneg_dividend
% 0.30/0.56 thf(fact_3350_Euclidean__Division_Opos__mod__bound,axiom,
% 0.30/0.56 ! [L: int,K: int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.56 => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% 0.30/0.56
% 0.30/0.56 % Euclidean_Division.pos_mod_bound
% 0.30/0.56 thf(fact_3351_neg__mod__bound,axiom,
% 0.30/0.56 ! [L: int,K: int] :
% 0.30/0.56 ( ( ord_less_int @ L @ zero_zero_int )
% 0.30/0.56 => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % neg_mod_bound
% 0.30/0.56 thf(fact_3352_zmod__eq__0__iff,axiom,
% 0.30/0.56 ! [M: int,D: int] :
% 0.30/0.56 ( ( ( modulo_modulo_int @ M @ D )
% 0.30/0.56 = zero_zero_int )
% 0.30/0.56 = ( ? [Q4: int] :
% 0.30/0.56 ( M
% 0.30/0.56 = ( times_times_int @ D @ Q4 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % zmod_eq_0_iff
% 0.30/0.56 thf(fact_3353_zmod__eq__0D,axiom,
% 0.30/0.56 ! [M: int,D: int] :
% 0.30/0.56 ( ( ( modulo_modulo_int @ M @ D )
% 0.30/0.56 = zero_zero_int )
% 0.30/0.56 => ? [Q2: int] :
% 0.30/0.56 ( M
% 0.30/0.56 = ( times_times_int @ D @ Q2 ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % zmod_eq_0D
% 0.30/0.56 thf(fact_3354_infinite__Icc,axiom,
% 0.30/0.56 ! [A: rat,B: rat] :
% 0.30/0.56 ( ( ord_less_rat @ A @ B )
% 0.30/0.56 => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % infinite_Icc
% 0.30/0.56 thf(fact_3355_infinite__Icc,axiom,
% 0.30/0.56 ! [A: real,B: real] :
% 0.30/0.56 ( ( ord_less_real @ A @ B )
% 0.30/0.56 => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % infinite_Icc
% 0.30/0.56 thf(fact_3356_finite__lists__length__le,axiom,
% 0.30/0.56 ! [A2: set_complex,N: nat] :
% 0.30/0.56 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.56 => ( finite8712137658972009173omplex
% 0.30/0.56 @ ( collect_list_complex
% 0.30/0.56 @ ^ [Xs: list_complex] :
% 0.30/0.56 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
% 0.30/0.56 & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_lists_length_le
% 0.30/0.56 thf(fact_3357_finite__lists__length__le,axiom,
% 0.30/0.56 ! [A2: set_VEBT_VEBT,N: nat] :
% 0.30/0.56 ( ( finite5795047828879050333T_VEBT @ A2 )
% 0.30/0.56 => ( finite3004134309566078307T_VEBT
% 0.30/0.56 @ ( collec5608196760682091941T_VEBT
% 0.30/0.56 @ ^ [Xs: list_VEBT_VEBT] :
% 0.30/0.56 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
% 0.30/0.56 & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_lists_length_le
% 0.30/0.56 thf(fact_3358_finite__lists__length__le,axiom,
% 0.30/0.56 ! [A2: set_o,N: nat] :
% 0.30/0.56 ( ( finite_finite_o @ A2 )
% 0.30/0.56 => ( finite_finite_list_o
% 0.30/0.56 @ ( collect_list_o
% 0.30/0.56 @ ^ [Xs: list_o] :
% 0.30/0.56 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
% 0.30/0.56 & ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_lists_length_le
% 0.30/0.56 thf(fact_3359_finite__lists__length__le,axiom,
% 0.30/0.56 ! [A2: set_nat,N: nat] :
% 0.30/0.56 ( ( finite_finite_nat @ A2 )
% 0.30/0.56 => ( finite8100373058378681591st_nat
% 0.30/0.56 @ ( collect_list_nat
% 0.30/0.56 @ ^ [Xs: list_nat] :
% 0.30/0.56 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
% 0.30/0.56 & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_lists_length_le
% 0.30/0.56 thf(fact_3360_finite__lists__length__le,axiom,
% 0.30/0.56 ! [A2: set_int,N: nat] :
% 0.30/0.56 ( ( finite_finite_int @ A2 )
% 0.30/0.56 => ( finite3922522038869484883st_int
% 0.30/0.56 @ ( collect_list_int
% 0.30/0.56 @ ^ [Xs: list_int] :
% 0.30/0.56 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
% 0.30/0.56 & ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % finite_lists_length_le
% 0.30/0.56 thf(fact_3361_div__mod__decomp__int,axiom,
% 0.30/0.56 ! [A2: int,N: int] :
% 0.30/0.56 ( A2
% 0.30/0.56 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N ) @ N ) @ ( modulo_modulo_int @ A2 @ N ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % div_mod_decomp_int
% 0.30/0.56 thf(fact_3362_neg__mod__conj,axiom,
% 0.30/0.56 ! [B: int,A: int] :
% 0.30/0.56 ( ( ord_less_int @ B @ zero_zero_int )
% 0.30/0.56 => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
% 0.30/0.56 & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % neg_mod_conj
% 0.30/0.56 thf(fact_3363_pos__mod__conj,axiom,
% 0.30/0.56 ! [B: int,A: int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.56 & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % pos_mod_conj
% 0.30/0.56 thf(fact_3364_zmod__trivial__iff,axiom,
% 0.30/0.56 ! [I2: int,K: int] :
% 0.30/0.56 ( ( ( modulo_modulo_int @ I2 @ K )
% 0.30/0.56 = I2 )
% 0.30/0.56 = ( ( K = zero_zero_int )
% 0.30/0.56 | ( ( ord_less_eq_int @ zero_zero_int @ I2 )
% 0.30/0.56 & ( ord_less_int @ I2 @ K ) )
% 0.30/0.56 | ( ( ord_less_eq_int @ I2 @ zero_zero_int )
% 0.30/0.56 & ( ord_less_int @ K @ I2 ) ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % zmod_trivial_iff
% 0.30/0.56 thf(fact_3365_neg__mod__sign,axiom,
% 0.30/0.56 ! [L: int,K: int] :
% 0.30/0.56 ( ( ord_less_int @ L @ zero_zero_int )
% 0.30/0.56 => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% 0.30/0.56
% 0.30/0.56 % neg_mod_sign
% 0.30/0.56 thf(fact_3366_Euclidean__Division_Opos__mod__sign,axiom,
% 0.30/0.56 ! [L: int,K: int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.56 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.30/0.56
% 0.30/0.56 % Euclidean_Division.pos_mod_sign
% 0.30/0.56 thf(fact_3367_verit__la__disequality,axiom,
% 0.30/0.56 ! [A: rat,B: rat] :
% 0.30/0.56 ( ( A = B )
% 0.30/0.56 | ~ ( ord_less_eq_rat @ A @ B )
% 0.30/0.56 | ~ ( ord_less_eq_rat @ B @ A ) ) ).
% 0.30/0.56
% 0.30/0.56 % verit_la_disequality
% 0.30/0.56 thf(fact_3368_verit__la__disequality,axiom,
% 0.30/0.56 ! [A: num,B: num] :
% 0.30/0.56 ( ( A = B )
% 0.30/0.56 | ~ ( ord_less_eq_num @ A @ B )
% 0.30/0.56 | ~ ( ord_less_eq_num @ B @ A ) ) ).
% 0.30/0.56
% 0.30/0.56 % verit_la_disequality
% 0.30/0.56 thf(fact_3369_verit__la__disequality,axiom,
% 0.30/0.56 ! [A: nat,B: nat] :
% 0.30/0.56 ( ( A = B )
% 0.30/0.56 | ~ ( ord_less_eq_nat @ A @ B )
% 0.30/0.56 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 0.30/0.56
% 0.30/0.56 % verit_la_disequality
% 0.30/0.56 thf(fact_3370_verit__la__disequality,axiom,
% 0.30/0.56 ! [A: int,B: int] :
% 0.30/0.56 ( ( A = B )
% 0.30/0.56 | ~ ( ord_less_eq_int @ A @ B )
% 0.30/0.56 | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 0.30/0.56
% 0.30/0.56 % verit_la_disequality
% 0.30/0.56 thf(fact_3371_verit__comp__simplify1_I2_J,axiom,
% 0.30/0.56 ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 0.30/0.56
% 0.30/0.56 % verit_comp_simplify1(2)
% 0.30/0.56 thf(fact_3372_verit__comp__simplify1_I2_J,axiom,
% 0.30/0.56 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 0.30/0.56
% 0.30/0.56 % verit_comp_simplify1(2)
% 0.30/0.56 thf(fact_3373_verit__comp__simplify1_I2_J,axiom,
% 0.30/0.56 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 0.30/0.56
% 0.30/0.56 % verit_comp_simplify1(2)
% 0.30/0.56 thf(fact_3374_verit__comp__simplify1_I2_J,axiom,
% 0.30/0.56 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.30/0.56
% 0.30/0.56 % verit_comp_simplify1(2)
% 0.30/0.56 thf(fact_3375_verit__comp__simplify1_I2_J,axiom,
% 0.30/0.56 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.30/0.56
% 0.30/0.56 % verit_comp_simplify1(2)
% 0.30/0.56 thf(fact_3376_verit__comp__simplify1_I1_J,axiom,
% 0.30/0.56 ! [A: real] :
% 0.30/0.56 ~ ( ord_less_real @ A @ A ) ).
% 0.30/0.56
% 0.30/0.56 % verit_comp_simplify1(1)
% 0.30/0.56 thf(fact_3377_verit__comp__simplify1_I1_J,axiom,
% 0.30/0.56 ! [A: rat] :
% 0.30/0.56 ~ ( ord_less_rat @ A @ A ) ).
% 0.30/0.56
% 0.30/0.56 % verit_comp_simplify1(1)
% 0.30/0.56 thf(fact_3378_verit__comp__simplify1_I1_J,axiom,
% 0.30/0.56 ! [A: num] :
% 0.30/0.56 ~ ( ord_less_num @ A @ A ) ).
% 0.30/0.56
% 0.30/0.56 % verit_comp_simplify1(1)
% 0.30/0.56 thf(fact_3379_verit__comp__simplify1_I1_J,axiom,
% 0.30/0.56 ! [A: nat] :
% 0.30/0.56 ~ ( ord_less_nat @ A @ A ) ).
% 0.30/0.56
% 0.30/0.56 % verit_comp_simplify1(1)
% 0.30/0.56 thf(fact_3380_verit__comp__simplify1_I1_J,axiom,
% 0.30/0.56 ! [A: int] :
% 0.30/0.56 ~ ( ord_less_int @ A @ A ) ).
% 0.30/0.56
% 0.30/0.56 % verit_comp_simplify1(1)
% 0.30/0.56 thf(fact_3381_verit__la__generic,axiom,
% 0.30/0.56 ! [A: int,X2: int] :
% 0.30/0.56 ( ( ord_less_eq_int @ A @ X2 )
% 0.30/0.56 | ( A = X2 )
% 0.30/0.56 | ( ord_less_eq_int @ X2 @ A ) ) ).
% 0.30/0.56
% 0.30/0.56 % verit_la_generic
% 0.30/0.56 thf(fact_3382_subset__eq__atLeast0__atMost__finite,axiom,
% 0.30/0.56 ! [N5: set_nat,N: nat] :
% 0.30/0.56 ( ( ord_less_eq_set_nat @ N5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.56 => ( finite_finite_nat @ N5 ) ) ).
% 0.30/0.56
% 0.30/0.56 % subset_eq_atLeast0_atMost_finite
% 0.30/0.56 thf(fact_3383_mod__pos__neg__trivial,axiom,
% 0.30/0.56 ! [K: int,L: int] :
% 0.30/0.56 ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.56 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 0.30/0.56 => ( ( modulo_modulo_int @ K @ L )
% 0.30/0.56 = ( plus_plus_int @ K @ L ) ) ) ) ).
% 0.30/0.56
% 0.30/0.57 % mod_pos_neg_trivial
% 0.30/0.57 thf(fact_3384_mod__pos__geq,axiom,
% 0.30/0.57 ! [L: int,K: int] :
% 0.30/0.57 ( ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.57 => ( ( ord_less_eq_int @ L @ K )
% 0.30/0.57 => ( ( modulo_modulo_int @ K @ L )
% 0.30/0.57 = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % mod_pos_geq
% 0.30/0.57 thf(fact_3385_verit__le__mono__div__int,axiom,
% 0.30/0.57 ! [A2: int,B3: int,N: int] :
% 0.30/0.57 ( ( ord_less_int @ A2 @ B3 )
% 0.30/0.57 => ( ( ord_less_int @ zero_zero_int @ N )
% 0.30/0.57 => ( ord_less_eq_int
% 0.30/0.57 @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N )
% 0.30/0.57 @ ( if_int
% 0.30/0.57 @ ( ( modulo_modulo_int @ B3 @ N )
% 0.30/0.57 = zero_zero_int )
% 0.30/0.57 @ one_one_int
% 0.30/0.57 @ zero_zero_int ) )
% 0.30/0.57 @ ( divide_divide_int @ B3 @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % verit_le_mono_div_int
% 0.30/0.57 thf(fact_3386_int__mod__pos__eq,axiom,
% 0.30/0.57 ! [A: int,B: int,Q3: int,R2: int] :
% 0.30/0.57 ( ( A
% 0.30/0.57 = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 0.30/0.57 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.30/0.57 => ( ( ord_less_int @ R2 @ B )
% 0.30/0.57 => ( ( modulo_modulo_int @ A @ B )
% 0.30/0.57 = R2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % int_mod_pos_eq
% 0.30/0.57 thf(fact_3387_int__mod__neg__eq,axiom,
% 0.30/0.57 ! [A: int,B: int,Q3: int,R2: int] :
% 0.30/0.57 ( ( A
% 0.30/0.57 = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 0.30/0.57 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.30/0.57 => ( ( ord_less_int @ B @ R2 )
% 0.30/0.57 => ( ( modulo_modulo_int @ A @ B )
% 0.30/0.57 = R2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % int_mod_neg_eq
% 0.30/0.57 thf(fact_3388_split__zmod,axiom,
% 0.30/0.57 ! [P: int > $o,N: int,K: int] :
% 0.30/0.57 ( ( P @ ( modulo_modulo_int @ N @ K ) )
% 0.30/0.57 = ( ( ( K = zero_zero_int )
% 0.30/0.57 => ( P @ N ) )
% 0.30/0.57 & ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.57 => ! [I4: int,J3: int] :
% 0.30/0.57 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.30/0.57 & ( ord_less_int @ J3 @ K )
% 0.30/0.57 & ( N
% 0.30/0.57 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.30/0.57 => ( P @ J3 ) ) )
% 0.30/0.57 & ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.57 => ! [I4: int,J3: int] :
% 0.30/0.57 ( ( ( ord_less_int @ K @ J3 )
% 0.30/0.57 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.30/0.57 & ( N
% 0.30/0.57 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.30/0.57 => ( P @ J3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % split_zmod
% 0.30/0.57 thf(fact_3389_zmod__zmult2__eq,axiom,
% 0.30/0.57 ! [C: int,A: int,B: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.57 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 0.30/0.57 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % zmod_zmult2_eq
% 0.30/0.57 thf(fact_3390_split__pos__lemma,axiom,
% 0.30/0.57 ! [K: int,P: int > int > $o,N: int] :
% 0.30/0.57 ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.57 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 0.30/0.57 = ( ! [I4: int,J3: int] :
% 0.30/0.57 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.30/0.57 & ( ord_less_int @ J3 @ K )
% 0.30/0.57 & ( N
% 0.30/0.57 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.30/0.57 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % split_pos_lemma
% 0.30/0.57 thf(fact_3391_split__neg__lemma,axiom,
% 0.30/0.57 ! [K: int,P: int > int > $o,N: int] :
% 0.30/0.57 ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.57 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 0.30/0.57 = ( ! [I4: int,J3: int] :
% 0.30/0.57 ( ( ( ord_less_int @ K @ J3 )
% 0.30/0.57 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.30/0.57 & ( N
% 0.30/0.57 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.30/0.57 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % split_neg_lemma
% 0.30/0.57 thf(fact_3392_verit__comp__simplify1_I3_J,axiom,
% 0.30/0.57 ! [B5: real,A5: real] :
% 0.30/0.57 ( ( ~ ( ord_less_eq_real @ B5 @ A5 ) )
% 0.30/0.57 = ( ord_less_real @ A5 @ B5 ) ) ).
% 0.30/0.57
% 0.30/0.57 % verit_comp_simplify1(3)
% 0.30/0.57 thf(fact_3393_verit__comp__simplify1_I3_J,axiom,
% 0.30/0.57 ! [B5: rat,A5: rat] :
% 0.30/0.57 ( ( ~ ( ord_less_eq_rat @ B5 @ A5 ) )
% 0.30/0.57 = ( ord_less_rat @ A5 @ B5 ) ) ).
% 0.30/0.57
% 0.30/0.57 % verit_comp_simplify1(3)
% 0.30/0.57 thf(fact_3394_verit__comp__simplify1_I3_J,axiom,
% 0.30/0.57 ! [B5: num,A5: num] :
% 0.30/0.57 ( ( ~ ( ord_less_eq_num @ B5 @ A5 ) )
% 0.30/0.57 = ( ord_less_num @ A5 @ B5 ) ) ).
% 0.30/0.57
% 0.30/0.57 % verit_comp_simplify1(3)
% 0.30/0.57 thf(fact_3395_verit__comp__simplify1_I3_J,axiom,
% 0.30/0.57 ! [B5: nat,A5: nat] :
% 0.30/0.57 ( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
% 0.30/0.57 = ( ord_less_nat @ A5 @ B5 ) ) ).
% 0.30/0.57
% 0.30/0.57 % verit_comp_simplify1(3)
% 0.30/0.57 thf(fact_3396_verit__comp__simplify1_I3_J,axiom,
% 0.30/0.57 ! [B5: int,A5: int] :
% 0.30/0.57 ( ( ~ ( ord_less_eq_int @ B5 @ A5 ) )
% 0.30/0.57 = ( ord_less_int @ A5 @ B5 ) ) ).
% 0.30/0.57
% 0.30/0.57 % verit_comp_simplify1(3)
% 0.30/0.57 thf(fact_3397_verit__sum__simplify,axiom,
% 0.30/0.57 ! [A: complex] :
% 0.30/0.57 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.30/0.57 = A ) ).
% 0.30/0.57
% 0.30/0.57 % verit_sum_simplify
% 0.30/0.57 thf(fact_3398_verit__sum__simplify,axiom,
% 0.30/0.57 ! [A: real] :
% 0.30/0.57 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.30/0.57 = A ) ).
% 0.30/0.57
% 0.30/0.57 % verit_sum_simplify
% 0.30/0.57 thf(fact_3399_verit__sum__simplify,axiom,
% 0.30/0.57 ! [A: rat] :
% 0.30/0.57 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.30/0.57 = A ) ).
% 0.30/0.57
% 0.30/0.57 % verit_sum_simplify
% 0.30/0.57 thf(fact_3400_verit__sum__simplify,axiom,
% 0.30/0.57 ! [A: nat] :
% 0.30/0.57 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.30/0.57 = A ) ).
% 0.30/0.57
% 0.30/0.57 % verit_sum_simplify
% 0.30/0.57 thf(fact_3401_verit__sum__simplify,axiom,
% 0.30/0.57 ! [A: int] :
% 0.30/0.57 ( ( plus_plus_int @ A @ zero_zero_int )
% 0.30/0.57 = A ) ).
% 0.30/0.57
% 0.30/0.57 % verit_sum_simplify
% 0.30/0.57 thf(fact_3402_verit__eq__simplify_I10_J,axiom,
% 0.30/0.57 ! [X22: num] :
% 0.30/0.57 ( one
% 0.30/0.57 != ( bit0 @ X22 ) ) ).
% 0.30/0.57
% 0.30/0.57 % verit_eq_simplify(10)
% 0.30/0.57 thf(fact_3403_pos__zmod__mult__2,axiom,
% 0.30/0.57 ! [A: int,B: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.57 => ( ( 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 ) )
% 0.30/0.57 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % pos_zmod_mult_2
% 0.30/0.57 thf(fact_3404_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
% 0.30/0.57 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
% 0.30/0.57 ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X2 )
% 0.30/0.57 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
% 0.30/0.57 thf(fact_3405_neg__zmod__mult__2,axiom,
% 0.30/0.57 ! [A: int,B: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.57 => ( ( 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 ) )
% 0.30/0.57 = ( 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 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % neg_zmod_mult_2
% 0.30/0.57 thf(fact_3406_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
% 0.30/0.57 ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
% 0.30/0.57 ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X2 )
% 0.30/0.57 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
% 0.30/0.57 thf(fact_3407_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
% 0.30/0.57 ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
% 0.30/0.57 ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X2 )
% 0.30/0.57 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
% 0.30/0.57 thf(fact_3408_finite__Collect__le__nat,axiom,
% 0.30/0.57 ! [K: nat] :
% 0.30/0.57 ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_Collect_le_nat
% 0.30/0.57 thf(fact_3409_finite__Collect__less__nat,axiom,
% 0.30/0.57 ! [K: nat] :
% 0.30/0.57 ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_Collect_less_nat
% 0.30/0.57 thf(fact_3410_finite__Collect__subsets,axiom,
% 0.30/0.57 ! [A2: set_nat] :
% 0.30/0.57 ( ( finite_finite_nat @ A2 )
% 0.30/0.57 => ( finite1152437895449049373et_nat
% 0.30/0.57 @ ( collect_set_nat
% 0.30/0.57 @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_Collect_subsets
% 0.30/0.57 thf(fact_3411_finite__Collect__subsets,axiom,
% 0.30/0.57 ! [A2: set_complex] :
% 0.30/0.57 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.57 => ( finite6551019134538273531omplex
% 0.30/0.57 @ ( collect_set_complex
% 0.30/0.57 @ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_Collect_subsets
% 0.30/0.57 thf(fact_3412_finite__Collect__subsets,axiom,
% 0.30/0.57 ! [A2: set_int] :
% 0.30/0.57 ( ( finite_finite_int @ A2 )
% 0.30/0.57 => ( finite6197958912794628473et_int
% 0.30/0.57 @ ( collect_set_int
% 0.30/0.57 @ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_Collect_subsets
% 0.30/0.57 thf(fact_3413_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
% 0.30/0.57 ! [X2: vEBT_VEBT,Xa2: nat,Y3: nat] :
% 0.30/0.57 ( ( ( vEBT_T_m_e_m_b_e_r @ X2 @ Xa2 )
% 0.30/0.57 = Y3 )
% 0.30/0.57 => ( ( ? [A4: $o,B4: $o] :
% 0.30/0.57 ( X2
% 0.30/0.57 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.57 => ( Y3
% 0.30/0.57 != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
% 0.30/0.57 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.57 ( X2
% 0.30/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.57 => ( Y3
% 0.30/0.57 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.57 => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.57 ( X2
% 0.30/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.30/0.57 => ( Y3
% 0.30/0.57 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.57 => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.57 ( X2
% 0.30/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.30/0.57 => ( Y3
% 0.30/0.57 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.57 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
% 0.30/0.57 ( ? [Summary2: vEBT_VEBT] :
% 0.30/0.57 ( X2
% 0.30/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.57 => ( Y3
% 0.30/0.57 != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa2 = Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa2 = Ma2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa2 ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
% 0.30/0.57 thf(fact_3414_finite__roots__unity,axiom,
% 0.30/0.57 ! [N: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.30/0.57 => ( finite_finite_real
% 0.30/0.57 @ ( collect_real
% 0.30/0.57 @ ^ [Z3: real] :
% 0.30/0.57 ( ( power_power_real @ Z3 @ N )
% 0.30/0.57 = one_one_real ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_roots_unity
% 0.30/0.57 thf(fact_3415_finite__roots__unity,axiom,
% 0.30/0.57 ! [N: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.30/0.57 => ( finite3207457112153483333omplex
% 0.30/0.57 @ ( collect_complex
% 0.30/0.57 @ ^ [Z3: complex] :
% 0.30/0.57 ( ( power_power_complex @ Z3 @ N )
% 0.30/0.57 = one_one_complex ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_roots_unity
% 0.30/0.57 thf(fact_3416_finite__Diff,axiom,
% 0.30/0.57 ! [A2: set_int,B3: set_int] :
% 0.30/0.57 ( ( finite_finite_int @ A2 )
% 0.30/0.57 => ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_Diff
% 0.30/0.57 thf(fact_3417_finite__Diff,axiom,
% 0.30/0.57 ! [A2: set_complex,B3: set_complex] :
% 0.30/0.57 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.57 => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_Diff
% 0.30/0.57 thf(fact_3418_finite__Diff,axiom,
% 0.30/0.57 ! [A2: set_nat,B3: set_nat] :
% 0.30/0.57 ( ( finite_finite_nat @ A2 )
% 0.30/0.57 => ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_Diff
% 0.30/0.57 thf(fact_3419_finite__Diff2,axiom,
% 0.30/0.57 ! [B3: set_int,A2: set_int] :
% 0.30/0.57 ( ( finite_finite_int @ B3 )
% 0.30/0.57 => ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B3 ) )
% 0.30/0.57 = ( finite_finite_int @ A2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_Diff2
% 0.30/0.57 thf(fact_3420_finite__Diff2,axiom,
% 0.30/0.57 ! [B3: set_complex,A2: set_complex] :
% 0.30/0.57 ( ( finite3207457112153483333omplex @ B3 )
% 0.30/0.57 => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.30/0.57 = ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_Diff2
% 0.30/0.57 thf(fact_3421_finite__Diff2,axiom,
% 0.30/0.57 ! [B3: set_nat,A2: set_nat] :
% 0.30/0.57 ( ( finite_finite_nat @ B3 )
% 0.30/0.57 => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 0.30/0.57 = ( finite_finite_nat @ A2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_Diff2
% 0.30/0.57 thf(fact_3422_semiring__norm_I90_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( ( bit1 @ M )
% 0.30/0.57 = ( bit1 @ N ) )
% 0.30/0.57 = ( M = N ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(90)
% 0.30/0.57 thf(fact_3423_finite__interval__int1,axiom,
% 0.30/0.57 ! [A: int,B: int] :
% 0.30/0.57 ( finite_finite_int
% 0.30/0.57 @ ( collect_int
% 0.30/0.57 @ ^ [I4: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ A @ I4 )
% 0.30/0.57 & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_interval_int1
% 0.30/0.57 thf(fact_3424_semiring__norm_I88_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( bit0 @ M )
% 0.30/0.57 != ( bit1 @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(88)
% 0.30/0.57 thf(fact_3425_semiring__norm_I89_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( bit1 @ M )
% 0.30/0.57 != ( bit0 @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(89)
% 0.30/0.57 thf(fact_3426_semiring__norm_I84_J,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( one
% 0.30/0.57 != ( bit1 @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(84)
% 0.30/0.57 thf(fact_3427_semiring__norm_I86_J,axiom,
% 0.30/0.57 ! [M: num] :
% 0.30/0.57 ( ( bit1 @ M )
% 0.30/0.57 != one ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(86)
% 0.30/0.57 thf(fact_3428_semiring__norm_I80_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( ord_less_num @ M @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(80)
% 0.30/0.57 thf(fact_3429_semiring__norm_I73_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(73)
% 0.30/0.57 thf(fact_3430_finite__interval__int3,axiom,
% 0.30/0.57 ! [A: int,B: int] :
% 0.30/0.57 ( finite_finite_int
% 0.30/0.57 @ ( collect_int
% 0.30/0.57 @ ^ [I4: int] :
% 0.30/0.57 ( ( ord_less_int @ A @ I4 )
% 0.30/0.57 & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_interval_int3
% 0.30/0.57 thf(fact_3431_finite__interval__int2,axiom,
% 0.30/0.57 ! [A: int,B: int] :
% 0.30/0.57 ( finite_finite_int
% 0.30/0.57 @ ( collect_int
% 0.30/0.57 @ ^ [I4: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ A @ I4 )
% 0.30/0.57 & ( ord_less_int @ I4 @ B ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_interval_int2
% 0.30/0.57 thf(fact_3432_semiring__norm_I7_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(7)
% 0.30/0.57 thf(fact_3433_semiring__norm_I9_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.57 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(9)
% 0.30/0.57 thf(fact_3434_semiring__norm_I15_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.57 = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(15)
% 0.30/0.57 thf(fact_3435_semiring__norm_I14_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(14)
% 0.30/0.57 thf(fact_3436_semiring__norm_I81_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.57 = ( ord_less_num @ M @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(81)
% 0.30/0.57 thf(fact_3437_semiring__norm_I72_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(72)
% 0.30/0.57 thf(fact_3438_semiring__norm_I77_J,axiom,
% 0.30/0.57 ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(77)
% 0.30/0.57 thf(fact_3439_semiring__norm_I70_J,axiom,
% 0.30/0.57 ! [M: num] :
% 0.30/0.57 ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(70)
% 0.30/0.57 thf(fact_3440_zdiv__numeral__Bit1,axiom,
% 0.30/0.57 ! [V: num,W: num] :
% 0.30/0.57 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.30/0.57 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % zdiv_numeral_Bit1
% 0.30/0.57 thf(fact_3441_semiring__norm_I3_J,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( plus_plus_num @ one @ ( bit0 @ N ) )
% 0.30/0.57 = ( bit1 @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(3)
% 0.30/0.57 thf(fact_3442_semiring__norm_I4_J,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( plus_plus_num @ one @ ( bit1 @ N ) )
% 0.30/0.57 = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(4)
% 0.30/0.57 thf(fact_3443_semiring__norm_I5_J,axiom,
% 0.30/0.57 ! [M: num] :
% 0.30/0.57 ( ( plus_plus_num @ ( bit0 @ M ) @ one )
% 0.30/0.57 = ( bit1 @ M ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(5)
% 0.30/0.57 thf(fact_3444_semiring__norm_I8_J,axiom,
% 0.30/0.57 ! [M: num] :
% 0.30/0.57 ( ( plus_plus_num @ ( bit1 @ M ) @ one )
% 0.30/0.57 = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(8)
% 0.30/0.57 thf(fact_3445_semiring__norm_I10_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(10)
% 0.30/0.57 thf(fact_3446_semiring__norm_I16_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(16)
% 0.30/0.57 thf(fact_3447_semiring__norm_I79_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(79)
% 0.30/0.57 thf(fact_3448_semiring__norm_I74_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.57 = ( ord_less_num @ M @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(74)
% 0.30/0.57 thf(fact_3449_Suc__div__eq__add3__div__numeral,axiom,
% 0.30/0.57 ! [M: nat,V: num] :
% 0.30/0.57 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.57 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Suc_div_eq_add3_div_numeral
% 0.30/0.57 thf(fact_3450_div__Suc__eq__div__add3,axiom,
% 0.30/0.57 ! [M: nat,N: nat] :
% 0.30/0.57 ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 0.30/0.57 = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % div_Suc_eq_div_add3
% 0.30/0.57 thf(fact_3451_Suc__mod__eq__add3__mod__numeral,axiom,
% 0.30/0.57 ! [M: nat,V: num] :
% 0.30/0.57 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.57 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Suc_mod_eq_add3_mod_numeral
% 0.30/0.57 thf(fact_3452_mod__Suc__eq__mod__add3,axiom,
% 0.30/0.57 ! [M: nat,N: nat] :
% 0.30/0.57 ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 0.30/0.57 = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % mod_Suc_eq_mod_add3
% 0.30/0.57 thf(fact_3453_zmod__numeral__Bit1,axiom,
% 0.30/0.57 ! [V: num,W: num] :
% 0.30/0.57 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.30/0.57 = ( 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 ) ) ).
% 0.30/0.57
% 0.30/0.57 % zmod_numeral_Bit1
% 0.30/0.57 thf(fact_3454_verit__eq__simplify_I14_J,axiom,
% 0.30/0.57 ! [X22: num,X32: num] :
% 0.30/0.57 ( ( bit0 @ X22 )
% 0.30/0.57 != ( bit1 @ X32 ) ) ).
% 0.30/0.57
% 0.30/0.57 % verit_eq_simplify(14)
% 0.30/0.57 thf(fact_3455_verit__eq__simplify_I12_J,axiom,
% 0.30/0.57 ! [X32: num] :
% 0.30/0.57 ( one
% 0.30/0.57 != ( bit1 @ X32 ) ) ).
% 0.30/0.57
% 0.30/0.57 % verit_eq_simplify(12)
% 0.30/0.57 thf(fact_3456_xor__num_Ocases,axiom,
% 0.30/0.57 ! [X2: product_prod_num_num] :
% 0.30/0.57 ( ( X2
% 0.30/0.57 != ( product_Pair_num_num @ one @ one ) )
% 0.30/0.57 => ( ! [N2: num] :
% 0.30/0.57 ( X2
% 0.30/0.57 != ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) )
% 0.30/0.57 => ( ! [N2: num] :
% 0.30/0.57 ( X2
% 0.30/0.57 != ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) )
% 0.30/0.57 => ( ! [M4: num] :
% 0.30/0.57 ( X2
% 0.30/0.57 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) )
% 0.30/0.57 => ( ! [M4: num,N2: num] :
% 0.30/0.57 ( X2
% 0.30/0.57 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) )
% 0.30/0.57 => ( ! [M4: num,N2: num] :
% 0.30/0.57 ( X2
% 0.30/0.57 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) )
% 0.30/0.57 => ( ! [M4: num] :
% 0.30/0.57 ( X2
% 0.30/0.57 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) )
% 0.30/0.57 => ( ! [M4: num,N2: num] :
% 0.30/0.57 ( X2
% 0.30/0.57 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) )
% 0.30/0.57 => ~ ! [M4: num,N2: num] :
% 0.30/0.57 ( X2
% 0.30/0.57 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % xor_num.cases
% 0.30/0.57 thf(fact_3457_num_Oexhaust,axiom,
% 0.30/0.57 ! [Y3: num] :
% 0.30/0.57 ( ( Y3 != one )
% 0.30/0.57 => ( ! [X23: num] :
% 0.30/0.57 ( Y3
% 0.30/0.57 != ( bit0 @ X23 ) )
% 0.30/0.57 => ~ ! [X33: num] :
% 0.30/0.57 ( Y3
% 0.30/0.57 != ( bit1 @ X33 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % num.exhaust
% 0.30/0.57 thf(fact_3458_finite__maxlen,axiom,
% 0.30/0.57 ! [M7: set_list_VEBT_VEBT] :
% 0.30/0.57 ( ( finite3004134309566078307T_VEBT @ M7 )
% 0.30/0.57 => ? [N2: nat] :
% 0.30/0.57 ! [X3: list_VEBT_VEBT] :
% 0.30/0.57 ( ( member2936631157270082147T_VEBT @ X3 @ M7 )
% 0.30/0.57 => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X3 ) @ N2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_maxlen
% 0.30/0.57 thf(fact_3459_finite__maxlen,axiom,
% 0.30/0.57 ! [M7: set_list_o] :
% 0.30/0.57 ( ( finite_finite_list_o @ M7 )
% 0.30/0.57 => ? [N2: nat] :
% 0.30/0.57 ! [X3: list_o] :
% 0.30/0.57 ( ( member_list_o @ X3 @ M7 )
% 0.30/0.57 => ( ord_less_nat @ ( size_size_list_o @ X3 ) @ N2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_maxlen
% 0.30/0.57 thf(fact_3460_finite__maxlen,axiom,
% 0.30/0.57 ! [M7: set_list_nat] :
% 0.30/0.57 ( ( finite8100373058378681591st_nat @ M7 )
% 0.30/0.57 => ? [N2: nat] :
% 0.30/0.57 ! [X3: list_nat] :
% 0.30/0.57 ( ( member_list_nat @ X3 @ M7 )
% 0.30/0.57 => ( ord_less_nat @ ( size_size_list_nat @ X3 ) @ N2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_maxlen
% 0.30/0.57 thf(fact_3461_finite__maxlen,axiom,
% 0.30/0.57 ! [M7: set_list_int] :
% 0.30/0.57 ( ( finite3922522038869484883st_int @ M7 )
% 0.30/0.57 => ? [N2: nat] :
% 0.30/0.57 ! [X3: list_int] :
% 0.30/0.57 ( ( member_list_int @ X3 @ M7 )
% 0.30/0.57 => ( ord_less_nat @ ( size_size_list_int @ X3 ) @ N2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_maxlen
% 0.30/0.57 thf(fact_3462_numeral__Bit1,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 0.30/0.57 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 0.30/0.57
% 0.30/0.57 % numeral_Bit1
% 0.30/0.57 thf(fact_3463_numeral__Bit1,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 0.30/0.57 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 0.30/0.57
% 0.30/0.57 % numeral_Bit1
% 0.30/0.57 thf(fact_3464_numeral__Bit1,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 0.30/0.57 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 0.30/0.57
% 0.30/0.57 % numeral_Bit1
% 0.30/0.57 thf(fact_3465_numeral__Bit1,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 0.30/0.57 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 0.30/0.57
% 0.30/0.57 % numeral_Bit1
% 0.30/0.57 thf(fact_3466_numeral__Bit1,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 0.30/0.57 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 0.30/0.57
% 0.30/0.57 % numeral_Bit1
% 0.30/0.57 thf(fact_3467_eval__nat__numeral_I3_J,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 0.30/0.57 = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % eval_nat_numeral(3)
% 0.30/0.57 thf(fact_3468_cong__exp__iff__simps_I10_J,axiom,
% 0.30/0.57 ! [M: num,Q3: num,N: num] :
% 0.30/0.57 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.30/0.57 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % cong_exp_iff_simps(10)
% 0.30/0.57 thf(fact_3469_cong__exp__iff__simps_I10_J,axiom,
% 0.30/0.57 ! [M: num,Q3: num,N: num] :
% 0.30/0.57 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.30/0.57 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % cong_exp_iff_simps(10)
% 0.30/0.57 thf(fact_3470_cong__exp__iff__simps_I10_J,axiom,
% 0.30/0.57 ! [M: num,Q3: num,N: num] :
% 0.30/0.57 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.30/0.57 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % cong_exp_iff_simps(10)
% 0.30/0.57 thf(fact_3471_cong__exp__iff__simps_I12_J,axiom,
% 0.30/0.57 ! [M: num,Q3: num,N: num] :
% 0.30/0.57 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.30/0.57 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % cong_exp_iff_simps(12)
% 0.30/0.57 thf(fact_3472_cong__exp__iff__simps_I12_J,axiom,
% 0.30/0.57 ! [M: num,Q3: num,N: num] :
% 0.30/0.57 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.30/0.57 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % cong_exp_iff_simps(12)
% 0.30/0.57 thf(fact_3473_cong__exp__iff__simps_I12_J,axiom,
% 0.30/0.57 ! [M: num,Q3: num,N: num] :
% 0.30/0.57 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.30/0.57 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % cong_exp_iff_simps(12)
% 0.30/0.57 thf(fact_3474_cong__exp__iff__simps_I13_J,axiom,
% 0.30/0.57 ! [M: num,Q3: num,N: num] :
% 0.30/0.57 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.30/0.57 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 0.30/0.57 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
% 0.30/0.57 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % cong_exp_iff_simps(13)
% 0.30/0.57 thf(fact_3475_cong__exp__iff__simps_I13_J,axiom,
% 0.30/0.57 ! [M: num,Q3: num,N: num] :
% 0.30/0.57 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.30/0.57 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 0.30/0.57 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
% 0.30/0.57 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % cong_exp_iff_simps(13)
% 0.30/0.57 thf(fact_3476_cong__exp__iff__simps_I13_J,axiom,
% 0.30/0.57 ! [M: num,Q3: num,N: num] :
% 0.30/0.57 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.30/0.57 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 0.30/0.57 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 0.30/0.57 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % cong_exp_iff_simps(13)
% 0.30/0.57 thf(fact_3477_numeral__code_I3_J,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 0.30/0.57 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 0.30/0.57
% 0.30/0.57 % numeral_code(3)
% 0.30/0.57 thf(fact_3478_numeral__code_I3_J,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 0.30/0.57 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 0.30/0.57
% 0.30/0.57 % numeral_code(3)
% 0.30/0.57 thf(fact_3479_numeral__code_I3_J,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 0.30/0.57 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 0.30/0.57
% 0.30/0.57 % numeral_code(3)
% 0.30/0.57 thf(fact_3480_numeral__code_I3_J,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 0.30/0.57 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 0.30/0.57
% 0.30/0.57 % numeral_code(3)
% 0.30/0.57 thf(fact_3481_numeral__code_I3_J,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 0.30/0.57 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 0.30/0.57
% 0.30/0.57 % numeral_code(3)
% 0.30/0.57 thf(fact_3482_power__numeral__odd,axiom,
% 0.30/0.57 ! [Z: complex,W: num] :
% 0.30/0.57 ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.30/0.57 = ( times_times_complex @ ( times_times_complex @ Z @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % power_numeral_odd
% 0.30/0.57 thf(fact_3483_power__numeral__odd,axiom,
% 0.30/0.57 ! [Z: real,W: num] :
% 0.30/0.57 ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.30/0.57 = ( times_times_real @ ( times_times_real @ Z @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % power_numeral_odd
% 0.30/0.57 thf(fact_3484_power__numeral__odd,axiom,
% 0.30/0.57 ! [Z: rat,W: num] :
% 0.30/0.57 ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.30/0.57 = ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % power_numeral_odd
% 0.30/0.57 thf(fact_3485_power__numeral__odd,axiom,
% 0.30/0.57 ! [Z: nat,W: num] :
% 0.30/0.57 ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.30/0.57 = ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % power_numeral_odd
% 0.30/0.57 thf(fact_3486_power__numeral__odd,axiom,
% 0.30/0.57 ! [Z: int,W: num] :
% 0.30/0.57 ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.30/0.57 = ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % power_numeral_odd
% 0.30/0.57 thf(fact_3487_numeral__Bit1__div__2,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.57 = ( numeral_numeral_nat @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % numeral_Bit1_div_2
% 0.30/0.57 thf(fact_3488_numeral__Bit1__div__2,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.57 = ( numeral_numeral_int @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % numeral_Bit1_div_2
% 0.30/0.57 thf(fact_3489_cong__exp__iff__simps_I3_J,axiom,
% 0.30/0.57 ! [N: num,Q3: num] :
% 0.30/0.57 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.30/0.57 != zero_zero_nat ) ).
% 0.30/0.57
% 0.30/0.57 % cong_exp_iff_simps(3)
% 0.30/0.57 thf(fact_3490_cong__exp__iff__simps_I3_J,axiom,
% 0.30/0.57 ! [N: num,Q3: num] :
% 0.30/0.57 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.30/0.57 != zero_zero_int ) ).
% 0.30/0.57
% 0.30/0.57 % cong_exp_iff_simps(3)
% 0.30/0.57 thf(fact_3491_cong__exp__iff__simps_I3_J,axiom,
% 0.30/0.57 ! [N: num,Q3: num] :
% 0.30/0.57 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.30/0.57 != zero_z3403309356797280102nteger ) ).
% 0.30/0.57
% 0.30/0.57 % cong_exp_iff_simps(3)
% 0.30/0.57 thf(fact_3492_power3__eq__cube,axiom,
% 0.30/0.57 ! [A: complex] :
% 0.30/0.57 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.30/0.57 = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).
% 0.30/0.57
% 0.30/0.57 % power3_eq_cube
% 0.30/0.57 thf(fact_3493_power3__eq__cube,axiom,
% 0.30/0.57 ! [A: real] :
% 0.30/0.57 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.30/0.57 = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).
% 0.30/0.57
% 0.30/0.57 % power3_eq_cube
% 0.30/0.57 thf(fact_3494_power3__eq__cube,axiom,
% 0.30/0.57 ! [A: rat] :
% 0.30/0.57 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.30/0.57 = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% 0.30/0.57
% 0.30/0.57 % power3_eq_cube
% 0.30/0.57 thf(fact_3495_power3__eq__cube,axiom,
% 0.30/0.57 ! [A: nat] :
% 0.30/0.57 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.30/0.57 = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% 0.30/0.57
% 0.30/0.57 % power3_eq_cube
% 0.30/0.57 thf(fact_3496_power3__eq__cube,axiom,
% 0.30/0.57 ! [A: int] :
% 0.30/0.57 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.30/0.57 = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% 0.30/0.57
% 0.30/0.57 % power3_eq_cube
% 0.30/0.57 thf(fact_3497_numeral__3__eq__3,axiom,
% 0.30/0.57 ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 0.30/0.57 = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % numeral_3_eq_3
% 0.30/0.57 thf(fact_3498_Suc3__eq__add__3,axiom,
% 0.30/0.57 ! [N: nat] :
% 0.30/0.57 ( ( suc @ ( suc @ ( suc @ N ) ) )
% 0.30/0.57 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % Suc3_eq_add_3
% 0.30/0.57 thf(fact_3499_maxt__bound,axiom,
% 0.30/0.57 ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_a_x_t @ T ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % maxt_bound
% 0.30/0.57 thf(fact_3500_mint__bound,axiom,
% 0.30/0.57 ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_i_n_t @ T ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % mint_bound
% 0.30/0.57 thf(fact_3501_num_Osize_I6_J,axiom,
% 0.30/0.57 ! [X32: num] :
% 0.30/0.57 ( ( size_size_num @ ( bit1 @ X32 ) )
% 0.30/0.57 = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % num.size(6)
% 0.30/0.57 thf(fact_3502_cong__exp__iff__simps_I7_J,axiom,
% 0.30/0.57 ! [Q3: num,N: num] :
% 0.30/0.57 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.30/0.57 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 0.30/0.57 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
% 0.30/0.57 = zero_zero_nat ) ) ).
% 0.30/0.57
% 0.30/0.57 % cong_exp_iff_simps(7)
% 0.30/0.57 thf(fact_3503_cong__exp__iff__simps_I7_J,axiom,
% 0.30/0.57 ! [Q3: num,N: num] :
% 0.30/0.57 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.30/0.57 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 0.30/0.57 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
% 0.30/0.57 = zero_zero_int ) ) ).
% 0.30/0.57
% 0.30/0.57 % cong_exp_iff_simps(7)
% 0.30/0.57 thf(fact_3504_cong__exp__iff__simps_I7_J,axiom,
% 0.30/0.57 ! [Q3: num,N: num] :
% 0.30/0.57 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.30/0.57 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 0.30/0.57 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 0.30/0.57 = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.57
% 0.30/0.57 % cong_exp_iff_simps(7)
% 0.30/0.57 thf(fact_3505_cong__exp__iff__simps_I11_J,axiom,
% 0.30/0.57 ! [M: num,Q3: num] :
% 0.30/0.57 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.30/0.57 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 0.30/0.57 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
% 0.30/0.57 = zero_zero_nat ) ) ).
% 0.30/0.57
% 0.30/0.57 % cong_exp_iff_simps(11)
% 0.30/0.57 thf(fact_3506_cong__exp__iff__simps_I11_J,axiom,
% 0.30/0.57 ! [M: num,Q3: num] :
% 0.30/0.57 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.30/0.57 = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 0.30/0.57 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
% 0.30/0.57 = zero_zero_int ) ) ).
% 0.30/0.57
% 0.30/0.57 % cong_exp_iff_simps(11)
% 0.30/0.57 thf(fact_3507_cong__exp__iff__simps_I11_J,axiom,
% 0.30/0.57 ! [M: num,Q3: num] :
% 0.30/0.57 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.30/0.57 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 0.30/0.57 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 0.30/0.57 = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.57
% 0.30/0.57 % cong_exp_iff_simps(11)
% 0.30/0.57 thf(fact_3508_Suc__div__eq__add3__div,axiom,
% 0.30/0.57 ! [M: nat,N: nat] :
% 0.30/0.57 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 0.30/0.57 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % Suc_div_eq_add3_div
% 0.30/0.57 thf(fact_3509_Suc__mod__eq__add3__mod,axiom,
% 0.30/0.57 ! [M: nat,N: nat] :
% 0.30/0.57 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 0.30/0.57 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 0.30/0.57
% 0.30/0.57 % Suc_mod_eq_add3_mod
% 0.30/0.57 thf(fact_3510_finite__has__maximal2,axiom,
% 0.30/0.57 ! [A2: set_real,A: real] :
% 0.30/0.57 ( ( finite_finite_real @ A2 )
% 0.30/0.57 => ( ( member_real @ A @ A2 )
% 0.30/0.57 => ? [X5: real] :
% 0.30/0.57 ( ( member_real @ X5 @ A2 )
% 0.30/0.57 & ( ord_less_eq_real @ A @ X5 )
% 0.30/0.57 & ! [Xa: real] :
% 0.30/0.57 ( ( member_real @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_real @ X5 @ Xa )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_maximal2
% 0.30/0.57 thf(fact_3511_finite__has__maximal2,axiom,
% 0.30/0.57 ! [A2: set_set_int,A: set_int] :
% 0.30/0.57 ( ( finite6197958912794628473et_int @ A2 )
% 0.30/0.57 => ( ( member_set_int @ A @ A2 )
% 0.30/0.57 => ? [X5: set_int] :
% 0.30/0.57 ( ( member_set_int @ X5 @ A2 )
% 0.30/0.57 & ( ord_less_eq_set_int @ A @ X5 )
% 0.30/0.57 & ! [Xa: set_int] :
% 0.30/0.57 ( ( member_set_int @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_set_int @ X5 @ Xa )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_maximal2
% 0.30/0.57 thf(fact_3512_finite__has__maximal2,axiom,
% 0.30/0.57 ! [A2: set_rat,A: rat] :
% 0.30/0.57 ( ( finite_finite_rat @ A2 )
% 0.30/0.57 => ( ( member_rat @ A @ A2 )
% 0.30/0.57 => ? [X5: rat] :
% 0.30/0.57 ( ( member_rat @ X5 @ A2 )
% 0.30/0.57 & ( ord_less_eq_rat @ A @ X5 )
% 0.30/0.57 & ! [Xa: rat] :
% 0.30/0.57 ( ( member_rat @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_rat @ X5 @ Xa )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_maximal2
% 0.30/0.57 thf(fact_3513_finite__has__maximal2,axiom,
% 0.30/0.57 ! [A2: set_num,A: num] :
% 0.30/0.57 ( ( finite_finite_num @ A2 )
% 0.30/0.57 => ( ( member_num @ A @ A2 )
% 0.30/0.57 => ? [X5: num] :
% 0.30/0.57 ( ( member_num @ X5 @ A2 )
% 0.30/0.57 & ( ord_less_eq_num @ A @ X5 )
% 0.30/0.57 & ! [Xa: num] :
% 0.30/0.57 ( ( member_num @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_num @ X5 @ Xa )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_maximal2
% 0.30/0.57 thf(fact_3514_finite__has__maximal2,axiom,
% 0.30/0.57 ! [A2: set_nat,A: nat] :
% 0.30/0.57 ( ( finite_finite_nat @ A2 )
% 0.30/0.57 => ( ( member_nat @ A @ A2 )
% 0.30/0.57 => ? [X5: nat] :
% 0.30/0.57 ( ( member_nat @ X5 @ A2 )
% 0.30/0.57 & ( ord_less_eq_nat @ A @ X5 )
% 0.30/0.57 & ! [Xa: nat] :
% 0.30/0.57 ( ( member_nat @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_nat @ X5 @ Xa )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_maximal2
% 0.30/0.57 thf(fact_3515_finite__has__maximal2,axiom,
% 0.30/0.57 ! [A2: set_int,A: int] :
% 0.30/0.57 ( ( finite_finite_int @ A2 )
% 0.30/0.57 => ( ( member_int @ A @ A2 )
% 0.30/0.57 => ? [X5: int] :
% 0.30/0.57 ( ( member_int @ X5 @ A2 )
% 0.30/0.57 & ( ord_less_eq_int @ A @ X5 )
% 0.30/0.57 & ! [Xa: int] :
% 0.30/0.57 ( ( member_int @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_int @ X5 @ Xa )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_maximal2
% 0.30/0.57 thf(fact_3516_finite__has__minimal2,axiom,
% 0.30/0.57 ! [A2: set_real,A: real] :
% 0.30/0.57 ( ( finite_finite_real @ A2 )
% 0.30/0.57 => ( ( member_real @ A @ A2 )
% 0.30/0.57 => ? [X5: real] :
% 0.30/0.57 ( ( member_real @ X5 @ A2 )
% 0.30/0.57 & ( ord_less_eq_real @ X5 @ A )
% 0.30/0.57 & ! [Xa: real] :
% 0.30/0.57 ( ( member_real @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_real @ Xa @ X5 )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_minimal2
% 0.30/0.57 thf(fact_3517_finite__has__minimal2,axiom,
% 0.30/0.57 ! [A2: set_set_int,A: set_int] :
% 0.30/0.57 ( ( finite6197958912794628473et_int @ A2 )
% 0.30/0.57 => ( ( member_set_int @ A @ A2 )
% 0.30/0.57 => ? [X5: set_int] :
% 0.30/0.57 ( ( member_set_int @ X5 @ A2 )
% 0.30/0.57 & ( ord_less_eq_set_int @ X5 @ A )
% 0.30/0.57 & ! [Xa: set_int] :
% 0.30/0.57 ( ( member_set_int @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_set_int @ Xa @ X5 )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_minimal2
% 0.30/0.57 thf(fact_3518_finite__has__minimal2,axiom,
% 0.30/0.57 ! [A2: set_rat,A: rat] :
% 0.30/0.57 ( ( finite_finite_rat @ A2 )
% 0.30/0.57 => ( ( member_rat @ A @ A2 )
% 0.30/0.57 => ? [X5: rat] :
% 0.30/0.57 ( ( member_rat @ X5 @ A2 )
% 0.30/0.57 & ( ord_less_eq_rat @ X5 @ A )
% 0.30/0.57 & ! [Xa: rat] :
% 0.30/0.57 ( ( member_rat @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_rat @ Xa @ X5 )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_minimal2
% 0.30/0.57 thf(fact_3519_finite__has__minimal2,axiom,
% 0.30/0.57 ! [A2: set_num,A: num] :
% 0.30/0.57 ( ( finite_finite_num @ A2 )
% 0.30/0.57 => ( ( member_num @ A @ A2 )
% 0.30/0.57 => ? [X5: num] :
% 0.30/0.57 ( ( member_num @ X5 @ A2 )
% 0.30/0.57 & ( ord_less_eq_num @ X5 @ A )
% 0.30/0.57 & ! [Xa: num] :
% 0.30/0.57 ( ( member_num @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_num @ Xa @ X5 )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_minimal2
% 0.30/0.57 thf(fact_3520_finite__has__minimal2,axiom,
% 0.30/0.57 ! [A2: set_nat,A: nat] :
% 0.30/0.57 ( ( finite_finite_nat @ A2 )
% 0.30/0.57 => ( ( member_nat @ A @ A2 )
% 0.30/0.57 => ? [X5: nat] :
% 0.30/0.57 ( ( member_nat @ X5 @ A2 )
% 0.30/0.57 & ( ord_less_eq_nat @ X5 @ A )
% 0.30/0.57 & ! [Xa: nat] :
% 0.30/0.57 ( ( member_nat @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_nat @ Xa @ X5 )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_minimal2
% 0.30/0.57 thf(fact_3521_finite__has__minimal2,axiom,
% 0.30/0.57 ! [A2: set_int,A: int] :
% 0.30/0.57 ( ( finite_finite_int @ A2 )
% 0.30/0.57 => ( ( member_int @ A @ A2 )
% 0.30/0.57 => ? [X5: int] :
% 0.30/0.57 ( ( member_int @ X5 @ A2 )
% 0.30/0.57 & ( ord_less_eq_int @ X5 @ A )
% 0.30/0.57 & ! [Xa: int] :
% 0.30/0.57 ( ( member_int @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_int @ Xa @ X5 )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_minimal2
% 0.30/0.57 thf(fact_3522_finite__subset,axiom,
% 0.30/0.57 ! [A2: set_nat,B3: set_nat] :
% 0.30/0.57 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.30/0.57 => ( ( finite_finite_nat @ B3 )
% 0.30/0.57 => ( finite_finite_nat @ A2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_subset
% 0.30/0.57 thf(fact_3523_finite__subset,axiom,
% 0.30/0.57 ! [A2: set_complex,B3: set_complex] :
% 0.30/0.57 ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.30/0.57 => ( ( finite3207457112153483333omplex @ B3 )
% 0.30/0.57 => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_subset
% 0.30/0.57 thf(fact_3524_finite__subset,axiom,
% 0.30/0.57 ! [A2: set_int,B3: set_int] :
% 0.30/0.57 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.30/0.57 => ( ( finite_finite_int @ B3 )
% 0.30/0.57 => ( finite_finite_int @ A2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_subset
% 0.30/0.57 thf(fact_3525_infinite__super,axiom,
% 0.30/0.57 ! [S3: set_nat,T3: set_nat] :
% 0.30/0.57 ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 0.30/0.57 => ( ~ ( finite_finite_nat @ S3 )
% 0.30/0.57 => ~ ( finite_finite_nat @ T3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % infinite_super
% 0.30/0.57 thf(fact_3526_infinite__super,axiom,
% 0.30/0.57 ! [S3: set_complex,T3: set_complex] :
% 0.30/0.57 ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.57 => ( ~ ( finite3207457112153483333omplex @ S3 )
% 0.30/0.57 => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % infinite_super
% 0.30/0.57 thf(fact_3527_infinite__super,axiom,
% 0.30/0.57 ! [S3: set_int,T3: set_int] :
% 0.30/0.57 ( ( ord_less_eq_set_int @ S3 @ T3 )
% 0.30/0.57 => ( ~ ( finite_finite_int @ S3 )
% 0.30/0.57 => ~ ( finite_finite_int @ T3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % infinite_super
% 0.30/0.57 thf(fact_3528_rev__finite__subset,axiom,
% 0.30/0.57 ! [B3: set_nat,A2: set_nat] :
% 0.30/0.57 ( ( finite_finite_nat @ B3 )
% 0.30/0.57 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.30/0.57 => ( finite_finite_nat @ A2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % rev_finite_subset
% 0.30/0.57 thf(fact_3529_rev__finite__subset,axiom,
% 0.30/0.57 ! [B3: set_complex,A2: set_complex] :
% 0.30/0.57 ( ( finite3207457112153483333omplex @ B3 )
% 0.30/0.57 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.30/0.57 => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % rev_finite_subset
% 0.30/0.57 thf(fact_3530_rev__finite__subset,axiom,
% 0.30/0.57 ! [B3: set_int,A2: set_int] :
% 0.30/0.57 ( ( finite_finite_int @ B3 )
% 0.30/0.57 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.30/0.57 => ( finite_finite_int @ A2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % rev_finite_subset
% 0.30/0.57 thf(fact_3531_Diff__infinite__finite,axiom,
% 0.30/0.57 ! [T3: set_int,S3: set_int] :
% 0.30/0.57 ( ( finite_finite_int @ T3 )
% 0.30/0.57 => ( ~ ( finite_finite_int @ S3 )
% 0.30/0.57 => ~ ( finite_finite_int @ ( minus_minus_set_int @ S3 @ T3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Diff_infinite_finite
% 0.30/0.57 thf(fact_3532_Diff__infinite__finite,axiom,
% 0.30/0.57 ! [T3: set_complex,S3: set_complex] :
% 0.30/0.57 ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.57 => ( ~ ( finite3207457112153483333omplex @ S3 )
% 0.30/0.57 => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S3 @ T3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Diff_infinite_finite
% 0.30/0.57 thf(fact_3533_Diff__infinite__finite,axiom,
% 0.30/0.57 ! [T3: set_nat,S3: set_nat] :
% 0.30/0.57 ( ( finite_finite_nat @ T3 )
% 0.30/0.57 => ( ~ ( finite_finite_nat @ S3 )
% 0.30/0.57 => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ T3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Diff_infinite_finite
% 0.30/0.57 thf(fact_3534_member__bound__height,axiom,
% 0.30/0.57 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.30/0.57 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.57 => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r @ T @ X2 ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % member_bound_height
% 0.30/0.57 thf(fact_3535_finite__has__maximal,axiom,
% 0.30/0.57 ! [A2: set_real] :
% 0.30/0.57 ( ( finite_finite_real @ A2 )
% 0.30/0.57 => ( ( A2 != bot_bot_set_real )
% 0.30/0.57 => ? [X5: real] :
% 0.30/0.57 ( ( member_real @ X5 @ A2 )
% 0.30/0.57 & ! [Xa: real] :
% 0.30/0.57 ( ( member_real @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_real @ X5 @ Xa )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_maximal
% 0.30/0.57 thf(fact_3536_finite__has__maximal,axiom,
% 0.30/0.57 ! [A2: set_set_int] :
% 0.30/0.57 ( ( finite6197958912794628473et_int @ A2 )
% 0.30/0.57 => ( ( A2 != bot_bot_set_set_int )
% 0.30/0.57 => ? [X5: set_int] :
% 0.30/0.57 ( ( member_set_int @ X5 @ A2 )
% 0.30/0.57 & ! [Xa: set_int] :
% 0.30/0.57 ( ( member_set_int @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_set_int @ X5 @ Xa )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_maximal
% 0.30/0.57 thf(fact_3537_finite__has__maximal,axiom,
% 0.30/0.57 ! [A2: set_rat] :
% 0.30/0.57 ( ( finite_finite_rat @ A2 )
% 0.30/0.57 => ( ( A2 != bot_bot_set_rat )
% 0.30/0.57 => ? [X5: rat] :
% 0.30/0.57 ( ( member_rat @ X5 @ A2 )
% 0.30/0.57 & ! [Xa: rat] :
% 0.30/0.57 ( ( member_rat @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_rat @ X5 @ Xa )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_maximal
% 0.30/0.57 thf(fact_3538_finite__has__maximal,axiom,
% 0.30/0.57 ! [A2: set_num] :
% 0.30/0.57 ( ( finite_finite_num @ A2 )
% 0.30/0.57 => ( ( A2 != bot_bot_set_num )
% 0.30/0.57 => ? [X5: num] :
% 0.30/0.57 ( ( member_num @ X5 @ A2 )
% 0.30/0.57 & ! [Xa: num] :
% 0.30/0.57 ( ( member_num @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_num @ X5 @ Xa )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_maximal
% 0.30/0.57 thf(fact_3539_finite__has__maximal,axiom,
% 0.30/0.57 ! [A2: set_nat] :
% 0.30/0.57 ( ( finite_finite_nat @ A2 )
% 0.30/0.57 => ( ( A2 != bot_bot_set_nat )
% 0.30/0.57 => ? [X5: nat] :
% 0.30/0.57 ( ( member_nat @ X5 @ A2 )
% 0.30/0.57 & ! [Xa: nat] :
% 0.30/0.57 ( ( member_nat @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_nat @ X5 @ Xa )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_maximal
% 0.30/0.57 thf(fact_3540_finite__has__maximal,axiom,
% 0.30/0.57 ! [A2: set_int] :
% 0.30/0.57 ( ( finite_finite_int @ A2 )
% 0.30/0.57 => ( ( A2 != bot_bot_set_int )
% 0.30/0.57 => ? [X5: int] :
% 0.30/0.57 ( ( member_int @ X5 @ A2 )
% 0.30/0.57 & ! [Xa: int] :
% 0.30/0.57 ( ( member_int @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_int @ X5 @ Xa )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_maximal
% 0.30/0.57 thf(fact_3541_finite__has__minimal,axiom,
% 0.30/0.57 ! [A2: set_real] :
% 0.30/0.57 ( ( finite_finite_real @ A2 )
% 0.30/0.57 => ( ( A2 != bot_bot_set_real )
% 0.30/0.57 => ? [X5: real] :
% 0.30/0.57 ( ( member_real @ X5 @ A2 )
% 0.30/0.57 & ! [Xa: real] :
% 0.30/0.57 ( ( member_real @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_real @ Xa @ X5 )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_minimal
% 0.30/0.57 thf(fact_3542_finite__has__minimal,axiom,
% 0.30/0.57 ! [A2: set_set_int] :
% 0.30/0.57 ( ( finite6197958912794628473et_int @ A2 )
% 0.30/0.57 => ( ( A2 != bot_bot_set_set_int )
% 0.30/0.57 => ? [X5: set_int] :
% 0.30/0.57 ( ( member_set_int @ X5 @ A2 )
% 0.30/0.57 & ! [Xa: set_int] :
% 0.30/0.57 ( ( member_set_int @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_set_int @ Xa @ X5 )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_minimal
% 0.30/0.57 thf(fact_3543_finite__has__minimal,axiom,
% 0.30/0.57 ! [A2: set_rat] :
% 0.30/0.57 ( ( finite_finite_rat @ A2 )
% 0.30/0.57 => ( ( A2 != bot_bot_set_rat )
% 0.30/0.57 => ? [X5: rat] :
% 0.30/0.57 ( ( member_rat @ X5 @ A2 )
% 0.30/0.57 & ! [Xa: rat] :
% 0.30/0.57 ( ( member_rat @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_rat @ Xa @ X5 )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_minimal
% 0.30/0.57 thf(fact_3544_finite__has__minimal,axiom,
% 0.30/0.57 ! [A2: set_num] :
% 0.30/0.57 ( ( finite_finite_num @ A2 )
% 0.30/0.57 => ( ( A2 != bot_bot_set_num )
% 0.30/0.57 => ? [X5: num] :
% 0.30/0.57 ( ( member_num @ X5 @ A2 )
% 0.30/0.57 & ! [Xa: num] :
% 0.30/0.57 ( ( member_num @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_num @ Xa @ X5 )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_minimal
% 0.30/0.57 thf(fact_3545_finite__has__minimal,axiom,
% 0.30/0.57 ! [A2: set_nat] :
% 0.30/0.57 ( ( finite_finite_nat @ A2 )
% 0.30/0.57 => ( ( A2 != bot_bot_set_nat )
% 0.30/0.57 => ? [X5: nat] :
% 0.30/0.57 ( ( member_nat @ X5 @ A2 )
% 0.30/0.57 & ! [Xa: nat] :
% 0.30/0.57 ( ( member_nat @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_nat @ Xa @ X5 )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_minimal
% 0.30/0.57 thf(fact_3546_finite__has__minimal,axiom,
% 0.30/0.57 ! [A2: set_int] :
% 0.30/0.57 ( ( finite_finite_int @ A2 )
% 0.30/0.57 => ( ( A2 != bot_bot_set_int )
% 0.30/0.57 => ? [X5: int] :
% 0.30/0.57 ( ( member_int @ X5 @ A2 )
% 0.30/0.57 & ! [Xa: int] :
% 0.30/0.57 ( ( member_int @ Xa @ A2 )
% 0.30/0.57 => ( ( ord_less_eq_int @ Xa @ X5 )
% 0.30/0.57 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_has_minimal
% 0.30/0.57 thf(fact_3547_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
% 0.30/0.57 ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 0.30/0.57 ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 0.30/0.57 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( X2 = Mi ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( X2 = Ma ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma @ X2 ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
% 0.30/0.57 thf(fact_3548_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
% 0.30/0.57 ! [X2: vEBT_VEBT,Xa2: nat,Y3: nat] :
% 0.30/0.57 ( ( ( vEBT_T_m_e_m_b_e_r @ X2 @ Xa2 )
% 0.30/0.57 = Y3 )
% 0.30/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.30/0.57 => ( ! [A4: $o,B4: $o] :
% 0.30/0.57 ( ( X2
% 0.30/0.57 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.57 => ( ( Y3
% 0.30/0.57 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
% 0.30/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
% 0.30/0.57 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.57 ( ( X2
% 0.30/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.57 => ( ( Y3
% 0.30/0.57 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 0.30/0.57 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.57 ( ( X2
% 0.30/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.30/0.57 => ( ( Y3
% 0.30/0.57 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 0.30/0.57 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.57 ( ( X2
% 0.30/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.30/0.57 => ( ( Y3
% 0.30/0.57 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
% 0.30/0.57 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.57 ( ( X2
% 0.30/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.57 => ( ( Y3
% 0.30/0.57 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa2 = Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa2 = Ma2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa2 ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
% 0.30/0.57 thf(fact_3549_mod__exhaust__less__4,axiom,
% 0.30/0.57 ! [M: nat] :
% 0.30/0.57 ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.57 = zero_zero_nat )
% 0.30/0.57 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.57 = one_one_nat )
% 0.30/0.57 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.57 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.57 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.57 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % mod_exhaust_less_4
% 0.30/0.57 thf(fact_3550_arcosh__1,axiom,
% 0.30/0.57 ( ( arcosh_real @ one_one_real )
% 0.30/0.57 = zero_zero_real ) ).
% 0.30/0.57
% 0.30/0.57 % arcosh_1
% 0.30/0.57 thf(fact_3551_finite__nth__roots,axiom,
% 0.30/0.57 ! [N: nat,C: complex] :
% 0.30/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.57 => ( finite3207457112153483333omplex
% 0.30/0.57 @ ( collect_complex
% 0.30/0.57 @ ^ [Z3: complex] :
% 0.30/0.57 ( ( power_power_complex @ Z3 @ N )
% 0.30/0.57 = C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % finite_nth_roots
% 0.30/0.57 thf(fact_3552_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
% 0.30/0.57 ! [X2: vEBT_VEBT,Xa2: nat,Y3: nat] :
% 0.30/0.57 ( ( ( vEBT_T_i_n_s_e_r_t @ X2 @ Xa2 )
% 0.30/0.57 = Y3 )
% 0.30/0.57 => ( ( ? [A4: $o,B4: $o] :
% 0.30/0.57 ( X2
% 0.30/0.57 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.57 => ( Y3
% 0.30/0.57 != ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
% 0.30/0.57 => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.57 ( X2
% 0.30/0.57 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 0.30/0.57 => ( Y3 != one_one_nat ) )
% 0.30/0.57 => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.57 ( X2
% 0.30/0.57 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 0.30/0.57 => ( Y3 != one_one_nat ) )
% 0.30/0.57 => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.57 ( X2
% 0.30/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.57 => ( Y3
% 0.30/0.57 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.57 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.57 ( ( X2
% 0.30/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.57 => ( Y3
% 0.30/0.57 != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.30/0.57 @ ( if_nat
% 0.30/0.57 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.57 & ~ ( ( Xa2 = Mi2 )
% 0.30/0.57 | ( Xa2 = Ma2 ) ) )
% 0.30/0.57 @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
% 0.30/0.57 @ one_one_nat ) ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
% 0.30/0.57 thf(fact_3553_insert__bound__height,axiom,
% 0.30/0.57 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.30/0.57 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.57 => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ X2 ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % insert_bound_height
% 0.30/0.57 thf(fact_3554_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
% 0.30/0.57 ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 0.30/0.57 ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 0.30/0.57 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.30/0.57 @ ( if_nat
% 0.30/0.57 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.30/0.57 & ~ ( ( X2 = Mi )
% 0.30/0.57 | ( X2 = Ma ) ) )
% 0.30/0.57 @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
% 0.30/0.57 @ one_one_nat ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
% 0.30/0.57 thf(fact_3555_divmod__algorithm__code_I8_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( ( ord_less_num @ M @ N )
% 0.30/0.57 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
% 0.30/0.57 & ( ~ ( ord_less_num @ M @ N )
% 0.30/0.57 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_algorithm_code(8)
% 0.30/0.57 thf(fact_3556_divmod__algorithm__code_I8_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( ( ord_less_num @ M @ N )
% 0.30/0.57 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
% 0.30/0.57 & ( ~ ( ord_less_num @ M @ N )
% 0.30/0.57 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_algorithm_code(8)
% 0.30/0.57 thf(fact_3557_divmod__algorithm__code_I8_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( ( ord_less_num @ M @ N )
% 0.30/0.57 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
% 0.30/0.57 & ( ~ ( ord_less_num @ M @ N )
% 0.30/0.57 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_algorithm_code(8)
% 0.30/0.57 thf(fact_3558_divmod__algorithm__code_I2_J,axiom,
% 0.30/0.57 ! [M: num] :
% 0.30/0.57 ( ( unique5052692396658037445od_int @ M @ one )
% 0.30/0.57 = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_algorithm_code(2)
% 0.30/0.57 thf(fact_3559_divmod__algorithm__code_I2_J,axiom,
% 0.30/0.57 ! [M: num] :
% 0.30/0.57 ( ( unique5055182867167087721od_nat @ M @ one )
% 0.30/0.57 = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_algorithm_code(2)
% 0.30/0.57 thf(fact_3560_divmod__algorithm__code_I2_J,axiom,
% 0.30/0.57 ! [M: num] :
% 0.30/0.57 ( ( unique3479559517661332726nteger @ M @ one )
% 0.30/0.57 = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_algorithm_code(2)
% 0.30/0.57 thf(fact_3561_divmod__algorithm__code_I3_J,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
% 0.30/0.57 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_algorithm_code(3)
% 0.30/0.57 thf(fact_3562_divmod__algorithm__code_I3_J,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
% 0.30/0.57 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_algorithm_code(3)
% 0.30/0.57 thf(fact_3563_divmod__algorithm__code_I3_J,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
% 0.30/0.57 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_algorithm_code(3)
% 0.30/0.57 thf(fact_3564_divmod__algorithm__code_I4_J,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
% 0.30/0.57 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_algorithm_code(4)
% 0.30/0.57 thf(fact_3565_divmod__algorithm__code_I4_J,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
% 0.30/0.57 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_algorithm_code(4)
% 0.30/0.57 thf(fact_3566_divmod__algorithm__code_I4_J,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
% 0.30/0.57 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_algorithm_code(4)
% 0.30/0.57 thf(fact_3567_divmod__algorithm__code_I7_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( ( ord_less_eq_num @ M @ N )
% 0.30/0.57 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
% 0.30/0.57 & ( ~ ( ord_less_eq_num @ M @ N )
% 0.30/0.57 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_algorithm_code(7)
% 0.30/0.57 thf(fact_3568_divmod__algorithm__code_I7_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( ( ord_less_eq_num @ M @ N )
% 0.30/0.57 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
% 0.30/0.57 & ( ~ ( ord_less_eq_num @ M @ N )
% 0.30/0.57 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_algorithm_code(7)
% 0.30/0.57 thf(fact_3569_divmod__algorithm__code_I7_J,axiom,
% 0.30/0.57 ! [M: num,N: num] :
% 0.30/0.57 ( ( ( ord_less_eq_num @ M @ N )
% 0.30/0.57 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
% 0.30/0.57 & ( ~ ( ord_less_eq_num @ M @ N )
% 0.30/0.57 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.57 = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_algorithm_code(7)
% 0.30/0.57 thf(fact_3570_minNull__bound,axiom,
% 0.30/0.57 ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_i_n_N_u_l_l @ T ) @ one_one_nat ) ).
% 0.30/0.57
% 0.30/0.57 % minNull_bound
% 0.30/0.57 thf(fact_3571_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I1_J,axiom,
% 0.30/0.57 ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.57 = one_one_nat ) ).
% 0.30/0.57
% 0.30/0.57 % T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(1)
% 0.30/0.57 thf(fact_3572_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I2_J,axiom,
% 0.30/0.57 ! [Uv: $o] :
% 0.30/0.57 ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $true @ Uv ) )
% 0.30/0.57 = one_one_nat ) ).
% 0.30/0.57
% 0.30/0.57 % T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
% 0.30/0.57 thf(fact_3573_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
% 0.30/0.57 ! [Uu: $o] :
% 0.30/0.57 ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ Uu @ $true ) )
% 0.30/0.57 = one_one_nat ) ).
% 0.30/0.57
% 0.30/0.57 % T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
% 0.30/0.57 thf(fact_3574_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
% 0.30/0.57 ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
% 0.30/0.57 ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S ) @ X2 )
% 0.30/0.57 = one_one_nat ) ).
% 0.30/0.57
% 0.30/0.57 % T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
% 0.30/0.57 thf(fact_3575_divmod__int__def,axiom,
% 0.30/0.57 ( unique5052692396658037445od_int
% 0.30/0.57 = ( ^ [M3: num,N3: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_int_def
% 0.30/0.57 thf(fact_3576_divmod__def,axiom,
% 0.30/0.57 ( unique5052692396658037445od_int
% 0.30/0.57 = ( ^ [M3: num,N3: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_def
% 0.30/0.57 thf(fact_3577_divmod__def,axiom,
% 0.30/0.57 ( unique5055182867167087721od_nat
% 0.30/0.57 = ( ^ [M3: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_def
% 0.30/0.57 thf(fact_3578_divmod__def,axiom,
% 0.30/0.57 ( unique3479559517661332726nteger
% 0.30/0.57 = ( ^ [M3: num,N3: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N3 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_def
% 0.30/0.57 thf(fact_3579_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
% 0.30/0.57 ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 0.30/0.57 ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) )
% 0.30/0.57 = one_one_nat ) ).
% 0.30/0.57
% 0.30/0.57 % T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
% 0.30/0.57 thf(fact_3580_divmod_H__nat__def,axiom,
% 0.30/0.57 ( unique5055182867167087721od_nat
% 0.30/0.57 = ( ^ [M3: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod'_nat_def
% 0.30/0.57 thf(fact_3581_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
% 0.30/0.57 ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
% 0.30/0.57 ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
% 0.30/0.57 = one_one_nat ) ).
% 0.30/0.57
% 0.30/0.57 % T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
% 0.30/0.57 thf(fact_3582_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
% 0.30/0.57 ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
% 0.30/0.57 ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ X2 )
% 0.30/0.57 = one_one_nat ) ).
% 0.30/0.57
% 0.30/0.57 % T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
% 0.30/0.57 thf(fact_3583_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
% 0.30/0.57 ! [A: $o,B: $o,X2: nat] :
% 0.30/0.57 ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Leaf @ A @ B ) @ X2 )
% 0.30/0.57 = ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( X2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
% 0.30/0.57 thf(fact_3584_eq__diff__eq_H,axiom,
% 0.30/0.57 ! [X2: real,Y3: real,Z: real] :
% 0.30/0.57 ( ( X2
% 0.30/0.57 = ( minus_minus_real @ Y3 @ Z ) )
% 0.30/0.57 = ( Y3
% 0.30/0.57 = ( plus_plus_real @ X2 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % eq_diff_eq'
% 0.30/0.57 thf(fact_3585_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
% 0.30/0.57 ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 0.30/0.57 ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X2 )
% 0.30/0.57 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
% 0.30/0.57 thf(fact_3586_divmod__divmod__step,axiom,
% 0.30/0.57 ( unique5055182867167087721od_nat
% 0.30/0.57 = ( ^ [M3: num,N3: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M3 @ N3 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M3 ) ) @ ( unique5026877609467782581ep_nat @ N3 @ ( unique5055182867167087721od_nat @ M3 @ ( bit0 @ N3 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_divmod_step
% 0.30/0.57 thf(fact_3587_divmod__divmod__step,axiom,
% 0.30/0.57 ( unique5052692396658037445od_int
% 0.30/0.57 = ( ^ [M3: num,N3: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M3 @ N3 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M3 ) ) @ ( unique5024387138958732305ep_int @ N3 @ ( unique5052692396658037445od_int @ M3 @ ( bit0 @ N3 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_divmod_step
% 0.30/0.57 thf(fact_3588_divmod__divmod__step,axiom,
% 0.30/0.57 ( unique3479559517661332726nteger
% 0.30/0.57 = ( ^ [M3: num,N3: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M3 @ N3 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M3 ) ) @ ( unique4921790084139445826nteger @ N3 @ ( unique3479559517661332726nteger @ M3 @ ( bit0 @ N3 ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_divmod_step
% 0.30/0.57 thf(fact_3589_insersimp,axiom,
% 0.30/0.57 ! [T: vEBT_VEBT,N: nat,Y3: nat] :
% 0.30/0.57 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.57 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 )
% 0.30/0.57 => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ Y3 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % insersimp
% 0.30/0.57 thf(fact_3590_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
% 0.30/0.57 ! [X2: vEBT_VEBT,Y3: nat] :
% 0.30/0.57 ( ( ( vEBT_T_m_i_n_N_u_l_l @ X2 )
% 0.30/0.57 = Y3 )
% 0.30/0.57 => ( ( ( X2
% 0.30/0.57 = ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.57 => ( Y3 != one_one_nat ) )
% 0.30/0.57 => ( ( ? [Uv2: $o] :
% 0.30/0.57 ( X2
% 0.30/0.57 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.30/0.57 => ( Y3 != one_one_nat ) )
% 0.30/0.57 => ( ( ? [Uu2: $o] :
% 0.30/0.57 ( X2
% 0.30/0.57 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.30/0.57 => ( Y3 != one_one_nat ) )
% 0.30/0.57 => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.57 ( X2
% 0.30/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.30/0.57 => ( Y3 != one_one_nat ) )
% 0.30/0.57 => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.57 ( X2
% 0.30/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
% 0.30/0.57 => ( Y3 != one_one_nat ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
% 0.30/0.57 thf(fact_3591_insertsimp,axiom,
% 0.30/0.57 ! [T: vEBT_VEBT,N: nat,L: nat] :
% 0.30/0.57 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.57 => ( ( vEBT_VEBT_minNull @ T )
% 0.30/0.57 => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ L ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % insertsimp
% 0.30/0.57 thf(fact_3592_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
% 0.30/0.57 ! [X2: vEBT_VEBT,Xa2: nat,Y3: nat] :
% 0.30/0.57 ( ( ( vEBT_T_i_n_s_e_r_t @ X2 @ Xa2 )
% 0.30/0.57 = Y3 )
% 0.30/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.30/0.57 => ( ! [A4: $o,B4: $o] :
% 0.30/0.57 ( ( X2
% 0.30/0.57 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.57 => ( ( Y3
% 0.30/0.57 = ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
% 0.30/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
% 0.30/0.57 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.57 ( ( X2
% 0.30/0.57 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 0.30/0.57 => ( ( Y3 = one_one_nat )
% 0.30/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) @ Xa2 ) ) ) )
% 0.30/0.57 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.57 ( ( X2
% 0.30/0.57 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 0.30/0.57 => ( ( Y3 = one_one_nat )
% 0.30/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) @ Xa2 ) ) ) )
% 0.30/0.57 => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.57 ( ( X2
% 0.30/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.57 => ( ( Y3
% 0.30/0.57 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
% 0.30/0.57 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.57 ( ( X2
% 0.30/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.57 => ( ( Y3
% 0.30/0.57 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.30/0.57 @ ( if_nat
% 0.30/0.57 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.57 & ~ ( ( Xa2 = Mi2 )
% 0.30/0.57 | ( Xa2 = Ma2 ) ) )
% 0.30/0.57 @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
% 0.30/0.57 @ one_one_nat ) ) )
% 0.30/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
% 0.30/0.57 thf(fact_3593_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_real,X2: real > complex,Y3: real > complex] :
% 0.30/0.57 ( ( finite_finite_real
% 0.30/0.57 @ ( collect_real
% 0.30/0.57 @ ^ [I4: real] :
% 0.30/0.57 ( ( member_real @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != one_one_complex ) ) ) )
% 0.30/0.57 => ( ( finite_finite_real
% 0.30/0.57 @ ( collect_real
% 0.30/0.57 @ ^ [I4: real] :
% 0.30/0.57 ( ( member_real @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != one_one_complex ) ) ) )
% 0.30/0.57 => ( finite_finite_real
% 0.30/0.57 @ ( collect_real
% 0.30/0.57 @ ^ [I4: real] :
% 0.30/0.57 ( ( member_real @ I4 @ I5 )
% 0.30/0.57 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != one_one_complex ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.finite_Collect_op
% 0.30/0.57 thf(fact_3594_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_nat,X2: nat > complex,Y3: nat > complex] :
% 0.30/0.57 ( ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [I4: nat] :
% 0.30/0.57 ( ( member_nat @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != one_one_complex ) ) ) )
% 0.30/0.57 => ( ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [I4: nat] :
% 0.30/0.57 ( ( member_nat @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != one_one_complex ) ) ) )
% 0.30/0.57 => ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [I4: nat] :
% 0.30/0.57 ( ( member_nat @ I4 @ I5 )
% 0.30/0.57 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != one_one_complex ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.finite_Collect_op
% 0.30/0.57 thf(fact_3595_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_int,X2: int > complex,Y3: int > complex] :
% 0.30/0.57 ( ( finite_finite_int
% 0.30/0.57 @ ( collect_int
% 0.30/0.57 @ ^ [I4: int] :
% 0.30/0.57 ( ( member_int @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != one_one_complex ) ) ) )
% 0.30/0.57 => ( ( finite_finite_int
% 0.30/0.57 @ ( collect_int
% 0.30/0.57 @ ^ [I4: int] :
% 0.30/0.57 ( ( member_int @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != one_one_complex ) ) ) )
% 0.30/0.57 => ( finite_finite_int
% 0.30/0.57 @ ( collect_int
% 0.30/0.57 @ ^ [I4: int] :
% 0.30/0.57 ( ( member_int @ I4 @ I5 )
% 0.30/0.57 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != one_one_complex ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.finite_Collect_op
% 0.30/0.57 thf(fact_3596_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_complex,X2: complex > complex,Y3: complex > complex] :
% 0.30/0.57 ( ( finite3207457112153483333omplex
% 0.30/0.57 @ ( collect_complex
% 0.30/0.57 @ ^ [I4: complex] :
% 0.30/0.57 ( ( member_complex @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != one_one_complex ) ) ) )
% 0.30/0.57 => ( ( finite3207457112153483333omplex
% 0.30/0.57 @ ( collect_complex
% 0.30/0.57 @ ^ [I4: complex] :
% 0.30/0.57 ( ( member_complex @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != one_one_complex ) ) ) )
% 0.30/0.57 => ( finite3207457112153483333omplex
% 0.30/0.57 @ ( collect_complex
% 0.30/0.57 @ ^ [I4: complex] :
% 0.30/0.57 ( ( member_complex @ I4 @ I5 )
% 0.30/0.57 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != one_one_complex ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.finite_Collect_op
% 0.30/0.57 thf(fact_3597_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_real,X2: real > real,Y3: real > real] :
% 0.30/0.57 ( ( finite_finite_real
% 0.30/0.57 @ ( collect_real
% 0.30/0.57 @ ^ [I4: real] :
% 0.30/0.57 ( ( member_real @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != one_one_real ) ) ) )
% 0.30/0.57 => ( ( finite_finite_real
% 0.30/0.57 @ ( collect_real
% 0.30/0.57 @ ^ [I4: real] :
% 0.30/0.57 ( ( member_real @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != one_one_real ) ) ) )
% 0.30/0.57 => ( finite_finite_real
% 0.30/0.57 @ ( collect_real
% 0.30/0.57 @ ^ [I4: real] :
% 0.30/0.57 ( ( member_real @ I4 @ I5 )
% 0.30/0.57 & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != one_one_real ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.finite_Collect_op
% 0.30/0.57 thf(fact_3598_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_nat,X2: nat > real,Y3: nat > real] :
% 0.30/0.57 ( ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [I4: nat] :
% 0.30/0.57 ( ( member_nat @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != one_one_real ) ) ) )
% 0.30/0.57 => ( ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [I4: nat] :
% 0.30/0.57 ( ( member_nat @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != one_one_real ) ) ) )
% 0.30/0.57 => ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [I4: nat] :
% 0.30/0.57 ( ( member_nat @ I4 @ I5 )
% 0.30/0.57 & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != one_one_real ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.finite_Collect_op
% 0.30/0.57 thf(fact_3599_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_int,X2: int > real,Y3: int > real] :
% 0.30/0.57 ( ( finite_finite_int
% 0.30/0.57 @ ( collect_int
% 0.30/0.57 @ ^ [I4: int] :
% 0.30/0.57 ( ( member_int @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != one_one_real ) ) ) )
% 0.30/0.57 => ( ( finite_finite_int
% 0.30/0.57 @ ( collect_int
% 0.30/0.57 @ ^ [I4: int] :
% 0.30/0.57 ( ( member_int @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != one_one_real ) ) ) )
% 0.30/0.57 => ( finite_finite_int
% 0.30/0.57 @ ( collect_int
% 0.30/0.57 @ ^ [I4: int] :
% 0.30/0.57 ( ( member_int @ I4 @ I5 )
% 0.30/0.57 & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != one_one_real ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.finite_Collect_op
% 0.30/0.57 thf(fact_3600_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_complex,X2: complex > real,Y3: complex > real] :
% 0.30/0.57 ( ( finite3207457112153483333omplex
% 0.30/0.57 @ ( collect_complex
% 0.30/0.57 @ ^ [I4: complex] :
% 0.30/0.57 ( ( member_complex @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != one_one_real ) ) ) )
% 0.30/0.57 => ( ( finite3207457112153483333omplex
% 0.30/0.57 @ ( collect_complex
% 0.30/0.57 @ ^ [I4: complex] :
% 0.30/0.57 ( ( member_complex @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != one_one_real ) ) ) )
% 0.30/0.57 => ( finite3207457112153483333omplex
% 0.30/0.57 @ ( collect_complex
% 0.30/0.57 @ ^ [I4: complex] :
% 0.30/0.57 ( ( member_complex @ I4 @ I5 )
% 0.30/0.57 & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != one_one_real ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.finite_Collect_op
% 0.30/0.57 thf(fact_3601_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_real,X2: real > rat,Y3: real > rat] :
% 0.30/0.57 ( ( finite_finite_real
% 0.30/0.57 @ ( collect_real
% 0.30/0.57 @ ^ [I4: real] :
% 0.30/0.57 ( ( member_real @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != one_one_rat ) ) ) )
% 0.30/0.57 => ( ( finite_finite_real
% 0.30/0.57 @ ( collect_real
% 0.30/0.57 @ ^ [I4: real] :
% 0.30/0.57 ( ( member_real @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != one_one_rat ) ) ) )
% 0.30/0.57 => ( finite_finite_real
% 0.30/0.57 @ ( collect_real
% 0.30/0.57 @ ^ [I4: real] :
% 0.30/0.57 ( ( member_real @ I4 @ I5 )
% 0.30/0.57 & ( ( times_times_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != one_one_rat ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.finite_Collect_op
% 0.30/0.57 thf(fact_3602_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_nat,X2: nat > rat,Y3: nat > rat] :
% 0.30/0.57 ( ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [I4: nat] :
% 0.30/0.57 ( ( member_nat @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != one_one_rat ) ) ) )
% 0.30/0.57 => ( ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [I4: nat] :
% 0.30/0.57 ( ( member_nat @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != one_one_rat ) ) ) )
% 0.30/0.57 => ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [I4: nat] :
% 0.30/0.57 ( ( member_nat @ I4 @ I5 )
% 0.30/0.57 & ( ( times_times_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != one_one_rat ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.finite_Collect_op
% 0.30/0.57 thf(fact_3603_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_real,X2: real > complex,Y3: real > complex] :
% 0.30/0.57 ( ( finite_finite_real
% 0.30/0.57 @ ( collect_real
% 0.30/0.57 @ ^ [I4: real] :
% 0.30/0.57 ( ( member_real @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != zero_zero_complex ) ) ) )
% 0.30/0.57 => ( ( finite_finite_real
% 0.30/0.57 @ ( collect_real
% 0.30/0.57 @ ^ [I4: real] :
% 0.30/0.57 ( ( member_real @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != zero_zero_complex ) ) ) )
% 0.30/0.57 => ( finite_finite_real
% 0.30/0.57 @ ( collect_real
% 0.30/0.57 @ ^ [I4: real] :
% 0.30/0.57 ( ( member_real @ I4 @ I5 )
% 0.30/0.57 & ( ( plus_plus_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != zero_zero_complex ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.finite_Collect_op
% 0.30/0.57 thf(fact_3604_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_nat,X2: nat > complex,Y3: nat > complex] :
% 0.30/0.57 ( ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [I4: nat] :
% 0.30/0.57 ( ( member_nat @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != zero_zero_complex ) ) ) )
% 0.30/0.57 => ( ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [I4: nat] :
% 0.30/0.57 ( ( member_nat @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != zero_zero_complex ) ) ) )
% 0.30/0.57 => ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [I4: nat] :
% 0.30/0.57 ( ( member_nat @ I4 @ I5 )
% 0.30/0.57 & ( ( plus_plus_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != zero_zero_complex ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.finite_Collect_op
% 0.30/0.57 thf(fact_3605_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_int,X2: int > complex,Y3: int > complex] :
% 0.30/0.57 ( ( finite_finite_int
% 0.30/0.57 @ ( collect_int
% 0.30/0.57 @ ^ [I4: int] :
% 0.30/0.57 ( ( member_int @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != zero_zero_complex ) ) ) )
% 0.30/0.57 => ( ( finite_finite_int
% 0.30/0.57 @ ( collect_int
% 0.30/0.57 @ ^ [I4: int] :
% 0.30/0.57 ( ( member_int @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != zero_zero_complex ) ) ) )
% 0.30/0.57 => ( finite_finite_int
% 0.30/0.57 @ ( collect_int
% 0.30/0.57 @ ^ [I4: int] :
% 0.30/0.57 ( ( member_int @ I4 @ I5 )
% 0.30/0.57 & ( ( plus_plus_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != zero_zero_complex ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.finite_Collect_op
% 0.30/0.57 thf(fact_3606_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_complex,X2: complex > complex,Y3: complex > complex] :
% 0.30/0.57 ( ( finite3207457112153483333omplex
% 0.30/0.57 @ ( collect_complex
% 0.30/0.57 @ ^ [I4: complex] :
% 0.30/0.57 ( ( member_complex @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != zero_zero_complex ) ) ) )
% 0.30/0.57 => ( ( finite3207457112153483333omplex
% 0.30/0.57 @ ( collect_complex
% 0.30/0.57 @ ^ [I4: complex] :
% 0.30/0.57 ( ( member_complex @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != zero_zero_complex ) ) ) )
% 0.30/0.57 => ( finite3207457112153483333omplex
% 0.30/0.57 @ ( collect_complex
% 0.30/0.57 @ ^ [I4: complex] :
% 0.30/0.57 ( ( member_complex @ I4 @ I5 )
% 0.30/0.57 & ( ( plus_plus_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != zero_zero_complex ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.finite_Collect_op
% 0.30/0.57 thf(fact_3607_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_real,X2: real > real,Y3: real > real] :
% 0.30/0.57 ( ( finite_finite_real
% 0.30/0.57 @ ( collect_real
% 0.30/0.57 @ ^ [I4: real] :
% 0.30/0.57 ( ( member_real @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != zero_zero_real ) ) ) )
% 0.30/0.57 => ( ( finite_finite_real
% 0.30/0.57 @ ( collect_real
% 0.30/0.57 @ ^ [I4: real] :
% 0.30/0.57 ( ( member_real @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != zero_zero_real ) ) ) )
% 0.30/0.57 => ( finite_finite_real
% 0.30/0.57 @ ( collect_real
% 0.30/0.57 @ ^ [I4: real] :
% 0.30/0.57 ( ( member_real @ I4 @ I5 )
% 0.30/0.57 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.finite_Collect_op
% 0.30/0.57 thf(fact_3608_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_nat,X2: nat > real,Y3: nat > real] :
% 0.30/0.57 ( ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [I4: nat] :
% 0.30/0.57 ( ( member_nat @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != zero_zero_real ) ) ) )
% 0.30/0.57 => ( ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [I4: nat] :
% 0.30/0.57 ( ( member_nat @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != zero_zero_real ) ) ) )
% 0.30/0.57 => ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [I4: nat] :
% 0.30/0.57 ( ( member_nat @ I4 @ I5 )
% 0.30/0.57 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.finite_Collect_op
% 0.30/0.57 thf(fact_3609_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_int,X2: int > real,Y3: int > real] :
% 0.30/0.57 ( ( finite_finite_int
% 0.30/0.57 @ ( collect_int
% 0.30/0.57 @ ^ [I4: int] :
% 0.30/0.57 ( ( member_int @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != zero_zero_real ) ) ) )
% 0.30/0.57 => ( ( finite_finite_int
% 0.30/0.57 @ ( collect_int
% 0.30/0.57 @ ^ [I4: int] :
% 0.30/0.57 ( ( member_int @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != zero_zero_real ) ) ) )
% 0.30/0.57 => ( finite_finite_int
% 0.30/0.57 @ ( collect_int
% 0.30/0.57 @ ^ [I4: int] :
% 0.30/0.57 ( ( member_int @ I4 @ I5 )
% 0.30/0.57 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.finite_Collect_op
% 0.30/0.57 thf(fact_3610_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_complex,X2: complex > real,Y3: complex > real] :
% 0.30/0.57 ( ( finite3207457112153483333omplex
% 0.30/0.57 @ ( collect_complex
% 0.30/0.57 @ ^ [I4: complex] :
% 0.30/0.57 ( ( member_complex @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != zero_zero_real ) ) ) )
% 0.30/0.57 => ( ( finite3207457112153483333omplex
% 0.30/0.57 @ ( collect_complex
% 0.30/0.57 @ ^ [I4: complex] :
% 0.30/0.57 ( ( member_complex @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != zero_zero_real ) ) ) )
% 0.30/0.57 => ( finite3207457112153483333omplex
% 0.30/0.57 @ ( collect_complex
% 0.30/0.57 @ ^ [I4: complex] :
% 0.30/0.57 ( ( member_complex @ I4 @ I5 )
% 0.30/0.57 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.finite_Collect_op
% 0.30/0.57 thf(fact_3611_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_real,X2: real > rat,Y3: real > rat] :
% 0.30/0.57 ( ( finite_finite_real
% 0.30/0.57 @ ( collect_real
% 0.30/0.57 @ ^ [I4: real] :
% 0.30/0.57 ( ( member_real @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != zero_zero_rat ) ) ) )
% 0.30/0.57 => ( ( finite_finite_real
% 0.30/0.57 @ ( collect_real
% 0.30/0.57 @ ^ [I4: real] :
% 0.30/0.57 ( ( member_real @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != zero_zero_rat ) ) ) )
% 0.30/0.57 => ( finite_finite_real
% 0.30/0.57 @ ( collect_real
% 0.30/0.57 @ ^ [I4: real] :
% 0.30/0.57 ( ( member_real @ I4 @ I5 )
% 0.30/0.57 & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.finite_Collect_op
% 0.30/0.57 thf(fact_3612_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57 ! [I5: set_nat,X2: nat > rat,Y3: nat > rat] :
% 0.30/0.57 ( ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [I4: nat] :
% 0.30/0.57 ( ( member_nat @ I4 @ I5 )
% 0.30/0.57 & ( ( X2 @ I4 )
% 0.30/0.57 != zero_zero_rat ) ) ) )
% 0.30/0.57 => ( ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [I4: nat] :
% 0.30/0.57 ( ( member_nat @ I4 @ I5 )
% 0.30/0.57 & ( ( Y3 @ I4 )
% 0.30/0.57 != zero_zero_rat ) ) ) )
% 0.30/0.57 => ( finite_finite_nat
% 0.30/0.57 @ ( collect_nat
% 0.30/0.57 @ ^ [I4: nat] :
% 0.30/0.57 ( ( member_nat @ I4 @ I5 )
% 0.30/0.57 & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57 != zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % sum.finite_Collect_op
% 0.30/0.57 thf(fact_3613_dbl__inc__simps_I3_J,axiom,
% 0.30/0.57 ( ( neg_nu8557863876264182079omplex @ one_one_complex )
% 0.30/0.57 = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dbl_inc_simps(3)
% 0.30/0.57 thf(fact_3614_dbl__inc__simps_I3_J,axiom,
% 0.30/0.57 ( ( neg_nu8295874005876285629c_real @ one_one_real )
% 0.30/0.57 = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dbl_inc_simps(3)
% 0.30/0.57 thf(fact_3615_dbl__inc__simps_I3_J,axiom,
% 0.30/0.57 ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
% 0.30/0.57 = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dbl_inc_simps(3)
% 0.30/0.57 thf(fact_3616_dbl__inc__simps_I3_J,axiom,
% 0.30/0.57 ( ( neg_nu5851722552734809277nc_int @ one_one_int )
% 0.30/0.57 = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dbl_inc_simps(3)
% 0.30/0.57 thf(fact_3617_prod_Oinject,axiom,
% 0.30/0.57 ! [X1: code_integer,X22: $o,Y1: code_integer,Y2: $o] :
% 0.30/0.57 ( ( ( produc6677183202524767010eger_o @ X1 @ X22 )
% 0.30/0.57 = ( produc6677183202524767010eger_o @ Y1 @ Y2 ) )
% 0.30/0.57 = ( ( X1 = Y1 )
% 0.30/0.57 & ( X22 = Y2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.inject
% 0.30/0.57 thf(fact_3618_prod_Oinject,axiom,
% 0.30/0.57 ! [X1: num,X22: num,Y1: num,Y2: num] :
% 0.30/0.57 ( ( ( product_Pair_num_num @ X1 @ X22 )
% 0.30/0.57 = ( product_Pair_num_num @ Y1 @ Y2 ) )
% 0.30/0.57 = ( ( X1 = Y1 )
% 0.30/0.57 & ( X22 = Y2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.inject
% 0.30/0.57 thf(fact_3619_prod_Oinject,axiom,
% 0.30/0.57 ! [X1: nat,X22: num,Y1: nat,Y2: num] :
% 0.30/0.57 ( ( ( product_Pair_nat_num @ X1 @ X22 )
% 0.30/0.57 = ( product_Pair_nat_num @ Y1 @ Y2 ) )
% 0.30/0.57 = ( ( X1 = Y1 )
% 0.30/0.57 & ( X22 = Y2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.inject
% 0.30/0.57 thf(fact_3620_prod_Oinject,axiom,
% 0.30/0.57 ! [X1: nat,X22: nat,Y1: nat,Y2: nat] :
% 0.30/0.57 ( ( ( product_Pair_nat_nat @ X1 @ X22 )
% 0.30/0.57 = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
% 0.30/0.57 = ( ( X1 = Y1 )
% 0.30/0.57 & ( X22 = Y2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.inject
% 0.30/0.57 thf(fact_3621_prod_Oinject,axiom,
% 0.30/0.57 ! [X1: int,X22: int,Y1: int,Y2: int] :
% 0.30/0.57 ( ( ( product_Pair_int_int @ X1 @ X22 )
% 0.30/0.57 = ( product_Pair_int_int @ Y1 @ Y2 ) )
% 0.30/0.57 = ( ( X1 = Y1 )
% 0.30/0.57 & ( X22 = Y2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod.inject
% 0.30/0.57 thf(fact_3622_old_Oprod_Oinject,axiom,
% 0.30/0.57 ! [A: code_integer,B: $o,A5: code_integer,B5: $o] :
% 0.30/0.57 ( ( ( produc6677183202524767010eger_o @ A @ B )
% 0.30/0.57 = ( produc6677183202524767010eger_o @ A5 @ B5 ) )
% 0.30/0.57 = ( ( A = A5 )
% 0.30/0.57 & ( B = B5 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % old.prod.inject
% 0.30/0.57 thf(fact_3623_old_Oprod_Oinject,axiom,
% 0.30/0.57 ! [A: num,B: num,A5: num,B5: num] :
% 0.30/0.57 ( ( ( product_Pair_num_num @ A @ B )
% 0.30/0.57 = ( product_Pair_num_num @ A5 @ B5 ) )
% 0.30/0.57 = ( ( A = A5 )
% 0.30/0.57 & ( B = B5 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % old.prod.inject
% 0.30/0.57 thf(fact_3624_old_Oprod_Oinject,axiom,
% 0.30/0.57 ! [A: nat,B: num,A5: nat,B5: num] :
% 0.30/0.57 ( ( ( product_Pair_nat_num @ A @ B )
% 0.30/0.57 = ( product_Pair_nat_num @ A5 @ B5 ) )
% 0.30/0.57 = ( ( A = A5 )
% 0.30/0.57 & ( B = B5 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % old.prod.inject
% 0.30/0.57 thf(fact_3625_old_Oprod_Oinject,axiom,
% 0.30/0.57 ! [A: nat,B: nat,A5: nat,B5: nat] :
% 0.30/0.57 ( ( ( product_Pair_nat_nat @ A @ B )
% 0.30/0.57 = ( product_Pair_nat_nat @ A5 @ B5 ) )
% 0.30/0.57 = ( ( A = A5 )
% 0.30/0.57 & ( B = B5 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % old.prod.inject
% 0.30/0.57 thf(fact_3626_old_Oprod_Oinject,axiom,
% 0.30/0.57 ! [A: int,B: int,A5: int,B5: int] :
% 0.30/0.57 ( ( ( product_Pair_int_int @ A @ B )
% 0.30/0.57 = ( product_Pair_int_int @ A5 @ B5 ) )
% 0.30/0.57 = ( ( A = A5 )
% 0.30/0.57 & ( B = B5 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % old.prod.inject
% 0.30/0.57 thf(fact_3627_dbl__inc__simps_I2_J,axiom,
% 0.30/0.57 ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
% 0.30/0.57 = one_one_complex ) ).
% 0.30/0.57
% 0.30/0.57 % dbl_inc_simps(2)
% 0.30/0.57 thf(fact_3628_dbl__inc__simps_I2_J,axiom,
% 0.30/0.57 ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
% 0.30/0.57 = one_one_real ) ).
% 0.30/0.57
% 0.30/0.57 % dbl_inc_simps(2)
% 0.30/0.57 thf(fact_3629_dbl__inc__simps_I2_J,axiom,
% 0.30/0.57 ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
% 0.30/0.57 = one_one_rat ) ).
% 0.30/0.57
% 0.30/0.57 % dbl_inc_simps(2)
% 0.30/0.57 thf(fact_3630_dbl__inc__simps_I2_J,axiom,
% 0.30/0.57 ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 0.30/0.57 = one_one_int ) ).
% 0.30/0.57
% 0.30/0.57 % dbl_inc_simps(2)
% 0.30/0.57 thf(fact_3631_dbl__inc__simps_I5_J,axiom,
% 0.30/0.57 ! [K: num] :
% 0.30/0.57 ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.30/0.57 = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dbl_inc_simps(5)
% 0.30/0.57 thf(fact_3632_dbl__inc__simps_I5_J,axiom,
% 0.30/0.57 ! [K: num] :
% 0.30/0.57 ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
% 0.30/0.57 = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dbl_inc_simps(5)
% 0.30/0.57 thf(fact_3633_dbl__inc__simps_I5_J,axiom,
% 0.30/0.57 ! [K: num] :
% 0.30/0.57 ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
% 0.30/0.57 = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dbl_inc_simps(5)
% 0.30/0.57 thf(fact_3634_dbl__inc__simps_I5_J,axiom,
% 0.30/0.57 ! [K: num] :
% 0.30/0.57 ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
% 0.30/0.57 = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dbl_inc_simps(5)
% 0.30/0.57 thf(fact_3635_old_Oprod_Oexhaust,axiom,
% 0.30/0.57 ! [Y3: produc6271795597528267376eger_o] :
% 0.30/0.57 ~ ! [A4: code_integer,B4: $o] :
% 0.30/0.57 ( Y3
% 0.30/0.57 != ( produc6677183202524767010eger_o @ A4 @ B4 ) ) ).
% 0.30/0.57
% 0.30/0.57 % old.prod.exhaust
% 0.30/0.57 thf(fact_3636_old_Oprod_Oexhaust,axiom,
% 0.30/0.57 ! [Y3: product_prod_num_num] :
% 0.30/0.57 ~ ! [A4: num,B4: num] :
% 0.30/0.57 ( Y3
% 0.30/0.57 != ( product_Pair_num_num @ A4 @ B4 ) ) ).
% 0.30/0.57
% 0.30/0.57 % old.prod.exhaust
% 0.30/0.57 thf(fact_3637_old_Oprod_Oexhaust,axiom,
% 0.30/0.57 ! [Y3: product_prod_nat_num] :
% 0.30/0.57 ~ ! [A4: nat,B4: num] :
% 0.30/0.57 ( Y3
% 0.30/0.57 != ( product_Pair_nat_num @ A4 @ B4 ) ) ).
% 0.30/0.57
% 0.30/0.57 % old.prod.exhaust
% 0.30/0.57 thf(fact_3638_old_Oprod_Oexhaust,axiom,
% 0.30/0.57 ! [Y3: product_prod_nat_nat] :
% 0.30/0.57 ~ ! [A4: nat,B4: nat] :
% 0.30/0.57 ( Y3
% 0.30/0.57 != ( product_Pair_nat_nat @ A4 @ B4 ) ) ).
% 0.30/0.57
% 0.30/0.57 % old.prod.exhaust
% 0.30/0.57 thf(fact_3639_old_Oprod_Oexhaust,axiom,
% 0.30/0.57 ! [Y3: product_prod_int_int] :
% 0.30/0.57 ~ ! [A4: int,B4: int] :
% 0.30/0.57 ( Y3
% 0.30/0.57 != ( product_Pair_int_int @ A4 @ B4 ) ) ).
% 0.30/0.57
% 0.30/0.57 % old.prod.exhaust
% 0.30/0.57 thf(fact_3640_surj__pair,axiom,
% 0.30/0.57 ! [P6: produc6271795597528267376eger_o] :
% 0.30/0.57 ? [X5: code_integer,Y4: $o] :
% 0.30/0.57 ( P6
% 0.30/0.57 = ( produc6677183202524767010eger_o @ X5 @ Y4 ) ) ).
% 0.30/0.57
% 0.30/0.57 % surj_pair
% 0.30/0.57 thf(fact_3641_surj__pair,axiom,
% 0.30/0.57 ! [P6: product_prod_num_num] :
% 0.30/0.57 ? [X5: num,Y4: num] :
% 0.30/0.57 ( P6
% 0.30/0.57 = ( product_Pair_num_num @ X5 @ Y4 ) ) ).
% 0.30/0.57
% 0.30/0.57 % surj_pair
% 0.30/0.57 thf(fact_3642_surj__pair,axiom,
% 0.30/0.57 ! [P6: product_prod_nat_num] :
% 0.30/0.57 ? [X5: nat,Y4: num] :
% 0.30/0.57 ( P6
% 0.30/0.57 = ( product_Pair_nat_num @ X5 @ Y4 ) ) ).
% 0.30/0.57
% 0.30/0.57 % surj_pair
% 0.30/0.57 thf(fact_3643_surj__pair,axiom,
% 0.30/0.57 ! [P6: product_prod_nat_nat] :
% 0.30/0.57 ? [X5: nat,Y4: nat] :
% 0.30/0.57 ( P6
% 0.30/0.57 = ( product_Pair_nat_nat @ X5 @ Y4 ) ) ).
% 0.30/0.57
% 0.30/0.57 % surj_pair
% 0.30/0.57 thf(fact_3644_surj__pair,axiom,
% 0.30/0.57 ! [P6: product_prod_int_int] :
% 0.30/0.57 ? [X5: int,Y4: int] :
% 0.30/0.57 ( P6
% 0.30/0.57 = ( product_Pair_int_int @ X5 @ Y4 ) ) ).
% 0.30/0.57
% 0.30/0.57 % surj_pair
% 0.30/0.57 thf(fact_3645_prod__cases,axiom,
% 0.30/0.57 ! [P: produc6271795597528267376eger_o > $o,P6: produc6271795597528267376eger_o] :
% 0.30/0.57 ( ! [A4: code_integer,B4: $o] : ( P @ ( produc6677183202524767010eger_o @ A4 @ B4 ) )
% 0.30/0.57 => ( P @ P6 ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_cases
% 0.30/0.57 thf(fact_3646_prod__cases,axiom,
% 0.30/0.57 ! [P: product_prod_num_num > $o,P6: product_prod_num_num] :
% 0.30/0.57 ( ! [A4: num,B4: num] : ( P @ ( product_Pair_num_num @ A4 @ B4 ) )
% 0.30/0.57 => ( P @ P6 ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_cases
% 0.30/0.57 thf(fact_3647_prod__cases,axiom,
% 0.30/0.57 ! [P: product_prod_nat_num > $o,P6: product_prod_nat_num] :
% 0.30/0.57 ( ! [A4: nat,B4: num] : ( P @ ( product_Pair_nat_num @ A4 @ B4 ) )
% 0.30/0.57 => ( P @ P6 ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_cases
% 0.30/0.57 thf(fact_3648_prod__cases,axiom,
% 0.30/0.57 ! [P: product_prod_nat_nat > $o,P6: product_prod_nat_nat] :
% 0.30/0.57 ( ! [A4: nat,B4: nat] : ( P @ ( product_Pair_nat_nat @ A4 @ B4 ) )
% 0.30/0.57 => ( P @ P6 ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_cases
% 0.30/0.57 thf(fact_3649_prod__cases,axiom,
% 0.30/0.57 ! [P: product_prod_int_int > $o,P6: product_prod_int_int] :
% 0.30/0.57 ( ! [A4: int,B4: int] : ( P @ ( product_Pair_int_int @ A4 @ B4 ) )
% 0.30/0.57 => ( P @ P6 ) ) ).
% 0.30/0.57
% 0.30/0.57 % prod_cases
% 0.30/0.57 thf(fact_3650_Pair__inject,axiom,
% 0.30/0.57 ! [A: code_integer,B: $o,A5: code_integer,B5: $o] :
% 0.30/0.57 ( ( ( produc6677183202524767010eger_o @ A @ B )
% 0.30/0.57 = ( produc6677183202524767010eger_o @ A5 @ B5 ) )
% 0.30/0.57 => ~ ( ( A = A5 )
% 0.30/0.57 => ( B = ( ~ B5 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Pair_inject
% 0.30/0.57 thf(fact_3651_Pair__inject,axiom,
% 0.30/0.57 ! [A: num,B: num,A5: num,B5: num] :
% 0.30/0.57 ( ( ( product_Pair_num_num @ A @ B )
% 0.30/0.57 = ( product_Pair_num_num @ A5 @ B5 ) )
% 0.30/0.57 => ~ ( ( A = A5 )
% 0.30/0.57 => ( B != B5 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Pair_inject
% 0.30/0.57 thf(fact_3652_Pair__inject,axiom,
% 0.30/0.57 ! [A: nat,B: num,A5: nat,B5: num] :
% 0.30/0.57 ( ( ( product_Pair_nat_num @ A @ B )
% 0.30/0.57 = ( product_Pair_nat_num @ A5 @ B5 ) )
% 0.30/0.57 => ~ ( ( A = A5 )
% 0.30/0.57 => ( B != B5 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Pair_inject
% 0.30/0.57 thf(fact_3653_Pair__inject,axiom,
% 0.30/0.57 ! [A: nat,B: nat,A5: nat,B5: nat] :
% 0.30/0.57 ( ( ( product_Pair_nat_nat @ A @ B )
% 0.30/0.57 = ( product_Pair_nat_nat @ A5 @ B5 ) )
% 0.30/0.57 => ~ ( ( A = A5 )
% 0.30/0.57 => ( B != B5 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Pair_inject
% 0.30/0.57 thf(fact_3654_Pair__inject,axiom,
% 0.30/0.57 ! [A: int,B: int,A5: int,B5: int] :
% 0.30/0.57 ( ( ( product_Pair_int_int @ A @ B )
% 0.30/0.57 = ( product_Pair_int_int @ A5 @ B5 ) )
% 0.30/0.57 => ~ ( ( A = A5 )
% 0.30/0.57 => ( B != B5 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Pair_inject
% 0.30/0.57 thf(fact_3655_dbl__inc__def,axiom,
% 0.30/0.57 ( neg_nu8557863876264182079omplex
% 0.30/0.57 = ( ^ [X: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dbl_inc_def
% 0.30/0.57 thf(fact_3656_dbl__inc__def,axiom,
% 0.30/0.57 ( neg_nu8295874005876285629c_real
% 0.30/0.57 = ( ^ [X: real] : ( plus_plus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dbl_inc_def
% 0.30/0.57 thf(fact_3657_dbl__inc__def,axiom,
% 0.30/0.57 ( neg_nu5219082963157363817nc_rat
% 0.30/0.57 = ( ^ [X: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dbl_inc_def
% 0.30/0.57 thf(fact_3658_dbl__inc__def,axiom,
% 0.30/0.57 ( neg_nu5851722552734809277nc_int
% 0.30/0.57 = ( ^ [X: int] : ( plus_plus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dbl_inc_def
% 0.30/0.57 thf(fact_3659_VEBT__internal_Oheight_Osimps_I1_J,axiom,
% 0.30/0.57 ! [A: $o,B: $o] :
% 0.30/0.57 ( ( vEBT_VEBT_height @ ( vEBT_Leaf @ A @ B ) )
% 0.30/0.57 = zero_zero_nat ) ).
% 0.30/0.57
% 0.30/0.57 % VEBT_internal.height.simps(1)
% 0.30/0.57 thf(fact_3660_divmod__BitM__2__eq,axiom,
% 0.30/0.57 ! [M: num] :
% 0.30/0.57 ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 0.30/0.57 = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 0.30/0.57
% 0.30/0.57 % divmod_BitM_2_eq
% 0.30/0.57 thf(fact_3661_insert__simp__excp,axiom,
% 0.30/0.57 ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X2: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.30/0.57 ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.30/0.57 => ( ( ord_less_nat @ X2 @ Mi )
% 0.30/0.57 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.57 => ( ( X2 != Ma )
% 0.30/0.57 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 0.30/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X2 @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( 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 ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % insert_simp_excp
% 0.30/0.57 thf(fact_3662_insert__simp__norm,axiom,
% 0.30/0.57 ! [X2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.30/0.57 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.30/0.57 => ( ( ord_less_nat @ Mi @ X2 )
% 0.30/0.57 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.57 => ( ( X2 != Ma )
% 0.30/0.57 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 0.30/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X2 @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % insert_simp_norm
% 0.30/0.57 thf(fact_3663_dual__order_Orefl,axiom,
% 0.30/0.57 ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.refl
% 0.30/0.57 thf(fact_3664_dual__order_Orefl,axiom,
% 0.30/0.57 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.refl
% 0.30/0.57 thf(fact_3665_dual__order_Orefl,axiom,
% 0.30/0.57 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.refl
% 0.30/0.57 thf(fact_3666_dual__order_Orefl,axiom,
% 0.30/0.57 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.refl
% 0.30/0.57 thf(fact_3667_dual__order_Orefl,axiom,
% 0.30/0.57 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.refl
% 0.30/0.57 thf(fact_3668_order__refl,axiom,
% 0.30/0.57 ! [X2: set_int] : ( ord_less_eq_set_int @ X2 @ X2 ) ).
% 0.30/0.57
% 0.30/0.57 % order_refl
% 0.30/0.57 thf(fact_3669_order__refl,axiom,
% 0.30/0.57 ! [X2: rat] : ( ord_less_eq_rat @ X2 @ X2 ) ).
% 0.30/0.57
% 0.30/0.57 % order_refl
% 0.30/0.57 thf(fact_3670_order__refl,axiom,
% 0.30/0.57 ! [X2: num] : ( ord_less_eq_num @ X2 @ X2 ) ).
% 0.30/0.57
% 0.30/0.57 % order_refl
% 0.30/0.57 thf(fact_3671_order__refl,axiom,
% 0.30/0.57 ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% 0.30/0.57
% 0.30/0.57 % order_refl
% 0.30/0.57 thf(fact_3672_order__refl,axiom,
% 0.30/0.57 ! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% 0.30/0.57
% 0.30/0.57 % order_refl
% 0.30/0.57 thf(fact_3673_max_Oabsorb1,axiom,
% 0.30/0.57 ! [B: extended_enat,A: extended_enat] :
% 0.30/0.57 ( ( ord_le2932123472753598470d_enat @ B @ A )
% 0.30/0.57 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 0.30/0.57 = A ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb1
% 0.30/0.57 thf(fact_3674_max_Oabsorb1,axiom,
% 0.30/0.57 ! [B: rat,A: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.57 => ( ( ord_max_rat @ A @ B )
% 0.30/0.57 = A ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb1
% 0.30/0.57 thf(fact_3675_max_Oabsorb1,axiom,
% 0.30/0.57 ! [B: num,A: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ B @ A )
% 0.30/0.57 => ( ( ord_max_num @ A @ B )
% 0.30/0.57 = A ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb1
% 0.30/0.57 thf(fact_3676_max_Oabsorb1,axiom,
% 0.30/0.57 ! [B: nat,A: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.57 => ( ( ord_max_nat @ A @ B )
% 0.30/0.57 = A ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb1
% 0.30/0.57 thf(fact_3677_max_Oabsorb1,axiom,
% 0.30/0.57 ! [B: int,A: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ B @ A )
% 0.30/0.57 => ( ( ord_max_int @ A @ B )
% 0.30/0.57 = A ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb1
% 0.30/0.57 thf(fact_3678_max_Oabsorb2,axiom,
% 0.30/0.57 ! [A: extended_enat,B: extended_enat] :
% 0.30/0.57 ( ( ord_le2932123472753598470d_enat @ A @ B )
% 0.30/0.57 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 0.30/0.57 = B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb2
% 0.30/0.57 thf(fact_3679_max_Oabsorb2,axiom,
% 0.30/0.57 ! [A: rat,B: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.57 => ( ( ord_max_rat @ A @ B )
% 0.30/0.57 = B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb2
% 0.30/0.57 thf(fact_3680_max_Oabsorb2,axiom,
% 0.30/0.57 ! [A: num,B: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.57 => ( ( ord_max_num @ A @ B )
% 0.30/0.57 = B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb2
% 0.30/0.57 thf(fact_3681_max_Oabsorb2,axiom,
% 0.30/0.57 ! [A: nat,B: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.57 => ( ( ord_max_nat @ A @ B )
% 0.30/0.57 = B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb2
% 0.30/0.57 thf(fact_3682_max_Oabsorb2,axiom,
% 0.30/0.57 ! [A: int,B: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.57 => ( ( ord_max_int @ A @ B )
% 0.30/0.57 = B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb2
% 0.30/0.57 thf(fact_3683_max_Obounded__iff,axiom,
% 0.30/0.57 ! [B: extended_enat,C: extended_enat,A: extended_enat] :
% 0.30/0.57 ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
% 0.30/0.57 = ( ( ord_le2932123472753598470d_enat @ B @ A )
% 0.30/0.57 & ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.bounded_iff
% 0.30/0.57 thf(fact_3684_max_Obounded__iff,axiom,
% 0.30/0.57 ! [B: rat,C: rat,A: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 0.30/0.57 = ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.57 & ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.bounded_iff
% 0.30/0.57 thf(fact_3685_max_Obounded__iff,axiom,
% 0.30/0.57 ! [B: num,C: num,A: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 0.30/0.57 = ( ( ord_less_eq_num @ B @ A )
% 0.30/0.57 & ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.bounded_iff
% 0.30/0.57 thf(fact_3686_max_Obounded__iff,axiom,
% 0.30/0.57 ! [B: nat,C: nat,A: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 0.30/0.57 = ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.57 & ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.bounded_iff
% 0.30/0.57 thf(fact_3687_max_Obounded__iff,axiom,
% 0.30/0.57 ! [B: int,C: int,A: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 0.30/0.57 = ( ( ord_less_eq_int @ B @ A )
% 0.30/0.57 & ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.bounded_iff
% 0.30/0.57 thf(fact_3688_max__less__iff__conj,axiom,
% 0.30/0.57 ! [X2: extended_enat,Y3: extended_enat,Z: extended_enat] :
% 0.30/0.57 ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ X2 @ Y3 ) @ Z )
% 0.30/0.57 = ( ( ord_le72135733267957522d_enat @ X2 @ Z )
% 0.30/0.57 & ( ord_le72135733267957522d_enat @ Y3 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_less_iff_conj
% 0.30/0.57 thf(fact_3689_max__less__iff__conj,axiom,
% 0.30/0.57 ! [X2: real,Y3: real,Z: real] :
% 0.30/0.57 ( ( ord_less_real @ ( ord_max_real @ X2 @ Y3 ) @ Z )
% 0.30/0.57 = ( ( ord_less_real @ X2 @ Z )
% 0.30/0.57 & ( ord_less_real @ Y3 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_less_iff_conj
% 0.30/0.57 thf(fact_3690_max__less__iff__conj,axiom,
% 0.30/0.57 ! [X2: rat,Y3: rat,Z: rat] :
% 0.30/0.57 ( ( ord_less_rat @ ( ord_max_rat @ X2 @ Y3 ) @ Z )
% 0.30/0.57 = ( ( ord_less_rat @ X2 @ Z )
% 0.30/0.57 & ( ord_less_rat @ Y3 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_less_iff_conj
% 0.30/0.57 thf(fact_3691_max__less__iff__conj,axiom,
% 0.30/0.57 ! [X2: num,Y3: num,Z: num] :
% 0.30/0.57 ( ( ord_less_num @ ( ord_max_num @ X2 @ Y3 ) @ Z )
% 0.30/0.57 = ( ( ord_less_num @ X2 @ Z )
% 0.30/0.57 & ( ord_less_num @ Y3 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_less_iff_conj
% 0.30/0.57 thf(fact_3692_max__less__iff__conj,axiom,
% 0.30/0.57 ! [X2: nat,Y3: nat,Z: nat] :
% 0.30/0.57 ( ( ord_less_nat @ ( ord_max_nat @ X2 @ Y3 ) @ Z )
% 0.30/0.57 = ( ( ord_less_nat @ X2 @ Z )
% 0.30/0.57 & ( ord_less_nat @ Y3 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_less_iff_conj
% 0.30/0.57 thf(fact_3693_max__less__iff__conj,axiom,
% 0.30/0.57 ! [X2: int,Y3: int,Z: int] :
% 0.30/0.57 ( ( ord_less_int @ ( ord_max_int @ X2 @ Y3 ) @ Z )
% 0.30/0.57 = ( ( ord_less_int @ X2 @ Z )
% 0.30/0.57 & ( ord_less_int @ Y3 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_less_iff_conj
% 0.30/0.57 thf(fact_3694_max_Oabsorb4,axiom,
% 0.30/0.57 ! [A: extended_enat,B: extended_enat] :
% 0.30/0.57 ( ( ord_le72135733267957522d_enat @ A @ B )
% 0.30/0.57 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 0.30/0.57 = B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb4
% 0.30/0.57 thf(fact_3695_max_Oabsorb4,axiom,
% 0.30/0.57 ! [A: real,B: real] :
% 0.30/0.57 ( ( ord_less_real @ A @ B )
% 0.30/0.57 => ( ( ord_max_real @ A @ B )
% 0.30/0.57 = B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb4
% 0.30/0.57 thf(fact_3696_max_Oabsorb4,axiom,
% 0.30/0.57 ! [A: rat,B: rat] :
% 0.30/0.57 ( ( ord_less_rat @ A @ B )
% 0.30/0.57 => ( ( ord_max_rat @ A @ B )
% 0.30/0.57 = B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb4
% 0.30/0.57 thf(fact_3697_max_Oabsorb4,axiom,
% 0.30/0.57 ! [A: num,B: num] :
% 0.30/0.57 ( ( ord_less_num @ A @ B )
% 0.30/0.57 => ( ( ord_max_num @ A @ B )
% 0.30/0.57 = B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb4
% 0.30/0.57 thf(fact_3698_max_Oabsorb4,axiom,
% 0.30/0.57 ! [A: nat,B: nat] :
% 0.30/0.57 ( ( ord_less_nat @ A @ B )
% 0.30/0.57 => ( ( ord_max_nat @ A @ B )
% 0.30/0.57 = B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb4
% 0.30/0.57 thf(fact_3699_max_Oabsorb4,axiom,
% 0.30/0.57 ! [A: int,B: int] :
% 0.30/0.57 ( ( ord_less_int @ A @ B )
% 0.30/0.57 => ( ( ord_max_int @ A @ B )
% 0.30/0.57 = B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb4
% 0.30/0.57 thf(fact_3700_max_Oabsorb3,axiom,
% 0.30/0.57 ! [B: extended_enat,A: extended_enat] :
% 0.30/0.57 ( ( ord_le72135733267957522d_enat @ B @ A )
% 0.30/0.57 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 0.30/0.57 = A ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb3
% 0.30/0.57 thf(fact_3701_max_Oabsorb3,axiom,
% 0.30/0.57 ! [B: real,A: real] :
% 0.30/0.57 ( ( ord_less_real @ B @ A )
% 0.30/0.57 => ( ( ord_max_real @ A @ B )
% 0.30/0.57 = A ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb3
% 0.30/0.57 thf(fact_3702_max_Oabsorb3,axiom,
% 0.30/0.57 ! [B: rat,A: rat] :
% 0.30/0.57 ( ( ord_less_rat @ B @ A )
% 0.30/0.57 => ( ( ord_max_rat @ A @ B )
% 0.30/0.57 = A ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb3
% 0.30/0.57 thf(fact_3703_max_Oabsorb3,axiom,
% 0.30/0.57 ! [B: num,A: num] :
% 0.30/0.57 ( ( ord_less_num @ B @ A )
% 0.30/0.57 => ( ( ord_max_num @ A @ B )
% 0.30/0.57 = A ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb3
% 0.30/0.57 thf(fact_3704_max_Oabsorb3,axiom,
% 0.30/0.57 ! [B: nat,A: nat] :
% 0.30/0.57 ( ( ord_less_nat @ B @ A )
% 0.30/0.57 => ( ( ord_max_nat @ A @ B )
% 0.30/0.57 = A ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb3
% 0.30/0.57 thf(fact_3705_max_Oabsorb3,axiom,
% 0.30/0.57 ! [B: int,A: int] :
% 0.30/0.57 ( ( ord_less_int @ B @ A )
% 0.30/0.57 => ( ( ord_max_int @ A @ B )
% 0.30/0.57 = A ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb3
% 0.30/0.57 thf(fact_3706_length__list__update,axiom,
% 0.30/0.57 ! [Xs2: list_VEBT_VEBT,I2: nat,X2: vEBT_VEBT] :
% 0.30/0.57 ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X2 ) )
% 0.30/0.57 = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % length_list_update
% 0.30/0.57 thf(fact_3707_length__list__update,axiom,
% 0.30/0.57 ! [Xs2: list_o,I2: nat,X2: $o] :
% 0.30/0.57 ( ( size_size_list_o @ ( list_update_o @ Xs2 @ I2 @ X2 ) )
% 0.30/0.57 = ( size_size_list_o @ Xs2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % length_list_update
% 0.30/0.57 thf(fact_3708_length__list__update,axiom,
% 0.30/0.57 ! [Xs2: list_nat,I2: nat,X2: nat] :
% 0.30/0.57 ( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I2 @ X2 ) )
% 0.30/0.57 = ( size_size_list_nat @ Xs2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % length_list_update
% 0.30/0.57 thf(fact_3709_length__list__update,axiom,
% 0.30/0.57 ! [Xs2: list_int,I2: nat,X2: int] :
% 0.30/0.57 ( ( size_size_list_int @ ( list_update_int @ Xs2 @ I2 @ X2 ) )
% 0.30/0.57 = ( size_size_list_int @ Xs2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % length_list_update
% 0.30/0.57 thf(fact_3710_max__Suc__Suc,axiom,
% 0.30/0.57 ! [M: nat,N: nat] :
% 0.30/0.57 ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.57 = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_Suc_Suc
% 0.30/0.57 thf(fact_3711_max__0R,axiom,
% 0.30/0.57 ! [N: nat] :
% 0.30/0.57 ( ( ord_max_nat @ N @ zero_zero_nat )
% 0.30/0.57 = N ) ).
% 0.30/0.57
% 0.30/0.57 % max_0R
% 0.30/0.57 thf(fact_3712_max__0L,axiom,
% 0.30/0.57 ! [N: nat] :
% 0.30/0.57 ( ( ord_max_nat @ zero_zero_nat @ N )
% 0.30/0.57 = N ) ).
% 0.30/0.57
% 0.30/0.57 % max_0L
% 0.30/0.57 thf(fact_3713_max__nat_Oright__neutral,axiom,
% 0.30/0.57 ! [A: nat] :
% 0.30/0.57 ( ( ord_max_nat @ A @ zero_zero_nat )
% 0.30/0.57 = A ) ).
% 0.30/0.57
% 0.30/0.57 % max_nat.right_neutral
% 0.30/0.57 thf(fact_3714_max__nat_Oneutr__eq__iff,axiom,
% 0.30/0.57 ! [A: nat,B: nat] :
% 0.30/0.57 ( ( zero_zero_nat
% 0.30/0.57 = ( ord_max_nat @ A @ B ) )
% 0.30/0.57 = ( ( A = zero_zero_nat )
% 0.30/0.57 & ( B = zero_zero_nat ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_nat.neutr_eq_iff
% 0.30/0.57 thf(fact_3715_max__nat_Oleft__neutral,axiom,
% 0.30/0.57 ! [A: nat] :
% 0.30/0.57 ( ( ord_max_nat @ zero_zero_nat @ A )
% 0.30/0.57 = A ) ).
% 0.30/0.57
% 0.30/0.57 % max_nat.left_neutral
% 0.30/0.57 thf(fact_3716_max__nat_Oeq__neutr__iff,axiom,
% 0.30/0.57 ! [A: nat,B: nat] :
% 0.30/0.57 ( ( ( ord_max_nat @ A @ B )
% 0.30/0.57 = zero_zero_nat )
% 0.30/0.57 = ( ( A = zero_zero_nat )
% 0.30/0.57 & ( B = zero_zero_nat ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_nat.eq_neutr_iff
% 0.30/0.57 thf(fact_3717_list__update__id,axiom,
% 0.30/0.57 ! [Xs2: list_int,I2: nat] :
% 0.30/0.57 ( ( list_update_int @ Xs2 @ I2 @ ( nth_int @ Xs2 @ I2 ) )
% 0.30/0.57 = Xs2 ) ).
% 0.30/0.57
% 0.30/0.57 % list_update_id
% 0.30/0.57 thf(fact_3718_list__update__id,axiom,
% 0.30/0.57 ! [Xs2: list_nat,I2: nat] :
% 0.30/0.57 ( ( list_update_nat @ Xs2 @ I2 @ ( nth_nat @ Xs2 @ I2 ) )
% 0.30/0.57 = Xs2 ) ).
% 0.30/0.57
% 0.30/0.57 % list_update_id
% 0.30/0.57 thf(fact_3719_list__update__id,axiom,
% 0.30/0.57 ! [Xs2: list_VEBT_VEBT,I2: nat] :
% 0.30/0.57 ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) )
% 0.30/0.57 = Xs2 ) ).
% 0.30/0.57
% 0.30/0.57 % list_update_id
% 0.30/0.57 thf(fact_3720_nth__list__update__neq,axiom,
% 0.30/0.57 ! [I2: nat,J: nat,Xs2: list_int,X2: int] :
% 0.30/0.57 ( ( I2 != J )
% 0.30/0.57 => ( ( nth_int @ ( list_update_int @ Xs2 @ I2 @ X2 ) @ J )
% 0.30/0.57 = ( nth_int @ Xs2 @ J ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % nth_list_update_neq
% 0.30/0.57 thf(fact_3721_nth__list__update__neq,axiom,
% 0.30/0.57 ! [I2: nat,J: nat,Xs2: list_nat,X2: nat] :
% 0.30/0.57 ( ( I2 != J )
% 0.30/0.57 => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I2 @ X2 ) @ J )
% 0.30/0.57 = ( nth_nat @ Xs2 @ J ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % nth_list_update_neq
% 0.30/0.57 thf(fact_3722_nth__list__update__neq,axiom,
% 0.30/0.57 ! [I2: nat,J: nat,Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 0.30/0.57 ( ( I2 != J )
% 0.30/0.57 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X2 ) @ J )
% 0.30/0.57 = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % nth_list_update_neq
% 0.30/0.57 thf(fact_3723_max__number__of_I1_J,axiom,
% 0.30/0.57 ! [U: num,V: num] :
% 0.30/0.57 ( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.30/0.57 => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.30/0.57 = ( numera1916890842035813515d_enat @ V ) ) )
% 0.30/0.57 & ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.30/0.57 => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.30/0.57 = ( numera1916890842035813515d_enat @ U ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_number_of(1)
% 0.30/0.57 thf(fact_3724_max__number__of_I1_J,axiom,
% 0.30/0.57 ! [U: num,V: num] :
% 0.30/0.57 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.57 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.57 = ( numeral_numeral_real @ V ) ) )
% 0.30/0.57 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.57 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.57 = ( numeral_numeral_real @ U ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_number_of(1)
% 0.30/0.57 thf(fact_3725_max__number__of_I1_J,axiom,
% 0.30/0.57 ! [U: num,V: num] :
% 0.30/0.57 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.57 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.57 = ( numeral_numeral_rat @ V ) ) )
% 0.30/0.57 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.57 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.57 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_number_of(1)
% 0.30/0.57 thf(fact_3726_max__number__of_I1_J,axiom,
% 0.30/0.57 ! [U: num,V: num] :
% 0.30/0.57 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.57 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.57 = ( numeral_numeral_nat @ V ) ) )
% 0.30/0.57 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.57 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.57 = ( numeral_numeral_nat @ U ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_number_of(1)
% 0.30/0.57 thf(fact_3727_max__number__of_I1_J,axiom,
% 0.30/0.57 ! [U: num,V: num] :
% 0.30/0.57 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.57 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.57 = ( numeral_numeral_int @ V ) ) )
% 0.30/0.57 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.57 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.57 = ( numeral_numeral_int @ U ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_number_of(1)
% 0.30/0.57 thf(fact_3728_max__0__1_I3_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X2 ) )
% 0.30/0.57 = ( numera1916890842035813515d_enat @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(3)
% 0.30/0.57 thf(fact_3729_max__0__1_I3_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X2 ) )
% 0.30/0.57 = ( numeral_numeral_real @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(3)
% 0.30/0.57 thf(fact_3730_max__0__1_I3_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X2 ) )
% 0.30/0.57 = ( numeral_numeral_rat @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(3)
% 0.30/0.57 thf(fact_3731_max__0__1_I3_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X2 ) )
% 0.30/0.57 = ( numeral_numeral_nat @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(3)
% 0.30/0.57 thf(fact_3732_max__0__1_I3_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X2 ) )
% 0.30/0.57 = ( numeral_numeral_int @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(3)
% 0.30/0.57 thf(fact_3733_max__0__1_I4_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X2 ) @ zero_z5237406670263579293d_enat )
% 0.30/0.57 = ( numera1916890842035813515d_enat @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(4)
% 0.30/0.57 thf(fact_3734_max__0__1_I4_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_max_real @ ( numeral_numeral_real @ X2 ) @ zero_zero_real )
% 0.30/0.57 = ( numeral_numeral_real @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(4)
% 0.30/0.57 thf(fact_3735_max__0__1_I4_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_max_rat @ ( numeral_numeral_rat @ X2 ) @ zero_zero_rat )
% 0.30/0.57 = ( numeral_numeral_rat @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(4)
% 0.30/0.57 thf(fact_3736_max__0__1_I4_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_max_nat @ ( numeral_numeral_nat @ X2 ) @ zero_zero_nat )
% 0.30/0.57 = ( numeral_numeral_nat @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(4)
% 0.30/0.57 thf(fact_3737_max__0__1_I4_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_max_int @ ( numeral_numeral_int @ X2 ) @ zero_zero_int )
% 0.30/0.57 = ( numeral_numeral_int @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(4)
% 0.30/0.57 thf(fact_3738_max__0__1_I2_J,axiom,
% 0.30/0.57 ( ( ord_max_real @ one_one_real @ zero_zero_real )
% 0.30/0.57 = one_one_real ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(2)
% 0.30/0.57 thf(fact_3739_max__0__1_I2_J,axiom,
% 0.30/0.57 ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
% 0.30/0.57 = one_one_rat ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(2)
% 0.30/0.57 thf(fact_3740_max__0__1_I2_J,axiom,
% 0.30/0.57 ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
% 0.30/0.57 = one_one_nat ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(2)
% 0.30/0.57 thf(fact_3741_max__0__1_I2_J,axiom,
% 0.30/0.57 ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
% 0.30/0.57 = one_on7984719198319812577d_enat ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(2)
% 0.30/0.57 thf(fact_3742_max__0__1_I2_J,axiom,
% 0.30/0.57 ( ( ord_max_int @ one_one_int @ zero_zero_int )
% 0.30/0.57 = one_one_int ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(2)
% 0.30/0.57 thf(fact_3743_max__0__1_I1_J,axiom,
% 0.30/0.57 ( ( ord_max_real @ zero_zero_real @ one_one_real )
% 0.30/0.57 = one_one_real ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(1)
% 0.30/0.57 thf(fact_3744_max__0__1_I1_J,axiom,
% 0.30/0.57 ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
% 0.30/0.57 = one_one_rat ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(1)
% 0.30/0.57 thf(fact_3745_max__0__1_I1_J,axiom,
% 0.30/0.57 ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
% 0.30/0.57 = one_one_nat ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(1)
% 0.30/0.57 thf(fact_3746_max__0__1_I1_J,axiom,
% 0.30/0.57 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
% 0.30/0.57 = one_on7984719198319812577d_enat ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(1)
% 0.30/0.57 thf(fact_3747_max__0__1_I1_J,axiom,
% 0.30/0.57 ( ( ord_max_int @ zero_zero_int @ one_one_int )
% 0.30/0.57 = one_one_int ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(1)
% 0.30/0.57 thf(fact_3748_max__0__1_I5_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X2 ) )
% 0.30/0.57 = ( numera1916890842035813515d_enat @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(5)
% 0.30/0.57 thf(fact_3749_max__0__1_I5_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X2 ) )
% 0.30/0.57 = ( numeral_numeral_real @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(5)
% 0.30/0.57 thf(fact_3750_max__0__1_I5_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X2 ) )
% 0.30/0.57 = ( numeral_numeral_rat @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(5)
% 0.30/0.57 thf(fact_3751_max__0__1_I5_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X2 ) )
% 0.30/0.57 = ( numeral_numeral_nat @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(5)
% 0.30/0.57 thf(fact_3752_max__0__1_I5_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X2 ) )
% 0.30/0.57 = ( numeral_numeral_int @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(5)
% 0.30/0.57 thf(fact_3753_max__0__1_I6_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X2 ) @ one_on7984719198319812577d_enat )
% 0.30/0.57 = ( numera1916890842035813515d_enat @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(6)
% 0.30/0.57 thf(fact_3754_max__0__1_I6_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_max_real @ ( numeral_numeral_real @ X2 ) @ one_one_real )
% 0.30/0.57 = ( numeral_numeral_real @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(6)
% 0.30/0.57 thf(fact_3755_max__0__1_I6_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_max_rat @ ( numeral_numeral_rat @ X2 ) @ one_one_rat )
% 0.30/0.57 = ( numeral_numeral_rat @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(6)
% 0.30/0.57 thf(fact_3756_max__0__1_I6_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_max_nat @ ( numeral_numeral_nat @ X2 ) @ one_one_nat )
% 0.30/0.57 = ( numeral_numeral_nat @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(6)
% 0.30/0.57 thf(fact_3757_max__0__1_I6_J,axiom,
% 0.30/0.57 ! [X2: num] :
% 0.30/0.57 ( ( ord_max_int @ ( numeral_numeral_int @ X2 ) @ one_one_int )
% 0.30/0.57 = ( numeral_numeral_int @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_0_1(6)
% 0.30/0.57 thf(fact_3758_list__update__beyond,axiom,
% 0.30/0.57 ! [Xs2: list_VEBT_VEBT,I2: nat,X2: vEBT_VEBT] :
% 0.30/0.57 ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ I2 )
% 0.30/0.57 => ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X2 )
% 0.30/0.57 = Xs2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % list_update_beyond
% 0.30/0.57 thf(fact_3759_list__update__beyond,axiom,
% 0.30/0.57 ! [Xs2: list_o,I2: nat,X2: $o] :
% 0.30/0.57 ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I2 )
% 0.30/0.57 => ( ( list_update_o @ Xs2 @ I2 @ X2 )
% 0.30/0.57 = Xs2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % list_update_beyond
% 0.30/0.57 thf(fact_3760_list__update__beyond,axiom,
% 0.30/0.57 ! [Xs2: list_nat,I2: nat,X2: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I2 )
% 0.30/0.57 => ( ( list_update_nat @ Xs2 @ I2 @ X2 )
% 0.30/0.57 = Xs2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % list_update_beyond
% 0.30/0.57 thf(fact_3761_list__update__beyond,axiom,
% 0.30/0.57 ! [Xs2: list_int,I2: nat,X2: int] :
% 0.30/0.57 ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ I2 )
% 0.30/0.57 => ( ( list_update_int @ Xs2 @ I2 @ X2 )
% 0.30/0.57 = Xs2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % list_update_beyond
% 0.30/0.57 thf(fact_3762_nth__list__update__eq,axiom,
% 0.30/0.57 ! [I2: nat,Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 0.30/0.57 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.57 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X2 ) @ I2 )
% 0.30/0.57 = X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % nth_list_update_eq
% 0.30/0.57 thf(fact_3763_nth__list__update__eq,axiom,
% 0.30/0.57 ! [I2: nat,Xs2: list_o,X2: $o] :
% 0.30/0.57 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.57 => ( ( nth_o @ ( list_update_o @ Xs2 @ I2 @ X2 ) @ I2 )
% 0.30/0.57 = X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % nth_list_update_eq
% 0.30/0.57 thf(fact_3764_nth__list__update__eq,axiom,
% 0.30/0.57 ! [I2: nat,Xs2: list_nat,X2: nat] :
% 0.30/0.57 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.57 => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I2 @ X2 ) @ I2 )
% 0.30/0.57 = X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % nth_list_update_eq
% 0.30/0.57 thf(fact_3765_nth__list__update__eq,axiom,
% 0.30/0.57 ! [I2: nat,Xs2: list_int,X2: int] :
% 0.30/0.57 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.57 => ( ( nth_int @ ( list_update_int @ Xs2 @ I2 @ X2 ) @ I2 )
% 0.30/0.57 = X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % nth_list_update_eq
% 0.30/0.57 thf(fact_3766_set__swap,axiom,
% 0.30/0.57 ! [I2: nat,Xs2: list_VEBT_VEBT,J: nat] :
% 0.30/0.57 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.57 => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.57 => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) )
% 0.30/0.57 = ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % set_swap
% 0.30/0.57 thf(fact_3767_set__swap,axiom,
% 0.30/0.57 ! [I2: nat,Xs2: list_o,J: nat] :
% 0.30/0.57 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.57 => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.57 => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs2 @ I2 @ ( nth_o @ Xs2 @ J ) ) @ J @ ( nth_o @ Xs2 @ I2 ) ) )
% 0.30/0.57 = ( set_o2 @ Xs2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % set_swap
% 0.30/0.57 thf(fact_3768_set__swap,axiom,
% 0.30/0.57 ! [I2: nat,Xs2: list_nat,J: nat] :
% 0.30/0.57 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.57 => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.57 => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I2 @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I2 ) ) )
% 0.30/0.57 = ( set_nat2 @ Xs2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % set_swap
% 0.30/0.57 thf(fact_3769_set__swap,axiom,
% 0.30/0.57 ! [I2: nat,Xs2: list_int,J: nat] :
% 0.30/0.57 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.57 => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.57 => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs2 @ I2 @ ( nth_int @ Xs2 @ J ) ) @ J @ ( nth_int @ Xs2 @ I2 ) ) )
% 0.30/0.57 = ( set_int2 @ Xs2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % set_swap
% 0.30/0.57 thf(fact_3770_max__def,axiom,
% 0.30/0.57 ( ord_ma741700101516333627d_enat
% 0.30/0.57 = ( ^ [A3: extended_enat,B2: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_def
% 0.30/0.57 thf(fact_3771_max__def,axiom,
% 0.30/0.57 ( ord_max_set_int
% 0.30/0.57 = ( ^ [A3: set_int,B2: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_def
% 0.30/0.57 thf(fact_3772_max__def,axiom,
% 0.30/0.57 ( ord_max_rat
% 0.30/0.57 = ( ^ [A3: rat,B2: rat] : ( if_rat @ ( ord_less_eq_rat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_def
% 0.30/0.57 thf(fact_3773_max__def,axiom,
% 0.30/0.57 ( ord_max_num
% 0.30/0.57 = ( ^ [A3: num,B2: num] : ( if_num @ ( ord_less_eq_num @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_def
% 0.30/0.57 thf(fact_3774_max__def,axiom,
% 0.30/0.57 ( ord_max_nat
% 0.30/0.57 = ( ^ [A3: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_def
% 0.30/0.57 thf(fact_3775_max__def,axiom,
% 0.30/0.57 ( ord_max_int
% 0.30/0.57 = ( ^ [A3: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_def
% 0.30/0.57 thf(fact_3776_max__absorb1,axiom,
% 0.30/0.57 ! [Y3: extended_enat,X2: extended_enat] :
% 0.30/0.57 ( ( ord_le2932123472753598470d_enat @ Y3 @ X2 )
% 0.30/0.57 => ( ( ord_ma741700101516333627d_enat @ X2 @ Y3 )
% 0.30/0.57 = X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_absorb1
% 0.30/0.57 thf(fact_3777_max__absorb1,axiom,
% 0.30/0.57 ! [Y3: set_int,X2: set_int] :
% 0.30/0.57 ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 0.30/0.57 => ( ( ord_max_set_int @ X2 @ Y3 )
% 0.30/0.57 = X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_absorb1
% 0.30/0.57 thf(fact_3778_max__absorb1,axiom,
% 0.30/0.57 ! [Y3: rat,X2: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 0.30/0.57 => ( ( ord_max_rat @ X2 @ Y3 )
% 0.30/0.57 = X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_absorb1
% 0.30/0.57 thf(fact_3779_max__absorb1,axiom,
% 0.30/0.57 ! [Y3: num,X2: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ Y3 @ X2 )
% 0.30/0.57 => ( ( ord_max_num @ X2 @ Y3 )
% 0.30/0.57 = X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_absorb1
% 0.30/0.57 thf(fact_3780_max__absorb1,axiom,
% 0.30/0.57 ! [Y3: nat,X2: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.30/0.57 => ( ( ord_max_nat @ X2 @ Y3 )
% 0.30/0.57 = X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_absorb1
% 0.30/0.57 thf(fact_3781_max__absorb1,axiom,
% 0.30/0.57 ! [Y3: int,X2: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ Y3 @ X2 )
% 0.30/0.57 => ( ( ord_max_int @ X2 @ Y3 )
% 0.30/0.57 = X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_absorb1
% 0.30/0.57 thf(fact_3782_max__absorb2,axiom,
% 0.30/0.57 ! [X2: extended_enat,Y3: extended_enat] :
% 0.30/0.57 ( ( ord_le2932123472753598470d_enat @ X2 @ Y3 )
% 0.30/0.57 => ( ( ord_ma741700101516333627d_enat @ X2 @ Y3 )
% 0.30/0.57 = Y3 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_absorb2
% 0.30/0.57 thf(fact_3783_max__absorb2,axiom,
% 0.30/0.57 ! [X2: set_int,Y3: set_int] :
% 0.30/0.57 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.30/0.57 => ( ( ord_max_set_int @ X2 @ Y3 )
% 0.30/0.57 = Y3 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_absorb2
% 0.30/0.57 thf(fact_3784_max__absorb2,axiom,
% 0.30/0.57 ! [X2: rat,Y3: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.30/0.57 => ( ( ord_max_rat @ X2 @ Y3 )
% 0.30/0.57 = Y3 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_absorb2
% 0.30/0.57 thf(fact_3785_max__absorb2,axiom,
% 0.30/0.57 ! [X2: num,Y3: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.30/0.57 => ( ( ord_max_num @ X2 @ Y3 )
% 0.30/0.57 = Y3 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_absorb2
% 0.30/0.57 thf(fact_3786_max__absorb2,axiom,
% 0.30/0.57 ! [X2: nat,Y3: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.30/0.57 => ( ( ord_max_nat @ X2 @ Y3 )
% 0.30/0.57 = Y3 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_absorb2
% 0.30/0.57 thf(fact_3787_max__absorb2,axiom,
% 0.30/0.57 ! [X2: int,Y3: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.30/0.57 => ( ( ord_max_int @ X2 @ Y3 )
% 0.30/0.57 = Y3 ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_absorb2
% 0.30/0.57 thf(fact_3788_max_Omono,axiom,
% 0.30/0.57 ! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
% 0.30/0.57 ( ( ord_le2932123472753598470d_enat @ C @ A )
% 0.30/0.57 => ( ( ord_le2932123472753598470d_enat @ D @ B )
% 0.30/0.57 => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ C @ D ) @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.mono
% 0.30/0.57 thf(fact_3789_max_Omono,axiom,
% 0.30/0.57 ! [C: rat,A: rat,D: rat,B: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ C @ A )
% 0.30/0.57 => ( ( ord_less_eq_rat @ D @ B )
% 0.30/0.57 => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.mono
% 0.30/0.57 thf(fact_3790_max_Omono,axiom,
% 0.30/0.57 ! [C: num,A: num,D: num,B: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ C @ A )
% 0.30/0.57 => ( ( ord_less_eq_num @ D @ B )
% 0.30/0.57 => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.mono
% 0.30/0.57 thf(fact_3791_max_Omono,axiom,
% 0.30/0.57 ! [C: nat,A: nat,D: nat,B: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ C @ A )
% 0.30/0.57 => ( ( ord_less_eq_nat @ D @ B )
% 0.30/0.57 => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.mono
% 0.30/0.57 thf(fact_3792_max_Omono,axiom,
% 0.30/0.57 ! [C: int,A: int,D: int,B: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ C @ A )
% 0.30/0.57 => ( ( ord_less_eq_int @ D @ B )
% 0.30/0.57 => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.mono
% 0.30/0.57 thf(fact_3793_max_OorderE,axiom,
% 0.30/0.57 ! [B: extended_enat,A: extended_enat] :
% 0.30/0.57 ( ( ord_le2932123472753598470d_enat @ B @ A )
% 0.30/0.57 => ( A
% 0.30/0.57 = ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.orderE
% 0.30/0.57 thf(fact_3794_max_OorderE,axiom,
% 0.30/0.57 ! [B: rat,A: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.57 => ( A
% 0.30/0.57 = ( ord_max_rat @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.orderE
% 0.30/0.57 thf(fact_3795_max_OorderE,axiom,
% 0.30/0.57 ! [B: num,A: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ B @ A )
% 0.30/0.57 => ( A
% 0.30/0.57 = ( ord_max_num @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.orderE
% 0.30/0.57 thf(fact_3796_max_OorderE,axiom,
% 0.30/0.57 ! [B: nat,A: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.57 => ( A
% 0.30/0.57 = ( ord_max_nat @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.orderE
% 0.30/0.57 thf(fact_3797_max_OorderE,axiom,
% 0.30/0.57 ! [B: int,A: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ B @ A )
% 0.30/0.57 => ( A
% 0.30/0.57 = ( ord_max_int @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.orderE
% 0.30/0.57 thf(fact_3798_max_OorderI,axiom,
% 0.30/0.57 ! [A: extended_enat,B: extended_enat] :
% 0.30/0.57 ( ( A
% 0.30/0.57 = ( ord_ma741700101516333627d_enat @ A @ B ) )
% 0.30/0.57 => ( ord_le2932123472753598470d_enat @ B @ A ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.orderI
% 0.30/0.57 thf(fact_3799_max_OorderI,axiom,
% 0.30/0.57 ! [A: rat,B: rat] :
% 0.30/0.57 ( ( A
% 0.30/0.57 = ( ord_max_rat @ A @ B ) )
% 0.30/0.57 => ( ord_less_eq_rat @ B @ A ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.orderI
% 0.30/0.57 thf(fact_3800_max_OorderI,axiom,
% 0.30/0.57 ! [A: num,B: num] :
% 0.30/0.57 ( ( A
% 0.30/0.57 = ( ord_max_num @ A @ B ) )
% 0.30/0.57 => ( ord_less_eq_num @ B @ A ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.orderI
% 0.30/0.57 thf(fact_3801_max_OorderI,axiom,
% 0.30/0.57 ! [A: nat,B: nat] :
% 0.30/0.57 ( ( A
% 0.30/0.57 = ( ord_max_nat @ A @ B ) )
% 0.30/0.57 => ( ord_less_eq_nat @ B @ A ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.orderI
% 0.30/0.57 thf(fact_3802_max_OorderI,axiom,
% 0.30/0.57 ! [A: int,B: int] :
% 0.30/0.57 ( ( A
% 0.30/0.57 = ( ord_max_int @ A @ B ) )
% 0.30/0.57 => ( ord_less_eq_int @ B @ A ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.orderI
% 0.30/0.57 thf(fact_3803_max_OboundedE,axiom,
% 0.30/0.57 ! [B: extended_enat,C: extended_enat,A: extended_enat] :
% 0.30/0.57 ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
% 0.30/0.57 => ~ ( ( ord_le2932123472753598470d_enat @ B @ A )
% 0.30/0.57 => ~ ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.boundedE
% 0.30/0.57 thf(fact_3804_max_OboundedE,axiom,
% 0.30/0.57 ! [B: rat,C: rat,A: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 0.30/0.57 => ~ ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.57 => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.boundedE
% 0.30/0.57 thf(fact_3805_max_OboundedE,axiom,
% 0.30/0.57 ! [B: num,C: num,A: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 0.30/0.57 => ~ ( ( ord_less_eq_num @ B @ A )
% 0.30/0.57 => ~ ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.boundedE
% 0.30/0.57 thf(fact_3806_max_OboundedE,axiom,
% 0.30/0.57 ! [B: nat,C: nat,A: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 0.30/0.57 => ~ ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.57 => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.boundedE
% 0.30/0.57 thf(fact_3807_max_OboundedE,axiom,
% 0.30/0.57 ! [B: int,C: int,A: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 0.30/0.57 => ~ ( ( ord_less_eq_int @ B @ A )
% 0.30/0.57 => ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.boundedE
% 0.30/0.57 thf(fact_3808_max_OboundedI,axiom,
% 0.30/0.57 ! [B: extended_enat,A: extended_enat,C: extended_enat] :
% 0.30/0.57 ( ( ord_le2932123472753598470d_enat @ B @ A )
% 0.30/0.57 => ( ( ord_le2932123472753598470d_enat @ C @ A )
% 0.30/0.57 => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.boundedI
% 0.30/0.57 thf(fact_3809_max_OboundedI,axiom,
% 0.30/0.57 ! [B: rat,A: rat,C: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.57 => ( ( ord_less_eq_rat @ C @ A )
% 0.30/0.57 => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.boundedI
% 0.30/0.57 thf(fact_3810_max_OboundedI,axiom,
% 0.30/0.57 ! [B: num,A: num,C: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ B @ A )
% 0.30/0.57 => ( ( ord_less_eq_num @ C @ A )
% 0.30/0.57 => ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.boundedI
% 0.30/0.57 thf(fact_3811_max_OboundedI,axiom,
% 0.30/0.57 ! [B: nat,A: nat,C: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.57 => ( ( ord_less_eq_nat @ C @ A )
% 0.30/0.57 => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.boundedI
% 0.30/0.57 thf(fact_3812_max_OboundedI,axiom,
% 0.30/0.57 ! [B: int,A: int,C: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ B @ A )
% 0.30/0.57 => ( ( ord_less_eq_int @ C @ A )
% 0.30/0.57 => ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.boundedI
% 0.30/0.57 thf(fact_3813_max_Oorder__iff,axiom,
% 0.30/0.57 ( ord_le2932123472753598470d_enat
% 0.30/0.57 = ( ^ [B2: extended_enat,A3: extended_enat] :
% 0.30/0.57 ( A3
% 0.30/0.57 = ( ord_ma741700101516333627d_enat @ A3 @ B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.order_iff
% 0.30/0.57 thf(fact_3814_max_Oorder__iff,axiom,
% 0.30/0.57 ( ord_less_eq_rat
% 0.30/0.57 = ( ^ [B2: rat,A3: rat] :
% 0.30/0.57 ( A3
% 0.30/0.57 = ( ord_max_rat @ A3 @ B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.order_iff
% 0.30/0.57 thf(fact_3815_max_Oorder__iff,axiom,
% 0.30/0.57 ( ord_less_eq_num
% 0.30/0.57 = ( ^ [B2: num,A3: num] :
% 0.30/0.57 ( A3
% 0.30/0.57 = ( ord_max_num @ A3 @ B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.order_iff
% 0.30/0.57 thf(fact_3816_max_Oorder__iff,axiom,
% 0.30/0.57 ( ord_less_eq_nat
% 0.30/0.57 = ( ^ [B2: nat,A3: nat] :
% 0.30/0.57 ( A3
% 0.30/0.57 = ( ord_max_nat @ A3 @ B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.order_iff
% 0.30/0.57 thf(fact_3817_max_Oorder__iff,axiom,
% 0.30/0.57 ( ord_less_eq_int
% 0.30/0.57 = ( ^ [B2: int,A3: int] :
% 0.30/0.57 ( A3
% 0.30/0.57 = ( ord_max_int @ A3 @ B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.order_iff
% 0.30/0.57 thf(fact_3818_max_Ocobounded1,axiom,
% 0.30/0.57 ! [A: extended_enat,B: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.cobounded1
% 0.30/0.57 thf(fact_3819_max_Ocobounded1,axiom,
% 0.30/0.57 ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.cobounded1
% 0.30/0.57 thf(fact_3820_max_Ocobounded1,axiom,
% 0.30/0.57 ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.cobounded1
% 0.30/0.57 thf(fact_3821_max_Ocobounded1,axiom,
% 0.30/0.57 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.cobounded1
% 0.30/0.57 thf(fact_3822_max_Ocobounded1,axiom,
% 0.30/0.57 ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.cobounded1
% 0.30/0.57 thf(fact_3823_max_Ocobounded2,axiom,
% 0.30/0.57 ! [B: extended_enat,A: extended_enat] : ( ord_le2932123472753598470d_enat @ B @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.cobounded2
% 0.30/0.57 thf(fact_3824_max_Ocobounded2,axiom,
% 0.30/0.57 ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.cobounded2
% 0.30/0.57 thf(fact_3825_max_Ocobounded2,axiom,
% 0.30/0.57 ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.cobounded2
% 0.30/0.57 thf(fact_3826_max_Ocobounded2,axiom,
% 0.30/0.57 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.cobounded2
% 0.30/0.57 thf(fact_3827_max_Ocobounded2,axiom,
% 0.30/0.57 ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.cobounded2
% 0.30/0.57 thf(fact_3828_le__max__iff__disj,axiom,
% 0.30/0.57 ! [Z: extended_enat,X2: extended_enat,Y3: extended_enat] :
% 0.30/0.57 ( ( ord_le2932123472753598470d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X2 @ Y3 ) )
% 0.30/0.57 = ( ( ord_le2932123472753598470d_enat @ Z @ X2 )
% 0.30/0.57 | ( ord_le2932123472753598470d_enat @ Z @ Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % le_max_iff_disj
% 0.30/0.57 thf(fact_3829_le__max__iff__disj,axiom,
% 0.30/0.57 ! [Z: rat,X2: rat,Y3: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X2 @ Y3 ) )
% 0.30/0.57 = ( ( ord_less_eq_rat @ Z @ X2 )
% 0.30/0.57 | ( ord_less_eq_rat @ Z @ Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % le_max_iff_disj
% 0.30/0.57 thf(fact_3830_le__max__iff__disj,axiom,
% 0.30/0.57 ! [Z: num,X2: num,Y3: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X2 @ Y3 ) )
% 0.30/0.57 = ( ( ord_less_eq_num @ Z @ X2 )
% 0.30/0.57 | ( ord_less_eq_num @ Z @ Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % le_max_iff_disj
% 0.30/0.57 thf(fact_3831_le__max__iff__disj,axiom,
% 0.30/0.57 ! [Z: nat,X2: nat,Y3: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X2 @ Y3 ) )
% 0.30/0.57 = ( ( ord_less_eq_nat @ Z @ X2 )
% 0.30/0.57 | ( ord_less_eq_nat @ Z @ Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % le_max_iff_disj
% 0.30/0.57 thf(fact_3832_le__max__iff__disj,axiom,
% 0.30/0.57 ! [Z: int,X2: int,Y3: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X2 @ Y3 ) )
% 0.30/0.57 = ( ( ord_less_eq_int @ Z @ X2 )
% 0.30/0.57 | ( ord_less_eq_int @ Z @ Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % le_max_iff_disj
% 0.30/0.57 thf(fact_3833_max_Oabsorb__iff1,axiom,
% 0.30/0.57 ( ord_le2932123472753598470d_enat
% 0.30/0.57 = ( ^ [B2: extended_enat,A3: extended_enat] :
% 0.30/0.57 ( ( ord_ma741700101516333627d_enat @ A3 @ B2 )
% 0.30/0.57 = A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb_iff1
% 0.30/0.57 thf(fact_3834_max_Oabsorb__iff1,axiom,
% 0.30/0.57 ( ord_less_eq_rat
% 0.30/0.57 = ( ^ [B2: rat,A3: rat] :
% 0.30/0.57 ( ( ord_max_rat @ A3 @ B2 )
% 0.30/0.57 = A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb_iff1
% 0.30/0.57 thf(fact_3835_max_Oabsorb__iff1,axiom,
% 0.30/0.57 ( ord_less_eq_num
% 0.30/0.57 = ( ^ [B2: num,A3: num] :
% 0.30/0.57 ( ( ord_max_num @ A3 @ B2 )
% 0.30/0.57 = A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb_iff1
% 0.30/0.57 thf(fact_3836_max_Oabsorb__iff1,axiom,
% 0.30/0.57 ( ord_less_eq_nat
% 0.30/0.57 = ( ^ [B2: nat,A3: nat] :
% 0.30/0.57 ( ( ord_max_nat @ A3 @ B2 )
% 0.30/0.57 = A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb_iff1
% 0.30/0.57 thf(fact_3837_max_Oabsorb__iff1,axiom,
% 0.30/0.57 ( ord_less_eq_int
% 0.30/0.57 = ( ^ [B2: int,A3: int] :
% 0.30/0.57 ( ( ord_max_int @ A3 @ B2 )
% 0.30/0.57 = A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb_iff1
% 0.30/0.57 thf(fact_3838_max_Oabsorb__iff2,axiom,
% 0.30/0.57 ( ord_le2932123472753598470d_enat
% 0.30/0.57 = ( ^ [A3: extended_enat,B2: extended_enat] :
% 0.30/0.57 ( ( ord_ma741700101516333627d_enat @ A3 @ B2 )
% 0.30/0.57 = B2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb_iff2
% 0.30/0.57 thf(fact_3839_max_Oabsorb__iff2,axiom,
% 0.30/0.57 ( ord_less_eq_rat
% 0.30/0.57 = ( ^ [A3: rat,B2: rat] :
% 0.30/0.57 ( ( ord_max_rat @ A3 @ B2 )
% 0.30/0.57 = B2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb_iff2
% 0.30/0.57 thf(fact_3840_max_Oabsorb__iff2,axiom,
% 0.30/0.57 ( ord_less_eq_num
% 0.30/0.57 = ( ^ [A3: num,B2: num] :
% 0.30/0.57 ( ( ord_max_num @ A3 @ B2 )
% 0.30/0.57 = B2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb_iff2
% 0.30/0.57 thf(fact_3841_max_Oabsorb__iff2,axiom,
% 0.30/0.57 ( ord_less_eq_nat
% 0.30/0.57 = ( ^ [A3: nat,B2: nat] :
% 0.30/0.57 ( ( ord_max_nat @ A3 @ B2 )
% 0.30/0.57 = B2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb_iff2
% 0.30/0.57 thf(fact_3842_max_Oabsorb__iff2,axiom,
% 0.30/0.57 ( ord_less_eq_int
% 0.30/0.57 = ( ^ [A3: int,B2: int] :
% 0.30/0.57 ( ( ord_max_int @ A3 @ B2 )
% 0.30/0.57 = B2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.absorb_iff2
% 0.30/0.57 thf(fact_3843_max_OcoboundedI1,axiom,
% 0.30/0.57 ! [C: extended_enat,A: extended_enat,B: extended_enat] :
% 0.30/0.57 ( ( ord_le2932123472753598470d_enat @ C @ A )
% 0.30/0.57 => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.coboundedI1
% 0.30/0.57 thf(fact_3844_max_OcoboundedI1,axiom,
% 0.30/0.57 ! [C: rat,A: rat,B: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ C @ A )
% 0.30/0.57 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.coboundedI1
% 0.30/0.57 thf(fact_3845_max_OcoboundedI1,axiom,
% 0.30/0.57 ! [C: num,A: num,B: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ C @ A )
% 0.30/0.57 => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.coboundedI1
% 0.30/0.57 thf(fact_3846_max_OcoboundedI1,axiom,
% 0.30/0.57 ! [C: nat,A: nat,B: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ C @ A )
% 0.30/0.57 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.coboundedI1
% 0.30/0.57 thf(fact_3847_max_OcoboundedI1,axiom,
% 0.30/0.57 ! [C: int,A: int,B: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ C @ A )
% 0.30/0.57 => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.coboundedI1
% 0.30/0.57 thf(fact_3848_max_OcoboundedI2,axiom,
% 0.30/0.57 ! [C: extended_enat,B: extended_enat,A: extended_enat] :
% 0.30/0.57 ( ( ord_le2932123472753598470d_enat @ C @ B )
% 0.30/0.57 => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.coboundedI2
% 0.30/0.57 thf(fact_3849_max_OcoboundedI2,axiom,
% 0.30/0.57 ! [C: rat,B: rat,A: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ C @ B )
% 0.30/0.57 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.coboundedI2
% 0.30/0.57 thf(fact_3850_max_OcoboundedI2,axiom,
% 0.30/0.57 ! [C: num,B: num,A: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ C @ B )
% 0.30/0.57 => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.coboundedI2
% 0.30/0.57 thf(fact_3851_max_OcoboundedI2,axiom,
% 0.30/0.57 ! [C: nat,B: nat,A: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ C @ B )
% 0.30/0.57 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.coboundedI2
% 0.30/0.57 thf(fact_3852_max_OcoboundedI2,axiom,
% 0.30/0.57 ! [C: int,B: int,A: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ C @ B )
% 0.30/0.57 => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.coboundedI2
% 0.30/0.57 thf(fact_3853_max_Ostrict__coboundedI2,axiom,
% 0.30/0.57 ! [C: extended_enat,B: extended_enat,A: extended_enat] :
% 0.30/0.57 ( ( ord_le72135733267957522d_enat @ C @ B )
% 0.30/0.57 => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_coboundedI2
% 0.30/0.57 thf(fact_3854_max_Ostrict__coboundedI2,axiom,
% 0.30/0.57 ! [C: real,B: real,A: real] :
% 0.30/0.57 ( ( ord_less_real @ C @ B )
% 0.30/0.57 => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_coboundedI2
% 0.30/0.57 thf(fact_3855_max_Ostrict__coboundedI2,axiom,
% 0.30/0.57 ! [C: rat,B: rat,A: rat] :
% 0.30/0.57 ( ( ord_less_rat @ C @ B )
% 0.30/0.57 => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_coboundedI2
% 0.30/0.57 thf(fact_3856_max_Ostrict__coboundedI2,axiom,
% 0.30/0.57 ! [C: num,B: num,A: num] :
% 0.30/0.57 ( ( ord_less_num @ C @ B )
% 0.30/0.57 => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_coboundedI2
% 0.30/0.57 thf(fact_3857_max_Ostrict__coboundedI2,axiom,
% 0.30/0.57 ! [C: nat,B: nat,A: nat] :
% 0.30/0.57 ( ( ord_less_nat @ C @ B )
% 0.30/0.57 => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_coboundedI2
% 0.30/0.57 thf(fact_3858_max_Ostrict__coboundedI2,axiom,
% 0.30/0.57 ! [C: int,B: int,A: int] :
% 0.30/0.57 ( ( ord_less_int @ C @ B )
% 0.30/0.57 => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_coboundedI2
% 0.30/0.57 thf(fact_3859_max_Ostrict__coboundedI1,axiom,
% 0.30/0.57 ! [C: extended_enat,A: extended_enat,B: extended_enat] :
% 0.30/0.57 ( ( ord_le72135733267957522d_enat @ C @ A )
% 0.30/0.57 => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_coboundedI1
% 0.30/0.57 thf(fact_3860_max_Ostrict__coboundedI1,axiom,
% 0.30/0.57 ! [C: real,A: real,B: real] :
% 0.30/0.57 ( ( ord_less_real @ C @ A )
% 0.30/0.57 => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_coboundedI1
% 0.30/0.57 thf(fact_3861_max_Ostrict__coboundedI1,axiom,
% 0.30/0.57 ! [C: rat,A: rat,B: rat] :
% 0.30/0.57 ( ( ord_less_rat @ C @ A )
% 0.30/0.57 => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_coboundedI1
% 0.30/0.57 thf(fact_3862_max_Ostrict__coboundedI1,axiom,
% 0.30/0.57 ! [C: num,A: num,B: num] :
% 0.30/0.57 ( ( ord_less_num @ C @ A )
% 0.30/0.57 => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_coboundedI1
% 0.30/0.57 thf(fact_3863_max_Ostrict__coboundedI1,axiom,
% 0.30/0.57 ! [C: nat,A: nat,B: nat] :
% 0.30/0.57 ( ( ord_less_nat @ C @ A )
% 0.30/0.57 => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_coboundedI1
% 0.30/0.57 thf(fact_3864_max_Ostrict__coboundedI1,axiom,
% 0.30/0.57 ! [C: int,A: int,B: int] :
% 0.30/0.57 ( ( ord_less_int @ C @ A )
% 0.30/0.57 => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_coboundedI1
% 0.30/0.57 thf(fact_3865_max_Ostrict__order__iff,axiom,
% 0.30/0.57 ( ord_le72135733267957522d_enat
% 0.30/0.57 = ( ^ [B2: extended_enat,A3: extended_enat] :
% 0.30/0.57 ( ( A3
% 0.30/0.57 = ( ord_ma741700101516333627d_enat @ A3 @ B2 ) )
% 0.30/0.57 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_order_iff
% 0.30/0.57 thf(fact_3866_max_Ostrict__order__iff,axiom,
% 0.30/0.57 ( ord_less_real
% 0.30/0.57 = ( ^ [B2: real,A3: real] :
% 0.30/0.57 ( ( A3
% 0.30/0.57 = ( ord_max_real @ A3 @ B2 ) )
% 0.30/0.57 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_order_iff
% 0.30/0.57 thf(fact_3867_max_Ostrict__order__iff,axiom,
% 0.30/0.57 ( ord_less_rat
% 0.30/0.57 = ( ^ [B2: rat,A3: rat] :
% 0.30/0.57 ( ( A3
% 0.30/0.57 = ( ord_max_rat @ A3 @ B2 ) )
% 0.30/0.57 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_order_iff
% 0.30/0.57 thf(fact_3868_max_Ostrict__order__iff,axiom,
% 0.30/0.57 ( ord_less_num
% 0.30/0.57 = ( ^ [B2: num,A3: num] :
% 0.30/0.57 ( ( A3
% 0.30/0.57 = ( ord_max_num @ A3 @ B2 ) )
% 0.30/0.57 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_order_iff
% 0.30/0.57 thf(fact_3869_max_Ostrict__order__iff,axiom,
% 0.30/0.57 ( ord_less_nat
% 0.30/0.57 = ( ^ [B2: nat,A3: nat] :
% 0.30/0.57 ( ( A3
% 0.30/0.57 = ( ord_max_nat @ A3 @ B2 ) )
% 0.30/0.57 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_order_iff
% 0.30/0.57 thf(fact_3870_max_Ostrict__order__iff,axiom,
% 0.30/0.57 ( ord_less_int
% 0.30/0.57 = ( ^ [B2: int,A3: int] :
% 0.30/0.57 ( ( A3
% 0.30/0.57 = ( ord_max_int @ A3 @ B2 ) )
% 0.30/0.57 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_order_iff
% 0.30/0.57 thf(fact_3871_max_Ostrict__boundedE,axiom,
% 0.30/0.57 ! [B: extended_enat,C: extended_enat,A: extended_enat] :
% 0.30/0.57 ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
% 0.30/0.57 => ~ ( ( ord_le72135733267957522d_enat @ B @ A )
% 0.30/0.57 => ~ ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_boundedE
% 0.30/0.57 thf(fact_3872_max_Ostrict__boundedE,axiom,
% 0.30/0.57 ! [B: real,C: real,A: real] :
% 0.30/0.57 ( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
% 0.30/0.57 => ~ ( ( ord_less_real @ B @ A )
% 0.30/0.57 => ~ ( ord_less_real @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_boundedE
% 0.30/0.57 thf(fact_3873_max_Ostrict__boundedE,axiom,
% 0.30/0.57 ! [B: rat,C: rat,A: rat] :
% 0.30/0.57 ( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
% 0.30/0.57 => ~ ( ( ord_less_rat @ B @ A )
% 0.30/0.57 => ~ ( ord_less_rat @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_boundedE
% 0.30/0.57 thf(fact_3874_max_Ostrict__boundedE,axiom,
% 0.30/0.57 ! [B: num,C: num,A: num] :
% 0.30/0.57 ( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
% 0.30/0.57 => ~ ( ( ord_less_num @ B @ A )
% 0.30/0.57 => ~ ( ord_less_num @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_boundedE
% 0.30/0.57 thf(fact_3875_max_Ostrict__boundedE,axiom,
% 0.30/0.57 ! [B: nat,C: nat,A: nat] :
% 0.30/0.57 ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
% 0.30/0.57 => ~ ( ( ord_less_nat @ B @ A )
% 0.30/0.57 => ~ ( ord_less_nat @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_boundedE
% 0.30/0.57 thf(fact_3876_max_Ostrict__boundedE,axiom,
% 0.30/0.57 ! [B: int,C: int,A: int] :
% 0.30/0.57 ( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
% 0.30/0.57 => ~ ( ( ord_less_int @ B @ A )
% 0.30/0.57 => ~ ( ord_less_int @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max.strict_boundedE
% 0.30/0.57 thf(fact_3877_less__max__iff__disj,axiom,
% 0.30/0.57 ! [Z: extended_enat,X2: extended_enat,Y3: extended_enat] :
% 0.30/0.57 ( ( ord_le72135733267957522d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X2 @ Y3 ) )
% 0.30/0.57 = ( ( ord_le72135733267957522d_enat @ Z @ X2 )
% 0.30/0.57 | ( ord_le72135733267957522d_enat @ Z @ Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % less_max_iff_disj
% 0.30/0.57 thf(fact_3878_less__max__iff__disj,axiom,
% 0.30/0.57 ! [Z: real,X2: real,Y3: real] :
% 0.30/0.57 ( ( ord_less_real @ Z @ ( ord_max_real @ X2 @ Y3 ) )
% 0.30/0.57 = ( ( ord_less_real @ Z @ X2 )
% 0.30/0.57 | ( ord_less_real @ Z @ Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % less_max_iff_disj
% 0.30/0.57 thf(fact_3879_less__max__iff__disj,axiom,
% 0.30/0.57 ! [Z: rat,X2: rat,Y3: rat] :
% 0.30/0.57 ( ( ord_less_rat @ Z @ ( ord_max_rat @ X2 @ Y3 ) )
% 0.30/0.57 = ( ( ord_less_rat @ Z @ X2 )
% 0.30/0.57 | ( ord_less_rat @ Z @ Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % less_max_iff_disj
% 0.30/0.57 thf(fact_3880_less__max__iff__disj,axiom,
% 0.30/0.57 ! [Z: num,X2: num,Y3: num] :
% 0.30/0.57 ( ( ord_less_num @ Z @ ( ord_max_num @ X2 @ Y3 ) )
% 0.30/0.57 = ( ( ord_less_num @ Z @ X2 )
% 0.30/0.57 | ( ord_less_num @ Z @ Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % less_max_iff_disj
% 0.30/0.57 thf(fact_3881_less__max__iff__disj,axiom,
% 0.30/0.57 ! [Z: nat,X2: nat,Y3: nat] :
% 0.30/0.57 ( ( ord_less_nat @ Z @ ( ord_max_nat @ X2 @ Y3 ) )
% 0.30/0.57 = ( ( ord_less_nat @ Z @ X2 )
% 0.30/0.57 | ( ord_less_nat @ Z @ Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % less_max_iff_disj
% 0.30/0.57 thf(fact_3882_less__max__iff__disj,axiom,
% 0.30/0.57 ! [Z: int,X2: int,Y3: int] :
% 0.30/0.57 ( ( ord_less_int @ Z @ ( ord_max_int @ X2 @ Y3 ) )
% 0.30/0.57 = ( ( ord_less_int @ Z @ X2 )
% 0.30/0.57 | ( ord_less_int @ Z @ Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % less_max_iff_disj
% 0.30/0.57 thf(fact_3883_max__add__distrib__right,axiom,
% 0.30/0.57 ! [X2: real,Y3: real,Z: real] :
% 0.30/0.57 ( ( plus_plus_real @ X2 @ ( ord_max_real @ Y3 @ Z ) )
% 0.30/0.57 = ( ord_max_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( plus_plus_real @ X2 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_add_distrib_right
% 0.30/0.57 thf(fact_3884_max__add__distrib__right,axiom,
% 0.30/0.57 ! [X2: rat,Y3: rat,Z: rat] :
% 0.30/0.57 ( ( plus_plus_rat @ X2 @ ( ord_max_rat @ Y3 @ Z ) )
% 0.30/0.57 = ( ord_max_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ ( plus_plus_rat @ X2 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_add_distrib_right
% 0.30/0.57 thf(fact_3885_max__add__distrib__right,axiom,
% 0.30/0.57 ! [X2: nat,Y3: nat,Z: nat] :
% 0.30/0.57 ( ( plus_plus_nat @ X2 @ ( ord_max_nat @ Y3 @ Z ) )
% 0.30/0.57 = ( ord_max_nat @ ( plus_plus_nat @ X2 @ Y3 ) @ ( plus_plus_nat @ X2 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_add_distrib_right
% 0.30/0.57 thf(fact_3886_max__add__distrib__right,axiom,
% 0.30/0.57 ! [X2: int,Y3: int,Z: int] :
% 0.30/0.57 ( ( plus_plus_int @ X2 @ ( ord_max_int @ Y3 @ Z ) )
% 0.30/0.57 = ( ord_max_int @ ( plus_plus_int @ X2 @ Y3 ) @ ( plus_plus_int @ X2 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_add_distrib_right
% 0.30/0.57 thf(fact_3887_max__add__distrib__left,axiom,
% 0.30/0.57 ! [X2: real,Y3: real,Z: real] :
% 0.30/0.57 ( ( plus_plus_real @ ( ord_max_real @ X2 @ Y3 ) @ Z )
% 0.30/0.57 = ( ord_max_real @ ( plus_plus_real @ X2 @ Z ) @ ( plus_plus_real @ Y3 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_add_distrib_left
% 0.30/0.57 thf(fact_3888_max__add__distrib__left,axiom,
% 0.30/0.57 ! [X2: rat,Y3: rat,Z: rat] :
% 0.30/0.57 ( ( plus_plus_rat @ ( ord_max_rat @ X2 @ Y3 ) @ Z )
% 0.30/0.57 = ( ord_max_rat @ ( plus_plus_rat @ X2 @ Z ) @ ( plus_plus_rat @ Y3 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_add_distrib_left
% 0.30/0.57 thf(fact_3889_max__add__distrib__left,axiom,
% 0.30/0.57 ! [X2: nat,Y3: nat,Z: nat] :
% 0.30/0.57 ( ( plus_plus_nat @ ( ord_max_nat @ X2 @ Y3 ) @ Z )
% 0.30/0.57 = ( ord_max_nat @ ( plus_plus_nat @ X2 @ Z ) @ ( plus_plus_nat @ Y3 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_add_distrib_left
% 0.30/0.57 thf(fact_3890_max__add__distrib__left,axiom,
% 0.30/0.57 ! [X2: int,Y3: int,Z: int] :
% 0.30/0.57 ( ( plus_plus_int @ ( ord_max_int @ X2 @ Y3 ) @ Z )
% 0.30/0.57 = ( ord_max_int @ ( plus_plus_int @ X2 @ Z ) @ ( plus_plus_int @ Y3 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_add_distrib_left
% 0.30/0.57 thf(fact_3891_max__diff__distrib__left,axiom,
% 0.30/0.57 ! [X2: real,Y3: real,Z: real] :
% 0.30/0.57 ( ( minus_minus_real @ ( ord_max_real @ X2 @ Y3 ) @ Z )
% 0.30/0.57 = ( ord_max_real @ ( minus_minus_real @ X2 @ Z ) @ ( minus_minus_real @ Y3 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_diff_distrib_left
% 0.30/0.57 thf(fact_3892_max__diff__distrib__left,axiom,
% 0.30/0.57 ! [X2: rat,Y3: rat,Z: rat] :
% 0.30/0.57 ( ( minus_minus_rat @ ( ord_max_rat @ X2 @ Y3 ) @ Z )
% 0.30/0.57 = ( ord_max_rat @ ( minus_minus_rat @ X2 @ Z ) @ ( minus_minus_rat @ Y3 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_diff_distrib_left
% 0.30/0.57 thf(fact_3893_max__diff__distrib__left,axiom,
% 0.30/0.57 ! [X2: int,Y3: int,Z: int] :
% 0.30/0.57 ( ( minus_minus_int @ ( ord_max_int @ X2 @ Y3 ) @ Z )
% 0.30/0.57 = ( ord_max_int @ ( minus_minus_int @ X2 @ Z ) @ ( minus_minus_int @ Y3 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_diff_distrib_left
% 0.30/0.57 thf(fact_3894_nat__add__max__right,axiom,
% 0.30/0.57 ! [M: nat,N: nat,Q3: nat] :
% 0.30/0.57 ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
% 0.30/0.57 = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % nat_add_max_right
% 0.30/0.57 thf(fact_3895_nat__add__max__left,axiom,
% 0.30/0.57 ! [M: nat,N: nat,Q3: nat] :
% 0.30/0.57 ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
% 0.30/0.57 = ( ord_max_nat @ ( plus_plus_nat @ M @ Q3 ) @ ( plus_plus_nat @ N @ Q3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % nat_add_max_left
% 0.30/0.57 thf(fact_3896_nat__mult__max__right,axiom,
% 0.30/0.57 ! [M: nat,N: nat,Q3: nat] :
% 0.30/0.57 ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
% 0.30/0.57 = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % nat_mult_max_right
% 0.30/0.57 thf(fact_3897_nat__mult__max__left,axiom,
% 0.30/0.57 ! [M: nat,N: nat,Q3: nat] :
% 0.30/0.57 ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
% 0.30/0.57 = ( ord_max_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % nat_mult_max_left
% 0.30/0.57 thf(fact_3898_semiring__norm_I26_J,axiom,
% 0.30/0.57 ( ( bitM @ one )
% 0.30/0.57 = one ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(26)
% 0.30/0.57 thf(fact_3899_max__def__raw,axiom,
% 0.30/0.57 ( ord_ma741700101516333627d_enat
% 0.30/0.57 = ( ^ [A3: extended_enat,B2: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_def_raw
% 0.30/0.57 thf(fact_3900_max__def__raw,axiom,
% 0.30/0.57 ( ord_max_set_int
% 0.30/0.57 = ( ^ [A3: set_int,B2: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_def_raw
% 0.30/0.57 thf(fact_3901_max__def__raw,axiom,
% 0.30/0.57 ( ord_max_rat
% 0.30/0.57 = ( ^ [A3: rat,B2: rat] : ( if_rat @ ( ord_less_eq_rat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_def_raw
% 0.30/0.57 thf(fact_3902_max__def__raw,axiom,
% 0.30/0.57 ( ord_max_num
% 0.30/0.57 = ( ^ [A3: num,B2: num] : ( if_num @ ( ord_less_eq_num @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_def_raw
% 0.30/0.57 thf(fact_3903_max__def__raw,axiom,
% 0.30/0.57 ( ord_max_nat
% 0.30/0.57 = ( ^ [A3: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_def_raw
% 0.30/0.57 thf(fact_3904_max__def__raw,axiom,
% 0.30/0.57 ( ord_max_int
% 0.30/0.57 = ( ^ [A3: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % max_def_raw
% 0.30/0.57 thf(fact_3905_nat__minus__add__max,axiom,
% 0.30/0.57 ! [N: nat,M: nat] :
% 0.30/0.57 ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 0.30/0.57 = ( ord_max_nat @ N @ M ) ) ).
% 0.30/0.57
% 0.30/0.57 % nat_minus_add_max
% 0.30/0.57 thf(fact_3906_set__update__subsetI,axiom,
% 0.30/0.57 ! [Xs2: list_P6011104703257516679at_nat,A2: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,I2: nat] :
% 0.30/0.57 ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 )
% 0.30/0.57 => ( ( member8440522571783428010at_nat @ X2 @ A2 )
% 0.30/0.57 => ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs2 @ I2 @ X2 ) ) @ A2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % set_update_subsetI
% 0.30/0.57 thf(fact_3907_set__update__subsetI,axiom,
% 0.30/0.57 ! [Xs2: list_complex,A2: set_complex,X2: complex,I2: nat] :
% 0.30/0.57 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
% 0.30/0.57 => ( ( member_complex @ X2 @ A2 )
% 0.30/0.57 => ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs2 @ I2 @ X2 ) ) @ A2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % set_update_subsetI
% 0.30/0.57 thf(fact_3908_set__update__subsetI,axiom,
% 0.30/0.57 ! [Xs2: list_real,A2: set_real,X2: real,I2: nat] :
% 0.30/0.57 ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A2 )
% 0.30/0.57 => ( ( member_real @ X2 @ A2 )
% 0.30/0.57 => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I2 @ X2 ) ) @ A2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % set_update_subsetI
% 0.30/0.57 thf(fact_3909_set__update__subsetI,axiom,
% 0.30/0.57 ! [Xs2: list_nat,A2: set_nat,X2: nat,I2: nat] :
% 0.30/0.57 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
% 0.30/0.57 => ( ( member_nat @ X2 @ A2 )
% 0.30/0.57 => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I2 @ X2 ) ) @ A2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % set_update_subsetI
% 0.30/0.57 thf(fact_3910_set__update__subsetI,axiom,
% 0.30/0.57 ! [Xs2: list_VEBT_VEBT,A2: set_VEBT_VEBT,X2: vEBT_VEBT,I2: nat] :
% 0.30/0.57 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
% 0.30/0.57 => ( ( member_VEBT_VEBT @ X2 @ A2 )
% 0.30/0.57 => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X2 ) ) @ A2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % set_update_subsetI
% 0.30/0.57 thf(fact_3911_set__update__subsetI,axiom,
% 0.30/0.57 ! [Xs2: list_int,A2: set_int,X2: int,I2: nat] :
% 0.30/0.57 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
% 0.30/0.57 => ( ( member_int @ X2 @ A2 )
% 0.30/0.57 => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I2 @ X2 ) ) @ A2 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % set_update_subsetI
% 0.30/0.57 thf(fact_3912_semiring__norm_I28_J,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( bitM @ ( bit1 @ N ) )
% 0.30/0.57 = ( bit1 @ ( bit0 @ N ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(28)
% 0.30/0.57 thf(fact_3913_semiring__norm_I27_J,axiom,
% 0.30/0.57 ! [N: num] :
% 0.30/0.57 ( ( bitM @ ( bit0 @ N ) )
% 0.30/0.57 = ( bit1 @ ( bitM @ N ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % semiring_norm(27)
% 0.30/0.57 thf(fact_3914_set__update__memI,axiom,
% 0.30/0.57 ! [N: nat,Xs2: list_P6011104703257516679at_nat,X2: product_prod_nat_nat] :
% 0.30/0.57 ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
% 0.30/0.57 => ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs2 @ N @ X2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % set_update_memI
% 0.30/0.57 thf(fact_3915_set__update__memI,axiom,
% 0.30/0.57 ! [N: nat,Xs2: list_complex,X2: complex] :
% 0.30/0.57 ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 0.30/0.57 => ( member_complex @ X2 @ ( set_complex2 @ ( list_update_complex @ Xs2 @ N @ X2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % set_update_memI
% 0.30/0.57 thf(fact_3916_set__update__memI,axiom,
% 0.30/0.57 ! [N: nat,Xs2: list_real,X2: real] :
% 0.30/0.57 ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 0.30/0.57 => ( member_real @ X2 @ ( set_real2 @ ( list_update_real @ Xs2 @ N @ X2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % set_update_memI
% 0.30/0.57 thf(fact_3917_set__update__memI,axiom,
% 0.30/0.57 ! [N: nat,Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 0.30/0.57 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.57 => ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N @ X2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % set_update_memI
% 0.30/0.57 thf(fact_3918_set__update__memI,axiom,
% 0.30/0.57 ! [N: nat,Xs2: list_o,X2: $o] :
% 0.30/0.57 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.57 => ( member_o @ X2 @ ( set_o2 @ ( list_update_o @ Xs2 @ N @ X2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % set_update_memI
% 0.30/0.57 thf(fact_3919_set__update__memI,axiom,
% 0.30/0.57 ! [N: nat,Xs2: list_nat,X2: nat] :
% 0.30/0.57 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.57 => ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % set_update_memI
% 0.30/0.57 thf(fact_3920_set__update__memI,axiom,
% 0.30/0.57 ! [N: nat,Xs2: list_int,X2: int] :
% 0.30/0.57 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.57 => ( member_int @ X2 @ ( set_int2 @ ( list_update_int @ Xs2 @ N @ X2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % set_update_memI
% 0.30/0.57 thf(fact_3921_order__antisym__conv,axiom,
% 0.30/0.57 ! [Y3: set_int,X2: set_int] :
% 0.30/0.57 ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 0.30/0.57 => ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.30/0.57 = ( X2 = Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_antisym_conv
% 0.30/0.57 thf(fact_3922_order__antisym__conv,axiom,
% 0.30/0.57 ! [Y3: rat,X2: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 0.30/0.57 => ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.30/0.57 = ( X2 = Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_antisym_conv
% 0.30/0.57 thf(fact_3923_order__antisym__conv,axiom,
% 0.30/0.57 ! [Y3: num,X2: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ Y3 @ X2 )
% 0.30/0.57 => ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.30/0.57 = ( X2 = Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_antisym_conv
% 0.30/0.57 thf(fact_3924_order__antisym__conv,axiom,
% 0.30/0.57 ! [Y3: nat,X2: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.30/0.57 => ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.30/0.57 = ( X2 = Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_antisym_conv
% 0.30/0.57 thf(fact_3925_order__antisym__conv,axiom,
% 0.30/0.57 ! [Y3: int,X2: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ Y3 @ X2 )
% 0.30/0.57 => ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.30/0.57 = ( X2 = Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_antisym_conv
% 0.30/0.57 thf(fact_3926_linorder__le__cases,axiom,
% 0.30/0.57 ! [X2: rat,Y3: rat] :
% 0.30/0.57 ( ~ ( ord_less_eq_rat @ X2 @ Y3 )
% 0.30/0.57 => ( ord_less_eq_rat @ Y3 @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % linorder_le_cases
% 0.30/0.57 thf(fact_3927_linorder__le__cases,axiom,
% 0.30/0.57 ! [X2: num,Y3: num] :
% 0.30/0.57 ( ~ ( ord_less_eq_num @ X2 @ Y3 )
% 0.30/0.57 => ( ord_less_eq_num @ Y3 @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % linorder_le_cases
% 0.30/0.57 thf(fact_3928_linorder__le__cases,axiom,
% 0.30/0.57 ! [X2: nat,Y3: nat] :
% 0.30/0.57 ( ~ ( ord_less_eq_nat @ X2 @ Y3 )
% 0.30/0.57 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % linorder_le_cases
% 0.30/0.57 thf(fact_3929_linorder__le__cases,axiom,
% 0.30/0.57 ! [X2: int,Y3: int] :
% 0.30/0.57 ( ~ ( ord_less_eq_int @ X2 @ Y3 )
% 0.30/0.57 => ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % linorder_le_cases
% 0.30/0.57 thf(fact_3930_ord__le__eq__subst,axiom,
% 0.30/0.57 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.57 => ( ( ( F @ B )
% 0.30/0.57 = C )
% 0.30/0.57 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_le_eq_subst
% 0.30/0.57 thf(fact_3931_ord__le__eq__subst,axiom,
% 0.30/0.57 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.30/0.57 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.57 => ( ( ( F @ B )
% 0.30/0.57 = C )
% 0.30/0.57 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_le_eq_subst
% 0.30/0.57 thf(fact_3932_ord__le__eq__subst,axiom,
% 0.30/0.57 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.57 => ( ( ( F @ B )
% 0.30/0.57 = C )
% 0.30/0.57 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_le_eq_subst
% 0.30/0.57 thf(fact_3933_ord__le__eq__subst,axiom,
% 0.30/0.57 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.30/0.57 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.57 => ( ( ( F @ B )
% 0.30/0.57 = C )
% 0.30/0.57 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_le_eq_subst
% 0.30/0.57 thf(fact_3934_ord__le__eq__subst,axiom,
% 0.30/0.57 ! [A: num,B: num,F: num > rat,C: rat] :
% 0.30/0.57 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.57 => ( ( ( F @ B )
% 0.30/0.57 = C )
% 0.30/0.57 => ( ! [X5: num,Y4: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_le_eq_subst
% 0.30/0.57 thf(fact_3935_ord__le__eq__subst,axiom,
% 0.30/0.57 ! [A: num,B: num,F: num > num,C: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.57 => ( ( ( F @ B )
% 0.30/0.57 = C )
% 0.30/0.57 => ( ! [X5: num,Y4: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_le_eq_subst
% 0.30/0.57 thf(fact_3936_ord__le__eq__subst,axiom,
% 0.30/0.57 ! [A: num,B: num,F: num > nat,C: nat] :
% 0.30/0.57 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.57 => ( ( ( F @ B )
% 0.30/0.57 = C )
% 0.30/0.57 => ( ! [X5: num,Y4: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_le_eq_subst
% 0.30/0.57 thf(fact_3937_ord__le__eq__subst,axiom,
% 0.30/0.57 ! [A: num,B: num,F: num > int,C: int] :
% 0.30/0.57 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.57 => ( ( ( F @ B )
% 0.30/0.57 = C )
% 0.30/0.57 => ( ! [X5: num,Y4: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_le_eq_subst
% 0.30/0.57 thf(fact_3938_ord__le__eq__subst,axiom,
% 0.30/0.57 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.57 => ( ( ( F @ B )
% 0.30/0.57 = C )
% 0.30/0.57 => ( ! [X5: nat,Y4: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_le_eq_subst
% 0.30/0.57 thf(fact_3939_ord__le__eq__subst,axiom,
% 0.30/0.57 ! [A: nat,B: nat,F: nat > num,C: num] :
% 0.30/0.57 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.57 => ( ( ( F @ B )
% 0.30/0.57 = C )
% 0.30/0.57 => ( ! [X5: nat,Y4: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_le_eq_subst
% 0.30/0.57 thf(fact_3940_ord__eq__le__subst,axiom,
% 0.30/0.57 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.30/0.57 ( ( A
% 0.30/0.57 = ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_rat @ B @ C )
% 0.30/0.57 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_eq_le_subst
% 0.30/0.57 thf(fact_3941_ord__eq__le__subst,axiom,
% 0.30/0.57 ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.30/0.57 ( ( A
% 0.30/0.57 = ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_rat @ B @ C )
% 0.30/0.57 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_eq_le_subst
% 0.30/0.57 thf(fact_3942_ord__eq__le__subst,axiom,
% 0.30/0.57 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.30/0.57 ( ( A
% 0.30/0.57 = ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_rat @ B @ C )
% 0.30/0.57 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_eq_le_subst
% 0.30/0.57 thf(fact_3943_ord__eq__le__subst,axiom,
% 0.30/0.57 ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.30/0.57 ( ( A
% 0.30/0.57 = ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_rat @ B @ C )
% 0.30/0.57 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_eq_le_subst
% 0.30/0.57 thf(fact_3944_ord__eq__le__subst,axiom,
% 0.30/0.57 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.30/0.57 ( ( A
% 0.30/0.57 = ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_num @ B @ C )
% 0.30/0.57 => ( ! [X5: num,Y4: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_eq_le_subst
% 0.30/0.57 thf(fact_3945_ord__eq__le__subst,axiom,
% 0.30/0.57 ! [A: num,F: num > num,B: num,C: num] :
% 0.30/0.57 ( ( A
% 0.30/0.57 = ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_num @ B @ C )
% 0.30/0.57 => ( ! [X5: num,Y4: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_eq_le_subst
% 0.30/0.57 thf(fact_3946_ord__eq__le__subst,axiom,
% 0.30/0.57 ! [A: nat,F: num > nat,B: num,C: num] :
% 0.30/0.57 ( ( A
% 0.30/0.57 = ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_num @ B @ C )
% 0.30/0.57 => ( ! [X5: num,Y4: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_eq_le_subst
% 0.30/0.57 thf(fact_3947_ord__eq__le__subst,axiom,
% 0.30/0.57 ! [A: int,F: num > int,B: num,C: num] :
% 0.30/0.57 ( ( A
% 0.30/0.57 = ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_num @ B @ C )
% 0.30/0.57 => ( ! [X5: num,Y4: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_eq_le_subst
% 0.30/0.57 thf(fact_3948_ord__eq__le__subst,axiom,
% 0.30/0.57 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.30/0.57 ( ( A
% 0.30/0.57 = ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_nat @ B @ C )
% 0.30/0.57 => ( ! [X5: nat,Y4: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_eq_le_subst
% 0.30/0.57 thf(fact_3949_ord__eq__le__subst,axiom,
% 0.30/0.57 ! [A: num,F: nat > num,B: nat,C: nat] :
% 0.30/0.57 ( ( A
% 0.30/0.57 = ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_nat @ B @ C )
% 0.30/0.57 => ( ! [X5: nat,Y4: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_eq_le_subst
% 0.30/0.57 thf(fact_3950_linorder__linear,axiom,
% 0.30/0.57 ! [X2: rat,Y3: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.30/0.57 | ( ord_less_eq_rat @ Y3 @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % linorder_linear
% 0.30/0.57 thf(fact_3951_linorder__linear,axiom,
% 0.30/0.57 ! [X2: num,Y3: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.30/0.57 | ( ord_less_eq_num @ Y3 @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % linorder_linear
% 0.30/0.57 thf(fact_3952_linorder__linear,axiom,
% 0.30/0.57 ! [X2: nat,Y3: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.30/0.57 | ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % linorder_linear
% 0.30/0.57 thf(fact_3953_linorder__linear,axiom,
% 0.30/0.57 ! [X2: int,Y3: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.30/0.57 | ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% 0.30/0.57
% 0.30/0.57 % linorder_linear
% 0.30/0.57 thf(fact_3954_order__eq__refl,axiom,
% 0.30/0.57 ! [X2: set_int,Y3: set_int] :
% 0.30/0.57 ( ( X2 = Y3 )
% 0.30/0.57 => ( ord_less_eq_set_int @ X2 @ Y3 ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_eq_refl
% 0.30/0.57 thf(fact_3955_order__eq__refl,axiom,
% 0.30/0.57 ! [X2: rat,Y3: rat] :
% 0.30/0.57 ( ( X2 = Y3 )
% 0.30/0.57 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_eq_refl
% 0.30/0.57 thf(fact_3956_order__eq__refl,axiom,
% 0.30/0.57 ! [X2: num,Y3: num] :
% 0.30/0.57 ( ( X2 = Y3 )
% 0.30/0.57 => ( ord_less_eq_num @ X2 @ Y3 ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_eq_refl
% 0.30/0.57 thf(fact_3957_order__eq__refl,axiom,
% 0.30/0.57 ! [X2: nat,Y3: nat] :
% 0.30/0.57 ( ( X2 = Y3 )
% 0.30/0.57 => ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_eq_refl
% 0.30/0.57 thf(fact_3958_order__eq__refl,axiom,
% 0.30/0.57 ! [X2: int,Y3: int] :
% 0.30/0.57 ( ( X2 = Y3 )
% 0.30/0.57 => ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_eq_refl
% 0.30/0.57 thf(fact_3959_order__subst2,axiom,
% 0.30/0.57 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.30/0.57 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst2
% 0.30/0.57 thf(fact_3960_order__subst2,axiom,
% 0.30/0.57 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.30/0.57 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.30/0.57 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst2
% 0.30/0.57 thf(fact_3961_order__subst2,axiom,
% 0.30/0.57 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.30/0.57 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst2
% 0.30/0.57 thf(fact_3962_order__subst2,axiom,
% 0.30/0.57 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.30/0.57 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.30/0.57 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst2
% 0.30/0.57 thf(fact_3963_order__subst2,axiom,
% 0.30/0.57 ! [A: num,B: num,F: num > rat,C: rat] :
% 0.30/0.57 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.30/0.57 => ( ! [X5: num,Y4: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst2
% 0.30/0.57 thf(fact_3964_order__subst2,axiom,
% 0.30/0.57 ! [A: num,B: num,F: num > num,C: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.30/0.57 => ( ! [X5: num,Y4: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst2
% 0.30/0.57 thf(fact_3965_order__subst2,axiom,
% 0.30/0.57 ! [A: num,B: num,F: num > nat,C: nat] :
% 0.30/0.57 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.30/0.57 => ( ! [X5: num,Y4: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst2
% 0.30/0.57 thf(fact_3966_order__subst2,axiom,
% 0.30/0.57 ! [A: num,B: num,F: num > int,C: int] :
% 0.30/0.57 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.30/0.57 => ( ! [X5: num,Y4: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst2
% 0.30/0.57 thf(fact_3967_order__subst2,axiom,
% 0.30/0.57 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.30/0.57 => ( ! [X5: nat,Y4: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst2
% 0.30/0.57 thf(fact_3968_order__subst2,axiom,
% 0.30/0.57 ! [A: nat,B: nat,F: nat > num,C: num] :
% 0.30/0.57 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.30/0.57 => ( ! [X5: nat,Y4: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst2
% 0.30/0.57 thf(fact_3969_order__subst1,axiom,
% 0.30/0.57 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_rat @ B @ C )
% 0.30/0.57 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst1
% 0.30/0.57 thf(fact_3970_order__subst1,axiom,
% 0.30/0.57 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.30/0.57 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_num @ B @ C )
% 0.30/0.57 => ( ! [X5: num,Y4: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst1
% 0.30/0.57 thf(fact_3971_order__subst1,axiom,
% 0.30/0.57 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_nat @ B @ C )
% 0.30/0.57 => ( ! [X5: nat,Y4: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst1
% 0.30/0.57 thf(fact_3972_order__subst1,axiom,
% 0.30/0.57 ! [A: rat,F: int > rat,B: int,C: int] :
% 0.30/0.57 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_int @ B @ C )
% 0.30/0.57 => ( ! [X5: int,Y4: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst1
% 0.30/0.57 thf(fact_3973_order__subst1,axiom,
% 0.30/0.57 ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.30/0.57 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_rat @ B @ C )
% 0.30/0.57 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst1
% 0.30/0.57 thf(fact_3974_order__subst1,axiom,
% 0.30/0.57 ! [A: num,F: num > num,B: num,C: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_num @ B @ C )
% 0.30/0.57 => ( ! [X5: num,Y4: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst1
% 0.30/0.57 thf(fact_3975_order__subst1,axiom,
% 0.30/0.57 ! [A: num,F: nat > num,B: nat,C: nat] :
% 0.30/0.57 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_nat @ B @ C )
% 0.30/0.57 => ( ! [X5: nat,Y4: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst1
% 0.30/0.57 thf(fact_3976_order__subst1,axiom,
% 0.30/0.57 ! [A: num,F: int > num,B: int,C: int] :
% 0.30/0.57 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_int @ B @ C )
% 0.30/0.57 => ( ! [X5: int,Y4: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst1
% 0.30/0.57 thf(fact_3977_order__subst1,axiom,
% 0.30/0.57 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_rat @ B @ C )
% 0.30/0.57 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst1
% 0.30/0.57 thf(fact_3978_order__subst1,axiom,
% 0.30/0.57 ! [A: nat,F: num > nat,B: num,C: num] :
% 0.30/0.57 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.30/0.57 => ( ( ord_less_eq_num @ B @ C )
% 0.30/0.57 => ( ! [X5: num,Y4: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.57 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.57 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_subst1
% 0.30/0.57 thf(fact_3979_Orderings_Oorder__eq__iff,axiom,
% 0.30/0.57 ( ( ^ [Y6: set_int,Z5: set_int] : ( Y6 = Z5 ) )
% 0.30/0.57 = ( ^ [A3: set_int,B2: set_int] :
% 0.30/0.57 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.30/0.57 & ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Orderings.order_eq_iff
% 0.30/0.57 thf(fact_3980_Orderings_Oorder__eq__iff,axiom,
% 0.30/0.57 ( ( ^ [Y6: rat,Z5: rat] : ( Y6 = Z5 ) )
% 0.30/0.57 = ( ^ [A3: rat,B2: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ A3 @ B2 )
% 0.30/0.57 & ( ord_less_eq_rat @ B2 @ A3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Orderings.order_eq_iff
% 0.30/0.57 thf(fact_3981_Orderings_Oorder__eq__iff,axiom,
% 0.30/0.57 ( ( ^ [Y6: num,Z5: num] : ( Y6 = Z5 ) )
% 0.30/0.57 = ( ^ [A3: num,B2: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ A3 @ B2 )
% 0.30/0.57 & ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Orderings.order_eq_iff
% 0.30/0.57 thf(fact_3982_Orderings_Oorder__eq__iff,axiom,
% 0.30/0.57 ( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 0.30/0.57 = ( ^ [A3: nat,B2: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ A3 @ B2 )
% 0.30/0.57 & ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Orderings.order_eq_iff
% 0.30/0.57 thf(fact_3983_Orderings_Oorder__eq__iff,axiom,
% 0.30/0.57 ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 0.30/0.57 = ( ^ [A3: int,B2: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ A3 @ B2 )
% 0.30/0.57 & ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % Orderings.order_eq_iff
% 0.30/0.57 thf(fact_3984_antisym,axiom,
% 0.30/0.57 ! [A: set_int,B: set_int] :
% 0.30/0.57 ( ( ord_less_eq_set_int @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_set_int @ B @ A )
% 0.30/0.57 => ( A = B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % antisym
% 0.30/0.57 thf(fact_3985_antisym,axiom,
% 0.30/0.57 ! [A: rat,B: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.57 => ( A = B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % antisym
% 0.30/0.57 thf(fact_3986_antisym,axiom,
% 0.30/0.57 ! [A: num,B: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_num @ B @ A )
% 0.30/0.57 => ( A = B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % antisym
% 0.30/0.57 thf(fact_3987_antisym,axiom,
% 0.30/0.57 ! [A: nat,B: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.57 => ( A = B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % antisym
% 0.30/0.57 thf(fact_3988_antisym,axiom,
% 0.30/0.57 ! [A: int,B: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_int @ B @ A )
% 0.30/0.57 => ( A = B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % antisym
% 0.30/0.57 thf(fact_3989_dual__order_Otrans,axiom,
% 0.30/0.57 ! [B: set_int,A: set_int,C: set_int] :
% 0.30/0.57 ( ( ord_less_eq_set_int @ B @ A )
% 0.30/0.57 => ( ( ord_less_eq_set_int @ C @ B )
% 0.30/0.57 => ( ord_less_eq_set_int @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.trans
% 0.30/0.57 thf(fact_3990_dual__order_Otrans,axiom,
% 0.30/0.57 ! [B: rat,A: rat,C: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.57 => ( ( ord_less_eq_rat @ C @ B )
% 0.30/0.57 => ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.trans
% 0.30/0.57 thf(fact_3991_dual__order_Otrans,axiom,
% 0.30/0.57 ! [B: num,A: num,C: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ B @ A )
% 0.30/0.57 => ( ( ord_less_eq_num @ C @ B )
% 0.30/0.57 => ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.trans
% 0.30/0.57 thf(fact_3992_dual__order_Otrans,axiom,
% 0.30/0.57 ! [B: nat,A: nat,C: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.57 => ( ( ord_less_eq_nat @ C @ B )
% 0.30/0.57 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.trans
% 0.30/0.57 thf(fact_3993_dual__order_Otrans,axiom,
% 0.30/0.57 ! [B: int,A: int,C: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ B @ A )
% 0.30/0.57 => ( ( ord_less_eq_int @ C @ B )
% 0.30/0.57 => ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.trans
% 0.30/0.57 thf(fact_3994_dual__order_Oantisym,axiom,
% 0.30/0.57 ! [B: set_int,A: set_int] :
% 0.30/0.57 ( ( ord_less_eq_set_int @ B @ A )
% 0.30/0.57 => ( ( ord_less_eq_set_int @ A @ B )
% 0.30/0.57 => ( A = B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.antisym
% 0.30/0.57 thf(fact_3995_dual__order_Oantisym,axiom,
% 0.30/0.57 ! [B: rat,A: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.57 => ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.57 => ( A = B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.antisym
% 0.30/0.57 thf(fact_3996_dual__order_Oantisym,axiom,
% 0.30/0.57 ! [B: num,A: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ B @ A )
% 0.30/0.57 => ( ( ord_less_eq_num @ A @ B )
% 0.30/0.57 => ( A = B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.antisym
% 0.30/0.57 thf(fact_3997_dual__order_Oantisym,axiom,
% 0.30/0.57 ! [B: nat,A: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.57 => ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.57 => ( A = B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.antisym
% 0.30/0.57 thf(fact_3998_dual__order_Oantisym,axiom,
% 0.30/0.57 ! [B: int,A: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ B @ A )
% 0.30/0.57 => ( ( ord_less_eq_int @ A @ B )
% 0.30/0.57 => ( A = B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.antisym
% 0.30/0.57 thf(fact_3999_dual__order_Oeq__iff,axiom,
% 0.30/0.57 ( ( ^ [Y6: set_int,Z5: set_int] : ( Y6 = Z5 ) )
% 0.30/0.57 = ( ^ [A3: set_int,B2: set_int] :
% 0.30/0.57 ( ( ord_less_eq_set_int @ B2 @ A3 )
% 0.30/0.57 & ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.eq_iff
% 0.30/0.57 thf(fact_4000_dual__order_Oeq__iff,axiom,
% 0.30/0.57 ( ( ^ [Y6: rat,Z5: rat] : ( Y6 = Z5 ) )
% 0.30/0.57 = ( ^ [A3: rat,B2: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ B2 @ A3 )
% 0.30/0.57 & ( ord_less_eq_rat @ A3 @ B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.eq_iff
% 0.30/0.57 thf(fact_4001_dual__order_Oeq__iff,axiom,
% 0.30/0.57 ( ( ^ [Y6: num,Z5: num] : ( Y6 = Z5 ) )
% 0.30/0.57 = ( ^ [A3: num,B2: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ B2 @ A3 )
% 0.30/0.57 & ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.eq_iff
% 0.30/0.57 thf(fact_4002_dual__order_Oeq__iff,axiom,
% 0.30/0.57 ( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 0.30/0.57 = ( ^ [A3: nat,B2: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ B2 @ A3 )
% 0.30/0.57 & ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.eq_iff
% 0.30/0.57 thf(fact_4003_dual__order_Oeq__iff,axiom,
% 0.30/0.57 ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 0.30/0.57 = ( ^ [A3: int,B2: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ B2 @ A3 )
% 0.30/0.57 & ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % dual_order.eq_iff
% 0.30/0.57 thf(fact_4004_linorder__wlog,axiom,
% 0.30/0.57 ! [P: rat > rat > $o,A: rat,B: rat] :
% 0.30/0.57 ( ! [A4: rat,B4: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ A4 @ B4 )
% 0.30/0.57 => ( P @ A4 @ B4 ) )
% 0.30/0.57 => ( ! [A4: rat,B4: rat] :
% 0.30/0.57 ( ( P @ B4 @ A4 )
% 0.30/0.57 => ( P @ A4 @ B4 ) )
% 0.30/0.57 => ( P @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % linorder_wlog
% 0.30/0.57 thf(fact_4005_linorder__wlog,axiom,
% 0.30/0.57 ! [P: num > num > $o,A: num,B: num] :
% 0.30/0.57 ( ! [A4: num,B4: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ A4 @ B4 )
% 0.30/0.57 => ( P @ A4 @ B4 ) )
% 0.30/0.57 => ( ! [A4: num,B4: num] :
% 0.30/0.57 ( ( P @ B4 @ A4 )
% 0.30/0.57 => ( P @ A4 @ B4 ) )
% 0.30/0.57 => ( P @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % linorder_wlog
% 0.30/0.57 thf(fact_4006_linorder__wlog,axiom,
% 0.30/0.57 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.30/0.57 ( ! [A4: nat,B4: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ A4 @ B4 )
% 0.30/0.57 => ( P @ A4 @ B4 ) )
% 0.30/0.57 => ( ! [A4: nat,B4: nat] :
% 0.30/0.57 ( ( P @ B4 @ A4 )
% 0.30/0.57 => ( P @ A4 @ B4 ) )
% 0.30/0.57 => ( P @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % linorder_wlog
% 0.30/0.57 thf(fact_4007_linorder__wlog,axiom,
% 0.30/0.57 ! [P: int > int > $o,A: int,B: int] :
% 0.30/0.57 ( ! [A4: int,B4: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ A4 @ B4 )
% 0.30/0.57 => ( P @ A4 @ B4 ) )
% 0.30/0.57 => ( ! [A4: int,B4: int] :
% 0.30/0.57 ( ( P @ B4 @ A4 )
% 0.30/0.57 => ( P @ A4 @ B4 ) )
% 0.30/0.57 => ( P @ A @ B ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % linorder_wlog
% 0.30/0.57 thf(fact_4008_order__trans,axiom,
% 0.30/0.57 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 0.30/0.57 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.30/0.57 => ( ( ord_less_eq_set_int @ Y3 @ Z )
% 0.30/0.57 => ( ord_less_eq_set_int @ X2 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_trans
% 0.30/0.57 thf(fact_4009_order__trans,axiom,
% 0.30/0.57 ! [X2: rat,Y3: rat,Z: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.30/0.57 => ( ( ord_less_eq_rat @ Y3 @ Z )
% 0.30/0.57 => ( ord_less_eq_rat @ X2 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_trans
% 0.30/0.57 thf(fact_4010_order__trans,axiom,
% 0.30/0.57 ! [X2: num,Y3: num,Z: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.30/0.57 => ( ( ord_less_eq_num @ Y3 @ Z )
% 0.30/0.57 => ( ord_less_eq_num @ X2 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_trans
% 0.30/0.57 thf(fact_4011_order__trans,axiom,
% 0.30/0.57 ! [X2: nat,Y3: nat,Z: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.30/0.57 => ( ( ord_less_eq_nat @ Y3 @ Z )
% 0.30/0.57 => ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_trans
% 0.30/0.57 thf(fact_4012_order__trans,axiom,
% 0.30/0.57 ! [X2: int,Y3: int,Z: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.30/0.57 => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.30/0.57 => ( ord_less_eq_int @ X2 @ Z ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_trans
% 0.30/0.57 thf(fact_4013_order_Otrans,axiom,
% 0.30/0.57 ! [A: set_int,B: set_int,C: set_int] :
% 0.30/0.57 ( ( ord_less_eq_set_int @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_set_int @ B @ C )
% 0.30/0.57 => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order.trans
% 0.30/0.57 thf(fact_4014_order_Otrans,axiom,
% 0.30/0.57 ! [A: rat,B: rat,C: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_rat @ B @ C )
% 0.30/0.57 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order.trans
% 0.30/0.57 thf(fact_4015_order_Otrans,axiom,
% 0.30/0.57 ! [A: num,B: num,C: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_num @ B @ C )
% 0.30/0.57 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order.trans
% 0.30/0.57 thf(fact_4016_order_Otrans,axiom,
% 0.30/0.57 ! [A: nat,B: nat,C: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_nat @ B @ C )
% 0.30/0.57 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order.trans
% 0.30/0.57 thf(fact_4017_order_Otrans,axiom,
% 0.30/0.57 ! [A: int,B: int,C: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.57 => ( ( ord_less_eq_int @ B @ C )
% 0.30/0.57 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order.trans
% 0.30/0.57 thf(fact_4018_order__antisym,axiom,
% 0.30/0.57 ! [X2: set_int,Y3: set_int] :
% 0.30/0.57 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.30/0.57 => ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 0.30/0.57 => ( X2 = Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_antisym
% 0.30/0.57 thf(fact_4019_order__antisym,axiom,
% 0.30/0.57 ! [X2: rat,Y3: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.30/0.57 => ( ( ord_less_eq_rat @ Y3 @ X2 )
% 0.30/0.57 => ( X2 = Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_antisym
% 0.30/0.57 thf(fact_4020_order__antisym,axiom,
% 0.30/0.57 ! [X2: num,Y3: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.30/0.57 => ( ( ord_less_eq_num @ Y3 @ X2 )
% 0.30/0.57 => ( X2 = Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_antisym
% 0.30/0.57 thf(fact_4021_order__antisym,axiom,
% 0.30/0.57 ! [X2: nat,Y3: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.30/0.57 => ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.30/0.57 => ( X2 = Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_antisym
% 0.30/0.57 thf(fact_4022_order__antisym,axiom,
% 0.30/0.57 ! [X2: int,Y3: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.30/0.57 => ( ( ord_less_eq_int @ Y3 @ X2 )
% 0.30/0.57 => ( X2 = Y3 ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_antisym
% 0.30/0.57 thf(fact_4023_ord__le__eq__trans,axiom,
% 0.30/0.57 ! [A: set_int,B: set_int,C: set_int] :
% 0.30/0.57 ( ( ord_less_eq_set_int @ A @ B )
% 0.30/0.57 => ( ( B = C )
% 0.30/0.57 => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_le_eq_trans
% 0.30/0.57 thf(fact_4024_ord__le__eq__trans,axiom,
% 0.30/0.57 ! [A: rat,B: rat,C: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.57 => ( ( B = C )
% 0.30/0.57 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_le_eq_trans
% 0.30/0.57 thf(fact_4025_ord__le__eq__trans,axiom,
% 0.30/0.57 ! [A: num,B: num,C: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.57 => ( ( B = C )
% 0.30/0.57 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_le_eq_trans
% 0.30/0.57 thf(fact_4026_ord__le__eq__trans,axiom,
% 0.30/0.57 ! [A: nat,B: nat,C: nat] :
% 0.30/0.57 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.57 => ( ( B = C )
% 0.30/0.57 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_le_eq_trans
% 0.30/0.57 thf(fact_4027_ord__le__eq__trans,axiom,
% 0.30/0.57 ! [A: int,B: int,C: int] :
% 0.30/0.57 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.57 => ( ( B = C )
% 0.30/0.57 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_le_eq_trans
% 0.30/0.57 thf(fact_4028_ord__eq__le__trans,axiom,
% 0.30/0.57 ! [A: set_int,B: set_int,C: set_int] :
% 0.30/0.57 ( ( A = B )
% 0.30/0.57 => ( ( ord_less_eq_set_int @ B @ C )
% 0.30/0.57 => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_eq_le_trans
% 0.30/0.57 thf(fact_4029_ord__eq__le__trans,axiom,
% 0.30/0.57 ! [A: rat,B: rat,C: rat] :
% 0.30/0.57 ( ( A = B )
% 0.30/0.57 => ( ( ord_less_eq_rat @ B @ C )
% 0.30/0.57 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_eq_le_trans
% 0.30/0.57 thf(fact_4030_ord__eq__le__trans,axiom,
% 0.30/0.57 ! [A: num,B: num,C: num] :
% 0.30/0.57 ( ( A = B )
% 0.30/0.57 => ( ( ord_less_eq_num @ B @ C )
% 0.30/0.57 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_eq_le_trans
% 0.30/0.57 thf(fact_4031_ord__eq__le__trans,axiom,
% 0.30/0.57 ! [A: nat,B: nat,C: nat] :
% 0.30/0.57 ( ( A = B )
% 0.30/0.57 => ( ( ord_less_eq_nat @ B @ C )
% 0.30/0.57 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_eq_le_trans
% 0.30/0.57 thf(fact_4032_ord__eq__le__trans,axiom,
% 0.30/0.57 ! [A: int,B: int,C: int] :
% 0.30/0.57 ( ( A = B )
% 0.30/0.57 => ( ( ord_less_eq_int @ B @ C )
% 0.30/0.57 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % ord_eq_le_trans
% 0.30/0.57 thf(fact_4033_order__class_Oorder__eq__iff,axiom,
% 0.30/0.57 ( ( ^ [Y6: set_int,Z5: set_int] : ( Y6 = Z5 ) )
% 0.30/0.57 = ( ^ [X: set_int,Y: set_int] :
% 0.30/0.57 ( ( ord_less_eq_set_int @ X @ Y )
% 0.30/0.57 & ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_class.order_eq_iff
% 0.30/0.57 thf(fact_4034_order__class_Oorder__eq__iff,axiom,
% 0.30/0.57 ( ( ^ [Y6: rat,Z5: rat] : ( Y6 = Z5 ) )
% 0.30/0.57 = ( ^ [X: rat,Y: rat] :
% 0.30/0.57 ( ( ord_less_eq_rat @ X @ Y )
% 0.30/0.57 & ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_class.order_eq_iff
% 0.30/0.57 thf(fact_4035_order__class_Oorder__eq__iff,axiom,
% 0.30/0.57 ( ( ^ [Y6: num,Z5: num] : ( Y6 = Z5 ) )
% 0.30/0.57 = ( ^ [X: num,Y: num] :
% 0.30/0.57 ( ( ord_less_eq_num @ X @ Y )
% 0.30/0.57 & ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 0.30/0.57
% 0.30/0.57 % order_class.order_eq_iff
% 0.30/0.57 thf(fact_4036_order__class_Oorder__eq__iff,axiom,
% 0.30/0.57 ( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 0.30/0.57 = ( ^ [X: nat,Y: nat] :
% 0.30/0.58 ( ( ord_less_eq_nat @ X @ Y )
% 0.30/0.58 & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_class.order_eq_iff
% 0.30/0.58 thf(fact_4037_order__class_Oorder__eq__iff,axiom,
% 0.30/0.58 ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 0.30/0.58 = ( ^ [X: int,Y: int] :
% 0.30/0.58 ( ( ord_less_eq_int @ X @ Y )
% 0.30/0.58 & ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_class.order_eq_iff
% 0.30/0.58 thf(fact_4038_le__cases3,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat,Z: rat] :
% 0.30/0.58 ( ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.30/0.58 => ~ ( ord_less_eq_rat @ Y3 @ Z ) )
% 0.30/0.58 => ( ( ( ord_less_eq_rat @ Y3 @ X2 )
% 0.30/0.58 => ~ ( ord_less_eq_rat @ X2 @ Z ) )
% 0.30/0.58 => ( ( ( ord_less_eq_rat @ X2 @ Z )
% 0.30/0.58 => ~ ( ord_less_eq_rat @ Z @ Y3 ) )
% 0.30/0.58 => ( ( ( ord_less_eq_rat @ Z @ Y3 )
% 0.30/0.58 => ~ ( ord_less_eq_rat @ Y3 @ X2 ) )
% 0.30/0.58 => ( ( ( ord_less_eq_rat @ Y3 @ Z )
% 0.30/0.58 => ~ ( ord_less_eq_rat @ Z @ X2 ) )
% 0.30/0.58 => ~ ( ( ord_less_eq_rat @ Z @ X2 )
% 0.30/0.58 => ~ ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % le_cases3
% 0.30/0.58 thf(fact_4039_le__cases3,axiom,
% 0.30/0.58 ! [X2: num,Y3: num,Z: num] :
% 0.30/0.58 ( ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.30/0.58 => ~ ( ord_less_eq_num @ Y3 @ Z ) )
% 0.30/0.58 => ( ( ( ord_less_eq_num @ Y3 @ X2 )
% 0.30/0.58 => ~ ( ord_less_eq_num @ X2 @ Z ) )
% 0.30/0.58 => ( ( ( ord_less_eq_num @ X2 @ Z )
% 0.30/0.58 => ~ ( ord_less_eq_num @ Z @ Y3 ) )
% 0.30/0.58 => ( ( ( ord_less_eq_num @ Z @ Y3 )
% 0.30/0.58 => ~ ( ord_less_eq_num @ Y3 @ X2 ) )
% 0.30/0.58 => ( ( ( ord_less_eq_num @ Y3 @ Z )
% 0.30/0.58 => ~ ( ord_less_eq_num @ Z @ X2 ) )
% 0.30/0.58 => ~ ( ( ord_less_eq_num @ Z @ X2 )
% 0.30/0.58 => ~ ( ord_less_eq_num @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % le_cases3
% 0.30/0.58 thf(fact_4040_le__cases3,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat,Z: nat] :
% 0.30/0.58 ( ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.30/0.58 => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
% 0.30/0.58 => ( ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.30/0.58 => ~ ( ord_less_eq_nat @ X2 @ Z ) )
% 0.30/0.58 => ( ( ( ord_less_eq_nat @ X2 @ Z )
% 0.30/0.58 => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
% 0.30/0.58 => ( ( ( ord_less_eq_nat @ Z @ Y3 )
% 0.30/0.58 => ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
% 0.30/0.58 => ( ( ( ord_less_eq_nat @ Y3 @ Z )
% 0.30/0.58 => ~ ( ord_less_eq_nat @ Z @ X2 ) )
% 0.30/0.58 => ~ ( ( ord_less_eq_nat @ Z @ X2 )
% 0.30/0.58 => ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % le_cases3
% 0.30/0.58 thf(fact_4041_le__cases3,axiom,
% 0.30/0.58 ! [X2: int,Y3: int,Z: int] :
% 0.30/0.58 ( ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.30/0.58 => ~ ( ord_less_eq_int @ Y3 @ Z ) )
% 0.30/0.58 => ( ( ( ord_less_eq_int @ Y3 @ X2 )
% 0.30/0.58 => ~ ( ord_less_eq_int @ X2 @ Z ) )
% 0.30/0.58 => ( ( ( ord_less_eq_int @ X2 @ Z )
% 0.30/0.58 => ~ ( ord_less_eq_int @ Z @ Y3 ) )
% 0.30/0.58 => ( ( ( ord_less_eq_int @ Z @ Y3 )
% 0.30/0.58 => ~ ( ord_less_eq_int @ Y3 @ X2 ) )
% 0.30/0.58 => ( ( ( ord_less_eq_int @ Y3 @ Z )
% 0.30/0.58 => ~ ( ord_less_eq_int @ Z @ X2 ) )
% 0.30/0.58 => ~ ( ( ord_less_eq_int @ Z @ X2 )
% 0.30/0.58 => ~ ( ord_less_eq_int @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % le_cases3
% 0.30/0.58 thf(fact_4042_nle__le,axiom,
% 0.30/0.58 ! [A: rat,B: rat] :
% 0.30/0.58 ( ( ~ ( ord_less_eq_rat @ A @ B ) )
% 0.30/0.58 = ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.58 & ( B != A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % nle_le
% 0.30/0.58 thf(fact_4043_nle__le,axiom,
% 0.30/0.58 ! [A: num,B: num] :
% 0.30/0.58 ( ( ~ ( ord_less_eq_num @ A @ B ) )
% 0.30/0.58 = ( ( ord_less_eq_num @ B @ A )
% 0.30/0.58 & ( B != A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % nle_le
% 0.30/0.58 thf(fact_4044_nle__le,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( ~ ( ord_less_eq_nat @ A @ B ) )
% 0.30/0.58 = ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.58 & ( B != A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % nle_le
% 0.30/0.58 thf(fact_4045_nle__le,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( ~ ( ord_less_eq_int @ A @ B ) )
% 0.30/0.58 = ( ( ord_less_eq_int @ B @ A )
% 0.30/0.58 & ( B != A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % nle_le
% 0.30/0.58 thf(fact_4046_nth__list__update,axiom,
% 0.30/0.58 ! [I2: nat,Xs2: list_VEBT_VEBT,J: nat,X2: vEBT_VEBT] :
% 0.30/0.58 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.58 => ( ( ( I2 = J )
% 0.30/0.58 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X2 ) @ J )
% 0.30/0.58 = X2 ) )
% 0.30/0.58 & ( ( I2 != J )
% 0.30/0.58 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X2 ) @ J )
% 0.30/0.58 = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % nth_list_update
% 0.30/0.58 thf(fact_4047_nth__list__update,axiom,
% 0.30/0.58 ! [I2: nat,Xs2: list_o,X2: $o,J: nat] :
% 0.30/0.58 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.58 => ( ( nth_o @ ( list_update_o @ Xs2 @ I2 @ X2 ) @ J )
% 0.30/0.58 = ( ( ( I2 = J )
% 0.30/0.58 => X2 )
% 0.30/0.58 & ( ( I2 != J )
% 0.30/0.58 => ( nth_o @ Xs2 @ J ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % nth_list_update
% 0.30/0.58 thf(fact_4048_nth__list__update,axiom,
% 0.30/0.58 ! [I2: nat,Xs2: list_nat,J: nat,X2: nat] :
% 0.30/0.58 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.58 => ( ( ( I2 = J )
% 0.30/0.58 => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I2 @ X2 ) @ J )
% 0.30/0.58 = X2 ) )
% 0.30/0.58 & ( ( I2 != J )
% 0.30/0.58 => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I2 @ X2 ) @ J )
% 0.30/0.58 = ( nth_nat @ Xs2 @ J ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % nth_list_update
% 0.30/0.58 thf(fact_4049_nth__list__update,axiom,
% 0.30/0.58 ! [I2: nat,Xs2: list_int,J: nat,X2: int] :
% 0.30/0.58 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.58 => ( ( ( I2 = J )
% 0.30/0.58 => ( ( nth_int @ ( list_update_int @ Xs2 @ I2 @ X2 ) @ J )
% 0.30/0.58 = X2 ) )
% 0.30/0.58 & ( ( I2 != J )
% 0.30/0.58 => ( ( nth_int @ ( list_update_int @ Xs2 @ I2 @ X2 ) @ J )
% 0.30/0.58 = ( nth_int @ Xs2 @ J ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % nth_list_update
% 0.30/0.58 thf(fact_4050_list__update__same__conv,axiom,
% 0.30/0.58 ! [I2: nat,Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 0.30/0.58 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.58 => ( ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X2 )
% 0.30/0.58 = Xs2 )
% 0.30/0.58 = ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
% 0.30/0.58 = X2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % list_update_same_conv
% 0.30/0.58 thf(fact_4051_list__update__same__conv,axiom,
% 0.30/0.58 ! [I2: nat,Xs2: list_o,X2: $o] :
% 0.30/0.58 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.58 => ( ( ( list_update_o @ Xs2 @ I2 @ X2 )
% 0.30/0.58 = Xs2 )
% 0.30/0.58 = ( ( nth_o @ Xs2 @ I2 )
% 0.30/0.58 = X2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % list_update_same_conv
% 0.30/0.58 thf(fact_4052_list__update__same__conv,axiom,
% 0.30/0.58 ! [I2: nat,Xs2: list_nat,X2: nat] :
% 0.30/0.58 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.58 => ( ( ( list_update_nat @ Xs2 @ I2 @ X2 )
% 0.30/0.58 = Xs2 )
% 0.30/0.58 = ( ( nth_nat @ Xs2 @ I2 )
% 0.30/0.58 = X2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % list_update_same_conv
% 0.30/0.58 thf(fact_4053_list__update__same__conv,axiom,
% 0.30/0.58 ! [I2: nat,Xs2: list_int,X2: int] :
% 0.30/0.58 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.58 => ( ( ( list_update_int @ Xs2 @ I2 @ X2 )
% 0.30/0.58 = Xs2 )
% 0.30/0.58 = ( ( nth_int @ Xs2 @ I2 )
% 0.30/0.58 = X2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % list_update_same_conv
% 0.30/0.58 thf(fact_4054_lt__ex,axiom,
% 0.30/0.58 ! [X2: real] :
% 0.30/0.58 ? [Y4: real] : ( ord_less_real @ Y4 @ X2 ) ).
% 0.30/0.58
% 0.30/0.58 % lt_ex
% 0.30/0.58 thf(fact_4055_lt__ex,axiom,
% 0.30/0.58 ! [X2: rat] :
% 0.30/0.58 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X2 ) ).
% 0.30/0.58
% 0.30/0.58 % lt_ex
% 0.30/0.58 thf(fact_4056_lt__ex,axiom,
% 0.30/0.58 ! [X2: int] :
% 0.30/0.58 ? [Y4: int] : ( ord_less_int @ Y4 @ X2 ) ).
% 0.30/0.58
% 0.30/0.58 % lt_ex
% 0.30/0.58 thf(fact_4057_gt__ex,axiom,
% 0.30/0.58 ! [X2: real] :
% 0.30/0.58 ? [X_12: real] : ( ord_less_real @ X2 @ X_12 ) ).
% 0.30/0.58
% 0.30/0.58 % gt_ex
% 0.30/0.58 thf(fact_4058_gt__ex,axiom,
% 0.30/0.58 ! [X2: rat] :
% 0.30/0.58 ? [X_12: rat] : ( ord_less_rat @ X2 @ X_12 ) ).
% 0.30/0.58
% 0.30/0.58 % gt_ex
% 0.30/0.58 thf(fact_4059_gt__ex,axiom,
% 0.30/0.58 ! [X2: nat] :
% 0.30/0.58 ? [X_12: nat] : ( ord_less_nat @ X2 @ X_12 ) ).
% 0.30/0.58
% 0.30/0.58 % gt_ex
% 0.30/0.58 thf(fact_4060_gt__ex,axiom,
% 0.30/0.58 ! [X2: int] :
% 0.30/0.58 ? [X_12: int] : ( ord_less_int @ X2 @ X_12 ) ).
% 0.30/0.58
% 0.30/0.58 % gt_ex
% 0.30/0.58 thf(fact_4061_dense,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.58 => ? [Z2: real] :
% 0.30/0.58 ( ( ord_less_real @ X2 @ Z2 )
% 0.30/0.58 & ( ord_less_real @ Z2 @ Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dense
% 0.30/0.58 thf(fact_4062_dense,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.58 => ? [Z2: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X2 @ Z2 )
% 0.30/0.58 & ( ord_less_rat @ Z2 @ Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dense
% 0.30/0.58 thf(fact_4063_less__imp__neq,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.58 => ( X2 != Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % less_imp_neq
% 0.30/0.58 thf(fact_4064_less__imp__neq,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.58 => ( X2 != Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % less_imp_neq
% 0.30/0.58 thf(fact_4065_less__imp__neq,axiom,
% 0.30/0.58 ! [X2: num,Y3: num] :
% 0.30/0.58 ( ( ord_less_num @ X2 @ Y3 )
% 0.30/0.58 => ( X2 != Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % less_imp_neq
% 0.30/0.58 thf(fact_4066_less__imp__neq,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat] :
% 0.30/0.58 ( ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.58 => ( X2 != Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % less_imp_neq
% 0.30/0.58 thf(fact_4067_less__imp__neq,axiom,
% 0.30/0.58 ! [X2: int,Y3: int] :
% 0.30/0.58 ( ( ord_less_int @ X2 @ Y3 )
% 0.30/0.58 => ( X2 != Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % less_imp_neq
% 0.30/0.58 thf(fact_4068_order_Oasym,axiom,
% 0.30/0.58 ! [A: real,B: real] :
% 0.30/0.58 ( ( ord_less_real @ A @ B )
% 0.30/0.58 => ~ ( ord_less_real @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.asym
% 0.30/0.58 thf(fact_4069_order_Oasym,axiom,
% 0.30/0.58 ! [A: rat,B: rat] :
% 0.30/0.58 ( ( ord_less_rat @ A @ B )
% 0.30/0.58 => ~ ( ord_less_rat @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.asym
% 0.30/0.58 thf(fact_4070_order_Oasym,axiom,
% 0.30/0.58 ! [A: num,B: num] :
% 0.30/0.58 ( ( ord_less_num @ A @ B )
% 0.30/0.58 => ~ ( ord_less_num @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.asym
% 0.30/0.58 thf(fact_4071_order_Oasym,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( ord_less_nat @ A @ B )
% 0.30/0.58 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.asym
% 0.30/0.58 thf(fact_4072_order_Oasym,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( ord_less_int @ A @ B )
% 0.30/0.58 => ~ ( ord_less_int @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.asym
% 0.30/0.58 thf(fact_4073_ord__eq__less__trans,axiom,
% 0.30/0.58 ! [A: real,B: real,C: real] :
% 0.30/0.58 ( ( A = B )
% 0.30/0.58 => ( ( ord_less_real @ B @ C )
% 0.30/0.58 => ( ord_less_real @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_eq_less_trans
% 0.30/0.58 thf(fact_4074_ord__eq__less__trans,axiom,
% 0.30/0.58 ! [A: rat,B: rat,C: rat] :
% 0.30/0.58 ( ( A = B )
% 0.30/0.58 => ( ( ord_less_rat @ B @ C )
% 0.30/0.58 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_eq_less_trans
% 0.30/0.58 thf(fact_4075_ord__eq__less__trans,axiom,
% 0.30/0.58 ! [A: num,B: num,C: num] :
% 0.30/0.58 ( ( A = B )
% 0.30/0.58 => ( ( ord_less_num @ B @ C )
% 0.30/0.58 => ( ord_less_num @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_eq_less_trans
% 0.30/0.58 thf(fact_4076_ord__eq__less__trans,axiom,
% 0.30/0.58 ! [A: nat,B: nat,C: nat] :
% 0.30/0.58 ( ( A = B )
% 0.30/0.58 => ( ( ord_less_nat @ B @ C )
% 0.30/0.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_eq_less_trans
% 0.30/0.58 thf(fact_4077_ord__eq__less__trans,axiom,
% 0.30/0.58 ! [A: int,B: int,C: int] :
% 0.30/0.58 ( ( A = B )
% 0.30/0.58 => ( ( ord_less_int @ B @ C )
% 0.30/0.58 => ( ord_less_int @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_eq_less_trans
% 0.30/0.58 thf(fact_4078_ord__less__eq__trans,axiom,
% 0.30/0.58 ! [A: real,B: real,C: real] :
% 0.30/0.58 ( ( ord_less_real @ A @ B )
% 0.30/0.58 => ( ( B = C )
% 0.30/0.58 => ( ord_less_real @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_less_eq_trans
% 0.30/0.58 thf(fact_4079_ord__less__eq__trans,axiom,
% 0.30/0.58 ! [A: rat,B: rat,C: rat] :
% 0.30/0.58 ( ( ord_less_rat @ A @ B )
% 0.30/0.58 => ( ( B = C )
% 0.30/0.58 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_less_eq_trans
% 0.30/0.58 thf(fact_4080_ord__less__eq__trans,axiom,
% 0.30/0.58 ! [A: num,B: num,C: num] :
% 0.30/0.58 ( ( ord_less_num @ A @ B )
% 0.30/0.58 => ( ( B = C )
% 0.30/0.58 => ( ord_less_num @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_less_eq_trans
% 0.30/0.58 thf(fact_4081_ord__less__eq__trans,axiom,
% 0.30/0.58 ! [A: nat,B: nat,C: nat] :
% 0.30/0.58 ( ( ord_less_nat @ A @ B )
% 0.30/0.58 => ( ( B = C )
% 0.30/0.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_less_eq_trans
% 0.30/0.58 thf(fact_4082_ord__less__eq__trans,axiom,
% 0.30/0.58 ! [A: int,B: int,C: int] :
% 0.30/0.58 ( ( ord_less_int @ A @ B )
% 0.30/0.58 => ( ( B = C )
% 0.30/0.58 => ( ord_less_int @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_less_eq_trans
% 0.30/0.58 thf(fact_4083_less__induct,axiom,
% 0.30/0.58 ! [P: nat > $o,A: nat] :
% 0.30/0.58 ( ! [X5: nat] :
% 0.30/0.58 ( ! [Y5: nat] :
% 0.30/0.58 ( ( ord_less_nat @ Y5 @ X5 )
% 0.30/0.58 => ( P @ Y5 ) )
% 0.30/0.58 => ( P @ X5 ) )
% 0.30/0.58 => ( P @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % less_induct
% 0.30/0.58 thf(fact_4084_antisym__conv3,axiom,
% 0.30/0.58 ! [Y3: real,X2: real] :
% 0.30/0.58 ( ~ ( ord_less_real @ Y3 @ X2 )
% 0.30/0.58 => ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
% 0.30/0.58 = ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % antisym_conv3
% 0.30/0.58 thf(fact_4085_antisym__conv3,axiom,
% 0.30/0.58 ! [Y3: rat,X2: rat] :
% 0.30/0.58 ( ~ ( ord_less_rat @ Y3 @ X2 )
% 0.30/0.58 => ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
% 0.30/0.58 = ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % antisym_conv3
% 0.30/0.58 thf(fact_4086_antisym__conv3,axiom,
% 0.30/0.58 ! [Y3: num,X2: num] :
% 0.30/0.58 ( ~ ( ord_less_num @ Y3 @ X2 )
% 0.30/0.58 => ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
% 0.30/0.58 = ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % antisym_conv3
% 0.30/0.58 thf(fact_4087_antisym__conv3,axiom,
% 0.30/0.58 ! [Y3: nat,X2: nat] :
% 0.30/0.58 ( ~ ( ord_less_nat @ Y3 @ X2 )
% 0.30/0.58 => ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
% 0.30/0.58 = ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % antisym_conv3
% 0.30/0.58 thf(fact_4088_antisym__conv3,axiom,
% 0.30/0.58 ! [Y3: int,X2: int] :
% 0.30/0.58 ( ~ ( ord_less_int @ Y3 @ X2 )
% 0.30/0.58 => ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
% 0.30/0.58 = ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % antisym_conv3
% 0.30/0.58 thf(fact_4089_linorder__cases,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ~ ( ord_less_real @ X2 @ Y3 )
% 0.30/0.58 => ( ( X2 != Y3 )
% 0.30/0.58 => ( ord_less_real @ Y3 @ X2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_cases
% 0.30/0.58 thf(fact_4090_linorder__cases,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ~ ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.58 => ( ( X2 != Y3 )
% 0.30/0.58 => ( ord_less_rat @ Y3 @ X2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_cases
% 0.30/0.58 thf(fact_4091_linorder__cases,axiom,
% 0.30/0.58 ! [X2: num,Y3: num] :
% 0.30/0.58 ( ~ ( ord_less_num @ X2 @ Y3 )
% 0.30/0.58 => ( ( X2 != Y3 )
% 0.30/0.58 => ( ord_less_num @ Y3 @ X2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_cases
% 0.30/0.58 thf(fact_4092_linorder__cases,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat] :
% 0.30/0.58 ( ~ ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.58 => ( ( X2 != Y3 )
% 0.30/0.58 => ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_cases
% 0.30/0.58 thf(fact_4093_linorder__cases,axiom,
% 0.30/0.58 ! [X2: int,Y3: int] :
% 0.30/0.58 ( ~ ( ord_less_int @ X2 @ Y3 )
% 0.30/0.58 => ( ( X2 != Y3 )
% 0.30/0.58 => ( ord_less_int @ Y3 @ X2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_cases
% 0.30/0.58 thf(fact_4094_dual__order_Oasym,axiom,
% 0.30/0.58 ! [B: real,A: real] :
% 0.30/0.58 ( ( ord_less_real @ B @ A )
% 0.30/0.58 => ~ ( ord_less_real @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.asym
% 0.30/0.58 thf(fact_4095_dual__order_Oasym,axiom,
% 0.30/0.58 ! [B: rat,A: rat] :
% 0.30/0.58 ( ( ord_less_rat @ B @ A )
% 0.30/0.58 => ~ ( ord_less_rat @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.asym
% 0.30/0.58 thf(fact_4096_dual__order_Oasym,axiom,
% 0.30/0.58 ! [B: num,A: num] :
% 0.30/0.58 ( ( ord_less_num @ B @ A )
% 0.30/0.58 => ~ ( ord_less_num @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.asym
% 0.30/0.58 thf(fact_4097_dual__order_Oasym,axiom,
% 0.30/0.58 ! [B: nat,A: nat] :
% 0.30/0.58 ( ( ord_less_nat @ B @ A )
% 0.30/0.58 => ~ ( ord_less_nat @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.asym
% 0.30/0.58 thf(fact_4098_dual__order_Oasym,axiom,
% 0.30/0.58 ! [B: int,A: int] :
% 0.30/0.58 ( ( ord_less_int @ B @ A )
% 0.30/0.58 => ~ ( ord_less_int @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.asym
% 0.30/0.58 thf(fact_4099_dual__order_Oirrefl,axiom,
% 0.30/0.58 ! [A: real] :
% 0.30/0.58 ~ ( ord_less_real @ A @ A ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.irrefl
% 0.30/0.58 thf(fact_4100_dual__order_Oirrefl,axiom,
% 0.30/0.58 ! [A: rat] :
% 0.30/0.58 ~ ( ord_less_rat @ A @ A ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.irrefl
% 0.30/0.58 thf(fact_4101_dual__order_Oirrefl,axiom,
% 0.30/0.58 ! [A: num] :
% 0.30/0.58 ~ ( ord_less_num @ A @ A ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.irrefl
% 0.30/0.58 thf(fact_4102_dual__order_Oirrefl,axiom,
% 0.30/0.58 ! [A: nat] :
% 0.30/0.58 ~ ( ord_less_nat @ A @ A ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.irrefl
% 0.30/0.58 thf(fact_4103_dual__order_Oirrefl,axiom,
% 0.30/0.58 ! [A: int] :
% 0.30/0.58 ~ ( ord_less_int @ A @ A ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.irrefl
% 0.30/0.58 thf(fact_4104_exists__least__iff,axiom,
% 0.30/0.58 ( ( ^ [P2: nat > $o] :
% 0.30/0.58 ? [X6: nat] : ( P2 @ X6 ) )
% 0.30/0.58 = ( ^ [P3: nat > $o] :
% 0.30/0.58 ? [N3: nat] :
% 0.30/0.58 ( ( P3 @ N3 )
% 0.30/0.58 & ! [M3: nat] :
% 0.30/0.58 ( ( ord_less_nat @ M3 @ N3 )
% 0.30/0.58 => ~ ( P3 @ M3 ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % exists_least_iff
% 0.30/0.58 thf(fact_4105_linorder__less__wlog,axiom,
% 0.30/0.58 ! [P: real > real > $o,A: real,B: real] :
% 0.30/0.58 ( ! [A4: real,B4: real] :
% 0.30/0.58 ( ( ord_less_real @ A4 @ B4 )
% 0.30/0.58 => ( P @ A4 @ B4 ) )
% 0.30/0.58 => ( ! [A4: real] : ( P @ A4 @ A4 )
% 0.30/0.58 => ( ! [A4: real,B4: real] :
% 0.30/0.58 ( ( P @ B4 @ A4 )
% 0.30/0.58 => ( P @ A4 @ B4 ) )
% 0.30/0.58 => ( P @ A @ B ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_less_wlog
% 0.30/0.58 thf(fact_4106_linorder__less__wlog,axiom,
% 0.30/0.58 ! [P: rat > rat > $o,A: rat,B: rat] :
% 0.30/0.58 ( ! [A4: rat,B4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ A4 @ B4 )
% 0.30/0.58 => ( P @ A4 @ B4 ) )
% 0.30/0.58 => ( ! [A4: rat] : ( P @ A4 @ A4 )
% 0.30/0.58 => ( ! [A4: rat,B4: rat] :
% 0.30/0.58 ( ( P @ B4 @ A4 )
% 0.30/0.58 => ( P @ A4 @ B4 ) )
% 0.30/0.58 => ( P @ A @ B ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_less_wlog
% 0.30/0.58 thf(fact_4107_linorder__less__wlog,axiom,
% 0.30/0.58 ! [P: num > num > $o,A: num,B: num] :
% 0.30/0.58 ( ! [A4: num,B4: num] :
% 0.30/0.58 ( ( ord_less_num @ A4 @ B4 )
% 0.30/0.58 => ( P @ A4 @ B4 ) )
% 0.30/0.58 => ( ! [A4: num] : ( P @ A4 @ A4 )
% 0.30/0.58 => ( ! [A4: num,B4: num] :
% 0.30/0.58 ( ( P @ B4 @ A4 )
% 0.30/0.58 => ( P @ A4 @ B4 ) )
% 0.30/0.58 => ( P @ A @ B ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_less_wlog
% 0.30/0.58 thf(fact_4108_linorder__less__wlog,axiom,
% 0.30/0.58 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.30/0.58 ( ! [A4: nat,B4: nat] :
% 0.30/0.58 ( ( ord_less_nat @ A4 @ B4 )
% 0.30/0.58 => ( P @ A4 @ B4 ) )
% 0.30/0.58 => ( ! [A4: nat] : ( P @ A4 @ A4 )
% 0.30/0.58 => ( ! [A4: nat,B4: nat] :
% 0.30/0.58 ( ( P @ B4 @ A4 )
% 0.30/0.58 => ( P @ A4 @ B4 ) )
% 0.30/0.58 => ( P @ A @ B ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_less_wlog
% 0.30/0.58 thf(fact_4109_linorder__less__wlog,axiom,
% 0.30/0.58 ! [P: int > int > $o,A: int,B: int] :
% 0.30/0.58 ( ! [A4: int,B4: int] :
% 0.30/0.58 ( ( ord_less_int @ A4 @ B4 )
% 0.30/0.58 => ( P @ A4 @ B4 ) )
% 0.30/0.58 => ( ! [A4: int] : ( P @ A4 @ A4 )
% 0.30/0.58 => ( ! [A4: int,B4: int] :
% 0.30/0.58 ( ( P @ B4 @ A4 )
% 0.30/0.58 => ( P @ A4 @ B4 ) )
% 0.30/0.58 => ( P @ A @ B ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_less_wlog
% 0.30/0.58 thf(fact_4110_order_Ostrict__trans,axiom,
% 0.30/0.58 ! [A: real,B: real,C: real] :
% 0.30/0.58 ( ( ord_less_real @ A @ B )
% 0.30/0.58 => ( ( ord_less_real @ B @ C )
% 0.30/0.58 => ( ord_less_real @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_trans
% 0.30/0.58 thf(fact_4111_order_Ostrict__trans,axiom,
% 0.30/0.58 ! [A: rat,B: rat,C: rat] :
% 0.30/0.58 ( ( ord_less_rat @ A @ B )
% 0.30/0.58 => ( ( ord_less_rat @ B @ C )
% 0.30/0.58 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_trans
% 0.30/0.58 thf(fact_4112_order_Ostrict__trans,axiom,
% 0.30/0.58 ! [A: num,B: num,C: num] :
% 0.30/0.58 ( ( ord_less_num @ A @ B )
% 0.30/0.58 => ( ( ord_less_num @ B @ C )
% 0.30/0.58 => ( ord_less_num @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_trans
% 0.30/0.58 thf(fact_4113_order_Ostrict__trans,axiom,
% 0.30/0.58 ! [A: nat,B: nat,C: nat] :
% 0.30/0.58 ( ( ord_less_nat @ A @ B )
% 0.30/0.58 => ( ( ord_less_nat @ B @ C )
% 0.30/0.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_trans
% 0.30/0.58 thf(fact_4114_order_Ostrict__trans,axiom,
% 0.30/0.58 ! [A: int,B: int,C: int] :
% 0.30/0.58 ( ( ord_less_int @ A @ B )
% 0.30/0.58 => ( ( ord_less_int @ B @ C )
% 0.30/0.58 => ( ord_less_int @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_trans
% 0.30/0.58 thf(fact_4115_not__less__iff__gr__or__eq,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
% 0.30/0.58 = ( ( ord_less_real @ Y3 @ X2 )
% 0.30/0.58 | ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % not_less_iff_gr_or_eq
% 0.30/0.58 thf(fact_4116_not__less__iff__gr__or__eq,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
% 0.30/0.58 = ( ( ord_less_rat @ Y3 @ X2 )
% 0.30/0.58 | ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % not_less_iff_gr_or_eq
% 0.30/0.58 thf(fact_4117_not__less__iff__gr__or__eq,axiom,
% 0.30/0.58 ! [X2: num,Y3: num] :
% 0.30/0.58 ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
% 0.30/0.58 = ( ( ord_less_num @ Y3 @ X2 )
% 0.30/0.58 | ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % not_less_iff_gr_or_eq
% 0.30/0.58 thf(fact_4118_not__less__iff__gr__or__eq,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat] :
% 0.30/0.58 ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
% 0.30/0.58 = ( ( ord_less_nat @ Y3 @ X2 )
% 0.30/0.58 | ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % not_less_iff_gr_or_eq
% 0.30/0.58 thf(fact_4119_not__less__iff__gr__or__eq,axiom,
% 0.30/0.58 ! [X2: int,Y3: int] :
% 0.30/0.58 ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
% 0.30/0.58 = ( ( ord_less_int @ Y3 @ X2 )
% 0.30/0.58 | ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % not_less_iff_gr_or_eq
% 0.30/0.58 thf(fact_4120_dual__order_Ostrict__trans,axiom,
% 0.30/0.58 ! [B: real,A: real,C: real] :
% 0.30/0.58 ( ( ord_less_real @ B @ A )
% 0.30/0.58 => ( ( ord_less_real @ C @ B )
% 0.30/0.58 => ( ord_less_real @ C @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_trans
% 0.30/0.58 thf(fact_4121_dual__order_Ostrict__trans,axiom,
% 0.30/0.58 ! [B: rat,A: rat,C: rat] :
% 0.30/0.58 ( ( ord_less_rat @ B @ A )
% 0.30/0.58 => ( ( ord_less_rat @ C @ B )
% 0.30/0.58 => ( ord_less_rat @ C @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_trans
% 0.30/0.58 thf(fact_4122_dual__order_Ostrict__trans,axiom,
% 0.30/0.58 ! [B: num,A: num,C: num] :
% 0.30/0.58 ( ( ord_less_num @ B @ A )
% 0.30/0.58 => ( ( ord_less_num @ C @ B )
% 0.30/0.58 => ( ord_less_num @ C @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_trans
% 0.30/0.58 thf(fact_4123_dual__order_Ostrict__trans,axiom,
% 0.30/0.58 ! [B: nat,A: nat,C: nat] :
% 0.30/0.58 ( ( ord_less_nat @ B @ A )
% 0.30/0.58 => ( ( ord_less_nat @ C @ B )
% 0.30/0.58 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_trans
% 0.30/0.58 thf(fact_4124_dual__order_Ostrict__trans,axiom,
% 0.30/0.58 ! [B: int,A: int,C: int] :
% 0.30/0.58 ( ( ord_less_int @ B @ A )
% 0.30/0.58 => ( ( ord_less_int @ C @ B )
% 0.30/0.58 => ( ord_less_int @ C @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_trans
% 0.30/0.58 thf(fact_4125_order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58 ! [A: real,B: real] :
% 0.30/0.58 ( ( ord_less_real @ A @ B )
% 0.30/0.58 => ( A != B ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_implies_not_eq
% 0.30/0.58 thf(fact_4126_order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58 ! [A: rat,B: rat] :
% 0.30/0.58 ( ( ord_less_rat @ A @ B )
% 0.30/0.58 => ( A != B ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_implies_not_eq
% 0.30/0.58 thf(fact_4127_order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58 ! [A: num,B: num] :
% 0.30/0.58 ( ( ord_less_num @ A @ B )
% 0.30/0.58 => ( A != B ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_implies_not_eq
% 0.30/0.58 thf(fact_4128_order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( ord_less_nat @ A @ B )
% 0.30/0.58 => ( A != B ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_implies_not_eq
% 0.30/0.58 thf(fact_4129_order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( ord_less_int @ A @ B )
% 0.30/0.58 => ( A != B ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_implies_not_eq
% 0.30/0.58 thf(fact_4130_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58 ! [B: real,A: real] :
% 0.30/0.58 ( ( ord_less_real @ B @ A )
% 0.30/0.58 => ( A != B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_implies_not_eq
% 0.30/0.58 thf(fact_4131_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58 ! [B: rat,A: rat] :
% 0.30/0.58 ( ( ord_less_rat @ B @ A )
% 0.30/0.58 => ( A != B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_implies_not_eq
% 0.30/0.58 thf(fact_4132_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58 ! [B: num,A: num] :
% 0.30/0.58 ( ( ord_less_num @ B @ A )
% 0.30/0.58 => ( A != B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_implies_not_eq
% 0.30/0.58 thf(fact_4133_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58 ! [B: nat,A: nat] :
% 0.30/0.58 ( ( ord_less_nat @ B @ A )
% 0.30/0.58 => ( A != B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_implies_not_eq
% 0.30/0.58 thf(fact_4134_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58 ! [B: int,A: int] :
% 0.30/0.58 ( ( ord_less_int @ B @ A )
% 0.30/0.58 => ( A != B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_implies_not_eq
% 0.30/0.58 thf(fact_4135_linorder__neqE,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ( X2 != Y3 )
% 0.30/0.58 => ( ~ ( ord_less_real @ X2 @ Y3 )
% 0.30/0.58 => ( ord_less_real @ Y3 @ X2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_neqE
% 0.30/0.58 thf(fact_4136_linorder__neqE,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ( X2 != Y3 )
% 0.30/0.58 => ( ~ ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.58 => ( ord_less_rat @ Y3 @ X2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_neqE
% 0.30/0.58 thf(fact_4137_linorder__neqE,axiom,
% 0.30/0.58 ! [X2: num,Y3: num] :
% 0.30/0.58 ( ( X2 != Y3 )
% 0.30/0.58 => ( ~ ( ord_less_num @ X2 @ Y3 )
% 0.30/0.58 => ( ord_less_num @ Y3 @ X2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_neqE
% 0.30/0.58 thf(fact_4138_linorder__neqE,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat] :
% 0.30/0.58 ( ( X2 != Y3 )
% 0.30/0.58 => ( ~ ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.58 => ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_neqE
% 0.30/0.58 thf(fact_4139_linorder__neqE,axiom,
% 0.30/0.58 ! [X2: int,Y3: int] :
% 0.30/0.58 ( ( X2 != Y3 )
% 0.30/0.58 => ( ~ ( ord_less_int @ X2 @ Y3 )
% 0.30/0.58 => ( ord_less_int @ Y3 @ X2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_neqE
% 0.30/0.58 thf(fact_4140_order__less__asym,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.58 => ~ ( ord_less_real @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_asym
% 0.30/0.58 thf(fact_4141_order__less__asym,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.58 => ~ ( ord_less_rat @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_asym
% 0.30/0.58 thf(fact_4142_order__less__asym,axiom,
% 0.30/0.58 ! [X2: num,Y3: num] :
% 0.30/0.58 ( ( ord_less_num @ X2 @ Y3 )
% 0.30/0.58 => ~ ( ord_less_num @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_asym
% 0.30/0.58 thf(fact_4143_order__less__asym,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat] :
% 0.30/0.58 ( ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.58 => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_asym
% 0.30/0.58 thf(fact_4144_order__less__asym,axiom,
% 0.30/0.58 ! [X2: int,Y3: int] :
% 0.30/0.58 ( ( ord_less_int @ X2 @ Y3 )
% 0.30/0.58 => ~ ( ord_less_int @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_asym
% 0.30/0.58 thf(fact_4145_linorder__neq__iff,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ( X2 != Y3 )
% 0.30/0.58 = ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.58 | ( ord_less_real @ Y3 @ X2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_neq_iff
% 0.30/0.58 thf(fact_4146_linorder__neq__iff,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ( X2 != Y3 )
% 0.30/0.58 = ( ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.58 | ( ord_less_rat @ Y3 @ X2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_neq_iff
% 0.30/0.58 thf(fact_4147_linorder__neq__iff,axiom,
% 0.30/0.58 ! [X2: num,Y3: num] :
% 0.30/0.58 ( ( X2 != Y3 )
% 0.30/0.58 = ( ( ord_less_num @ X2 @ Y3 )
% 0.30/0.58 | ( ord_less_num @ Y3 @ X2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_neq_iff
% 0.30/0.58 thf(fact_4148_linorder__neq__iff,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat] :
% 0.30/0.58 ( ( X2 != Y3 )
% 0.30/0.58 = ( ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.58 | ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_neq_iff
% 0.30/0.58 thf(fact_4149_linorder__neq__iff,axiom,
% 0.30/0.58 ! [X2: int,Y3: int] :
% 0.30/0.58 ( ( X2 != Y3 )
% 0.30/0.58 = ( ( ord_less_int @ X2 @ Y3 )
% 0.30/0.58 | ( ord_less_int @ Y3 @ X2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_neq_iff
% 0.30/0.58 thf(fact_4150_order__less__asym_H,axiom,
% 0.30/0.58 ! [A: real,B: real] :
% 0.30/0.58 ( ( ord_less_real @ A @ B )
% 0.30/0.58 => ~ ( ord_less_real @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_asym'
% 0.30/0.58 thf(fact_4151_order__less__asym_H,axiom,
% 0.30/0.58 ! [A: rat,B: rat] :
% 0.30/0.58 ( ( ord_less_rat @ A @ B )
% 0.30/0.58 => ~ ( ord_less_rat @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_asym'
% 0.30/0.58 thf(fact_4152_order__less__asym_H,axiom,
% 0.30/0.58 ! [A: num,B: num] :
% 0.30/0.58 ( ( ord_less_num @ A @ B )
% 0.30/0.58 => ~ ( ord_less_num @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_asym'
% 0.30/0.58 thf(fact_4153_order__less__asym_H,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( ord_less_nat @ A @ B )
% 0.30/0.58 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_asym'
% 0.30/0.58 thf(fact_4154_order__less__asym_H,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( ord_less_int @ A @ B )
% 0.30/0.58 => ~ ( ord_less_int @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_asym'
% 0.30/0.58 thf(fact_4155_order__less__trans,axiom,
% 0.30/0.58 ! [X2: real,Y3: real,Z: real] :
% 0.30/0.58 ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_real @ Y3 @ Z )
% 0.30/0.58 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_trans
% 0.30/0.58 thf(fact_4156_order__less__trans,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat,Z: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_rat @ Y3 @ Z )
% 0.30/0.58 => ( ord_less_rat @ X2 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_trans
% 0.30/0.58 thf(fact_4157_order__less__trans,axiom,
% 0.30/0.58 ! [X2: num,Y3: num,Z: num] :
% 0.30/0.58 ( ( ord_less_num @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_num @ Y3 @ Z )
% 0.30/0.58 => ( ord_less_num @ X2 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_trans
% 0.30/0.58 thf(fact_4158_order__less__trans,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat,Z: nat] :
% 0.30/0.58 ( ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_nat @ Y3 @ Z )
% 0.30/0.58 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_trans
% 0.30/0.58 thf(fact_4159_order__less__trans,axiom,
% 0.30/0.58 ! [X2: int,Y3: int,Z: int] :
% 0.30/0.58 ( ( ord_less_int @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_int @ Y3 @ Z )
% 0.30/0.58 => ( ord_less_int @ X2 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_trans
% 0.30/0.58 thf(fact_4160_ord__eq__less__subst,axiom,
% 0.30/0.58 ! [A: real,F: real > real,B: real,C: real] :
% 0.30/0.58 ( ( A
% 0.30/0.58 = ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_real @ B @ C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_eq_less_subst
% 0.30/0.58 thf(fact_4161_ord__eq__less__subst,axiom,
% 0.30/0.58 ! [A: rat,F: real > rat,B: real,C: real] :
% 0.30/0.58 ( ( A
% 0.30/0.58 = ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_real @ B @ C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_eq_less_subst
% 0.30/0.58 thf(fact_4162_ord__eq__less__subst,axiom,
% 0.30/0.58 ! [A: num,F: real > num,B: real,C: real] :
% 0.30/0.58 ( ( A
% 0.30/0.58 = ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_real @ B @ C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_eq_less_subst
% 0.30/0.58 thf(fact_4163_ord__eq__less__subst,axiom,
% 0.30/0.58 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.30/0.58 ( ( A
% 0.30/0.58 = ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_real @ B @ C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_eq_less_subst
% 0.30/0.58 thf(fact_4164_ord__eq__less__subst,axiom,
% 0.30/0.58 ! [A: int,F: real > int,B: real,C: real] :
% 0.30/0.58 ( ( A
% 0.30/0.58 = ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_real @ B @ C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_eq_less_subst
% 0.30/0.58 thf(fact_4165_ord__eq__less__subst,axiom,
% 0.30/0.58 ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.30/0.58 ( ( A
% 0.30/0.58 = ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_rat @ B @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_eq_less_subst
% 0.30/0.58 thf(fact_4166_ord__eq__less__subst,axiom,
% 0.30/0.58 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.30/0.58 ( ( A
% 0.30/0.58 = ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_rat @ B @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_eq_less_subst
% 0.30/0.58 thf(fact_4167_ord__eq__less__subst,axiom,
% 0.30/0.58 ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.30/0.58 ( ( A
% 0.30/0.58 = ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_rat @ B @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_eq_less_subst
% 0.30/0.58 thf(fact_4168_ord__eq__less__subst,axiom,
% 0.30/0.58 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.30/0.58 ( ( A
% 0.30/0.58 = ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_rat @ B @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_eq_less_subst
% 0.30/0.58 thf(fact_4169_ord__eq__less__subst,axiom,
% 0.30/0.58 ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.30/0.58 ( ( A
% 0.30/0.58 = ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_rat @ B @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_eq_less_subst
% 0.30/0.58 thf(fact_4170_ord__less__eq__subst,axiom,
% 0.30/0.58 ! [A: real,B: real,F: real > real,C: real] :
% 0.30/0.58 ( ( ord_less_real @ A @ B )
% 0.30/0.58 => ( ( ( F @ B )
% 0.30/0.58 = C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_less_eq_subst
% 0.30/0.58 thf(fact_4171_ord__less__eq__subst,axiom,
% 0.30/0.58 ! [A: real,B: real,F: real > rat,C: rat] :
% 0.30/0.58 ( ( ord_less_real @ A @ B )
% 0.30/0.58 => ( ( ( F @ B )
% 0.30/0.58 = C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_less_eq_subst
% 0.30/0.58 thf(fact_4172_ord__less__eq__subst,axiom,
% 0.30/0.58 ! [A: real,B: real,F: real > num,C: num] :
% 0.30/0.58 ( ( ord_less_real @ A @ B )
% 0.30/0.58 => ( ( ( F @ B )
% 0.30/0.58 = C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_less_eq_subst
% 0.30/0.58 thf(fact_4173_ord__less__eq__subst,axiom,
% 0.30/0.58 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.30/0.58 ( ( ord_less_real @ A @ B )
% 0.30/0.58 => ( ( ( F @ B )
% 0.30/0.58 = C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_less_eq_subst
% 0.30/0.58 thf(fact_4174_ord__less__eq__subst,axiom,
% 0.30/0.58 ! [A: real,B: real,F: real > int,C: int] :
% 0.30/0.58 ( ( ord_less_real @ A @ B )
% 0.30/0.58 => ( ( ( F @ B )
% 0.30/0.58 = C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_less_eq_subst
% 0.30/0.58 thf(fact_4175_ord__less__eq__subst,axiom,
% 0.30/0.58 ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.30/0.58 ( ( ord_less_rat @ A @ B )
% 0.30/0.58 => ( ( ( F @ B )
% 0.30/0.58 = C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_less_eq_subst
% 0.30/0.58 thf(fact_4176_ord__less__eq__subst,axiom,
% 0.30/0.58 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.30/0.58 ( ( ord_less_rat @ A @ B )
% 0.30/0.58 => ( ( ( F @ B )
% 0.30/0.58 = C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_less_eq_subst
% 0.30/0.58 thf(fact_4177_ord__less__eq__subst,axiom,
% 0.30/0.58 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.30/0.58 ( ( ord_less_rat @ A @ B )
% 0.30/0.58 => ( ( ( F @ B )
% 0.30/0.58 = C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_less_eq_subst
% 0.30/0.58 thf(fact_4178_ord__less__eq__subst,axiom,
% 0.30/0.58 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.30/0.58 ( ( ord_less_rat @ A @ B )
% 0.30/0.58 => ( ( ( F @ B )
% 0.30/0.58 = C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_less_eq_subst
% 0.30/0.58 thf(fact_4179_ord__less__eq__subst,axiom,
% 0.30/0.58 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.30/0.58 ( ( ord_less_rat @ A @ B )
% 0.30/0.58 => ( ( ( F @ B )
% 0.30/0.58 = C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % ord_less_eq_subst
% 0.30/0.58 thf(fact_4180_order__less__irrefl,axiom,
% 0.30/0.58 ! [X2: real] :
% 0.30/0.58 ~ ( ord_less_real @ X2 @ X2 ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_irrefl
% 0.30/0.58 thf(fact_4181_order__less__irrefl,axiom,
% 0.30/0.58 ! [X2: rat] :
% 0.30/0.58 ~ ( ord_less_rat @ X2 @ X2 ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_irrefl
% 0.30/0.58 thf(fact_4182_order__less__irrefl,axiom,
% 0.30/0.58 ! [X2: num] :
% 0.30/0.58 ~ ( ord_less_num @ X2 @ X2 ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_irrefl
% 0.30/0.58 thf(fact_4183_order__less__irrefl,axiom,
% 0.30/0.58 ! [X2: nat] :
% 0.30/0.58 ~ ( ord_less_nat @ X2 @ X2 ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_irrefl
% 0.30/0.58 thf(fact_4184_order__less__irrefl,axiom,
% 0.30/0.58 ! [X2: int] :
% 0.30/0.58 ~ ( ord_less_int @ X2 @ X2 ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_irrefl
% 0.30/0.58 thf(fact_4185_order__less__subst1,axiom,
% 0.30/0.58 ! [A: real,F: real > real,B: real,C: real] :
% 0.30/0.58 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_real @ B @ C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst1
% 0.30/0.58 thf(fact_4186_order__less__subst1,axiom,
% 0.30/0.58 ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.30/0.58 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_rat @ B @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst1
% 0.30/0.58 thf(fact_4187_order__less__subst1,axiom,
% 0.30/0.58 ! [A: real,F: num > real,B: num,C: num] :
% 0.30/0.58 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_num @ B @ C )
% 0.30/0.58 => ( ! [X5: num,Y4: num] :
% 0.30/0.58 ( ( ord_less_num @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst1
% 0.30/0.58 thf(fact_4188_order__less__subst1,axiom,
% 0.30/0.58 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.30/0.58 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_nat @ B @ C )
% 0.30/0.58 => ( ! [X5: nat,Y4: nat] :
% 0.30/0.58 ( ( ord_less_nat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst1
% 0.30/0.58 thf(fact_4189_order__less__subst1,axiom,
% 0.30/0.58 ! [A: real,F: int > real,B: int,C: int] :
% 0.30/0.58 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_int @ B @ C )
% 0.30/0.58 => ( ! [X5: int,Y4: int] :
% 0.30/0.58 ( ( ord_less_int @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst1
% 0.30/0.58 thf(fact_4190_order__less__subst1,axiom,
% 0.30/0.58 ! [A: rat,F: real > rat,B: real,C: real] :
% 0.30/0.58 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_real @ B @ C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst1
% 0.30/0.58 thf(fact_4191_order__less__subst1,axiom,
% 0.30/0.58 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.30/0.58 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_rat @ B @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst1
% 0.30/0.58 thf(fact_4192_order__less__subst1,axiom,
% 0.30/0.58 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.30/0.58 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_num @ B @ C )
% 0.30/0.58 => ( ! [X5: num,Y4: num] :
% 0.30/0.58 ( ( ord_less_num @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst1
% 0.30/0.58 thf(fact_4193_order__less__subst1,axiom,
% 0.30/0.58 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.30/0.58 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_nat @ B @ C )
% 0.30/0.58 => ( ! [X5: nat,Y4: nat] :
% 0.30/0.58 ( ( ord_less_nat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst1
% 0.30/0.58 thf(fact_4194_order__less__subst1,axiom,
% 0.30/0.58 ! [A: rat,F: int > rat,B: int,C: int] :
% 0.30/0.58 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_int @ B @ C )
% 0.30/0.58 => ( ! [X5: int,Y4: int] :
% 0.30/0.58 ( ( ord_less_int @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst1
% 0.30/0.58 thf(fact_4195_order__less__subst2,axiom,
% 0.30/0.58 ! [A: real,B: real,F: real > real,C: real] :
% 0.30/0.58 ( ( ord_less_real @ A @ B )
% 0.30/0.58 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst2
% 0.30/0.58 thf(fact_4196_order__less__subst2,axiom,
% 0.30/0.58 ! [A: real,B: real,F: real > rat,C: rat] :
% 0.30/0.58 ( ( ord_less_real @ A @ B )
% 0.30/0.58 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst2
% 0.30/0.58 thf(fact_4197_order__less__subst2,axiom,
% 0.30/0.58 ! [A: real,B: real,F: real > num,C: num] :
% 0.30/0.58 ( ( ord_less_real @ A @ B )
% 0.30/0.58 => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst2
% 0.30/0.58 thf(fact_4198_order__less__subst2,axiom,
% 0.30/0.58 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.30/0.58 ( ( ord_less_real @ A @ B )
% 0.30/0.58 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst2
% 0.30/0.58 thf(fact_4199_order__less__subst2,axiom,
% 0.30/0.58 ! [A: real,B: real,F: real > int,C: int] :
% 0.30/0.58 ( ( ord_less_real @ A @ B )
% 0.30/0.58 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst2
% 0.30/0.58 thf(fact_4200_order__less__subst2,axiom,
% 0.30/0.58 ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.30/0.58 ( ( ord_less_rat @ A @ B )
% 0.30/0.58 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst2
% 0.30/0.58 thf(fact_4201_order__less__subst2,axiom,
% 0.30/0.58 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.30/0.58 ( ( ord_less_rat @ A @ B )
% 0.30/0.58 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst2
% 0.30/0.58 thf(fact_4202_order__less__subst2,axiom,
% 0.30/0.58 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.30/0.58 ( ( ord_less_rat @ A @ B )
% 0.30/0.58 => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst2
% 0.30/0.58 thf(fact_4203_order__less__subst2,axiom,
% 0.30/0.58 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.30/0.58 ( ( ord_less_rat @ A @ B )
% 0.30/0.58 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst2
% 0.30/0.58 thf(fact_4204_order__less__subst2,axiom,
% 0.30/0.58 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.30/0.58 ( ( ord_less_rat @ A @ B )
% 0.30/0.58 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_subst2
% 0.30/0.58 thf(fact_4205_order__less__not__sym,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.58 => ~ ( ord_less_real @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_not_sym
% 0.30/0.58 thf(fact_4206_order__less__not__sym,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.58 => ~ ( ord_less_rat @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_not_sym
% 0.30/0.58 thf(fact_4207_order__less__not__sym,axiom,
% 0.30/0.58 ! [X2: num,Y3: num] :
% 0.30/0.58 ( ( ord_less_num @ X2 @ Y3 )
% 0.30/0.58 => ~ ( ord_less_num @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_not_sym
% 0.30/0.58 thf(fact_4208_order__less__not__sym,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat] :
% 0.30/0.58 ( ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.58 => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_not_sym
% 0.30/0.58 thf(fact_4209_order__less__not__sym,axiom,
% 0.30/0.58 ! [X2: int,Y3: int] :
% 0.30/0.58 ( ( ord_less_int @ X2 @ Y3 )
% 0.30/0.58 => ~ ( ord_less_int @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_not_sym
% 0.30/0.58 thf(fact_4210_order__less__imp__triv,axiom,
% 0.30/0.58 ! [X2: real,Y3: real,P: $o] :
% 0.30/0.58 ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_real @ Y3 @ X2 )
% 0.30/0.58 => P ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_triv
% 0.30/0.58 thf(fact_4211_order__less__imp__triv,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat,P: $o] :
% 0.30/0.58 ( ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_rat @ Y3 @ X2 )
% 0.30/0.58 => P ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_triv
% 0.30/0.58 thf(fact_4212_order__less__imp__triv,axiom,
% 0.30/0.58 ! [X2: num,Y3: num,P: $o] :
% 0.30/0.58 ( ( ord_less_num @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_num @ Y3 @ X2 )
% 0.30/0.58 => P ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_triv
% 0.30/0.58 thf(fact_4213_order__less__imp__triv,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat,P: $o] :
% 0.30/0.58 ( ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_nat @ Y3 @ X2 )
% 0.30/0.58 => P ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_triv
% 0.30/0.58 thf(fact_4214_order__less__imp__triv,axiom,
% 0.30/0.58 ! [X2: int,Y3: int,P: $o] :
% 0.30/0.58 ( ( ord_less_int @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_int @ Y3 @ X2 )
% 0.30/0.58 => P ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_triv
% 0.30/0.58 thf(fact_4215_linorder__less__linear,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.58 | ( X2 = Y3 )
% 0.30/0.58 | ( ord_less_real @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_less_linear
% 0.30/0.58 thf(fact_4216_linorder__less__linear,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.58 | ( X2 = Y3 )
% 0.30/0.58 | ( ord_less_rat @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_less_linear
% 0.30/0.58 thf(fact_4217_linorder__less__linear,axiom,
% 0.30/0.58 ! [X2: num,Y3: num] :
% 0.30/0.58 ( ( ord_less_num @ X2 @ Y3 )
% 0.30/0.58 | ( X2 = Y3 )
% 0.30/0.58 | ( ord_less_num @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_less_linear
% 0.30/0.58 thf(fact_4218_linorder__less__linear,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat] :
% 0.30/0.58 ( ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.58 | ( X2 = Y3 )
% 0.30/0.58 | ( ord_less_nat @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_less_linear
% 0.30/0.58 thf(fact_4219_linorder__less__linear,axiom,
% 0.30/0.58 ! [X2: int,Y3: int] :
% 0.30/0.58 ( ( ord_less_int @ X2 @ Y3 )
% 0.30/0.58 | ( X2 = Y3 )
% 0.30/0.58 | ( ord_less_int @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_less_linear
% 0.30/0.58 thf(fact_4220_order__less__imp__not__eq,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.58 => ( X2 != Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_not_eq
% 0.30/0.58 thf(fact_4221_order__less__imp__not__eq,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.58 => ( X2 != Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_not_eq
% 0.30/0.58 thf(fact_4222_order__less__imp__not__eq,axiom,
% 0.30/0.58 ! [X2: num,Y3: num] :
% 0.30/0.58 ( ( ord_less_num @ X2 @ Y3 )
% 0.30/0.58 => ( X2 != Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_not_eq
% 0.30/0.58 thf(fact_4223_order__less__imp__not__eq,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat] :
% 0.30/0.58 ( ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.58 => ( X2 != Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_not_eq
% 0.30/0.58 thf(fact_4224_order__less__imp__not__eq,axiom,
% 0.30/0.58 ! [X2: int,Y3: int] :
% 0.30/0.58 ( ( ord_less_int @ X2 @ Y3 )
% 0.30/0.58 => ( X2 != Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_not_eq
% 0.30/0.58 thf(fact_4225_order__less__imp__not__eq2,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.58 => ( Y3 != X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_not_eq2
% 0.30/0.58 thf(fact_4226_order__less__imp__not__eq2,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.58 => ( Y3 != X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_not_eq2
% 0.30/0.58 thf(fact_4227_order__less__imp__not__eq2,axiom,
% 0.30/0.58 ! [X2: num,Y3: num] :
% 0.30/0.58 ( ( ord_less_num @ X2 @ Y3 )
% 0.30/0.58 => ( Y3 != X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_not_eq2
% 0.30/0.58 thf(fact_4228_order__less__imp__not__eq2,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat] :
% 0.30/0.58 ( ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.58 => ( Y3 != X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_not_eq2
% 0.30/0.58 thf(fact_4229_order__less__imp__not__eq2,axiom,
% 0.30/0.58 ! [X2: int,Y3: int] :
% 0.30/0.58 ( ( ord_less_int @ X2 @ Y3 )
% 0.30/0.58 => ( Y3 != X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_not_eq2
% 0.30/0.58 thf(fact_4230_order__less__imp__not__less,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.58 => ~ ( ord_less_real @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_not_less
% 0.30/0.58 thf(fact_4231_order__less__imp__not__less,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.58 => ~ ( ord_less_rat @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_not_less
% 0.30/0.58 thf(fact_4232_order__less__imp__not__less,axiom,
% 0.30/0.58 ! [X2: num,Y3: num] :
% 0.30/0.58 ( ( ord_less_num @ X2 @ Y3 )
% 0.30/0.58 => ~ ( ord_less_num @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_not_less
% 0.30/0.58 thf(fact_4233_order__less__imp__not__less,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat] :
% 0.30/0.58 ( ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.58 => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_not_less
% 0.30/0.58 thf(fact_4234_order__less__imp__not__less,axiom,
% 0.30/0.58 ! [X2: int,Y3: int] :
% 0.30/0.58 ( ( ord_less_int @ X2 @ Y3 )
% 0.30/0.58 => ~ ( ord_less_int @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_not_less
% 0.30/0.58 thf(fact_4235_eval__nat__numeral_I2_J,axiom,
% 0.30/0.58 ! [N: num] :
% 0.30/0.58 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 0.30/0.58 = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % eval_nat_numeral(2)
% 0.30/0.58 thf(fact_4236_BitM__plus__one,axiom,
% 0.30/0.58 ! [N: num] :
% 0.30/0.58 ( ( plus_plus_num @ ( bitM @ N ) @ one )
% 0.30/0.58 = ( bit0 @ N ) ) ).
% 0.30/0.58
% 0.30/0.58 % BitM_plus_one
% 0.30/0.58 thf(fact_4237_one__plus__BitM,axiom,
% 0.30/0.58 ! [N: num] :
% 0.30/0.58 ( ( plus_plus_num @ one @ ( bitM @ N ) )
% 0.30/0.58 = ( bit0 @ N ) ) ).
% 0.30/0.58
% 0.30/0.58 % one_plus_BitM
% 0.30/0.58 thf(fact_4238_numeral__BitM,axiom,
% 0.30/0.58 ! [N: num] :
% 0.30/0.58 ( ( numera6690914467698888265omplex @ ( bitM @ N ) )
% 0.30/0.58 = ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N ) ) @ one_one_complex ) ) ).
% 0.30/0.58
% 0.30/0.58 % numeral_BitM
% 0.30/0.58 thf(fact_4239_numeral__BitM,axiom,
% 0.30/0.58 ! [N: num] :
% 0.30/0.58 ( ( numeral_numeral_real @ ( bitM @ N ) )
% 0.30/0.58 = ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N ) ) @ one_one_real ) ) ).
% 0.30/0.58
% 0.30/0.58 % numeral_BitM
% 0.30/0.58 thf(fact_4240_numeral__BitM,axiom,
% 0.30/0.58 ! [N: num] :
% 0.30/0.58 ( ( numeral_numeral_rat @ ( bitM @ N ) )
% 0.30/0.58 = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N ) ) @ one_one_rat ) ) ).
% 0.30/0.58
% 0.30/0.58 % numeral_BitM
% 0.30/0.58 thf(fact_4241_numeral__BitM,axiom,
% 0.30/0.58 ! [N: num] :
% 0.30/0.58 ( ( numeral_numeral_int @ ( bitM @ N ) )
% 0.30/0.58 = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).
% 0.30/0.58
% 0.30/0.58 % numeral_BitM
% 0.30/0.58 thf(fact_4242_order__le__imp__less__or__eq,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.58 | ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_imp_less_or_eq
% 0.30/0.58 thf(fact_4243_order__le__imp__less__or__eq,axiom,
% 0.30/0.58 ! [X2: set_int,Y3: set_int] :
% 0.30/0.58 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_set_int @ X2 @ Y3 )
% 0.30/0.58 | ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_imp_less_or_eq
% 0.30/0.58 thf(fact_4244_order__le__imp__less__or__eq,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.58 | ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_imp_less_or_eq
% 0.30/0.58 thf(fact_4245_order__le__imp__less__or__eq,axiom,
% 0.30/0.58 ! [X2: num,Y3: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_num @ X2 @ Y3 )
% 0.30/0.58 | ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_imp_less_or_eq
% 0.30/0.58 thf(fact_4246_order__le__imp__less__or__eq,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat] :
% 0.30/0.58 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.58 | ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_imp_less_or_eq
% 0.30/0.58 thf(fact_4247_order__le__imp__less__or__eq,axiom,
% 0.30/0.58 ! [X2: int,Y3: int] :
% 0.30/0.58 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_int @ X2 @ Y3 )
% 0.30/0.58 | ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_imp_less_or_eq
% 0.30/0.58 thf(fact_4248_linorder__le__less__linear,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.30/0.58 | ( ord_less_real @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_le_less_linear
% 0.30/0.58 thf(fact_4249_linorder__le__less__linear,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.30/0.58 | ( ord_less_rat @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_le_less_linear
% 0.30/0.58 thf(fact_4250_linorder__le__less__linear,axiom,
% 0.30/0.58 ! [X2: num,Y3: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.30/0.58 | ( ord_less_num @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_le_less_linear
% 0.30/0.58 thf(fact_4251_linorder__le__less__linear,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat] :
% 0.30/0.58 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.30/0.58 | ( ord_less_nat @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_le_less_linear
% 0.30/0.58 thf(fact_4252_linorder__le__less__linear,axiom,
% 0.30/0.58 ! [X2: int,Y3: int] :
% 0.30/0.58 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.30/0.58 | ( ord_less_int @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_le_less_linear
% 0.30/0.58 thf(fact_4253_order__less__le__subst2,axiom,
% 0.30/0.58 ! [A: real,B: real,F: real > real,C: real] :
% 0.30/0.58 ( ( ord_less_real @ A @ B )
% 0.30/0.58 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst2
% 0.30/0.58 thf(fact_4254_order__less__le__subst2,axiom,
% 0.30/0.58 ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.30/0.58 ( ( ord_less_rat @ A @ B )
% 0.30/0.58 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst2
% 0.30/0.58 thf(fact_4255_order__less__le__subst2,axiom,
% 0.30/0.58 ! [A: num,B: num,F: num > real,C: real] :
% 0.30/0.58 ( ( ord_less_num @ A @ B )
% 0.30/0.58 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: num,Y4: num] :
% 0.30/0.58 ( ( ord_less_num @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst2
% 0.30/0.58 thf(fact_4256_order__less__le__subst2,axiom,
% 0.30/0.58 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.30/0.58 ( ( ord_less_nat @ A @ B )
% 0.30/0.58 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: nat,Y4: nat] :
% 0.30/0.58 ( ( ord_less_nat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst2
% 0.30/0.58 thf(fact_4257_order__less__le__subst2,axiom,
% 0.30/0.58 ! [A: int,B: int,F: int > real,C: real] :
% 0.30/0.58 ( ( ord_less_int @ A @ B )
% 0.30/0.58 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: int,Y4: int] :
% 0.30/0.58 ( ( ord_less_int @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst2
% 0.30/0.58 thf(fact_4258_order__less__le__subst2,axiom,
% 0.30/0.58 ! [A: real,B: real,F: real > rat,C: rat] :
% 0.30/0.58 ( ( ord_less_real @ A @ B )
% 0.30/0.58 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst2
% 0.30/0.58 thf(fact_4259_order__less__le__subst2,axiom,
% 0.30/0.58 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.30/0.58 ( ( ord_less_rat @ A @ B )
% 0.30/0.58 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst2
% 0.30/0.58 thf(fact_4260_order__less__le__subst2,axiom,
% 0.30/0.58 ! [A: num,B: num,F: num > rat,C: rat] :
% 0.30/0.58 ( ( ord_less_num @ A @ B )
% 0.30/0.58 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: num,Y4: num] :
% 0.30/0.58 ( ( ord_less_num @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst2
% 0.30/0.58 thf(fact_4261_order__less__le__subst2,axiom,
% 0.30/0.58 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 0.30/0.58 ( ( ord_less_nat @ A @ B )
% 0.30/0.58 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: nat,Y4: nat] :
% 0.30/0.58 ( ( ord_less_nat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst2
% 0.30/0.58 thf(fact_4262_order__less__le__subst2,axiom,
% 0.30/0.58 ! [A: int,B: int,F: int > rat,C: rat] :
% 0.30/0.58 ( ( ord_less_int @ A @ B )
% 0.30/0.58 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: int,Y4: int] :
% 0.30/0.58 ( ( ord_less_int @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst2
% 0.30/0.58 thf(fact_4263_order__less__le__subst1,axiom,
% 0.30/0.58 ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.30/0.58 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_eq_rat @ B @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst1
% 0.30/0.58 thf(fact_4264_order__less__le__subst1,axiom,
% 0.30/0.58 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.30/0.58 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_eq_rat @ B @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst1
% 0.30/0.58 thf(fact_4265_order__less__le__subst1,axiom,
% 0.30/0.58 ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.30/0.58 ( ( ord_less_num @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_eq_rat @ B @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst1
% 0.30/0.58 thf(fact_4266_order__less__le__subst1,axiom,
% 0.30/0.58 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.30/0.58 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_eq_rat @ B @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst1
% 0.30/0.58 thf(fact_4267_order__less__le__subst1,axiom,
% 0.30/0.58 ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.30/0.58 ( ( ord_less_int @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_eq_rat @ B @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst1
% 0.30/0.58 thf(fact_4268_order__less__le__subst1,axiom,
% 0.30/0.58 ! [A: real,F: num > real,B: num,C: num] :
% 0.30/0.58 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_eq_num @ B @ C )
% 0.30/0.58 => ( ! [X5: num,Y4: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst1
% 0.30/0.58 thf(fact_4269_order__less__le__subst1,axiom,
% 0.30/0.58 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.30/0.58 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_eq_num @ B @ C )
% 0.30/0.58 => ( ! [X5: num,Y4: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst1
% 0.30/0.58 thf(fact_4270_order__less__le__subst1,axiom,
% 0.30/0.58 ! [A: num,F: num > num,B: num,C: num] :
% 0.30/0.58 ( ( ord_less_num @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_eq_num @ B @ C )
% 0.30/0.58 => ( ! [X5: num,Y4: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst1
% 0.30/0.58 thf(fact_4271_order__less__le__subst1,axiom,
% 0.30/0.58 ! [A: nat,F: num > nat,B: num,C: num] :
% 0.30/0.58 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_eq_num @ B @ C )
% 0.30/0.58 => ( ! [X5: num,Y4: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst1
% 0.30/0.58 thf(fact_4272_order__less__le__subst1,axiom,
% 0.30/0.58 ! [A: int,F: num > int,B: num,C: num] :
% 0.30/0.58 ( ( ord_less_int @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_eq_num @ B @ C )
% 0.30/0.58 => ( ! [X5: num,Y4: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_subst1
% 0.30/0.58 thf(fact_4273_order__le__less__subst2,axiom,
% 0.30/0.58 ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.30/0.58 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.58 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst2
% 0.30/0.58 thf(fact_4274_order__le__less__subst2,axiom,
% 0.30/0.58 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.58 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst2
% 0.30/0.58 thf(fact_4275_order__le__less__subst2,axiom,
% 0.30/0.58 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.30/0.58 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.58 => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst2
% 0.30/0.58 thf(fact_4276_order__le__less__subst2,axiom,
% 0.30/0.58 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.58 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst2
% 0.30/0.58 thf(fact_4277_order__le__less__subst2,axiom,
% 0.30/0.58 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.30/0.58 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.58 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst2
% 0.30/0.58 thf(fact_4278_order__le__less__subst2,axiom,
% 0.30/0.58 ! [A: num,B: num,F: num > real,C: real] :
% 0.30/0.58 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.58 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: num,Y4: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst2
% 0.30/0.58 thf(fact_4279_order__le__less__subst2,axiom,
% 0.30/0.58 ! [A: num,B: num,F: num > rat,C: rat] :
% 0.30/0.58 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.58 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: num,Y4: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst2
% 0.30/0.58 thf(fact_4280_order__le__less__subst2,axiom,
% 0.30/0.58 ! [A: num,B: num,F: num > num,C: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.58 => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: num,Y4: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst2
% 0.30/0.58 thf(fact_4281_order__le__less__subst2,axiom,
% 0.30/0.58 ! [A: num,B: num,F: num > nat,C: nat] :
% 0.30/0.58 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.58 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: num,Y4: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst2
% 0.30/0.58 thf(fact_4282_order__le__less__subst2,axiom,
% 0.30/0.58 ! [A: num,B: num,F: num > int,C: int] :
% 0.30/0.58 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.58 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.30/0.58 => ( ! [X5: num,Y4: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst2
% 0.30/0.58 thf(fact_4283_order__le__less__subst1,axiom,
% 0.30/0.58 ! [A: real,F: real > real,B: real,C: real] :
% 0.30/0.58 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_real @ B @ C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst1
% 0.30/0.58 thf(fact_4284_order__le__less__subst1,axiom,
% 0.30/0.58 ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.30/0.58 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_rat @ B @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst1
% 0.30/0.58 thf(fact_4285_order__le__less__subst1,axiom,
% 0.30/0.58 ! [A: real,F: num > real,B: num,C: num] :
% 0.30/0.58 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_num @ B @ C )
% 0.30/0.58 => ( ! [X5: num,Y4: num] :
% 0.30/0.58 ( ( ord_less_num @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst1
% 0.30/0.58 thf(fact_4286_order__le__less__subst1,axiom,
% 0.30/0.58 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.30/0.58 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_nat @ B @ C )
% 0.30/0.58 => ( ! [X5: nat,Y4: nat] :
% 0.30/0.58 ( ( ord_less_nat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst1
% 0.30/0.58 thf(fact_4287_order__le__less__subst1,axiom,
% 0.30/0.58 ! [A: real,F: int > real,B: int,C: int] :
% 0.30/0.58 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_int @ B @ C )
% 0.30/0.58 => ( ! [X5: int,Y4: int] :
% 0.30/0.58 ( ( ord_less_int @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst1
% 0.30/0.58 thf(fact_4288_order__le__less__subst1,axiom,
% 0.30/0.58 ! [A: rat,F: real > rat,B: real,C: real] :
% 0.30/0.58 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_real @ B @ C )
% 0.30/0.58 => ( ! [X5: real,Y4: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst1
% 0.30/0.58 thf(fact_4289_order__le__less__subst1,axiom,
% 0.30/0.58 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_rat @ B @ C )
% 0.30/0.58 => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst1
% 0.30/0.58 thf(fact_4290_order__le__less__subst1,axiom,
% 0.30/0.58 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.30/0.58 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_num @ B @ C )
% 0.30/0.58 => ( ! [X5: num,Y4: num] :
% 0.30/0.58 ( ( ord_less_num @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst1
% 0.30/0.58 thf(fact_4291_order__le__less__subst1,axiom,
% 0.30/0.58 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_nat @ B @ C )
% 0.30/0.58 => ( ! [X5: nat,Y4: nat] :
% 0.30/0.58 ( ( ord_less_nat @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst1
% 0.30/0.58 thf(fact_4292_order__le__less__subst1,axiom,
% 0.30/0.58 ! [A: rat,F: int > rat,B: int,C: int] :
% 0.30/0.58 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.30/0.58 => ( ( ord_less_int @ B @ C )
% 0.30/0.58 => ( ! [X5: int,Y4: int] :
% 0.30/0.58 ( ( ord_less_int @ X5 @ Y4 )
% 0.30/0.58 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_subst1
% 0.30/0.58 thf(fact_4293_order__less__le__trans,axiom,
% 0.30/0.58 ! [X2: real,Y3: real,Z: real] :
% 0.30/0.58 ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_eq_real @ Y3 @ Z )
% 0.30/0.58 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_trans
% 0.30/0.58 thf(fact_4294_order__less__le__trans,axiom,
% 0.30/0.58 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 0.30/0.58 ( ( ord_less_set_int @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_eq_set_int @ Y3 @ Z )
% 0.30/0.58 => ( ord_less_set_int @ X2 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_trans
% 0.30/0.58 thf(fact_4295_order__less__le__trans,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat,Z: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_eq_rat @ Y3 @ Z )
% 0.30/0.58 => ( ord_less_rat @ X2 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_trans
% 0.30/0.58 thf(fact_4296_order__less__le__trans,axiom,
% 0.30/0.58 ! [X2: num,Y3: num,Z: num] :
% 0.30/0.58 ( ( ord_less_num @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_eq_num @ Y3 @ Z )
% 0.30/0.58 => ( ord_less_num @ X2 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_trans
% 0.30/0.58 thf(fact_4297_order__less__le__trans,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat,Z: nat] :
% 0.30/0.58 ( ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_eq_nat @ Y3 @ Z )
% 0.30/0.58 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_trans
% 0.30/0.58 thf(fact_4298_order__less__le__trans,axiom,
% 0.30/0.58 ! [X2: int,Y3: int,Z: int] :
% 0.30/0.58 ( ( ord_less_int @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.30/0.58 => ( ord_less_int @ X2 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le_trans
% 0.30/0.58 thf(fact_4299_order__le__less__trans,axiom,
% 0.30/0.58 ! [X2: real,Y3: real,Z: real] :
% 0.30/0.58 ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_real @ Y3 @ Z )
% 0.30/0.58 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_trans
% 0.30/0.58 thf(fact_4300_order__le__less__trans,axiom,
% 0.30/0.58 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 0.30/0.58 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_set_int @ Y3 @ Z )
% 0.30/0.58 => ( ord_less_set_int @ X2 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_trans
% 0.30/0.58 thf(fact_4301_order__le__less__trans,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat,Z: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_rat @ Y3 @ Z )
% 0.30/0.58 => ( ord_less_rat @ X2 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_trans
% 0.30/0.58 thf(fact_4302_order__le__less__trans,axiom,
% 0.30/0.58 ! [X2: num,Y3: num,Z: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_num @ Y3 @ Z )
% 0.30/0.58 => ( ord_less_num @ X2 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_trans
% 0.30/0.58 thf(fact_4303_order__le__less__trans,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat,Z: nat] :
% 0.30/0.58 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_nat @ Y3 @ Z )
% 0.30/0.58 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_trans
% 0.30/0.58 thf(fact_4304_order__le__less__trans,axiom,
% 0.30/0.58 ! [X2: int,Y3: int,Z: int] :
% 0.30/0.58 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_int @ Y3 @ Z )
% 0.30/0.58 => ( ord_less_int @ X2 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less_trans
% 0.30/0.58 thf(fact_4305_order__neq__le__trans,axiom,
% 0.30/0.58 ! [A: real,B: real] :
% 0.30/0.58 ( ( A != B )
% 0.30/0.58 => ( ( ord_less_eq_real @ A @ B )
% 0.30/0.58 => ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_neq_le_trans
% 0.30/0.58 thf(fact_4306_order__neq__le__trans,axiom,
% 0.30/0.58 ! [A: set_int,B: set_int] :
% 0.30/0.58 ( ( A != B )
% 0.30/0.58 => ( ( ord_less_eq_set_int @ A @ B )
% 0.30/0.58 => ( ord_less_set_int @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_neq_le_trans
% 0.30/0.58 thf(fact_4307_order__neq__le__trans,axiom,
% 0.30/0.58 ! [A: rat,B: rat] :
% 0.30/0.58 ( ( A != B )
% 0.30/0.58 => ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.58 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_neq_le_trans
% 0.30/0.58 thf(fact_4308_order__neq__le__trans,axiom,
% 0.30/0.58 ! [A: num,B: num] :
% 0.30/0.58 ( ( A != B )
% 0.30/0.58 => ( ( ord_less_eq_num @ A @ B )
% 0.30/0.58 => ( ord_less_num @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_neq_le_trans
% 0.30/0.58 thf(fact_4309_order__neq__le__trans,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( A != B )
% 0.30/0.58 => ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.58 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_neq_le_trans
% 0.30/0.58 thf(fact_4310_order__neq__le__trans,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( A != B )
% 0.30/0.58 => ( ( ord_less_eq_int @ A @ B )
% 0.30/0.58 => ( ord_less_int @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_neq_le_trans
% 0.30/0.58 thf(fact_4311_order__le__neq__trans,axiom,
% 0.30/0.58 ! [A: real,B: real] :
% 0.30/0.58 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.58 => ( ( A != B )
% 0.30/0.58 => ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_neq_trans
% 0.30/0.58 thf(fact_4312_order__le__neq__trans,axiom,
% 0.30/0.58 ! [A: set_int,B: set_int] :
% 0.30/0.58 ( ( ord_less_eq_set_int @ A @ B )
% 0.30/0.58 => ( ( A != B )
% 0.30/0.58 => ( ord_less_set_int @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_neq_trans
% 0.30/0.58 thf(fact_4313_order__le__neq__trans,axiom,
% 0.30/0.58 ! [A: rat,B: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.58 => ( ( A != B )
% 0.30/0.58 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_neq_trans
% 0.30/0.58 thf(fact_4314_order__le__neq__trans,axiom,
% 0.30/0.58 ! [A: num,B: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.58 => ( ( A != B )
% 0.30/0.58 => ( ord_less_num @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_neq_trans
% 0.30/0.58 thf(fact_4315_order__le__neq__trans,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.58 => ( ( A != B )
% 0.30/0.58 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_neq_trans
% 0.30/0.58 thf(fact_4316_order__le__neq__trans,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.58 => ( ( A != B )
% 0.30/0.58 => ( ord_less_int @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_neq_trans
% 0.30/0.58 thf(fact_4317_order__less__imp__le,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.58 => ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_le
% 0.30/0.58 thf(fact_4318_order__less__imp__le,axiom,
% 0.30/0.58 ! [X2: set_int,Y3: set_int] :
% 0.30/0.58 ( ( ord_less_set_int @ X2 @ Y3 )
% 0.30/0.58 => ( ord_less_eq_set_int @ X2 @ Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_le
% 0.30/0.58 thf(fact_4319_order__less__imp__le,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.58 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_le
% 0.30/0.58 thf(fact_4320_order__less__imp__le,axiom,
% 0.30/0.58 ! [X2: num,Y3: num] :
% 0.30/0.58 ( ( ord_less_num @ X2 @ Y3 )
% 0.30/0.58 => ( ord_less_eq_num @ X2 @ Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_le
% 0.30/0.58 thf(fact_4321_order__less__imp__le,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat] :
% 0.30/0.58 ( ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.58 => ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_le
% 0.30/0.58 thf(fact_4322_order__less__imp__le,axiom,
% 0.30/0.58 ! [X2: int,Y3: int] :
% 0.30/0.58 ( ( ord_less_int @ X2 @ Y3 )
% 0.30/0.58 => ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_imp_le
% 0.30/0.58 thf(fact_4323_linorder__not__less,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
% 0.30/0.58 = ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_not_less
% 0.30/0.58 thf(fact_4324_linorder__not__less,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
% 0.30/0.58 = ( ord_less_eq_rat @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_not_less
% 0.30/0.58 thf(fact_4325_linorder__not__less,axiom,
% 0.30/0.58 ! [X2: num,Y3: num] :
% 0.30/0.58 ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
% 0.30/0.58 = ( ord_less_eq_num @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_not_less
% 0.30/0.58 thf(fact_4326_linorder__not__less,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat] :
% 0.30/0.58 ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
% 0.30/0.58 = ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_not_less
% 0.30/0.58 thf(fact_4327_linorder__not__less,axiom,
% 0.30/0.58 ! [X2: int,Y3: int] :
% 0.30/0.58 ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
% 0.30/0.58 = ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_not_less
% 0.30/0.58 thf(fact_4328_linorder__not__le,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ( ~ ( ord_less_eq_real @ X2 @ Y3 ) )
% 0.30/0.58 = ( ord_less_real @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_not_le
% 0.30/0.58 thf(fact_4329_linorder__not__le,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ( ~ ( ord_less_eq_rat @ X2 @ Y3 ) )
% 0.30/0.58 = ( ord_less_rat @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_not_le
% 0.30/0.58 thf(fact_4330_linorder__not__le,axiom,
% 0.30/0.58 ! [X2: num,Y3: num] :
% 0.30/0.58 ( ( ~ ( ord_less_eq_num @ X2 @ Y3 ) )
% 0.30/0.58 = ( ord_less_num @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_not_le
% 0.30/0.58 thf(fact_4331_linorder__not__le,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat] :
% 0.30/0.58 ( ( ~ ( ord_less_eq_nat @ X2 @ Y3 ) )
% 0.30/0.58 = ( ord_less_nat @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_not_le
% 0.30/0.58 thf(fact_4332_linorder__not__le,axiom,
% 0.30/0.58 ! [X2: int,Y3: int] :
% 0.30/0.58 ( ( ~ ( ord_less_eq_int @ X2 @ Y3 ) )
% 0.30/0.58 = ( ord_less_int @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % linorder_not_le
% 0.30/0.58 thf(fact_4333_order__less__le,axiom,
% 0.30/0.58 ( ord_less_real
% 0.30/0.58 = ( ^ [X: real,Y: real] :
% 0.30/0.58 ( ( ord_less_eq_real @ X @ Y )
% 0.30/0.58 & ( X != Y ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le
% 0.30/0.58 thf(fact_4334_order__less__le,axiom,
% 0.30/0.58 ( ord_less_set_int
% 0.30/0.58 = ( ^ [X: set_int,Y: set_int] :
% 0.30/0.58 ( ( ord_less_eq_set_int @ X @ Y )
% 0.30/0.58 & ( X != Y ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le
% 0.30/0.58 thf(fact_4335_order__less__le,axiom,
% 0.30/0.58 ( ord_less_rat
% 0.30/0.58 = ( ^ [X: rat,Y: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ X @ Y )
% 0.30/0.58 & ( X != Y ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le
% 0.30/0.58 thf(fact_4336_order__less__le,axiom,
% 0.30/0.58 ( ord_less_num
% 0.30/0.58 = ( ^ [X: num,Y: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ X @ Y )
% 0.30/0.58 & ( X != Y ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le
% 0.30/0.58 thf(fact_4337_order__less__le,axiom,
% 0.30/0.58 ( ord_less_nat
% 0.30/0.58 = ( ^ [X: nat,Y: nat] :
% 0.30/0.58 ( ( ord_less_eq_nat @ X @ Y )
% 0.30/0.58 & ( X != Y ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le
% 0.30/0.58 thf(fact_4338_order__less__le,axiom,
% 0.30/0.58 ( ord_less_int
% 0.30/0.58 = ( ^ [X: int,Y: int] :
% 0.30/0.58 ( ( ord_less_eq_int @ X @ Y )
% 0.30/0.58 & ( X != Y ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_less_le
% 0.30/0.58 thf(fact_4339_order__le__less,axiom,
% 0.30/0.58 ( ord_less_eq_real
% 0.30/0.58 = ( ^ [X: real,Y: real] :
% 0.30/0.58 ( ( ord_less_real @ X @ Y )
% 0.30/0.58 | ( X = Y ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less
% 0.30/0.58 thf(fact_4340_order__le__less,axiom,
% 0.30/0.58 ( ord_less_eq_set_int
% 0.30/0.58 = ( ^ [X: set_int,Y: set_int] :
% 0.30/0.58 ( ( ord_less_set_int @ X @ Y )
% 0.30/0.58 | ( X = Y ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less
% 0.30/0.58 thf(fact_4341_order__le__less,axiom,
% 0.30/0.58 ( ord_less_eq_rat
% 0.30/0.58 = ( ^ [X: rat,Y: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X @ Y )
% 0.30/0.58 | ( X = Y ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less
% 0.30/0.58 thf(fact_4342_order__le__less,axiom,
% 0.30/0.58 ( ord_less_eq_num
% 0.30/0.58 = ( ^ [X: num,Y: num] :
% 0.30/0.58 ( ( ord_less_num @ X @ Y )
% 0.30/0.58 | ( X = Y ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less
% 0.30/0.58 thf(fact_4343_order__le__less,axiom,
% 0.30/0.58 ( ord_less_eq_nat
% 0.30/0.58 = ( ^ [X: nat,Y: nat] :
% 0.30/0.58 ( ( ord_less_nat @ X @ Y )
% 0.30/0.58 | ( X = Y ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less
% 0.30/0.58 thf(fact_4344_order__le__less,axiom,
% 0.30/0.58 ( ord_less_eq_int
% 0.30/0.58 = ( ^ [X: int,Y: int] :
% 0.30/0.58 ( ( ord_less_int @ X @ Y )
% 0.30/0.58 | ( X = Y ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order_le_less
% 0.30/0.58 thf(fact_4345_dual__order_Ostrict__implies__order,axiom,
% 0.30/0.58 ! [B: real,A: real] :
% 0.30/0.58 ( ( ord_less_real @ B @ A )
% 0.30/0.58 => ( ord_less_eq_real @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_implies_order
% 0.30/0.58 thf(fact_4346_dual__order_Ostrict__implies__order,axiom,
% 0.30/0.58 ! [B: set_int,A: set_int] :
% 0.30/0.58 ( ( ord_less_set_int @ B @ A )
% 0.30/0.58 => ( ord_less_eq_set_int @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_implies_order
% 0.30/0.58 thf(fact_4347_dual__order_Ostrict__implies__order,axiom,
% 0.30/0.58 ! [B: rat,A: rat] :
% 0.30/0.58 ( ( ord_less_rat @ B @ A )
% 0.30/0.58 => ( ord_less_eq_rat @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_implies_order
% 0.30/0.58 thf(fact_4348_dual__order_Ostrict__implies__order,axiom,
% 0.30/0.58 ! [B: num,A: num] :
% 0.30/0.58 ( ( ord_less_num @ B @ A )
% 0.30/0.58 => ( ord_less_eq_num @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_implies_order
% 0.30/0.58 thf(fact_4349_dual__order_Ostrict__implies__order,axiom,
% 0.30/0.58 ! [B: nat,A: nat] :
% 0.30/0.58 ( ( ord_less_nat @ B @ A )
% 0.30/0.58 => ( ord_less_eq_nat @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_implies_order
% 0.30/0.58 thf(fact_4350_dual__order_Ostrict__implies__order,axiom,
% 0.30/0.58 ! [B: int,A: int] :
% 0.30/0.58 ( ( ord_less_int @ B @ A )
% 0.30/0.58 => ( ord_less_eq_int @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_implies_order
% 0.30/0.58 thf(fact_4351_order_Ostrict__implies__order,axiom,
% 0.30/0.58 ! [A: real,B: real] :
% 0.30/0.58 ( ( ord_less_real @ A @ B )
% 0.30/0.58 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_implies_order
% 0.30/0.58 thf(fact_4352_order_Ostrict__implies__order,axiom,
% 0.30/0.58 ! [A: set_int,B: set_int] :
% 0.30/0.58 ( ( ord_less_set_int @ A @ B )
% 0.30/0.58 => ( ord_less_eq_set_int @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_implies_order
% 0.30/0.58 thf(fact_4353_order_Ostrict__implies__order,axiom,
% 0.30/0.58 ! [A: rat,B: rat] :
% 0.30/0.58 ( ( ord_less_rat @ A @ B )
% 0.30/0.58 => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_implies_order
% 0.30/0.58 thf(fact_4354_order_Ostrict__implies__order,axiom,
% 0.30/0.58 ! [A: num,B: num] :
% 0.30/0.58 ( ( ord_less_num @ A @ B )
% 0.30/0.58 => ( ord_less_eq_num @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_implies_order
% 0.30/0.58 thf(fact_4355_order_Ostrict__implies__order,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( ord_less_nat @ A @ B )
% 0.30/0.58 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_implies_order
% 0.30/0.58 thf(fact_4356_order_Ostrict__implies__order,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( ord_less_int @ A @ B )
% 0.30/0.58 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_implies_order
% 0.30/0.58 thf(fact_4357_dual__order_Ostrict__iff__not,axiom,
% 0.30/0.58 ( ord_less_real
% 0.30/0.58 = ( ^ [B2: real,A3: real] :
% 0.30/0.58 ( ( ord_less_eq_real @ B2 @ A3 )
% 0.30/0.58 & ~ ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_iff_not
% 0.30/0.58 thf(fact_4358_dual__order_Ostrict__iff__not,axiom,
% 0.30/0.58 ( ord_less_set_int
% 0.30/0.58 = ( ^ [B2: set_int,A3: set_int] :
% 0.30/0.58 ( ( ord_less_eq_set_int @ B2 @ A3 )
% 0.30/0.58 & ~ ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_iff_not
% 0.30/0.58 thf(fact_4359_dual__order_Ostrict__iff__not,axiom,
% 0.30/0.58 ( ord_less_rat
% 0.30/0.58 = ( ^ [B2: rat,A3: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ B2 @ A3 )
% 0.30/0.58 & ~ ( ord_less_eq_rat @ A3 @ B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_iff_not
% 0.30/0.58 thf(fact_4360_dual__order_Ostrict__iff__not,axiom,
% 0.30/0.58 ( ord_less_num
% 0.30/0.58 = ( ^ [B2: num,A3: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ B2 @ A3 )
% 0.30/0.58 & ~ ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_iff_not
% 0.30/0.58 thf(fact_4361_dual__order_Ostrict__iff__not,axiom,
% 0.30/0.58 ( ord_less_nat
% 0.30/0.58 = ( ^ [B2: nat,A3: nat] :
% 0.30/0.58 ( ( ord_less_eq_nat @ B2 @ A3 )
% 0.30/0.58 & ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_iff_not
% 0.30/0.58 thf(fact_4362_dual__order_Ostrict__iff__not,axiom,
% 0.30/0.58 ( ord_less_int
% 0.30/0.58 = ( ^ [B2: int,A3: int] :
% 0.30/0.58 ( ( ord_less_eq_int @ B2 @ A3 )
% 0.30/0.58 & ~ ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_iff_not
% 0.30/0.58 thf(fact_4363_dual__order_Ostrict__trans2,axiom,
% 0.30/0.58 ! [B: real,A: real,C: real] :
% 0.30/0.58 ( ( ord_less_real @ B @ A )
% 0.30/0.58 => ( ( ord_less_eq_real @ C @ B )
% 0.30/0.58 => ( ord_less_real @ C @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_trans2
% 0.30/0.58 thf(fact_4364_dual__order_Ostrict__trans2,axiom,
% 0.30/0.58 ! [B: set_int,A: set_int,C: set_int] :
% 0.30/0.58 ( ( ord_less_set_int @ B @ A )
% 0.30/0.58 => ( ( ord_less_eq_set_int @ C @ B )
% 0.30/0.58 => ( ord_less_set_int @ C @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_trans2
% 0.30/0.58 thf(fact_4365_dual__order_Ostrict__trans2,axiom,
% 0.30/0.58 ! [B: rat,A: rat,C: rat] :
% 0.30/0.58 ( ( ord_less_rat @ B @ A )
% 0.30/0.58 => ( ( ord_less_eq_rat @ C @ B )
% 0.30/0.58 => ( ord_less_rat @ C @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_trans2
% 0.30/0.58 thf(fact_4366_dual__order_Ostrict__trans2,axiom,
% 0.30/0.58 ! [B: num,A: num,C: num] :
% 0.30/0.58 ( ( ord_less_num @ B @ A )
% 0.30/0.58 => ( ( ord_less_eq_num @ C @ B )
% 0.30/0.58 => ( ord_less_num @ C @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_trans2
% 0.30/0.58 thf(fact_4367_dual__order_Ostrict__trans2,axiom,
% 0.30/0.58 ! [B: nat,A: nat,C: nat] :
% 0.30/0.58 ( ( ord_less_nat @ B @ A )
% 0.30/0.58 => ( ( ord_less_eq_nat @ C @ B )
% 0.30/0.58 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_trans2
% 0.30/0.58 thf(fact_4368_dual__order_Ostrict__trans2,axiom,
% 0.30/0.58 ! [B: int,A: int,C: int] :
% 0.30/0.58 ( ( ord_less_int @ B @ A )
% 0.30/0.58 => ( ( ord_less_eq_int @ C @ B )
% 0.30/0.58 => ( ord_less_int @ C @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_trans2
% 0.30/0.58 thf(fact_4369_dual__order_Ostrict__trans1,axiom,
% 0.30/0.58 ! [B: real,A: real,C: real] :
% 0.30/0.58 ( ( ord_less_eq_real @ B @ A )
% 0.30/0.58 => ( ( ord_less_real @ C @ B )
% 0.30/0.58 => ( ord_less_real @ C @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_trans1
% 0.30/0.58 thf(fact_4370_dual__order_Ostrict__trans1,axiom,
% 0.30/0.58 ! [B: set_int,A: set_int,C: set_int] :
% 0.30/0.58 ( ( ord_less_eq_set_int @ B @ A )
% 0.30/0.58 => ( ( ord_less_set_int @ C @ B )
% 0.30/0.58 => ( ord_less_set_int @ C @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_trans1
% 0.30/0.58 thf(fact_4371_dual__order_Ostrict__trans1,axiom,
% 0.30/0.58 ! [B: rat,A: rat,C: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.58 => ( ( ord_less_rat @ C @ B )
% 0.30/0.58 => ( ord_less_rat @ C @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_trans1
% 0.30/0.58 thf(fact_4372_dual__order_Ostrict__trans1,axiom,
% 0.30/0.58 ! [B: num,A: num,C: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ B @ A )
% 0.30/0.58 => ( ( ord_less_num @ C @ B )
% 0.30/0.58 => ( ord_less_num @ C @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_trans1
% 0.30/0.58 thf(fact_4373_dual__order_Ostrict__trans1,axiom,
% 0.30/0.58 ! [B: nat,A: nat,C: nat] :
% 0.30/0.58 ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.58 => ( ( ord_less_nat @ C @ B )
% 0.30/0.58 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_trans1
% 0.30/0.58 thf(fact_4374_dual__order_Ostrict__trans1,axiom,
% 0.30/0.58 ! [B: int,A: int,C: int] :
% 0.30/0.58 ( ( ord_less_eq_int @ B @ A )
% 0.30/0.58 => ( ( ord_less_int @ C @ B )
% 0.30/0.58 => ( ord_less_int @ C @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_trans1
% 0.30/0.58 thf(fact_4375_dual__order_Ostrict__iff__order,axiom,
% 0.30/0.58 ( ord_less_real
% 0.30/0.58 = ( ^ [B2: real,A3: real] :
% 0.30/0.58 ( ( ord_less_eq_real @ B2 @ A3 )
% 0.30/0.58 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_iff_order
% 0.30/0.58 thf(fact_4376_dual__order_Ostrict__iff__order,axiom,
% 0.30/0.58 ( ord_less_set_int
% 0.30/0.58 = ( ^ [B2: set_int,A3: set_int] :
% 0.30/0.58 ( ( ord_less_eq_set_int @ B2 @ A3 )
% 0.30/0.58 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_iff_order
% 0.30/0.58 thf(fact_4377_dual__order_Ostrict__iff__order,axiom,
% 0.30/0.58 ( ord_less_rat
% 0.30/0.58 = ( ^ [B2: rat,A3: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ B2 @ A3 )
% 0.30/0.58 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_iff_order
% 0.30/0.58 thf(fact_4378_dual__order_Ostrict__iff__order,axiom,
% 0.30/0.58 ( ord_less_num
% 0.30/0.58 = ( ^ [B2: num,A3: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ B2 @ A3 )
% 0.30/0.58 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_iff_order
% 0.30/0.58 thf(fact_4379_dual__order_Ostrict__iff__order,axiom,
% 0.30/0.58 ( ord_less_nat
% 0.30/0.58 = ( ^ [B2: nat,A3: nat] :
% 0.30/0.58 ( ( ord_less_eq_nat @ B2 @ A3 )
% 0.30/0.58 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_iff_order
% 0.30/0.58 thf(fact_4380_dual__order_Ostrict__iff__order,axiom,
% 0.30/0.58 ( ord_less_int
% 0.30/0.58 = ( ^ [B2: int,A3: int] :
% 0.30/0.58 ( ( ord_less_eq_int @ B2 @ A3 )
% 0.30/0.58 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.strict_iff_order
% 0.30/0.58 thf(fact_4381_dual__order_Oorder__iff__strict,axiom,
% 0.30/0.58 ( ord_less_eq_real
% 0.30/0.58 = ( ^ [B2: real,A3: real] :
% 0.30/0.58 ( ( ord_less_real @ B2 @ A3 )
% 0.30/0.58 | ( A3 = B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.order_iff_strict
% 0.30/0.58 thf(fact_4382_dual__order_Oorder__iff__strict,axiom,
% 0.30/0.58 ( ord_less_eq_set_int
% 0.30/0.58 = ( ^ [B2: set_int,A3: set_int] :
% 0.30/0.58 ( ( ord_less_set_int @ B2 @ A3 )
% 0.30/0.58 | ( A3 = B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.order_iff_strict
% 0.30/0.58 thf(fact_4383_dual__order_Oorder__iff__strict,axiom,
% 0.30/0.58 ( ord_less_eq_rat
% 0.30/0.58 = ( ^ [B2: rat,A3: rat] :
% 0.30/0.58 ( ( ord_less_rat @ B2 @ A3 )
% 0.30/0.58 | ( A3 = B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.order_iff_strict
% 0.30/0.58 thf(fact_4384_dual__order_Oorder__iff__strict,axiom,
% 0.30/0.58 ( ord_less_eq_num
% 0.30/0.58 = ( ^ [B2: num,A3: num] :
% 0.30/0.58 ( ( ord_less_num @ B2 @ A3 )
% 0.30/0.58 | ( A3 = B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.order_iff_strict
% 0.30/0.58 thf(fact_4385_dual__order_Oorder__iff__strict,axiom,
% 0.30/0.58 ( ord_less_eq_nat
% 0.30/0.58 = ( ^ [B2: nat,A3: nat] :
% 0.30/0.58 ( ( ord_less_nat @ B2 @ A3 )
% 0.30/0.58 | ( A3 = B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.order_iff_strict
% 0.30/0.58 thf(fact_4386_dual__order_Oorder__iff__strict,axiom,
% 0.30/0.58 ( ord_less_eq_int
% 0.30/0.58 = ( ^ [B2: int,A3: int] :
% 0.30/0.58 ( ( ord_less_int @ B2 @ A3 )
% 0.30/0.58 | ( A3 = B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dual_order.order_iff_strict
% 0.30/0.58 thf(fact_4387_dense__le__bounded,axiom,
% 0.30/0.58 ! [X2: real,Y3: real,Z: real] :
% 0.30/0.58 ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.58 => ( ! [W2: real] :
% 0.30/0.58 ( ( ord_less_real @ X2 @ W2 )
% 0.30/0.58 => ( ( ord_less_real @ W2 @ Y3 )
% 0.30/0.58 => ( ord_less_eq_real @ W2 @ Z ) ) )
% 0.30/0.58 => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dense_le_bounded
% 0.30/0.58 thf(fact_4388_dense__le__bounded,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat,Z: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.58 => ( ! [W2: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X2 @ W2 )
% 0.30/0.58 => ( ( ord_less_rat @ W2 @ Y3 )
% 0.30/0.58 => ( ord_less_eq_rat @ W2 @ Z ) ) )
% 0.30/0.58 => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dense_le_bounded
% 0.30/0.58 thf(fact_4389_dense__ge__bounded,axiom,
% 0.30/0.58 ! [Z: real,X2: real,Y3: real] :
% 0.30/0.58 ( ( ord_less_real @ Z @ X2 )
% 0.30/0.58 => ( ! [W2: real] :
% 0.30/0.58 ( ( ord_less_real @ Z @ W2 )
% 0.30/0.58 => ( ( ord_less_real @ W2 @ X2 )
% 0.30/0.58 => ( ord_less_eq_real @ Y3 @ W2 ) ) )
% 0.30/0.58 => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dense_ge_bounded
% 0.30/0.58 thf(fact_4390_dense__ge__bounded,axiom,
% 0.30/0.58 ! [Z: rat,X2: rat,Y3: rat] :
% 0.30/0.58 ( ( ord_less_rat @ Z @ X2 )
% 0.30/0.58 => ( ! [W2: rat] :
% 0.30/0.58 ( ( ord_less_rat @ Z @ W2 )
% 0.30/0.58 => ( ( ord_less_rat @ W2 @ X2 )
% 0.30/0.58 => ( ord_less_eq_rat @ Y3 @ W2 ) ) )
% 0.30/0.58 => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dense_ge_bounded
% 0.30/0.58 thf(fact_4391_order_Ostrict__iff__not,axiom,
% 0.30/0.58 ( ord_less_real
% 0.30/0.58 = ( ^ [A3: real,B2: real] :
% 0.30/0.58 ( ( ord_less_eq_real @ A3 @ B2 )
% 0.30/0.58 & ~ ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_iff_not
% 0.30/0.58 thf(fact_4392_order_Ostrict__iff__not,axiom,
% 0.30/0.58 ( ord_less_set_int
% 0.30/0.58 = ( ^ [A3: set_int,B2: set_int] :
% 0.30/0.58 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.30/0.58 & ~ ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_iff_not
% 0.30/0.58 thf(fact_4393_order_Ostrict__iff__not,axiom,
% 0.30/0.58 ( ord_less_rat
% 0.30/0.58 = ( ^ [A3: rat,B2: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ A3 @ B2 )
% 0.30/0.58 & ~ ( ord_less_eq_rat @ B2 @ A3 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_iff_not
% 0.30/0.58 thf(fact_4394_order_Ostrict__iff__not,axiom,
% 0.30/0.58 ( ord_less_num
% 0.30/0.58 = ( ^ [A3: num,B2: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ A3 @ B2 )
% 0.30/0.58 & ~ ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_iff_not
% 0.30/0.58 thf(fact_4395_order_Ostrict__iff__not,axiom,
% 0.30/0.58 ( ord_less_nat
% 0.30/0.58 = ( ^ [A3: nat,B2: nat] :
% 0.30/0.58 ( ( ord_less_eq_nat @ A3 @ B2 )
% 0.30/0.58 & ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_iff_not
% 0.30/0.58 thf(fact_4396_order_Ostrict__iff__not,axiom,
% 0.30/0.58 ( ord_less_int
% 0.30/0.58 = ( ^ [A3: int,B2: int] :
% 0.30/0.58 ( ( ord_less_eq_int @ A3 @ B2 )
% 0.30/0.58 & ~ ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_iff_not
% 0.30/0.58 thf(fact_4397_order_Ostrict__trans2,axiom,
% 0.30/0.58 ! [A: real,B: real,C: real] :
% 0.30/0.58 ( ( ord_less_real @ A @ B )
% 0.30/0.58 => ( ( ord_less_eq_real @ B @ C )
% 0.30/0.58 => ( ord_less_real @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_trans2
% 0.30/0.58 thf(fact_4398_order_Ostrict__trans2,axiom,
% 0.30/0.58 ! [A: set_int,B: set_int,C: set_int] :
% 0.30/0.58 ( ( ord_less_set_int @ A @ B )
% 0.30/0.58 => ( ( ord_less_eq_set_int @ B @ C )
% 0.30/0.58 => ( ord_less_set_int @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_trans2
% 0.30/0.58 thf(fact_4399_order_Ostrict__trans2,axiom,
% 0.30/0.58 ! [A: rat,B: rat,C: rat] :
% 0.30/0.58 ( ( ord_less_rat @ A @ B )
% 0.30/0.58 => ( ( ord_less_eq_rat @ B @ C )
% 0.30/0.58 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_trans2
% 0.30/0.58 thf(fact_4400_order_Ostrict__trans2,axiom,
% 0.30/0.58 ! [A: num,B: num,C: num] :
% 0.30/0.58 ( ( ord_less_num @ A @ B )
% 0.30/0.58 => ( ( ord_less_eq_num @ B @ C )
% 0.30/0.58 => ( ord_less_num @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_trans2
% 0.30/0.58 thf(fact_4401_order_Ostrict__trans2,axiom,
% 0.30/0.58 ! [A: nat,B: nat,C: nat] :
% 0.30/0.58 ( ( ord_less_nat @ A @ B )
% 0.30/0.58 => ( ( ord_less_eq_nat @ B @ C )
% 0.30/0.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_trans2
% 0.30/0.58 thf(fact_4402_order_Ostrict__trans2,axiom,
% 0.30/0.58 ! [A: int,B: int,C: int] :
% 0.30/0.58 ( ( ord_less_int @ A @ B )
% 0.30/0.58 => ( ( ord_less_eq_int @ B @ C )
% 0.30/0.58 => ( ord_less_int @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_trans2
% 0.30/0.58 thf(fact_4403_order_Ostrict__trans1,axiom,
% 0.30/0.58 ! [A: real,B: real,C: real] :
% 0.30/0.58 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.58 => ( ( ord_less_real @ B @ C )
% 0.30/0.58 => ( ord_less_real @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_trans1
% 0.30/0.58 thf(fact_4404_order_Ostrict__trans1,axiom,
% 0.30/0.58 ! [A: set_int,B: set_int,C: set_int] :
% 0.30/0.58 ( ( ord_less_eq_set_int @ A @ B )
% 0.30/0.58 => ( ( ord_less_set_int @ B @ C )
% 0.30/0.58 => ( ord_less_set_int @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_trans1
% 0.30/0.58 thf(fact_4405_order_Ostrict__trans1,axiom,
% 0.30/0.58 ! [A: rat,B: rat,C: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.58 => ( ( ord_less_rat @ B @ C )
% 0.30/0.58 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_trans1
% 0.30/0.58 thf(fact_4406_order_Ostrict__trans1,axiom,
% 0.30/0.58 ! [A: num,B: num,C: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.58 => ( ( ord_less_num @ B @ C )
% 0.30/0.58 => ( ord_less_num @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_trans1
% 0.30/0.58 thf(fact_4407_order_Ostrict__trans1,axiom,
% 0.30/0.58 ! [A: nat,B: nat,C: nat] :
% 0.30/0.58 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.58 => ( ( ord_less_nat @ B @ C )
% 0.30/0.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_trans1
% 0.30/0.58 thf(fact_4408_order_Ostrict__trans1,axiom,
% 0.30/0.58 ! [A: int,B: int,C: int] :
% 0.30/0.58 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.58 => ( ( ord_less_int @ B @ C )
% 0.30/0.58 => ( ord_less_int @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_trans1
% 0.30/0.58 thf(fact_4409_order_Ostrict__iff__order,axiom,
% 0.30/0.58 ( ord_less_real
% 0.30/0.58 = ( ^ [A3: real,B2: real] :
% 0.30/0.58 ( ( ord_less_eq_real @ A3 @ B2 )
% 0.30/0.58 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_iff_order
% 0.30/0.58 thf(fact_4410_order_Ostrict__iff__order,axiom,
% 0.30/0.58 ( ord_less_set_int
% 0.30/0.58 = ( ^ [A3: set_int,B2: set_int] :
% 0.30/0.58 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.30/0.58 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_iff_order
% 0.30/0.58 thf(fact_4411_order_Ostrict__iff__order,axiom,
% 0.30/0.58 ( ord_less_rat
% 0.30/0.58 = ( ^ [A3: rat,B2: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ A3 @ B2 )
% 0.30/0.58 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_iff_order
% 0.30/0.58 thf(fact_4412_order_Ostrict__iff__order,axiom,
% 0.30/0.58 ( ord_less_num
% 0.30/0.58 = ( ^ [A3: num,B2: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ A3 @ B2 )
% 0.30/0.58 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_iff_order
% 0.30/0.58 thf(fact_4413_order_Ostrict__iff__order,axiom,
% 0.30/0.58 ( ord_less_nat
% 0.30/0.58 = ( ^ [A3: nat,B2: nat] :
% 0.30/0.58 ( ( ord_less_eq_nat @ A3 @ B2 )
% 0.30/0.58 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_iff_order
% 0.30/0.58 thf(fact_4414_order_Ostrict__iff__order,axiom,
% 0.30/0.58 ( ord_less_int
% 0.30/0.58 = ( ^ [A3: int,B2: int] :
% 0.30/0.58 ( ( ord_less_eq_int @ A3 @ B2 )
% 0.30/0.58 & ( A3 != B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.strict_iff_order
% 0.30/0.58 thf(fact_4415_order_Oorder__iff__strict,axiom,
% 0.30/0.58 ( ord_less_eq_real
% 0.30/0.58 = ( ^ [A3: real,B2: real] :
% 0.30/0.58 ( ( ord_less_real @ A3 @ B2 )
% 0.30/0.58 | ( A3 = B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.order_iff_strict
% 0.30/0.58 thf(fact_4416_order_Oorder__iff__strict,axiom,
% 0.30/0.58 ( ord_less_eq_set_int
% 0.30/0.58 = ( ^ [A3: set_int,B2: set_int] :
% 0.30/0.58 ( ( ord_less_set_int @ A3 @ B2 )
% 0.30/0.58 | ( A3 = B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.order_iff_strict
% 0.30/0.58 thf(fact_4417_order_Oorder__iff__strict,axiom,
% 0.30/0.58 ( ord_less_eq_rat
% 0.30/0.58 = ( ^ [A3: rat,B2: rat] :
% 0.30/0.58 ( ( ord_less_rat @ A3 @ B2 )
% 0.30/0.58 | ( A3 = B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.order_iff_strict
% 0.30/0.58 thf(fact_4418_order_Oorder__iff__strict,axiom,
% 0.30/0.58 ( ord_less_eq_num
% 0.30/0.58 = ( ^ [A3: num,B2: num] :
% 0.30/0.58 ( ( ord_less_num @ A3 @ B2 )
% 0.30/0.58 | ( A3 = B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.order_iff_strict
% 0.30/0.58 thf(fact_4419_order_Oorder__iff__strict,axiom,
% 0.30/0.58 ( ord_less_eq_nat
% 0.30/0.58 = ( ^ [A3: nat,B2: nat] :
% 0.30/0.58 ( ( ord_less_nat @ A3 @ B2 )
% 0.30/0.58 | ( A3 = B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.order_iff_strict
% 0.30/0.58 thf(fact_4420_order_Oorder__iff__strict,axiom,
% 0.30/0.58 ( ord_less_eq_int
% 0.30/0.58 = ( ^ [A3: int,B2: int] :
% 0.30/0.58 ( ( ord_less_int @ A3 @ B2 )
% 0.30/0.58 | ( A3 = B2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % order.order_iff_strict
% 0.30/0.58 thf(fact_4421_not__le__imp__less,axiom,
% 0.30/0.58 ! [Y3: real,X2: real] :
% 0.30/0.58 ( ~ ( ord_less_eq_real @ Y3 @ X2 )
% 0.30/0.58 => ( ord_less_real @ X2 @ Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % not_le_imp_less
% 0.30/0.58 thf(fact_4422_not__le__imp__less,axiom,
% 0.30/0.58 ! [Y3: rat,X2: rat] :
% 0.30/0.58 ( ~ ( ord_less_eq_rat @ Y3 @ X2 )
% 0.30/0.58 => ( ord_less_rat @ X2 @ Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % not_le_imp_less
% 0.30/0.58 thf(fact_4423_not__le__imp__less,axiom,
% 0.30/0.58 ! [Y3: num,X2: num] :
% 0.30/0.58 ( ~ ( ord_less_eq_num @ Y3 @ X2 )
% 0.30/0.58 => ( ord_less_num @ X2 @ Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % not_le_imp_less
% 0.30/0.58 thf(fact_4424_not__le__imp__less,axiom,
% 0.30/0.58 ! [Y3: nat,X2: nat] :
% 0.30/0.58 ( ~ ( ord_less_eq_nat @ Y3 @ X2 )
% 0.30/0.58 => ( ord_less_nat @ X2 @ Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % not_le_imp_less
% 0.30/0.58 thf(fact_4425_not__le__imp__less,axiom,
% 0.30/0.58 ! [Y3: int,X2: int] :
% 0.30/0.58 ( ~ ( ord_less_eq_int @ Y3 @ X2 )
% 0.30/0.58 => ( ord_less_int @ X2 @ Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % not_le_imp_less
% 0.30/0.58 thf(fact_4426_less__le__not__le,axiom,
% 0.30/0.58 ( ord_less_real
% 0.30/0.58 = ( ^ [X: real,Y: real] :
% 0.30/0.58 ( ( ord_less_eq_real @ X @ Y )
% 0.30/0.58 & ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % less_le_not_le
% 0.30/0.58 thf(fact_4427_less__le__not__le,axiom,
% 0.30/0.58 ( ord_less_set_int
% 0.30/0.58 = ( ^ [X: set_int,Y: set_int] :
% 0.30/0.58 ( ( ord_less_eq_set_int @ X @ Y )
% 0.30/0.58 & ~ ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % less_le_not_le
% 0.30/0.58 thf(fact_4428_less__le__not__le,axiom,
% 0.30/0.58 ( ord_less_rat
% 0.30/0.58 = ( ^ [X: rat,Y: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ X @ Y )
% 0.30/0.58 & ~ ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % less_le_not_le
% 0.30/0.58 thf(fact_4429_less__le__not__le,axiom,
% 0.30/0.58 ( ord_less_num
% 0.30/0.58 = ( ^ [X: num,Y: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ X @ Y )
% 0.30/0.58 & ~ ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % less_le_not_le
% 0.30/0.58 thf(fact_4430_less__le__not__le,axiom,
% 0.30/0.58 ( ord_less_nat
% 0.30/0.58 = ( ^ [X: nat,Y: nat] :
% 0.30/0.58 ( ( ord_less_eq_nat @ X @ Y )
% 0.30/0.58 & ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % less_le_not_le
% 0.30/0.58 thf(fact_4431_less__le__not__le,axiom,
% 0.30/0.58 ( ord_less_int
% 0.30/0.58 = ( ^ [X: int,Y: int] :
% 0.30/0.58 ( ( ord_less_eq_int @ X @ Y )
% 0.30/0.58 & ~ ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % less_le_not_le
% 0.30/0.58 thf(fact_4432_dense__le,axiom,
% 0.30/0.58 ! [Y3: real,Z: real] :
% 0.30/0.58 ( ! [X5: real] :
% 0.30/0.58 ( ( ord_less_real @ X5 @ Y3 )
% 0.30/0.58 => ( ord_less_eq_real @ X5 @ Z ) )
% 0.30/0.58 => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 0.30/0.58
% 0.30/0.58 % dense_le
% 0.30/0.58 thf(fact_4433_dense__le,axiom,
% 0.30/0.58 ! [Y3: rat,Z: rat] :
% 0.30/0.58 ( ! [X5: rat] :
% 0.30/0.58 ( ( ord_less_rat @ X5 @ Y3 )
% 0.30/0.58 => ( ord_less_eq_rat @ X5 @ Z ) )
% 0.30/0.58 => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 0.30/0.58
% 0.30/0.58 % dense_le
% 0.30/0.58 thf(fact_4434_dense__ge,axiom,
% 0.30/0.58 ! [Z: real,Y3: real] :
% 0.30/0.58 ( ! [X5: real] :
% 0.30/0.58 ( ( ord_less_real @ Z @ X5 )
% 0.30/0.58 => ( ord_less_eq_real @ Y3 @ X5 ) )
% 0.30/0.58 => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 0.30/0.58
% 0.30/0.58 % dense_ge
% 0.30/0.58 thf(fact_4435_dense__ge,axiom,
% 0.30/0.58 ! [Z: rat,Y3: rat] :
% 0.30/0.58 ( ! [X5: rat] :
% 0.30/0.58 ( ( ord_less_rat @ Z @ X5 )
% 0.30/0.58 => ( ord_less_eq_rat @ Y3 @ X5 ) )
% 0.30/0.58 => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 0.30/0.58
% 0.30/0.58 % dense_ge
% 0.30/0.58 thf(fact_4436_antisym__conv2,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.30/0.58 => ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
% 0.30/0.58 = ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % antisym_conv2
% 0.30/0.58 thf(fact_4437_antisym__conv2,axiom,
% 0.30/0.58 ! [X2: set_int,Y3: set_int] :
% 0.30/0.58 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.30/0.58 => ( ( ~ ( ord_less_set_int @ X2 @ Y3 ) )
% 0.30/0.58 = ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % antisym_conv2
% 0.30/0.58 thf(fact_4438_antisym__conv2,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.30/0.58 => ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
% 0.30/0.58 = ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % antisym_conv2
% 0.30/0.58 thf(fact_4439_antisym__conv2,axiom,
% 0.30/0.58 ! [X2: num,Y3: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.30/0.58 => ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
% 0.30/0.58 = ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % antisym_conv2
% 0.30/0.58 thf(fact_4440_antisym__conv2,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat] :
% 0.30/0.58 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.30/0.58 => ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
% 0.30/0.58 = ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % antisym_conv2
% 0.30/0.58 thf(fact_4441_antisym__conv2,axiom,
% 0.30/0.58 ! [X2: int,Y3: int] :
% 0.30/0.58 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.30/0.58 => ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
% 0.30/0.58 = ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % antisym_conv2
% 0.30/0.58 thf(fact_4442_antisym__conv1,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ~ ( ord_less_real @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.30/0.58 = ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % antisym_conv1
% 0.30/0.58 thf(fact_4443_antisym__conv1,axiom,
% 0.30/0.58 ! [X2: set_int,Y3: set_int] :
% 0.30/0.58 ( ~ ( ord_less_set_int @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.30/0.58 = ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % antisym_conv1
% 0.30/0.58 thf(fact_4444_antisym__conv1,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ~ ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.30/0.58 = ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % antisym_conv1
% 0.30/0.58 thf(fact_4445_antisym__conv1,axiom,
% 0.30/0.58 ! [X2: num,Y3: num] :
% 0.30/0.58 ( ~ ( ord_less_num @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.30/0.58 = ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % antisym_conv1
% 0.30/0.58 thf(fact_4446_antisym__conv1,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat] :
% 0.30/0.58 ( ~ ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.30/0.58 = ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % antisym_conv1
% 0.30/0.58 thf(fact_4447_antisym__conv1,axiom,
% 0.30/0.58 ! [X2: int,Y3: int] :
% 0.30/0.58 ( ~ ( ord_less_int @ X2 @ Y3 )
% 0.30/0.58 => ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.30/0.58 = ( X2 = Y3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % antisym_conv1
% 0.30/0.58 thf(fact_4448_nless__le,axiom,
% 0.30/0.58 ! [A: real,B: real] :
% 0.30/0.58 ( ( ~ ( ord_less_real @ A @ B ) )
% 0.30/0.58 = ( ~ ( ord_less_eq_real @ A @ B )
% 0.30/0.58 | ( A = B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % nless_le
% 0.30/0.58 thf(fact_4449_nless__le,axiom,
% 0.30/0.58 ! [A: set_int,B: set_int] :
% 0.30/0.58 ( ( ~ ( ord_less_set_int @ A @ B ) )
% 0.30/0.58 = ( ~ ( ord_less_eq_set_int @ A @ B )
% 0.30/0.58 | ( A = B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % nless_le
% 0.30/0.58 thf(fact_4450_nless__le,axiom,
% 0.30/0.58 ! [A: rat,B: rat] :
% 0.30/0.58 ( ( ~ ( ord_less_rat @ A @ B ) )
% 0.30/0.58 = ( ~ ( ord_less_eq_rat @ A @ B )
% 0.30/0.58 | ( A = B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % nless_le
% 0.30/0.58 thf(fact_4451_nless__le,axiom,
% 0.30/0.58 ! [A: num,B: num] :
% 0.30/0.58 ( ( ~ ( ord_less_num @ A @ B ) )
% 0.30/0.58 = ( ~ ( ord_less_eq_num @ A @ B )
% 0.30/0.58 | ( A = B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % nless_le
% 0.30/0.58 thf(fact_4452_nless__le,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( ~ ( ord_less_nat @ A @ B ) )
% 0.30/0.58 = ( ~ ( ord_less_eq_nat @ A @ B )
% 0.30/0.58 | ( A = B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % nless_le
% 0.30/0.58 thf(fact_4453_nless__le,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( ~ ( ord_less_int @ A @ B ) )
% 0.30/0.58 = ( ~ ( ord_less_eq_int @ A @ B )
% 0.30/0.58 | ( A = B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % nless_le
% 0.30/0.58 thf(fact_4454_leI,axiom,
% 0.30/0.58 ! [X2: real,Y3: real] :
% 0.30/0.58 ( ~ ( ord_less_real @ X2 @ Y3 )
% 0.30/0.58 => ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % leI
% 0.30/0.58 thf(fact_4455_leI,axiom,
% 0.30/0.58 ! [X2: rat,Y3: rat] :
% 0.30/0.58 ( ~ ( ord_less_rat @ X2 @ Y3 )
% 0.30/0.58 => ( ord_less_eq_rat @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % leI
% 0.30/0.58 thf(fact_4456_leI,axiom,
% 0.30/0.58 ! [X2: num,Y3: num] :
% 0.30/0.58 ( ~ ( ord_less_num @ X2 @ Y3 )
% 0.30/0.58 => ( ord_less_eq_num @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % leI
% 0.30/0.58 thf(fact_4457_leI,axiom,
% 0.30/0.58 ! [X2: nat,Y3: nat] :
% 0.30/0.58 ( ~ ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.58 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % leI
% 0.30/0.58 thf(fact_4458_leI,axiom,
% 0.30/0.58 ! [X2: int,Y3: int] :
% 0.30/0.58 ( ~ ( ord_less_int @ X2 @ Y3 )
% 0.30/0.58 => ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% 0.30/0.58
% 0.30/0.58 % leI
% 0.30/0.58 thf(fact_4459_leD,axiom,
% 0.30/0.58 ! [Y3: real,X2: real] :
% 0.30/0.58 ( ( ord_less_eq_real @ Y3 @ X2 )
% 0.30/0.58 => ~ ( ord_less_real @ X2 @ Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % leD
% 0.30/0.58 thf(fact_4460_leD,axiom,
% 0.30/0.58 ! [Y3: set_int,X2: set_int] :
% 0.30/0.58 ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 0.30/0.58 => ~ ( ord_less_set_int @ X2 @ Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % leD
% 0.30/0.58 thf(fact_4461_leD,axiom,
% 0.30/0.58 ! [Y3: rat,X2: rat] :
% 0.30/0.58 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 0.30/0.58 => ~ ( ord_less_rat @ X2 @ Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % leD
% 0.30/0.58 thf(fact_4462_leD,axiom,
% 0.30/0.58 ! [Y3: num,X2: num] :
% 0.30/0.58 ( ( ord_less_eq_num @ Y3 @ X2 )
% 0.30/0.58 => ~ ( ord_less_num @ X2 @ Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % leD
% 0.30/0.58 thf(fact_4463_leD,axiom,
% 0.30/0.58 ! [Y3: nat,X2: nat] :
% 0.30/0.58 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.30/0.58 => ~ ( ord_less_nat @ X2 @ Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % leD
% 0.30/0.58 thf(fact_4464_leD,axiom,
% 0.30/0.58 ! [Y3: int,X2: int] :
% 0.30/0.58 ( ( ord_less_eq_int @ Y3 @ X2 )
% 0.30/0.58 => ~ ( ord_less_int @ X2 @ Y3 ) ) ).
% 0.30/0.58
% 0.30/0.58 % leD
% 0.30/0.58 thf(fact_4465_bot_Oextremum__uniqueI,axiom,
% 0.30/0.58 ! [A: set_nat] :
% 0.30/0.58 ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 0.30/0.58 => ( A = bot_bot_set_nat ) ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum_uniqueI
% 0.30/0.58 thf(fact_4466_bot_Oextremum__uniqueI,axiom,
% 0.30/0.58 ! [A: extended_enat] :
% 0.30/0.58 ( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
% 0.30/0.58 => ( A = bot_bo4199563552545308370d_enat ) ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum_uniqueI
% 0.30/0.58 thf(fact_4467_bot_Oextremum__uniqueI,axiom,
% 0.30/0.58 ! [A: set_real] :
% 0.30/0.58 ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 0.30/0.58 => ( A = bot_bot_set_real ) ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum_uniqueI
% 0.30/0.58 thf(fact_4468_bot_Oextremum__uniqueI,axiom,
% 0.30/0.58 ! [A: set_int] :
% 0.30/0.58 ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 0.30/0.58 => ( A = bot_bot_set_int ) ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum_uniqueI
% 0.30/0.58 thf(fact_4469_bot_Oextremum__uniqueI,axiom,
% 0.30/0.58 ! [A: nat] :
% 0.30/0.58 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 0.30/0.58 => ( A = bot_bot_nat ) ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum_uniqueI
% 0.30/0.58 thf(fact_4470_bot_Oextremum__unique,axiom,
% 0.30/0.58 ! [A: set_nat] :
% 0.30/0.58 ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 0.30/0.58 = ( A = bot_bot_set_nat ) ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum_unique
% 0.30/0.58 thf(fact_4471_bot_Oextremum__unique,axiom,
% 0.30/0.58 ! [A: extended_enat] :
% 0.30/0.58 ( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
% 0.30/0.58 = ( A = bot_bo4199563552545308370d_enat ) ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum_unique
% 0.30/0.58 thf(fact_4472_bot_Oextremum__unique,axiom,
% 0.30/0.58 ! [A: set_real] :
% 0.30/0.58 ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 0.30/0.58 = ( A = bot_bot_set_real ) ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum_unique
% 0.30/0.58 thf(fact_4473_bot_Oextremum__unique,axiom,
% 0.30/0.58 ! [A: set_int] :
% 0.30/0.58 ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 0.30/0.58 = ( A = bot_bot_set_int ) ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum_unique
% 0.30/0.58 thf(fact_4474_bot_Oextremum__unique,axiom,
% 0.30/0.58 ! [A: nat] :
% 0.30/0.58 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 0.30/0.58 = ( A = bot_bot_nat ) ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum_unique
% 0.30/0.58 thf(fact_4475_bot_Oextremum,axiom,
% 0.30/0.58 ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum
% 0.30/0.58 thf(fact_4476_bot_Oextremum,axiom,
% 0.30/0.58 ! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ bot_bo4199563552545308370d_enat @ A ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum
% 0.30/0.58 thf(fact_4477_bot_Oextremum,axiom,
% 0.30/0.58 ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum
% 0.30/0.58 thf(fact_4478_bot_Oextremum,axiom,
% 0.30/0.58 ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum
% 0.30/0.58 thf(fact_4479_bot_Oextremum,axiom,
% 0.30/0.58 ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum
% 0.30/0.58 thf(fact_4480_bot_Oextremum__strict,axiom,
% 0.30/0.58 ! [A: set_nat] :
% 0.30/0.58 ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum_strict
% 0.30/0.58 thf(fact_4481_bot_Oextremum__strict,axiom,
% 0.30/0.58 ! [A: extended_enat] :
% 0.30/0.58 ~ ( ord_le72135733267957522d_enat @ A @ bot_bo4199563552545308370d_enat ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum_strict
% 0.30/0.58 thf(fact_4482_bot_Oextremum__strict,axiom,
% 0.30/0.58 ! [A: set_int] :
% 0.30/0.58 ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum_strict
% 0.30/0.58 thf(fact_4483_bot_Oextremum__strict,axiom,
% 0.30/0.58 ! [A: set_real] :
% 0.30/0.58 ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum_strict
% 0.30/0.58 thf(fact_4484_bot_Oextremum__strict,axiom,
% 0.30/0.58 ! [A: nat] :
% 0.30/0.58 ~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% 0.30/0.58
% 0.30/0.58 % bot.extremum_strict
% 0.30/0.58 thf(fact_4485_bot_Onot__eq__extremum,axiom,
% 0.30/0.58 ! [A: set_nat] :
% 0.30/0.58 ( ( A != bot_bot_set_nat )
% 0.30/0.58 = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % bot.not_eq_extremum
% 0.30/0.58 thf(fact_4486_bot_Onot__eq__extremum,axiom,
% 0.30/0.58 ! [A: extended_enat] :
% 0.30/0.58 ( ( A != bot_bo4199563552545308370d_enat )
% 0.30/0.58 = ( ord_le72135733267957522d_enat @ bot_bo4199563552545308370d_enat @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % bot.not_eq_extremum
% 0.30/0.58 thf(fact_4487_bot_Onot__eq__extremum,axiom,
% 0.30/0.58 ! [A: set_int] :
% 0.30/0.58 ( ( A != bot_bot_set_int )
% 0.30/0.58 = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % bot.not_eq_extremum
% 0.30/0.58 thf(fact_4488_bot_Onot__eq__extremum,axiom,
% 0.30/0.58 ! [A: set_real] :
% 0.30/0.58 ( ( A != bot_bot_set_real )
% 0.30/0.58 = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % bot.not_eq_extremum
% 0.30/0.58 thf(fact_4489_bot_Onot__eq__extremum,axiom,
% 0.30/0.58 ! [A: nat] :
% 0.30/0.58 ( ( A != bot_bot_nat )
% 0.30/0.58 = ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % bot.not_eq_extremum
% 0.30/0.58 thf(fact_4490_VEBT__internal_Oheight_Ocases,axiom,
% 0.30/0.58 ! [X2: vEBT_VEBT] :
% 0.30/0.58 ( ! [A4: $o,B4: $o] :
% 0.30/0.58 ( X2
% 0.30/0.58 != ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.58 => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.58 ( X2
% 0.30/0.58 != ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % VEBT_internal.height.cases
% 0.30/0.58 thf(fact_4491_vebt__insert_Osimps_I5_J,axiom,
% 0.30/0.58 ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 0.30/0.58 ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 0.30/0.58 = ( if_VEBT_VEBT
% 0.30/0.58 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.30/0.58 & ~ ( ( X2 = Mi )
% 0.30/0.58 | ( X2 = Ma ) ) )
% 0.30/0.58 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ X2 @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
% 0.30/0.58 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % vebt_insert.simps(5)
% 0.30/0.58 thf(fact_4492_vebt__insert_Oelims,axiom,
% 0.30/0.58 ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 0.30/0.58 ( ( ( vEBT_vebt_insert @ X2 @ Xa2 )
% 0.30/0.58 = Y3 )
% 0.30/0.58 => ( ! [A4: $o,B4: $o] :
% 0.30/0.58 ( ( X2
% 0.30/0.58 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.58 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.58 => ( Y3
% 0.30/0.58 = ( vEBT_Leaf @ $true @ B4 ) ) )
% 0.30/0.58 & ( ( Xa2 != zero_zero_nat )
% 0.30/0.58 => ( ( ( Xa2 = one_one_nat )
% 0.30/0.58 => ( Y3
% 0.30/0.58 = ( vEBT_Leaf @ A4 @ $true ) ) )
% 0.30/0.58 & ( ( Xa2 != one_one_nat )
% 0.30/0.58 => ( Y3
% 0.30/0.58 = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ) ) )
% 0.30/0.58 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.58 ( ( X2
% 0.30/0.58 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 0.30/0.58 => ( Y3
% 0.30/0.58 != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) ) )
% 0.30/0.58 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.58 ( ( X2
% 0.30/0.58 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 0.30/0.58 => ( Y3
% 0.30/0.58 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) ) )
% 0.30/0.58 => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.58 ( ( X2
% 0.30/0.58 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.58 => ( Y3
% 0.30/0.58 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) ) )
% 0.30/0.58 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.58 ( ( X2
% 0.30/0.58 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.58 => ( Y3
% 0.30/0.58 != ( if_VEBT_VEBT
% 0.30/0.58 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.58 & ~ ( ( Xa2 = Mi2 )
% 0.30/0.58 | ( Xa2 = Ma2 ) ) )
% 0.30/0.58 @ ( 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 @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 0.30/0.58 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % vebt_insert.elims
% 0.30/0.58 thf(fact_4493_vebt__insert_Opelims,axiom,
% 0.30/0.58 ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 0.30/0.58 ( ( ( vEBT_vebt_insert @ X2 @ Xa2 )
% 0.30/0.58 = Y3 )
% 0.30/0.58 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.30/0.58 => ( ! [A4: $o,B4: $o] :
% 0.30/0.58 ( ( X2
% 0.30/0.58 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.58 => ( ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.58 => ( Y3
% 0.30/0.58 = ( vEBT_Leaf @ $true @ B4 ) ) )
% 0.30/0.58 & ( ( Xa2 != zero_zero_nat )
% 0.30/0.58 => ( ( ( Xa2 = one_one_nat )
% 0.30/0.58 => ( Y3
% 0.30/0.58 = ( vEBT_Leaf @ A4 @ $true ) ) )
% 0.30/0.58 & ( ( Xa2 != one_one_nat )
% 0.30/0.58 => ( Y3
% 0.30/0.58 = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ) )
% 0.30/0.58 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
% 0.30/0.58 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.58 ( ( X2
% 0.30/0.58 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 0.30/0.58 => ( ( Y3
% 0.30/0.58 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 0.30/0.58 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) @ Xa2 ) ) ) )
% 0.30/0.58 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.58 ( ( X2
% 0.30/0.58 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 0.30/0.58 => ( ( Y3
% 0.30/0.58 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 0.30/0.58 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) @ Xa2 ) ) ) )
% 0.30/0.58 => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.58 ( ( X2
% 0.30/0.58 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.58 => ( ( Y3
% 0.30/0.58 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.58 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
% 0.30/0.58 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.58 ( ( X2
% 0.30/0.58 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 0.30/0.58 => ( ( Y3
% 0.30/0.58 = ( if_VEBT_VEBT
% 0.30/0.58 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.58 & ~ ( ( Xa2 = Mi2 )
% 0.30/0.58 | ( Xa2 = Ma2 ) ) )
% 0.30/0.58 @ ( 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 @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 0.30/0.58 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
% 0.30/0.58 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % vebt_insert.pelims
% 0.30/0.58 thf(fact_4494_gcd__nat__induct,axiom,
% 0.30/0.58 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.30/0.58 ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
% 0.30/0.58 => ( ! [M4: nat,N2: nat] :
% 0.30/0.58 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.58 => ( ( P @ N2 @ ( modulo_modulo_nat @ M4 @ N2 ) )
% 0.30/0.58 => ( P @ M4 @ N2 ) ) )
% 0.30/0.58 => ( P @ M @ N ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % gcd_nat_induct
% 0.30/0.58 thf(fact_4495_concat__bit__Suc,axiom,
% 0.30/0.58 ! [N: nat,K: int,L: int] :
% 0.30/0.58 ( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
% 0.30/0.58 = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % concat_bit_Suc
% 0.30/0.58 thf(fact_4496_dbl__simps_I3_J,axiom,
% 0.30/0.58 ( ( neg_nu7009210354673126013omplex @ one_one_complex )
% 0.30/0.58 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dbl_simps(3)
% 0.30/0.58 thf(fact_4497_dbl__simps_I3_J,axiom,
% 0.30/0.58 ( ( neg_numeral_dbl_real @ one_one_real )
% 0.30/0.58 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dbl_simps(3)
% 0.30/0.58 thf(fact_4498_dbl__simps_I3_J,axiom,
% 0.30/0.58 ( ( neg_numeral_dbl_rat @ one_one_rat )
% 0.30/0.58 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dbl_simps(3)
% 0.30/0.58 thf(fact_4499_dbl__simps_I3_J,axiom,
% 0.30/0.58 ( ( neg_numeral_dbl_int @ one_one_int )
% 0.30/0.58 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dbl_simps(3)
% 0.30/0.58 thf(fact_4500_even__succ__mod__exp,axiom,
% 0.30/0.58 ! [A: nat,N: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.58 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.58 = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_succ_mod_exp
% 0.30/0.58 thf(fact_4501_even__succ__mod__exp,axiom,
% 0.30/0.58 ! [A: int,N: nat] :
% 0.30/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.58 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.58 = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_succ_mod_exp
% 0.30/0.58 thf(fact_4502_even__succ__mod__exp,axiom,
% 0.30/0.58 ! [A: code_integer,N: nat] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.58 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.58 = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_succ_mod_exp
% 0.30/0.58 thf(fact_4503_nat__dvd__1__iff__1,axiom,
% 0.30/0.58 ! [M: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ M @ one_one_nat )
% 0.30/0.58 = ( M = one_one_nat ) ) ).
% 0.30/0.58
% 0.30/0.58 % nat_dvd_1_iff_1
% 0.30/0.58 thf(fact_4504_max__enat__simps_I3_J,axiom,
% 0.30/0.58 ! [Q3: extended_enat] :
% 0.30/0.58 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q3 )
% 0.30/0.58 = Q3 ) ).
% 0.30/0.58
% 0.30/0.58 % max_enat_simps(3)
% 0.30/0.58 thf(fact_4505_max__enat__simps_I2_J,axiom,
% 0.30/0.58 ! [Q3: extended_enat] :
% 0.30/0.58 ( ( ord_ma741700101516333627d_enat @ Q3 @ zero_z5237406670263579293d_enat )
% 0.30/0.58 = Q3 ) ).
% 0.30/0.58
% 0.30/0.58 % max_enat_simps(2)
% 0.30/0.58 thf(fact_4506_dvd__0__right,axiom,
% 0.30/0.58 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_0_right
% 0.30/0.58 thf(fact_4507_dvd__0__right,axiom,
% 0.30/0.58 ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_0_right
% 0.30/0.58 thf(fact_4508_dvd__0__right,axiom,
% 0.30/0.58 ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_0_right
% 0.30/0.58 thf(fact_4509_dvd__0__right,axiom,
% 0.30/0.58 ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_0_right
% 0.30/0.58 thf(fact_4510_dvd__0__right,axiom,
% 0.30/0.58 ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_0_right
% 0.30/0.58 thf(fact_4511_dvd__0__right,axiom,
% 0.30/0.58 ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_0_right
% 0.30/0.58 thf(fact_4512_dvd__0__left__iff,axiom,
% 0.30/0.58 ! [A: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 0.30/0.58 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_0_left_iff
% 0.30/0.58 thf(fact_4513_dvd__0__left__iff,axiom,
% 0.30/0.58 ! [A: complex] :
% 0.30/0.58 ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 0.30/0.58 = ( A = zero_zero_complex ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_0_left_iff
% 0.30/0.58 thf(fact_4514_dvd__0__left__iff,axiom,
% 0.30/0.58 ! [A: real] :
% 0.30/0.58 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 0.30/0.58 = ( A = zero_zero_real ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_0_left_iff
% 0.30/0.58 thf(fact_4515_dvd__0__left__iff,axiom,
% 0.30/0.58 ! [A: rat] :
% 0.30/0.58 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 0.30/0.58 = ( A = zero_zero_rat ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_0_left_iff
% 0.30/0.58 thf(fact_4516_dvd__0__left__iff,axiom,
% 0.30/0.58 ! [A: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.30/0.58 = ( A = zero_zero_nat ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_0_left_iff
% 0.30/0.58 thf(fact_4517_dvd__0__left__iff,axiom,
% 0.30/0.58 ! [A: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 0.30/0.58 = ( A = zero_zero_int ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_0_left_iff
% 0.30/0.58 thf(fact_4518_dvd__1__left,axiom,
% 0.30/0.58 ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_1_left
% 0.30/0.58 thf(fact_4519_dvd__1__iff__1,axiom,
% 0.30/0.58 ! [M: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.30/0.58 = ( M
% 0.30/0.58 = ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_1_iff_1
% 0.30/0.58 thf(fact_4520_dvd__add__triv__left__iff,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.30/0.58 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_triv_left_iff
% 0.30/0.58 thf(fact_4521_dvd__add__triv__left__iff,axiom,
% 0.30/0.58 ! [A: real,B: real] :
% 0.30/0.58 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.30/0.58 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_triv_left_iff
% 0.30/0.58 thf(fact_4522_dvd__add__triv__left__iff,axiom,
% 0.30/0.58 ! [A: rat,B: rat] :
% 0.30/0.58 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.30/0.58 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_triv_left_iff
% 0.30/0.58 thf(fact_4523_dvd__add__triv__left__iff,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.30/0.58 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_triv_left_iff
% 0.30/0.58 thf(fact_4524_dvd__add__triv__left__iff,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.30/0.58 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_triv_left_iff
% 0.30/0.58 thf(fact_4525_dvd__add__triv__right__iff,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
% 0.30/0.58 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_triv_right_iff
% 0.30/0.58 thf(fact_4526_dvd__add__triv__right__iff,axiom,
% 0.30/0.58 ! [A: real,B: real] :
% 0.30/0.58 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.30/0.58 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_triv_right_iff
% 0.30/0.58 thf(fact_4527_dvd__add__triv__right__iff,axiom,
% 0.30/0.58 ! [A: rat,B: rat] :
% 0.30/0.58 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.30/0.58 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_triv_right_iff
% 0.30/0.58 thf(fact_4528_dvd__add__triv__right__iff,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.30/0.58 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_triv_right_iff
% 0.30/0.58 thf(fact_4529_dvd__add__triv__right__iff,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.30/0.58 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_triv_right_iff
% 0.30/0.58 thf(fact_4530_div__dvd__div,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.58 => ( ( dvd_dvd_Code_integer @ A @ C )
% 0.30/0.58 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
% 0.30/0.58 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % div_dvd_div
% 0.30/0.58 thf(fact_4531_div__dvd__div,axiom,
% 0.30/0.58 ! [A: nat,B: nat,C: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58 => ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.58 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
% 0.30/0.58 = ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % div_dvd_div
% 0.30/0.58 thf(fact_4532_div__dvd__div,axiom,
% 0.30/0.58 ! [A: int,B: int,C: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ A @ B )
% 0.30/0.58 => ( ( dvd_dvd_int @ A @ C )
% 0.30/0.58 => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
% 0.30/0.58 = ( dvd_dvd_int @ B @ C ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % div_dvd_div
% 0.30/0.58 thf(fact_4533_nat__mult__dvd__cancel__disj,axiom,
% 0.30/0.58 ! [K: nat,M: nat,N: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.58 = ( ( K = zero_zero_nat )
% 0.30/0.58 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % nat_mult_dvd_cancel_disj
% 0.30/0.58 thf(fact_4534_concat__bit__0,axiom,
% 0.30/0.58 ! [K: int,L: int] :
% 0.30/0.58 ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
% 0.30/0.58 = L ) ).
% 0.30/0.58
% 0.30/0.58 % concat_bit_0
% 0.30/0.58 thf(fact_4535_dbl__simps_I2_J,axiom,
% 0.30/0.58 ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
% 0.30/0.58 = zero_zero_complex ) ).
% 0.30/0.58
% 0.30/0.58 % dbl_simps(2)
% 0.30/0.58 thf(fact_4536_dbl__simps_I2_J,axiom,
% 0.30/0.58 ( ( neg_numeral_dbl_real @ zero_zero_real )
% 0.30/0.58 = zero_zero_real ) ).
% 0.30/0.58
% 0.30/0.58 % dbl_simps(2)
% 0.30/0.58 thf(fact_4537_dbl__simps_I2_J,axiom,
% 0.30/0.58 ( ( neg_numeral_dbl_rat @ zero_zero_rat )
% 0.30/0.58 = zero_zero_rat ) ).
% 0.30/0.58
% 0.30/0.58 % dbl_simps(2)
% 0.30/0.58 thf(fact_4538_dbl__simps_I2_J,axiom,
% 0.30/0.58 ( ( neg_numeral_dbl_int @ zero_zero_int )
% 0.30/0.58 = zero_zero_int ) ).
% 0.30/0.58
% 0.30/0.58 % dbl_simps(2)
% 0.30/0.58 thf(fact_4539_dvd__mult__cancel__left,axiom,
% 0.30/0.58 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.30/0.58 = ( ( C = zero_z3403309356797280102nteger )
% 0.30/0.58 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_cancel_left
% 0.30/0.58 thf(fact_4540_dvd__mult__cancel__left,axiom,
% 0.30/0.58 ! [C: complex,A: complex,B: complex] :
% 0.30/0.58 ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.30/0.58 = ( ( C = zero_zero_complex )
% 0.30/0.58 | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_cancel_left
% 0.30/0.58 thf(fact_4541_dvd__mult__cancel__left,axiom,
% 0.30/0.58 ! [C: real,A: real,B: real] :
% 0.30/0.58 ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.58 = ( ( C = zero_zero_real )
% 0.30/0.58 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_cancel_left
% 0.30/0.58 thf(fact_4542_dvd__mult__cancel__left,axiom,
% 0.30/0.58 ! [C: rat,A: rat,B: rat] :
% 0.30/0.58 ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.58 = ( ( C = zero_zero_rat )
% 0.30/0.58 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_cancel_left
% 0.30/0.58 thf(fact_4543_dvd__mult__cancel__left,axiom,
% 0.30/0.58 ! [C: int,A: int,B: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.58 = ( ( C = zero_zero_int )
% 0.30/0.58 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_cancel_left
% 0.30/0.58 thf(fact_4544_dvd__mult__cancel__right,axiom,
% 0.30/0.58 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.30/0.58 = ( ( C = zero_z3403309356797280102nteger )
% 0.30/0.58 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_cancel_right
% 0.30/0.58 thf(fact_4545_dvd__mult__cancel__right,axiom,
% 0.30/0.58 ! [A: complex,C: complex,B: complex] :
% 0.30/0.58 ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 0.30/0.58 = ( ( C = zero_zero_complex )
% 0.30/0.58 | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_cancel_right
% 0.30/0.58 thf(fact_4546_dvd__mult__cancel__right,axiom,
% 0.30/0.58 ! [A: real,C: real,B: real] :
% 0.30/0.58 ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.30/0.58 = ( ( C = zero_zero_real )
% 0.30/0.58 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_cancel_right
% 0.30/0.58 thf(fact_4547_dvd__mult__cancel__right,axiom,
% 0.30/0.58 ! [A: rat,C: rat,B: rat] :
% 0.30/0.58 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.30/0.58 = ( ( C = zero_zero_rat )
% 0.30/0.58 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_cancel_right
% 0.30/0.58 thf(fact_4548_dvd__mult__cancel__right,axiom,
% 0.30/0.58 ! [A: int,C: int,B: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.30/0.58 = ( ( C = zero_zero_int )
% 0.30/0.58 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_cancel_right
% 0.30/0.58 thf(fact_4549_dvd__times__left__cancel__iff,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.58 ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.58 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 0.30/0.58 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_times_left_cancel_iff
% 0.30/0.58 thf(fact_4550_dvd__times__left__cancel__iff,axiom,
% 0.30/0.58 ! [A: nat,B: nat,C: nat] :
% 0.30/0.58 ( ( A != zero_zero_nat )
% 0.30/0.58 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
% 0.30/0.58 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_times_left_cancel_iff
% 0.30/0.58 thf(fact_4551_dvd__times__left__cancel__iff,axiom,
% 0.30/0.58 ! [A: int,B: int,C: int] :
% 0.30/0.58 ( ( A != zero_zero_int )
% 0.30/0.58 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
% 0.30/0.58 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_times_left_cancel_iff
% 0.30/0.58 thf(fact_4552_dvd__times__right__cancel__iff,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.58 ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.58 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 0.30/0.58 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_times_right_cancel_iff
% 0.30/0.58 thf(fact_4553_dvd__times__right__cancel__iff,axiom,
% 0.30/0.58 ! [A: nat,B: nat,C: nat] :
% 0.30/0.58 ( ( A != zero_zero_nat )
% 0.30/0.58 => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
% 0.30/0.58 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_times_right_cancel_iff
% 0.30/0.58 thf(fact_4554_dvd__times__right__cancel__iff,axiom,
% 0.30/0.58 ! [A: int,B: int,C: int] :
% 0.30/0.58 ( ( A != zero_zero_int )
% 0.30/0.58 => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
% 0.30/0.58 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_times_right_cancel_iff
% 0.30/0.58 thf(fact_4555_unit__prod,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.58 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.58 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_prod
% 0.30/0.58 thf(fact_4556_unit__prod,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.58 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.58 => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_prod
% 0.30/0.58 thf(fact_4557_unit__prod,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.58 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.58 => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_prod
% 0.30/0.58 thf(fact_4558_dvd__add__times__triv__right__iff,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
% 0.30/0.58 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_times_triv_right_iff
% 0.30/0.58 thf(fact_4559_dvd__add__times__triv__right__iff,axiom,
% 0.30/0.58 ! [A: real,B: real,C: real] :
% 0.30/0.58 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
% 0.30/0.58 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_times_triv_right_iff
% 0.30/0.58 thf(fact_4560_dvd__add__times__triv__right__iff,axiom,
% 0.30/0.58 ! [A: rat,B: rat,C: rat] :
% 0.30/0.58 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
% 0.30/0.58 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_times_triv_right_iff
% 0.30/0.58 thf(fact_4561_dvd__add__times__triv__right__iff,axiom,
% 0.30/0.58 ! [A: nat,B: nat,C: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
% 0.30/0.58 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_times_triv_right_iff
% 0.30/0.58 thf(fact_4562_dvd__add__times__triv__right__iff,axiom,
% 0.30/0.58 ! [A: int,B: int,C: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
% 0.30/0.58 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_times_triv_right_iff
% 0.30/0.58 thf(fact_4563_dvd__add__times__triv__left__iff,axiom,
% 0.30/0.58 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
% 0.30/0.58 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_times_triv_left_iff
% 0.30/0.58 thf(fact_4564_dvd__add__times__triv__left__iff,axiom,
% 0.30/0.58 ! [A: real,C: real,B: real] :
% 0.30/0.58 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
% 0.30/0.58 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_times_triv_left_iff
% 0.30/0.58 thf(fact_4565_dvd__add__times__triv__left__iff,axiom,
% 0.30/0.58 ! [A: rat,C: rat,B: rat] :
% 0.30/0.58 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
% 0.30/0.58 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_times_triv_left_iff
% 0.30/0.58 thf(fact_4566_dvd__add__times__triv__left__iff,axiom,
% 0.30/0.58 ! [A: nat,C: nat,B: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
% 0.30/0.58 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_times_triv_left_iff
% 0.30/0.58 thf(fact_4567_dvd__add__times__triv__left__iff,axiom,
% 0.30/0.58 ! [A: int,C: int,B: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
% 0.30/0.58 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_add_times_triv_left_iff
% 0.30/0.58 thf(fact_4568_dvd__mult__div__cancel,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.58 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
% 0.30/0.58 = B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_div_cancel
% 0.30/0.58 thf(fact_4569_dvd__mult__div__cancel,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
% 0.30/0.58 = B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_div_cancel
% 0.30/0.58 thf(fact_4570_dvd__mult__div__cancel,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ A @ B )
% 0.30/0.58 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
% 0.30/0.58 = B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_div_cancel
% 0.30/0.58 thf(fact_4571_dvd__div__mult__self,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.58 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 0.30/0.58 = B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_div_mult_self
% 0.30/0.58 thf(fact_4572_dvd__div__mult__self,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 0.30/0.58 = B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_div_mult_self
% 0.30/0.58 thf(fact_4573_dvd__div__mult__self,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ A @ B )
% 0.30/0.58 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 0.30/0.58 = B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_div_mult_self
% 0.30/0.58 thf(fact_4574_unit__div,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.58 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.58 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_div
% 0.30/0.58 thf(fact_4575_unit__div,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.58 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.58 => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_div
% 0.30/0.58 thf(fact_4576_unit__div,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.58 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.58 => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_div
% 0.30/0.58 thf(fact_4577_unit__div__1__unit,axiom,
% 0.30/0.58 ! [A: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.58 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_div_1_unit
% 0.30/0.58 thf(fact_4578_unit__div__1__unit,axiom,
% 0.30/0.58 ! [A: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.58 => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_div_1_unit
% 0.30/0.58 thf(fact_4579_unit__div__1__unit,axiom,
% 0.30/0.58 ! [A: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.58 => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_div_1_unit
% 0.30/0.58 thf(fact_4580_unit__div__1__div__1,axiom,
% 0.30/0.58 ! [A: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.58 => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 0.30/0.58 = A ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_div_1_div_1
% 0.30/0.58 thf(fact_4581_unit__div__1__div__1,axiom,
% 0.30/0.58 ! [A: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.58 => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
% 0.30/0.58 = A ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_div_1_div_1
% 0.30/0.58 thf(fact_4582_unit__div__1__div__1,axiom,
% 0.30/0.58 ! [A: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.58 => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
% 0.30/0.58 = A ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_div_1_div_1
% 0.30/0.58 thf(fact_4583_div__add,axiom,
% 0.30/0.58 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.30/0.58 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.58 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.30/0.58 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % div_add
% 0.30/0.58 thf(fact_4584_div__add,axiom,
% 0.30/0.58 ! [C: nat,A: nat,B: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ C @ A )
% 0.30/0.58 => ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.58 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.58 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % div_add
% 0.30/0.58 thf(fact_4585_div__add,axiom,
% 0.30/0.58 ! [C: int,A: int,B: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ C @ A )
% 0.30/0.58 => ( ( dvd_dvd_int @ C @ B )
% 0.30/0.58 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.58 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % div_add
% 0.30/0.58 thf(fact_4586_div__diff,axiom,
% 0.30/0.58 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.30/0.58 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.58 => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 0.30/0.58 = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % div_diff
% 0.30/0.58 thf(fact_4587_div__diff,axiom,
% 0.30/0.58 ! [C: int,A: int,B: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ C @ A )
% 0.30/0.58 => ( ( dvd_dvd_int @ C @ B )
% 0.30/0.58 => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.58 = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % div_diff
% 0.30/0.58 thf(fact_4588_dvd__imp__mod__0,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58 => ( ( modulo_modulo_nat @ B @ A )
% 0.30/0.58 = zero_zero_nat ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_imp_mod_0
% 0.30/0.58 thf(fact_4589_dvd__imp__mod__0,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ A @ B )
% 0.30/0.58 => ( ( modulo_modulo_int @ B @ A )
% 0.30/0.58 = zero_zero_int ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_imp_mod_0
% 0.30/0.58 thf(fact_4590_dvd__imp__mod__0,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.58 => ( ( modulo364778990260209775nteger @ B @ A )
% 0.30/0.58 = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_imp_mod_0
% 0.30/0.58 thf(fact_4591_concat__bit__nonnegative__iff,axiom,
% 0.30/0.58 ! [N: nat,K: int,L: int] :
% 0.30/0.58 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
% 0.30/0.58 = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).
% 0.30/0.58
% 0.30/0.58 % concat_bit_nonnegative_iff
% 0.30/0.58 thf(fact_4592_concat__bit__negative__iff,axiom,
% 0.30/0.58 ! [N: nat,K: int,L: int] :
% 0.30/0.58 ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
% 0.30/0.58 = ( ord_less_int @ L @ zero_zero_int ) ) ).
% 0.30/0.58
% 0.30/0.58 % concat_bit_negative_iff
% 0.30/0.58 thf(fact_4593_dbl__simps_I5_J,axiom,
% 0.30/0.58 ! [K: num] :
% 0.30/0.58 ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.30/0.58 = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dbl_simps(5)
% 0.30/0.58 thf(fact_4594_dbl__simps_I5_J,axiom,
% 0.30/0.58 ! [K: num] :
% 0.30/0.58 ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
% 0.30/0.58 = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dbl_simps(5)
% 0.30/0.58 thf(fact_4595_dbl__simps_I5_J,axiom,
% 0.30/0.58 ! [K: num] :
% 0.30/0.58 ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
% 0.30/0.58 = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dbl_simps(5)
% 0.30/0.58 thf(fact_4596_dbl__simps_I5_J,axiom,
% 0.30/0.58 ! [K: num] :
% 0.30/0.58 ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
% 0.30/0.58 = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dbl_simps(5)
% 0.30/0.58 thf(fact_4597_even__Suc__Suc__iff,axiom,
% 0.30/0.58 ! [N: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
% 0.30/0.58 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_Suc_Suc_iff
% 0.30/0.58 thf(fact_4598_even__Suc,axiom,
% 0.30/0.58 ! [N: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
% 0.30/0.58 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_Suc
% 0.30/0.58 thf(fact_4599_unit__div__mult__self,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.58 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 0.30/0.58 = B ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_div_mult_self
% 0.30/0.58 thf(fact_4600_unit__div__mult__self,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.58 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 0.30/0.58 = B ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_div_mult_self
% 0.30/0.58 thf(fact_4601_unit__div__mult__self,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.58 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 0.30/0.58 = B ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_div_mult_self
% 0.30/0.58 thf(fact_4602_unit__mult__div__div,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.58 => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 0.30/0.58 = ( divide6298287555418463151nteger @ B @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_mult_div_div
% 0.30/0.58 thf(fact_4603_unit__mult__div__div,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.58 => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
% 0.30/0.58 = ( divide_divide_nat @ B @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_mult_div_div
% 0.30/0.58 thf(fact_4604_unit__mult__div__div,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.58 => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
% 0.30/0.58 = ( divide_divide_int @ B @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_mult_div_div
% 0.30/0.58 thf(fact_4605_pow__divides__pow__iff,axiom,
% 0.30/0.58 ! [N: nat,A: nat,B: nat] :
% 0.30/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.58 => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.30/0.58 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % pow_divides_pow_iff
% 0.30/0.58 thf(fact_4606_pow__divides__pow__iff,axiom,
% 0.30/0.58 ! [N: nat,A: int,B: int] :
% 0.30/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.58 => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.30/0.58 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % pow_divides_pow_iff
% 0.30/0.58 thf(fact_4607_even__mult__iff,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.30/0.58 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_mult_iff
% 0.30/0.58 thf(fact_4608_even__mult__iff,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
% 0.30/0.58 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_mult_iff
% 0.30/0.58 thf(fact_4609_even__mult__iff,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
% 0.30/0.58 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_mult_iff
% 0.30/0.58 thf(fact_4610_even__add,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.30/0.58 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_add
% 0.30/0.58 thf(fact_4611_even__add,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
% 0.30/0.58 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_add
% 0.30/0.58 thf(fact_4612_even__add,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
% 0.30/0.58 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_add
% 0.30/0.58 thf(fact_4613_odd__add,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer] :
% 0.30/0.58 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
% 0.30/0.58 = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.58 != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % odd_add
% 0.30/0.58 thf(fact_4614_odd__add,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
% 0.30/0.58 = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.58 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % odd_add
% 0.30/0.58 thf(fact_4615_odd__add,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
% 0.30/0.58 = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.58 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % odd_add
% 0.30/0.58 thf(fact_4616_even__mod__2__iff,axiom,
% 0.30/0.58 ! [A: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.58 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_mod_2_iff
% 0.30/0.58 thf(fact_4617_even__mod__2__iff,axiom,
% 0.30/0.58 ! [A: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.30/0.58 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_mod_2_iff
% 0.30/0.58 thf(fact_4618_even__mod__2__iff,axiom,
% 0.30/0.58 ! [A: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.30/0.58 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_mod_2_iff
% 0.30/0.58 thf(fact_4619_even__Suc__div__two,axiom,
% 0.30/0.58 ! [N: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.58 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.58 = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_Suc_div_two
% 0.30/0.58 thf(fact_4620_odd__Suc__div__two,axiom,
% 0.30/0.58 ! [N: nat] :
% 0.30/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.58 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.58 = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % odd_Suc_div_two
% 0.30/0.58 thf(fact_4621_dvd__numeral__simp,axiom,
% 0.30/0.58 ! [M: num,N: num] :
% 0.30/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.58 = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N @ M ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_numeral_simp
% 0.30/0.58 thf(fact_4622_dvd__numeral__simp,axiom,
% 0.30/0.58 ! [M: num,N: num] :
% 0.30/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.58 = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N @ M ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_numeral_simp
% 0.30/0.58 thf(fact_4623_dvd__numeral__simp,axiom,
% 0.30/0.58 ! [M: num,N: num] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
% 0.30/0.58 = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N @ M ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_numeral_simp
% 0.30/0.58 thf(fact_4624_zero__le__power__eq__numeral,axiom,
% 0.30/0.58 ! [A: real,W: num] :
% 0.30/0.58 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.30/0.58 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % zero_le_power_eq_numeral
% 0.30/0.58 thf(fact_4625_zero__le__power__eq__numeral,axiom,
% 0.30/0.58 ! [A: rat,W: num] :
% 0.30/0.58 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.30/0.58 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % zero_le_power_eq_numeral
% 0.30/0.58 thf(fact_4626_zero__le__power__eq__numeral,axiom,
% 0.30/0.58 ! [A: int,W: num] :
% 0.30/0.58 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.30/0.58 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % zero_le_power_eq_numeral
% 0.30/0.58 thf(fact_4627_power__less__zero__eq__numeral,axiom,
% 0.30/0.58 ! [A: real,W: num] :
% 0.30/0.58 ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 0.30/0.58 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % power_less_zero_eq_numeral
% 0.30/0.58 thf(fact_4628_power__less__zero__eq__numeral,axiom,
% 0.30/0.58 ! [A: rat,W: num] :
% 0.30/0.58 ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 0.30/0.58 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % power_less_zero_eq_numeral
% 0.30/0.58 thf(fact_4629_power__less__zero__eq__numeral,axiom,
% 0.30/0.58 ! [A: int,W: num] :
% 0.30/0.58 ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 0.30/0.58 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % power_less_zero_eq_numeral
% 0.30/0.58 thf(fact_4630_power__less__zero__eq,axiom,
% 0.30/0.58 ! [A: real,N: nat] :
% 0.30/0.58 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 0.30/0.58 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.58 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % power_less_zero_eq
% 0.30/0.58 thf(fact_4631_power__less__zero__eq,axiom,
% 0.30/0.58 ! [A: rat,N: nat] :
% 0.30/0.58 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 0.30/0.58 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.58 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % power_less_zero_eq
% 0.30/0.58 thf(fact_4632_power__less__zero__eq,axiom,
% 0.30/0.58 ! [A: int,N: nat] :
% 0.30/0.58 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 0.30/0.58 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.58 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % power_less_zero_eq
% 0.30/0.58 thf(fact_4633_even__plus__one__iff,axiom,
% 0.30/0.58 ! [A: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
% 0.30/0.58 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_plus_one_iff
% 0.30/0.58 thf(fact_4634_even__plus__one__iff,axiom,
% 0.30/0.58 ! [A: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
% 0.30/0.58 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_plus_one_iff
% 0.30/0.58 thf(fact_4635_even__plus__one__iff,axiom,
% 0.30/0.58 ! [A: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
% 0.30/0.58 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_plus_one_iff
% 0.30/0.58 thf(fact_4636_even__diff,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.30/0.58 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_diff
% 0.30/0.58 thf(fact_4637_even__diff,axiom,
% 0.30/0.58 ! [A: int,B: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
% 0.30/0.58 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_diff
% 0.30/0.58 thf(fact_4638_odd__Suc__minus__one,axiom,
% 0.30/0.58 ! [N: nat] :
% 0.30/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.58 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.30/0.58 = N ) ) ).
% 0.30/0.58
% 0.30/0.58 % odd_Suc_minus_one
% 0.30/0.58 thf(fact_4639_even__diff__nat,axiom,
% 0.30/0.58 ! [M: nat,N: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.58 = ( ( ord_less_nat @ M @ N )
% 0.30/0.58 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_diff_nat
% 0.30/0.58 thf(fact_4640_zero__less__power__eq__numeral,axiom,
% 0.30/0.58 ! [A: real,W: num] :
% 0.30/0.58 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.30/0.58 = ( ( ( numeral_numeral_nat @ W )
% 0.30/0.58 = zero_zero_nat )
% 0.30/0.58 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( A != zero_zero_real ) )
% 0.30/0.58 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % zero_less_power_eq_numeral
% 0.30/0.58 thf(fact_4641_zero__less__power__eq__numeral,axiom,
% 0.30/0.58 ! [A: rat,W: num] :
% 0.30/0.58 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.30/0.58 = ( ( ( numeral_numeral_nat @ W )
% 0.30/0.58 = zero_zero_nat )
% 0.30/0.58 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( A != zero_zero_rat ) )
% 0.30/0.58 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % zero_less_power_eq_numeral
% 0.30/0.58 thf(fact_4642_zero__less__power__eq__numeral,axiom,
% 0.30/0.58 ! [A: int,W: num] :
% 0.30/0.58 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.30/0.58 = ( ( ( numeral_numeral_nat @ W )
% 0.30/0.58 = zero_zero_nat )
% 0.30/0.58 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( A != zero_zero_int ) )
% 0.30/0.58 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % zero_less_power_eq_numeral
% 0.30/0.58 thf(fact_4643_odd__succ__div__two,axiom,
% 0.30/0.58 ! [A: code_integer] :
% 0.30/0.58 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.58 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % odd_succ_div_two
% 0.30/0.58 thf(fact_4644_odd__succ__div__two,axiom,
% 0.30/0.58 ! [A: nat] :
% 0.30/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.58 = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % odd_succ_div_two
% 0.30/0.58 thf(fact_4645_odd__succ__div__two,axiom,
% 0.30/0.58 ! [A: int] :
% 0.30/0.58 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.58 = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % odd_succ_div_two
% 0.30/0.58 thf(fact_4646_even__succ__div__two,axiom,
% 0.30/0.58 ! [A: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.58 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_succ_div_two
% 0.30/0.58 thf(fact_4647_even__succ__div__two,axiom,
% 0.30/0.58 ! [A: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.58 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_succ_div_two
% 0.30/0.58 thf(fact_4648_even__succ__div__two,axiom,
% 0.30/0.58 ! [A: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.58 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_succ_div_two
% 0.30/0.58 thf(fact_4649_even__succ__div__2,axiom,
% 0.30/0.58 ! [A: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.58 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_succ_div_2
% 0.30/0.58 thf(fact_4650_even__succ__div__2,axiom,
% 0.30/0.58 ! [A: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.58 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_succ_div_2
% 0.30/0.58 thf(fact_4651_even__succ__div__2,axiom,
% 0.30/0.58 ! [A: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.58 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_succ_div_2
% 0.30/0.58 thf(fact_4652_even__power,axiom,
% 0.30/0.58 ! [A: code_integer,N: nat] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 0.30/0.58 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_power
% 0.30/0.58 thf(fact_4653_even__power,axiom,
% 0.30/0.58 ! [A: nat,N: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
% 0.30/0.58 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_power
% 0.30/0.58 thf(fact_4654_even__power,axiom,
% 0.30/0.58 ! [A: int,N: nat] :
% 0.30/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
% 0.30/0.58 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_power
% 0.30/0.58 thf(fact_4655_odd__two__times__div__two__nat,axiom,
% 0.30/0.58 ! [N: nat] :
% 0.30/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.58 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.58 = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % odd_two_times_div_two_nat
% 0.30/0.58 thf(fact_4656_odd__two__times__div__two__succ,axiom,
% 0.30/0.58 ! [A: code_integer] :
% 0.30/0.58 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
% 0.30/0.58 = A ) ) ).
% 0.30/0.58
% 0.30/0.58 % odd_two_times_div_two_succ
% 0.30/0.58 thf(fact_4657_odd__two__times__div__two__succ,axiom,
% 0.30/0.58 ! [A: nat] :
% 0.30/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
% 0.30/0.58 = A ) ) ).
% 0.30/0.58
% 0.30/0.58 % odd_two_times_div_two_succ
% 0.30/0.58 thf(fact_4658_odd__two__times__div__two__succ,axiom,
% 0.30/0.58 ! [A: int] :
% 0.30/0.58 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
% 0.30/0.58 = A ) ) ).
% 0.30/0.58
% 0.30/0.58 % odd_two_times_div_two_succ
% 0.30/0.58 thf(fact_4659_power__le__zero__eq__numeral,axiom,
% 0.30/0.58 ! [A: real,W: num] :
% 0.30/0.58 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 0.30/0.58 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.30/0.58 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( A = zero_zero_real ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % power_le_zero_eq_numeral
% 0.30/0.58 thf(fact_4660_power__le__zero__eq__numeral,axiom,
% 0.30/0.58 ! [A: rat,W: num] :
% 0.30/0.58 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 0.30/0.58 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.30/0.58 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % power_le_zero_eq_numeral
% 0.30/0.58 thf(fact_4661_power__le__zero__eq__numeral,axiom,
% 0.30/0.58 ! [A: int,W: num] :
% 0.30/0.58 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 0.30/0.58 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.30/0.58 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.58 & ( A = zero_zero_int ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % power_le_zero_eq_numeral
% 0.30/0.58 thf(fact_4662_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.30/0.58 ! [N: nat] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
% 0.30/0.58 = ( N = zero_zero_nat ) ) ).
% 0.30/0.58
% 0.30/0.58 % semiring_parity_class.even_mask_iff
% 0.30/0.58 thf(fact_4663_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.30/0.58 ! [N: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
% 0.30/0.58 = ( N = zero_zero_nat ) ) ).
% 0.30/0.58
% 0.30/0.58 % semiring_parity_class.even_mask_iff
% 0.30/0.58 thf(fact_4664_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.30/0.58 ! [N: nat] :
% 0.30/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 0.30/0.58 = ( N = zero_zero_nat ) ) ).
% 0.30/0.58
% 0.30/0.58 % semiring_parity_class.even_mask_iff
% 0.30/0.58 thf(fact_4665_even__succ__div__exp,axiom,
% 0.30/0.58 ! [A: code_integer,N: nat] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.58 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.58 = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_succ_div_exp
% 0.30/0.58 thf(fact_4666_even__succ__div__exp,axiom,
% 0.30/0.58 ! [A: nat,N: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.58 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.58 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_succ_div_exp
% 0.30/0.58 thf(fact_4667_even__succ__div__exp,axiom,
% 0.30/0.58 ! [A: int,N: nat] :
% 0.30/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.58 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.58 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.58 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % even_succ_div_exp
% 0.30/0.58 thf(fact_4668_dvd__refl,axiom,
% 0.30/0.58 ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_refl
% 0.30/0.58 thf(fact_4669_dvd__refl,axiom,
% 0.30/0.58 ! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_refl
% 0.30/0.58 thf(fact_4670_dvd__refl,axiom,
% 0.30/0.58 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_refl
% 0.30/0.58 thf(fact_4671_dvd__trans,axiom,
% 0.30/0.58 ! [A: nat,B: nat,C: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58 => ( ( dvd_dvd_nat @ B @ C )
% 0.30/0.58 => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_trans
% 0.30/0.58 thf(fact_4672_dvd__trans,axiom,
% 0.30/0.58 ! [A: int,B: int,C: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ A @ B )
% 0.30/0.58 => ( ( dvd_dvd_int @ B @ C )
% 0.30/0.58 => ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_trans
% 0.30/0.58 thf(fact_4673_dvd__trans,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.58 => ( ( dvd_dvd_Code_integer @ B @ C )
% 0.30/0.58 => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_trans
% 0.30/0.58 thf(fact_4674_division__decomp,axiom,
% 0.30/0.58 ! [A: nat,B: nat,C: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.30/0.58 => ? [B7: nat,C5: nat] :
% 0.30/0.58 ( ( A
% 0.30/0.58 = ( times_times_nat @ B7 @ C5 ) )
% 0.30/0.58 & ( dvd_dvd_nat @ B7 @ B )
% 0.30/0.58 & ( dvd_dvd_nat @ C5 @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % division_decomp
% 0.30/0.58 thf(fact_4675_division__decomp,axiom,
% 0.30/0.58 ! [A: int,B: int,C: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 0.30/0.58 => ? [B7: int,C5: int] :
% 0.30/0.58 ( ( A
% 0.30/0.58 = ( times_times_int @ B7 @ C5 ) )
% 0.30/0.58 & ( dvd_dvd_int @ B7 @ B )
% 0.30/0.58 & ( dvd_dvd_int @ C5 @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % division_decomp
% 0.30/0.58 thf(fact_4676_dvd__productE,axiom,
% 0.30/0.58 ! [P6: nat,A: nat,B: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ P6 @ ( times_times_nat @ A @ B ) )
% 0.30/0.58 => ~ ! [X5: nat,Y4: nat] :
% 0.30/0.58 ( ( P6
% 0.30/0.58 = ( times_times_nat @ X5 @ Y4 ) )
% 0.30/0.58 => ( ( dvd_dvd_nat @ X5 @ A )
% 0.30/0.58 => ~ ( dvd_dvd_nat @ Y4 @ B ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_productE
% 0.30/0.58 thf(fact_4677_dvd__productE,axiom,
% 0.30/0.58 ! [P6: int,A: int,B: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ P6 @ ( times_times_int @ A @ B ) )
% 0.30/0.58 => ~ ! [X5: int,Y4: int] :
% 0.30/0.58 ( ( P6
% 0.30/0.58 = ( times_times_int @ X5 @ Y4 ) )
% 0.30/0.58 => ( ( dvd_dvd_int @ X5 @ A )
% 0.30/0.58 => ~ ( dvd_dvd_int @ Y4 @ B ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_productE
% 0.30/0.58 thf(fact_4678_bot__nat__def,axiom,
% 0.30/0.58 bot_bot_nat = zero_zero_nat ).
% 0.30/0.58
% 0.30/0.58 % bot_nat_def
% 0.30/0.58 thf(fact_4679_dvd__0__left,axiom,
% 0.30/0.58 ! [A: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 0.30/0.58 => ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_0_left
% 0.30/0.58 thf(fact_4680_dvd__0__left,axiom,
% 0.30/0.58 ! [A: complex] :
% 0.30/0.58 ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 0.30/0.58 => ( A = zero_zero_complex ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_0_left
% 0.30/0.58 thf(fact_4681_dvd__0__left,axiom,
% 0.30/0.58 ! [A: real] :
% 0.30/0.58 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 0.30/0.58 => ( A = zero_zero_real ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_0_left
% 0.30/0.58 thf(fact_4682_dvd__0__left,axiom,
% 0.30/0.58 ! [A: rat] :
% 0.30/0.58 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 0.30/0.58 => ( A = zero_zero_rat ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_0_left
% 0.30/0.58 thf(fact_4683_dvd__0__left,axiom,
% 0.30/0.58 ! [A: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.30/0.58 => ( A = zero_zero_nat ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_0_left
% 0.30/0.58 thf(fact_4684_dvd__0__left,axiom,
% 0.30/0.58 ! [A: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 0.30/0.58 => ( A = zero_zero_int ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_0_left
% 0.30/0.58 thf(fact_4685_dvd__triv__right,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_triv_right
% 0.30/0.58 thf(fact_4686_dvd__triv__right,axiom,
% 0.30/0.58 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_triv_right
% 0.30/0.58 thf(fact_4687_dvd__triv__right,axiom,
% 0.30/0.58 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_triv_right
% 0.30/0.58 thf(fact_4688_dvd__triv__right,axiom,
% 0.30/0.58 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_triv_right
% 0.30/0.58 thf(fact_4689_dvd__triv__right,axiom,
% 0.30/0.58 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_triv_right
% 0.30/0.58 thf(fact_4690_dvd__mult__right,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.30/0.58 => ( dvd_dvd_Code_integer @ B @ C ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_right
% 0.30/0.58 thf(fact_4691_dvd__mult__right,axiom,
% 0.30/0.58 ! [A: real,B: real,C: real] :
% 0.30/0.58 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 0.30/0.58 => ( dvd_dvd_real @ B @ C ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_right
% 0.30/0.58 thf(fact_4692_dvd__mult__right,axiom,
% 0.30/0.58 ! [A: rat,B: rat,C: rat] :
% 0.30/0.58 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.30/0.58 => ( dvd_dvd_rat @ B @ C ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_right
% 0.30/0.58 thf(fact_4693_dvd__mult__right,axiom,
% 0.30/0.58 ! [A: nat,B: nat,C: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.30/0.58 => ( dvd_dvd_nat @ B @ C ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_right
% 0.30/0.58 thf(fact_4694_dvd__mult__right,axiom,
% 0.30/0.58 ! [A: int,B: int,C: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.30/0.58 => ( dvd_dvd_int @ B @ C ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_right
% 0.30/0.58 thf(fact_4695_mult__dvd__mono,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.58 => ( ( dvd_dvd_Code_integer @ C @ D )
% 0.30/0.58 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % mult_dvd_mono
% 0.30/0.58 thf(fact_4696_mult__dvd__mono,axiom,
% 0.30/0.58 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.58 ( ( dvd_dvd_real @ A @ B )
% 0.30/0.58 => ( ( dvd_dvd_real @ C @ D )
% 0.30/0.58 => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % mult_dvd_mono
% 0.30/0.58 thf(fact_4697_mult__dvd__mono,axiom,
% 0.30/0.58 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.58 ( ( dvd_dvd_rat @ A @ B )
% 0.30/0.58 => ( ( dvd_dvd_rat @ C @ D )
% 0.30/0.58 => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % mult_dvd_mono
% 0.30/0.58 thf(fact_4698_mult__dvd__mono,axiom,
% 0.30/0.58 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58 => ( ( dvd_dvd_nat @ C @ D )
% 0.30/0.58 => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % mult_dvd_mono
% 0.30/0.58 thf(fact_4699_mult__dvd__mono,axiom,
% 0.30/0.58 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ A @ B )
% 0.30/0.58 => ( ( dvd_dvd_int @ C @ D )
% 0.30/0.58 => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % mult_dvd_mono
% 0.30/0.58 thf(fact_4700_dvd__triv__left,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_triv_left
% 0.30/0.58 thf(fact_4701_dvd__triv__left,axiom,
% 0.30/0.58 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_triv_left
% 0.30/0.58 thf(fact_4702_dvd__triv__left,axiom,
% 0.30/0.58 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_triv_left
% 0.30/0.58 thf(fact_4703_dvd__triv__left,axiom,
% 0.30/0.58 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_triv_left
% 0.30/0.58 thf(fact_4704_dvd__triv__left,axiom,
% 0.30/0.58 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_triv_left
% 0.30/0.58 thf(fact_4705_dvd__mult__left,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.30/0.58 => ( dvd_dvd_Code_integer @ A @ C ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_left
% 0.30/0.58 thf(fact_4706_dvd__mult__left,axiom,
% 0.30/0.58 ! [A: real,B: real,C: real] :
% 0.30/0.58 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 0.30/0.58 => ( dvd_dvd_real @ A @ C ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_left
% 0.30/0.58 thf(fact_4707_dvd__mult__left,axiom,
% 0.30/0.58 ! [A: rat,B: rat,C: rat] :
% 0.30/0.58 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.30/0.58 => ( dvd_dvd_rat @ A @ C ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_left
% 0.30/0.58 thf(fact_4708_dvd__mult__left,axiom,
% 0.30/0.58 ! [A: nat,B: nat,C: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.30/0.58 => ( dvd_dvd_nat @ A @ C ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_left
% 0.30/0.58 thf(fact_4709_dvd__mult__left,axiom,
% 0.30/0.58 ! [A: int,B: int,C: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.30/0.58 => ( dvd_dvd_int @ A @ C ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult_left
% 0.30/0.58 thf(fact_4710_dvd__mult2,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.58 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult2
% 0.30/0.58 thf(fact_4711_dvd__mult2,axiom,
% 0.30/0.58 ! [A: real,B: real,C: real] :
% 0.30/0.58 ( ( dvd_dvd_real @ A @ B )
% 0.30/0.58 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult2
% 0.30/0.58 thf(fact_4712_dvd__mult2,axiom,
% 0.30/0.58 ! [A: rat,B: rat,C: rat] :
% 0.30/0.58 ( ( dvd_dvd_rat @ A @ B )
% 0.30/0.58 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult2
% 0.30/0.58 thf(fact_4713_dvd__mult2,axiom,
% 0.30/0.58 ! [A: nat,B: nat,C: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult2
% 0.30/0.58 thf(fact_4714_dvd__mult2,axiom,
% 0.30/0.58 ! [A: int,B: int,C: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ A @ B )
% 0.30/0.58 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult2
% 0.30/0.58 thf(fact_4715_dvd__mult,axiom,
% 0.30/0.58 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ A @ C )
% 0.30/0.58 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult
% 0.30/0.58 thf(fact_4716_dvd__mult,axiom,
% 0.30/0.58 ! [A: real,C: real,B: real] :
% 0.30/0.58 ( ( dvd_dvd_real @ A @ C )
% 0.30/0.58 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult
% 0.30/0.58 thf(fact_4717_dvd__mult,axiom,
% 0.30/0.58 ! [A: rat,C: rat,B: rat] :
% 0.30/0.58 ( ( dvd_dvd_rat @ A @ C )
% 0.30/0.58 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult
% 0.30/0.58 thf(fact_4718_dvd__mult,axiom,
% 0.30/0.58 ! [A: nat,C: nat,B: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.58 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult
% 0.30/0.58 thf(fact_4719_dvd__mult,axiom,
% 0.30/0.58 ! [A: int,C: int,B: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ A @ C )
% 0.30/0.58 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_mult
% 0.30/0.58 thf(fact_4720_dvd__def,axiom,
% 0.30/0.58 ( dvd_dvd_Code_integer
% 0.30/0.58 = ( ^ [B2: code_integer,A3: code_integer] :
% 0.30/0.58 ? [K2: code_integer] :
% 0.30/0.58 ( A3
% 0.30/0.58 = ( times_3573771949741848930nteger @ B2 @ K2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_def
% 0.30/0.58 thf(fact_4721_dvd__def,axiom,
% 0.30/0.58 ( dvd_dvd_real
% 0.30/0.58 = ( ^ [B2: real,A3: real] :
% 0.30/0.58 ? [K2: real] :
% 0.30/0.58 ( A3
% 0.30/0.58 = ( times_times_real @ B2 @ K2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_def
% 0.30/0.58 thf(fact_4722_dvd__def,axiom,
% 0.30/0.58 ( dvd_dvd_rat
% 0.30/0.58 = ( ^ [B2: rat,A3: rat] :
% 0.30/0.58 ? [K2: rat] :
% 0.30/0.58 ( A3
% 0.30/0.58 = ( times_times_rat @ B2 @ K2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_def
% 0.30/0.58 thf(fact_4723_dvd__def,axiom,
% 0.30/0.58 ( dvd_dvd_nat
% 0.30/0.58 = ( ^ [B2: nat,A3: nat] :
% 0.30/0.58 ? [K2: nat] :
% 0.30/0.58 ( A3
% 0.30/0.58 = ( times_times_nat @ B2 @ K2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_def
% 0.30/0.58 thf(fact_4724_dvd__def,axiom,
% 0.30/0.58 ( dvd_dvd_int
% 0.30/0.58 = ( ^ [B2: int,A3: int] :
% 0.30/0.58 ? [K2: int] :
% 0.30/0.58 ( A3
% 0.30/0.58 = ( times_times_int @ B2 @ K2 ) ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_def
% 0.30/0.58 thf(fact_4725_dvdI,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer,K: code_integer] :
% 0.30/0.58 ( ( A
% 0.30/0.58 = ( times_3573771949741848930nteger @ B @ K ) )
% 0.30/0.58 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvdI
% 0.30/0.58 thf(fact_4726_dvdI,axiom,
% 0.30/0.58 ! [A: real,B: real,K: real] :
% 0.30/0.58 ( ( A
% 0.30/0.58 = ( times_times_real @ B @ K ) )
% 0.30/0.58 => ( dvd_dvd_real @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvdI
% 0.30/0.58 thf(fact_4727_dvdI,axiom,
% 0.30/0.58 ! [A: rat,B: rat,K: rat] :
% 0.30/0.58 ( ( A
% 0.30/0.58 = ( times_times_rat @ B @ K ) )
% 0.30/0.58 => ( dvd_dvd_rat @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvdI
% 0.30/0.58 thf(fact_4728_dvdI,axiom,
% 0.30/0.58 ! [A: nat,B: nat,K: nat] :
% 0.30/0.58 ( ( A
% 0.30/0.58 = ( times_times_nat @ B @ K ) )
% 0.30/0.58 => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvdI
% 0.30/0.58 thf(fact_4729_dvdI,axiom,
% 0.30/0.58 ! [A: int,B: int,K: int] :
% 0.30/0.58 ( ( A
% 0.30/0.58 = ( times_times_int @ B @ K ) )
% 0.30/0.58 => ( dvd_dvd_int @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvdI
% 0.30/0.58 thf(fact_4730_dvdE,axiom,
% 0.30/0.58 ! [B: code_integer,A: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.30/0.58 => ~ ! [K3: code_integer] :
% 0.30/0.58 ( A
% 0.30/0.58 != ( times_3573771949741848930nteger @ B @ K3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvdE
% 0.30/0.58 thf(fact_4731_dvdE,axiom,
% 0.30/0.58 ! [B: real,A: real] :
% 0.30/0.58 ( ( dvd_dvd_real @ B @ A )
% 0.30/0.58 => ~ ! [K3: real] :
% 0.30/0.58 ( A
% 0.30/0.58 != ( times_times_real @ B @ K3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvdE
% 0.30/0.58 thf(fact_4732_dvdE,axiom,
% 0.30/0.58 ! [B: rat,A: rat] :
% 0.30/0.58 ( ( dvd_dvd_rat @ B @ A )
% 0.30/0.58 => ~ ! [K3: rat] :
% 0.30/0.58 ( A
% 0.30/0.58 != ( times_times_rat @ B @ K3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvdE
% 0.30/0.58 thf(fact_4733_dvdE,axiom,
% 0.30/0.58 ! [B: nat,A: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ B @ A )
% 0.30/0.58 => ~ ! [K3: nat] :
% 0.30/0.58 ( A
% 0.30/0.58 != ( times_times_nat @ B @ K3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvdE
% 0.30/0.58 thf(fact_4734_dvdE,axiom,
% 0.30/0.58 ! [B: int,A: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ B @ A )
% 0.30/0.58 => ~ ! [K3: int] :
% 0.30/0.58 ( A
% 0.30/0.58 != ( times_times_int @ B @ K3 ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvdE
% 0.30/0.58 thf(fact_4735_one__dvd,axiom,
% 0.30/0.58 ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% 0.30/0.58
% 0.30/0.58 % one_dvd
% 0.30/0.58 thf(fact_4736_one__dvd,axiom,
% 0.30/0.58 ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% 0.30/0.58
% 0.30/0.58 % one_dvd
% 0.30/0.58 thf(fact_4737_one__dvd,axiom,
% 0.30/0.58 ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% 0.30/0.58
% 0.30/0.58 % one_dvd
% 0.30/0.58 thf(fact_4738_one__dvd,axiom,
% 0.30/0.58 ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% 0.30/0.58
% 0.30/0.58 % one_dvd
% 0.30/0.58 thf(fact_4739_one__dvd,axiom,
% 0.30/0.58 ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% 0.30/0.58
% 0.30/0.58 % one_dvd
% 0.30/0.58 thf(fact_4740_one__dvd,axiom,
% 0.30/0.58 ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% 0.30/0.58
% 0.30/0.58 % one_dvd
% 0.30/0.58 thf(fact_4741_unit__imp__dvd,axiom,
% 0.30/0.58 ! [B: code_integer,A: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.58 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_imp_dvd
% 0.30/0.58 thf(fact_4742_unit__imp__dvd,axiom,
% 0.30/0.58 ! [B: nat,A: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.58 => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_imp_dvd
% 0.30/0.58 thf(fact_4743_unit__imp__dvd,axiom,
% 0.30/0.58 ! [B: int,A: int] :
% 0.30/0.58 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.58 => ( dvd_dvd_int @ B @ A ) ) ).
% 0.30/0.58
% 0.30/0.58 % unit_imp_dvd
% 0.30/0.58 thf(fact_4744_dvd__unit__imp__unit,axiom,
% 0.30/0.58 ! [A: code_integer,B: code_integer] :
% 0.30/0.58 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.58 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.58 => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% 0.30/0.58
% 0.30/0.58 % dvd_unit_imp_unit
% 0.30/0.58 thf(fact_4745_dvd__unit__imp__unit,axiom,
% 0.30/0.58 ! [A: nat,B: nat] :
% 0.30/0.58 ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.59 => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_unit_imp_unit
% 0.30/0.59 thf(fact_4746_dvd__unit__imp__unit,axiom,
% 0.30/0.59 ! [A: int,B: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ A @ B )
% 0.30/0.59 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.59 => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_unit_imp_unit
% 0.30/0.59 thf(fact_4747_dvd__add,axiom,
% 0.30/0.59 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ A @ C )
% 0.30/0.59 => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_add
% 0.30/0.59 thf(fact_4748_dvd__add,axiom,
% 0.30/0.59 ! [A: real,B: real,C: real] :
% 0.30/0.59 ( ( dvd_dvd_real @ A @ B )
% 0.30/0.59 => ( ( dvd_dvd_real @ A @ C )
% 0.30/0.59 => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_add
% 0.30/0.59 thf(fact_4749_dvd__add,axiom,
% 0.30/0.59 ! [A: rat,B: rat,C: rat] :
% 0.30/0.59 ( ( dvd_dvd_rat @ A @ B )
% 0.30/0.59 => ( ( dvd_dvd_rat @ A @ C )
% 0.30/0.59 => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_add
% 0.30/0.59 thf(fact_4750_dvd__add,axiom,
% 0.30/0.59 ! [A: nat,B: nat,C: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.59 => ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.59 => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_add
% 0.30/0.59 thf(fact_4751_dvd__add,axiom,
% 0.30/0.59 ! [A: int,B: int,C: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ A @ B )
% 0.30/0.59 => ( ( dvd_dvd_int @ A @ C )
% 0.30/0.59 => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_add
% 0.30/0.59 thf(fact_4752_dvd__add__left__iff,axiom,
% 0.30/0.59 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ A @ C )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_add_left_iff
% 0.30/0.59 thf(fact_4753_dvd__add__left__iff,axiom,
% 0.30/0.59 ! [A: real,C: real,B: real] :
% 0.30/0.59 ( ( dvd_dvd_real @ A @ C )
% 0.30/0.59 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.30/0.59 = ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_add_left_iff
% 0.30/0.59 thf(fact_4754_dvd__add__left__iff,axiom,
% 0.30/0.59 ! [A: rat,C: rat,B: rat] :
% 0.30/0.59 ( ( dvd_dvd_rat @ A @ C )
% 0.30/0.59 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.59 = ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_add_left_iff
% 0.30/0.59 thf(fact_4755_dvd__add__left__iff,axiom,
% 0.30/0.59 ! [A: nat,C: nat,B: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.59 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.30/0.59 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_add_left_iff
% 0.30/0.59 thf(fact_4756_dvd__add__left__iff,axiom,
% 0.30/0.59 ! [A: int,C: int,B: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ A @ C )
% 0.30/0.59 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.30/0.59 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_add_left_iff
% 0.30/0.59 thf(fact_4757_dvd__add__right__iff,axiom,
% 0.30/0.59 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_add_right_iff
% 0.30/0.59 thf(fact_4758_dvd__add__right__iff,axiom,
% 0.30/0.59 ! [A: real,B: real,C: real] :
% 0.30/0.59 ( ( dvd_dvd_real @ A @ B )
% 0.30/0.59 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.30/0.59 = ( dvd_dvd_real @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_add_right_iff
% 0.30/0.59 thf(fact_4759_dvd__add__right__iff,axiom,
% 0.30/0.59 ! [A: rat,B: rat,C: rat] :
% 0.30/0.59 ( ( dvd_dvd_rat @ A @ B )
% 0.30/0.59 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.59 = ( dvd_dvd_rat @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_add_right_iff
% 0.30/0.59 thf(fact_4760_dvd__add__right__iff,axiom,
% 0.30/0.59 ! [A: nat,B: nat,C: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.59 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.30/0.59 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_add_right_iff
% 0.30/0.59 thf(fact_4761_dvd__add__right__iff,axiom,
% 0.30/0.59 ! [A: int,B: int,C: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ A @ B )
% 0.30/0.59 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.30/0.59 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_add_right_iff
% 0.30/0.59 thf(fact_4762_dvd__diff__commute,axiom,
% 0.30/0.59 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_diff_commute
% 0.30/0.59 thf(fact_4763_dvd__diff__commute,axiom,
% 0.30/0.59 ! [A: int,C: int,B: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.30/0.59 = ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_diff_commute
% 0.30/0.59 thf(fact_4764_dvd__diff,axiom,
% 0.30/0.59 ! [X2: code_integer,Y3: code_integer,Z: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ X2 @ Y3 )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ X2 @ Z )
% 0.30/0.59 => ( dvd_dvd_Code_integer @ X2 @ ( minus_8373710615458151222nteger @ Y3 @ Z ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_diff
% 0.30/0.59 thf(fact_4765_dvd__diff,axiom,
% 0.30/0.59 ! [X2: real,Y3: real,Z: real] :
% 0.30/0.59 ( ( dvd_dvd_real @ X2 @ Y3 )
% 0.30/0.59 => ( ( dvd_dvd_real @ X2 @ Z )
% 0.30/0.59 => ( dvd_dvd_real @ X2 @ ( minus_minus_real @ Y3 @ Z ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_diff
% 0.30/0.59 thf(fact_4766_dvd__diff,axiom,
% 0.30/0.59 ! [X2: rat,Y3: rat,Z: rat] :
% 0.30/0.59 ( ( dvd_dvd_rat @ X2 @ Y3 )
% 0.30/0.59 => ( ( dvd_dvd_rat @ X2 @ Z )
% 0.30/0.59 => ( dvd_dvd_rat @ X2 @ ( minus_minus_rat @ Y3 @ Z ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_diff
% 0.30/0.59 thf(fact_4767_dvd__diff,axiom,
% 0.30/0.59 ! [X2: int,Y3: int,Z: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ X2 @ Y3 )
% 0.30/0.59 => ( ( dvd_dvd_int @ X2 @ Z )
% 0.30/0.59 => ( dvd_dvd_int @ X2 @ ( minus_minus_int @ Y3 @ Z ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_diff
% 0.30/0.59 thf(fact_4768_div__div__div__same,axiom,
% 0.30/0.59 ! [D: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ D @ B )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.30/0.59 => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
% 0.30/0.59 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_div_div_same
% 0.30/0.59 thf(fact_4769_div__div__div__same,axiom,
% 0.30/0.59 ! [D: nat,B: nat,A: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ D @ B )
% 0.30/0.59 => ( ( dvd_dvd_nat @ B @ A )
% 0.30/0.59 => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
% 0.30/0.59 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_div_div_same
% 0.30/0.59 thf(fact_4770_div__div__div__same,axiom,
% 0.30/0.59 ! [D: int,B: int,A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ D @ B )
% 0.30/0.59 => ( ( dvd_dvd_int @ B @ A )
% 0.30/0.59 => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
% 0.30/0.59 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_div_div_same
% 0.30/0.59 thf(fact_4771_dvd__div__eq__cancel,axiom,
% 0.30/0.59 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.59 ( ( ( divide6298287555418463151nteger @ A @ C )
% 0.30/0.59 = ( divide6298287555418463151nteger @ B @ C ) )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ C @ A )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.59 => ( A = B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_cancel
% 0.30/0.59 thf(fact_4772_dvd__div__eq__cancel,axiom,
% 0.30/0.59 ! [A: complex,C: complex,B: complex] :
% 0.30/0.59 ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.30/0.59 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.30/0.59 => ( ( dvd_dvd_complex @ C @ A )
% 0.30/0.59 => ( ( dvd_dvd_complex @ C @ B )
% 0.30/0.59 => ( A = B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_cancel
% 0.30/0.59 thf(fact_4773_dvd__div__eq__cancel,axiom,
% 0.30/0.59 ! [A: real,C: real,B: real] :
% 0.30/0.59 ( ( ( divide_divide_real @ A @ C )
% 0.30/0.59 = ( divide_divide_real @ B @ C ) )
% 0.30/0.59 => ( ( dvd_dvd_real @ C @ A )
% 0.30/0.59 => ( ( dvd_dvd_real @ C @ B )
% 0.30/0.59 => ( A = B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_cancel
% 0.30/0.59 thf(fact_4774_dvd__div__eq__cancel,axiom,
% 0.30/0.59 ! [A: rat,C: rat,B: rat] :
% 0.30/0.59 ( ( ( divide_divide_rat @ A @ C )
% 0.30/0.59 = ( divide_divide_rat @ B @ C ) )
% 0.30/0.59 => ( ( dvd_dvd_rat @ C @ A )
% 0.30/0.59 => ( ( dvd_dvd_rat @ C @ B )
% 0.30/0.59 => ( A = B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_cancel
% 0.30/0.59 thf(fact_4775_dvd__div__eq__cancel,axiom,
% 0.30/0.59 ! [A: nat,C: nat,B: nat] :
% 0.30/0.59 ( ( ( divide_divide_nat @ A @ C )
% 0.30/0.59 = ( divide_divide_nat @ B @ C ) )
% 0.30/0.59 => ( ( dvd_dvd_nat @ C @ A )
% 0.30/0.59 => ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.59 => ( A = B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_cancel
% 0.30/0.59 thf(fact_4776_dvd__div__eq__cancel,axiom,
% 0.30/0.59 ! [A: int,C: int,B: int] :
% 0.30/0.59 ( ( ( divide_divide_int @ A @ C )
% 0.30/0.59 = ( divide_divide_int @ B @ C ) )
% 0.30/0.59 => ( ( dvd_dvd_int @ C @ A )
% 0.30/0.59 => ( ( dvd_dvd_int @ C @ B )
% 0.30/0.59 => ( A = B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_cancel
% 0.30/0.59 thf(fact_4777_dvd__div__eq__iff,axiom,
% 0.30/0.59 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.59 => ( ( ( divide6298287555418463151nteger @ A @ C )
% 0.30/0.59 = ( divide6298287555418463151nteger @ B @ C ) )
% 0.30/0.59 = ( A = B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_iff
% 0.30/0.59 thf(fact_4778_dvd__div__eq__iff,axiom,
% 0.30/0.59 ! [C: complex,A: complex,B: complex] :
% 0.30/0.59 ( ( dvd_dvd_complex @ C @ A )
% 0.30/0.59 => ( ( dvd_dvd_complex @ C @ B )
% 0.30/0.59 => ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.30/0.59 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.30/0.59 = ( A = B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_iff
% 0.30/0.59 thf(fact_4779_dvd__div__eq__iff,axiom,
% 0.30/0.59 ! [C: real,A: real,B: real] :
% 0.30/0.59 ( ( dvd_dvd_real @ C @ A )
% 0.30/0.59 => ( ( dvd_dvd_real @ C @ B )
% 0.30/0.59 => ( ( ( divide_divide_real @ A @ C )
% 0.30/0.59 = ( divide_divide_real @ B @ C ) )
% 0.30/0.59 = ( A = B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_iff
% 0.30/0.59 thf(fact_4780_dvd__div__eq__iff,axiom,
% 0.30/0.59 ! [C: rat,A: rat,B: rat] :
% 0.30/0.59 ( ( dvd_dvd_rat @ C @ A )
% 0.30/0.59 => ( ( dvd_dvd_rat @ C @ B )
% 0.30/0.59 => ( ( ( divide_divide_rat @ A @ C )
% 0.30/0.59 = ( divide_divide_rat @ B @ C ) )
% 0.30/0.59 = ( A = B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_iff
% 0.30/0.59 thf(fact_4781_dvd__div__eq__iff,axiom,
% 0.30/0.59 ! [C: nat,A: nat,B: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ C @ A )
% 0.30/0.59 => ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.59 => ( ( ( divide_divide_nat @ A @ C )
% 0.30/0.59 = ( divide_divide_nat @ B @ C ) )
% 0.30/0.59 = ( A = B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_iff
% 0.30/0.59 thf(fact_4782_dvd__div__eq__iff,axiom,
% 0.30/0.59 ! [C: int,A: int,B: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ C @ A )
% 0.30/0.59 => ( ( dvd_dvd_int @ C @ B )
% 0.30/0.59 => ( ( ( divide_divide_int @ A @ C )
% 0.30/0.59 = ( divide_divide_int @ B @ C ) )
% 0.30/0.59 = ( A = B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_iff
% 0.30/0.59 thf(fact_4783_dvd__power__same,axiom,
% 0.30/0.59 ! [X2: code_integer,Y3: code_integer,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ X2 @ Y3 )
% 0.30/0.59 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X2 @ N ) @ ( power_8256067586552552935nteger @ Y3 @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power_same
% 0.30/0.59 thf(fact_4784_dvd__power__same,axiom,
% 0.30/0.59 ! [X2: nat,Y3: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ X2 @ Y3 )
% 0.30/0.59 => ( dvd_dvd_nat @ ( power_power_nat @ X2 @ N ) @ ( power_power_nat @ Y3 @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power_same
% 0.30/0.59 thf(fact_4785_dvd__power__same,axiom,
% 0.30/0.59 ! [X2: real,Y3: real,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_real @ X2 @ Y3 )
% 0.30/0.59 => ( dvd_dvd_real @ ( power_power_real @ X2 @ N ) @ ( power_power_real @ Y3 @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power_same
% 0.30/0.59 thf(fact_4786_dvd__power__same,axiom,
% 0.30/0.59 ! [X2: int,Y3: int,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_int @ X2 @ Y3 )
% 0.30/0.59 => ( dvd_dvd_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y3 @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power_same
% 0.30/0.59 thf(fact_4787_dvd__power__same,axiom,
% 0.30/0.59 ! [X2: complex,Y3: complex,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_complex @ X2 @ Y3 )
% 0.30/0.59 => ( dvd_dvd_complex @ ( power_power_complex @ X2 @ N ) @ ( power_power_complex @ Y3 @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power_same
% 0.30/0.59 thf(fact_4788_dvd__mod__iff,axiom,
% 0.30/0.59 ! [C: nat,B: nat,A: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.59 => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 0.30/0.59 = ( dvd_dvd_nat @ C @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mod_iff
% 0.30/0.59 thf(fact_4789_dvd__mod__iff,axiom,
% 0.30/0.59 ! [C: int,B: int,A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ C @ B )
% 0.30/0.59 => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.59 = ( dvd_dvd_int @ C @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mod_iff
% 0.30/0.59 thf(fact_4790_dvd__mod__iff,axiom,
% 0.30/0.59 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mod_iff
% 0.30/0.59 thf(fact_4791_dvd__mod__imp__dvd,axiom,
% 0.30/0.59 ! [C: nat,A: nat,B: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 0.30/0.59 => ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.59 => ( dvd_dvd_nat @ C @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mod_imp_dvd
% 0.30/0.59 thf(fact_4792_dvd__mod__imp__dvd,axiom,
% 0.30/0.59 ! [C: int,A: int,B: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.59 => ( ( dvd_dvd_int @ C @ B )
% 0.30/0.59 => ( dvd_dvd_int @ C @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mod_imp_dvd
% 0.30/0.59 thf(fact_4793_dvd__mod__imp__dvd,axiom,
% 0.30/0.59 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.59 => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mod_imp_dvd
% 0.30/0.59 thf(fact_4794_dvd__mod,axiom,
% 0.30/0.59 ! [K: nat,M: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ K @ M )
% 0.30/0.59 => ( ( dvd_dvd_nat @ K @ N )
% 0.30/0.59 => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mod
% 0.30/0.59 thf(fact_4795_dvd__mod,axiom,
% 0.30/0.59 ! [K: int,M: int,N: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ K @ M )
% 0.30/0.59 => ( ( dvd_dvd_int @ K @ N )
% 0.30/0.59 => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mod
% 0.30/0.59 thf(fact_4796_dvd__mod,axiom,
% 0.30/0.59 ! [K: code_integer,M: code_integer,N: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ K @ M )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ K @ N )
% 0.30/0.59 => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mod
% 0.30/0.59 thf(fact_4797_mod__mod__cancel,axiom,
% 0.30/0.59 ! [C: nat,B: nat,A: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.59 => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ C )
% 0.30/0.59 = ( modulo_modulo_nat @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mod_mod_cancel
% 0.30/0.59 thf(fact_4798_mod__mod__cancel,axiom,
% 0.30/0.59 ! [C: int,B: int,A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ C @ B )
% 0.30/0.59 => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ C )
% 0.30/0.59 = ( modulo_modulo_int @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mod_mod_cancel
% 0.30/0.59 thf(fact_4799_mod__mod__cancel,axiom,
% 0.30/0.59 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.59 => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C )
% 0.30/0.59 = ( modulo364778990260209775nteger @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mod_mod_cancel
% 0.30/0.59 thf(fact_4800_dvd__diff__nat,axiom,
% 0.30/0.59 ! [K: nat,M: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ K @ M )
% 0.30/0.59 => ( ( dvd_dvd_nat @ K @ N )
% 0.30/0.59 => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_diff_nat
% 0.30/0.59 thf(fact_4801_zdvd__zdiffD,axiom,
% 0.30/0.59 ! [K: int,M: int,N: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
% 0.30/0.59 => ( ( dvd_dvd_int @ K @ N )
% 0.30/0.59 => ( dvd_dvd_int @ K @ M ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % zdvd_zdiffD
% 0.30/0.59 thf(fact_4802_bot__enat__def,axiom,
% 0.30/0.59 bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% 0.30/0.59
% 0.30/0.59 % bot_enat_def
% 0.30/0.59 thf(fact_4803_dvd__pos__nat,axiom,
% 0.30/0.59 ! [N: nat,M: nat] :
% 0.30/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.59 => ( ( dvd_dvd_nat @ M @ N )
% 0.30/0.59 => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_pos_nat
% 0.30/0.59 thf(fact_4804_bezout__lemma__nat,axiom,
% 0.30/0.59 ! [D: nat,A: nat,B: nat,X2: nat,Y3: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ D @ A )
% 0.30/0.59 => ( ( dvd_dvd_nat @ D @ B )
% 0.30/0.59 => ( ( ( ( times_times_nat @ A @ X2 )
% 0.30/0.59 = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D ) )
% 0.30/0.59 | ( ( times_times_nat @ B @ X2 )
% 0.30/0.59 = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) )
% 0.30/0.59 => ? [X5: nat,Y4: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ D @ A )
% 0.30/0.59 & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
% 0.30/0.59 & ( ( ( times_times_nat @ A @ X5 )
% 0.30/0.59 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y4 ) @ D ) )
% 0.30/0.59 | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X5 )
% 0.30/0.59 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % bezout_lemma_nat
% 0.30/0.59 thf(fact_4805_bezout__add__nat,axiom,
% 0.30/0.59 ! [A: nat,B: nat] :
% 0.30/0.59 ? [D3: nat,X5: nat,Y4: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ D3 @ A )
% 0.30/0.59 & ( dvd_dvd_nat @ D3 @ B )
% 0.30/0.59 & ( ( ( times_times_nat @ A @ X5 )
% 0.30/0.59 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D3 ) )
% 0.30/0.59 | ( ( times_times_nat @ B @ X5 )
% 0.30/0.59 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D3 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % bezout_add_nat
% 0.30/0.59 thf(fact_4806_bezout1__nat,axiom,
% 0.30/0.59 ! [A: nat,B: nat] :
% 0.30/0.59 ? [D3: nat,X5: nat,Y4: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ D3 @ A )
% 0.30/0.59 & ( dvd_dvd_nat @ D3 @ B )
% 0.30/0.59 & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X5 ) @ ( times_times_nat @ B @ Y4 ) )
% 0.30/0.59 = D3 )
% 0.30/0.59 | ( ( minus_minus_nat @ ( times_times_nat @ B @ X5 ) @ ( times_times_nat @ A @ Y4 ) )
% 0.30/0.59 = D3 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % bezout1_nat
% 0.30/0.59 thf(fact_4807_subset__divisors__dvd,axiom,
% 0.30/0.59 ! [A: complex,B: complex] :
% 0.30/0.59 ( ( ord_le211207098394363844omplex
% 0.30/0.59 @ ( collect_complex
% 0.30/0.59 @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ A ) )
% 0.30/0.59 @ ( collect_complex
% 0.30/0.59 @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ B ) ) )
% 0.30/0.59 = ( dvd_dvd_complex @ A @ B ) ) ).
% 0.30/0.59
% 0.30/0.59 % subset_divisors_dvd
% 0.30/0.59 thf(fact_4808_subset__divisors__dvd,axiom,
% 0.30/0.59 ! [A: real,B: real] :
% 0.30/0.59 ( ( ord_less_eq_set_real
% 0.30/0.59 @ ( collect_real
% 0.30/0.59 @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ A ) )
% 0.30/0.59 @ ( collect_real
% 0.30/0.59 @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ B ) ) )
% 0.30/0.59 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.30/0.59
% 0.30/0.59 % subset_divisors_dvd
% 0.30/0.59 thf(fact_4809_subset__divisors__dvd,axiom,
% 0.30/0.59 ! [A: nat,B: nat] :
% 0.30/0.59 ( ( ord_less_eq_set_nat
% 0.30/0.59 @ ( collect_nat
% 0.30/0.59 @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
% 0.30/0.59 @ ( collect_nat
% 0.30/0.59 @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B ) ) )
% 0.30/0.59 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.30/0.59
% 0.30/0.59 % subset_divisors_dvd
% 0.30/0.59 thf(fact_4810_subset__divisors__dvd,axiom,
% 0.30/0.59 ! [A: code_integer,B: code_integer] :
% 0.30/0.59 ( ( ord_le7084787975880047091nteger
% 0.30/0.59 @ ( collect_Code_integer
% 0.30/0.59 @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ A ) )
% 0.30/0.59 @ ( collect_Code_integer
% 0.30/0.59 @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ B ) ) )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.30/0.59
% 0.30/0.59 % subset_divisors_dvd
% 0.30/0.59 thf(fact_4811_subset__divisors__dvd,axiom,
% 0.30/0.59 ! [A: int,B: int] :
% 0.30/0.59 ( ( ord_less_eq_set_int
% 0.30/0.59 @ ( collect_int
% 0.30/0.59 @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
% 0.30/0.59 @ ( collect_int
% 0.30/0.59 @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B ) ) )
% 0.30/0.59 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.30/0.59
% 0.30/0.59 % subset_divisors_dvd
% 0.30/0.59 thf(fact_4812_concat__bit__assoc,axiom,
% 0.30/0.59 ! [N: nat,K: int,M: nat,L: int,R2: int] :
% 0.30/0.59 ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L @ R2 ) )
% 0.30/0.59 = ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R2 ) ) ).
% 0.30/0.59
% 0.30/0.59 % concat_bit_assoc
% 0.30/0.59 thf(fact_4813_strict__subset__divisors__dvd,axiom,
% 0.30/0.59 ! [A: complex,B: complex] :
% 0.30/0.59 ( ( ord_less_set_complex
% 0.30/0.59 @ ( collect_complex
% 0.30/0.59 @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ A ) )
% 0.30/0.59 @ ( collect_complex
% 0.30/0.59 @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ B ) ) )
% 0.30/0.59 = ( ( dvd_dvd_complex @ A @ B )
% 0.30/0.59 & ~ ( dvd_dvd_complex @ B @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % strict_subset_divisors_dvd
% 0.30/0.59 thf(fact_4814_strict__subset__divisors__dvd,axiom,
% 0.30/0.59 ! [A: real,B: real] :
% 0.30/0.59 ( ( ord_less_set_real
% 0.30/0.59 @ ( collect_real
% 0.30/0.59 @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ A ) )
% 0.30/0.59 @ ( collect_real
% 0.30/0.59 @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ B ) ) )
% 0.30/0.59 = ( ( dvd_dvd_real @ A @ B )
% 0.30/0.59 & ~ ( dvd_dvd_real @ B @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % strict_subset_divisors_dvd
% 0.30/0.59 thf(fact_4815_strict__subset__divisors__dvd,axiom,
% 0.30/0.59 ! [A: nat,B: nat] :
% 0.30/0.59 ( ( ord_less_set_nat
% 0.30/0.59 @ ( collect_nat
% 0.30/0.59 @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
% 0.30/0.59 @ ( collect_nat
% 0.30/0.59 @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B ) ) )
% 0.30/0.59 = ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.59 & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % strict_subset_divisors_dvd
% 0.30/0.59 thf(fact_4816_strict__subset__divisors__dvd,axiom,
% 0.30/0.59 ! [A: int,B: int] :
% 0.30/0.59 ( ( ord_less_set_int
% 0.30/0.59 @ ( collect_int
% 0.30/0.59 @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
% 0.30/0.59 @ ( collect_int
% 0.30/0.59 @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B ) ) )
% 0.30/0.59 = ( ( dvd_dvd_int @ A @ B )
% 0.30/0.59 & ~ ( dvd_dvd_int @ B @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % strict_subset_divisors_dvd
% 0.30/0.59 thf(fact_4817_strict__subset__divisors__dvd,axiom,
% 0.30/0.59 ! [A: code_integer,B: code_integer] :
% 0.30/0.59 ( ( ord_le1307284697595431911nteger
% 0.30/0.59 @ ( collect_Code_integer
% 0.30/0.59 @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ A ) )
% 0.30/0.59 @ ( collect_Code_integer
% 0.30/0.59 @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ B ) ) )
% 0.30/0.59 = ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.59 & ~ ( dvd_dvd_Code_integer @ B @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % strict_subset_divisors_dvd
% 0.30/0.59 thf(fact_4818_not__is__unit__0,axiom,
% 0.30/0.59 ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 0.30/0.59
% 0.30/0.59 % not_is_unit_0
% 0.30/0.59 thf(fact_4819_not__is__unit__0,axiom,
% 0.30/0.59 ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 0.30/0.59
% 0.30/0.59 % not_is_unit_0
% 0.30/0.59 thf(fact_4820_not__is__unit__0,axiom,
% 0.30/0.59 ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 0.30/0.59
% 0.30/0.59 % not_is_unit_0
% 0.30/0.59 thf(fact_4821_minf_I10_J,axiom,
% 0.30/0.59 ! [D: code_integer,S: code_integer] :
% 0.30/0.59 ? [Z2: code_integer] :
% 0.30/0.59 ! [X3: code_integer] :
% 0.30/0.59 ( ( ord_le6747313008572928689nteger @ X3 @ Z2 )
% 0.30/0.59 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) ) )
% 0.30/0.59 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % minf(10)
% 0.30/0.59 thf(fact_4822_minf_I10_J,axiom,
% 0.30/0.59 ! [D: real,S: real] :
% 0.30/0.59 ? [Z2: real] :
% 0.30/0.59 ! [X3: real] :
% 0.30/0.59 ( ( ord_less_real @ X3 @ Z2 )
% 0.30/0.59 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) )
% 0.30/0.59 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % minf(10)
% 0.30/0.59 thf(fact_4823_minf_I10_J,axiom,
% 0.30/0.59 ! [D: rat,S: rat] :
% 0.30/0.59 ? [Z2: rat] :
% 0.30/0.59 ! [X3: rat] :
% 0.30/0.59 ( ( ord_less_rat @ X3 @ Z2 )
% 0.30/0.59 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) )
% 0.30/0.59 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % minf(10)
% 0.30/0.59 thf(fact_4824_minf_I10_J,axiom,
% 0.30/0.59 ! [D: nat,S: nat] :
% 0.30/0.59 ? [Z2: nat] :
% 0.30/0.59 ! [X3: nat] :
% 0.30/0.59 ( ( ord_less_nat @ X3 @ Z2 )
% 0.30/0.59 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) )
% 0.30/0.59 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % minf(10)
% 0.30/0.59 thf(fact_4825_minf_I10_J,axiom,
% 0.30/0.59 ! [D: int,S: int] :
% 0.30/0.59 ? [Z2: int] :
% 0.30/0.59 ! [X3: int] :
% 0.30/0.59 ( ( ord_less_int @ X3 @ Z2 )
% 0.30/0.59 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) )
% 0.30/0.59 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % minf(10)
% 0.30/0.59 thf(fact_4826_minf_I9_J,axiom,
% 0.30/0.59 ! [D: code_integer,S: code_integer] :
% 0.30/0.59 ? [Z2: code_integer] :
% 0.30/0.59 ! [X3: code_integer] :
% 0.30/0.59 ( ( ord_le6747313008572928689nteger @ X3 @ Z2 )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % minf(9)
% 0.30/0.59 thf(fact_4827_minf_I9_J,axiom,
% 0.30/0.59 ! [D: real,S: real] :
% 0.30/0.59 ? [Z2: real] :
% 0.30/0.59 ! [X3: real] :
% 0.30/0.59 ( ( ord_less_real @ X3 @ Z2 )
% 0.30/0.59 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) )
% 0.30/0.59 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % minf(9)
% 0.30/0.59 thf(fact_4828_minf_I9_J,axiom,
% 0.30/0.59 ! [D: rat,S: rat] :
% 0.30/0.59 ? [Z2: rat] :
% 0.30/0.59 ! [X3: rat] :
% 0.30/0.59 ( ( ord_less_rat @ X3 @ Z2 )
% 0.30/0.59 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) )
% 0.30/0.59 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % minf(9)
% 0.30/0.59 thf(fact_4829_minf_I9_J,axiom,
% 0.30/0.59 ! [D: nat,S: nat] :
% 0.30/0.59 ? [Z2: nat] :
% 0.30/0.59 ! [X3: nat] :
% 0.30/0.59 ( ( ord_less_nat @ X3 @ Z2 )
% 0.30/0.59 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) )
% 0.30/0.59 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % minf(9)
% 0.30/0.59 thf(fact_4830_minf_I9_J,axiom,
% 0.30/0.59 ! [D: int,S: int] :
% 0.30/0.59 ? [Z2: int] :
% 0.30/0.59 ! [X3: int] :
% 0.30/0.59 ( ( ord_less_int @ X3 @ Z2 )
% 0.30/0.59 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) )
% 0.30/0.59 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % minf(9)
% 0.30/0.59 thf(fact_4831_pinf_I10_J,axiom,
% 0.30/0.59 ! [D: code_integer,S: code_integer] :
% 0.30/0.59 ? [Z2: code_integer] :
% 0.30/0.59 ! [X3: code_integer] :
% 0.30/0.59 ( ( ord_le6747313008572928689nteger @ Z2 @ X3 )
% 0.30/0.59 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) ) )
% 0.30/0.59 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % pinf(10)
% 0.30/0.59 thf(fact_4832_pinf_I10_J,axiom,
% 0.30/0.59 ! [D: real,S: real] :
% 0.30/0.59 ? [Z2: real] :
% 0.30/0.59 ! [X3: real] :
% 0.30/0.59 ( ( ord_less_real @ Z2 @ X3 )
% 0.30/0.59 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) )
% 0.30/0.59 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % pinf(10)
% 0.30/0.59 thf(fact_4833_pinf_I10_J,axiom,
% 0.30/0.59 ! [D: rat,S: rat] :
% 0.30/0.59 ? [Z2: rat] :
% 0.30/0.59 ! [X3: rat] :
% 0.30/0.59 ( ( ord_less_rat @ Z2 @ X3 )
% 0.30/0.59 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) )
% 0.30/0.59 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % pinf(10)
% 0.30/0.59 thf(fact_4834_pinf_I10_J,axiom,
% 0.30/0.59 ! [D: nat,S: nat] :
% 0.30/0.59 ? [Z2: nat] :
% 0.30/0.59 ! [X3: nat] :
% 0.30/0.59 ( ( ord_less_nat @ Z2 @ X3 )
% 0.30/0.59 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) )
% 0.30/0.59 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % pinf(10)
% 0.30/0.59 thf(fact_4835_pinf_I10_J,axiom,
% 0.30/0.59 ! [D: int,S: int] :
% 0.30/0.59 ? [Z2: int] :
% 0.30/0.59 ! [X3: int] :
% 0.30/0.59 ( ( ord_less_int @ Z2 @ X3 )
% 0.30/0.59 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) )
% 0.30/0.59 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % pinf(10)
% 0.30/0.59 thf(fact_4836_pinf_I9_J,axiom,
% 0.30/0.59 ! [D: code_integer,S: code_integer] :
% 0.30/0.59 ? [Z2: code_integer] :
% 0.30/0.59 ! [X3: code_integer] :
% 0.30/0.59 ( ( ord_le6747313008572928689nteger @ Z2 @ X3 )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % pinf(9)
% 0.30/0.59 thf(fact_4837_pinf_I9_J,axiom,
% 0.30/0.59 ! [D: real,S: real] :
% 0.30/0.59 ? [Z2: real] :
% 0.30/0.59 ! [X3: real] :
% 0.30/0.59 ( ( ord_less_real @ Z2 @ X3 )
% 0.30/0.59 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) )
% 0.30/0.59 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % pinf(9)
% 0.30/0.59 thf(fact_4838_pinf_I9_J,axiom,
% 0.30/0.59 ! [D: rat,S: rat] :
% 0.30/0.59 ? [Z2: rat] :
% 0.30/0.59 ! [X3: rat] :
% 0.30/0.59 ( ( ord_less_rat @ Z2 @ X3 )
% 0.30/0.59 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) )
% 0.30/0.59 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % pinf(9)
% 0.30/0.59 thf(fact_4839_pinf_I9_J,axiom,
% 0.30/0.59 ! [D: nat,S: nat] :
% 0.30/0.59 ? [Z2: nat] :
% 0.30/0.59 ! [X3: nat] :
% 0.30/0.59 ( ( ord_less_nat @ Z2 @ X3 )
% 0.30/0.59 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) )
% 0.30/0.59 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % pinf(9)
% 0.30/0.59 thf(fact_4840_pinf_I9_J,axiom,
% 0.30/0.59 ! [D: int,S: int] :
% 0.30/0.59 ? [Z2: int] :
% 0.30/0.59 ! [X3: int] :
% 0.30/0.59 ( ( ord_less_int @ Z2 @ X3 )
% 0.30/0.59 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) )
% 0.30/0.59 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % pinf(9)
% 0.30/0.59 thf(fact_4841_dvd__div__eq__0__iff,axiom,
% 0.30/0.59 ! [B: code_integer,A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.30/0.59 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.30/0.59 = zero_z3403309356797280102nteger )
% 0.30/0.59 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_0_iff
% 0.30/0.59 thf(fact_4842_dvd__div__eq__0__iff,axiom,
% 0.30/0.59 ! [B: complex,A: complex] :
% 0.30/0.59 ( ( dvd_dvd_complex @ B @ A )
% 0.30/0.59 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.30/0.59 = zero_zero_complex )
% 0.30/0.59 = ( A = zero_zero_complex ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_0_iff
% 0.30/0.59 thf(fact_4843_dvd__div__eq__0__iff,axiom,
% 0.30/0.59 ! [B: real,A: real] :
% 0.30/0.59 ( ( dvd_dvd_real @ B @ A )
% 0.30/0.59 => ( ( ( divide_divide_real @ A @ B )
% 0.30/0.59 = zero_zero_real )
% 0.30/0.59 = ( A = zero_zero_real ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_0_iff
% 0.30/0.59 thf(fact_4844_dvd__div__eq__0__iff,axiom,
% 0.30/0.59 ! [B: rat,A: rat] :
% 0.30/0.59 ( ( dvd_dvd_rat @ B @ A )
% 0.30/0.59 => ( ( ( divide_divide_rat @ A @ B )
% 0.30/0.59 = zero_zero_rat )
% 0.30/0.59 = ( A = zero_zero_rat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_0_iff
% 0.30/0.59 thf(fact_4845_dvd__div__eq__0__iff,axiom,
% 0.30/0.59 ! [B: nat,A: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ B @ A )
% 0.30/0.59 => ( ( ( divide_divide_nat @ A @ B )
% 0.30/0.59 = zero_zero_nat )
% 0.30/0.59 = ( A = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_0_iff
% 0.30/0.59 thf(fact_4846_dvd__div__eq__0__iff,axiom,
% 0.30/0.59 ! [B: int,A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ B @ A )
% 0.30/0.59 => ( ( ( divide_divide_int @ A @ B )
% 0.30/0.59 = zero_zero_int )
% 0.30/0.59 = ( A = zero_zero_int ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_0_iff
% 0.30/0.59 thf(fact_4847_is__unit__mult__iff,axiom,
% 0.30/0.59 ! [A: code_integer,B: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
% 0.30/0.59 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.59 & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % is_unit_mult_iff
% 0.30/0.59 thf(fact_4848_is__unit__mult__iff,axiom,
% 0.30/0.59 ! [A: nat,B: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
% 0.30/0.59 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.59 & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % is_unit_mult_iff
% 0.30/0.59 thf(fact_4849_is__unit__mult__iff,axiom,
% 0.30/0.59 ! [A: int,B: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
% 0.30/0.59 = ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.59 & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % is_unit_mult_iff
% 0.30/0.59 thf(fact_4850_dvd__mult__unit__iff,axiom,
% 0.30/0.59 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mult_unit_iff
% 0.30/0.59 thf(fact_4851_dvd__mult__unit__iff,axiom,
% 0.30/0.59 ! [B: nat,A: nat,C: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.59 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
% 0.30/0.59 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mult_unit_iff
% 0.30/0.59 thf(fact_4852_dvd__mult__unit__iff,axiom,
% 0.30/0.59 ! [B: int,A: int,C: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.59 => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
% 0.30/0.59 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mult_unit_iff
% 0.30/0.59 thf(fact_4853_mult__unit__dvd__iff,axiom,
% 0.30/0.59 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mult_unit_dvd_iff
% 0.30/0.59 thf(fact_4854_mult__unit__dvd__iff,axiom,
% 0.30/0.59 ! [B: nat,A: nat,C: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.59 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.30/0.59 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mult_unit_dvd_iff
% 0.30/0.59 thf(fact_4855_mult__unit__dvd__iff,axiom,
% 0.30/0.59 ! [B: int,A: int,C: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.59 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.30/0.59 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mult_unit_dvd_iff
% 0.30/0.59 thf(fact_4856_dvd__mult__unit__iff_H,axiom,
% 0.30/0.59 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mult_unit_iff'
% 0.30/0.59 thf(fact_4857_dvd__mult__unit__iff_H,axiom,
% 0.30/0.59 ! [B: nat,A: nat,C: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.59 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.30/0.59 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mult_unit_iff'
% 0.30/0.59 thf(fact_4858_dvd__mult__unit__iff_H,axiom,
% 0.30/0.59 ! [B: int,A: int,C: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.59 => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 0.30/0.59 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mult_unit_iff'
% 0.30/0.59 thf(fact_4859_mult__unit__dvd__iff_H,axiom,
% 0.30/0.59 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mult_unit_dvd_iff'
% 0.30/0.59 thf(fact_4860_mult__unit__dvd__iff_H,axiom,
% 0.30/0.59 ! [A: nat,B: nat,C: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.59 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.30/0.59 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mult_unit_dvd_iff'
% 0.30/0.59 thf(fact_4861_mult__unit__dvd__iff_H,axiom,
% 0.30/0.59 ! [A: int,B: int,C: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.59 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.30/0.59 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mult_unit_dvd_iff'
% 0.30/0.59 thf(fact_4862_unit__mult__left__cancel,axiom,
% 0.30/0.59 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.59 => ( ( ( times_3573771949741848930nteger @ A @ B )
% 0.30/0.59 = ( times_3573771949741848930nteger @ A @ C ) )
% 0.30/0.59 = ( B = C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_mult_left_cancel
% 0.30/0.59 thf(fact_4863_unit__mult__left__cancel,axiom,
% 0.30/0.59 ! [A: nat,B: nat,C: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.59 => ( ( ( times_times_nat @ A @ B )
% 0.30/0.59 = ( times_times_nat @ A @ C ) )
% 0.30/0.59 = ( B = C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_mult_left_cancel
% 0.30/0.59 thf(fact_4864_unit__mult__left__cancel,axiom,
% 0.30/0.59 ! [A: int,B: int,C: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.59 => ( ( ( times_times_int @ A @ B )
% 0.30/0.59 = ( times_times_int @ A @ C ) )
% 0.30/0.59 = ( B = C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_mult_left_cancel
% 0.30/0.59 thf(fact_4865_unit__mult__right__cancel,axiom,
% 0.30/0.59 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.59 => ( ( ( times_3573771949741848930nteger @ B @ A )
% 0.30/0.59 = ( times_3573771949741848930nteger @ C @ A ) )
% 0.30/0.59 = ( B = C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_mult_right_cancel
% 0.30/0.59 thf(fact_4866_unit__mult__right__cancel,axiom,
% 0.30/0.59 ! [A: nat,B: nat,C: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.59 => ( ( ( times_times_nat @ B @ A )
% 0.30/0.59 = ( times_times_nat @ C @ A ) )
% 0.30/0.59 = ( B = C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_mult_right_cancel
% 0.30/0.59 thf(fact_4867_unit__mult__right__cancel,axiom,
% 0.30/0.59 ! [A: int,B: int,C: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.59 => ( ( ( times_times_int @ B @ A )
% 0.30/0.59 = ( times_times_int @ C @ A ) )
% 0.30/0.59 = ( B = C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_mult_right_cancel
% 0.30/0.59 thf(fact_4868_div__mult__div__if__dvd,axiom,
% 0.30/0.59 ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ D @ C )
% 0.30/0.59 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
% 0.30/0.59 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_mult_div_if_dvd
% 0.30/0.59 thf(fact_4869_div__mult__div__if__dvd,axiom,
% 0.30/0.59 ! [B: nat,A: nat,D: nat,C: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ B @ A )
% 0.30/0.59 => ( ( dvd_dvd_nat @ D @ C )
% 0.30/0.59 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
% 0.30/0.59 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_mult_div_if_dvd
% 0.30/0.59 thf(fact_4870_div__mult__div__if__dvd,axiom,
% 0.30/0.59 ! [B: int,A: int,D: int,C: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ B @ A )
% 0.30/0.59 => ( ( dvd_dvd_int @ D @ C )
% 0.30/0.59 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
% 0.30/0.59 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_mult_div_if_dvd
% 0.30/0.59 thf(fact_4871_dvd__mult__imp__div,axiom,
% 0.30/0.59 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
% 0.30/0.59 => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mult_imp_div
% 0.30/0.59 thf(fact_4872_dvd__mult__imp__div,axiom,
% 0.30/0.59 ! [A: nat,C: nat,B: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
% 0.30/0.59 => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mult_imp_div
% 0.30/0.59 thf(fact_4873_dvd__mult__imp__div,axiom,
% 0.30/0.59 ! [A: int,C: int,B: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
% 0.30/0.59 => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mult_imp_div
% 0.30/0.59 thf(fact_4874_dvd__div__mult2__eq,axiom,
% 0.30/0.59 ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
% 0.30/0.59 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.30/0.59 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_mult2_eq
% 0.30/0.59 thf(fact_4875_dvd__div__mult2__eq,axiom,
% 0.30/0.59 ! [B: nat,C: nat,A: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
% 0.30/0.59 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.30/0.59 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_mult2_eq
% 0.30/0.59 thf(fact_4876_dvd__div__mult2__eq,axiom,
% 0.30/0.59 ! [B: int,C: int,A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
% 0.30/0.59 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.30/0.59 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_mult2_eq
% 0.30/0.59 thf(fact_4877_div__div__eq__right,axiom,
% 0.30/0.59 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.30/0.59 => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.30/0.59 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_div_eq_right
% 0.30/0.59 thf(fact_4878_div__div__eq__right,axiom,
% 0.30/0.59 ! [C: nat,B: nat,A: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.59 => ( ( dvd_dvd_nat @ B @ A )
% 0.30/0.59 => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.30/0.59 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_div_eq_right
% 0.30/0.59 thf(fact_4879_div__div__eq__right,axiom,
% 0.30/0.59 ! [C: int,B: int,A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ C @ B )
% 0.30/0.59 => ( ( dvd_dvd_int @ B @ A )
% 0.30/0.59 => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.30/0.59 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_div_eq_right
% 0.30/0.59 thf(fact_4880_div__mult__swap,axiom,
% 0.30/0.59 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.59 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.30/0.59 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_mult_swap
% 0.30/0.59 thf(fact_4881_div__mult__swap,axiom,
% 0.30/0.59 ! [C: nat,B: nat,A: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.59 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.30/0.59 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_mult_swap
% 0.30/0.59 thf(fact_4882_div__mult__swap,axiom,
% 0.30/0.59 ! [C: int,B: int,A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ C @ B )
% 0.30/0.59 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.30/0.59 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_mult_swap
% 0.30/0.59 thf(fact_4883_dvd__div__mult,axiom,
% 0.30/0.59 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.59 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
% 0.30/0.59 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_mult
% 0.30/0.59 thf(fact_4884_dvd__div__mult,axiom,
% 0.30/0.59 ! [C: nat,B: nat,A: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.59 => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
% 0.30/0.59 = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_mult
% 0.30/0.59 thf(fact_4885_dvd__div__mult,axiom,
% 0.30/0.59 ! [C: int,B: int,A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ C @ B )
% 0.30/0.59 => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
% 0.30/0.59 = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_mult
% 0.30/0.59 thf(fact_4886_unit__div__cancel,axiom,
% 0.30/0.59 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.59 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.30/0.59 = ( divide6298287555418463151nteger @ C @ A ) )
% 0.30/0.59 = ( B = C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_div_cancel
% 0.30/0.59 thf(fact_4887_unit__div__cancel,axiom,
% 0.30/0.59 ! [A: nat,B: nat,C: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.59 => ( ( ( divide_divide_nat @ B @ A )
% 0.30/0.59 = ( divide_divide_nat @ C @ A ) )
% 0.30/0.59 = ( B = C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_div_cancel
% 0.30/0.59 thf(fact_4888_unit__div__cancel,axiom,
% 0.30/0.59 ! [A: int,B: int,C: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.59 => ( ( ( divide_divide_int @ B @ A )
% 0.30/0.59 = ( divide_divide_int @ C @ A ) )
% 0.30/0.59 = ( B = C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_div_cancel
% 0.30/0.59 thf(fact_4889_div__unit__dvd__iff,axiom,
% 0.30/0.59 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_unit_dvd_iff
% 0.30/0.59 thf(fact_4890_div__unit__dvd__iff,axiom,
% 0.30/0.59 ! [B: nat,A: nat,C: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.59 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.30/0.59 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_unit_dvd_iff
% 0.30/0.59 thf(fact_4891_div__unit__dvd__iff,axiom,
% 0.30/0.59 ! [B: int,A: int,C: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.59 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.30/0.59 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_unit_dvd_iff
% 0.30/0.59 thf(fact_4892_dvd__div__unit__iff,axiom,
% 0.30/0.59 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_unit_iff
% 0.30/0.59 thf(fact_4893_dvd__div__unit__iff,axiom,
% 0.30/0.59 ! [B: nat,A: nat,C: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.59 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
% 0.30/0.59 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_unit_iff
% 0.30/0.59 thf(fact_4894_dvd__div__unit__iff,axiom,
% 0.30/0.59 ! [B: int,A: int,C: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.59 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
% 0.30/0.59 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_unit_iff
% 0.30/0.59 thf(fact_4895_div__plus__div__distrib__dvd__left,axiom,
% 0.30/0.59 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.30/0.59 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.30/0.59 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_plus_div_distrib_dvd_left
% 0.30/0.59 thf(fact_4896_div__plus__div__distrib__dvd__left,axiom,
% 0.30/0.59 ! [C: nat,A: nat,B: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ C @ A )
% 0.30/0.59 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.59 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_plus_div_distrib_dvd_left
% 0.30/0.59 thf(fact_4897_div__plus__div__distrib__dvd__left,axiom,
% 0.30/0.59 ! [C: int,A: int,B: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ C @ A )
% 0.30/0.59 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.59 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_plus_div_distrib_dvd_left
% 0.30/0.59 thf(fact_4898_div__plus__div__distrib__dvd__right,axiom,
% 0.30/0.59 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.59 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.30/0.59 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_plus_div_distrib_dvd_right
% 0.30/0.59 thf(fact_4899_div__plus__div__distrib__dvd__right,axiom,
% 0.30/0.59 ! [C: nat,B: nat,A: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.59 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.59 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_plus_div_distrib_dvd_right
% 0.30/0.59 thf(fact_4900_div__plus__div__distrib__dvd__right,axiom,
% 0.30/0.59 ! [C: int,B: int,A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ C @ B )
% 0.30/0.59 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.59 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_plus_div_distrib_dvd_right
% 0.30/0.59 thf(fact_4901_div__power,axiom,
% 0.30/0.59 ! [B: code_integer,A: code_integer,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.30/0.59 => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N )
% 0.30/0.59 = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_power
% 0.30/0.59 thf(fact_4902_div__power,axiom,
% 0.30/0.59 ! [B: nat,A: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ B @ A )
% 0.30/0.59 => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N )
% 0.30/0.59 = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_power
% 0.30/0.59 thf(fact_4903_div__power,axiom,
% 0.30/0.59 ! [B: int,A: int,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_int @ B @ A )
% 0.30/0.59 => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N )
% 0.30/0.59 = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_power
% 0.30/0.59 thf(fact_4904_mod__eq__0__iff__dvd,axiom,
% 0.30/0.59 ! [A: nat,B: nat] :
% 0.30/0.59 ( ( ( modulo_modulo_nat @ A @ B )
% 0.30/0.59 = zero_zero_nat )
% 0.30/0.59 = ( dvd_dvd_nat @ B @ A ) ) ).
% 0.30/0.59
% 0.30/0.59 % mod_eq_0_iff_dvd
% 0.30/0.59 thf(fact_4905_mod__eq__0__iff__dvd,axiom,
% 0.30/0.59 ! [A: int,B: int] :
% 0.30/0.59 ( ( ( modulo_modulo_int @ A @ B )
% 0.30/0.59 = zero_zero_int )
% 0.30/0.59 = ( dvd_dvd_int @ B @ A ) ) ).
% 0.30/0.59
% 0.30/0.59 % mod_eq_0_iff_dvd
% 0.30/0.59 thf(fact_4906_mod__eq__0__iff__dvd,axiom,
% 0.30/0.59 ! [A: code_integer,B: code_integer] :
% 0.30/0.59 ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.30/0.59 = zero_z3403309356797280102nteger )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.30/0.59
% 0.30/0.59 % mod_eq_0_iff_dvd
% 0.30/0.59 thf(fact_4907_dvd__eq__mod__eq__0,axiom,
% 0.30/0.59 ( dvd_dvd_nat
% 0.30/0.59 = ( ^ [A3: nat,B2: nat] :
% 0.30/0.59 ( ( modulo_modulo_nat @ B2 @ A3 )
% 0.30/0.59 = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_eq_mod_eq_0
% 0.30/0.59 thf(fact_4908_dvd__eq__mod__eq__0,axiom,
% 0.30/0.59 ( dvd_dvd_int
% 0.30/0.59 = ( ^ [A3: int,B2: int] :
% 0.30/0.59 ( ( modulo_modulo_int @ B2 @ A3 )
% 0.30/0.59 = zero_zero_int ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_eq_mod_eq_0
% 0.30/0.59 thf(fact_4909_dvd__eq__mod__eq__0,axiom,
% 0.30/0.59 ( dvd_dvd_Code_integer
% 0.30/0.59 = ( ^ [A3: code_integer,B2: code_integer] :
% 0.30/0.59 ( ( modulo364778990260209775nteger @ B2 @ A3 )
% 0.30/0.59 = zero_z3403309356797280102nteger ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_eq_mod_eq_0
% 0.30/0.59 thf(fact_4910_mod__0__imp__dvd,axiom,
% 0.30/0.59 ! [A: nat,B: nat] :
% 0.30/0.59 ( ( ( modulo_modulo_nat @ A @ B )
% 0.30/0.59 = zero_zero_nat )
% 0.30/0.59 => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.30/0.59
% 0.30/0.59 % mod_0_imp_dvd
% 0.30/0.59 thf(fact_4911_mod__0__imp__dvd,axiom,
% 0.30/0.59 ! [A: int,B: int] :
% 0.30/0.59 ( ( ( modulo_modulo_int @ A @ B )
% 0.30/0.59 = zero_zero_int )
% 0.30/0.59 => ( dvd_dvd_int @ B @ A ) ) ).
% 0.30/0.59
% 0.30/0.59 % mod_0_imp_dvd
% 0.30/0.59 thf(fact_4912_mod__0__imp__dvd,axiom,
% 0.30/0.59 ! [A: code_integer,B: code_integer] :
% 0.30/0.59 ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.30/0.59 = zero_z3403309356797280102nteger )
% 0.30/0.59 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.30/0.59
% 0.30/0.59 % mod_0_imp_dvd
% 0.30/0.59 thf(fact_4913_dvd__power__le,axiom,
% 0.30/0.59 ! [X2: code_integer,Y3: code_integer,N: nat,M: nat] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ X2 @ Y3 )
% 0.30/0.59 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.59 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X2 @ N ) @ ( power_8256067586552552935nteger @ Y3 @ M ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power_le
% 0.30/0.59 thf(fact_4914_dvd__power__le,axiom,
% 0.30/0.59 ! [X2: nat,Y3: nat,N: nat,M: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ X2 @ Y3 )
% 0.30/0.59 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.59 => ( dvd_dvd_nat @ ( power_power_nat @ X2 @ N ) @ ( power_power_nat @ Y3 @ M ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power_le
% 0.30/0.59 thf(fact_4915_dvd__power__le,axiom,
% 0.30/0.59 ! [X2: real,Y3: real,N: nat,M: nat] :
% 0.30/0.59 ( ( dvd_dvd_real @ X2 @ Y3 )
% 0.30/0.59 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.59 => ( dvd_dvd_real @ ( power_power_real @ X2 @ N ) @ ( power_power_real @ Y3 @ M ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power_le
% 0.30/0.59 thf(fact_4916_dvd__power__le,axiom,
% 0.30/0.59 ! [X2: int,Y3: int,N: nat,M: nat] :
% 0.30/0.59 ( ( dvd_dvd_int @ X2 @ Y3 )
% 0.30/0.59 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.59 => ( dvd_dvd_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y3 @ M ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power_le
% 0.30/0.59 thf(fact_4917_dvd__power__le,axiom,
% 0.30/0.59 ! [X2: complex,Y3: complex,N: nat,M: nat] :
% 0.30/0.59 ( ( dvd_dvd_complex @ X2 @ Y3 )
% 0.30/0.59 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.59 => ( dvd_dvd_complex @ ( power_power_complex @ X2 @ N ) @ ( power_power_complex @ Y3 @ M ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power_le
% 0.30/0.59 thf(fact_4918_power__le__dvd,axiom,
% 0.30/0.59 ! [A: code_integer,N: nat,B: code_integer,M: nat] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
% 0.30/0.59 => ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.59 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % power_le_dvd
% 0.30/0.59 thf(fact_4919_power__le__dvd,axiom,
% 0.30/0.59 ! [A: nat,N: nat,B: nat,M: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
% 0.30/0.59 => ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.59 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % power_le_dvd
% 0.30/0.59 thf(fact_4920_power__le__dvd,axiom,
% 0.30/0.59 ! [A: real,N: nat,B: real,M: nat] :
% 0.30/0.59 ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
% 0.30/0.59 => ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.59 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % power_le_dvd
% 0.30/0.59 thf(fact_4921_power__le__dvd,axiom,
% 0.30/0.59 ! [A: int,N: nat,B: int,M: nat] :
% 0.30/0.59 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
% 0.30/0.59 => ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.59 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % power_le_dvd
% 0.30/0.59 thf(fact_4922_power__le__dvd,axiom,
% 0.30/0.59 ! [A: complex,N: nat,B: complex,M: nat] :
% 0.30/0.59 ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
% 0.30/0.59 => ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.59 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % power_le_dvd
% 0.30/0.59 thf(fact_4923_le__imp__power__dvd,axiom,
% 0.30/0.59 ! [M: nat,N: nat,A: code_integer] :
% 0.30/0.59 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.59 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % le_imp_power_dvd
% 0.30/0.59 thf(fact_4924_le__imp__power__dvd,axiom,
% 0.30/0.59 ! [M: nat,N: nat,A: nat] :
% 0.30/0.59 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.59 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % le_imp_power_dvd
% 0.30/0.59 thf(fact_4925_le__imp__power__dvd,axiom,
% 0.30/0.59 ! [M: nat,N: nat,A: real] :
% 0.30/0.59 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.59 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % le_imp_power_dvd
% 0.30/0.59 thf(fact_4926_le__imp__power__dvd,axiom,
% 0.30/0.59 ! [M: nat,N: nat,A: int] :
% 0.30/0.59 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.59 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % le_imp_power_dvd
% 0.30/0.59 thf(fact_4927_le__imp__power__dvd,axiom,
% 0.30/0.59 ! [M: nat,N: nat,A: complex] :
% 0.30/0.59 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.59 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % le_imp_power_dvd
% 0.30/0.59 thf(fact_4928_dvd__minus__mod,axiom,
% 0.30/0.59 ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_minus_mod
% 0.30/0.59 thf(fact_4929_dvd__minus__mod,axiom,
% 0.30/0.59 ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_minus_mod
% 0.30/0.59 thf(fact_4930_dvd__minus__mod,axiom,
% 0.30/0.59 ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_minus_mod
% 0.30/0.59 thf(fact_4931_mod__eq__dvd__iff,axiom,
% 0.30/0.59 ! [A: int,C: int,B: int] :
% 0.30/0.59 ( ( ( modulo_modulo_int @ A @ C )
% 0.30/0.59 = ( modulo_modulo_int @ B @ C ) )
% 0.30/0.59 = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mod_eq_dvd_iff
% 0.30/0.59 thf(fact_4932_mod__eq__dvd__iff,axiom,
% 0.30/0.59 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.59 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.30/0.59 = ( modulo364778990260209775nteger @ B @ C ) )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mod_eq_dvd_iff
% 0.30/0.59 thf(fact_4933_nat__dvd__not__less,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.59 => ( ( ord_less_nat @ M @ N )
% 0.30/0.59 => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % nat_dvd_not_less
% 0.30/0.59 thf(fact_4934_bezout__add__strong__nat,axiom,
% 0.30/0.59 ! [A: nat,B: nat] :
% 0.30/0.59 ( ( A != zero_zero_nat )
% 0.30/0.59 => ? [D3: nat,X5: nat,Y4: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ D3 @ A )
% 0.30/0.59 & ( dvd_dvd_nat @ D3 @ B )
% 0.30/0.59 & ( ( times_times_nat @ A @ X5 )
% 0.30/0.59 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D3 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % bezout_add_strong_nat
% 0.30/0.59 thf(fact_4935_dvd__minus__self,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
% 0.30/0.59 = ( ( ord_less_nat @ N @ M )
% 0.30/0.59 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_minus_self
% 0.30/0.59 thf(fact_4936_zdvd__antisym__nonneg,axiom,
% 0.30/0.59 ! [M: int,N: int] :
% 0.30/0.59 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 0.30/0.59 => ( ( ord_less_eq_int @ zero_zero_int @ N )
% 0.30/0.59 => ( ( dvd_dvd_int @ M @ N )
% 0.30/0.59 => ( ( dvd_dvd_int @ N @ M )
% 0.30/0.59 => ( M = N ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % zdvd_antisym_nonneg
% 0.30/0.59 thf(fact_4937_less__eq__dvd__minus,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.59 => ( ( dvd_dvd_nat @ M @ N )
% 0.30/0.59 = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % less_eq_dvd_minus
% 0.30/0.59 thf(fact_4938_dvd__diffD1,axiom,
% 0.30/0.59 ! [K: nat,M: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.59 => ( ( dvd_dvd_nat @ K @ M )
% 0.30/0.59 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.59 => ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_diffD1
% 0.30/0.59 thf(fact_4939_dvd__diffD,axiom,
% 0.30/0.59 ! [K: nat,M: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.59 => ( ( dvd_dvd_nat @ K @ N )
% 0.30/0.59 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.59 => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_diffD
% 0.30/0.59 thf(fact_4940_zdvd__mult__cancel,axiom,
% 0.30/0.59 ! [K: int,M: int,N: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
% 0.30/0.59 => ( ( K != zero_zero_int )
% 0.30/0.59 => ( dvd_dvd_int @ M @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % zdvd_mult_cancel
% 0.30/0.59 thf(fact_4941_zdvd__mono,axiom,
% 0.30/0.59 ! [K: int,M: int,T: int] :
% 0.30/0.59 ( ( K != zero_zero_int )
% 0.30/0.59 => ( ( dvd_dvd_int @ M @ T )
% 0.30/0.59 = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % zdvd_mono
% 0.30/0.59 thf(fact_4942_dbl__def,axiom,
% 0.30/0.59 ( neg_numeral_dbl_real
% 0.30/0.59 = ( ^ [X: real] : ( plus_plus_real @ X @ X ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dbl_def
% 0.30/0.59 thf(fact_4943_dbl__def,axiom,
% 0.30/0.59 ( neg_numeral_dbl_rat
% 0.30/0.59 = ( ^ [X: rat] : ( plus_plus_rat @ X @ X ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dbl_def
% 0.30/0.59 thf(fact_4944_dbl__def,axiom,
% 0.30/0.59 ( neg_numeral_dbl_int
% 0.30/0.59 = ( ^ [X: int] : ( plus_plus_int @ X @ X ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dbl_def
% 0.30/0.59 thf(fact_4945_zdvd__reduce,axiom,
% 0.30/0.59 ! [K: int,N: int,M: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
% 0.30/0.59 = ( dvd_dvd_int @ K @ N ) ) ).
% 0.30/0.59
% 0.30/0.59 % zdvd_reduce
% 0.30/0.59 thf(fact_4946_zdvd__period,axiom,
% 0.30/0.59 ! [A: int,D: int,X2: int,T: int,C: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ A @ D )
% 0.30/0.59 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X2 @ T ) )
% 0.30/0.59 = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X2 @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % zdvd_period
% 0.30/0.59 thf(fact_4947_finite__divisors__nat,axiom,
% 0.30/0.59 ! [M: nat] :
% 0.30/0.59 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.59 => ( finite_finite_nat
% 0.30/0.59 @ ( collect_nat
% 0.30/0.59 @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % finite_divisors_nat
% 0.30/0.59 thf(fact_4948_div2__even__ext__nat,axiom,
% 0.30/0.59 ! [X2: nat,Y3: nat] :
% 0.30/0.59 ( ( ( divide_divide_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59 = ( divide_divide_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.59 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 )
% 0.30/0.59 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y3 ) )
% 0.30/0.59 => ( X2 = Y3 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div2_even_ext_nat
% 0.30/0.59 thf(fact_4949_unit__dvdE,axiom,
% 0.30/0.59 ! [A: code_integer,B: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.59 => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.59 => ! [C3: code_integer] :
% 0.30/0.59 ( B
% 0.30/0.59 != ( times_3573771949741848930nteger @ A @ C3 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_dvdE
% 0.30/0.59 thf(fact_4950_unit__dvdE,axiom,
% 0.30/0.59 ! [A: nat,B: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.59 => ~ ( ( A != zero_zero_nat )
% 0.30/0.59 => ! [C3: nat] :
% 0.30/0.59 ( B
% 0.30/0.59 != ( times_times_nat @ A @ C3 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_dvdE
% 0.30/0.59 thf(fact_4951_unit__dvdE,axiom,
% 0.30/0.59 ! [A: int,B: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.59 => ~ ( ( A != zero_zero_int )
% 0.30/0.59 => ! [C3: int] :
% 0.30/0.59 ( B
% 0.30/0.59 != ( times_times_int @ A @ C3 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_dvdE
% 0.30/0.59 thf(fact_4952_unity__coeff__ex,axiom,
% 0.30/0.59 ! [P: code_integer > $o,L: code_integer] :
% 0.30/0.59 ( ( ? [X: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X ) ) )
% 0.30/0.59 = ( ? [X: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X @ zero_z3403309356797280102nteger ) )
% 0.30/0.59 & ( P @ X ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unity_coeff_ex
% 0.30/0.59 thf(fact_4953_unity__coeff__ex,axiom,
% 0.30/0.59 ! [P: complex > $o,L: complex] :
% 0.30/0.59 ( ( ? [X: complex] : ( P @ ( times_times_complex @ L @ X ) ) )
% 0.30/0.59 = ( ? [X: complex] :
% 0.30/0.59 ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X @ zero_zero_complex ) )
% 0.30/0.59 & ( P @ X ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unity_coeff_ex
% 0.30/0.59 thf(fact_4954_unity__coeff__ex,axiom,
% 0.30/0.59 ! [P: real > $o,L: real] :
% 0.30/0.59 ( ( ? [X: real] : ( P @ ( times_times_real @ L @ X ) ) )
% 0.30/0.59 = ( ? [X: real] :
% 0.30/0.59 ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X @ zero_zero_real ) )
% 0.30/0.59 & ( P @ X ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unity_coeff_ex
% 0.30/0.59 thf(fact_4955_unity__coeff__ex,axiom,
% 0.30/0.59 ! [P: rat > $o,L: rat] :
% 0.30/0.59 ( ( ? [X: rat] : ( P @ ( times_times_rat @ L @ X ) ) )
% 0.30/0.59 = ( ? [X: rat] :
% 0.30/0.59 ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X @ zero_zero_rat ) )
% 0.30/0.59 & ( P @ X ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unity_coeff_ex
% 0.30/0.59 thf(fact_4956_unity__coeff__ex,axiom,
% 0.30/0.59 ! [P: nat > $o,L: nat] :
% 0.30/0.59 ( ( ? [X: nat] : ( P @ ( times_times_nat @ L @ X ) ) )
% 0.30/0.59 = ( ? [X: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X @ zero_zero_nat ) )
% 0.30/0.59 & ( P @ X ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unity_coeff_ex
% 0.30/0.59 thf(fact_4957_unity__coeff__ex,axiom,
% 0.30/0.59 ! [P: int > $o,L: int] :
% 0.30/0.59 ( ( ? [X: int] : ( P @ ( times_times_int @ L @ X ) ) )
% 0.30/0.59 = ( ? [X: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X @ zero_zero_int ) )
% 0.30/0.59 & ( P @ X ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unity_coeff_ex
% 0.30/0.59 thf(fact_4958_dvd__div__div__eq__mult,axiom,
% 0.30/0.59 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 0.30/0.59 ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.59 => ( ( C != zero_z3403309356797280102nteger )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ C @ D )
% 0.30/0.59 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.30/0.59 = ( divide6298287555418463151nteger @ D @ C ) )
% 0.30/0.59 = ( ( times_3573771949741848930nteger @ B @ C )
% 0.30/0.59 = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_div_eq_mult
% 0.30/0.59 thf(fact_4959_dvd__div__div__eq__mult,axiom,
% 0.30/0.59 ! [A: nat,C: nat,B: nat,D: nat] :
% 0.30/0.59 ( ( A != zero_zero_nat )
% 0.30/0.59 => ( ( C != zero_zero_nat )
% 0.30/0.59 => ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.59 => ( ( dvd_dvd_nat @ C @ D )
% 0.30/0.59 => ( ( ( divide_divide_nat @ B @ A )
% 0.30/0.59 = ( divide_divide_nat @ D @ C ) )
% 0.30/0.59 = ( ( times_times_nat @ B @ C )
% 0.30/0.59 = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_div_eq_mult
% 0.30/0.59 thf(fact_4960_dvd__div__div__eq__mult,axiom,
% 0.30/0.59 ! [A: int,C: int,B: int,D: int] :
% 0.30/0.59 ( ( A != zero_zero_int )
% 0.30/0.59 => ( ( C != zero_zero_int )
% 0.30/0.59 => ( ( dvd_dvd_int @ A @ B )
% 0.30/0.59 => ( ( dvd_dvd_int @ C @ D )
% 0.30/0.59 => ( ( ( divide_divide_int @ B @ A )
% 0.30/0.59 = ( divide_divide_int @ D @ C ) )
% 0.30/0.59 = ( ( times_times_int @ B @ C )
% 0.30/0.59 = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_div_eq_mult
% 0.30/0.59 thf(fact_4961_dvd__div__iff__mult,axiom,
% 0.30/0.59 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.59 ( ( C != zero_z3403309356797280102nteger )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_iff_mult
% 0.30/0.59 thf(fact_4962_dvd__div__iff__mult,axiom,
% 0.30/0.59 ! [C: nat,B: nat,A: nat] :
% 0.30/0.59 ( ( C != zero_zero_nat )
% 0.30/0.59 => ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.59 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.30/0.59 = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_iff_mult
% 0.30/0.59 thf(fact_4963_dvd__div__iff__mult,axiom,
% 0.30/0.59 ! [C: int,B: int,A: int] :
% 0.30/0.59 ( ( C != zero_zero_int )
% 0.30/0.59 => ( ( dvd_dvd_int @ C @ B )
% 0.30/0.59 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.30/0.59 = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_iff_mult
% 0.30/0.59 thf(fact_4964_div__dvd__iff__mult,axiom,
% 0.30/0.59 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.59 ( ( B != zero_z3403309356797280102nteger )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_dvd_iff_mult
% 0.30/0.59 thf(fact_4965_div__dvd__iff__mult,axiom,
% 0.30/0.59 ! [B: nat,A: nat,C: nat] :
% 0.30/0.59 ( ( B != zero_zero_nat )
% 0.30/0.59 => ( ( dvd_dvd_nat @ B @ A )
% 0.30/0.59 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.30/0.59 = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_dvd_iff_mult
% 0.30/0.59 thf(fact_4966_div__dvd__iff__mult,axiom,
% 0.30/0.59 ! [B: int,A: int,C: int] :
% 0.30/0.59 ( ( B != zero_zero_int )
% 0.30/0.59 => ( ( dvd_dvd_int @ B @ A )
% 0.30/0.59 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.30/0.59 = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_dvd_iff_mult
% 0.30/0.59 thf(fact_4967_dvd__div__eq__mult,axiom,
% 0.30/0.59 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.59 ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.59 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.30/0.59 = C )
% 0.30/0.59 = ( B
% 0.30/0.59 = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_mult
% 0.30/0.59 thf(fact_4968_dvd__div__eq__mult,axiom,
% 0.30/0.59 ! [A: nat,B: nat,C: nat] :
% 0.30/0.59 ( ( A != zero_zero_nat )
% 0.30/0.59 => ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.59 => ( ( ( divide_divide_nat @ B @ A )
% 0.30/0.59 = C )
% 0.30/0.59 = ( B
% 0.30/0.59 = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_mult
% 0.30/0.59 thf(fact_4969_dvd__div__eq__mult,axiom,
% 0.30/0.59 ! [A: int,B: int,C: int] :
% 0.30/0.59 ( ( A != zero_zero_int )
% 0.30/0.59 => ( ( dvd_dvd_int @ A @ B )
% 0.30/0.59 => ( ( ( divide_divide_int @ B @ A )
% 0.30/0.59 = C )
% 0.30/0.59 = ( B
% 0.30/0.59 = ( times_times_int @ C @ A ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_div_eq_mult
% 0.30/0.59 thf(fact_4970_unit__div__eq__0__iff,axiom,
% 0.30/0.59 ! [B: code_integer,A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.59 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.30/0.59 = zero_z3403309356797280102nteger )
% 0.30/0.59 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_div_eq_0_iff
% 0.30/0.59 thf(fact_4971_unit__div__eq__0__iff,axiom,
% 0.30/0.59 ! [B: nat,A: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.59 => ( ( ( divide_divide_nat @ A @ B )
% 0.30/0.59 = zero_zero_nat )
% 0.30/0.59 = ( A = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_div_eq_0_iff
% 0.30/0.59 thf(fact_4972_unit__div__eq__0__iff,axiom,
% 0.30/0.59 ! [B: int,A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.59 => ( ( ( divide_divide_int @ A @ B )
% 0.30/0.59 = zero_zero_int )
% 0.30/0.59 = ( A = zero_zero_int ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_div_eq_0_iff
% 0.30/0.59 thf(fact_4973_even__numeral,axiom,
% 0.30/0.59 ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_numeral
% 0.30/0.59 thf(fact_4974_even__numeral,axiom,
% 0.30/0.59 ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_numeral
% 0.30/0.59 thf(fact_4975_even__numeral,axiom,
% 0.30/0.59 ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_numeral
% 0.30/0.59 thf(fact_4976_inf__period_I3_J,axiom,
% 0.30/0.59 ! [D: code_integer,D4: code_integer,T: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ D @ D4 )
% 0.30/0.59 => ! [X3: code_integer,K4: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ T ) )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % inf_period(3)
% 0.30/0.59 thf(fact_4977_inf__period_I3_J,axiom,
% 0.30/0.59 ! [D: real,D4: real,T: real] :
% 0.30/0.59 ( ( dvd_dvd_real @ D @ D4 )
% 0.30/0.59 => ! [X3: real,K4: real] :
% 0.30/0.59 ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ T ) )
% 0.30/0.59 = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % inf_period(3)
% 0.30/0.59 thf(fact_4978_inf__period_I3_J,axiom,
% 0.30/0.59 ! [D: rat,D4: rat,T: rat] :
% 0.30/0.59 ( ( dvd_dvd_rat @ D @ D4 )
% 0.30/0.59 => ! [X3: rat,K4: rat] :
% 0.30/0.59 ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ T ) )
% 0.30/0.59 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % inf_period(3)
% 0.30/0.59 thf(fact_4979_inf__period_I3_J,axiom,
% 0.30/0.59 ! [D: int,D4: int,T: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ D @ D4 )
% 0.30/0.59 => ! [X3: int,K4: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) )
% 0.30/0.59 = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % inf_period(3)
% 0.30/0.59 thf(fact_4980_inf__period_I4_J,axiom,
% 0.30/0.59 ! [D: code_integer,D4: code_integer,T: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ D @ D4 )
% 0.30/0.59 => ! [X3: code_integer,K4: code_integer] :
% 0.30/0.59 ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ T ) ) )
% 0.30/0.59 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % inf_period(4)
% 0.30/0.59 thf(fact_4981_inf__period_I4_J,axiom,
% 0.30/0.59 ! [D: real,D4: real,T: real] :
% 0.30/0.59 ( ( dvd_dvd_real @ D @ D4 )
% 0.30/0.59 => ! [X3: real,K4: real] :
% 0.30/0.59 ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ T ) ) )
% 0.30/0.59 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % inf_period(4)
% 0.30/0.59 thf(fact_4982_inf__period_I4_J,axiom,
% 0.30/0.59 ! [D: rat,D4: rat,T: rat] :
% 0.30/0.59 ( ( dvd_dvd_rat @ D @ D4 )
% 0.30/0.59 => ! [X3: rat,K4: rat] :
% 0.30/0.59 ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ T ) ) )
% 0.30/0.59 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % inf_period(4)
% 0.30/0.59 thf(fact_4983_inf__period_I4_J,axiom,
% 0.30/0.59 ! [D: int,D4: int,T: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ D @ D4 )
% 0.30/0.59 => ! [X3: int,K4: int] :
% 0.30/0.59 ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) ) )
% 0.30/0.59 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % inf_period(4)
% 0.30/0.59 thf(fact_4984_is__unit__div__mult2__eq,axiom,
% 0.30/0.59 ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.30/0.59 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.30/0.59 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % is_unit_div_mult2_eq
% 0.30/0.59 thf(fact_4985_is__unit__div__mult2__eq,axiom,
% 0.30/0.59 ! [B: nat,C: nat,A: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.59 => ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.30/0.59 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.30/0.59 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % is_unit_div_mult2_eq
% 0.30/0.59 thf(fact_4986_is__unit__div__mult2__eq,axiom,
% 0.30/0.59 ! [B: int,C: int,A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.59 => ( ( dvd_dvd_int @ C @ one_one_int )
% 0.30/0.59 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.30/0.59 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % is_unit_div_mult2_eq
% 0.30/0.59 thf(fact_4987_unit__div__mult__swap,axiom,
% 0.30/0.59 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.30/0.59 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.30/0.59 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_div_mult_swap
% 0.30/0.59 thf(fact_4988_unit__div__mult__swap,axiom,
% 0.30/0.59 ! [C: nat,A: nat,B: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.30/0.59 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.30/0.59 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_div_mult_swap
% 0.30/0.59 thf(fact_4989_unit__div__mult__swap,axiom,
% 0.30/0.59 ! [C: int,A: int,B: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ C @ one_one_int )
% 0.30/0.59 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.30/0.59 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_div_mult_swap
% 0.30/0.59 thf(fact_4990_unit__div__commute,axiom,
% 0.30/0.59 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.59 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.30/0.59 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_div_commute
% 0.30/0.59 thf(fact_4991_unit__div__commute,axiom,
% 0.30/0.59 ! [B: nat,A: nat,C: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.59 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.30/0.59 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_div_commute
% 0.30/0.59 thf(fact_4992_unit__div__commute,axiom,
% 0.30/0.59 ! [B: int,A: int,C: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.59 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.30/0.59 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_div_commute
% 0.30/0.59 thf(fact_4993_div__mult__unit2,axiom,
% 0.30/0.59 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.30/0.59 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.30/0.59 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_mult_unit2
% 0.30/0.59 thf(fact_4994_div__mult__unit2,axiom,
% 0.30/0.59 ! [C: nat,B: nat,A: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.30/0.59 => ( ( dvd_dvd_nat @ B @ A )
% 0.30/0.59 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.30/0.59 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_mult_unit2
% 0.30/0.59 thf(fact_4995_div__mult__unit2,axiom,
% 0.30/0.59 ! [C: int,B: int,A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ C @ one_one_int )
% 0.30/0.59 => ( ( dvd_dvd_int @ B @ A )
% 0.30/0.59 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.30/0.59 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % div_mult_unit2
% 0.30/0.59 thf(fact_4996_unit__eq__div2,axiom,
% 0.30/0.59 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.59 => ( ( A
% 0.30/0.59 = ( divide6298287555418463151nteger @ C @ B ) )
% 0.30/0.59 = ( ( times_3573771949741848930nteger @ A @ B )
% 0.30/0.59 = C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_eq_div2
% 0.30/0.59 thf(fact_4997_unit__eq__div2,axiom,
% 0.30/0.59 ! [B: nat,A: nat,C: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.59 => ( ( A
% 0.30/0.59 = ( divide_divide_nat @ C @ B ) )
% 0.30/0.59 = ( ( times_times_nat @ A @ B )
% 0.30/0.59 = C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_eq_div2
% 0.30/0.59 thf(fact_4998_unit__eq__div2,axiom,
% 0.30/0.59 ! [B: int,A: int,C: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.59 => ( ( A
% 0.30/0.59 = ( divide_divide_int @ C @ B ) )
% 0.30/0.59 = ( ( times_times_int @ A @ B )
% 0.30/0.59 = C ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_eq_div2
% 0.30/0.59 thf(fact_4999_unit__eq__div1,axiom,
% 0.30/0.59 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.59 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.30/0.59 = C )
% 0.30/0.59 = ( A
% 0.30/0.59 = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_eq_div1
% 0.30/0.59 thf(fact_5000_unit__eq__div1,axiom,
% 0.30/0.59 ! [B: nat,A: nat,C: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.59 => ( ( ( divide_divide_nat @ A @ B )
% 0.30/0.59 = C )
% 0.30/0.59 = ( A
% 0.30/0.59 = ( times_times_nat @ C @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_eq_div1
% 0.30/0.59 thf(fact_5001_unit__eq__div1,axiom,
% 0.30/0.59 ! [B: int,A: int,C: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.59 => ( ( ( divide_divide_int @ A @ B )
% 0.30/0.59 = C )
% 0.30/0.59 = ( A
% 0.30/0.59 = ( times_times_int @ C @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_eq_div1
% 0.30/0.59 thf(fact_5002_is__unit__power__iff,axiom,
% 0.30/0.59 ! [A: code_integer,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
% 0.30/0.59 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.59 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % is_unit_power_iff
% 0.30/0.59 thf(fact_5003_is__unit__power__iff,axiom,
% 0.30/0.59 ! [A: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
% 0.30/0.59 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.59 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % is_unit_power_iff
% 0.30/0.59 thf(fact_5004_is__unit__power__iff,axiom,
% 0.30/0.59 ! [A: int,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
% 0.30/0.59 = ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.59 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % is_unit_power_iff
% 0.30/0.59 thf(fact_5005_unit__imp__mod__eq__0,axiom,
% 0.30/0.59 ! [B: nat,A: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.59 => ( ( modulo_modulo_nat @ A @ B )
% 0.30/0.59 = zero_zero_nat ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_imp_mod_eq_0
% 0.30/0.59 thf(fact_5006_unit__imp__mod__eq__0,axiom,
% 0.30/0.59 ! [B: int,A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.59 => ( ( modulo_modulo_int @ A @ B )
% 0.30/0.59 = zero_zero_int ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_imp_mod_eq_0
% 0.30/0.59 thf(fact_5007_unit__imp__mod__eq__0,axiom,
% 0.30/0.59 ! [B: code_integer,A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.59 => ( ( modulo364778990260209775nteger @ A @ B )
% 0.30/0.59 = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.59
% 0.30/0.59 % unit_imp_mod_eq_0
% 0.30/0.59 thf(fact_5008_dvd__imp__le,axiom,
% 0.30/0.59 ! [K: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ K @ N )
% 0.30/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.59 => ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_imp_le
% 0.30/0.59 thf(fact_5009_nat__mult__dvd__cancel1,axiom,
% 0.30/0.59 ! [K: nat,M: nat,N: nat] :
% 0.30/0.59 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.59 => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.59 = ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % nat_mult_dvd_cancel1
% 0.30/0.59 thf(fact_5010_dvd__mult__cancel,axiom,
% 0.30/0.59 ! [K: nat,M: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.59 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.59 => ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mult_cancel
% 0.30/0.59 thf(fact_5011_zdvd__imp__le,axiom,
% 0.30/0.59 ! [Z: int,N: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ Z @ N )
% 0.30/0.59 => ( ( ord_less_int @ zero_zero_int @ N )
% 0.30/0.59 => ( ord_less_eq_int @ Z @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % zdvd_imp_le
% 0.30/0.59 thf(fact_5012_mod__greater__zero__iff__not__dvd,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
% 0.30/0.59 = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mod_greater_zero_iff_not_dvd
% 0.30/0.59 thf(fact_5013_mod__eq__dvd__iff__nat,axiom,
% 0.30/0.59 ! [N: nat,M: nat,Q3: nat] :
% 0.30/0.59 ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.59 => ( ( ( modulo_modulo_nat @ M @ Q3 )
% 0.30/0.59 = ( modulo_modulo_nat @ N @ Q3 ) )
% 0.30/0.59 = ( dvd_dvd_nat @ Q3 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mod_eq_dvd_iff_nat
% 0.30/0.59 thf(fact_5014_prod__decode__aux_Ocases,axiom,
% 0.30/0.59 ! [X2: product_prod_nat_nat] :
% 0.30/0.59 ~ ! [K3: nat,M4: nat] :
% 0.30/0.59 ( X2
% 0.30/0.59 != ( product_Pair_nat_nat @ K3 @ M4 ) ) ).
% 0.30/0.59
% 0.30/0.59 % prod_decode_aux.cases
% 0.30/0.59 thf(fact_5015_even__zero,axiom,
% 0.30/0.59 dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 0.30/0.59
% 0.30/0.59 % even_zero
% 0.30/0.59 thf(fact_5016_even__zero,axiom,
% 0.30/0.59 dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 0.30/0.59
% 0.30/0.59 % even_zero
% 0.30/0.59 thf(fact_5017_even__zero,axiom,
% 0.30/0.59 dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 0.30/0.59
% 0.30/0.59 % even_zero
% 0.30/0.59 thf(fact_5018_is__unitE,axiom,
% 0.30/0.59 ! [A: code_integer,C: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.59 => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.59 => ! [B4: code_integer] :
% 0.30/0.59 ( ( B4 != zero_z3403309356797280102nteger )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ B4 @ one_one_Code_integer )
% 0.30/0.59 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 0.30/0.59 = B4 )
% 0.30/0.59 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B4 )
% 0.30/0.59 = A )
% 0.30/0.59 => ( ( ( times_3573771949741848930nteger @ A @ B4 )
% 0.30/0.59 = one_one_Code_integer )
% 0.30/0.59 => ( ( divide6298287555418463151nteger @ C @ A )
% 0.30/0.59 != ( times_3573771949741848930nteger @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % is_unitE
% 0.30/0.59 thf(fact_5019_is__unitE,axiom,
% 0.30/0.59 ! [A: nat,C: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.59 => ~ ( ( A != zero_zero_nat )
% 0.30/0.59 => ! [B4: nat] :
% 0.30/0.59 ( ( B4 != zero_zero_nat )
% 0.30/0.59 => ( ( dvd_dvd_nat @ B4 @ one_one_nat )
% 0.30/0.59 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 0.30/0.59 = B4 )
% 0.30/0.59 => ( ( ( divide_divide_nat @ one_one_nat @ B4 )
% 0.30/0.59 = A )
% 0.30/0.59 => ( ( ( times_times_nat @ A @ B4 )
% 0.30/0.59 = one_one_nat )
% 0.30/0.59 => ( ( divide_divide_nat @ C @ A )
% 0.30/0.59 != ( times_times_nat @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % is_unitE
% 0.30/0.59 thf(fact_5020_is__unitE,axiom,
% 0.30/0.59 ! [A: int,C: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.59 => ~ ( ( A != zero_zero_int )
% 0.30/0.59 => ! [B4: int] :
% 0.30/0.59 ( ( B4 != zero_zero_int )
% 0.30/0.59 => ( ( dvd_dvd_int @ B4 @ one_one_int )
% 0.30/0.59 => ( ( ( divide_divide_int @ one_one_int @ A )
% 0.30/0.59 = B4 )
% 0.30/0.59 => ( ( ( divide_divide_int @ one_one_int @ B4 )
% 0.30/0.59 = A )
% 0.30/0.59 => ( ( ( times_times_int @ A @ B4 )
% 0.30/0.59 = one_one_int )
% 0.30/0.59 => ( ( divide_divide_int @ C @ A )
% 0.30/0.59 != ( times_times_int @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % is_unitE
% 0.30/0.59 thf(fact_5021_is__unit__div__mult__cancel__left,axiom,
% 0.30/0.59 ! [A: code_integer,B: code_integer] :
% 0.30/0.59 ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.59 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.30/0.59 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % is_unit_div_mult_cancel_left
% 0.30/0.59 thf(fact_5022_is__unit__div__mult__cancel__left,axiom,
% 0.30/0.59 ! [A: nat,B: nat] :
% 0.30/0.59 ( ( A != zero_zero_nat )
% 0.30/0.59 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.59 => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
% 0.30/0.59 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % is_unit_div_mult_cancel_left
% 0.30/0.59 thf(fact_5023_is__unit__div__mult__cancel__left,axiom,
% 0.30/0.59 ! [A: int,B: int] :
% 0.30/0.59 ( ( A != zero_zero_int )
% 0.30/0.59 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.59 => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
% 0.30/0.59 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % is_unit_div_mult_cancel_left
% 0.30/0.59 thf(fact_5024_is__unit__div__mult__cancel__right,axiom,
% 0.30/0.59 ! [A: code_integer,B: code_integer] :
% 0.30/0.59 ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.59 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
% 0.30/0.59 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % is_unit_div_mult_cancel_right
% 0.30/0.59 thf(fact_5025_is__unit__div__mult__cancel__right,axiom,
% 0.30/0.59 ! [A: nat,B: nat] :
% 0.30/0.59 ( ( A != zero_zero_nat )
% 0.30/0.59 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.59 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
% 0.30/0.59 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % is_unit_div_mult_cancel_right
% 0.30/0.59 thf(fact_5026_is__unit__div__mult__cancel__right,axiom,
% 0.30/0.59 ! [A: int,B: int] :
% 0.30/0.59 ( ( A != zero_zero_int )
% 0.30/0.59 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.59 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
% 0.30/0.59 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % is_unit_div_mult_cancel_right
% 0.30/0.59 thf(fact_5027_evenE,axiom,
% 0.30/0.59 ! [A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ~ ! [B4: code_integer] :
% 0.30/0.59 ( A
% 0.30/0.59 != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % evenE
% 0.30/0.59 thf(fact_5028_evenE,axiom,
% 0.30/0.59 ! [A: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ~ ! [B4: nat] :
% 0.30/0.59 ( A
% 0.30/0.59 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % evenE
% 0.30/0.59 thf(fact_5029_evenE,axiom,
% 0.30/0.59 ! [A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ~ ! [B4: int] :
% 0.30/0.59 ( A
% 0.30/0.59 != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % evenE
% 0.30/0.59 thf(fact_5030_odd__one,axiom,
% 0.30/0.59 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% 0.30/0.59
% 0.30/0.59 % odd_one
% 0.30/0.59 thf(fact_5031_odd__one,axiom,
% 0.30/0.59 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% 0.30/0.59
% 0.30/0.59 % odd_one
% 0.30/0.59 thf(fact_5032_odd__one,axiom,
% 0.30/0.59 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% 0.30/0.59
% 0.30/0.59 % odd_one
% 0.30/0.59 thf(fact_5033_odd__even__add,axiom,
% 0.30/0.59 ! [A: code_integer,B: code_integer] :
% 0.30/0.59 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
% 0.30/0.59 => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % odd_even_add
% 0.30/0.59 thf(fact_5034_odd__even__add,axiom,
% 0.30/0.59 ! [A: nat,B: nat] :
% 0.30/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.30/0.59 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % odd_even_add
% 0.30/0.59 thf(fact_5035_odd__even__add,axiom,
% 0.30/0.59 ! [A: int,B: int] :
% 0.30/0.59 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
% 0.30/0.59 => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % odd_even_add
% 0.30/0.59 thf(fact_5036_bit__eq__rec,axiom,
% 0.30/0.59 ( ( ^ [Y6: code_integer,Z5: code_integer] : ( Y6 = Z5 ) )
% 0.30/0.59 = ( ^ [A3: code_integer,B2: code_integer] :
% 0.30/0.59 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) )
% 0.30/0.59 & ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.59 = ( divide6298287555418463151nteger @ B2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % bit_eq_rec
% 0.30/0.59 thf(fact_5037_bit__eq__rec,axiom,
% 0.30/0.59 ( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 0.30/0.59 = ( ^ [A3: nat,B2: nat] :
% 0.30/0.59 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
% 0.30/0.59 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
% 0.30/0.59 & ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59 = ( divide_divide_nat @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % bit_eq_rec
% 0.30/0.59 thf(fact_5038_bit__eq__rec,axiom,
% 0.30/0.59 ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 0.30/0.59 = ( ^ [A3: int,B2: int] :
% 0.30/0.59 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
% 0.30/0.59 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
% 0.30/0.59 & ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.59 = ( divide_divide_int @ B2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % bit_eq_rec
% 0.30/0.59 thf(fact_5039_dvd__power__iff,axiom,
% 0.30/0.59 ! [X2: code_integer,M: nat,N: nat] :
% 0.30/0.59 ( ( X2 != zero_z3403309356797280102nteger )
% 0.30/0.59 => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X2 @ M ) @ ( power_8256067586552552935nteger @ X2 @ N ) )
% 0.30/0.59 = ( ( dvd_dvd_Code_integer @ X2 @ one_one_Code_integer )
% 0.30/0.59 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power_iff
% 0.30/0.59 thf(fact_5040_dvd__power__iff,axiom,
% 0.30/0.59 ! [X2: nat,M: nat,N: nat] :
% 0.30/0.59 ( ( X2 != zero_zero_nat )
% 0.30/0.59 => ( ( dvd_dvd_nat @ ( power_power_nat @ X2 @ M ) @ ( power_power_nat @ X2 @ N ) )
% 0.30/0.59 = ( ( dvd_dvd_nat @ X2 @ one_one_nat )
% 0.30/0.59 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power_iff
% 0.30/0.59 thf(fact_5041_dvd__power__iff,axiom,
% 0.30/0.59 ! [X2: int,M: nat,N: nat] :
% 0.30/0.59 ( ( X2 != zero_zero_int )
% 0.30/0.59 => ( ( dvd_dvd_int @ ( power_power_int @ X2 @ M ) @ ( power_power_int @ X2 @ N ) )
% 0.30/0.59 = ( ( dvd_dvd_int @ X2 @ one_one_int )
% 0.30/0.59 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power_iff
% 0.30/0.59 thf(fact_5042_odd__numeral,axiom,
% 0.30/0.59 ! [N: num] :
% 0.30/0.59 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % odd_numeral
% 0.30/0.59 thf(fact_5043_odd__numeral,axiom,
% 0.30/0.59 ! [N: num] :
% 0.30/0.59 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % odd_numeral
% 0.30/0.59 thf(fact_5044_odd__numeral,axiom,
% 0.30/0.59 ! [N: num] :
% 0.30/0.59 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % odd_numeral
% 0.30/0.59 thf(fact_5045_dvd__power,axiom,
% 0.30/0.59 ! [N: nat,X2: code_integer] :
% 0.30/0.59 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.59 | ( X2 = one_one_Code_integer ) )
% 0.30/0.59 => ( dvd_dvd_Code_integer @ X2 @ ( power_8256067586552552935nteger @ X2 @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power
% 0.30/0.59 thf(fact_5046_dvd__power,axiom,
% 0.30/0.59 ! [N: nat,X2: rat] :
% 0.30/0.59 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.59 | ( X2 = one_one_rat ) )
% 0.30/0.59 => ( dvd_dvd_rat @ X2 @ ( power_power_rat @ X2 @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power
% 0.30/0.59 thf(fact_5047_dvd__power,axiom,
% 0.30/0.59 ! [N: nat,X2: nat] :
% 0.30/0.59 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.59 | ( X2 = one_one_nat ) )
% 0.30/0.59 => ( dvd_dvd_nat @ X2 @ ( power_power_nat @ X2 @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power
% 0.30/0.59 thf(fact_5048_dvd__power,axiom,
% 0.30/0.59 ! [N: nat,X2: real] :
% 0.30/0.59 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.59 | ( X2 = one_one_real ) )
% 0.30/0.59 => ( dvd_dvd_real @ X2 @ ( power_power_real @ X2 @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power
% 0.30/0.59 thf(fact_5049_dvd__power,axiom,
% 0.30/0.59 ! [N: nat,X2: int] :
% 0.30/0.59 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.59 | ( X2 = one_one_int ) )
% 0.30/0.59 => ( dvd_dvd_int @ X2 @ ( power_power_int @ X2 @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power
% 0.30/0.59 thf(fact_5050_dvd__power,axiom,
% 0.30/0.59 ! [N: nat,X2: complex] :
% 0.30/0.59 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.59 | ( X2 = one_one_complex ) )
% 0.30/0.59 => ( dvd_dvd_complex @ X2 @ ( power_power_complex @ X2 @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power
% 0.30/0.59 thf(fact_5051_even__even__mod__4__iff,axiom,
% 0.30/0.59 ! [N: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_even_mod_4_iff
% 0.30/0.59 thf(fact_5052_dvd__mult__cancel2,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.59 => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
% 0.30/0.59 = ( N = one_one_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mult_cancel2
% 0.30/0.59 thf(fact_5053_dvd__mult__cancel1,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.59 => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
% 0.30/0.59 = ( N = one_one_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_mult_cancel1
% 0.30/0.59 thf(fact_5054_odd__numeral__BitM,axiom,
% 0.30/0.59 ! [W: num] :
% 0.30/0.59 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % odd_numeral_BitM
% 0.30/0.59 thf(fact_5055_odd__numeral__BitM,axiom,
% 0.30/0.59 ! [W: num] :
% 0.30/0.59 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % odd_numeral_BitM
% 0.30/0.59 thf(fact_5056_odd__numeral__BitM,axiom,
% 0.30/0.59 ! [W: num] :
% 0.30/0.59 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % odd_numeral_BitM
% 0.30/0.59 thf(fact_5057_dvd__minus__add,axiom,
% 0.30/0.59 ! [Q3: nat,N: nat,R2: nat,M: nat] :
% 0.30/0.59 ( ( ord_less_eq_nat @ Q3 @ N )
% 0.30/0.59 => ( ( ord_less_eq_nat @ Q3 @ ( times_times_nat @ R2 @ M ) )
% 0.30/0.59 => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q3 ) )
% 0.30/0.59 = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q3 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_minus_add
% 0.30/0.59 thf(fact_5058_power__dvd__imp__le,axiom,
% 0.30/0.59 ! [I2: nat,M: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N ) )
% 0.30/0.59 => ( ( ord_less_nat @ one_one_nat @ I2 )
% 0.30/0.59 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % power_dvd_imp_le
% 0.30/0.59 thf(fact_5059_mod__nat__eqI,axiom,
% 0.30/0.59 ! [R2: nat,N: nat,M: nat] :
% 0.30/0.59 ( ( ord_less_nat @ R2 @ N )
% 0.30/0.59 => ( ( ord_less_eq_nat @ R2 @ M )
% 0.30/0.59 => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R2 ) )
% 0.30/0.59 => ( ( modulo_modulo_nat @ M @ N )
% 0.30/0.59 = R2 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mod_nat_eqI
% 0.30/0.59 thf(fact_5060_mod__int__pos__iff,axiom,
% 0.30/0.59 ! [K: int,L: int] :
% 0.30/0.59 ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
% 0.30/0.59 = ( ( dvd_dvd_int @ L @ K )
% 0.30/0.59 | ( ( L = zero_zero_int )
% 0.30/0.59 & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 0.30/0.59 | ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mod_int_pos_iff
% 0.30/0.59 thf(fact_5061_bset_I9_J,axiom,
% 0.30/0.59 ! [D: int,D4: int,B3: set_int,T: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ D @ D4 )
% 0.30/0.59 => ! [X3: int] :
% 0.30/0.59 ( ! [Xa3: int] :
% 0.30/0.59 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.59 => ! [Xb3: int] :
% 0.30/0.59 ( ( member_int @ Xb3 @ B3 )
% 0.30/0.59 => ( X3
% 0.30/0.59 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.59 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) )
% 0.30/0.59 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % bset(9)
% 0.30/0.59 thf(fact_5062_bset_I10_J,axiom,
% 0.30/0.59 ! [D: int,D4: int,B3: set_int,T: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ D @ D4 )
% 0.30/0.59 => ! [X3: int] :
% 0.30/0.59 ( ! [Xa3: int] :
% 0.30/0.59 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.59 => ! [Xb3: int] :
% 0.30/0.59 ( ( member_int @ Xb3 @ B3 )
% 0.30/0.59 => ( X3
% 0.30/0.59 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.59 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) )
% 0.30/0.59 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % bset(10)
% 0.30/0.59 thf(fact_5063_aset_I9_J,axiom,
% 0.30/0.59 ! [D: int,D4: int,A2: set_int,T: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ D @ D4 )
% 0.30/0.59 => ! [X3: int] :
% 0.30/0.59 ( ! [Xa3: int] :
% 0.30/0.59 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.59 => ! [Xb3: int] :
% 0.30/0.59 ( ( member_int @ Xb3 @ A2 )
% 0.30/0.59 => ( X3
% 0.30/0.59 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.59 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) )
% 0.30/0.59 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % aset(9)
% 0.30/0.59 thf(fact_5064_aset_I10_J,axiom,
% 0.30/0.59 ! [D: int,D4: int,A2: set_int,T: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ D @ D4 )
% 0.30/0.59 => ! [X3: int] :
% 0.30/0.59 ( ! [Xa3: int] :
% 0.30/0.59 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.59 => ! [Xb3: int] :
% 0.30/0.59 ( ( member_int @ Xb3 @ A2 )
% 0.30/0.59 => ( X3
% 0.30/0.59 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.59 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) )
% 0.30/0.59 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % aset(10)
% 0.30/0.59 thf(fact_5065_even__two__times__div__two,axiom,
% 0.30/0.59 ! [A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.30/0.59 = A ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_two_times_div_two
% 0.30/0.59 thf(fact_5066_even__two__times__div__two,axiom,
% 0.30/0.59 ! [A: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.59 = A ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_two_times_div_two
% 0.30/0.59 thf(fact_5067_even__two__times__div__two,axiom,
% 0.30/0.59 ! [A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.30/0.59 = A ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_two_times_div_two
% 0.30/0.59 thf(fact_5068_even__iff__mod__2__eq__zero,axiom,
% 0.30/0.59 ! [A: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59 = zero_zero_nat ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_iff_mod_2_eq_zero
% 0.30/0.59 thf(fact_5069_even__iff__mod__2__eq__zero,axiom,
% 0.30/0.59 ! [A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.59 = zero_zero_int ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_iff_mod_2_eq_zero
% 0.30/0.59 thf(fact_5070_even__iff__mod__2__eq__zero,axiom,
% 0.30/0.59 ! [A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.59 = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_iff_mod_2_eq_zero
% 0.30/0.59 thf(fact_5071_odd__iff__mod__2__eq__one,axiom,
% 0.30/0.59 ! [A: nat] :
% 0.30/0.59 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.59 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59 = one_one_nat ) ) ).
% 0.30/0.59
% 0.30/0.59 % odd_iff_mod_2_eq_one
% 0.30/0.59 thf(fact_5072_odd__iff__mod__2__eq__one,axiom,
% 0.30/0.59 ! [A: int] :
% 0.30/0.59 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.59 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.59 = one_one_int ) ) ).
% 0.30/0.59
% 0.30/0.59 % odd_iff_mod_2_eq_one
% 0.30/0.59 thf(fact_5073_odd__iff__mod__2__eq__one,axiom,
% 0.30/0.59 ! [A: code_integer] :
% 0.30/0.59 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.59 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.59 = one_one_Code_integer ) ) ).
% 0.30/0.59
% 0.30/0.59 % odd_iff_mod_2_eq_one
% 0.30/0.59 thf(fact_5074_power__mono__odd,axiom,
% 0.30/0.59 ! [N: nat,A: real,B: real] :
% 0.30/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 => ( ( ord_less_eq_real @ A @ B )
% 0.30/0.59 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % power_mono_odd
% 0.30/0.59 thf(fact_5075_power__mono__odd,axiom,
% 0.30/0.59 ! [N: nat,A: rat,B: rat] :
% 0.30/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 => ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.59 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % power_mono_odd
% 0.30/0.59 thf(fact_5076_power__mono__odd,axiom,
% 0.30/0.59 ! [N: nat,A: int,B: int] :
% 0.30/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 => ( ( ord_less_eq_int @ A @ B )
% 0.30/0.59 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % power_mono_odd
% 0.30/0.59 thf(fact_5077_odd__pos,axiom,
% 0.30/0.59 ! [N: nat] :
% 0.30/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.59
% 0.30/0.59 % odd_pos
% 0.30/0.59 thf(fact_5078_dvd__power__iff__le,axiom,
% 0.30/0.59 ! [K: nat,M: nat,N: nat] :
% 0.30/0.59 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.30/0.59 => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
% 0.30/0.59 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_power_iff_le
% 0.30/0.59 thf(fact_5079_even__unset__bit__iff,axiom,
% 0.30/0.59 ! [M: nat,A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 0.30/0.59 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 | ( M = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_unset_bit_iff
% 0.30/0.59 thf(fact_5080_even__unset__bit__iff,axiom,
% 0.30/0.59 ! [M: nat,A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.30/0.59 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 | ( M = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_unset_bit_iff
% 0.30/0.59 thf(fact_5081_even__unset__bit__iff,axiom,
% 0.30/0.59 ! [M: nat,A: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.30/0.59 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 | ( M = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_unset_bit_iff
% 0.30/0.59 thf(fact_5082_even__set__bit__iff,axiom,
% 0.30/0.59 ! [M: nat,A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 0.30/0.59 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 & ( M != zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_set_bit_iff
% 0.30/0.59 thf(fact_5083_even__set__bit__iff,axiom,
% 0.30/0.59 ! [M: nat,A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.30/0.59 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 & ( M != zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_set_bit_iff
% 0.30/0.59 thf(fact_5084_even__set__bit__iff,axiom,
% 0.30/0.59 ! [M: nat,A: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.30/0.59 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 & ( M != zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_set_bit_iff
% 0.30/0.59 thf(fact_5085_even__flip__bit__iff,axiom,
% 0.30/0.59 ! [M: nat,A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 0.30/0.59 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 != ( M = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_flip_bit_iff
% 0.30/0.59 thf(fact_5086_even__flip__bit__iff,axiom,
% 0.30/0.59 ! [M: nat,A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.30/0.59 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 != ( M = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_flip_bit_iff
% 0.30/0.59 thf(fact_5087_even__flip__bit__iff,axiom,
% 0.30/0.59 ! [M: nat,A: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.30/0.59 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 != ( M = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_flip_bit_iff
% 0.30/0.59 thf(fact_5088_even__diff__iff,axiom,
% 0.30/0.59 ! [K: int,L: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
% 0.30/0.59 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_diff_iff
% 0.30/0.59 thf(fact_5089_oddE,axiom,
% 0.30/0.59 ! [A: code_integer] :
% 0.30/0.59 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ~ ! [B4: code_integer] :
% 0.30/0.59 ( A
% 0.30/0.59 != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) @ one_one_Code_integer ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % oddE
% 0.30/0.59 thf(fact_5090_oddE,axiom,
% 0.30/0.59 ! [A: nat] :
% 0.30/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ~ ! [B4: nat] :
% 0.30/0.59 ( A
% 0.30/0.59 != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) @ one_one_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % oddE
% 0.30/0.59 thf(fact_5091_oddE,axiom,
% 0.30/0.59 ! [A: int] :
% 0.30/0.59 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ~ ! [B4: int] :
% 0.30/0.59 ( A
% 0.30/0.59 != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) @ one_one_int ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % oddE
% 0.30/0.59 thf(fact_5092_parity__cases,axiom,
% 0.30/0.59 ! [A: nat] :
% 0.30/0.59 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59 != zero_zero_nat ) )
% 0.30/0.59 => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59 != one_one_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % parity_cases
% 0.30/0.59 thf(fact_5093_parity__cases,axiom,
% 0.30/0.59 ! [A: int] :
% 0.30/0.59 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.59 != zero_zero_int ) )
% 0.30/0.59 => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.59 != one_one_int ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % parity_cases
% 0.30/0.59 thf(fact_5094_parity__cases,axiom,
% 0.30/0.59 ! [A: code_integer] :
% 0.30/0.59 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.59 != zero_z3403309356797280102nteger ) )
% 0.30/0.59 => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.59 != one_one_Code_integer ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % parity_cases
% 0.30/0.59 thf(fact_5095_mod2__eq__if,axiom,
% 0.30/0.59 ! [A: nat] :
% 0.30/0.59 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59 = zero_zero_nat ) )
% 0.30/0.59 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59 = one_one_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mod2_eq_if
% 0.30/0.59 thf(fact_5096_mod2__eq__if,axiom,
% 0.30/0.59 ! [A: int] :
% 0.30/0.59 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.59 = zero_zero_int ) )
% 0.30/0.59 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.59 = one_one_int ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mod2_eq_if
% 0.30/0.59 thf(fact_5097_mod2__eq__if,axiom,
% 0.30/0.59 ! [A: code_integer] :
% 0.30/0.59 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.59 = zero_z3403309356797280102nteger ) )
% 0.30/0.59 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.59 = one_one_Code_integer ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mod2_eq_if
% 0.30/0.59 thf(fact_5098_zero__le__even__power,axiom,
% 0.30/0.59 ! [N: nat,A: real] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % zero_le_even_power
% 0.30/0.59 thf(fact_5099_zero__le__even__power,axiom,
% 0.30/0.59 ! [N: nat,A: rat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % zero_le_even_power
% 0.30/0.59 thf(fact_5100_zero__le__even__power,axiom,
% 0.30/0.59 ! [N: nat,A: int] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % zero_le_even_power
% 0.30/0.59 thf(fact_5101_zero__le__odd__power,axiom,
% 0.30/0.59 ! [N: nat,A: real] :
% 0.30/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.30/0.59 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % zero_le_odd_power
% 0.30/0.59 thf(fact_5102_zero__le__odd__power,axiom,
% 0.30/0.59 ! [N: nat,A: rat] :
% 0.30/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.30/0.59 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % zero_le_odd_power
% 0.30/0.59 thf(fact_5103_zero__le__odd__power,axiom,
% 0.30/0.59 ! [N: nat,A: int] :
% 0.30/0.59 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.30/0.59 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % zero_le_odd_power
% 0.30/0.59 thf(fact_5104_zero__le__power__eq,axiom,
% 0.30/0.59 ! [A: real,N: nat] :
% 0.30/0.59 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.30/0.59 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % zero_le_power_eq
% 0.30/0.59 thf(fact_5105_zero__le__power__eq,axiom,
% 0.30/0.59 ! [A: rat,N: nat] :
% 0.30/0.59 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.30/0.59 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % zero_le_power_eq
% 0.30/0.59 thf(fact_5106_zero__le__power__eq,axiom,
% 0.30/0.59 ! [A: int,N: nat] :
% 0.30/0.59 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.30/0.59 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % zero_le_power_eq
% 0.30/0.59 thf(fact_5107_list__decode_Ocases,axiom,
% 0.30/0.59 ! [X2: nat] :
% 0.30/0.59 ( ( X2 != zero_zero_nat )
% 0.30/0.59 => ~ ! [N2: nat] :
% 0.30/0.59 ( X2
% 0.30/0.59 != ( suc @ N2 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % list_decode.cases
% 0.30/0.59 thf(fact_5108_zero__less__power__eq,axiom,
% 0.30/0.59 ! [A: real,N: nat] :
% 0.30/0.59 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.30/0.59 = ( ( N = zero_zero_nat )
% 0.30/0.59 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 & ( A != zero_zero_real ) )
% 0.30/0.59 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % zero_less_power_eq
% 0.30/0.59 thf(fact_5109_zero__less__power__eq,axiom,
% 0.30/0.59 ! [A: rat,N: nat] :
% 0.30/0.59 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.30/0.59 = ( ( N = zero_zero_nat )
% 0.30/0.59 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 & ( A != zero_zero_rat ) )
% 0.30/0.59 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % zero_less_power_eq
% 0.30/0.59 thf(fact_5110_zero__less__power__eq,axiom,
% 0.30/0.59 ! [A: int,N: nat] :
% 0.30/0.59 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.30/0.59 = ( ( N = zero_zero_nat )
% 0.30/0.59 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 & ( A != zero_zero_int ) )
% 0.30/0.59 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % zero_less_power_eq
% 0.30/0.59 thf(fact_5111_Euclid__induct,axiom,
% 0.30/0.59 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.30/0.59 ( ! [A4: nat,B4: nat] :
% 0.30/0.59 ( ( P @ A4 @ B4 )
% 0.30/0.59 = ( P @ B4 @ A4 ) )
% 0.30/0.59 => ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
% 0.30/0.59 => ( ! [A4: nat,B4: nat] :
% 0.30/0.59 ( ( P @ A4 @ B4 )
% 0.30/0.59 => ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
% 0.30/0.59 => ( P @ A @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Euclid_induct
% 0.30/0.59 thf(fact_5112_even__mask__div__iff_H,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.59 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_mask_div_iff'
% 0.30/0.59 thf(fact_5113_even__mask__div__iff_H,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.59 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_mask_div_iff'
% 0.30/0.59 thf(fact_5114_even__mask__div__iff_H,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.59 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_mask_div_iff'
% 0.30/0.59 thf(fact_5115_power__le__zero__eq,axiom,
% 0.30/0.59 ! [A: real,N: nat] :
% 0.30/0.59 ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 0.30/0.59 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.59 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.30/0.59 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 & ( A = zero_zero_real ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % power_le_zero_eq
% 0.30/0.59 thf(fact_5116_power__le__zero__eq,axiom,
% 0.30/0.59 ! [A: rat,N: nat] :
% 0.30/0.59 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 0.30/0.59 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.59 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.30/0.59 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % power_le_zero_eq
% 0.30/0.59 thf(fact_5117_power__le__zero__eq,axiom,
% 0.30/0.59 ! [A: int,N: nat] :
% 0.30/0.59 ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 0.30/0.59 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.59 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.30/0.59 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 & ( A = zero_zero_int ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % power_le_zero_eq
% 0.30/0.59 thf(fact_5118_even__mod__4__div__2,axiom,
% 0.30/0.59 ! [N: nat] :
% 0.30/0.59 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.59 = ( suc @ zero_zero_nat ) )
% 0.30/0.59 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_mod_4_div_2
% 0.30/0.59 thf(fact_5119_even__mask__div__iff,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.59 = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 = zero_z3403309356797280102nteger )
% 0.30/0.59 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_mask_div_iff
% 0.30/0.59 thf(fact_5120_even__mask__div__iff,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.59 = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 = zero_zero_nat )
% 0.30/0.59 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_mask_div_iff
% 0.30/0.59 thf(fact_5121_even__mask__div__iff,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.59 = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 = zero_zero_int )
% 0.30/0.59 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_mask_div_iff
% 0.30/0.59 thf(fact_5122_odd__mod__4__div__2,axiom,
% 0.30/0.59 ! [N: nat] :
% 0.30/0.59 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.59 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.30/0.59 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % odd_mod_4_div_2
% 0.30/0.59 thf(fact_5123_even__mult__exp__div__exp__iff,axiom,
% 0.30/0.59 ! [A: code_integer,M: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.59 = ( ( ord_less_nat @ N @ M )
% 0.30/0.59 | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 = zero_z3403309356797280102nteger )
% 0.30/0.59 | ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.59 & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_mult_exp_div_exp_iff
% 0.30/0.59 thf(fact_5124_even__mult__exp__div__exp__iff,axiom,
% 0.30/0.59 ! [A: nat,M: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.59 = ( ( ord_less_nat @ N @ M )
% 0.30/0.59 | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 = zero_zero_nat )
% 0.30/0.59 | ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.59 & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_mult_exp_div_exp_iff
% 0.30/0.59 thf(fact_5125_even__mult__exp__div__exp__iff,axiom,
% 0.30/0.59 ! [A: int,M: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.59 = ( ( ord_less_nat @ N @ M )
% 0.30/0.59 | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.30/0.59 = zero_zero_int )
% 0.30/0.59 | ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.59 & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_mult_exp_div_exp_iff
% 0.30/0.59 thf(fact_5126_infinite__growing,axiom,
% 0.30/0.59 ! [X8: set_real] :
% 0.30/0.59 ( ( X8 != bot_bot_set_real )
% 0.30/0.59 => ( ! [X5: real] :
% 0.30/0.59 ( ( member_real @ X5 @ X8 )
% 0.30/0.59 => ? [Xa: real] :
% 0.30/0.59 ( ( member_real @ Xa @ X8 )
% 0.30/0.59 & ( ord_less_real @ X5 @ Xa ) ) )
% 0.30/0.59 => ~ ( finite_finite_real @ X8 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % infinite_growing
% 0.30/0.59 thf(fact_5127_infinite__growing,axiom,
% 0.30/0.59 ! [X8: set_rat] :
% 0.30/0.59 ( ( X8 != bot_bot_set_rat )
% 0.30/0.59 => ( ! [X5: rat] :
% 0.30/0.59 ( ( member_rat @ X5 @ X8 )
% 0.30/0.59 => ? [Xa: rat] :
% 0.30/0.59 ( ( member_rat @ Xa @ X8 )
% 0.30/0.59 & ( ord_less_rat @ X5 @ Xa ) ) )
% 0.30/0.59 => ~ ( finite_finite_rat @ X8 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % infinite_growing
% 0.30/0.59 thf(fact_5128_infinite__growing,axiom,
% 0.30/0.59 ! [X8: set_num] :
% 0.30/0.59 ( ( X8 != bot_bot_set_num )
% 0.30/0.59 => ( ! [X5: num] :
% 0.30/0.59 ( ( member_num @ X5 @ X8 )
% 0.30/0.59 => ? [Xa: num] :
% 0.30/0.59 ( ( member_num @ Xa @ X8 )
% 0.30/0.59 & ( ord_less_num @ X5 @ Xa ) ) )
% 0.30/0.59 => ~ ( finite_finite_num @ X8 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % infinite_growing
% 0.30/0.59 thf(fact_5129_infinite__growing,axiom,
% 0.30/0.59 ! [X8: set_nat] :
% 0.30/0.59 ( ( X8 != bot_bot_set_nat )
% 0.30/0.59 => ( ! [X5: nat] :
% 0.30/0.59 ( ( member_nat @ X5 @ X8 )
% 0.30/0.59 => ? [Xa: nat] :
% 0.30/0.59 ( ( member_nat @ Xa @ X8 )
% 0.30/0.59 & ( ord_less_nat @ X5 @ Xa ) ) )
% 0.30/0.59 => ~ ( finite_finite_nat @ X8 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % infinite_growing
% 0.30/0.59 thf(fact_5130_infinite__growing,axiom,
% 0.30/0.59 ! [X8: set_int] :
% 0.30/0.59 ( ( X8 != bot_bot_set_int )
% 0.30/0.59 => ( ! [X5: int] :
% 0.30/0.59 ( ( member_int @ X5 @ X8 )
% 0.30/0.59 => ? [Xa: int] :
% 0.30/0.59 ( ( member_int @ Xa @ X8 )
% 0.30/0.59 & ( ord_less_int @ X5 @ Xa ) ) )
% 0.30/0.59 => ~ ( finite_finite_int @ X8 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % infinite_growing
% 0.30/0.59 thf(fact_5131_ex__min__if__finite,axiom,
% 0.30/0.59 ! [S3: set_real] :
% 0.30/0.59 ( ( finite_finite_real @ S3 )
% 0.30/0.59 => ( ( S3 != bot_bot_set_real )
% 0.30/0.59 => ? [X5: real] :
% 0.30/0.59 ( ( member_real @ X5 @ S3 )
% 0.30/0.59 & ~ ? [Xa: real] :
% 0.30/0.59 ( ( member_real @ Xa @ S3 )
% 0.30/0.59 & ( ord_less_real @ Xa @ X5 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % ex_min_if_finite
% 0.30/0.59 thf(fact_5132_ex__min__if__finite,axiom,
% 0.30/0.59 ! [S3: set_rat] :
% 0.30/0.59 ( ( finite_finite_rat @ S3 )
% 0.30/0.59 => ( ( S3 != bot_bot_set_rat )
% 0.30/0.59 => ? [X5: rat] :
% 0.30/0.59 ( ( member_rat @ X5 @ S3 )
% 0.30/0.59 & ~ ? [Xa: rat] :
% 0.30/0.59 ( ( member_rat @ Xa @ S3 )
% 0.30/0.59 & ( ord_less_rat @ Xa @ X5 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % ex_min_if_finite
% 0.30/0.59 thf(fact_5133_ex__min__if__finite,axiom,
% 0.30/0.59 ! [S3: set_num] :
% 0.30/0.59 ( ( finite_finite_num @ S3 )
% 0.30/0.59 => ( ( S3 != bot_bot_set_num )
% 0.30/0.59 => ? [X5: num] :
% 0.30/0.59 ( ( member_num @ X5 @ S3 )
% 0.30/0.59 & ~ ? [Xa: num] :
% 0.30/0.59 ( ( member_num @ Xa @ S3 )
% 0.30/0.59 & ( ord_less_num @ Xa @ X5 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % ex_min_if_finite
% 0.30/0.59 thf(fact_5134_ex__min__if__finite,axiom,
% 0.30/0.59 ! [S3: set_nat] :
% 0.30/0.59 ( ( finite_finite_nat @ S3 )
% 0.30/0.59 => ( ( S3 != bot_bot_set_nat )
% 0.30/0.59 => ? [X5: nat] :
% 0.30/0.59 ( ( member_nat @ X5 @ S3 )
% 0.30/0.59 & ~ ? [Xa: nat] :
% 0.30/0.59 ( ( member_nat @ Xa @ S3 )
% 0.30/0.59 & ( ord_less_nat @ Xa @ X5 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % ex_min_if_finite
% 0.30/0.59 thf(fact_5135_ex__min__if__finite,axiom,
% 0.30/0.59 ! [S3: set_int] :
% 0.30/0.59 ( ( finite_finite_int @ S3 )
% 0.30/0.59 => ( ( S3 != bot_bot_set_int )
% 0.30/0.59 => ? [X5: int] :
% 0.30/0.59 ( ( member_int @ X5 @ S3 )
% 0.30/0.59 & ~ ? [Xa: int] :
% 0.30/0.59 ( ( member_int @ Xa @ S3 )
% 0.30/0.59 & ( ord_less_int @ Xa @ X5 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % ex_min_if_finite
% 0.30/0.59 thf(fact_5136_triangle__def,axiom,
% 0.30/0.59 ( nat_triangle
% 0.30/0.59 = ( ^ [N3: nat] : ( divide_divide_nat @ ( times_times_nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % triangle_def
% 0.30/0.59 thf(fact_5137_vebt__buildup_Oelims,axiom,
% 0.30/0.59 ! [X2: nat,Y3: vEBT_VEBT] :
% 0.30/0.59 ( ( ( vEBT_vebt_buildup @ X2 )
% 0.30/0.59 = Y3 )
% 0.30/0.59 => ( ( ( X2 = zero_zero_nat )
% 0.30/0.59 => ( Y3
% 0.30/0.59 != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.30/0.59 => ( ( ( X2
% 0.30/0.59 = ( suc @ zero_zero_nat ) )
% 0.30/0.59 => ( Y3
% 0.30/0.59 != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.30/0.59 => ~ ! [Va3: nat] :
% 0.30/0.59 ( ( X2
% 0.30/0.59 = ( suc @ ( suc @ Va3 ) ) )
% 0.30/0.59 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 0.30/0.59 => ( Y3
% 0.30/0.59 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 0.30/0.59 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 0.30/0.59 => ( Y3
% 0.30/0.59 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % vebt_buildup.elims
% 0.30/0.59 thf(fact_5138_divmod__algorithm__code_I6_J,axiom,
% 0.30/0.59 ! [M: num,N: num] :
% 0.30/0.59 ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.59 = ( produc4245557441103728435nt_int
% 0.30/0.59 @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
% 0.30/0.59 @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % divmod_algorithm_code(6)
% 0.30/0.59 thf(fact_5139_divmod__algorithm__code_I6_J,axiom,
% 0.30/0.59 ! [M: num,N: num] :
% 0.30/0.59 ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.59 = ( produc2626176000494625587at_nat
% 0.30/0.59 @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
% 0.30/0.59 @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % divmod_algorithm_code(6)
% 0.30/0.59 thf(fact_5140_divmod__algorithm__code_I6_J,axiom,
% 0.30/0.59 ! [M: num,N: num] :
% 0.30/0.59 ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.59 = ( produc6916734918728496179nteger
% 0.30/0.59 @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
% 0.30/0.59 @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % divmod_algorithm_code(6)
% 0.30/0.59 thf(fact_5141_flip__bit__0,axiom,
% 0.30/0.59 ! [A: code_integer] :
% 0.30/0.59 ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 0.30/0.59 = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % flip_bit_0
% 0.30/0.59 thf(fact_5142_flip__bit__0,axiom,
% 0.30/0.59 ! [A: int] :
% 0.30/0.59 ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 0.30/0.59 = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % flip_bit_0
% 0.30/0.59 thf(fact_5143_flip__bit__0,axiom,
% 0.30/0.59 ! [A: nat] :
% 0.30/0.59 ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 0.30/0.59 = ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % flip_bit_0
% 0.30/0.59 thf(fact_5144_option_Osize__gen_I2_J,axiom,
% 0.30/0.59 ! [X2: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
% 0.30/0.59 ( ( size_o8335143837870341156at_nat @ X2 @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.30/0.59 = ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % option.size_gen(2)
% 0.30/0.59 thf(fact_5145_option_Osize__gen_I2_J,axiom,
% 0.30/0.59 ! [X2: nat > nat,X22: nat] :
% 0.30/0.59 ( ( size_option_nat @ X2 @ ( some_nat @ X22 ) )
% 0.30/0.59 = ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % option.size_gen(2)
% 0.30/0.59 thf(fact_5146_option_Osize__gen_I2_J,axiom,
% 0.30/0.59 ! [X2: num > nat,X22: num] :
% 0.30/0.59 ( ( size_option_num @ X2 @ ( some_num @ X22 ) )
% 0.30/0.59 = ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % option.size_gen(2)
% 0.30/0.59 thf(fact_5147_signed__take__bit__Suc,axiom,
% 0.30/0.59 ! [N: nat,A: code_integer] :
% 0.30/0.59 ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
% 0.30/0.59 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % signed_take_bit_Suc
% 0.30/0.59 thf(fact_5148_signed__take__bit__Suc,axiom,
% 0.30/0.59 ! [N: nat,A: int] :
% 0.30/0.59 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
% 0.30/0.59 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % signed_take_bit_Suc
% 0.30/0.59 thf(fact_5149_set__decode__Suc,axiom,
% 0.30/0.59 ! [N: nat,X2: nat] :
% 0.30/0.59 ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X2 ) )
% 0.30/0.59 = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % set_decode_Suc
% 0.30/0.59 thf(fact_5150_intind,axiom,
% 0.30/0.59 ! [I2: nat,N: nat,P: int > $o,X2: int] :
% 0.30/0.59 ( ( ord_less_nat @ I2 @ N )
% 0.30/0.59 => ( ( P @ X2 )
% 0.30/0.59 => ( P @ ( nth_int @ ( replicate_int @ N @ X2 ) @ I2 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % intind
% 0.30/0.59 thf(fact_5151_intind,axiom,
% 0.30/0.59 ! [I2: nat,N: nat,P: nat > $o,X2: nat] :
% 0.30/0.59 ( ( ord_less_nat @ I2 @ N )
% 0.30/0.59 => ( ( P @ X2 )
% 0.30/0.59 => ( P @ ( nth_nat @ ( replicate_nat @ N @ X2 ) @ I2 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % intind
% 0.30/0.59 thf(fact_5152_intind,axiom,
% 0.30/0.59 ! [I2: nat,N: nat,P: vEBT_VEBT > $o,X2: vEBT_VEBT] :
% 0.30/0.59 ( ( ord_less_nat @ I2 @ N )
% 0.30/0.59 => ( ( P @ X2 )
% 0.30/0.59 => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X2 ) @ I2 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % intind
% 0.30/0.59 thf(fact_5153_of__bool__less__eq__iff,axiom,
% 0.30/0.59 ! [P: $o,Q: $o] :
% 0.30/0.59 ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.30/0.59 = ( P
% 0.30/0.59 => Q ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_less_eq_iff
% 0.30/0.59 thf(fact_5154_of__bool__less__eq__iff,axiom,
% 0.30/0.59 ! [P: $o,Q: $o] :
% 0.30/0.59 ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.30/0.59 = ( P
% 0.30/0.59 => Q ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_less_eq_iff
% 0.30/0.59 thf(fact_5155_of__bool__less__eq__iff,axiom,
% 0.30/0.59 ! [P: $o,Q: $o] :
% 0.30/0.59 ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.30/0.59 = ( P
% 0.30/0.59 => Q ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_less_eq_iff
% 0.30/0.59 thf(fact_5156_of__bool__less__eq__iff,axiom,
% 0.30/0.59 ! [P: $o,Q: $o] :
% 0.30/0.59 ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 0.30/0.59 = ( P
% 0.30/0.59 => Q ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_less_eq_iff
% 0.30/0.59 thf(fact_5157_of__bool__eq__0__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ( zero_n1201886186963655149omplex @ P )
% 0.30/0.59 = zero_zero_complex )
% 0.30/0.59 = ( ~ P ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq_0_iff
% 0.30/0.59 thf(fact_5158_of__bool__eq__0__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ( zero_n3304061248610475627l_real @ P )
% 0.30/0.59 = zero_zero_real )
% 0.30/0.59 = ( ~ P ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq_0_iff
% 0.30/0.59 thf(fact_5159_of__bool__eq__0__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 0.30/0.59 = zero_zero_rat )
% 0.30/0.59 = ( ~ P ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq_0_iff
% 0.30/0.59 thf(fact_5160_of__bool__eq__0__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 0.30/0.59 = zero_zero_nat )
% 0.30/0.59 = ( ~ P ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq_0_iff
% 0.30/0.59 thf(fact_5161_of__bool__eq__0__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ( zero_n2684676970156552555ol_int @ P )
% 0.30/0.59 = zero_zero_int )
% 0.30/0.59 = ( ~ P ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq_0_iff
% 0.30/0.59 thf(fact_5162_of__bool__eq__0__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ( zero_n356916108424825756nteger @ P )
% 0.30/0.59 = zero_z3403309356797280102nteger )
% 0.30/0.59 = ( ~ P ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq_0_iff
% 0.30/0.59 thf(fact_5163_of__bool__eq_I1_J,axiom,
% 0.30/0.59 ( ( zero_n1201886186963655149omplex @ $false )
% 0.30/0.59 = zero_zero_complex ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq(1)
% 0.30/0.59 thf(fact_5164_of__bool__eq_I1_J,axiom,
% 0.30/0.59 ( ( zero_n3304061248610475627l_real @ $false )
% 0.30/0.59 = zero_zero_real ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq(1)
% 0.30/0.59 thf(fact_5165_of__bool__eq_I1_J,axiom,
% 0.30/0.59 ( ( zero_n2052037380579107095ol_rat @ $false )
% 0.30/0.59 = zero_zero_rat ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq(1)
% 0.30/0.59 thf(fact_5166_of__bool__eq_I1_J,axiom,
% 0.30/0.59 ( ( zero_n2687167440665602831ol_nat @ $false )
% 0.30/0.59 = zero_zero_nat ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq(1)
% 0.30/0.59 thf(fact_5167_of__bool__eq_I1_J,axiom,
% 0.30/0.59 ( ( zero_n2684676970156552555ol_int @ $false )
% 0.30/0.59 = zero_zero_int ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq(1)
% 0.30/0.59 thf(fact_5168_of__bool__eq_I1_J,axiom,
% 0.30/0.59 ( ( zero_n356916108424825756nteger @ $false )
% 0.30/0.59 = zero_z3403309356797280102nteger ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq(1)
% 0.30/0.59 thf(fact_5169_of__bool__less__iff,axiom,
% 0.30/0.59 ! [P: $o,Q: $o] :
% 0.30/0.59 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 0.30/0.59 = ( ~ P
% 0.30/0.59 & Q ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_less_iff
% 0.30/0.59 thf(fact_5170_of__bool__less__iff,axiom,
% 0.30/0.59 ! [P: $o,Q: $o] :
% 0.30/0.59 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.30/0.59 = ( ~ P
% 0.30/0.59 & Q ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_less_iff
% 0.30/0.59 thf(fact_5171_of__bool__less__iff,axiom,
% 0.30/0.59 ! [P: $o,Q: $o] :
% 0.30/0.59 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.30/0.59 = ( ~ P
% 0.30/0.59 & Q ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_less_iff
% 0.30/0.59 thf(fact_5172_of__bool__less__iff,axiom,
% 0.30/0.59 ! [P: $o,Q: $o] :
% 0.30/0.59 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.30/0.59 = ( ~ P
% 0.30/0.59 & Q ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_less_iff
% 0.30/0.59 thf(fact_5173_of__bool__less__iff,axiom,
% 0.30/0.59 ! [P: $o,Q: $o] :
% 0.30/0.59 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 0.30/0.59 = ( ~ P
% 0.30/0.59 & Q ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_less_iff
% 0.30/0.59 thf(fact_5174_of__bool__eq__1__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ( zero_n1201886186963655149omplex @ P )
% 0.30/0.59 = one_one_complex )
% 0.30/0.59 = P ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq_1_iff
% 0.30/0.59 thf(fact_5175_of__bool__eq__1__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ( zero_n3304061248610475627l_real @ P )
% 0.30/0.59 = one_one_real )
% 0.30/0.59 = P ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq_1_iff
% 0.30/0.59 thf(fact_5176_of__bool__eq__1__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 0.30/0.59 = one_one_rat )
% 0.30/0.59 = P ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq_1_iff
% 0.30/0.59 thf(fact_5177_of__bool__eq__1__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 0.30/0.59 = one_one_nat )
% 0.30/0.59 = P ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq_1_iff
% 0.30/0.59 thf(fact_5178_of__bool__eq__1__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ( zero_n2684676970156552555ol_int @ P )
% 0.30/0.59 = one_one_int )
% 0.30/0.59 = P ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq_1_iff
% 0.30/0.59 thf(fact_5179_of__bool__eq__1__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ( zero_n356916108424825756nteger @ P )
% 0.30/0.59 = one_one_Code_integer )
% 0.30/0.59 = P ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq_1_iff
% 0.30/0.59 thf(fact_5180_of__bool__eq_I2_J,axiom,
% 0.30/0.59 ( ( zero_n1201886186963655149omplex @ $true )
% 0.30/0.59 = one_one_complex ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq(2)
% 0.30/0.59 thf(fact_5181_of__bool__eq_I2_J,axiom,
% 0.30/0.59 ( ( zero_n3304061248610475627l_real @ $true )
% 0.30/0.59 = one_one_real ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq(2)
% 0.30/0.59 thf(fact_5182_of__bool__eq_I2_J,axiom,
% 0.30/0.59 ( ( zero_n2052037380579107095ol_rat @ $true )
% 0.30/0.59 = one_one_rat ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq(2)
% 0.30/0.59 thf(fact_5183_of__bool__eq_I2_J,axiom,
% 0.30/0.59 ( ( zero_n2687167440665602831ol_nat @ $true )
% 0.30/0.59 = one_one_nat ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq(2)
% 0.30/0.59 thf(fact_5184_of__bool__eq_I2_J,axiom,
% 0.30/0.59 ( ( zero_n2684676970156552555ol_int @ $true )
% 0.30/0.59 = one_one_int ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq(2)
% 0.30/0.59 thf(fact_5185_of__bool__eq_I2_J,axiom,
% 0.30/0.59 ( ( zero_n356916108424825756nteger @ $true )
% 0.30/0.59 = one_one_Code_integer ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq(2)
% 0.30/0.59 thf(fact_5186_signed__take__bit__of__0,axiom,
% 0.30/0.59 ! [N: nat] :
% 0.30/0.59 ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
% 0.30/0.59 = zero_zero_int ) ).
% 0.30/0.59
% 0.30/0.59 % signed_take_bit_of_0
% 0.30/0.59 thf(fact_5187_length__replicate,axiom,
% 0.30/0.59 ! [N: nat,X2: vEBT_VEBT] :
% 0.30/0.59 ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N @ X2 ) )
% 0.30/0.59 = N ) ).
% 0.30/0.59
% 0.30/0.59 % length_replicate
% 0.30/0.59 thf(fact_5188_length__replicate,axiom,
% 0.30/0.59 ! [N: nat,X2: $o] :
% 0.30/0.59 ( ( size_size_list_o @ ( replicate_o @ N @ X2 ) )
% 0.30/0.59 = N ) ).
% 0.30/0.59
% 0.30/0.59 % length_replicate
% 0.30/0.59 thf(fact_5189_length__replicate,axiom,
% 0.30/0.59 ! [N: nat,X2: nat] :
% 0.30/0.59 ( ( size_size_list_nat @ ( replicate_nat @ N @ X2 ) )
% 0.30/0.59 = N ) ).
% 0.30/0.59
% 0.30/0.59 % length_replicate
% 0.30/0.59 thf(fact_5190_length__replicate,axiom,
% 0.30/0.59 ! [N: nat,X2: int] :
% 0.30/0.59 ( ( size_size_list_int @ ( replicate_int @ N @ X2 ) )
% 0.30/0.59 = N ) ).
% 0.30/0.59
% 0.30/0.59 % length_replicate
% 0.30/0.59 thf(fact_5191_of__bool__or__iff,axiom,
% 0.30/0.59 ! [P: $o,Q: $o] :
% 0.30/0.59 ( ( zero_n2687167440665602831ol_nat
% 0.30/0.59 @ ( P
% 0.30/0.59 | Q ) )
% 0.30/0.59 = ( ord_max_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_or_iff
% 0.30/0.59 thf(fact_5192_of__bool__or__iff,axiom,
% 0.30/0.59 ! [P: $o,Q: $o] :
% 0.30/0.59 ( ( zero_n2684676970156552555ol_int
% 0.30/0.59 @ ( P
% 0.30/0.59 | Q ) )
% 0.30/0.59 = ( ord_max_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_or_iff
% 0.30/0.59 thf(fact_5193_of__bool__or__iff,axiom,
% 0.30/0.59 ! [P: $o,Q: $o] :
% 0.30/0.59 ( ( zero_n356916108424825756nteger
% 0.30/0.59 @ ( P
% 0.30/0.59 | Q ) )
% 0.30/0.59 = ( ord_max_Code_integer @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_or_iff
% 0.30/0.59 thf(fact_5194_case__prod__conv,axiom,
% 0.30/0.59 ! [F: nat > nat > nat,A: nat,B: nat] :
% 0.30/0.59 ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 0.30/0.59 = ( F @ A @ B ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prod_conv
% 0.30/0.59 thf(fact_5195_case__prod__conv,axiom,
% 0.30/0.59 ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
% 0.30/0.59 ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 0.30/0.59 = ( F @ A @ B ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prod_conv
% 0.30/0.59 thf(fact_5196_case__prod__conv,axiom,
% 0.30/0.59 ! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
% 0.30/0.59 ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 0.30/0.59 = ( F @ A @ B ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prod_conv
% 0.30/0.59 thf(fact_5197_case__prod__conv,axiom,
% 0.30/0.59 ! [F: int > int > product_prod_int_int,A: int,B: int] :
% 0.30/0.59 ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
% 0.30/0.59 = ( F @ A @ B ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prod_conv
% 0.30/0.59 thf(fact_5198_case__prod__conv,axiom,
% 0.30/0.59 ! [F: int > int > $o,A: int,B: int] :
% 0.30/0.59 ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
% 0.30/0.59 = ( F @ A @ B ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prod_conv
% 0.30/0.59 thf(fact_5199_zero__less__of__bool__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 0.30/0.59 = P ) ).
% 0.30/0.59
% 0.30/0.59 % zero_less_of_bool_iff
% 0.30/0.59 thf(fact_5200_zero__less__of__bool__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 0.30/0.59 = P ) ).
% 0.30/0.59
% 0.30/0.59 % zero_less_of_bool_iff
% 0.30/0.59 thf(fact_5201_zero__less__of__bool__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.30/0.59 = P ) ).
% 0.30/0.59
% 0.30/0.59 % zero_less_of_bool_iff
% 0.30/0.59 thf(fact_5202_zero__less__of__bool__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.30/0.59 = P ) ).
% 0.30/0.59
% 0.30/0.59 % zero_less_of_bool_iff
% 0.30/0.59 thf(fact_5203_zero__less__of__bool__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
% 0.30/0.59 = P ) ).
% 0.30/0.59
% 0.30/0.59 % zero_less_of_bool_iff
% 0.30/0.59 thf(fact_5204_of__bool__less__one__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 0.30/0.59 = ( ~ P ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_less_one_iff
% 0.30/0.59 thf(fact_5205_of__bool__less__one__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 0.30/0.59 = ( ~ P ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_less_one_iff
% 0.30/0.59 thf(fact_5206_of__bool__less__one__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 0.30/0.59 = ( ~ P ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_less_one_iff
% 0.30/0.59 thf(fact_5207_of__bool__less__one__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 0.30/0.59 = ( ~ P ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_less_one_iff
% 0.30/0.59 thf(fact_5208_of__bool__less__one__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
% 0.30/0.59 = ( ~ P ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_less_one_iff
% 0.30/0.59 thf(fact_5209_of__bool__not__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( zero_n1201886186963655149omplex @ ~ P )
% 0.30/0.59 = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_not_iff
% 0.30/0.59 thf(fact_5210_of__bool__not__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( zero_n3304061248610475627l_real @ ~ P )
% 0.30/0.59 = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_not_iff
% 0.30/0.59 thf(fact_5211_of__bool__not__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( zero_n2052037380579107095ol_rat @ ~ P )
% 0.30/0.59 = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_not_iff
% 0.30/0.59 thf(fact_5212_of__bool__not__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( zero_n2684676970156552555ol_int @ ~ P )
% 0.30/0.59 = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_not_iff
% 0.30/0.59 thf(fact_5213_of__bool__not__iff,axiom,
% 0.30/0.59 ! [P: $o] :
% 0.30/0.59 ( ( zero_n356916108424825756nteger @ ~ P )
% 0.30/0.59 = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_not_iff
% 0.30/0.59 thf(fact_5214_Suc__0__mod__eq,axiom,
% 0.30/0.59 ! [N: nat] :
% 0.30/0.59 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.59 = ( zero_n2687167440665602831ol_nat
% 0.30/0.59 @ ( N
% 0.30/0.59 != ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Suc_0_mod_eq
% 0.30/0.59 thf(fact_5215_signed__take__bit__Suc__1,axiom,
% 0.30/0.59 ! [N: nat] :
% 0.30/0.59 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
% 0.30/0.59 = one_one_int ) ).
% 0.30/0.59
% 0.30/0.59 % signed_take_bit_Suc_1
% 0.30/0.59 thf(fact_5216_signed__take__bit__numeral__of__1,axiom,
% 0.30/0.59 ! [K: num] :
% 0.30/0.59 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
% 0.30/0.59 = one_one_int ) ).
% 0.30/0.59
% 0.30/0.59 % signed_take_bit_numeral_of_1
% 0.30/0.59 thf(fact_5217_in__set__replicate,axiom,
% 0.30/0.59 ! [X2: product_prod_nat_nat,N: nat,Y3: product_prod_nat_nat] :
% 0.30/0.59 ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ Y3 ) ) )
% 0.30/0.59 = ( ( X2 = Y3 )
% 0.30/0.59 & ( N != zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % in_set_replicate
% 0.30/0.59 thf(fact_5218_in__set__replicate,axiom,
% 0.30/0.59 ! [X2: complex,N: nat,Y3: complex] :
% 0.30/0.59 ( ( member_complex @ X2 @ ( set_complex2 @ ( replicate_complex @ N @ Y3 ) ) )
% 0.30/0.59 = ( ( X2 = Y3 )
% 0.30/0.59 & ( N != zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % in_set_replicate
% 0.30/0.59 thf(fact_5219_in__set__replicate,axiom,
% 0.30/0.59 ! [X2: real,N: nat,Y3: real] :
% 0.30/0.59 ( ( member_real @ X2 @ ( set_real2 @ ( replicate_real @ N @ Y3 ) ) )
% 0.30/0.59 = ( ( X2 = Y3 )
% 0.30/0.59 & ( N != zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % in_set_replicate
% 0.30/0.59 thf(fact_5220_in__set__replicate,axiom,
% 0.30/0.59 ! [X2: int,N: nat,Y3: int] :
% 0.30/0.59 ( ( member_int @ X2 @ ( set_int2 @ ( replicate_int @ N @ Y3 ) ) )
% 0.30/0.59 = ( ( X2 = Y3 )
% 0.30/0.59 & ( N != zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % in_set_replicate
% 0.30/0.59 thf(fact_5221_in__set__replicate,axiom,
% 0.30/0.59 ! [X2: nat,N: nat,Y3: nat] :
% 0.30/0.59 ( ( member_nat @ X2 @ ( set_nat2 @ ( replicate_nat @ N @ Y3 ) ) )
% 0.30/0.59 = ( ( X2 = Y3 )
% 0.30/0.59 & ( N != zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % in_set_replicate
% 0.30/0.59 thf(fact_5222_in__set__replicate,axiom,
% 0.30/0.59 ! [X2: vEBT_VEBT,N: nat,Y3: vEBT_VEBT] :
% 0.30/0.59 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y3 ) ) )
% 0.30/0.59 = ( ( X2 = Y3 )
% 0.30/0.59 & ( N != zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % in_set_replicate
% 0.30/0.59 thf(fact_5223_Bex__set__replicate,axiom,
% 0.30/0.59 ! [N: nat,A: int,P: int > $o] :
% 0.30/0.59 ( ( ? [X: int] :
% 0.30/0.59 ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
% 0.30/0.59 & ( P @ X ) ) )
% 0.30/0.59 = ( ( P @ A )
% 0.30/0.59 & ( N != zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Bex_set_replicate
% 0.30/0.59 thf(fact_5224_Bex__set__replicate,axiom,
% 0.30/0.59 ! [N: nat,A: nat,P: nat > $o] :
% 0.30/0.59 ( ( ? [X: nat] :
% 0.30/0.59 ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
% 0.30/0.59 & ( P @ X ) ) )
% 0.30/0.59 = ( ( P @ A )
% 0.30/0.59 & ( N != zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Bex_set_replicate
% 0.30/0.59 thf(fact_5225_Bex__set__replicate,axiom,
% 0.30/0.59 ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.30/0.59 ( ( ? [X: vEBT_VEBT] :
% 0.30/0.59 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
% 0.30/0.59 & ( P @ X ) ) )
% 0.30/0.59 = ( ( P @ A )
% 0.30/0.59 & ( N != zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Bex_set_replicate
% 0.30/0.59 thf(fact_5226_Ball__set__replicate,axiom,
% 0.30/0.59 ! [N: nat,A: int,P: int > $o] :
% 0.30/0.59 ( ( ! [X: int] :
% 0.30/0.59 ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
% 0.30/0.59 => ( P @ X ) ) )
% 0.30/0.59 = ( ( P @ A )
% 0.30/0.59 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Ball_set_replicate
% 0.30/0.59 thf(fact_5227_Ball__set__replicate,axiom,
% 0.30/0.59 ! [N: nat,A: nat,P: nat > $o] :
% 0.30/0.59 ( ( ! [X: nat] :
% 0.30/0.59 ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
% 0.30/0.59 => ( P @ X ) ) )
% 0.30/0.59 = ( ( P @ A )
% 0.30/0.59 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Ball_set_replicate
% 0.30/0.59 thf(fact_5228_Ball__set__replicate,axiom,
% 0.30/0.59 ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.30/0.59 ( ( ! [X: vEBT_VEBT] :
% 0.30/0.59 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
% 0.30/0.59 => ( P @ X ) ) )
% 0.30/0.59 = ( ( P @ A )
% 0.30/0.59 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Ball_set_replicate
% 0.30/0.59 thf(fact_5229_nth__replicate,axiom,
% 0.30/0.59 ! [I2: nat,N: nat,X2: int] :
% 0.30/0.59 ( ( ord_less_nat @ I2 @ N )
% 0.30/0.59 => ( ( nth_int @ ( replicate_int @ N @ X2 ) @ I2 )
% 0.30/0.59 = X2 ) ) ).
% 0.30/0.59
% 0.30/0.59 % nth_replicate
% 0.30/0.59 thf(fact_5230_nth__replicate,axiom,
% 0.30/0.59 ! [I2: nat,N: nat,X2: nat] :
% 0.30/0.59 ( ( ord_less_nat @ I2 @ N )
% 0.30/0.59 => ( ( nth_nat @ ( replicate_nat @ N @ X2 ) @ I2 )
% 0.30/0.59 = X2 ) ) ).
% 0.30/0.59
% 0.30/0.59 % nth_replicate
% 0.30/0.59 thf(fact_5231_nth__replicate,axiom,
% 0.30/0.59 ! [I2: nat,N: nat,X2: vEBT_VEBT] :
% 0.30/0.59 ( ( ord_less_nat @ I2 @ N )
% 0.30/0.59 => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X2 ) @ I2 )
% 0.30/0.59 = X2 ) ) ).
% 0.30/0.59
% 0.30/0.59 % nth_replicate
% 0.30/0.59 thf(fact_5232_triangle__Suc,axiom,
% 0.30/0.59 ! [N: nat] :
% 0.30/0.59 ( ( nat_triangle @ ( suc @ N ) )
% 0.30/0.59 = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % triangle_Suc
% 0.30/0.59 thf(fact_5233_signed__take__bit__Suc__bit0,axiom,
% 0.30/0.59 ! [N: nat,K: num] :
% 0.30/0.59 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.30/0.59 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % signed_take_bit_Suc_bit0
% 0.30/0.59 thf(fact_5234_odd__of__bool__self,axiom,
% 0.30/0.59 ! [P6: $o] :
% 0.30/0.59 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P6 ) ) )
% 0.30/0.59 = P6 ) ).
% 0.30/0.59
% 0.30/0.59 % odd_of_bool_self
% 0.30/0.59 thf(fact_5235_odd__of__bool__self,axiom,
% 0.30/0.59 ! [P6: $o] :
% 0.30/0.59 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P6 ) ) )
% 0.30/0.59 = P6 ) ).
% 0.30/0.59
% 0.30/0.59 % odd_of_bool_self
% 0.30/0.59 thf(fact_5236_odd__of__bool__self,axiom,
% 0.30/0.59 ! [P6: $o] :
% 0.30/0.59 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P6 ) ) )
% 0.30/0.59 = P6 ) ).
% 0.30/0.59
% 0.30/0.59 % odd_of_bool_self
% 0.30/0.59 thf(fact_5237_of__bool__half__eq__0,axiom,
% 0.30/0.59 ! [B: $o] :
% 0.30/0.59 ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59 = zero_zero_nat ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_half_eq_0
% 0.30/0.59 thf(fact_5238_of__bool__half__eq__0,axiom,
% 0.30/0.59 ! [B: $o] :
% 0.30/0.59 ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.59 = zero_zero_int ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_half_eq_0
% 0.30/0.59 thf(fact_5239_of__bool__half__eq__0,axiom,
% 0.30/0.59 ! [B: $o] :
% 0.30/0.59 ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.59 = zero_z3403309356797280102nteger ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_half_eq_0
% 0.30/0.59 thf(fact_5240_set__decode__0,axiom,
% 0.30/0.59 ! [X2: nat] :
% 0.30/0.59 ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X2 ) )
% 0.30/0.59 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % set_decode_0
% 0.30/0.59 thf(fact_5241_divmod__algorithm__code_I5_J,axiom,
% 0.30/0.59 ! [M: num,N: num] :
% 0.30/0.59 ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.59 = ( produc4245557441103728435nt_int
% 0.30/0.59 @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
% 0.30/0.59 @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % divmod_algorithm_code(5)
% 0.30/0.59 thf(fact_5242_divmod__algorithm__code_I5_J,axiom,
% 0.30/0.59 ! [M: num,N: num] :
% 0.30/0.59 ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.59 = ( produc2626176000494625587at_nat
% 0.30/0.59 @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
% 0.30/0.59 @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % divmod_algorithm_code(5)
% 0.30/0.59 thf(fact_5243_divmod__algorithm__code_I5_J,axiom,
% 0.30/0.59 ! [M: num,N: num] :
% 0.30/0.59 ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.59 = ( produc6916734918728496179nteger
% 0.30/0.59 @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
% 0.30/0.59 @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % divmod_algorithm_code(5)
% 0.30/0.59 thf(fact_5244_one__div__2__pow__eq,axiom,
% 0.30/0.59 ! [N: nat] :
% 0.30/0.59 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % one_div_2_pow_eq
% 0.30/0.59 thf(fact_5245_one__div__2__pow__eq,axiom,
% 0.30/0.59 ! [N: nat] :
% 0.30/0.59 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % one_div_2_pow_eq
% 0.30/0.59 thf(fact_5246_one__div__2__pow__eq,axiom,
% 0.30/0.59 ! [N: nat] :
% 0.30/0.59 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % one_div_2_pow_eq
% 0.30/0.59 thf(fact_5247_bits__1__div__exp,axiom,
% 0.30/0.59 ! [N: nat] :
% 0.30/0.59 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % bits_1_div_exp
% 0.30/0.59 thf(fact_5248_bits__1__div__exp,axiom,
% 0.30/0.59 ! [N: nat] :
% 0.30/0.59 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % bits_1_div_exp
% 0.30/0.59 thf(fact_5249_bits__1__div__exp,axiom,
% 0.30/0.59 ! [N: nat] :
% 0.30/0.59 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % bits_1_div_exp
% 0.30/0.59 thf(fact_5250_signed__take__bit__Suc__bit1,axiom,
% 0.30/0.59 ! [N: nat,K: num] :
% 0.30/0.59 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.30/0.59 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.30/0.59
% 0.30/0.59 % signed_take_bit_Suc_bit1
% 0.30/0.59 thf(fact_5251_one__mod__2__pow__eq,axiom,
% 0.30/0.59 ! [N: nat] :
% 0.30/0.59 ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % one_mod_2_pow_eq
% 0.30/0.59 thf(fact_5252_one__mod__2__pow__eq,axiom,
% 0.30/0.59 ! [N: nat] :
% 0.30/0.59 ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % one_mod_2_pow_eq
% 0.30/0.59 thf(fact_5253_one__mod__2__pow__eq,axiom,
% 0.30/0.59 ! [N: nat] :
% 0.30/0.59 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % one_mod_2_pow_eq
% 0.30/0.59 thf(fact_5254_dvd__antisym,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( dvd_dvd_nat @ M @ N )
% 0.30/0.59 => ( ( dvd_dvd_nat @ N @ M )
% 0.30/0.59 => ( M = N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % dvd_antisym
% 0.30/0.59 thf(fact_5255_of__bool__eq__iff,axiom,
% 0.30/0.59 ! [P6: $o,Q3: $o] :
% 0.30/0.59 ( ( ( zero_n2687167440665602831ol_nat @ P6 )
% 0.30/0.59 = ( zero_n2687167440665602831ol_nat @ Q3 ) )
% 0.30/0.59 = ( P6 = Q3 ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq_iff
% 0.30/0.59 thf(fact_5256_of__bool__eq__iff,axiom,
% 0.30/0.59 ! [P6: $o,Q3: $o] :
% 0.30/0.59 ( ( ( zero_n2684676970156552555ol_int @ P6 )
% 0.30/0.59 = ( zero_n2684676970156552555ol_int @ Q3 ) )
% 0.30/0.59 = ( P6 = Q3 ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq_iff
% 0.30/0.59 thf(fact_5257_of__bool__eq__iff,axiom,
% 0.30/0.59 ! [P6: $o,Q3: $o] :
% 0.30/0.59 ( ( ( zero_n356916108424825756nteger @ P6 )
% 0.30/0.59 = ( zero_n356916108424825756nteger @ Q3 ) )
% 0.30/0.59 = ( P6 = Q3 ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_eq_iff
% 0.30/0.59 thf(fact_5258_of__bool__conj,axiom,
% 0.30/0.59 ! [P: $o,Q: $o] :
% 0.30/0.59 ( ( zero_n3304061248610475627l_real
% 0.30/0.59 @ ( P
% 0.30/0.59 & Q ) )
% 0.30/0.59 = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_conj
% 0.30/0.59 thf(fact_5259_of__bool__conj,axiom,
% 0.30/0.59 ! [P: $o,Q: $o] :
% 0.30/0.59 ( ( zero_n2052037380579107095ol_rat
% 0.30/0.59 @ ( P
% 0.30/0.59 & Q ) )
% 0.30/0.59 = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_conj
% 0.30/0.59 thf(fact_5260_of__bool__conj,axiom,
% 0.30/0.59 ! [P: $o,Q: $o] :
% 0.30/0.59 ( ( zero_n2687167440665602831ol_nat
% 0.30/0.59 @ ( P
% 0.30/0.59 & Q ) )
% 0.30/0.59 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_conj
% 0.30/0.59 thf(fact_5261_of__bool__conj,axiom,
% 0.30/0.59 ! [P: $o,Q: $o] :
% 0.30/0.59 ( ( zero_n2684676970156552555ol_int
% 0.30/0.59 @ ( P
% 0.30/0.59 & Q ) )
% 0.30/0.59 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_conj
% 0.30/0.59 thf(fact_5262_of__bool__conj,axiom,
% 0.30/0.59 ! [P: $o,Q: $o] :
% 0.30/0.59 ( ( zero_n356916108424825756nteger
% 0.30/0.59 @ ( P
% 0.30/0.59 & Q ) )
% 0.30/0.59 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_conj
% 0.30/0.59 thf(fact_5263_old_Oprod_Ocase,axiom,
% 0.30/0.59 ! [F: nat > nat > nat,X1: nat,X22: nat] :
% 0.30/0.59 ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 0.30/0.59 = ( F @ X1 @ X22 ) ) ).
% 0.30/0.59
% 0.30/0.59 % old.prod.case
% 0.30/0.59 thf(fact_5264_old_Oprod_Ocase,axiom,
% 0.30/0.59 ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X22: nat] :
% 0.30/0.59 ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 0.30/0.59 = ( F @ X1 @ X22 ) ) ).
% 0.30/0.59
% 0.30/0.59 % old.prod.case
% 0.30/0.59 thf(fact_5265_old_Oprod_Ocase,axiom,
% 0.30/0.59 ! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X22: nat] :
% 0.30/0.59 ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 0.30/0.59 = ( F @ X1 @ X22 ) ) ).
% 0.30/0.59
% 0.30/0.59 % old.prod.case
% 0.30/0.59 thf(fact_5266_old_Oprod_Ocase,axiom,
% 0.30/0.59 ! [F: int > int > product_prod_int_int,X1: int,X22: int] :
% 0.30/0.59 ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
% 0.30/0.59 = ( F @ X1 @ X22 ) ) ).
% 0.30/0.59
% 0.30/0.59 % old.prod.case
% 0.30/0.59 thf(fact_5267_old_Oprod_Ocase,axiom,
% 0.30/0.59 ! [F: int > int > $o,X1: int,X22: int] :
% 0.30/0.59 ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
% 0.30/0.59 = ( F @ X1 @ X22 ) ) ).
% 0.30/0.59
% 0.30/0.59 % old.prod.case
% 0.30/0.59 thf(fact_5268_signed__take__bit__mult,axiom,
% 0.30/0.59 ! [N: nat,K: int,L: int] :
% 0.30/0.59 ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 0.30/0.59 = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % signed_take_bit_mult
% 0.30/0.59 thf(fact_5269_signed__take__bit__add,axiom,
% 0.30/0.59 ! [N: nat,K: int,L: int] :
% 0.30/0.59 ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 0.30/0.59 = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % signed_take_bit_add
% 0.30/0.59 thf(fact_5270_signed__take__bit__diff,axiom,
% 0.30/0.59 ! [N: nat,K: int,L: int] :
% 0.30/0.59 ( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 0.30/0.59 = ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % signed_take_bit_diff
% 0.30/0.59 thf(fact_5271_case__prodE2,axiom,
% 0.30/0.59 ! [Q: nat > $o,P: nat > nat > nat,Z: product_prod_nat_nat] :
% 0.30/0.59 ( ( Q @ ( produc6842872674320459806at_nat @ P @ Z ) )
% 0.30/0.59 => ~ ! [X5: nat,Y4: nat] :
% 0.30/0.59 ( ( Z
% 0.30/0.59 = ( product_Pair_nat_nat @ X5 @ Y4 ) )
% 0.30/0.59 => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prodE2
% 0.30/0.59 thf(fact_5272_case__prodE2,axiom,
% 0.30/0.59 ! [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] :
% 0.30/0.59 ( ( Q @ ( produc27273713700761075at_nat @ P @ Z ) )
% 0.30/0.59 => ~ ! [X5: nat,Y4: nat] :
% 0.30/0.59 ( ( Z
% 0.30/0.59 = ( product_Pair_nat_nat @ X5 @ Y4 ) )
% 0.30/0.59 => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prodE2
% 0.30/0.59 thf(fact_5273_case__prodE2,axiom,
% 0.30/0.59 ! [Q: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat] :
% 0.30/0.59 ( ( Q @ ( produc8739625826339149834_nat_o @ P @ Z ) )
% 0.30/0.59 => ~ ! [X5: nat,Y4: nat] :
% 0.30/0.59 ( ( Z
% 0.30/0.59 = ( product_Pair_nat_nat @ X5 @ Y4 ) )
% 0.30/0.59 => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prodE2
% 0.30/0.59 thf(fact_5274_case__prodE2,axiom,
% 0.30/0.59 ! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z: product_prod_int_int] :
% 0.30/0.59 ( ( Q @ ( produc4245557441103728435nt_int @ P @ Z ) )
% 0.30/0.59 => ~ ! [X5: int,Y4: int] :
% 0.30/0.59 ( ( Z
% 0.30/0.59 = ( product_Pair_int_int @ X5 @ Y4 ) )
% 0.30/0.59 => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prodE2
% 0.30/0.59 thf(fact_5275_case__prodE2,axiom,
% 0.30/0.59 ! [Q: $o > $o,P: int > int > $o,Z: product_prod_int_int] :
% 0.30/0.59 ( ( Q @ ( produc4947309494688390418_int_o @ P @ Z ) )
% 0.30/0.59 => ~ ! [X5: int,Y4: int] :
% 0.30/0.59 ( ( Z
% 0.30/0.59 = ( product_Pair_int_int @ X5 @ Y4 ) )
% 0.30/0.59 => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prodE2
% 0.30/0.59 thf(fact_5276_case__prod__eta,axiom,
% 0.30/0.59 ! [F: product_prod_nat_nat > nat] :
% 0.30/0.59 ( ( produc6842872674320459806at_nat
% 0.30/0.59 @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
% 0.30/0.59 = F ) ).
% 0.30/0.59
% 0.30/0.59 % case_prod_eta
% 0.30/0.59 thf(fact_5277_case__prod__eta,axiom,
% 0.30/0.59 ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 0.30/0.59 ( ( produc27273713700761075at_nat
% 0.30/0.59 @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
% 0.30/0.59 = F ) ).
% 0.30/0.59
% 0.30/0.59 % case_prod_eta
% 0.30/0.59 thf(fact_5278_case__prod__eta,axiom,
% 0.30/0.59 ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 0.30/0.59 ( ( produc8739625826339149834_nat_o
% 0.30/0.59 @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
% 0.30/0.59 = F ) ).
% 0.30/0.59
% 0.30/0.59 % case_prod_eta
% 0.30/0.59 thf(fact_5279_case__prod__eta,axiom,
% 0.30/0.59 ! [F: product_prod_int_int > product_prod_int_int] :
% 0.30/0.59 ( ( produc4245557441103728435nt_int
% 0.30/0.59 @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
% 0.30/0.59 = F ) ).
% 0.30/0.59
% 0.30/0.59 % case_prod_eta
% 0.30/0.59 thf(fact_5280_case__prod__eta,axiom,
% 0.30/0.59 ! [F: product_prod_int_int > $o] :
% 0.30/0.59 ( ( produc4947309494688390418_int_o
% 0.30/0.59 @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
% 0.30/0.59 = F ) ).
% 0.30/0.59
% 0.30/0.59 % case_prod_eta
% 0.30/0.59 thf(fact_5281_cond__case__prod__eta,axiom,
% 0.30/0.59 ! [F: nat > nat > nat,G: product_prod_nat_nat > nat] :
% 0.30/0.59 ( ! [X5: nat,Y4: nat] :
% 0.30/0.59 ( ( F @ X5 @ Y4 )
% 0.30/0.59 = ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
% 0.30/0.59 => ( ( produc6842872674320459806at_nat @ F )
% 0.30/0.59 = G ) ) ).
% 0.30/0.59
% 0.30/0.59 % cond_case_prod_eta
% 0.30/0.59 thf(fact_5282_cond__case__prod__eta,axiom,
% 0.30/0.59 ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 0.30/0.59 ( ! [X5: nat,Y4: nat] :
% 0.30/0.59 ( ( F @ X5 @ Y4 )
% 0.30/0.59 = ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
% 0.30/0.59 => ( ( produc27273713700761075at_nat @ F )
% 0.30/0.59 = G ) ) ).
% 0.30/0.59
% 0.30/0.59 % cond_case_prod_eta
% 0.30/0.59 thf(fact_5283_cond__case__prod__eta,axiom,
% 0.30/0.59 ! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 0.30/0.59 ( ! [X5: nat,Y4: nat] :
% 0.30/0.59 ( ( F @ X5 @ Y4 )
% 0.30/0.59 = ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
% 0.30/0.59 => ( ( produc8739625826339149834_nat_o @ F )
% 0.30/0.59 = G ) ) ).
% 0.30/0.59
% 0.30/0.59 % cond_case_prod_eta
% 0.30/0.59 thf(fact_5284_cond__case__prod__eta,axiom,
% 0.30/0.59 ! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
% 0.30/0.59 ( ! [X5: int,Y4: int] :
% 0.30/0.59 ( ( F @ X5 @ Y4 )
% 0.30/0.59 = ( G @ ( product_Pair_int_int @ X5 @ Y4 ) ) )
% 0.30/0.59 => ( ( produc4245557441103728435nt_int @ F )
% 0.30/0.59 = G ) ) ).
% 0.30/0.59
% 0.30/0.59 % cond_case_prod_eta
% 0.30/0.59 thf(fact_5285_cond__case__prod__eta,axiom,
% 0.30/0.59 ! [F: int > int > $o,G: product_prod_int_int > $o] :
% 0.30/0.59 ( ! [X5: int,Y4: int] :
% 0.30/0.59 ( ( F @ X5 @ Y4 )
% 0.30/0.59 = ( G @ ( product_Pair_int_int @ X5 @ Y4 ) ) )
% 0.30/0.59 => ( ( produc4947309494688390418_int_o @ F )
% 0.30/0.59 = G ) ) ).
% 0.30/0.59
% 0.30/0.59 % cond_case_prod_eta
% 0.30/0.59 thf(fact_5286_zero__less__eq__of__bool,axiom,
% 0.30/0.59 ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.30/0.59
% 0.30/0.59 % zero_less_eq_of_bool
% 0.30/0.59 thf(fact_5287_zero__less__eq__of__bool,axiom,
% 0.30/0.59 ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.30/0.59
% 0.30/0.59 % zero_less_eq_of_bool
% 0.30/0.59 thf(fact_5288_zero__less__eq__of__bool,axiom,
% 0.30/0.59 ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.30/0.59
% 0.30/0.59 % zero_less_eq_of_bool
% 0.30/0.59 thf(fact_5289_zero__less__eq__of__bool,axiom,
% 0.30/0.59 ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.30/0.59
% 0.30/0.59 % zero_less_eq_of_bool
% 0.30/0.59 thf(fact_5290_zero__less__eq__of__bool,axiom,
% 0.30/0.59 ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).
% 0.30/0.59
% 0.30/0.59 % zero_less_eq_of_bool
% 0.30/0.59 thf(fact_5291_of__bool__less__eq__one,axiom,
% 0.30/0.59 ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_less_eq_one
% 0.30/0.59 thf(fact_5292_of__bool__less__eq__one,axiom,
% 0.30/0.59 ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_less_eq_one
% 0.30/0.59 thf(fact_5293_of__bool__less__eq__one,axiom,
% 0.30/0.59 ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_less_eq_one
% 0.30/0.59 thf(fact_5294_of__bool__less__eq__one,axiom,
% 0.30/0.59 ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_less_eq_one
% 0.30/0.59 thf(fact_5295_of__bool__less__eq__one,axiom,
% 0.30/0.59 ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_less_eq_one
% 0.30/0.59 thf(fact_5296_of__bool__def,axiom,
% 0.30/0.59 ( zero_n1201886186963655149omplex
% 0.30/0.59 = ( ^ [P4: $o] : ( if_complex @ P4 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_def
% 0.30/0.59 thf(fact_5297_of__bool__def,axiom,
% 0.30/0.59 ( zero_n3304061248610475627l_real
% 0.30/0.59 = ( ^ [P4: $o] : ( if_real @ P4 @ one_one_real @ zero_zero_real ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_def
% 0.30/0.59 thf(fact_5298_of__bool__def,axiom,
% 0.30/0.59 ( zero_n2052037380579107095ol_rat
% 0.30/0.59 = ( ^ [P4: $o] : ( if_rat @ P4 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_def
% 0.30/0.59 thf(fact_5299_of__bool__def,axiom,
% 0.30/0.59 ( zero_n2687167440665602831ol_nat
% 0.30/0.59 = ( ^ [P4: $o] : ( if_nat @ P4 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_def
% 0.30/0.59 thf(fact_5300_of__bool__def,axiom,
% 0.30/0.59 ( zero_n2684676970156552555ol_int
% 0.30/0.59 = ( ^ [P4: $o] : ( if_int @ P4 @ one_one_int @ zero_zero_int ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_def
% 0.30/0.59 thf(fact_5301_of__bool__def,axiom,
% 0.30/0.59 ( zero_n356916108424825756nteger
% 0.30/0.59 = ( ^ [P4: $o] : ( if_Code_integer @ P4 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_def
% 0.30/0.59 thf(fact_5302_split__of__bool,axiom,
% 0.30/0.59 ! [P: complex > $o,P6: $o] :
% 0.30/0.59 ( ( P @ ( zero_n1201886186963655149omplex @ P6 ) )
% 0.30/0.59 = ( ( P6
% 0.30/0.59 => ( P @ one_one_complex ) )
% 0.30/0.59 & ( ~ P6
% 0.30/0.59 => ( P @ zero_zero_complex ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % split_of_bool
% 0.30/0.59 thf(fact_5303_split__of__bool,axiom,
% 0.30/0.59 ! [P: real > $o,P6: $o] :
% 0.30/0.59 ( ( P @ ( zero_n3304061248610475627l_real @ P6 ) )
% 0.30/0.59 = ( ( P6
% 0.30/0.59 => ( P @ one_one_real ) )
% 0.30/0.59 & ( ~ P6
% 0.30/0.59 => ( P @ zero_zero_real ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % split_of_bool
% 0.30/0.59 thf(fact_5304_split__of__bool,axiom,
% 0.30/0.59 ! [P: rat > $o,P6: $o] :
% 0.30/0.59 ( ( P @ ( zero_n2052037380579107095ol_rat @ P6 ) )
% 0.30/0.59 = ( ( P6
% 0.30/0.59 => ( P @ one_one_rat ) )
% 0.30/0.59 & ( ~ P6
% 0.30/0.59 => ( P @ zero_zero_rat ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % split_of_bool
% 0.30/0.59 thf(fact_5305_split__of__bool,axiom,
% 0.30/0.59 ! [P: nat > $o,P6: $o] :
% 0.30/0.59 ( ( P @ ( zero_n2687167440665602831ol_nat @ P6 ) )
% 0.30/0.59 = ( ( P6
% 0.30/0.59 => ( P @ one_one_nat ) )
% 0.30/0.59 & ( ~ P6
% 0.30/0.59 => ( P @ zero_zero_nat ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % split_of_bool
% 0.30/0.59 thf(fact_5306_split__of__bool,axiom,
% 0.30/0.59 ! [P: int > $o,P6: $o] :
% 0.30/0.59 ( ( P @ ( zero_n2684676970156552555ol_int @ P6 ) )
% 0.30/0.59 = ( ( P6
% 0.30/0.59 => ( P @ one_one_int ) )
% 0.30/0.59 & ( ~ P6
% 0.30/0.59 => ( P @ zero_zero_int ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % split_of_bool
% 0.30/0.59 thf(fact_5307_split__of__bool,axiom,
% 0.30/0.59 ! [P: code_integer > $o,P6: $o] :
% 0.30/0.59 ( ( P @ ( zero_n356916108424825756nteger @ P6 ) )
% 0.30/0.59 = ( ( P6
% 0.30/0.59 => ( P @ one_one_Code_integer ) )
% 0.30/0.59 & ( ~ P6
% 0.30/0.59 => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % split_of_bool
% 0.30/0.59 thf(fact_5308_split__of__bool__asm,axiom,
% 0.30/0.59 ! [P: complex > $o,P6: $o] :
% 0.30/0.59 ( ( P @ ( zero_n1201886186963655149omplex @ P6 ) )
% 0.30/0.59 = ( ~ ( ( P6
% 0.30/0.59 & ~ ( P @ one_one_complex ) )
% 0.30/0.59 | ( ~ P6
% 0.30/0.59 & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % split_of_bool_asm
% 0.30/0.59 thf(fact_5309_split__of__bool__asm,axiom,
% 0.30/0.59 ! [P: real > $o,P6: $o] :
% 0.30/0.59 ( ( P @ ( zero_n3304061248610475627l_real @ P6 ) )
% 0.30/0.59 = ( ~ ( ( P6
% 0.30/0.59 & ~ ( P @ one_one_real ) )
% 0.30/0.59 | ( ~ P6
% 0.30/0.59 & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % split_of_bool_asm
% 0.30/0.59 thf(fact_5310_split__of__bool__asm,axiom,
% 0.30/0.59 ! [P: rat > $o,P6: $o] :
% 0.30/0.59 ( ( P @ ( zero_n2052037380579107095ol_rat @ P6 ) )
% 0.30/0.59 = ( ~ ( ( P6
% 0.30/0.59 & ~ ( P @ one_one_rat ) )
% 0.30/0.59 | ( ~ P6
% 0.30/0.59 & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % split_of_bool_asm
% 0.30/0.59 thf(fact_5311_split__of__bool__asm,axiom,
% 0.30/0.59 ! [P: nat > $o,P6: $o] :
% 0.30/0.59 ( ( P @ ( zero_n2687167440665602831ol_nat @ P6 ) )
% 0.30/0.59 = ( ~ ( ( P6
% 0.30/0.59 & ~ ( P @ one_one_nat ) )
% 0.30/0.59 | ( ~ P6
% 0.30/0.59 & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % split_of_bool_asm
% 0.30/0.59 thf(fact_5312_split__of__bool__asm,axiom,
% 0.30/0.59 ! [P: int > $o,P6: $o] :
% 0.30/0.59 ( ( P @ ( zero_n2684676970156552555ol_int @ P6 ) )
% 0.30/0.59 = ( ~ ( ( P6
% 0.30/0.59 & ~ ( P @ one_one_int ) )
% 0.30/0.59 | ( ~ P6
% 0.30/0.59 & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % split_of_bool_asm
% 0.30/0.59 thf(fact_5313_split__of__bool__asm,axiom,
% 0.30/0.59 ! [P: code_integer > $o,P6: $o] :
% 0.30/0.59 ( ( P @ ( zero_n356916108424825756nteger @ P6 ) )
% 0.30/0.59 = ( ~ ( ( P6
% 0.30/0.59 & ~ ( P @ one_one_Code_integer ) )
% 0.30/0.59 | ( ~ P6
% 0.30/0.59 & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % split_of_bool_asm
% 0.30/0.59 thf(fact_5314_replicate__length__same,axiom,
% 0.30/0.59 ! [Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 0.30/0.59 ( ! [X5: vEBT_VEBT] :
% 0.30/0.59 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.30/0.59 => ( X5 = X2 ) )
% 0.30/0.59 => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ X2 )
% 0.30/0.59 = Xs2 ) ) ).
% 0.30/0.59
% 0.30/0.59 % replicate_length_same
% 0.30/0.59 thf(fact_5315_replicate__length__same,axiom,
% 0.30/0.59 ! [Xs2: list_o,X2: $o] :
% 0.30/0.59 ( ! [X5: $o] :
% 0.30/0.59 ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
% 0.30/0.59 => ( X5 = X2 ) )
% 0.30/0.59 => ( ( replicate_o @ ( size_size_list_o @ Xs2 ) @ X2 )
% 0.30/0.59 = Xs2 ) ) ).
% 0.30/0.59
% 0.30/0.59 % replicate_length_same
% 0.30/0.59 thf(fact_5316_replicate__length__same,axiom,
% 0.30/0.59 ! [Xs2: list_nat,X2: nat] :
% 0.30/0.59 ( ! [X5: nat] :
% 0.30/0.59 ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
% 0.30/0.59 => ( X5 = X2 ) )
% 0.30/0.59 => ( ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X2 )
% 0.30/0.59 = Xs2 ) ) ).
% 0.30/0.59
% 0.30/0.59 % replicate_length_same
% 0.30/0.59 thf(fact_5317_replicate__length__same,axiom,
% 0.30/0.59 ! [Xs2: list_int,X2: int] :
% 0.30/0.59 ( ! [X5: int] :
% 0.30/0.59 ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
% 0.30/0.59 => ( X5 = X2 ) )
% 0.30/0.59 => ( ( replicate_int @ ( size_size_list_int @ Xs2 ) @ X2 )
% 0.30/0.59 = Xs2 ) ) ).
% 0.30/0.59
% 0.30/0.59 % replicate_length_same
% 0.30/0.59 thf(fact_5318_replicate__eqI,axiom,
% 0.30/0.59 ! [Xs2: list_P6011104703257516679at_nat,N: nat,X2: product_prod_nat_nat] :
% 0.30/0.59 ( ( ( size_s5460976970255530739at_nat @ Xs2 )
% 0.30/0.59 = N )
% 0.30/0.59 => ( ! [Y4: product_prod_nat_nat] :
% 0.30/0.59 ( ( member8440522571783428010at_nat @ Y4 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
% 0.30/0.59 => ( Y4 = X2 ) )
% 0.30/0.59 => ( Xs2
% 0.30/0.59 = ( replic4235873036481779905at_nat @ N @ X2 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % replicate_eqI
% 0.30/0.59 thf(fact_5319_replicate__eqI,axiom,
% 0.30/0.59 ! [Xs2: list_complex,N: nat,X2: complex] :
% 0.30/0.59 ( ( ( size_s3451745648224563538omplex @ Xs2 )
% 0.30/0.59 = N )
% 0.30/0.59 => ( ! [Y4: complex] :
% 0.30/0.59 ( ( member_complex @ Y4 @ ( set_complex2 @ Xs2 ) )
% 0.30/0.59 => ( Y4 = X2 ) )
% 0.30/0.59 => ( Xs2
% 0.30/0.59 = ( replicate_complex @ N @ X2 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % replicate_eqI
% 0.30/0.59 thf(fact_5320_replicate__eqI,axiom,
% 0.30/0.59 ! [Xs2: list_real,N: nat,X2: real] :
% 0.30/0.59 ( ( ( size_size_list_real @ Xs2 )
% 0.30/0.59 = N )
% 0.30/0.59 => ( ! [Y4: real] :
% 0.30/0.59 ( ( member_real @ Y4 @ ( set_real2 @ Xs2 ) )
% 0.30/0.59 => ( Y4 = X2 ) )
% 0.30/0.59 => ( Xs2
% 0.30/0.59 = ( replicate_real @ N @ X2 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % replicate_eqI
% 0.30/0.59 thf(fact_5321_replicate__eqI,axiom,
% 0.30/0.59 ! [Xs2: list_VEBT_VEBT,N: nat,X2: vEBT_VEBT] :
% 0.30/0.59 ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.30/0.59 = N )
% 0.30/0.59 => ( ! [Y4: vEBT_VEBT] :
% 0.30/0.59 ( ( member_VEBT_VEBT @ Y4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.30/0.59 => ( Y4 = X2 ) )
% 0.30/0.59 => ( Xs2
% 0.30/0.59 = ( replicate_VEBT_VEBT @ N @ X2 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % replicate_eqI
% 0.30/0.59 thf(fact_5322_replicate__eqI,axiom,
% 0.30/0.59 ! [Xs2: list_o,N: nat,X2: $o] :
% 0.30/0.59 ( ( ( size_size_list_o @ Xs2 )
% 0.30/0.59 = N )
% 0.30/0.59 => ( ! [Y4: $o] :
% 0.30/0.59 ( ( member_o @ Y4 @ ( set_o2 @ Xs2 ) )
% 0.30/0.59 => ( Y4 = X2 ) )
% 0.30/0.59 => ( Xs2
% 0.30/0.59 = ( replicate_o @ N @ X2 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % replicate_eqI
% 0.30/0.59 thf(fact_5323_replicate__eqI,axiom,
% 0.30/0.59 ! [Xs2: list_nat,N: nat,X2: nat] :
% 0.30/0.59 ( ( ( size_size_list_nat @ Xs2 )
% 0.30/0.59 = N )
% 0.30/0.59 => ( ! [Y4: nat] :
% 0.30/0.59 ( ( member_nat @ Y4 @ ( set_nat2 @ Xs2 ) )
% 0.30/0.59 => ( Y4 = X2 ) )
% 0.30/0.59 => ( Xs2
% 0.30/0.59 = ( replicate_nat @ N @ X2 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % replicate_eqI
% 0.30/0.59 thf(fact_5324_replicate__eqI,axiom,
% 0.30/0.59 ! [Xs2: list_int,N: nat,X2: int] :
% 0.30/0.59 ( ( ( size_size_list_int @ Xs2 )
% 0.30/0.59 = N )
% 0.30/0.59 => ( ! [Y4: int] :
% 0.30/0.59 ( ( member_int @ Y4 @ ( set_int2 @ Xs2 ) )
% 0.30/0.59 => ( Y4 = X2 ) )
% 0.30/0.59 => ( Xs2
% 0.30/0.59 = ( replicate_int @ N @ X2 ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % replicate_eqI
% 0.30/0.59 thf(fact_5325_subset__decode__imp__le,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 0.30/0.59 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.59
% 0.30/0.59 % subset_decode_imp_le
% 0.30/0.59 thf(fact_5326_of__bool__odd__eq__mod__2,axiom,
% 0.30/0.59 ! [A: nat] :
% 0.30/0.59 ( ( zero_n2687167440665602831ol_nat
% 0.30/0.59 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.59 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_odd_eq_mod_2
% 0.30/0.59 thf(fact_5327_of__bool__odd__eq__mod__2,axiom,
% 0.30/0.59 ! [A: int] :
% 0.30/0.59 ( ( zero_n2684676970156552555ol_int
% 0.30/0.59 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.59 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_odd_eq_mod_2
% 0.30/0.59 thf(fact_5328_of__bool__odd__eq__mod__2,axiom,
% 0.30/0.59 ! [A: code_integer] :
% 0.30/0.59 ( ( zero_n356916108424825756nteger
% 0.30/0.59 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.59 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % of_bool_odd_eq_mod_2
% 0.30/0.59 thf(fact_5329_signed__take__bit__int__less__exp,axiom,
% 0.30/0.59 ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.59
% 0.30/0.59 % signed_take_bit_int_less_exp
% 0.30/0.59 thf(fact_5330_even__signed__take__bit__iff,axiom,
% 0.30/0.59 ! [M: nat,A: code_integer] :
% 0.30/0.59 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
% 0.30/0.59 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_signed_take_bit_iff
% 0.30/0.59 thf(fact_5331_even__signed__take__bit__iff,axiom,
% 0.30/0.59 ! [M: nat,A: int] :
% 0.30/0.59 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
% 0.30/0.59 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.30/0.59
% 0.30/0.59 % even_signed_take_bit_iff
% 0.30/0.59 thf(fact_5332_bits__induct,axiom,
% 0.30/0.59 ! [P: nat > $o,A: nat] :
% 0.30/0.59 ( ! [A4: nat] :
% 0.30/0.59 ( ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59 = A4 )
% 0.30/0.59 => ( P @ A4 ) )
% 0.30/0.59 => ( ! [A4: nat,B4: $o] :
% 0.30/0.59 ( ( P @ A4 )
% 0.30/0.59 => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59 = A4 )
% 0.30/0.59 => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 0.30/0.59 => ( P @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % bits_induct
% 0.30/0.59 thf(fact_5333_bits__induct,axiom,
% 0.30/0.59 ! [P: int > $o,A: int] :
% 0.30/0.59 ( ! [A4: int] :
% 0.30/0.59 ( ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.59 = A4 )
% 0.30/0.59 => ( P @ A4 ) )
% 0.30/0.59 => ( ! [A4: int,B4: $o] :
% 0.30/0.59 ( ( P @ A4 )
% 0.30/0.59 => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.59 = A4 )
% 0.30/0.59 => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 0.30/0.59 => ( P @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % bits_induct
% 0.30/0.59 thf(fact_5334_bits__induct,axiom,
% 0.30/0.59 ! [P: code_integer > $o,A: code_integer] :
% 0.30/0.59 ( ! [A4: code_integer] :
% 0.30/0.59 ( ( ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.59 = A4 )
% 0.30/0.59 => ( P @ A4 ) )
% 0.30/0.59 => ( ! [A4: code_integer,B4: $o] :
% 0.30/0.59 ( ( P @ A4 )
% 0.30/0.59 => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B4 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.59 = A4 )
% 0.30/0.59 => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B4 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 0.30/0.59 => ( P @ A ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % bits_induct
% 0.30/0.59 thf(fact_5335_signed__take__bit__int__greater__eq__self__iff,axiom,
% 0.30/0.59 ! [K: int,N: nat] :
% 0.30/0.59 ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.30/0.59 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % signed_take_bit_int_greater_eq_self_iff
% 0.30/0.59 thf(fact_5336_signed__take__bit__int__less__self__iff,axiom,
% 0.30/0.59 ! [N: nat,K: int] :
% 0.30/0.59 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 0.30/0.59 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 0.30/0.59
% 0.30/0.59 % signed_take_bit_int_less_self_iff
% 0.30/0.59 thf(fact_5337_exp__mod__exp,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % exp_mod_exp
% 0.30/0.59 thf(fact_5338_exp__mod__exp,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % exp_mod_exp
% 0.30/0.59 thf(fact_5339_exp__mod__exp,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % exp_mod_exp
% 0.30/0.59 thf(fact_5340_signed__take__bit__int__less__eq,axiom,
% 0.30/0.59 ! [N: nat,K: int] :
% 0.30/0.59 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 0.30/0.59 => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % signed_take_bit_int_less_eq
% 0.30/0.59 thf(fact_5341_option_Osize__gen_I1_J,axiom,
% 0.30/0.59 ! [X2: nat > nat] :
% 0.30/0.59 ( ( size_option_nat @ X2 @ none_nat )
% 0.30/0.59 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.59
% 0.30/0.59 % option.size_gen(1)
% 0.30/0.59 thf(fact_5342_option_Osize__gen_I1_J,axiom,
% 0.30/0.59 ! [X2: product_prod_nat_nat > nat] :
% 0.30/0.59 ( ( size_o8335143837870341156at_nat @ X2 @ none_P5556105721700978146at_nat )
% 0.30/0.59 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.59
% 0.30/0.59 % option.size_gen(1)
% 0.30/0.59 thf(fact_5343_option_Osize__gen_I1_J,axiom,
% 0.30/0.59 ! [X2: num > nat] :
% 0.30/0.59 ( ( size_option_num @ X2 @ none_num )
% 0.30/0.59 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.59
% 0.30/0.59 % option.size_gen(1)
% 0.30/0.59 thf(fact_5344_divmod__step__nat__def,axiom,
% 0.30/0.59 ( unique5026877609467782581ep_nat
% 0.30/0.59 = ( ^ [L2: num] :
% 0.30/0.59 ( produc2626176000494625587at_nat
% 0.30/0.59 @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % divmod_step_nat_def
% 0.30/0.59 thf(fact_5345_set__decode__def,axiom,
% 0.30/0.59 ( nat_set_decode
% 0.30/0.59 = ( ^ [X: nat] :
% 0.30/0.59 ( collect_nat
% 0.30/0.59 @ ^ [N3: nat] :
% 0.30/0.59 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % set_decode_def
% 0.30/0.59 thf(fact_5346_exp__div__exp__eq,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59 = ( times_times_nat
% 0.30/0.59 @ ( zero_n2687167440665602831ol_nat
% 0.30/0.59 @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.30/0.59 != zero_zero_nat )
% 0.30/0.59 & ( ord_less_eq_nat @ N @ M ) ) )
% 0.30/0.59 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % exp_div_exp_eq
% 0.30/0.59 thf(fact_5347_exp__div__exp__eq,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59 = ( times_times_int
% 0.30/0.59 @ ( zero_n2684676970156552555ol_int
% 0.30/0.59 @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.30/0.59 != zero_zero_int )
% 0.30/0.59 & ( ord_less_eq_nat @ N @ M ) ) )
% 0.30/0.59 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % exp_div_exp_eq
% 0.30/0.59 thf(fact_5348_exp__div__exp__eq,axiom,
% 0.30/0.59 ! [M: nat,N: nat] :
% 0.30/0.59 ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59 = ( times_3573771949741848930nteger
% 0.30/0.59 @ ( zero_n356916108424825756nteger
% 0.30/0.59 @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 0.30/0.59 != zero_z3403309356797280102nteger )
% 0.30/0.59 & ( ord_less_eq_nat @ N @ M ) ) )
% 0.30/0.59 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % exp_div_exp_eq
% 0.30/0.59 thf(fact_5349_divmod__step__int__def,axiom,
% 0.30/0.59 ( unique5024387138958732305ep_int
% 0.30/0.59 = ( ^ [L2: num] :
% 0.30/0.59 ( produc4245557441103728435nt_int
% 0.30/0.59 @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % divmod_step_int_def
% 0.30/0.59 thf(fact_5350_vebt__buildup_Osimps_I3_J,axiom,
% 0.30/0.59 ! [Va: nat] :
% 0.30/0.59 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.59 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.59 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 0.30/0.59 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.59 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.59 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % vebt_buildup.simps(3)
% 0.30/0.59 thf(fact_5351_divmod__step__def,axiom,
% 0.30/0.59 ( unique5026877609467782581ep_nat
% 0.30/0.59 = ( ^ [L2: num] :
% 0.30/0.59 ( produc2626176000494625587at_nat
% 0.30/0.59 @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % divmod_step_def
% 0.30/0.59 thf(fact_5352_divmod__step__def,axiom,
% 0.30/0.59 ( unique5024387138958732305ep_int
% 0.30/0.59 = ( ^ [L2: num] :
% 0.30/0.59 ( produc4245557441103728435nt_int
% 0.30/0.59 @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % divmod_step_def
% 0.30/0.59 thf(fact_5353_divmod__step__def,axiom,
% 0.30/0.59 ( unique4921790084139445826nteger
% 0.30/0.59 = ( ^ [L2: num] :
% 0.30/0.59 ( produc6916734918728496179nteger
% 0.30/0.59 @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % divmod_step_def
% 0.30/0.59 thf(fact_5354_divmod__nat__if,axiom,
% 0.30/0.59 ( divmod_nat
% 0.30/0.59 = ( ^ [M3: nat,N3: nat] :
% 0.30/0.59 ( if_Pro6206227464963214023at_nat
% 0.30/0.59 @ ( ( N3 = zero_zero_nat )
% 0.30/0.59 | ( ord_less_nat @ M3 @ N3 ) )
% 0.30/0.59 @ ( product_Pair_nat_nat @ zero_zero_nat @ M3 )
% 0.30/0.59 @ ( produc2626176000494625587at_nat
% 0.30/0.59 @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
% 0.30/0.59 @ ( divmod_nat @ ( minus_minus_nat @ M3 @ N3 ) @ N3 ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % divmod_nat_if
% 0.30/0.59 thf(fact_5355_signed__take__bit__Suc__minus__bit1,axiom,
% 0.30/0.59 ! [N: nat,K: num] :
% 0.30/0.59 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.30/0.59 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.30/0.59
% 0.30/0.59 % signed_take_bit_Suc_minus_bit1
% 0.30/0.59 thf(fact_5356_signed__take__bit__rec,axiom,
% 0.30/0.59 ( bit_ri6519982836138164636nteger
% 0.30/0.59 = ( ^ [N3: nat,A3: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % signed_take_bit_rec
% 0.30/0.59 thf(fact_5357_signed__take__bit__rec,axiom,
% 0.30/0.59 ( bit_ri631733984087533419it_int
% 0.30/0.59 = ( ^ [N3: nat,A3: int] : ( if_int @ ( N3 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % signed_take_bit_rec
% 0.30/0.59 thf(fact_5358_signed__take__bit__numeral__bit1,axiom,
% 0.30/0.59 ! [L: num,K: num] :
% 0.30/0.59 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.30/0.59 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.30/0.59
% 0.30/0.59 % signed_take_bit_numeral_bit1
% 0.30/0.59 thf(fact_5359_vebt__buildup_Opelims,axiom,
% 0.30/0.59 ! [X2: nat,Y3: vEBT_VEBT] :
% 0.30/0.59 ( ( ( vEBT_vebt_buildup @ X2 )
% 0.30/0.59 = Y3 )
% 0.30/0.59 => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X2 )
% 0.30/0.59 => ( ( ( X2 = zero_zero_nat )
% 0.30/0.59 => ( ( Y3
% 0.30/0.59 = ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.59 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 0.30/0.59 => ( ( ( X2
% 0.30/0.59 = ( suc @ zero_zero_nat ) )
% 0.30/0.59 => ( ( Y3
% 0.30/0.59 = ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.59 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 0.30/0.59 => ~ ! [Va3: nat] :
% 0.30/0.59 ( ( X2
% 0.30/0.59 = ( suc @ ( suc @ Va3 ) ) )
% 0.30/0.59 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 0.30/0.59 => ( Y3
% 0.30/0.59 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 0.30/0.59 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 0.30/0.59 => ( Y3
% 0.30/0.59 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.30/0.59 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % vebt_buildup.pelims
% 0.30/0.59 thf(fact_5360_diff__shunt__var,axiom,
% 0.30/0.59 ! [X2: set_real,Y3: set_real] :
% 0.30/0.59 ( ( ( minus_minus_set_real @ X2 @ Y3 )
% 0.30/0.59 = bot_bot_set_real )
% 0.30/0.59 = ( ord_less_eq_set_real @ X2 @ Y3 ) ) ).
% 0.30/0.59
% 0.30/0.59 % diff_shunt_var
% 0.30/0.59 thf(fact_5361_diff__shunt__var,axiom,
% 0.30/0.59 ! [X2: set_nat,Y3: set_nat] :
% 0.30/0.59 ( ( ( minus_minus_set_nat @ X2 @ Y3 )
% 0.30/0.59 = bot_bot_set_nat )
% 0.30/0.59 = ( ord_less_eq_set_nat @ X2 @ Y3 ) ) ).
% 0.30/0.59
% 0.30/0.59 % diff_shunt_var
% 0.30/0.59 thf(fact_5362_diff__shunt__var,axiom,
% 0.30/0.59 ! [X2: set_int,Y3: set_int] :
% 0.30/0.59 ( ( ( minus_minus_set_int @ X2 @ Y3 )
% 0.30/0.59 = bot_bot_set_int )
% 0.30/0.59 = ( ord_less_eq_set_int @ X2 @ Y3 ) ) ).
% 0.30/0.59
% 0.30/0.59 % diff_shunt_var
% 0.30/0.59 thf(fact_5363_add__scale__eq__noteq,axiom,
% 0.30/0.59 ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
% 0.30/0.59 ( ( R2 != zero_zero_complex )
% 0.30/0.59 => ( ( ( A = B )
% 0.30/0.59 & ( C != D ) )
% 0.30/0.59 => ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C ) )
% 0.30/0.59 != ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % add_scale_eq_noteq
% 0.30/0.59 thf(fact_5364_add__scale__eq__noteq,axiom,
% 0.30/0.59 ! [R2: real,A: real,B: real,C: real,D: real] :
% 0.30/0.59 ( ( R2 != zero_zero_real )
% 0.30/0.59 => ( ( ( A = B )
% 0.30/0.59 & ( C != D ) )
% 0.30/0.59 => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
% 0.30/0.59 != ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % add_scale_eq_noteq
% 0.30/0.59 thf(fact_5365_add__scale__eq__noteq,axiom,
% 0.30/0.59 ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.59 ( ( R2 != zero_zero_rat )
% 0.30/0.59 => ( ( ( A = B )
% 0.30/0.59 & ( C != D ) )
% 0.30/0.59 => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
% 0.30/0.59 != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % add_scale_eq_noteq
% 0.30/0.59 thf(fact_5366_add__scale__eq__noteq,axiom,
% 0.30/0.59 ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.59 ( ( R2 != zero_zero_nat )
% 0.30/0.59 => ( ( ( A = B )
% 0.30/0.59 & ( C != D ) )
% 0.30/0.59 => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
% 0.30/0.59 != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % add_scale_eq_noteq
% 0.30/0.59 thf(fact_5367_add__scale__eq__noteq,axiom,
% 0.30/0.59 ! [R2: int,A: int,B: int,C: int,D: int] :
% 0.30/0.59 ( ( R2 != zero_zero_int )
% 0.30/0.59 => ( ( ( A = B )
% 0.30/0.59 & ( C != D ) )
% 0.30/0.59 => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
% 0.30/0.59 != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % add_scale_eq_noteq
% 0.30/0.59 thf(fact_5368_add_Oinverse__inverse,axiom,
% 0.30/0.59 ! [A: int] :
% 0.30/0.59 ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 0.30/0.59 = A ) ).
% 0.30/0.59
% 0.30/0.59 % add.inverse_inverse
% 0.30/0.59 thf(fact_5369_add_Oinverse__inverse,axiom,
% 0.30/0.59 ! [A: real] :
% 0.30/0.59 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 0.30/0.59 = A ) ).
% 0.30/0.59
% 0.30/0.59 % add.inverse_inverse
% 0.30/0.59 thf(fact_5370_add_Oinverse__inverse,axiom,
% 0.30/0.59 ! [A: complex] :
% 0.30/0.59 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
% 0.30/0.59 = A ) ).
% 0.30/0.59
% 0.30/0.59 % add.inverse_inverse
% 0.30/0.59 thf(fact_5371_add_Oinverse__inverse,axiom,
% 0.30/0.59 ! [A: code_integer] :
% 0.30/0.59 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.59 = A ) ).
% 0.30/0.59
% 0.30/0.59 % add.inverse_inverse
% 0.30/0.59 thf(fact_5372_add_Oinverse__inverse,axiom,
% 0.30/0.59 ! [A: rat] :
% 0.30/0.59 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
% 0.30/0.59 = A ) ).
% 0.30/0.59
% 0.30/0.59 % add.inverse_inverse
% 0.30/0.59 thf(fact_5373_neg__equal__iff__equal,axiom,
% 0.30/0.59 ! [A: int,B: int] :
% 0.30/0.59 ( ( ( uminus_uminus_int @ A )
% 0.30/0.59 = ( uminus_uminus_int @ B ) )
% 0.30/0.59 = ( A = B ) ) ).
% 0.30/0.59
% 0.30/0.59 % neg_equal_iff_equal
% 0.30/0.59 thf(fact_5374_neg__equal__iff__equal,axiom,
% 0.30/0.59 ! [A: real,B: real] :
% 0.30/0.59 ( ( ( uminus_uminus_real @ A )
% 0.30/0.59 = ( uminus_uminus_real @ B ) )
% 0.30/0.59 = ( A = B ) ) ).
% 0.30/0.59
% 0.30/0.59 % neg_equal_iff_equal
% 0.30/0.59 thf(fact_5375_neg__equal__iff__equal,axiom,
% 0.30/0.59 ! [A: complex,B: complex] :
% 0.30/0.59 ( ( ( uminus1482373934393186551omplex @ A )
% 0.30/0.59 = ( uminus1482373934393186551omplex @ B ) )
% 0.30/0.59 = ( A = B ) ) ).
% 0.30/0.59
% 0.30/0.59 % neg_equal_iff_equal
% 0.30/0.59 thf(fact_5376_neg__equal__iff__equal,axiom,
% 0.30/0.59 ! [A: code_integer,B: code_integer] :
% 0.30/0.59 ( ( ( uminus1351360451143612070nteger @ A )
% 0.30/0.59 = ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.59 = ( A = B ) ) ).
% 0.30/0.59
% 0.30/0.59 % neg_equal_iff_equal
% 0.30/0.59 thf(fact_5377_neg__equal__iff__equal,axiom,
% 0.30/0.59 ! [A: rat,B: rat] :
% 0.30/0.59 ( ( ( uminus_uminus_rat @ A )
% 0.30/0.59 = ( uminus_uminus_rat @ B ) )
% 0.30/0.59 = ( A = B ) ) ).
% 0.30/0.59
% 0.30/0.59 % neg_equal_iff_equal
% 0.30/0.59 thf(fact_5378_Compl__anti__mono,axiom,
% 0.30/0.59 ! [A2: set_int,B3: set_int] :
% 0.30/0.59 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.30/0.59 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B3 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % Compl_anti_mono
% 0.30/0.59 thf(fact_5379_Compl__subset__Compl__iff,axiom,
% 0.30/0.59 ! [A2: set_int,B3: set_int] :
% 0.30/0.59 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B3 ) )
% 0.30/0.59 = ( ord_less_eq_set_int @ B3 @ A2 ) ) ).
% 0.30/0.59
% 0.30/0.59 % Compl_subset_Compl_iff
% 0.30/0.59 thf(fact_5380_case__prodI2,axiom,
% 0.30/0.59 ! [P6: produc6271795597528267376eger_o,C: code_integer > $o > $o] :
% 0.30/0.59 ( ! [A4: code_integer,B4: $o] :
% 0.30/0.59 ( ( P6
% 0.30/0.59 = ( produc6677183202524767010eger_o @ A4 @ B4 ) )
% 0.30/0.59 => ( C @ A4 @ B4 ) )
% 0.30/0.59 => ( produc7828578312038201481er_o_o @ C @ P6 ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prodI2
% 0.30/0.59 thf(fact_5381_case__prodI2,axiom,
% 0.30/0.59 ! [P6: product_prod_num_num,C: num > num > $o] :
% 0.30/0.59 ( ! [A4: num,B4: num] :
% 0.30/0.59 ( ( P6
% 0.30/0.59 = ( product_Pair_num_num @ A4 @ B4 ) )
% 0.30/0.59 => ( C @ A4 @ B4 ) )
% 0.30/0.59 => ( produc5703948589228662326_num_o @ C @ P6 ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prodI2
% 0.30/0.59 thf(fact_5382_case__prodI2,axiom,
% 0.30/0.59 ! [P6: product_prod_nat_num,C: nat > num > $o] :
% 0.30/0.59 ( ! [A4: nat,B4: num] :
% 0.30/0.59 ( ( P6
% 0.30/0.59 = ( product_Pair_nat_num @ A4 @ B4 ) )
% 0.30/0.59 => ( C @ A4 @ B4 ) )
% 0.30/0.59 => ( produc4927758841916487424_num_o @ C @ P6 ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prodI2
% 0.30/0.59 thf(fact_5383_case__prodI2,axiom,
% 0.30/0.59 ! [P6: product_prod_nat_nat,C: nat > nat > $o] :
% 0.30/0.59 ( ! [A4: nat,B4: nat] :
% 0.30/0.59 ( ( P6
% 0.30/0.59 = ( product_Pair_nat_nat @ A4 @ B4 ) )
% 0.30/0.59 => ( C @ A4 @ B4 ) )
% 0.30/0.59 => ( produc6081775807080527818_nat_o @ C @ P6 ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prodI2
% 0.30/0.59 thf(fact_5384_case__prodI2,axiom,
% 0.30/0.59 ! [P6: product_prod_int_int,C: int > int > $o] :
% 0.30/0.59 ( ! [A4: int,B4: int] :
% 0.30/0.59 ( ( P6
% 0.30/0.59 = ( product_Pair_int_int @ A4 @ B4 ) )
% 0.30/0.59 => ( C @ A4 @ B4 ) )
% 0.30/0.59 => ( produc4947309494688390418_int_o @ C @ P6 ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prodI2
% 0.30/0.59 thf(fact_5385_case__prodI,axiom,
% 0.30/0.59 ! [F: code_integer > $o > $o,A: code_integer,B: $o] :
% 0.30/0.59 ( ( F @ A @ B )
% 0.30/0.59 => ( produc7828578312038201481er_o_o @ F @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prodI
% 0.30/0.59 thf(fact_5386_case__prodI,axiom,
% 0.30/0.59 ! [F: num > num > $o,A: num,B: num] :
% 0.30/0.59 ( ( F @ A @ B )
% 0.30/0.59 => ( produc5703948589228662326_num_o @ F @ ( product_Pair_num_num @ A @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prodI
% 0.30/0.59 thf(fact_5387_case__prodI,axiom,
% 0.30/0.59 ! [F: nat > num > $o,A: nat,B: num] :
% 0.30/0.59 ( ( F @ A @ B )
% 0.30/0.59 => ( produc4927758841916487424_num_o @ F @ ( product_Pair_nat_num @ A @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prodI
% 0.30/0.59 thf(fact_5388_case__prodI,axiom,
% 0.30/0.59 ! [F: nat > nat > $o,A: nat,B: nat] :
% 0.30/0.59 ( ( F @ A @ B )
% 0.30/0.59 => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prodI
% 0.30/0.59 thf(fact_5389_case__prodI,axiom,
% 0.30/0.59 ! [F: int > int > $o,A: int,B: int] :
% 0.30/0.59 ( ( F @ A @ B )
% 0.30/0.59 => ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prodI
% 0.30/0.59 thf(fact_5390_mem__case__prodI2,axiom,
% 0.30/0.59 ! [P6: produc6271795597528267376eger_o,Z: complex,C: code_integer > $o > set_complex] :
% 0.30/0.59 ( ! [A4: code_integer,B4: $o] :
% 0.30/0.59 ( ( P6
% 0.30/0.59 = ( produc6677183202524767010eger_o @ A4 @ B4 ) )
% 0.30/0.59 => ( member_complex @ Z @ ( C @ A4 @ B4 ) ) )
% 0.30/0.59 => ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ P6 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI2
% 0.30/0.59 thf(fact_5391_mem__case__prodI2,axiom,
% 0.30/0.59 ! [P6: produc6271795597528267376eger_o,Z: real,C: code_integer > $o > set_real] :
% 0.30/0.59 ( ! [A4: code_integer,B4: $o] :
% 0.30/0.59 ( ( P6
% 0.30/0.59 = ( produc6677183202524767010eger_o @ A4 @ B4 ) )
% 0.30/0.59 => ( member_real @ Z @ ( C @ A4 @ B4 ) ) )
% 0.30/0.59 => ( member_real @ Z @ ( produc242741666403216561t_real @ C @ P6 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI2
% 0.30/0.59 thf(fact_5392_mem__case__prodI2,axiom,
% 0.30/0.59 ! [P6: produc6271795597528267376eger_o,Z: nat,C: code_integer > $o > set_nat] :
% 0.30/0.59 ( ! [A4: code_integer,B4: $o] :
% 0.30/0.59 ( ( P6
% 0.30/0.59 = ( produc6677183202524767010eger_o @ A4 @ B4 ) )
% 0.30/0.59 => ( member_nat @ Z @ ( C @ A4 @ B4 ) ) )
% 0.30/0.59 => ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ P6 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI2
% 0.30/0.59 thf(fact_5393_mem__case__prodI2,axiom,
% 0.30/0.59 ! [P6: produc6271795597528267376eger_o,Z: int,C: code_integer > $o > set_int] :
% 0.30/0.59 ( ! [A4: code_integer,B4: $o] :
% 0.30/0.59 ( ( P6
% 0.30/0.59 = ( produc6677183202524767010eger_o @ A4 @ B4 ) )
% 0.30/0.59 => ( member_int @ Z @ ( C @ A4 @ B4 ) ) )
% 0.30/0.59 => ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ P6 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI2
% 0.30/0.59 thf(fact_5394_mem__case__prodI2,axiom,
% 0.30/0.59 ! [P6: product_prod_num_num,Z: complex,C: num > num > set_complex] :
% 0.30/0.59 ( ! [A4: num,B4: num] :
% 0.30/0.59 ( ( P6
% 0.30/0.59 = ( product_Pair_num_num @ A4 @ B4 ) )
% 0.30/0.59 => ( member_complex @ Z @ ( C @ A4 @ B4 ) ) )
% 0.30/0.59 => ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ P6 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI2
% 0.30/0.59 thf(fact_5395_mem__case__prodI2,axiom,
% 0.30/0.59 ! [P6: product_prod_num_num,Z: real,C: num > num > set_real] :
% 0.30/0.59 ( ! [A4: num,B4: num] :
% 0.30/0.59 ( ( P6
% 0.30/0.59 = ( product_Pair_num_num @ A4 @ B4 ) )
% 0.30/0.59 => ( member_real @ Z @ ( C @ A4 @ B4 ) ) )
% 0.30/0.59 => ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ P6 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI2
% 0.30/0.59 thf(fact_5396_mem__case__prodI2,axiom,
% 0.30/0.59 ! [P6: product_prod_num_num,Z: nat,C: num > num > set_nat] :
% 0.30/0.59 ( ! [A4: num,B4: num] :
% 0.30/0.59 ( ( P6
% 0.30/0.59 = ( product_Pair_num_num @ A4 @ B4 ) )
% 0.30/0.59 => ( member_nat @ Z @ ( C @ A4 @ B4 ) ) )
% 0.30/0.59 => ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ P6 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI2
% 0.30/0.59 thf(fact_5397_mem__case__prodI2,axiom,
% 0.30/0.59 ! [P6: product_prod_num_num,Z: int,C: num > num > set_int] :
% 0.30/0.59 ( ! [A4: num,B4: num] :
% 0.30/0.59 ( ( P6
% 0.30/0.59 = ( product_Pair_num_num @ A4 @ B4 ) )
% 0.30/0.59 => ( member_int @ Z @ ( C @ A4 @ B4 ) ) )
% 0.30/0.59 => ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ P6 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI2
% 0.30/0.59 thf(fact_5398_mem__case__prodI2,axiom,
% 0.30/0.59 ! [P6: product_prod_nat_num,Z: complex,C: nat > num > set_complex] :
% 0.30/0.59 ( ! [A4: nat,B4: num] :
% 0.30/0.59 ( ( P6
% 0.30/0.59 = ( product_Pair_nat_num @ A4 @ B4 ) )
% 0.30/0.59 => ( member_complex @ Z @ ( C @ A4 @ B4 ) ) )
% 0.30/0.59 => ( member_complex @ Z @ ( produc6231982587499038204omplex @ C @ P6 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI2
% 0.30/0.59 thf(fact_5399_mem__case__prodI2,axiom,
% 0.30/0.59 ! [P6: product_prod_nat_num,Z: real,C: nat > num > set_real] :
% 0.30/0.59 ( ! [A4: nat,B4: num] :
% 0.30/0.59 ( ( P6
% 0.30/0.59 = ( product_Pair_nat_num @ A4 @ B4 ) )
% 0.30/0.59 => ( member_real @ Z @ ( C @ A4 @ B4 ) ) )
% 0.30/0.59 => ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ P6 ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI2
% 0.30/0.59 thf(fact_5400_mem__case__prodI,axiom,
% 0.30/0.59 ! [Z: complex,C: code_integer > $o > set_complex,A: code_integer,B: $o] :
% 0.30/0.59 ( ( member_complex @ Z @ ( C @ A @ B ) )
% 0.30/0.59 => ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI
% 0.30/0.59 thf(fact_5401_mem__case__prodI,axiom,
% 0.30/0.59 ! [Z: real,C: code_integer > $o > set_real,A: code_integer,B: $o] :
% 0.30/0.59 ( ( member_real @ Z @ ( C @ A @ B ) )
% 0.30/0.59 => ( member_real @ Z @ ( produc242741666403216561t_real @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI
% 0.30/0.59 thf(fact_5402_mem__case__prodI,axiom,
% 0.30/0.59 ! [Z: nat,C: code_integer > $o > set_nat,A: code_integer,B: $o] :
% 0.30/0.59 ( ( member_nat @ Z @ ( C @ A @ B ) )
% 0.30/0.59 => ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI
% 0.30/0.59 thf(fact_5403_mem__case__prodI,axiom,
% 0.30/0.59 ! [Z: int,C: code_integer > $o > set_int,A: code_integer,B: $o] :
% 0.30/0.59 ( ( member_int @ Z @ ( C @ A @ B ) )
% 0.30/0.59 => ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI
% 0.30/0.59 thf(fact_5404_mem__case__prodI,axiom,
% 0.30/0.59 ! [Z: complex,C: num > num > set_complex,A: num,B: num] :
% 0.30/0.59 ( ( member_complex @ Z @ ( C @ A @ B ) )
% 0.30/0.59 => ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI
% 0.30/0.59 thf(fact_5405_mem__case__prodI,axiom,
% 0.30/0.59 ! [Z: real,C: num > num > set_real,A: num,B: num] :
% 0.30/0.59 ( ( member_real @ Z @ ( C @ A @ B ) )
% 0.30/0.59 => ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI
% 0.30/0.59 thf(fact_5406_mem__case__prodI,axiom,
% 0.30/0.59 ! [Z: nat,C: num > num > set_nat,A: num,B: num] :
% 0.30/0.59 ( ( member_nat @ Z @ ( C @ A @ B ) )
% 0.30/0.59 => ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI
% 0.30/0.59 thf(fact_5407_mem__case__prodI,axiom,
% 0.30/0.59 ! [Z: int,C: num > num > set_int,A: num,B: num] :
% 0.30/0.59 ( ( member_int @ Z @ ( C @ A @ B ) )
% 0.30/0.59 => ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI
% 0.30/0.59 thf(fact_5408_mem__case__prodI,axiom,
% 0.30/0.59 ! [Z: complex,C: nat > num > set_complex,A: nat,B: num] :
% 0.30/0.59 ( ( member_complex @ Z @ ( C @ A @ B ) )
% 0.30/0.59 => ( member_complex @ Z @ ( produc6231982587499038204omplex @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI
% 0.30/0.59 thf(fact_5409_mem__case__prodI,axiom,
% 0.30/0.59 ! [Z: real,C: nat > num > set_real,A: nat,B: num] :
% 0.30/0.59 ( ( member_real @ Z @ ( C @ A @ B ) )
% 0.30/0.59 => ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).
% 0.30/0.59
% 0.30/0.59 % mem_case_prodI
% 0.30/0.59 thf(fact_5410_case__prodI2_H,axiom,
% 0.30/0.59 ! [P6: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X2: product_prod_nat_nat] :
% 0.30/0.59 ( ! [A4: nat,B4: nat] :
% 0.30/0.59 ( ( ( product_Pair_nat_nat @ A4 @ B4 )
% 0.30/0.59 = P6 )
% 0.30/0.59 => ( C @ A4 @ B4 @ X2 ) )
% 0.30/0.59 => ( produc8739625826339149834_nat_o @ C @ P6 @ X2 ) ) ).
% 0.30/0.59
% 0.30/0.59 % case_prodI2'
% 0.30/0.59 thf(fact_5411_compl__le__compl__iff,axiom,
% 0.30/0.59 ! [X2: set_int,Y3: set_int] :
% 0.30/0.59 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ ( uminus1532241313380277803et_int @ Y3 ) )
% 0.30/0.59 = ( ord_less_eq_set_int @ Y3 @ X2 ) ) ).
% 0.30/0.59
% 0.30/0.59 % compl_le_compl_iff
% 0.30/0.59 thf(fact_5412_neg__le__iff__le,axiom,
% 0.30/0.59 ! [B: real,A: real] :
% 0.30/0.59 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.30/0.59 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/0.59
% 0.30/0.59 % neg_le_iff_le
% 0.30/0.59 thf(fact_5413_neg__le__iff__le,axiom,
% 0.30/0.59 ! [B: code_integer,A: code_integer] :
% 0.30/0.59 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.59 = ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 0.30/0.59
% 0.30/0.59 % neg_le_iff_le
% 0.30/0.59 thf(fact_5414_neg__le__iff__le,axiom,
% 0.30/0.59 ! [B: rat,A: rat] :
% 0.30/0.59 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 0.30/0.59 = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.30/0.59
% 0.30/0.59 % neg_le_iff_le
% 0.30/0.59 thf(fact_5415_neg__le__iff__le,axiom,
% 0.30/0.59 ! [B: int,A: int] :
% 0.30/0.59 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 0.30/0.60 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_le_iff_le
% 0.30/0.60 thf(fact_5416_add_Oinverse__neutral,axiom,
% 0.30/0.60 ( ( uminus_uminus_int @ zero_zero_int )
% 0.30/0.60 = zero_zero_int ) ).
% 0.30/0.60
% 0.30/0.60 % add.inverse_neutral
% 0.30/0.60 thf(fact_5417_add_Oinverse__neutral,axiom,
% 0.30/0.60 ( ( uminus_uminus_real @ zero_zero_real )
% 0.30/0.60 = zero_zero_real ) ).
% 0.30/0.60
% 0.30/0.60 % add.inverse_neutral
% 0.30/0.60 thf(fact_5418_add_Oinverse__neutral,axiom,
% 0.30/0.60 ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 0.30/0.60 = zero_zero_complex ) ).
% 0.30/0.60
% 0.30/0.60 % add.inverse_neutral
% 0.30/0.60 thf(fact_5419_add_Oinverse__neutral,axiom,
% 0.30/0.60 ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 0.30/0.60 = zero_z3403309356797280102nteger ) ).
% 0.30/0.60
% 0.30/0.60 % add.inverse_neutral
% 0.30/0.60 thf(fact_5420_add_Oinverse__neutral,axiom,
% 0.30/0.60 ( ( uminus_uminus_rat @ zero_zero_rat )
% 0.30/0.60 = zero_zero_rat ) ).
% 0.30/0.60
% 0.30/0.60 % add.inverse_neutral
% 0.30/0.60 thf(fact_5421_neg__0__equal__iff__equal,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( zero_zero_int
% 0.30/0.60 = ( uminus_uminus_int @ A ) )
% 0.30/0.60 = ( zero_zero_int = A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_0_equal_iff_equal
% 0.30/0.60 thf(fact_5422_neg__0__equal__iff__equal,axiom,
% 0.30/0.60 ! [A: real] :
% 0.30/0.60 ( ( zero_zero_real
% 0.30/0.60 = ( uminus_uminus_real @ A ) )
% 0.30/0.60 = ( zero_zero_real = A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_0_equal_iff_equal
% 0.30/0.60 thf(fact_5423_neg__0__equal__iff__equal,axiom,
% 0.30/0.60 ! [A: complex] :
% 0.30/0.60 ( ( zero_zero_complex
% 0.30/0.60 = ( uminus1482373934393186551omplex @ A ) )
% 0.30/0.60 = ( zero_zero_complex = A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_0_equal_iff_equal
% 0.30/0.60 thf(fact_5424_neg__0__equal__iff__equal,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( zero_z3403309356797280102nteger
% 0.30/0.60 = ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.60 = ( zero_z3403309356797280102nteger = A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_0_equal_iff_equal
% 0.30/0.60 thf(fact_5425_neg__0__equal__iff__equal,axiom,
% 0.30/0.60 ! [A: rat] :
% 0.30/0.60 ( ( zero_zero_rat
% 0.30/0.60 = ( uminus_uminus_rat @ A ) )
% 0.30/0.60 = ( zero_zero_rat = A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_0_equal_iff_equal
% 0.30/0.60 thf(fact_5426_neg__equal__0__iff__equal,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( ( uminus_uminus_int @ A )
% 0.30/0.60 = zero_zero_int )
% 0.30/0.60 = ( A = zero_zero_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_equal_0_iff_equal
% 0.30/0.60 thf(fact_5427_neg__equal__0__iff__equal,axiom,
% 0.30/0.60 ! [A: real] :
% 0.30/0.60 ( ( ( uminus_uminus_real @ A )
% 0.30/0.60 = zero_zero_real )
% 0.30/0.60 = ( A = zero_zero_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_equal_0_iff_equal
% 0.30/0.60 thf(fact_5428_neg__equal__0__iff__equal,axiom,
% 0.30/0.60 ! [A: complex] :
% 0.30/0.60 ( ( ( uminus1482373934393186551omplex @ A )
% 0.30/0.60 = zero_zero_complex )
% 0.30/0.60 = ( A = zero_zero_complex ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_equal_0_iff_equal
% 0.30/0.60 thf(fact_5429_neg__equal__0__iff__equal,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( ( uminus1351360451143612070nteger @ A )
% 0.30/0.60 = zero_z3403309356797280102nteger )
% 0.30/0.60 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_equal_0_iff_equal
% 0.30/0.60 thf(fact_5430_neg__equal__0__iff__equal,axiom,
% 0.30/0.60 ! [A: rat] :
% 0.30/0.60 ( ( ( uminus_uminus_rat @ A )
% 0.30/0.60 = zero_zero_rat )
% 0.30/0.60 = ( A = zero_zero_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_equal_0_iff_equal
% 0.30/0.60 thf(fact_5431_equal__neg__zero,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( uminus_uminus_int @ A ) )
% 0.30/0.60 = ( A = zero_zero_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % equal_neg_zero
% 0.30/0.60 thf(fact_5432_equal__neg__zero,axiom,
% 0.30/0.60 ! [A: real] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( uminus_uminus_real @ A ) )
% 0.30/0.60 = ( A = zero_zero_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % equal_neg_zero
% 0.30/0.60 thf(fact_5433_equal__neg__zero,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.60 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60
% 0.30/0.60 % equal_neg_zero
% 0.30/0.60 thf(fact_5434_equal__neg__zero,axiom,
% 0.30/0.60 ! [A: rat] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( uminus_uminus_rat @ A ) )
% 0.30/0.60 = ( A = zero_zero_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % equal_neg_zero
% 0.30/0.60 thf(fact_5435_neg__equal__zero,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( ( uminus_uminus_int @ A )
% 0.30/0.60 = A )
% 0.30/0.60 = ( A = zero_zero_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_equal_zero
% 0.30/0.60 thf(fact_5436_neg__equal__zero,axiom,
% 0.30/0.60 ! [A: real] :
% 0.30/0.60 ( ( ( uminus_uminus_real @ A )
% 0.30/0.60 = A )
% 0.30/0.60 = ( A = zero_zero_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_equal_zero
% 0.30/0.60 thf(fact_5437_neg__equal__zero,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( ( uminus1351360451143612070nteger @ A )
% 0.30/0.60 = A )
% 0.30/0.60 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_equal_zero
% 0.30/0.60 thf(fact_5438_neg__equal__zero,axiom,
% 0.30/0.60 ! [A: rat] :
% 0.30/0.60 ( ( ( uminus_uminus_rat @ A )
% 0.30/0.60 = A )
% 0.30/0.60 = ( A = zero_zero_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_equal_zero
% 0.30/0.60 thf(fact_5439_neg__less__iff__less,axiom,
% 0.30/0.60 ! [B: int,A: int] :
% 0.30/0.60 ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 0.30/0.60 = ( ord_less_int @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_less_iff_less
% 0.30/0.60 thf(fact_5440_neg__less__iff__less,axiom,
% 0.30/0.60 ! [B: real,A: real] :
% 0.30/0.60 ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.30/0.60 = ( ord_less_real @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_less_iff_less
% 0.30/0.60 thf(fact_5441_neg__less__iff__less,axiom,
% 0.30/0.60 ! [B: code_integer,A: code_integer] :
% 0.30/0.60 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.60 = ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_less_iff_less
% 0.30/0.60 thf(fact_5442_neg__less__iff__less,axiom,
% 0.30/0.60 ! [B: rat,A: rat] :
% 0.30/0.60 ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 0.30/0.60 = ( ord_less_rat @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_less_iff_less
% 0.30/0.60 thf(fact_5443_neg__numeral__eq__iff,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 0.30/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.60 = ( M = N ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_eq_iff
% 0.30/0.60 thf(fact_5444_neg__numeral__eq__iff,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 0.30/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.60 = ( M = N ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_eq_iff
% 0.30/0.60 thf(fact_5445_neg__numeral__eq__iff,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.30/0.60 = ( M = N ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_eq_iff
% 0.30/0.60 thf(fact_5446_neg__numeral__eq__iff,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.60 = ( M = N ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_eq_iff
% 0.30/0.60 thf(fact_5447_neg__numeral__eq__iff,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 0.30/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.60 = ( M = N ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_eq_iff
% 0.30/0.60 thf(fact_5448_mult__minus__right,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
% 0.30/0.60 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus_right
% 0.30/0.60 thf(fact_5449_mult__minus__right,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
% 0.30/0.60 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus_right
% 0.30/0.60 thf(fact_5450_mult__minus__right,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus_right
% 0.30/0.60 thf(fact_5451_mult__minus__right,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus_right
% 0.30/0.60 thf(fact_5452_mult__minus__right,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.30/0.60 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus_right
% 0.30/0.60 thf(fact_5453_minus__mult__minus,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.30/0.60 = ( times_times_int @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_mult_minus
% 0.30/0.60 thf(fact_5454_minus__mult__minus,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.30/0.60 = ( times_times_real @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_mult_minus
% 0.30/0.60 thf(fact_5455_minus__mult__minus,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.30/0.60 = ( times_times_complex @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_mult_minus
% 0.30/0.60 thf(fact_5456_minus__mult__minus,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.60 = ( times_3573771949741848930nteger @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_mult_minus
% 0.30/0.60 thf(fact_5457_minus__mult__minus,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.30/0.60 = ( times_times_rat @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_mult_minus
% 0.30/0.60 thf(fact_5458_mult__minus__left,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.60 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus_left
% 0.30/0.60 thf(fact_5459_mult__minus__left,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 0.30/0.60 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus_left
% 0.30/0.60 thf(fact_5460_mult__minus__left,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus_left
% 0.30/0.60 thf(fact_5461_mult__minus__left,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus_left
% 0.30/0.60 thf(fact_5462_mult__minus__left,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.30/0.60 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus_left
% 0.30/0.60 thf(fact_5463_add__minus__cancel,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 0.30/0.60 = B ) ).
% 0.30/0.60
% 0.30/0.60 % add_minus_cancel
% 0.30/0.60 thf(fact_5464_add__minus__cancel,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
% 0.30/0.60 = B ) ).
% 0.30/0.60
% 0.30/0.60 % add_minus_cancel
% 0.30/0.60 thf(fact_5465_add__minus__cancel,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
% 0.30/0.60 = B ) ).
% 0.30/0.60
% 0.30/0.60 % add_minus_cancel
% 0.30/0.60 thf(fact_5466_add__minus__cancel,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
% 0.30/0.60 = B ) ).
% 0.30/0.60
% 0.30/0.60 % add_minus_cancel
% 0.30/0.60 thf(fact_5467_add__minus__cancel,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
% 0.30/0.60 = B ) ).
% 0.30/0.60
% 0.30/0.60 % add_minus_cancel
% 0.30/0.60 thf(fact_5468_minus__add__cancel,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 0.30/0.60 = B ) ).
% 0.30/0.60
% 0.30/0.60 % minus_add_cancel
% 0.30/0.60 thf(fact_5469_minus__add__cancel,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
% 0.30/0.60 = B ) ).
% 0.30/0.60
% 0.30/0.60 % minus_add_cancel
% 0.30/0.60 thf(fact_5470_minus__add__cancel,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
% 0.30/0.60 = B ) ).
% 0.30/0.60
% 0.30/0.60 % minus_add_cancel
% 0.30/0.60 thf(fact_5471_minus__add__cancel,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.30/0.60 = B ) ).
% 0.30/0.60
% 0.30/0.60 % minus_add_cancel
% 0.30/0.60 thf(fact_5472_minus__add__cancel,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
% 0.30/0.60 = B ) ).
% 0.30/0.60
% 0.30/0.60 % minus_add_cancel
% 0.30/0.60 thf(fact_5473_minus__add__distrib,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.30/0.60 = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_add_distrib
% 0.30/0.60 thf(fact_5474_minus__add__distrib,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.30/0.60 = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_add_distrib
% 0.30/0.60 thf(fact_5475_minus__add__distrib,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 0.30/0.60 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_add_distrib
% 0.30/0.60 thf(fact_5476_minus__add__distrib,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.30/0.60 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_add_distrib
% 0.30/0.60 thf(fact_5477_minus__add__distrib,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.30/0.60 = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_add_distrib
% 0.30/0.60 thf(fact_5478_minus__diff__eq,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 0.30/0.60 = ( minus_minus_int @ B @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_diff_eq
% 0.30/0.60 thf(fact_5479_minus__diff__eq,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
% 0.30/0.60 = ( minus_minus_real @ B @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_diff_eq
% 0.30/0.60 thf(fact_5480_minus__diff__eq,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
% 0.30/0.60 = ( minus_minus_complex @ B @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_diff_eq
% 0.30/0.60 thf(fact_5481_minus__diff__eq,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.30/0.60 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_diff_eq
% 0.30/0.60 thf(fact_5482_minus__diff__eq,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
% 0.30/0.60 = ( minus_minus_rat @ B @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_diff_eq
% 0.30/0.60 thf(fact_5483_div__minus__minus,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.30/0.60 = ( divide_divide_int @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % div_minus_minus
% 0.30/0.60 thf(fact_5484_div__minus__minus,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.60 = ( divide6298287555418463151nteger @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % div_minus_minus
% 0.30/0.60 thf(fact_5485_dvd__minus__iff,axiom,
% 0.30/0.60 ! [X2: int,Y3: int] :
% 0.30/0.60 ( ( dvd_dvd_int @ X2 @ ( uminus_uminus_int @ Y3 ) )
% 0.30/0.60 = ( dvd_dvd_int @ X2 @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % dvd_minus_iff
% 0.30/0.60 thf(fact_5486_dvd__minus__iff,axiom,
% 0.30/0.60 ! [X2: real,Y3: real] :
% 0.30/0.60 ( ( dvd_dvd_real @ X2 @ ( uminus_uminus_real @ Y3 ) )
% 0.30/0.60 = ( dvd_dvd_real @ X2 @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % dvd_minus_iff
% 0.30/0.60 thf(fact_5487_dvd__minus__iff,axiom,
% 0.30/0.60 ! [X2: complex,Y3: complex] :
% 0.30/0.60 ( ( dvd_dvd_complex @ X2 @ ( uminus1482373934393186551omplex @ Y3 ) )
% 0.30/0.60 = ( dvd_dvd_complex @ X2 @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % dvd_minus_iff
% 0.30/0.60 thf(fact_5488_dvd__minus__iff,axiom,
% 0.30/0.60 ! [X2: code_integer,Y3: code_integer] :
% 0.30/0.60 ( ( dvd_dvd_Code_integer @ X2 @ ( uminus1351360451143612070nteger @ Y3 ) )
% 0.30/0.60 = ( dvd_dvd_Code_integer @ X2 @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % dvd_minus_iff
% 0.30/0.60 thf(fact_5489_dvd__minus__iff,axiom,
% 0.30/0.60 ! [X2: rat,Y3: rat] :
% 0.30/0.60 ( ( dvd_dvd_rat @ X2 @ ( uminus_uminus_rat @ Y3 ) )
% 0.30/0.60 = ( dvd_dvd_rat @ X2 @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % dvd_minus_iff
% 0.30/0.60 thf(fact_5490_minus__dvd__iff,axiom,
% 0.30/0.60 ! [X2: int,Y3: int] :
% 0.30/0.60 ( ( dvd_dvd_int @ ( uminus_uminus_int @ X2 ) @ Y3 )
% 0.30/0.60 = ( dvd_dvd_int @ X2 @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_dvd_iff
% 0.30/0.60 thf(fact_5491_minus__dvd__iff,axiom,
% 0.30/0.60 ! [X2: real,Y3: real] :
% 0.30/0.60 ( ( dvd_dvd_real @ ( uminus_uminus_real @ X2 ) @ Y3 )
% 0.30/0.60 = ( dvd_dvd_real @ X2 @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_dvd_iff
% 0.30/0.60 thf(fact_5492_minus__dvd__iff,axiom,
% 0.30/0.60 ! [X2: complex,Y3: complex] :
% 0.30/0.60 ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X2 ) @ Y3 )
% 0.30/0.60 = ( dvd_dvd_complex @ X2 @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_dvd_iff
% 0.30/0.60 thf(fact_5493_minus__dvd__iff,axiom,
% 0.30/0.60 ! [X2: code_integer,Y3: code_integer] :
% 0.30/0.60 ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X2 ) @ Y3 )
% 0.30/0.60 = ( dvd_dvd_Code_integer @ X2 @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_dvd_iff
% 0.30/0.60 thf(fact_5494_minus__dvd__iff,axiom,
% 0.30/0.60 ! [X2: rat,Y3: rat] :
% 0.30/0.60 ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X2 ) @ Y3 )
% 0.30/0.60 = ( dvd_dvd_rat @ X2 @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_dvd_iff
% 0.30/0.60 thf(fact_5495_mod__minus__minus,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.30/0.60 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mod_minus_minus
% 0.30/0.60 thf(fact_5496_mod__minus__minus,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mod_minus_minus
% 0.30/0.60 thf(fact_5497_real__add__minus__iff,axiom,
% 0.30/0.60 ! [X2: real,A: real] :
% 0.30/0.60 ( ( ( plus_plus_real @ X2 @ ( uminus_uminus_real @ A ) )
% 0.30/0.60 = zero_zero_real )
% 0.30/0.60 = ( X2 = A ) ) ).
% 0.30/0.60
% 0.30/0.60 % real_add_minus_iff
% 0.30/0.60 thf(fact_5498_neg__less__eq__nonneg,axiom,
% 0.30/0.60 ! [A: real] :
% 0.30/0.60 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 0.30/0.60 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_less_eq_nonneg
% 0.30/0.60 thf(fact_5499_neg__less__eq__nonneg,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.30/0.60 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_less_eq_nonneg
% 0.30/0.60 thf(fact_5500_neg__less__eq__nonneg,axiom,
% 0.30/0.60 ! [A: rat] :
% 0.30/0.60 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.30/0.60 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_less_eq_nonneg
% 0.30/0.60 thf(fact_5501_neg__less__eq__nonneg,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 0.30/0.60 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_less_eq_nonneg
% 0.30/0.60 thf(fact_5502_less__eq__neg__nonpos,axiom,
% 0.30/0.60 ! [A: real] :
% 0.30/0.60 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 0.30/0.60 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_eq_neg_nonpos
% 0.30/0.60 thf(fact_5503_less__eq__neg__nonpos,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.60 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_eq_neg_nonpos
% 0.30/0.60 thf(fact_5504_less__eq__neg__nonpos,axiom,
% 0.30/0.60 ! [A: rat] :
% 0.30/0.60 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.30/0.60 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_eq_neg_nonpos
% 0.30/0.60 thf(fact_5505_less__eq__neg__nonpos,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 0.30/0.60 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_eq_neg_nonpos
% 0.30/0.60 thf(fact_5506_neg__le__0__iff__le,axiom,
% 0.30/0.60 ! [A: real] :
% 0.30/0.60 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.30/0.60 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_le_0_iff_le
% 0.30/0.60 thf(fact_5507_neg__le__0__iff__le,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.30/0.60 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_le_0_iff_le
% 0.30/0.60 thf(fact_5508_neg__le__0__iff__le,axiom,
% 0.30/0.60 ! [A: rat] :
% 0.30/0.60 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.30/0.60 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_le_0_iff_le
% 0.30/0.60 thf(fact_5509_neg__le__0__iff__le,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.30/0.60 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_le_0_iff_le
% 0.30/0.60 thf(fact_5510_neg__0__le__iff__le,axiom,
% 0.30/0.60 ! [A: real] :
% 0.30/0.60 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.30/0.60 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_0_le_iff_le
% 0.30/0.60 thf(fact_5511_neg__0__le__iff__le,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.60 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_0_le_iff_le
% 0.30/0.60 thf(fact_5512_neg__0__le__iff__le,axiom,
% 0.30/0.60 ! [A: rat] :
% 0.30/0.60 ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.30/0.60 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_0_le_iff_le
% 0.30/0.60 thf(fact_5513_neg__0__le__iff__le,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.30/0.60 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_0_le_iff_le
% 0.30/0.60 thf(fact_5514_neg__less__0__iff__less,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.30/0.60 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_less_0_iff_less
% 0.30/0.60 thf(fact_5515_neg__less__0__iff__less,axiom,
% 0.30/0.60 ! [A: real] :
% 0.30/0.60 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.30/0.60 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_less_0_iff_less
% 0.30/0.60 thf(fact_5516_neg__less__0__iff__less,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.30/0.60 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_less_0_iff_less
% 0.30/0.60 thf(fact_5517_neg__less__0__iff__less,axiom,
% 0.30/0.60 ! [A: rat] :
% 0.30/0.60 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.30/0.60 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_less_0_iff_less
% 0.30/0.60 thf(fact_5518_neg__0__less__iff__less,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.30/0.60 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_0_less_iff_less
% 0.30/0.60 thf(fact_5519_neg__0__less__iff__less,axiom,
% 0.30/0.60 ! [A: real] :
% 0.30/0.60 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.30/0.60 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_0_less_iff_less
% 0.30/0.60 thf(fact_5520_neg__0__less__iff__less,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.60 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_0_less_iff_less
% 0.30/0.60 thf(fact_5521_neg__0__less__iff__less,axiom,
% 0.30/0.60 ! [A: rat] :
% 0.30/0.60 ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.30/0.60 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_0_less_iff_less
% 0.30/0.60 thf(fact_5522_neg__less__pos,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 0.30/0.60 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_less_pos
% 0.30/0.60 thf(fact_5523_neg__less__pos,axiom,
% 0.30/0.60 ! [A: real] :
% 0.30/0.60 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 0.30/0.60 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_less_pos
% 0.30/0.60 thf(fact_5524_neg__less__pos,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.30/0.60 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_less_pos
% 0.30/0.60 thf(fact_5525_neg__less__pos,axiom,
% 0.30/0.60 ! [A: rat] :
% 0.30/0.60 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.30/0.60 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_less_pos
% 0.30/0.60 thf(fact_5526_less__neg__neg,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 0.30/0.60 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_neg_neg
% 0.30/0.60 thf(fact_5527_less__neg__neg,axiom,
% 0.30/0.60 ! [A: real] :
% 0.30/0.60 ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 0.30/0.60 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_neg_neg
% 0.30/0.60 thf(fact_5528_less__neg__neg,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.60 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_neg_neg
% 0.30/0.60 thf(fact_5529_less__neg__neg,axiom,
% 0.30/0.60 ! [A: rat] :
% 0.30/0.60 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.30/0.60 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_neg_neg
% 0.30/0.60 thf(fact_5530_ab__left__minus,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.30/0.60 = zero_zero_int ) ).
% 0.30/0.60
% 0.30/0.60 % ab_left_minus
% 0.30/0.60 thf(fact_5531_ab__left__minus,axiom,
% 0.30/0.60 ! [A: real] :
% 0.30/0.60 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.30/0.60 = zero_zero_real ) ).
% 0.30/0.60
% 0.30/0.60 % ab_left_minus
% 0.30/0.60 thf(fact_5532_ab__left__minus,axiom,
% 0.30/0.60 ! [A: complex] :
% 0.30/0.60 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.30/0.60 = zero_zero_complex ) ).
% 0.30/0.60
% 0.30/0.60 % ab_left_minus
% 0.30/0.60 thf(fact_5533_ab__left__minus,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.30/0.60 = zero_z3403309356797280102nteger ) ).
% 0.30/0.60
% 0.30/0.60 % ab_left_minus
% 0.30/0.60 thf(fact_5534_ab__left__minus,axiom,
% 0.30/0.60 ! [A: rat] :
% 0.30/0.60 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.30/0.60 = zero_zero_rat ) ).
% 0.30/0.60
% 0.30/0.60 % ab_left_minus
% 0.30/0.60 thf(fact_5535_add_Oright__inverse,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 0.30/0.60 = zero_zero_int ) ).
% 0.30/0.60
% 0.30/0.60 % add.right_inverse
% 0.30/0.60 thf(fact_5536_add_Oright__inverse,axiom,
% 0.30/0.60 ! [A: real] :
% 0.30/0.60 ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 0.30/0.60 = zero_zero_real ) ).
% 0.30/0.60
% 0.30/0.60 % add.right_inverse
% 0.30/0.60 thf(fact_5537_add_Oright__inverse,axiom,
% 0.30/0.60 ! [A: complex] :
% 0.30/0.60 ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 0.30/0.60 = zero_zero_complex ) ).
% 0.30/0.60
% 0.30/0.60 % add.right_inverse
% 0.30/0.60 thf(fact_5538_add_Oright__inverse,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.60 = zero_z3403309356797280102nteger ) ).
% 0.30/0.60
% 0.30/0.60 % add.right_inverse
% 0.30/0.60 thf(fact_5539_add_Oright__inverse,axiom,
% 0.30/0.60 ! [A: rat] :
% 0.30/0.60 ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.30/0.60 = zero_zero_rat ) ).
% 0.30/0.60
% 0.30/0.60 % add.right_inverse
% 0.30/0.60 thf(fact_5540_diff__0,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( minus_minus_int @ zero_zero_int @ A )
% 0.30/0.60 = ( uminus_uminus_int @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_0
% 0.30/0.60 thf(fact_5541_diff__0,axiom,
% 0.30/0.60 ! [A: real] :
% 0.30/0.60 ( ( minus_minus_real @ zero_zero_real @ A )
% 0.30/0.60 = ( uminus_uminus_real @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_0
% 0.30/0.60 thf(fact_5542_diff__0,axiom,
% 0.30/0.60 ! [A: complex] :
% 0.30/0.60 ( ( minus_minus_complex @ zero_zero_complex @ A )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_0
% 0.30/0.60 thf(fact_5543_diff__0,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_0
% 0.30/0.60 thf(fact_5544_diff__0,axiom,
% 0.30/0.60 ! [A: rat] :
% 0.30/0.60 ( ( minus_minus_rat @ zero_zero_rat @ A )
% 0.30/0.60 = ( uminus_uminus_rat @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_0
% 0.30/0.60 thf(fact_5545_verit__minus__simplify_I3_J,axiom,
% 0.30/0.60 ! [B: int] :
% 0.30/0.60 ( ( minus_minus_int @ zero_zero_int @ B )
% 0.30/0.60 = ( uminus_uminus_int @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % verit_minus_simplify(3)
% 0.30/0.60 thf(fact_5546_verit__minus__simplify_I3_J,axiom,
% 0.30/0.60 ! [B: real] :
% 0.30/0.60 ( ( minus_minus_real @ zero_zero_real @ B )
% 0.30/0.60 = ( uminus_uminus_real @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % verit_minus_simplify(3)
% 0.30/0.60 thf(fact_5547_verit__minus__simplify_I3_J,axiom,
% 0.30/0.60 ! [B: complex] :
% 0.30/0.60 ( ( minus_minus_complex @ zero_zero_complex @ B )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % verit_minus_simplify(3)
% 0.30/0.60 thf(fact_5548_verit__minus__simplify_I3_J,axiom,
% 0.30/0.60 ! [B: code_integer] :
% 0.30/0.60 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % verit_minus_simplify(3)
% 0.30/0.60 thf(fact_5549_verit__minus__simplify_I3_J,axiom,
% 0.30/0.60 ! [B: rat] :
% 0.30/0.60 ( ( minus_minus_rat @ zero_zero_rat @ B )
% 0.30/0.60 = ( uminus_uminus_rat @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % verit_minus_simplify(3)
% 0.30/0.60 thf(fact_5550_add__neg__numeral__simps_I3_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.60 = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_simps(3)
% 0.30/0.60 thf(fact_5551_add__neg__numeral__simps_I3_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.60 = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_simps(3)
% 0.30/0.60 thf(fact_5552_add__neg__numeral__simps_I3_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_simps(3)
% 0.30/0.60 thf(fact_5553_add__neg__numeral__simps_I3_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_simps(3)
% 0.30/0.60 thf(fact_5554_add__neg__numeral__simps_I3_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.60 = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_simps(3)
% 0.30/0.60 thf(fact_5555_mult__minus1__right,axiom,
% 0.30/0.60 ! [Z: int] :
% 0.30/0.60 ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60 = ( uminus_uminus_int @ Z ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus1_right
% 0.30/0.60 thf(fact_5556_mult__minus1__right,axiom,
% 0.30/0.60 ! [Z: real] :
% 0.30/0.60 ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.60 = ( uminus_uminus_real @ Z ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus1_right
% 0.30/0.60 thf(fact_5557_mult__minus1__right,axiom,
% 0.30/0.60 ! [Z: complex] :
% 0.30/0.60 ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus1_right
% 0.30/0.60 thf(fact_5558_mult__minus1__right,axiom,
% 0.30/0.60 ! [Z: code_integer] :
% 0.30/0.60 ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus1_right
% 0.30/0.60 thf(fact_5559_mult__minus1__right,axiom,
% 0.30/0.60 ! [Z: rat] :
% 0.30/0.60 ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.60 = ( uminus_uminus_rat @ Z ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus1_right
% 0.30/0.60 thf(fact_5560_mult__minus1,axiom,
% 0.30/0.60 ! [Z: int] :
% 0.30/0.60 ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
% 0.30/0.60 = ( uminus_uminus_int @ Z ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus1
% 0.30/0.60 thf(fact_5561_mult__minus1,axiom,
% 0.30/0.60 ! [Z: real] :
% 0.30/0.60 ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
% 0.30/0.60 = ( uminus_uminus_real @ Z ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus1
% 0.30/0.60 thf(fact_5562_mult__minus1,axiom,
% 0.30/0.60 ! [Z: complex] :
% 0.30/0.60 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus1
% 0.30/0.60 thf(fact_5563_mult__minus1,axiom,
% 0.30/0.60 ! [Z: code_integer] :
% 0.30/0.60 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus1
% 0.30/0.60 thf(fact_5564_mult__minus1,axiom,
% 0.30/0.60 ! [Z: rat] :
% 0.30/0.60 ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
% 0.30/0.60 = ( uminus_uminus_rat @ Z ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_minus1
% 0.30/0.60 thf(fact_5565_uminus__add__conv__diff,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.60 = ( minus_minus_int @ B @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % uminus_add_conv_diff
% 0.30/0.60 thf(fact_5566_uminus__add__conv__diff,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
% 0.30/0.60 = ( minus_minus_real @ B @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % uminus_add_conv_diff
% 0.30/0.60 thf(fact_5567_uminus__add__conv__diff,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.30/0.60 = ( minus_minus_complex @ B @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % uminus_add_conv_diff
% 0.30/0.60 thf(fact_5568_uminus__add__conv__diff,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.30/0.60 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % uminus_add_conv_diff
% 0.30/0.60 thf(fact_5569_uminus__add__conv__diff,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.30/0.60 = ( minus_minus_rat @ B @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % uminus_add_conv_diff
% 0.30/0.60 thf(fact_5570_diff__minus__eq__add,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 0.30/0.60 = ( plus_plus_int @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_minus_eq_add
% 0.30/0.60 thf(fact_5571_diff__minus__eq__add,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
% 0.30/0.60 = ( plus_plus_real @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_minus_eq_add
% 0.30/0.60 thf(fact_5572_diff__minus__eq__add,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 0.30/0.60 = ( plus_plus_complex @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_minus_eq_add
% 0.30/0.60 thf(fact_5573_diff__minus__eq__add,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.60 = ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_minus_eq_add
% 0.30/0.60 thf(fact_5574_diff__minus__eq__add,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.30/0.60 = ( plus_plus_rat @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_minus_eq_add
% 0.30/0.60 thf(fact_5575_divide__minus1,axiom,
% 0.30/0.60 ! [X2: real] :
% 0.30/0.60 ( ( divide_divide_real @ X2 @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.60 = ( uminus_uminus_real @ X2 ) ) ).
% 0.30/0.60
% 0.30/0.60 % divide_minus1
% 0.30/0.60 thf(fact_5576_divide__minus1,axiom,
% 0.30/0.60 ! [X2: complex] :
% 0.30/0.60 ( ( divide1717551699836669952omplex @ X2 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ X2 ) ) ).
% 0.30/0.60
% 0.30/0.60 % divide_minus1
% 0.30/0.60 thf(fact_5577_divide__minus1,axiom,
% 0.30/0.60 ! [X2: rat] :
% 0.30/0.60 ( ( divide_divide_rat @ X2 @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.60 = ( uminus_uminus_rat @ X2 ) ) ).
% 0.30/0.60
% 0.30/0.60 % divide_minus1
% 0.30/0.60 thf(fact_5578_div__minus1__right,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60 = ( uminus_uminus_int @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % div_minus1_right
% 0.30/0.60 thf(fact_5579_div__minus1__right,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % div_minus1_right
% 0.30/0.60 thf(fact_5580_minus__mod__self1,axiom,
% 0.30/0.60 ! [B: int,A: int] :
% 0.30/0.60 ( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
% 0.30/0.60 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_mod_self1
% 0.30/0.60 thf(fact_5581_minus__mod__self1,axiom,
% 0.30/0.60 ! [B: code_integer,A: code_integer] :
% 0.30/0.60 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ B )
% 0.30/0.60 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_mod_self1
% 0.30/0.60 thf(fact_5582_signed__take__bit__of__minus__1,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.60
% 0.30/0.60 % signed_take_bit_of_minus_1
% 0.30/0.60 thf(fact_5583_signed__take__bit__of__minus__1,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % signed_take_bit_of_minus_1
% 0.30/0.60 thf(fact_5584_pred__numeral__simps_I1_J,axiom,
% 0.30/0.60 ( ( pred_numeral @ one )
% 0.30/0.60 = zero_zero_nat ) ).
% 0.30/0.60
% 0.30/0.60 % pred_numeral_simps(1)
% 0.30/0.60 thf(fact_5585_Suc__eq__numeral,axiom,
% 0.30/0.60 ! [N: nat,K: num] :
% 0.30/0.60 ( ( ( suc @ N )
% 0.30/0.60 = ( numeral_numeral_nat @ K ) )
% 0.30/0.60 = ( N
% 0.30/0.60 = ( pred_numeral @ K ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Suc_eq_numeral
% 0.30/0.60 thf(fact_5586_eq__numeral__Suc,axiom,
% 0.30/0.60 ! [K: num,N: nat] :
% 0.30/0.60 ( ( ( numeral_numeral_nat @ K )
% 0.30/0.60 = ( suc @ N ) )
% 0.30/0.60 = ( ( pred_numeral @ K )
% 0.30/0.60 = N ) ) ).
% 0.30/0.60
% 0.30/0.60 % eq_numeral_Suc
% 0.30/0.60 thf(fact_5587_dbl__simps_I1_J,axiom,
% 0.30/0.60 ! [K: num] :
% 0.30/0.60 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.60 = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dbl_simps(1)
% 0.30/0.60 thf(fact_5588_dbl__simps_I1_J,axiom,
% 0.30/0.60 ! [K: num] :
% 0.30/0.60 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.30/0.60 = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dbl_simps(1)
% 0.30/0.60 thf(fact_5589_dbl__simps_I1_J,axiom,
% 0.30/0.60 ! [K: num] :
% 0.30/0.60 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dbl_simps(1)
% 0.30/0.60 thf(fact_5590_dbl__simps_I1_J,axiom,
% 0.30/0.60 ! [K: num] :
% 0.30/0.60 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dbl_simps(1)
% 0.30/0.60 thf(fact_5591_dbl__simps_I1_J,axiom,
% 0.30/0.60 ! [K: num] :
% 0.30/0.60 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.30/0.60 = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dbl_simps(1)
% 0.30/0.60 thf(fact_5592_dbl__inc__simps_I4_J,axiom,
% 0.30/0.60 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % dbl_inc_simps(4)
% 0.30/0.60 thf(fact_5593_dbl__inc__simps_I4_J,axiom,
% 0.30/0.60 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.60 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % dbl_inc_simps(4)
% 0.30/0.60 thf(fact_5594_dbl__inc__simps_I4_J,axiom,
% 0.30/0.60 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.60
% 0.30/0.60 % dbl_inc_simps(4)
% 0.30/0.60 thf(fact_5595_dbl__inc__simps_I4_J,axiom,
% 0.30/0.60 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.60
% 0.30/0.60 % dbl_inc_simps(4)
% 0.30/0.60 thf(fact_5596_dbl__inc__simps_I4_J,axiom,
% 0.30/0.60 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.60 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % dbl_inc_simps(4)
% 0.30/0.60 thf(fact_5597_add__neg__numeral__special_I8_J,axiom,
% 0.30/0.60 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.30/0.60 = zero_zero_int ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_special(8)
% 0.30/0.60 thf(fact_5598_add__neg__numeral__special_I8_J,axiom,
% 0.30/0.60 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.30/0.60 = zero_zero_real ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_special(8)
% 0.30/0.60 thf(fact_5599_add__neg__numeral__special_I8_J,axiom,
% 0.30/0.60 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 0.30/0.60 = zero_zero_complex ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_special(8)
% 0.30/0.60 thf(fact_5600_add__neg__numeral__special_I8_J,axiom,
% 0.30/0.60 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.30/0.60 = zero_z3403309356797280102nteger ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_special(8)
% 0.30/0.60 thf(fact_5601_add__neg__numeral__special_I8_J,axiom,
% 0.30/0.60 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.30/0.60 = zero_zero_rat ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_special(8)
% 0.30/0.60 thf(fact_5602_add__neg__numeral__special_I7_J,axiom,
% 0.30/0.60 ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60 = zero_zero_int ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_special(7)
% 0.30/0.60 thf(fact_5603_add__neg__numeral__special_I7_J,axiom,
% 0.30/0.60 ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.60 = zero_zero_real ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_special(7)
% 0.30/0.60 thf(fact_5604_add__neg__numeral__special_I7_J,axiom,
% 0.30/0.60 ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.60 = zero_zero_complex ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_special(7)
% 0.30/0.60 thf(fact_5605_add__neg__numeral__special_I7_J,axiom,
% 0.30/0.60 ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60 = zero_z3403309356797280102nteger ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_special(7)
% 0.30/0.60 thf(fact_5606_add__neg__numeral__special_I7_J,axiom,
% 0.30/0.60 ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.60 = zero_zero_rat ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_special(7)
% 0.30/0.60 thf(fact_5607_diff__numeral__special_I12_J,axiom,
% 0.30/0.60 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60 = zero_zero_int ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(12)
% 0.30/0.60 thf(fact_5608_diff__numeral__special_I12_J,axiom,
% 0.30/0.60 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.60 = zero_zero_real ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(12)
% 0.30/0.60 thf(fact_5609_diff__numeral__special_I12_J,axiom,
% 0.30/0.60 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.60 = zero_zero_complex ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(12)
% 0.30/0.60 thf(fact_5610_diff__numeral__special_I12_J,axiom,
% 0.30/0.60 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60 = zero_z3403309356797280102nteger ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(12)
% 0.30/0.60 thf(fact_5611_diff__numeral__special_I12_J,axiom,
% 0.30/0.60 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.60 = zero_zero_rat ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(12)
% 0.30/0.60 thf(fact_5612_neg__one__eq__numeral__iff,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( ( uminus_uminus_int @ one_one_int )
% 0.30/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.60 = ( N = one ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_eq_numeral_iff
% 0.30/0.60 thf(fact_5613_neg__one__eq__numeral__iff,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( ( uminus_uminus_real @ one_one_real )
% 0.30/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.60 = ( N = one ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_eq_numeral_iff
% 0.30/0.60 thf(fact_5614_neg__one__eq__numeral__iff,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( ( uminus1482373934393186551omplex @ one_one_complex )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.30/0.60 = ( N = one ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_eq_numeral_iff
% 0.30/0.60 thf(fact_5615_neg__one__eq__numeral__iff,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.60 = ( N = one ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_eq_numeral_iff
% 0.30/0.60 thf(fact_5616_neg__one__eq__numeral__iff,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( ( uminus_uminus_rat @ one_one_rat )
% 0.30/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.60 = ( N = one ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_eq_numeral_iff
% 0.30/0.60 thf(fact_5617_numeral__eq__neg__one__iff,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
% 0.30/0.60 = ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60 = ( N = one ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_eq_neg_one_iff
% 0.30/0.60 thf(fact_5618_numeral__eq__neg__one__iff,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
% 0.30/0.60 = ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.60 = ( N = one ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_eq_neg_one_iff
% 0.30/0.60 thf(fact_5619_numeral__eq__neg__one__iff,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.60 = ( N = one ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_eq_neg_one_iff
% 0.30/0.60 thf(fact_5620_numeral__eq__neg__one__iff,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60 = ( N = one ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_eq_neg_one_iff
% 0.30/0.60 thf(fact_5621_numeral__eq__neg__one__iff,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
% 0.30/0.60 = ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.60 = ( N = one ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_eq_neg_one_iff
% 0.30/0.60 thf(fact_5622_left__minus__one__mult__self,axiom,
% 0.30/0.60 ! [N: nat,A: int] :
% 0.30/0.60 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
% 0.30/0.60 = A ) ).
% 0.30/0.60
% 0.30/0.60 % left_minus_one_mult_self
% 0.30/0.60 thf(fact_5623_left__minus__one__mult__self,axiom,
% 0.30/0.60 ! [N: nat,A: real] :
% 0.30/0.60 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
% 0.30/0.60 = A ) ).
% 0.30/0.60
% 0.30/0.60 % left_minus_one_mult_self
% 0.30/0.60 thf(fact_5624_left__minus__one__mult__self,axiom,
% 0.30/0.60 ! [N: nat,A: complex] :
% 0.30/0.60 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
% 0.30/0.60 = A ) ).
% 0.30/0.60
% 0.30/0.60 % left_minus_one_mult_self
% 0.30/0.60 thf(fact_5625_left__minus__one__mult__self,axiom,
% 0.30/0.60 ! [N: nat,A: code_integer] :
% 0.30/0.60 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
% 0.30/0.60 = A ) ).
% 0.30/0.60
% 0.30/0.60 % left_minus_one_mult_self
% 0.30/0.60 thf(fact_5626_left__minus__one__mult__self,axiom,
% 0.30/0.60 ! [N: nat,A: rat] :
% 0.30/0.60 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ A ) )
% 0.30/0.60 = A ) ).
% 0.30/0.60
% 0.30/0.60 % left_minus_one_mult_self
% 0.30/0.60 thf(fact_5627_minus__one__mult__self,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
% 0.30/0.60 = one_one_int ) ).
% 0.30/0.60
% 0.30/0.60 % minus_one_mult_self
% 0.30/0.60 thf(fact_5628_minus__one__mult__self,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
% 0.30/0.60 = one_one_real ) ).
% 0.30/0.60
% 0.30/0.60 % minus_one_mult_self
% 0.30/0.60 thf(fact_5629_minus__one__mult__self,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
% 0.30/0.60 = one_one_complex ) ).
% 0.30/0.60
% 0.30/0.60 % minus_one_mult_self
% 0.30/0.60 thf(fact_5630_minus__one__mult__self,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
% 0.30/0.60 = one_one_Code_integer ) ).
% 0.30/0.60
% 0.30/0.60 % minus_one_mult_self
% 0.30/0.60 thf(fact_5631_minus__one__mult__self,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
% 0.30/0.60 = one_one_rat ) ).
% 0.30/0.60
% 0.30/0.60 % minus_one_mult_self
% 0.30/0.60 thf(fact_5632_mod__minus1__right,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60 = zero_zero_int ) ).
% 0.30/0.60
% 0.30/0.60 % mod_minus1_right
% 0.30/0.60 thf(fact_5633_mod__minus1__right,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60 = zero_z3403309356797280102nteger ) ).
% 0.30/0.60
% 0.30/0.60 % mod_minus1_right
% 0.30/0.60 thf(fact_5634_max__number__of_I2_J,axiom,
% 0.30/0.60 ! [U: num,V: num] :
% 0.30/0.60 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.60 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 0.30/0.60 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.60 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.60 = ( numeral_numeral_real @ U ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % max_number_of(2)
% 0.30/0.60 thf(fact_5635_max__number__of_I2_J,axiom,
% 0.30/0.60 ! [U: num,V: num] :
% 0.30/0.60 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.30/0.60 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 0.30/0.60 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.30/0.60 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.30/0.60 = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % max_number_of(2)
% 0.30/0.60 thf(fact_5636_max__number__of_I2_J,axiom,
% 0.30/0.60 ! [U: num,V: num] :
% 0.30/0.60 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.60 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 0.30/0.60 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.60 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.60 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % max_number_of(2)
% 0.30/0.60 thf(fact_5637_max__number__of_I2_J,axiom,
% 0.30/0.60 ! [U: num,V: num] :
% 0.30/0.60 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.60 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 0.30/0.60 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.60 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.60 = ( numeral_numeral_int @ U ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % max_number_of(2)
% 0.30/0.60 thf(fact_5638_max__number__of_I3_J,axiom,
% 0.30/0.60 ! [U: num,V: num] :
% 0.30/0.60 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.60 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.60 = ( numeral_numeral_real @ V ) ) )
% 0.30/0.60 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.60 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % max_number_of(3)
% 0.30/0.60 thf(fact_5639_max__number__of_I3_J,axiom,
% 0.30/0.60 ! [U: num,V: num] :
% 0.30/0.60 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.30/0.60 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.30/0.60 = ( numera6620942414471956472nteger @ V ) ) )
% 0.30/0.60 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.30/0.60 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % max_number_of(3)
% 0.30/0.60 thf(fact_5640_max__number__of_I3_J,axiom,
% 0.30/0.60 ! [U: num,V: num] :
% 0.30/0.60 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.60 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.60 = ( numeral_numeral_rat @ V ) ) )
% 0.30/0.60 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.60 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % max_number_of(3)
% 0.30/0.60 thf(fact_5641_max__number__of_I3_J,axiom,
% 0.30/0.60 ! [U: num,V: num] :
% 0.30/0.60 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.60 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.60 = ( numeral_numeral_int @ V ) ) )
% 0.30/0.60 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.60 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % max_number_of(3)
% 0.30/0.60 thf(fact_5642_max__number__of_I4_J,axiom,
% 0.30/0.60 ! [U: num,V: num] :
% 0.30/0.60 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.60 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 0.30/0.60 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.60 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % max_number_of(4)
% 0.30/0.60 thf(fact_5643_max__number__of_I4_J,axiom,
% 0.30/0.60 ! [U: num,V: num] :
% 0.30/0.60 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.30/0.60 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 0.30/0.60 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.30/0.60 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % max_number_of(4)
% 0.30/0.60 thf(fact_5644_max__number__of_I4_J,axiom,
% 0.30/0.60 ! [U: num,V: num] :
% 0.30/0.60 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.60 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 0.30/0.60 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.60 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % max_number_of(4)
% 0.30/0.60 thf(fact_5645_max__number__of_I4_J,axiom,
% 0.30/0.60 ! [U: num,V: num] :
% 0.30/0.60 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.60 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 0.30/0.60 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.60 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % max_number_of(4)
% 0.30/0.60 thf(fact_5646_semiring__norm_I168_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: int] :
% 0.30/0.60 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 0.30/0.60 = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(168)
% 0.30/0.60 thf(fact_5647_semiring__norm_I168_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: real] :
% 0.30/0.60 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 0.30/0.60 = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(168)
% 0.30/0.60 thf(fact_5648_semiring__norm_I168_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: complex] :
% 0.30/0.60 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 0.30/0.60 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(168)
% 0.30/0.60 thf(fact_5649_semiring__norm_I168_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: code_integer] :
% 0.30/0.60 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 0.30/0.60 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(168)
% 0.30/0.60 thf(fact_5650_semiring__norm_I168_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: rat] :
% 0.30/0.60 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 0.30/0.60 = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(168)
% 0.30/0.60 thf(fact_5651_diff__numeral__simps_I3_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_simps(3)
% 0.30/0.60 thf(fact_5652_diff__numeral__simps_I3_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 0.30/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_simps(3)
% 0.30/0.60 thf(fact_5653_diff__numeral__simps_I3_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_simps(3)
% 0.30/0.60 thf(fact_5654_diff__numeral__simps_I3_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_simps(3)
% 0.30/0.60 thf(fact_5655_diff__numeral__simps_I3_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 0.30/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_simps(3)
% 0.30/0.60 thf(fact_5656_diff__numeral__simps_I2_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.60 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_simps(2)
% 0.30/0.60 thf(fact_5657_diff__numeral__simps_I2_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.60 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_simps(2)
% 0.30/0.60 thf(fact_5658_diff__numeral__simps_I2_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.30/0.60 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_simps(2)
% 0.30/0.60 thf(fact_5659_diff__numeral__simps_I2_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.60 = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_simps(2)
% 0.30/0.60 thf(fact_5660_diff__numeral__simps_I2_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.60 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_simps(2)
% 0.30/0.60 thf(fact_5661_semiring__norm_I172_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: int] :
% 0.30/0.60 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 0.30/0.60 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(172)
% 0.30/0.60 thf(fact_5662_semiring__norm_I172_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: real] :
% 0.30/0.60 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 0.30/0.60 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(172)
% 0.30/0.60 thf(fact_5663_semiring__norm_I172_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: complex] :
% 0.30/0.60 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 0.30/0.60 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(172)
% 0.30/0.60 thf(fact_5664_semiring__norm_I172_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: code_integer] :
% 0.30/0.60 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 0.30/0.60 = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(172)
% 0.30/0.60 thf(fact_5665_semiring__norm_I172_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: rat] :
% 0.30/0.60 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 0.30/0.60 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(172)
% 0.30/0.60 thf(fact_5666_semiring__norm_I171_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: int] :
% 0.30/0.60 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 0.30/0.60 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(171)
% 0.30/0.60 thf(fact_5667_semiring__norm_I171_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: real] :
% 0.30/0.60 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 0.30/0.60 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(171)
% 0.30/0.60 thf(fact_5668_semiring__norm_I171_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: complex] :
% 0.30/0.60 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 0.30/0.60 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(171)
% 0.30/0.60 thf(fact_5669_semiring__norm_I171_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: code_integer] :
% 0.30/0.60 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 0.30/0.60 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(171)
% 0.30/0.60 thf(fact_5670_semiring__norm_I171_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: rat] :
% 0.30/0.60 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 0.30/0.60 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(171)
% 0.30/0.60 thf(fact_5671_semiring__norm_I170_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: int] :
% 0.30/0.60 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y3 ) )
% 0.30/0.60 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(170)
% 0.30/0.60 thf(fact_5672_semiring__norm_I170_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: real] :
% 0.30/0.60 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y3 ) )
% 0.30/0.60 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(170)
% 0.30/0.60 thf(fact_5673_semiring__norm_I170_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: complex] :
% 0.30/0.60 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y3 ) )
% 0.30/0.60 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(170)
% 0.30/0.60 thf(fact_5674_semiring__norm_I170_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: code_integer] :
% 0.30/0.60 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y3 ) )
% 0.30/0.60 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(170)
% 0.30/0.60 thf(fact_5675_semiring__norm_I170_J,axiom,
% 0.30/0.60 ! [V: num,W: num,Y3: rat] :
% 0.30/0.60 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y3 ) )
% 0.30/0.60 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % semiring_norm(170)
% 0.30/0.60 thf(fact_5676_mult__neg__numeral__simps_I1_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.60 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_neg_numeral_simps(1)
% 0.30/0.60 thf(fact_5677_mult__neg__numeral__simps_I1_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.60 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_neg_numeral_simps(1)
% 0.30/0.60 thf(fact_5678_mult__neg__numeral__simps_I1_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.30/0.60 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_neg_numeral_simps(1)
% 0.30/0.60 thf(fact_5679_mult__neg__numeral__simps_I1_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.60 = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_neg_numeral_simps(1)
% 0.30/0.60 thf(fact_5680_mult__neg__numeral__simps_I1_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.60 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_neg_numeral_simps(1)
% 0.30/0.60 thf(fact_5681_mult__neg__numeral__simps_I2_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_neg_numeral_simps(2)
% 0.30/0.60 thf(fact_5682_mult__neg__numeral__simps_I2_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 0.30/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_neg_numeral_simps(2)
% 0.30/0.60 thf(fact_5683_mult__neg__numeral__simps_I2_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_neg_numeral_simps(2)
% 0.30/0.60 thf(fact_5684_mult__neg__numeral__simps_I2_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_neg_numeral_simps(2)
% 0.30/0.60 thf(fact_5685_mult__neg__numeral__simps_I2_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 0.30/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_neg_numeral_simps(2)
% 0.30/0.60 thf(fact_5686_mult__neg__numeral__simps_I3_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_neg_numeral_simps(3)
% 0.30/0.60 thf(fact_5687_mult__neg__numeral__simps_I3_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_neg_numeral_simps(3)
% 0.30/0.60 thf(fact_5688_mult__neg__numeral__simps_I3_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_neg_numeral_simps(3)
% 0.30/0.60 thf(fact_5689_mult__neg__numeral__simps_I3_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_neg_numeral_simps(3)
% 0.30/0.60 thf(fact_5690_mult__neg__numeral__simps_I3_J,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_neg_numeral_simps(3)
% 0.30/0.60 thf(fact_5691_less__numeral__Suc,axiom,
% 0.30/0.60 ! [K: num,N: nat] :
% 0.30/0.60 ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.30/0.60 = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_numeral_Suc
% 0.30/0.60 thf(fact_5692_less__Suc__numeral,axiom,
% 0.30/0.60 ! [N: nat,K: num] :
% 0.30/0.60 ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.60 = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_Suc_numeral
% 0.30/0.60 thf(fact_5693_pred__numeral__simps_I3_J,axiom,
% 0.30/0.60 ! [K: num] :
% 0.30/0.60 ( ( pred_numeral @ ( bit1 @ K ) )
% 0.30/0.60 = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % pred_numeral_simps(3)
% 0.30/0.60 thf(fact_5694_le__numeral__Suc,axiom,
% 0.30/0.60 ! [K: num,N: nat] :
% 0.30/0.60 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.30/0.60 = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_numeral_Suc
% 0.30/0.60 thf(fact_5695_le__Suc__numeral,axiom,
% 0.30/0.60 ! [N: nat,K: num] :
% 0.30/0.60 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.60 = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_Suc_numeral
% 0.30/0.60 thf(fact_5696_neg__numeral__le__iff,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.60 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_iff
% 0.30/0.60 thf(fact_5697_neg__numeral__le__iff,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.60 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_iff
% 0.30/0.60 thf(fact_5698_neg__numeral__le__iff,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.60 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_iff
% 0.30/0.60 thf(fact_5699_neg__numeral__le__iff,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.60 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_iff
% 0.30/0.60 thf(fact_5700_diff__Suc__numeral,axiom,
% 0.30/0.60 ! [N: nat,K: num] :
% 0.30/0.60 ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.60 = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_Suc_numeral
% 0.30/0.60 thf(fact_5701_diff__numeral__Suc,axiom,
% 0.30/0.60 ! [K: num,N: nat] :
% 0.30/0.60 ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.30/0.60 = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_Suc
% 0.30/0.60 thf(fact_5702_neg__numeral__less__iff,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.60 = ( ord_less_num @ N @ M ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_iff
% 0.30/0.60 thf(fact_5703_neg__numeral__less__iff,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.60 = ( ord_less_num @ N @ M ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_iff
% 0.30/0.60 thf(fact_5704_neg__numeral__less__iff,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.60 = ( ord_less_num @ N @ M ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_iff
% 0.30/0.60 thf(fact_5705_neg__numeral__less__iff,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.60 = ( ord_less_num @ N @ M ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_iff
% 0.30/0.60 thf(fact_5706_max__Suc__numeral,axiom,
% 0.30/0.60 ! [N: nat,K: num] :
% 0.30/0.60 ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.60 = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % max_Suc_numeral
% 0.30/0.60 thf(fact_5707_max__numeral__Suc,axiom,
% 0.30/0.60 ! [K: num,N: nat] :
% 0.30/0.60 ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.30/0.60 = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % max_numeral_Suc
% 0.30/0.60 thf(fact_5708_pred__numeral__simps_I2_J,axiom,
% 0.30/0.60 ! [K: num] :
% 0.30/0.60 ( ( pred_numeral @ ( bit0 @ K ) )
% 0.30/0.60 = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % pred_numeral_simps(2)
% 0.30/0.60 thf(fact_5709_not__neg__one__le__neg__numeral__iff,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 0.30/0.60 = ( M != one ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_neg_one_le_neg_numeral_iff
% 0.30/0.60 thf(fact_5710_not__neg__one__le__neg__numeral__iff,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 0.30/0.60 = ( M != one ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_neg_one_le_neg_numeral_iff
% 0.30/0.60 thf(fact_5711_not__neg__one__le__neg__numeral__iff,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 0.30/0.60 = ( M != one ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_neg_one_le_neg_numeral_iff
% 0.30/0.60 thf(fact_5712_not__neg__one__le__neg__numeral__iff,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 0.30/0.60 = ( M != one ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_neg_one_le_neg_numeral_iff
% 0.30/0.60 thf(fact_5713_neg__numeral__less__neg__one__iff,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60 = ( M != one ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_neg_one_iff
% 0.30/0.60 thf(fact_5714_neg__numeral__less__neg__one__iff,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.60 = ( M != one ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_neg_one_iff
% 0.30/0.60 thf(fact_5715_neg__numeral__less__neg__one__iff,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60 = ( M != one ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_neg_one_iff
% 0.30/0.60 thf(fact_5716_neg__numeral__less__neg__one__iff,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.60 = ( M != one ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_neg_one_iff
% 0.30/0.60 thf(fact_5717_le__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.60 ! [A: real,B: real,W: num] :
% 0.30/0.60 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.30/0.60 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_divide_eq_numeral1(2)
% 0.30/0.60 thf(fact_5718_le__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.60 ! [A: rat,B: rat,W: num] :
% 0.30/0.60 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.30/0.60 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_divide_eq_numeral1(2)
% 0.30/0.60 thf(fact_5719_divide__le__eq__numeral1_I2_J,axiom,
% 0.30/0.60 ! [B: real,W: num,A: real] :
% 0.30/0.60 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 0.30/0.60 = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % divide_le_eq_numeral1(2)
% 0.30/0.60 thf(fact_5720_divide__le__eq__numeral1_I2_J,axiom,
% 0.30/0.60 ! [B: rat,W: num,A: rat] :
% 0.30/0.60 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 0.30/0.60 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % divide_le_eq_numeral1(2)
% 0.30/0.60 thf(fact_5721_eq__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.60 ! [A: real,B: real,W: num] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.30/0.60 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.30/0.60 != zero_zero_real )
% 0.30/0.60 => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.60 = B ) )
% 0.30/0.60 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.30/0.60 = zero_zero_real )
% 0.30/0.60 => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % eq_divide_eq_numeral1(2)
% 0.30/0.60 thf(fact_5722_eq__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.60 ! [A: complex,B: complex,W: num] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
% 0.30/0.60 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.60 != zero_zero_complex )
% 0.30/0.60 => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.60 = B ) )
% 0.30/0.60 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.60 = zero_zero_complex )
% 0.30/0.60 => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % eq_divide_eq_numeral1(2)
% 0.30/0.60 thf(fact_5723_eq__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.60 ! [A: rat,B: rat,W: num] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.30/0.60 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.30/0.60 != zero_zero_rat )
% 0.30/0.60 => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.30/0.60 = B ) )
% 0.30/0.60 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.30/0.60 = zero_zero_rat )
% 0.30/0.60 => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % eq_divide_eq_numeral1(2)
% 0.30/0.60 thf(fact_5724_divide__eq__eq__numeral1_I2_J,axiom,
% 0.30/0.60 ! [B: real,W: num,A: real] :
% 0.30/0.60 ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.60 = A )
% 0.30/0.60 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.30/0.60 != zero_zero_real )
% 0.30/0.60 => ( B
% 0.30/0.60 = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
% 0.30/0.60 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.30/0.60 = zero_zero_real )
% 0.30/0.60 => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % divide_eq_eq_numeral1(2)
% 0.30/0.60 thf(fact_5725_divide__eq__eq__numeral1_I2_J,axiom,
% 0.30/0.60 ! [B: complex,W: num,A: complex] :
% 0.30/0.60 ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.60 = A )
% 0.30/0.60 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.60 != zero_zero_complex )
% 0.30/0.60 => ( B
% 0.30/0.60 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
% 0.30/0.60 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.60 = zero_zero_complex )
% 0.30/0.60 => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % divide_eq_eq_numeral1(2)
% 0.30/0.60 thf(fact_5726_divide__eq__eq__numeral1_I2_J,axiom,
% 0.30/0.60 ! [B: rat,W: num,A: rat] :
% 0.30/0.60 ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.30/0.60 = A )
% 0.30/0.60 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.30/0.60 != zero_zero_rat )
% 0.30/0.60 => ( B
% 0.30/0.60 = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
% 0.30/0.60 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.30/0.60 = zero_zero_rat )
% 0.30/0.60 => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % divide_eq_eq_numeral1(2)
% 0.30/0.60 thf(fact_5727_less__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.60 ! [A: real,B: real,W: num] :
% 0.30/0.60 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.30/0.60 = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_divide_eq_numeral1(2)
% 0.30/0.60 thf(fact_5728_less__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.60 ! [A: rat,B: rat,W: num] :
% 0.30/0.60 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.30/0.60 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_divide_eq_numeral1(2)
% 0.30/0.60 thf(fact_5729_divide__less__eq__numeral1_I2_J,axiom,
% 0.30/0.60 ! [B: real,W: num,A: real] :
% 0.30/0.60 ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 0.30/0.60 = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % divide_less_eq_numeral1(2)
% 0.30/0.60 thf(fact_5730_divide__less__eq__numeral1_I2_J,axiom,
% 0.30/0.60 ! [B: rat,W: num,A: rat] :
% 0.30/0.60 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 0.30/0.60 = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % divide_less_eq_numeral1(2)
% 0.30/0.60 thf(fact_5731_power2__minus,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.60 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power2_minus
% 0.30/0.60 thf(fact_5732_power2__minus,axiom,
% 0.30/0.60 ! [A: real] :
% 0.30/0.60 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.60 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power2_minus
% 0.30/0.60 thf(fact_5733_power2__minus,axiom,
% 0.30/0.60 ! [A: complex] :
% 0.30/0.60 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.60 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power2_minus
% 0.30/0.60 thf(fact_5734_power2__minus,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.60 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power2_minus
% 0.30/0.60 thf(fact_5735_power2__minus,axiom,
% 0.30/0.60 ! [A: rat] :
% 0.30/0.60 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.60 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power2_minus
% 0.30/0.60 thf(fact_5736_add__neg__numeral__special_I9_J,axiom,
% 0.30/0.60 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_special(9)
% 0.30/0.60 thf(fact_5737_add__neg__numeral__special_I9_J,axiom,
% 0.30/0.60 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_special(9)
% 0.30/0.60 thf(fact_5738_add__neg__numeral__special_I9_J,axiom,
% 0.30/0.60 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_special(9)
% 0.30/0.60 thf(fact_5739_add__neg__numeral__special_I9_J,axiom,
% 0.30/0.60 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_special(9)
% 0.30/0.60 thf(fact_5740_add__neg__numeral__special_I9_J,axiom,
% 0.30/0.60 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add_neg_numeral_special(9)
% 0.30/0.60 thf(fact_5741_diff__numeral__special_I10_J,axiom,
% 0.30/0.60 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.30/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(10)
% 0.30/0.60 thf(fact_5742_diff__numeral__special_I10_J,axiom,
% 0.30/0.60 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.30/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(10)
% 0.30/0.60 thf(fact_5743_diff__numeral__special_I10_J,axiom,
% 0.30/0.60 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(10)
% 0.30/0.60 thf(fact_5744_diff__numeral__special_I10_J,axiom,
% 0.30/0.60 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(10)
% 0.30/0.60 thf(fact_5745_diff__numeral__special_I10_J,axiom,
% 0.30/0.60 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.30/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(10)
% 0.30/0.60 thf(fact_5746_diff__numeral__special_I11_J,axiom,
% 0.30/0.60 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(11)
% 0.30/0.60 thf(fact_5747_diff__numeral__special_I11_J,axiom,
% 0.30/0.60 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.60 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(11)
% 0.30/0.60 thf(fact_5748_diff__numeral__special_I11_J,axiom,
% 0.30/0.60 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.60 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(11)
% 0.30/0.60 thf(fact_5749_diff__numeral__special_I11_J,axiom,
% 0.30/0.60 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(11)
% 0.30/0.60 thf(fact_5750_diff__numeral__special_I11_J,axiom,
% 0.30/0.60 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.60 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(11)
% 0.30/0.60 thf(fact_5751_minus__1__div__2__eq,axiom,
% 0.30/0.60 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.60 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_1_div_2_eq
% 0.30/0.60 thf(fact_5752_minus__1__div__2__eq,axiom,
% 0.30/0.60 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_1_div_2_eq
% 0.30/0.60 thf(fact_5753_minus__1__mod__2__eq,axiom,
% 0.30/0.60 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.60 = one_one_int ) ).
% 0.30/0.60
% 0.30/0.60 % minus_1_mod_2_eq
% 0.30/0.60 thf(fact_5754_minus__1__mod__2__eq,axiom,
% 0.30/0.60 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.60 = one_one_Code_integer ) ).
% 0.30/0.60
% 0.30/0.60 % minus_1_mod_2_eq
% 0.30/0.60 thf(fact_5755_bits__minus__1__mod__2__eq,axiom,
% 0.30/0.60 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.60 = one_one_int ) ).
% 0.30/0.60
% 0.30/0.60 % bits_minus_1_mod_2_eq
% 0.30/0.60 thf(fact_5756_bits__minus__1__mod__2__eq,axiom,
% 0.30/0.60 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.60 = one_one_Code_integer ) ).
% 0.30/0.60
% 0.30/0.60 % bits_minus_1_mod_2_eq
% 0.30/0.60 thf(fact_5757_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.60 ! [A: int,N: nat] :
% 0.30/0.60 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.60 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Power.ring_1_class.power_minus_even
% 0.30/0.60 thf(fact_5758_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.60 ! [A: real,N: nat] :
% 0.30/0.60 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.60 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Power.ring_1_class.power_minus_even
% 0.30/0.60 thf(fact_5759_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.60 ! [A: complex,N: nat] :
% 0.30/0.60 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.60 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Power.ring_1_class.power_minus_even
% 0.30/0.60 thf(fact_5760_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.60 ! [A: code_integer,N: nat] :
% 0.30/0.60 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.60 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Power.ring_1_class.power_minus_even
% 0.30/0.60 thf(fact_5761_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.60 ! [A: rat,N: nat] :
% 0.30/0.60 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.60 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Power.ring_1_class.power_minus_even
% 0.30/0.60 thf(fact_5762_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.60 ! [N: nat,A: int] :
% 0.30/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.30/0.60 = ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Parity.ring_1_class.power_minus_even
% 0.30/0.60 thf(fact_5763_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.60 ! [N: nat,A: real] :
% 0.30/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.30/0.60 = ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Parity.ring_1_class.power_minus_even
% 0.30/0.60 thf(fact_5764_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.60 ! [N: nat,A: complex] :
% 0.30/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.30/0.60 = ( power_power_complex @ A @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Parity.ring_1_class.power_minus_even
% 0.30/0.60 thf(fact_5765_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.60 ! [N: nat,A: code_integer] :
% 0.30/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.30/0.60 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Parity.ring_1_class.power_minus_even
% 0.30/0.60 thf(fact_5766_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.60 ! [N: nat,A: rat] :
% 0.30/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.30/0.60 = ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % Parity.ring_1_class.power_minus_even
% 0.30/0.60 thf(fact_5767_power__minus__odd,axiom,
% 0.30/0.60 ! [N: nat,A: int] :
% 0.30/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.30/0.60 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus_odd
% 0.30/0.60 thf(fact_5768_power__minus__odd,axiom,
% 0.30/0.60 ! [N: nat,A: real] :
% 0.30/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.30/0.60 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus_odd
% 0.30/0.60 thf(fact_5769_power__minus__odd,axiom,
% 0.30/0.60 ! [N: nat,A: complex] :
% 0.30/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus_odd
% 0.30/0.60 thf(fact_5770_power__minus__odd,axiom,
% 0.30/0.60 ! [N: nat,A: code_integer] :
% 0.30/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus_odd
% 0.30/0.60 thf(fact_5771_power__minus__odd,axiom,
% 0.30/0.60 ! [N: nat,A: rat] :
% 0.30/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.30/0.60 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus_odd
% 0.30/0.60 thf(fact_5772_diff__numeral__special_I3_J,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.60 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(3)
% 0.30/0.60 thf(fact_5773_diff__numeral__special_I3_J,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.60 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(3)
% 0.30/0.60 thf(fact_5774_diff__numeral__special_I3_J,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.30/0.60 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(3)
% 0.30/0.60 thf(fact_5775_diff__numeral__special_I3_J,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.60 = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(3)
% 0.30/0.60 thf(fact_5776_diff__numeral__special_I3_J,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.60 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(3)
% 0.30/0.60 thf(fact_5777_diff__numeral__special_I4_J,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
% 0.30/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(4)
% 0.30/0.60 thf(fact_5778_diff__numeral__special_I4_J,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
% 0.30/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(4)
% 0.30/0.60 thf(fact_5779_diff__numeral__special_I4_J,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(4)
% 0.30/0.60 thf(fact_5780_diff__numeral__special_I4_J,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(4)
% 0.30/0.60 thf(fact_5781_diff__numeral__special_I4_J,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
% 0.30/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_numeral_special(4)
% 0.30/0.60 thf(fact_5782_dbl__simps_I4_J,axiom,
% 0.30/0.60 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dbl_simps(4)
% 0.30/0.60 thf(fact_5783_dbl__simps_I4_J,axiom,
% 0.30/0.60 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dbl_simps(4)
% 0.30/0.60 thf(fact_5784_dbl__simps_I4_J,axiom,
% 0.30/0.60 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dbl_simps(4)
% 0.30/0.60 thf(fact_5785_dbl__simps_I4_J,axiom,
% 0.30/0.60 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dbl_simps(4)
% 0.30/0.60 thf(fact_5786_dbl__simps_I4_J,axiom,
% 0.30/0.60 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dbl_simps(4)
% 0.30/0.60 thf(fact_5787_power__minus1__even,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.60 = one_one_int ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus1_even
% 0.30/0.60 thf(fact_5788_power__minus1__even,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.60 = one_one_real ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus1_even
% 0.30/0.60 thf(fact_5789_power__minus1__even,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.60 = one_one_complex ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus1_even
% 0.30/0.60 thf(fact_5790_power__minus1__even,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.60 = one_one_Code_integer ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus1_even
% 0.30/0.60 thf(fact_5791_power__minus1__even,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.60 = one_one_rat ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus1_even
% 0.30/0.60 thf(fact_5792_neg__one__even__power,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.30/0.60 = one_one_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_even_power
% 0.30/0.60 thf(fact_5793_neg__one__even__power,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.30/0.60 = one_one_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_even_power
% 0.30/0.60 thf(fact_5794_neg__one__even__power,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.30/0.60 = one_one_complex ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_even_power
% 0.30/0.60 thf(fact_5795_neg__one__even__power,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.30/0.60 = one_one_Code_integer ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_even_power
% 0.30/0.60 thf(fact_5796_neg__one__even__power,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.30/0.60 = one_one_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_even_power
% 0.30/0.60 thf(fact_5797_neg__one__odd__power,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.30/0.60 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_odd_power
% 0.30/0.60 thf(fact_5798_neg__one__odd__power,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.30/0.60 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_odd_power
% 0.30/0.60 thf(fact_5799_neg__one__odd__power,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_odd_power
% 0.30/0.60 thf(fact_5800_neg__one__odd__power,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_odd_power
% 0.30/0.60 thf(fact_5801_neg__one__odd__power,axiom,
% 0.30/0.60 ! [N: nat] :
% 0.30/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.30/0.60 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_odd_power
% 0.30/0.60 thf(fact_5802_signed__take__bit__0,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % signed_take_bit_0
% 0.30/0.60 thf(fact_5803_signed__take__bit__0,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 0.30/0.60 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % signed_take_bit_0
% 0.30/0.60 thf(fact_5804_signed__take__bit__Suc__minus__bit0,axiom,
% 0.30/0.60 ! [N: nat,K: num] :
% 0.30/0.60 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.30/0.60 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % signed_take_bit_Suc_minus_bit0
% 0.30/0.60 thf(fact_5805_signed__take__bit__numeral__bit0,axiom,
% 0.30/0.60 ! [L: num,K: num] :
% 0.30/0.60 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.30/0.60 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % signed_take_bit_numeral_bit0
% 0.30/0.60 thf(fact_5806_signed__take__bit__numeral__minus__bit0,axiom,
% 0.30/0.60 ! [L: num,K: num] :
% 0.30/0.60 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.30/0.60 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % signed_take_bit_numeral_minus_bit0
% 0.30/0.60 thf(fact_5807_signed__take__bit__numeral__minus__bit1,axiom,
% 0.30/0.60 ! [L: num,K: num] :
% 0.30/0.60 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.30/0.60 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % signed_take_bit_numeral_minus_bit1
% 0.30/0.60 thf(fact_5808_signed__take__bit__minus,axiom,
% 0.30/0.60 ! [N: nat,K: int] :
% 0.30/0.60 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
% 0.30/0.60 = ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % signed_take_bit_minus
% 0.30/0.60 thf(fact_5809_equation__minus__iff,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( uminus_uminus_int @ B ) )
% 0.30/0.60 = ( B
% 0.30/0.60 = ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % equation_minus_iff
% 0.30/0.60 thf(fact_5810_equation__minus__iff,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( uminus_uminus_real @ B ) )
% 0.30/0.60 = ( B
% 0.30/0.60 = ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % equation_minus_iff
% 0.30/0.60 thf(fact_5811_equation__minus__iff,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( uminus1482373934393186551omplex @ B ) )
% 0.30/0.60 = ( B
% 0.30/0.60 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % equation_minus_iff
% 0.30/0.60 thf(fact_5812_equation__minus__iff,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.60 = ( B
% 0.30/0.60 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % equation_minus_iff
% 0.30/0.60 thf(fact_5813_equation__minus__iff,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( uminus_uminus_rat @ B ) )
% 0.30/0.60 = ( B
% 0.30/0.60 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % equation_minus_iff
% 0.30/0.60 thf(fact_5814_minus__equation__iff,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( ( uminus_uminus_int @ A )
% 0.30/0.60 = B )
% 0.30/0.60 = ( ( uminus_uminus_int @ B )
% 0.30/0.60 = A ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_equation_iff
% 0.30/0.60 thf(fact_5815_minus__equation__iff,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( ( uminus_uminus_real @ A )
% 0.30/0.60 = B )
% 0.30/0.60 = ( ( uminus_uminus_real @ B )
% 0.30/0.60 = A ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_equation_iff
% 0.30/0.60 thf(fact_5816_minus__equation__iff,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( ( uminus1482373934393186551omplex @ A )
% 0.30/0.60 = B )
% 0.30/0.60 = ( ( uminus1482373934393186551omplex @ B )
% 0.30/0.60 = A ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_equation_iff
% 0.30/0.60 thf(fact_5817_minus__equation__iff,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( ( uminus1351360451143612070nteger @ A )
% 0.30/0.60 = B )
% 0.30/0.60 = ( ( uminus1351360451143612070nteger @ B )
% 0.30/0.60 = A ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_equation_iff
% 0.30/0.60 thf(fact_5818_minus__equation__iff,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( ( uminus_uminus_rat @ A )
% 0.30/0.60 = B )
% 0.30/0.60 = ( ( uminus_uminus_rat @ B )
% 0.30/0.60 = A ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_equation_iff
% 0.30/0.60 thf(fact_5819_compl__mono,axiom,
% 0.30/0.60 ! [X2: set_int,Y3: set_int] :
% 0.30/0.60 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.30/0.60 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ ( uminus1532241313380277803et_int @ X2 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % compl_mono
% 0.30/0.60 thf(fact_5820_compl__le__swap1,axiom,
% 0.30/0.60 ! [Y3: set_int,X2: set_int] :
% 0.30/0.60 ( ( ord_less_eq_set_int @ Y3 @ ( uminus1532241313380277803et_int @ X2 ) )
% 0.30/0.60 => ( ord_less_eq_set_int @ X2 @ ( uminus1532241313380277803et_int @ Y3 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % compl_le_swap1
% 0.30/0.60 thf(fact_5821_compl__le__swap2,axiom,
% 0.30/0.60 ! [Y3: set_int,X2: set_int] :
% 0.30/0.60 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ X2 )
% 0.30/0.60 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % compl_le_swap2
% 0.30/0.60 thf(fact_5822_mem__case__prodE,axiom,
% 0.30/0.60 ! [Z: complex,C: code_integer > $o > set_complex,P6: produc6271795597528267376eger_o] :
% 0.30/0.60 ( ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ P6 ) )
% 0.30/0.60 => ~ ! [X5: code_integer,Y4: $o] :
% 0.30/0.60 ( ( P6
% 0.30/0.60 = ( produc6677183202524767010eger_o @ X5 @ Y4 ) )
% 0.30/0.60 => ~ ( member_complex @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mem_case_prodE
% 0.30/0.60 thf(fact_5823_mem__case__prodE,axiom,
% 0.30/0.60 ! [Z: real,C: code_integer > $o > set_real,P6: produc6271795597528267376eger_o] :
% 0.30/0.60 ( ( member_real @ Z @ ( produc242741666403216561t_real @ C @ P6 ) )
% 0.30/0.60 => ~ ! [X5: code_integer,Y4: $o] :
% 0.30/0.60 ( ( P6
% 0.30/0.60 = ( produc6677183202524767010eger_o @ X5 @ Y4 ) )
% 0.30/0.60 => ~ ( member_real @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mem_case_prodE
% 0.30/0.60 thf(fact_5824_mem__case__prodE,axiom,
% 0.30/0.60 ! [Z: nat,C: code_integer > $o > set_nat,P6: produc6271795597528267376eger_o] :
% 0.30/0.60 ( ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ P6 ) )
% 0.30/0.60 => ~ ! [X5: code_integer,Y4: $o] :
% 0.30/0.60 ( ( P6
% 0.30/0.60 = ( produc6677183202524767010eger_o @ X5 @ Y4 ) )
% 0.30/0.60 => ~ ( member_nat @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mem_case_prodE
% 0.30/0.60 thf(fact_5825_mem__case__prodE,axiom,
% 0.30/0.60 ! [Z: int,C: code_integer > $o > set_int,P6: produc6271795597528267376eger_o] :
% 0.30/0.60 ( ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ P6 ) )
% 0.30/0.60 => ~ ! [X5: code_integer,Y4: $o] :
% 0.30/0.60 ( ( P6
% 0.30/0.60 = ( produc6677183202524767010eger_o @ X5 @ Y4 ) )
% 0.30/0.60 => ~ ( member_int @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mem_case_prodE
% 0.30/0.60 thf(fact_5826_mem__case__prodE,axiom,
% 0.30/0.60 ! [Z: complex,C: num > num > set_complex,P6: product_prod_num_num] :
% 0.30/0.60 ( ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ P6 ) )
% 0.30/0.60 => ~ ! [X5: num,Y4: num] :
% 0.30/0.60 ( ( P6
% 0.30/0.60 = ( product_Pair_num_num @ X5 @ Y4 ) )
% 0.30/0.60 => ~ ( member_complex @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mem_case_prodE
% 0.30/0.60 thf(fact_5827_mem__case__prodE,axiom,
% 0.30/0.60 ! [Z: real,C: num > num > set_real,P6: product_prod_num_num] :
% 0.30/0.60 ( ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ P6 ) )
% 0.30/0.60 => ~ ! [X5: num,Y4: num] :
% 0.30/0.60 ( ( P6
% 0.30/0.60 = ( product_Pair_num_num @ X5 @ Y4 ) )
% 0.30/0.60 => ~ ( member_real @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mem_case_prodE
% 0.30/0.60 thf(fact_5828_mem__case__prodE,axiom,
% 0.30/0.60 ! [Z: nat,C: num > num > set_nat,P6: product_prod_num_num] :
% 0.30/0.60 ( ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ P6 ) )
% 0.30/0.60 => ~ ! [X5: num,Y4: num] :
% 0.30/0.60 ( ( P6
% 0.30/0.60 = ( product_Pair_num_num @ X5 @ Y4 ) )
% 0.30/0.60 => ~ ( member_nat @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mem_case_prodE
% 0.30/0.60 thf(fact_5829_mem__case__prodE,axiom,
% 0.30/0.60 ! [Z: int,C: num > num > set_int,P6: product_prod_num_num] :
% 0.30/0.60 ( ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ P6 ) )
% 0.30/0.60 => ~ ! [X5: num,Y4: num] :
% 0.30/0.60 ( ( P6
% 0.30/0.60 = ( product_Pair_num_num @ X5 @ Y4 ) )
% 0.30/0.60 => ~ ( member_int @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mem_case_prodE
% 0.30/0.60 thf(fact_5830_mem__case__prodE,axiom,
% 0.30/0.60 ! [Z: complex,C: nat > num > set_complex,P6: product_prod_nat_num] :
% 0.30/0.60 ( ( member_complex @ Z @ ( produc6231982587499038204omplex @ C @ P6 ) )
% 0.30/0.60 => ~ ! [X5: nat,Y4: num] :
% 0.30/0.60 ( ( P6
% 0.30/0.60 = ( product_Pair_nat_num @ X5 @ Y4 ) )
% 0.30/0.60 => ~ ( member_complex @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mem_case_prodE
% 0.30/0.60 thf(fact_5831_mem__case__prodE,axiom,
% 0.30/0.60 ! [Z: real,C: nat > num > set_real,P6: product_prod_nat_num] :
% 0.30/0.60 ( ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ P6 ) )
% 0.30/0.60 => ~ ! [X5: nat,Y4: num] :
% 0.30/0.60 ( ( P6
% 0.30/0.60 = ( product_Pair_nat_num @ X5 @ Y4 ) )
% 0.30/0.60 => ~ ( member_real @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mem_case_prodE
% 0.30/0.60 thf(fact_5832_le__minus__iff,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 0.30/0.60 = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_minus_iff
% 0.30/0.60 thf(fact_5833_le__minus__iff,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.60 = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_minus_iff
% 0.30/0.60 thf(fact_5834_le__minus__iff,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.30/0.60 = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_minus_iff
% 0.30/0.60 thf(fact_5835_le__minus__iff,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 0.30/0.60 = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_minus_iff
% 0.30/0.60 thf(fact_5836_minus__le__iff,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.30/0.60 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_le_iff
% 0.30/0.60 thf(fact_5837_minus__le__iff,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.30/0.60 = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_le_iff
% 0.30/0.60 thf(fact_5838_minus__le__iff,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.30/0.60 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_le_iff
% 0.30/0.60 thf(fact_5839_minus__le__iff,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.60 = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_le_iff
% 0.30/0.60 thf(fact_5840_le__imp__neg__le,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.60 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_imp_neg_le
% 0.30/0.60 thf(fact_5841_le__imp__neg__le,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.30/0.60 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_imp_neg_le
% 0.30/0.60 thf(fact_5842_le__imp__neg__le,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.60 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_imp_neg_le
% 0.30/0.60 thf(fact_5843_le__imp__neg__le,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.60 => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_imp_neg_le
% 0.30/0.60 thf(fact_5844_minus__less__iff,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.60 = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_less_iff
% 0.30/0.60 thf(fact_5845_minus__less__iff,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 0.30/0.60 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_less_iff
% 0.30/0.60 thf(fact_5846_minus__less__iff,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.30/0.60 = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_less_iff
% 0.30/0.60 thf(fact_5847_minus__less__iff,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.30/0.60 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_less_iff
% 0.30/0.60 thf(fact_5848_less__minus__iff,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
% 0.30/0.60 = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_minus_iff
% 0.30/0.60 thf(fact_5849_less__minus__iff,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 0.30/0.60 = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_minus_iff
% 0.30/0.60 thf(fact_5850_less__minus__iff,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.60 = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_minus_iff
% 0.30/0.60 thf(fact_5851_less__minus__iff,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.30/0.60 = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_minus_iff
% 0.30/0.60 thf(fact_5852_verit__negate__coefficient_I2_J,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( ord_less_int @ A @ B )
% 0.30/0.60 => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % verit_negate_coefficient(2)
% 0.30/0.60 thf(fact_5853_verit__negate__coefficient_I2_J,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( ord_less_real @ A @ B )
% 0.30/0.60 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % verit_negate_coefficient(2)
% 0.30/0.60 thf(fact_5854_verit__negate__coefficient_I2_J,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.30/0.60 => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % verit_negate_coefficient(2)
% 0.30/0.60 thf(fact_5855_verit__negate__coefficient_I2_J,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( ord_less_rat @ A @ B )
% 0.30/0.60 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % verit_negate_coefficient(2)
% 0.30/0.60 thf(fact_5856_numeral__neq__neg__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( numeral_numeral_int @ M )
% 0.30/0.60 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_neq_neg_numeral
% 0.30/0.60 thf(fact_5857_numeral__neq__neg__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( numeral_numeral_real @ M )
% 0.30/0.60 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_neq_neg_numeral
% 0.30/0.60 thf(fact_5858_numeral__neq__neg__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( numera6690914467698888265omplex @ M )
% 0.30/0.60 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_neq_neg_numeral
% 0.30/0.60 thf(fact_5859_numeral__neq__neg__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( numera6620942414471956472nteger @ M )
% 0.30/0.60 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_neq_neg_numeral
% 0.30/0.60 thf(fact_5860_numeral__neq__neg__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( numeral_numeral_rat @ M )
% 0.30/0.60 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_neq_neg_numeral
% 0.30/0.60 thf(fact_5861_neg__numeral__neq__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 0.30/0.60 != ( numeral_numeral_int @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_neq_numeral
% 0.30/0.60 thf(fact_5862_neg__numeral__neq__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 0.30/0.60 != ( numeral_numeral_real @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_neq_numeral
% 0.30/0.60 thf(fact_5863_neg__numeral__neq__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 0.30/0.60 != ( numera6690914467698888265omplex @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_neq_numeral
% 0.30/0.60 thf(fact_5864_neg__numeral__neq__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 0.30/0.60 != ( numera6620942414471956472nteger @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_neq_numeral
% 0.30/0.60 thf(fact_5865_neg__numeral__neq__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 0.30/0.60 != ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_neq_numeral
% 0.30/0.60 thf(fact_5866_minus__mult__commute,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.60 = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_mult_commute
% 0.30/0.60 thf(fact_5867_minus__mult__commute,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 0.30/0.60 = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_mult_commute
% 0.30/0.60 thf(fact_5868_minus__mult__commute,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.30/0.60 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_mult_commute
% 0.30/0.60 thf(fact_5869_minus__mult__commute,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.30/0.60 = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_mult_commute
% 0.30/0.60 thf(fact_5870_minus__mult__commute,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.30/0.60 = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_mult_commute
% 0.30/0.60 thf(fact_5871_square__eq__iff,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( ( times_times_int @ A @ A )
% 0.30/0.60 = ( times_times_int @ B @ B ) )
% 0.30/0.60 = ( ( A = B )
% 0.30/0.60 | ( A
% 0.30/0.60 = ( uminus_uminus_int @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % square_eq_iff
% 0.30/0.60 thf(fact_5872_square__eq__iff,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( ( times_times_real @ A @ A )
% 0.30/0.60 = ( times_times_real @ B @ B ) )
% 0.30/0.60 = ( ( A = B )
% 0.30/0.60 | ( A
% 0.30/0.60 = ( uminus_uminus_real @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % square_eq_iff
% 0.30/0.60 thf(fact_5873_square__eq__iff,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( ( times_times_complex @ A @ A )
% 0.30/0.60 = ( times_times_complex @ B @ B ) )
% 0.30/0.60 = ( ( A = B )
% 0.30/0.60 | ( A
% 0.30/0.60 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % square_eq_iff
% 0.30/0.60 thf(fact_5874_square__eq__iff,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( ( times_3573771949741848930nteger @ A @ A )
% 0.30/0.60 = ( times_3573771949741848930nteger @ B @ B ) )
% 0.30/0.60 = ( ( A = B )
% 0.30/0.60 | ( A
% 0.30/0.60 = ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % square_eq_iff
% 0.30/0.60 thf(fact_5875_square__eq__iff,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( ( times_times_rat @ A @ A )
% 0.30/0.60 = ( times_times_rat @ B @ B ) )
% 0.30/0.60 = ( ( A = B )
% 0.30/0.60 | ( A
% 0.30/0.60 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % square_eq_iff
% 0.30/0.60 thf(fact_5876_one__neq__neg__one,axiom,
% 0.30/0.60 ( one_one_int
% 0.30/0.60 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % one_neq_neg_one
% 0.30/0.60 thf(fact_5877_one__neq__neg__one,axiom,
% 0.30/0.60 ( one_one_real
% 0.30/0.60 != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % one_neq_neg_one
% 0.30/0.60 thf(fact_5878_one__neq__neg__one,axiom,
% 0.30/0.60 ( one_one_complex
% 0.30/0.60 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.60
% 0.30/0.60 % one_neq_neg_one
% 0.30/0.60 thf(fact_5879_one__neq__neg__one,axiom,
% 0.30/0.60 ( one_one_Code_integer
% 0.30/0.60 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.60
% 0.30/0.60 % one_neq_neg_one
% 0.30/0.60 thf(fact_5880_one__neq__neg__one,axiom,
% 0.30/0.60 ( one_one_rat
% 0.30/0.60 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % one_neq_neg_one
% 0.30/0.60 thf(fact_5881_is__num__normalize_I8_J,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.30/0.60 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % is_num_normalize(8)
% 0.30/0.60 thf(fact_5882_is__num__normalize_I8_J,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.30/0.60 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % is_num_normalize(8)
% 0.30/0.60 thf(fact_5883_is__num__normalize_I8_J,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 0.30/0.60 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % is_num_normalize(8)
% 0.30/0.60 thf(fact_5884_is__num__normalize_I8_J,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.30/0.60 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % is_num_normalize(8)
% 0.30/0.60 thf(fact_5885_is__num__normalize_I8_J,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.30/0.60 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % is_num_normalize(8)
% 0.30/0.60 thf(fact_5886_group__cancel_Oneg1,axiom,
% 0.30/0.60 ! [A2: int,K: int,A: int] :
% 0.30/0.60 ( ( A2
% 0.30/0.60 = ( plus_plus_int @ K @ A ) )
% 0.30/0.60 => ( ( uminus_uminus_int @ A2 )
% 0.30/0.60 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % group_cancel.neg1
% 0.30/0.60 thf(fact_5887_group__cancel_Oneg1,axiom,
% 0.30/0.60 ! [A2: real,K: real,A: real] :
% 0.30/0.60 ( ( A2
% 0.30/0.60 = ( plus_plus_real @ K @ A ) )
% 0.30/0.60 => ( ( uminus_uminus_real @ A2 )
% 0.30/0.60 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % group_cancel.neg1
% 0.30/0.60 thf(fact_5888_group__cancel_Oneg1,axiom,
% 0.30/0.60 ! [A2: complex,K: complex,A: complex] :
% 0.30/0.60 ( ( A2
% 0.30/0.60 = ( plus_plus_complex @ K @ A ) )
% 0.30/0.60 => ( ( uminus1482373934393186551omplex @ A2 )
% 0.30/0.60 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % group_cancel.neg1
% 0.30/0.60 thf(fact_5889_group__cancel_Oneg1,axiom,
% 0.30/0.60 ! [A2: code_integer,K: code_integer,A: code_integer] :
% 0.30/0.60 ( ( A2
% 0.30/0.60 = ( plus_p5714425477246183910nteger @ K @ A ) )
% 0.30/0.60 => ( ( uminus1351360451143612070nteger @ A2 )
% 0.30/0.60 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % group_cancel.neg1
% 0.30/0.60 thf(fact_5890_group__cancel_Oneg1,axiom,
% 0.30/0.60 ! [A2: rat,K: rat,A: rat] :
% 0.30/0.60 ( ( A2
% 0.30/0.60 = ( plus_plus_rat @ K @ A ) )
% 0.30/0.60 => ( ( uminus_uminus_rat @ A2 )
% 0.30/0.60 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % group_cancel.neg1
% 0.30/0.60 thf(fact_5891_add_Oinverse__distrib__swap,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.30/0.60 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add.inverse_distrib_swap
% 0.30/0.60 thf(fact_5892_add_Oinverse__distrib__swap,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.30/0.60 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add.inverse_distrib_swap
% 0.30/0.60 thf(fact_5893_add_Oinverse__distrib__swap,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 0.30/0.60 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add.inverse_distrib_swap
% 0.30/0.60 thf(fact_5894_add_Oinverse__distrib__swap,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.30/0.60 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add.inverse_distrib_swap
% 0.30/0.60 thf(fact_5895_add_Oinverse__distrib__swap,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.30/0.60 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add.inverse_distrib_swap
% 0.30/0.60 thf(fact_5896_minus__diff__commute,axiom,
% 0.30/0.60 ! [B: int,A: int] :
% 0.30/0.60 ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 0.30/0.60 = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_diff_commute
% 0.30/0.60 thf(fact_5897_minus__diff__commute,axiom,
% 0.30/0.60 ! [B: real,A: real] :
% 0.30/0.60 ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
% 0.30/0.60 = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_diff_commute
% 0.30/0.60 thf(fact_5898_minus__diff__commute,axiom,
% 0.30/0.60 ! [B: complex,A: complex] :
% 0.30/0.60 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
% 0.30/0.60 = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_diff_commute
% 0.30/0.60 thf(fact_5899_minus__diff__commute,axiom,
% 0.30/0.60 ! [B: code_integer,A: code_integer] :
% 0.30/0.60 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
% 0.30/0.60 = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_diff_commute
% 0.30/0.60 thf(fact_5900_minus__diff__commute,axiom,
% 0.30/0.60 ! [B: rat,A: rat] :
% 0.30/0.60 ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
% 0.30/0.60 = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_diff_commute
% 0.30/0.60 thf(fact_5901_minus__diff__minus,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.30/0.60 = ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_diff_minus
% 0.30/0.60 thf(fact_5902_minus__diff__minus,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.30/0.60 = ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_diff_minus
% 0.30/0.60 thf(fact_5903_minus__diff__minus,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_diff_minus
% 0.30/0.60 thf(fact_5904_minus__diff__minus,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_diff_minus
% 0.30/0.60 thf(fact_5905_minus__diff__minus,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.30/0.60 = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_diff_minus
% 0.30/0.60 thf(fact_5906_minus__divide__left,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.60 = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_left
% 0.30/0.60 thf(fact_5907_minus__divide__left,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.30/0.60 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_left
% 0.30/0.60 thf(fact_5908_minus__divide__left,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.30/0.60 = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_left
% 0.30/0.60 thf(fact_5909_minus__divide__divide,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.30/0.60 = ( divide_divide_real @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_divide
% 0.30/0.60 thf(fact_5910_minus__divide__divide,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.30/0.60 = ( divide1717551699836669952omplex @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_divide
% 0.30/0.60 thf(fact_5911_minus__divide__divide,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.30/0.60 = ( divide_divide_rat @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_divide
% 0.30/0.60 thf(fact_5912_minus__divide__right,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.60 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_right
% 0.30/0.60 thf(fact_5913_minus__divide__right,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.30/0.60 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_right
% 0.30/0.60 thf(fact_5914_minus__divide__right,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.30/0.60 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_right
% 0.30/0.60 thf(fact_5915_div__minus__right,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.30/0.60 = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % div_minus_right
% 0.30/0.60 thf(fact_5916_div__minus__right,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.60 = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % div_minus_right
% 0.30/0.60 thf(fact_5917_mod__minus__eq,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
% 0.30/0.60 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % mod_minus_eq
% 0.30/0.60 thf(fact_5918_mod__minus__eq,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) @ B )
% 0.30/0.60 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % mod_minus_eq
% 0.30/0.60 thf(fact_5919_mod__minus__cong,axiom,
% 0.30/0.60 ! [A: int,B: int,A5: int] :
% 0.30/0.60 ( ( ( modulo_modulo_int @ A @ B )
% 0.30/0.60 = ( modulo_modulo_int @ A5 @ B ) )
% 0.30/0.60 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.60 = ( modulo_modulo_int @ ( uminus_uminus_int @ A5 ) @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mod_minus_cong
% 0.30/0.60 thf(fact_5920_mod__minus__cong,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer,A5: code_integer] :
% 0.30/0.60 ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.30/0.60 = ( modulo364778990260209775nteger @ A5 @ B ) )
% 0.30/0.60 => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.30/0.60 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A5 ) @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mod_minus_cong
% 0.30/0.60 thf(fact_5921_mod__minus__right,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 0.30/0.60 = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mod_minus_right
% 0.30/0.60 thf(fact_5922_mod__minus__right,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % mod_minus_right
% 0.30/0.60 thf(fact_5923_case__prodE,axiom,
% 0.30/0.60 ! [C: code_integer > $o > $o,P6: produc6271795597528267376eger_o] :
% 0.30/0.60 ( ( produc7828578312038201481er_o_o @ C @ P6 )
% 0.30/0.60 => ~ ! [X5: code_integer,Y4: $o] :
% 0.30/0.60 ( ( P6
% 0.30/0.60 = ( produc6677183202524767010eger_o @ X5 @ Y4 ) )
% 0.30/0.60 => ~ ( C @ X5 @ Y4 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % case_prodE
% 0.30/0.60 thf(fact_5924_case__prodE,axiom,
% 0.30/0.60 ! [C: num > num > $o,P6: product_prod_num_num] :
% 0.30/0.60 ( ( produc5703948589228662326_num_o @ C @ P6 )
% 0.30/0.60 => ~ ! [X5: num,Y4: num] :
% 0.30/0.60 ( ( P6
% 0.30/0.60 = ( product_Pair_num_num @ X5 @ Y4 ) )
% 0.30/0.60 => ~ ( C @ X5 @ Y4 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % case_prodE
% 0.30/0.60 thf(fact_5925_case__prodE,axiom,
% 0.30/0.60 ! [C: nat > num > $o,P6: product_prod_nat_num] :
% 0.30/0.60 ( ( produc4927758841916487424_num_o @ C @ P6 )
% 0.30/0.60 => ~ ! [X5: nat,Y4: num] :
% 0.30/0.60 ( ( P6
% 0.30/0.60 = ( product_Pair_nat_num @ X5 @ Y4 ) )
% 0.30/0.60 => ~ ( C @ X5 @ Y4 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % case_prodE
% 0.30/0.60 thf(fact_5926_case__prodE,axiom,
% 0.30/0.60 ! [C: nat > nat > $o,P6: product_prod_nat_nat] :
% 0.30/0.60 ( ( produc6081775807080527818_nat_o @ C @ P6 )
% 0.30/0.60 => ~ ! [X5: nat,Y4: nat] :
% 0.30/0.60 ( ( P6
% 0.30/0.60 = ( product_Pair_nat_nat @ X5 @ Y4 ) )
% 0.30/0.60 => ~ ( C @ X5 @ Y4 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % case_prodE
% 0.30/0.60 thf(fact_5927_case__prodE,axiom,
% 0.30/0.60 ! [C: int > int > $o,P6: product_prod_int_int] :
% 0.30/0.60 ( ( produc4947309494688390418_int_o @ C @ P6 )
% 0.30/0.60 => ~ ! [X5: int,Y4: int] :
% 0.30/0.60 ( ( P6
% 0.30/0.60 = ( product_Pair_int_int @ X5 @ Y4 ) )
% 0.30/0.60 => ~ ( C @ X5 @ Y4 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % case_prodE
% 0.30/0.60 thf(fact_5928_case__prodD,axiom,
% 0.30/0.60 ! [F: code_integer > $o > $o,A: code_integer,B: $o] :
% 0.30/0.60 ( ( produc7828578312038201481er_o_o @ F @ ( produc6677183202524767010eger_o @ A @ B ) )
% 0.30/0.60 => ( F @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % case_prodD
% 0.30/0.60 thf(fact_5929_case__prodD,axiom,
% 0.30/0.60 ! [F: num > num > $o,A: num,B: num] :
% 0.30/0.60 ( ( produc5703948589228662326_num_o @ F @ ( product_Pair_num_num @ A @ B ) )
% 0.30/0.60 => ( F @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % case_prodD
% 0.30/0.60 thf(fact_5930_case__prodD,axiom,
% 0.30/0.60 ! [F: nat > num > $o,A: nat,B: num] :
% 0.30/0.60 ( ( produc4927758841916487424_num_o @ F @ ( product_Pair_nat_num @ A @ B ) )
% 0.30/0.60 => ( F @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % case_prodD
% 0.30/0.60 thf(fact_5931_case__prodD,axiom,
% 0.30/0.60 ! [F: nat > nat > $o,A: nat,B: nat] :
% 0.30/0.60 ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 0.30/0.60 => ( F @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % case_prodD
% 0.30/0.60 thf(fact_5932_case__prodD,axiom,
% 0.30/0.60 ! [F: int > int > $o,A: int,B: int] :
% 0.30/0.60 ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
% 0.30/0.60 => ( F @ A @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % case_prodD
% 0.30/0.60 thf(fact_5933_case__prodE_H,axiom,
% 0.30/0.60 ! [C: nat > nat > product_prod_nat_nat > $o,P6: product_prod_nat_nat,Z: product_prod_nat_nat] :
% 0.30/0.60 ( ( produc8739625826339149834_nat_o @ C @ P6 @ Z )
% 0.30/0.60 => ~ ! [X5: nat,Y4: nat] :
% 0.30/0.60 ( ( P6
% 0.30/0.60 = ( product_Pair_nat_nat @ X5 @ Y4 ) )
% 0.30/0.60 => ~ ( C @ X5 @ Y4 @ Z ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % case_prodE'
% 0.30/0.60 thf(fact_5934_case__prodD_H,axiom,
% 0.30/0.60 ! [R: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
% 0.30/0.60 ( ( produc8739625826339149834_nat_o @ R @ ( product_Pair_nat_nat @ A @ B ) @ C )
% 0.30/0.60 => ( R @ A @ B @ C ) ) ).
% 0.30/0.60
% 0.30/0.60 % case_prodD'
% 0.30/0.60 thf(fact_5935_neg__numeral__le__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_numeral
% 0.30/0.60 thf(fact_5936_neg__numeral__le__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_numeral
% 0.30/0.60 thf(fact_5937_neg__numeral__le__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_numeral
% 0.30/0.60 thf(fact_5938_neg__numeral__le__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_numeral
% 0.30/0.60 thf(fact_5939_not__numeral__le__neg__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_numeral_le_neg_numeral
% 0.30/0.60 thf(fact_5940_not__numeral__le__neg__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_numeral_le_neg_numeral
% 0.30/0.60 thf(fact_5941_not__numeral__le__neg__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_numeral_le_neg_numeral
% 0.30/0.60 thf(fact_5942_not__numeral__le__neg__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_numeral_le_neg_numeral
% 0.30/0.60 thf(fact_5943_zero__neq__neg__numeral,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( zero_zero_int
% 0.30/0.60 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % zero_neq_neg_numeral
% 0.30/0.60 thf(fact_5944_zero__neq__neg__numeral,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( zero_zero_real
% 0.30/0.60 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % zero_neq_neg_numeral
% 0.30/0.60 thf(fact_5945_zero__neq__neg__numeral,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( zero_zero_complex
% 0.30/0.60 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % zero_neq_neg_numeral
% 0.30/0.60 thf(fact_5946_zero__neq__neg__numeral,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( zero_z3403309356797280102nteger
% 0.30/0.60 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % zero_neq_neg_numeral
% 0.30/0.60 thf(fact_5947_zero__neq__neg__numeral,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( zero_zero_rat
% 0.30/0.60 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % zero_neq_neg_numeral
% 0.30/0.60 thf(fact_5948_not__numeral__less__neg__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_numeral_less_neg_numeral
% 0.30/0.60 thf(fact_5949_not__numeral__less__neg__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_numeral_less_neg_numeral
% 0.30/0.60 thf(fact_5950_not__numeral__less__neg__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_numeral_less_neg_numeral
% 0.30/0.60 thf(fact_5951_not__numeral__less__neg__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] :
% 0.30/0.60 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_numeral_less_neg_numeral
% 0.30/0.60 thf(fact_5952_neg__numeral__less__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_numeral
% 0.30/0.60 thf(fact_5953_neg__numeral__less__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_numeral
% 0.30/0.60 thf(fact_5954_neg__numeral__less__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_numeral
% 0.30/0.60 thf(fact_5955_neg__numeral__less__numeral,axiom,
% 0.30/0.60 ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_numeral
% 0.30/0.60 thf(fact_5956_le__minus__one__simps_I4_J,axiom,
% 0.30/0.60 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_minus_one_simps(4)
% 0.30/0.60 thf(fact_5957_le__minus__one__simps_I4_J,axiom,
% 0.30/0.60 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_minus_one_simps(4)
% 0.30/0.60 thf(fact_5958_le__minus__one__simps_I4_J,axiom,
% 0.30/0.60 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_minus_one_simps(4)
% 0.30/0.60 thf(fact_5959_le__minus__one__simps_I4_J,axiom,
% 0.30/0.60 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_minus_one_simps(4)
% 0.30/0.60 thf(fact_5960_le__minus__one__simps_I2_J,axiom,
% 0.30/0.60 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.30/0.60
% 0.30/0.60 % le_minus_one_simps(2)
% 0.30/0.60 thf(fact_5961_le__minus__one__simps_I2_J,axiom,
% 0.30/0.60 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.30/0.60
% 0.30/0.60 % le_minus_one_simps(2)
% 0.30/0.60 thf(fact_5962_le__minus__one__simps_I2_J,axiom,
% 0.30/0.60 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.30/0.60
% 0.30/0.60 % le_minus_one_simps(2)
% 0.30/0.60 thf(fact_5963_le__minus__one__simps_I2_J,axiom,
% 0.30/0.60 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.30/0.60
% 0.30/0.60 % le_minus_one_simps(2)
% 0.30/0.60 thf(fact_5964_zero__neq__neg__one,axiom,
% 0.30/0.60 ( zero_zero_int
% 0.30/0.60 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % zero_neq_neg_one
% 0.30/0.60 thf(fact_5965_zero__neq__neg__one,axiom,
% 0.30/0.60 ( zero_zero_real
% 0.30/0.60 != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % zero_neq_neg_one
% 0.30/0.60 thf(fact_5966_zero__neq__neg__one,axiom,
% 0.30/0.60 ( zero_zero_complex
% 0.30/0.60 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.60
% 0.30/0.60 % zero_neq_neg_one
% 0.30/0.60 thf(fact_5967_zero__neq__neg__one,axiom,
% 0.30/0.60 ( zero_z3403309356797280102nteger
% 0.30/0.60 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.60
% 0.30/0.60 % zero_neq_neg_one
% 0.30/0.60 thf(fact_5968_zero__neq__neg__one,axiom,
% 0.30/0.60 ( zero_zero_rat
% 0.30/0.60 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % zero_neq_neg_one
% 0.30/0.60 thf(fact_5969_neg__eq__iff__add__eq__0,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( ( uminus_uminus_int @ A )
% 0.30/0.60 = B )
% 0.30/0.60 = ( ( plus_plus_int @ A @ B )
% 0.30/0.60 = zero_zero_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_eq_iff_add_eq_0
% 0.30/0.60 thf(fact_5970_neg__eq__iff__add__eq__0,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( ( uminus_uminus_real @ A )
% 0.30/0.60 = B )
% 0.30/0.60 = ( ( plus_plus_real @ A @ B )
% 0.30/0.60 = zero_zero_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_eq_iff_add_eq_0
% 0.30/0.60 thf(fact_5971_neg__eq__iff__add__eq__0,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( ( uminus1482373934393186551omplex @ A )
% 0.30/0.60 = B )
% 0.30/0.60 = ( ( plus_plus_complex @ A @ B )
% 0.30/0.60 = zero_zero_complex ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_eq_iff_add_eq_0
% 0.30/0.60 thf(fact_5972_neg__eq__iff__add__eq__0,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( ( uminus1351360451143612070nteger @ A )
% 0.30/0.60 = B )
% 0.30/0.60 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.30/0.60 = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_eq_iff_add_eq_0
% 0.30/0.60 thf(fact_5973_neg__eq__iff__add__eq__0,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( ( uminus_uminus_rat @ A )
% 0.30/0.60 = B )
% 0.30/0.60 = ( ( plus_plus_rat @ A @ B )
% 0.30/0.60 = zero_zero_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_eq_iff_add_eq_0
% 0.30/0.60 thf(fact_5974_eq__neg__iff__add__eq__0,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( uminus_uminus_int @ B ) )
% 0.30/0.60 = ( ( plus_plus_int @ A @ B )
% 0.30/0.60 = zero_zero_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % eq_neg_iff_add_eq_0
% 0.30/0.60 thf(fact_5975_eq__neg__iff__add__eq__0,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( uminus_uminus_real @ B ) )
% 0.30/0.60 = ( ( plus_plus_real @ A @ B )
% 0.30/0.60 = zero_zero_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % eq_neg_iff_add_eq_0
% 0.30/0.60 thf(fact_5976_eq__neg__iff__add__eq__0,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( uminus1482373934393186551omplex @ B ) )
% 0.30/0.60 = ( ( plus_plus_complex @ A @ B )
% 0.30/0.60 = zero_zero_complex ) ) ).
% 0.30/0.60
% 0.30/0.60 % eq_neg_iff_add_eq_0
% 0.30/0.60 thf(fact_5977_eq__neg__iff__add__eq__0,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.60 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.30/0.60 = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60
% 0.30/0.60 % eq_neg_iff_add_eq_0
% 0.30/0.60 thf(fact_5978_eq__neg__iff__add__eq__0,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( uminus_uminus_rat @ B ) )
% 0.30/0.60 = ( ( plus_plus_rat @ A @ B )
% 0.30/0.60 = zero_zero_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % eq_neg_iff_add_eq_0
% 0.30/0.60 thf(fact_5979_add_Oinverse__unique,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( ( plus_plus_int @ A @ B )
% 0.30/0.60 = zero_zero_int )
% 0.30/0.60 => ( ( uminus_uminus_int @ A )
% 0.30/0.60 = B ) ) ).
% 0.30/0.60
% 0.30/0.60 % add.inverse_unique
% 0.30/0.60 thf(fact_5980_add_Oinverse__unique,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( ( plus_plus_real @ A @ B )
% 0.30/0.60 = zero_zero_real )
% 0.30/0.60 => ( ( uminus_uminus_real @ A )
% 0.30/0.60 = B ) ) ).
% 0.30/0.60
% 0.30/0.60 % add.inverse_unique
% 0.30/0.60 thf(fact_5981_add_Oinverse__unique,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( ( plus_plus_complex @ A @ B )
% 0.30/0.60 = zero_zero_complex )
% 0.30/0.60 => ( ( uminus1482373934393186551omplex @ A )
% 0.30/0.60 = B ) ) ).
% 0.30/0.60
% 0.30/0.60 % add.inverse_unique
% 0.30/0.60 thf(fact_5982_add_Oinverse__unique,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.30/0.60 = zero_z3403309356797280102nteger )
% 0.30/0.60 => ( ( uminus1351360451143612070nteger @ A )
% 0.30/0.60 = B ) ) ).
% 0.30/0.60
% 0.30/0.60 % add.inverse_unique
% 0.30/0.60 thf(fact_5983_add_Oinverse__unique,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( ( plus_plus_rat @ A @ B )
% 0.30/0.60 = zero_zero_rat )
% 0.30/0.60 => ( ( uminus_uminus_rat @ A )
% 0.30/0.60 = B ) ) ).
% 0.30/0.60
% 0.30/0.60 % add.inverse_unique
% 0.30/0.60 thf(fact_5984_ab__group__add__class_Oab__left__minus,axiom,
% 0.30/0.60 ! [A: int] :
% 0.30/0.60 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.30/0.60 = zero_zero_int ) ).
% 0.30/0.60
% 0.30/0.60 % ab_group_add_class.ab_left_minus
% 0.30/0.60 thf(fact_5985_ab__group__add__class_Oab__left__minus,axiom,
% 0.30/0.60 ! [A: real] :
% 0.30/0.60 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.30/0.60 = zero_zero_real ) ).
% 0.30/0.60
% 0.30/0.60 % ab_group_add_class.ab_left_minus
% 0.30/0.60 thf(fact_5986_ab__group__add__class_Oab__left__minus,axiom,
% 0.30/0.60 ! [A: complex] :
% 0.30/0.60 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.30/0.60 = zero_zero_complex ) ).
% 0.30/0.60
% 0.30/0.60 % ab_group_add_class.ab_left_minus
% 0.30/0.60 thf(fact_5987_ab__group__add__class_Oab__left__minus,axiom,
% 0.30/0.60 ! [A: code_integer] :
% 0.30/0.60 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.30/0.60 = zero_z3403309356797280102nteger ) ).
% 0.30/0.60
% 0.30/0.60 % ab_group_add_class.ab_left_minus
% 0.30/0.60 thf(fact_5988_ab__group__add__class_Oab__left__minus,axiom,
% 0.30/0.60 ! [A: rat] :
% 0.30/0.60 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.30/0.60 = zero_zero_rat ) ).
% 0.30/0.60
% 0.30/0.60 % ab_group_add_class.ab_left_minus
% 0.30/0.60 thf(fact_5989_add__eq__0__iff,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( ( plus_plus_int @ A @ B )
% 0.30/0.60 = zero_zero_int )
% 0.30/0.60 = ( B
% 0.30/0.60 = ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add_eq_0_iff
% 0.30/0.60 thf(fact_5990_add__eq__0__iff,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( ( plus_plus_real @ A @ B )
% 0.30/0.60 = zero_zero_real )
% 0.30/0.60 = ( B
% 0.30/0.60 = ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add_eq_0_iff
% 0.30/0.60 thf(fact_5991_add__eq__0__iff,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( ( plus_plus_complex @ A @ B )
% 0.30/0.60 = zero_zero_complex )
% 0.30/0.60 = ( B
% 0.30/0.60 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add_eq_0_iff
% 0.30/0.60 thf(fact_5992_add__eq__0__iff,axiom,
% 0.30/0.60 ! [A: code_integer,B: code_integer] :
% 0.30/0.60 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.30/0.60 = zero_z3403309356797280102nteger )
% 0.30/0.60 = ( B
% 0.30/0.60 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add_eq_0_iff
% 0.30/0.60 thf(fact_5993_add__eq__0__iff,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( ( plus_plus_rat @ A @ B )
% 0.30/0.60 = zero_zero_rat )
% 0.30/0.60 = ( B
% 0.30/0.60 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add_eq_0_iff
% 0.30/0.60 thf(fact_5994_less__minus__one__simps_I4_J,axiom,
% 0.30/0.60 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_minus_one_simps(4)
% 0.30/0.60 thf(fact_5995_less__minus__one__simps_I4_J,axiom,
% 0.30/0.60 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_minus_one_simps(4)
% 0.30/0.60 thf(fact_5996_less__minus__one__simps_I4_J,axiom,
% 0.30/0.60 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_minus_one_simps(4)
% 0.30/0.60 thf(fact_5997_less__minus__one__simps_I4_J,axiom,
% 0.30/0.60 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_minus_one_simps(4)
% 0.30/0.60 thf(fact_5998_less__minus__one__simps_I2_J,axiom,
% 0.30/0.60 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.30/0.60
% 0.30/0.60 % less_minus_one_simps(2)
% 0.30/0.60 thf(fact_5999_less__minus__one__simps_I2_J,axiom,
% 0.30/0.60 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.30/0.60
% 0.30/0.60 % less_minus_one_simps(2)
% 0.30/0.60 thf(fact_6000_less__minus__one__simps_I2_J,axiom,
% 0.30/0.60 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.30/0.60
% 0.30/0.60 % less_minus_one_simps(2)
% 0.30/0.60 thf(fact_6001_less__minus__one__simps_I2_J,axiom,
% 0.30/0.60 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.30/0.60
% 0.30/0.60 % less_minus_one_simps(2)
% 0.30/0.60 thf(fact_6002_numeral__times__minus__swap,axiom,
% 0.30/0.60 ! [W: num,X2: int] :
% 0.30/0.60 ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X2 ) )
% 0.30/0.60 = ( times_times_int @ X2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_times_minus_swap
% 0.30/0.60 thf(fact_6003_numeral__times__minus__swap,axiom,
% 0.30/0.60 ! [W: num,X2: real] :
% 0.30/0.60 ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X2 ) )
% 0.30/0.60 = ( times_times_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_times_minus_swap
% 0.30/0.60 thf(fact_6004_numeral__times__minus__swap,axiom,
% 0.30/0.60 ! [W: num,X2: complex] :
% 0.30/0.60 ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X2 ) )
% 0.30/0.60 = ( times_times_complex @ X2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_times_minus_swap
% 0.30/0.60 thf(fact_6005_numeral__times__minus__swap,axiom,
% 0.30/0.60 ! [W: num,X2: code_integer] :
% 0.30/0.60 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X2 ) )
% 0.30/0.60 = ( times_3573771949741848930nteger @ X2 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_times_minus_swap
% 0.30/0.60 thf(fact_6006_numeral__times__minus__swap,axiom,
% 0.30/0.60 ! [W: num,X2: rat] :
% 0.30/0.60 ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X2 ) )
% 0.30/0.60 = ( times_times_rat @ X2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_times_minus_swap
% 0.30/0.60 thf(fact_6007_nonzero__minus__divide__divide,axiom,
% 0.30/0.60 ! [B: real,A: real] :
% 0.30/0.60 ( ( B != zero_zero_real )
% 0.30/0.60 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.30/0.60 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % nonzero_minus_divide_divide
% 0.30/0.60 thf(fact_6008_nonzero__minus__divide__divide,axiom,
% 0.30/0.60 ! [B: complex,A: complex] :
% 0.30/0.60 ( ( B != zero_zero_complex )
% 0.30/0.60 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.30/0.60 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % nonzero_minus_divide_divide
% 0.30/0.60 thf(fact_6009_nonzero__minus__divide__divide,axiom,
% 0.30/0.60 ! [B: rat,A: rat] :
% 0.30/0.60 ( ( B != zero_zero_rat )
% 0.30/0.60 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.30/0.60 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % nonzero_minus_divide_divide
% 0.30/0.60 thf(fact_6010_nonzero__minus__divide__right,axiom,
% 0.30/0.60 ! [B: real,A: real] :
% 0.30/0.60 ( ( B != zero_zero_real )
% 0.30/0.60 => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.60 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % nonzero_minus_divide_right
% 0.30/0.60 thf(fact_6011_nonzero__minus__divide__right,axiom,
% 0.30/0.60 ! [B: complex,A: complex] :
% 0.30/0.60 ( ( B != zero_zero_complex )
% 0.30/0.60 => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.30/0.60 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % nonzero_minus_divide_right
% 0.30/0.60 thf(fact_6012_nonzero__minus__divide__right,axiom,
% 0.30/0.60 ! [B: rat,A: rat] :
% 0.30/0.60 ( ( B != zero_zero_rat )
% 0.30/0.60 => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.30/0.60 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % nonzero_minus_divide_right
% 0.30/0.60 thf(fact_6013_numeral__neq__neg__one,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( numeral_numeral_int @ N )
% 0.30/0.60 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_neq_neg_one
% 0.30/0.60 thf(fact_6014_numeral__neq__neg__one,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( numeral_numeral_real @ N )
% 0.30/0.60 != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_neq_neg_one
% 0.30/0.60 thf(fact_6015_numeral__neq__neg__one,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( numera6690914467698888265omplex @ N )
% 0.30/0.60 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_neq_neg_one
% 0.30/0.60 thf(fact_6016_numeral__neq__neg__one,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( numera6620942414471956472nteger @ N )
% 0.30/0.60 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_neq_neg_one
% 0.30/0.60 thf(fact_6017_numeral__neq__neg__one,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( ( numeral_numeral_rat @ N )
% 0.30/0.60 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_neq_neg_one
% 0.30/0.60 thf(fact_6018_one__neq__neg__numeral,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( one_one_int
% 0.30/0.60 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % one_neq_neg_numeral
% 0.30/0.60 thf(fact_6019_one__neq__neg__numeral,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( one_one_real
% 0.30/0.60 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % one_neq_neg_numeral
% 0.30/0.60 thf(fact_6020_one__neq__neg__numeral,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( one_one_complex
% 0.30/0.60 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % one_neq_neg_numeral
% 0.30/0.60 thf(fact_6021_one__neq__neg__numeral,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( one_one_Code_integer
% 0.30/0.60 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % one_neq_neg_numeral
% 0.30/0.60 thf(fact_6022_one__neq__neg__numeral,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ( one_one_rat
% 0.30/0.60 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % one_neq_neg_numeral
% 0.30/0.60 thf(fact_6023_square__eq__1__iff,axiom,
% 0.30/0.60 ! [X2: int] :
% 0.30/0.60 ( ( ( times_times_int @ X2 @ X2 )
% 0.30/0.60 = one_one_int )
% 0.30/0.60 = ( ( X2 = one_one_int )
% 0.30/0.60 | ( X2
% 0.30/0.60 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % square_eq_1_iff
% 0.30/0.60 thf(fact_6024_square__eq__1__iff,axiom,
% 0.30/0.60 ! [X2: real] :
% 0.30/0.60 ( ( ( times_times_real @ X2 @ X2 )
% 0.30/0.60 = one_one_real )
% 0.30/0.60 = ( ( X2 = one_one_real )
% 0.30/0.60 | ( X2
% 0.30/0.60 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % square_eq_1_iff
% 0.30/0.60 thf(fact_6025_square__eq__1__iff,axiom,
% 0.30/0.60 ! [X2: complex] :
% 0.30/0.60 ( ( ( times_times_complex @ X2 @ X2 )
% 0.30/0.60 = one_one_complex )
% 0.30/0.60 = ( ( X2 = one_one_complex )
% 0.30/0.60 | ( X2
% 0.30/0.60 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % square_eq_1_iff
% 0.30/0.60 thf(fact_6026_square__eq__1__iff,axiom,
% 0.30/0.60 ! [X2: code_integer] :
% 0.30/0.60 ( ( ( times_3573771949741848930nteger @ X2 @ X2 )
% 0.30/0.60 = one_one_Code_integer )
% 0.30/0.60 = ( ( X2 = one_one_Code_integer )
% 0.30/0.60 | ( X2
% 0.30/0.60 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % square_eq_1_iff
% 0.30/0.60 thf(fact_6027_square__eq__1__iff,axiom,
% 0.30/0.60 ! [X2: rat] :
% 0.30/0.60 ( ( ( times_times_rat @ X2 @ X2 )
% 0.30/0.60 = one_one_rat )
% 0.30/0.60 = ( ( X2 = one_one_rat )
% 0.30/0.60 | ( X2
% 0.30/0.60 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % square_eq_1_iff
% 0.30/0.60 thf(fact_6028_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.30/0.60 ( minus_minus_int
% 0.30/0.60 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.30/0.60 thf(fact_6029_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.30/0.60 ( minus_minus_real
% 0.30/0.60 = ( ^ [A3: real,B2: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.30/0.60 thf(fact_6030_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.30/0.60 ( minus_minus_complex
% 0.30/0.60 = ( ^ [A3: complex,B2: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.30/0.60 thf(fact_6031_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.30/0.60 ( minus_8373710615458151222nteger
% 0.30/0.60 = ( ^ [A3: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.30/0.60 thf(fact_6032_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.30/0.60 ( minus_minus_rat
% 0.30/0.60 = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.30/0.60 thf(fact_6033_diff__conv__add__uminus,axiom,
% 0.30/0.60 ( minus_minus_int
% 0.30/0.60 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_conv_add_uminus
% 0.30/0.60 thf(fact_6034_diff__conv__add__uminus,axiom,
% 0.30/0.60 ( minus_minus_real
% 0.30/0.60 = ( ^ [A3: real,B2: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_conv_add_uminus
% 0.30/0.60 thf(fact_6035_diff__conv__add__uminus,axiom,
% 0.30/0.60 ( minus_minus_complex
% 0.30/0.60 = ( ^ [A3: complex,B2: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_conv_add_uminus
% 0.30/0.60 thf(fact_6036_diff__conv__add__uminus,axiom,
% 0.30/0.60 ( minus_8373710615458151222nteger
% 0.30/0.60 = ( ^ [A3: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_conv_add_uminus
% 0.30/0.60 thf(fact_6037_diff__conv__add__uminus,axiom,
% 0.30/0.60 ( minus_minus_rat
% 0.30/0.60 = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % diff_conv_add_uminus
% 0.30/0.60 thf(fact_6038_group__cancel_Osub2,axiom,
% 0.30/0.60 ! [B3: int,K: int,B: int,A: int] :
% 0.30/0.60 ( ( B3
% 0.30/0.60 = ( plus_plus_int @ K @ B ) )
% 0.30/0.60 => ( ( minus_minus_int @ A @ B3 )
% 0.30/0.60 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % group_cancel.sub2
% 0.30/0.60 thf(fact_6039_group__cancel_Osub2,axiom,
% 0.30/0.60 ! [B3: real,K: real,B: real,A: real] :
% 0.30/0.60 ( ( B3
% 0.30/0.60 = ( plus_plus_real @ K @ B ) )
% 0.30/0.60 => ( ( minus_minus_real @ A @ B3 )
% 0.30/0.60 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % group_cancel.sub2
% 0.30/0.60 thf(fact_6040_group__cancel_Osub2,axiom,
% 0.30/0.60 ! [B3: complex,K: complex,B: complex,A: complex] :
% 0.30/0.60 ( ( B3
% 0.30/0.60 = ( plus_plus_complex @ K @ B ) )
% 0.30/0.60 => ( ( minus_minus_complex @ A @ B3 )
% 0.30/0.60 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % group_cancel.sub2
% 0.30/0.60 thf(fact_6041_group__cancel_Osub2,axiom,
% 0.30/0.60 ! [B3: code_integer,K: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.60 ( ( B3
% 0.30/0.60 = ( plus_p5714425477246183910nteger @ K @ B ) )
% 0.30/0.60 => ( ( minus_8373710615458151222nteger @ A @ B3 )
% 0.30/0.60 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % group_cancel.sub2
% 0.30/0.60 thf(fact_6042_group__cancel_Osub2,axiom,
% 0.30/0.60 ! [B3: rat,K: rat,B: rat,A: rat] :
% 0.30/0.60 ( ( B3
% 0.30/0.60 = ( plus_plus_rat @ K @ B ) )
% 0.30/0.60 => ( ( minus_minus_rat @ A @ B3 )
% 0.30/0.60 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % group_cancel.sub2
% 0.30/0.60 thf(fact_6043_dvd__neg__div,axiom,
% 0.30/0.60 ! [B: int,A: int] :
% 0.30/0.60 ( ( dvd_dvd_int @ B @ A )
% 0.30/0.60 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.60 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dvd_neg_div
% 0.30/0.60 thf(fact_6044_dvd__neg__div,axiom,
% 0.30/0.60 ! [B: real,A: real] :
% 0.30/0.60 ( ( dvd_dvd_real @ B @ A )
% 0.30/0.60 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
% 0.30/0.60 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dvd_neg_div
% 0.30/0.60 thf(fact_6045_dvd__neg__div,axiom,
% 0.30/0.60 ! [B: complex,A: complex] :
% 0.30/0.60 ( ( dvd_dvd_complex @ B @ A )
% 0.30/0.60 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dvd_neg_div
% 0.30/0.60 thf(fact_6046_dvd__neg__div,axiom,
% 0.30/0.60 ! [B: code_integer,A: code_integer] :
% 0.30/0.60 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.30/0.60 => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dvd_neg_div
% 0.30/0.60 thf(fact_6047_dvd__neg__div,axiom,
% 0.30/0.60 ! [B: rat,A: rat] :
% 0.30/0.60 ( ( dvd_dvd_rat @ B @ A )
% 0.30/0.60 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.30/0.60 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dvd_neg_div
% 0.30/0.60 thf(fact_6048_dvd__div__neg,axiom,
% 0.30/0.60 ! [B: int,A: int] :
% 0.30/0.60 ( ( dvd_dvd_int @ B @ A )
% 0.30/0.60 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.30/0.60 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dvd_div_neg
% 0.30/0.60 thf(fact_6049_dvd__div__neg,axiom,
% 0.30/0.60 ! [B: real,A: real] :
% 0.30/0.60 ( ( dvd_dvd_real @ B @ A )
% 0.30/0.60 => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
% 0.30/0.60 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dvd_div_neg
% 0.30/0.60 thf(fact_6050_dvd__div__neg,axiom,
% 0.30/0.60 ! [B: complex,A: complex] :
% 0.30/0.60 ( ( dvd_dvd_complex @ B @ A )
% 0.30/0.60 => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dvd_div_neg
% 0.30/0.60 thf(fact_6051_dvd__div__neg,axiom,
% 0.30/0.60 ! [B: code_integer,A: code_integer] :
% 0.30/0.60 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.30/0.60 => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dvd_div_neg
% 0.30/0.60 thf(fact_6052_dvd__div__neg,axiom,
% 0.30/0.60 ! [B: rat,A: rat] :
% 0.30/0.60 ( ( dvd_dvd_rat @ B @ A )
% 0.30/0.60 => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.30/0.60 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % dvd_div_neg
% 0.30/0.60 thf(fact_6053_subset__Compl__self__eq,axiom,
% 0.30/0.60 ! [A2: set_nat] :
% 0.30/0.60 ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
% 0.30/0.60 = ( A2 = bot_bot_set_nat ) ) ).
% 0.30/0.60
% 0.30/0.60 % subset_Compl_self_eq
% 0.30/0.60 thf(fact_6054_subset__Compl__self__eq,axiom,
% 0.30/0.60 ! [A2: set_real] :
% 0.30/0.60 ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
% 0.30/0.60 = ( A2 = bot_bot_set_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % subset_Compl_self_eq
% 0.30/0.60 thf(fact_6055_subset__Compl__self__eq,axiom,
% 0.30/0.60 ! [A2: set_int] :
% 0.30/0.60 ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
% 0.30/0.60 = ( A2 = bot_bot_set_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % subset_Compl_self_eq
% 0.30/0.60 thf(fact_6056_real__minus__mult__self__le,axiom,
% 0.30/0.60 ! [U: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X2 @ X2 ) ) ).
% 0.30/0.60
% 0.30/0.60 % real_minus_mult_self_le
% 0.30/0.60 thf(fact_6057_zmult__eq__1__iff,axiom,
% 0.30/0.60 ! [M: int,N: int] :
% 0.30/0.60 ( ( ( times_times_int @ M @ N )
% 0.30/0.60 = one_one_int )
% 0.30/0.60 = ( ( ( M = one_one_int )
% 0.30/0.60 & ( N = one_one_int ) )
% 0.30/0.60 | ( ( M
% 0.30/0.60 = ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60 & ( N
% 0.30/0.60 = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % zmult_eq_1_iff
% 0.30/0.60 thf(fact_6058_pos__zmult__eq__1__iff__lemma,axiom,
% 0.30/0.60 ! [M: int,N: int] :
% 0.30/0.60 ( ( ( times_times_int @ M @ N )
% 0.30/0.60 = one_one_int )
% 0.30/0.60 => ( ( M = one_one_int )
% 0.30/0.60 | ( M
% 0.30/0.60 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % pos_zmult_eq_1_iff_lemma
% 0.30/0.60 thf(fact_6059_minus__int__code_I2_J,axiom,
% 0.30/0.60 ! [L: int] :
% 0.30/0.60 ( ( minus_minus_int @ zero_zero_int @ L )
% 0.30/0.60 = ( uminus_uminus_int @ L ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_int_code(2)
% 0.30/0.60 thf(fact_6060_numeral__eq__Suc,axiom,
% 0.30/0.60 ( numeral_numeral_nat
% 0.30/0.60 = ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % numeral_eq_Suc
% 0.30/0.60 thf(fact_6061_minus__real__def,axiom,
% 0.30/0.60 ( minus_minus_real
% 0.30/0.60 = ( ^ [X: real,Y: real] : ( plus_plus_real @ X @ ( uminus_uminus_real @ Y ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_real_def
% 0.30/0.60 thf(fact_6062_not__zero__le__neg__numeral,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_zero_le_neg_numeral
% 0.30/0.60 thf(fact_6063_not__zero__le__neg__numeral,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_zero_le_neg_numeral
% 0.30/0.60 thf(fact_6064_not__zero__le__neg__numeral,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_zero_le_neg_numeral
% 0.30/0.60 thf(fact_6065_not__zero__le__neg__numeral,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_zero_le_neg_numeral
% 0.30/0.60 thf(fact_6066_neg__numeral__le__zero,axiom,
% 0.30/0.60 ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_zero
% 0.30/0.60 thf(fact_6067_neg__numeral__le__zero,axiom,
% 0.30/0.60 ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_zero
% 0.30/0.60 thf(fact_6068_neg__numeral__le__zero,axiom,
% 0.30/0.60 ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_zero
% 0.30/0.60 thf(fact_6069_neg__numeral__le__zero,axiom,
% 0.30/0.60 ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_zero
% 0.30/0.60 thf(fact_6070_not__zero__less__neg__numeral,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_zero_less_neg_numeral
% 0.30/0.60 thf(fact_6071_not__zero__less__neg__numeral,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_zero_less_neg_numeral
% 0.30/0.60 thf(fact_6072_not__zero__less__neg__numeral,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_zero_less_neg_numeral
% 0.30/0.60 thf(fact_6073_not__zero__less__neg__numeral,axiom,
% 0.30/0.60 ! [N: num] :
% 0.30/0.60 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_zero_less_neg_numeral
% 0.30/0.60 thf(fact_6074_neg__numeral__less__zero,axiom,
% 0.30/0.60 ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_zero
% 0.30/0.60 thf(fact_6075_neg__numeral__less__zero,axiom,
% 0.30/0.60 ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_zero
% 0.30/0.60 thf(fact_6076_neg__numeral__less__zero,axiom,
% 0.30/0.60 ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_zero
% 0.30/0.60 thf(fact_6077_neg__numeral__less__zero,axiom,
% 0.30/0.60 ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_zero
% 0.30/0.60 thf(fact_6078_le__minus__one__simps_I3_J,axiom,
% 0.30/0.60 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_minus_one_simps(3)
% 0.30/0.60 thf(fact_6079_le__minus__one__simps_I3_J,axiom,
% 0.30/0.60 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_minus_one_simps(3)
% 0.30/0.60 thf(fact_6080_le__minus__one__simps_I3_J,axiom,
% 0.30/0.60 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_minus_one_simps(3)
% 0.30/0.60 thf(fact_6081_le__minus__one__simps_I3_J,axiom,
% 0.30/0.60 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % le_minus_one_simps(3)
% 0.30/0.60 thf(fact_6082_le__minus__one__simps_I1_J,axiom,
% 0.30/0.60 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.30/0.60
% 0.30/0.60 % le_minus_one_simps(1)
% 0.30/0.60 thf(fact_6083_le__minus__one__simps_I1_J,axiom,
% 0.30/0.60 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.30/0.60
% 0.30/0.60 % le_minus_one_simps(1)
% 0.30/0.60 thf(fact_6084_le__minus__one__simps_I1_J,axiom,
% 0.30/0.60 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.30/0.60
% 0.30/0.60 % le_minus_one_simps(1)
% 0.30/0.60 thf(fact_6085_le__minus__one__simps_I1_J,axiom,
% 0.30/0.60 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.30/0.60
% 0.30/0.60 % le_minus_one_simps(1)
% 0.30/0.60 thf(fact_6086_less__minus__one__simps_I1_J,axiom,
% 0.30/0.60 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.30/0.60
% 0.30/0.60 % less_minus_one_simps(1)
% 0.30/0.60 thf(fact_6087_less__minus__one__simps_I1_J,axiom,
% 0.30/0.60 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.30/0.60
% 0.30/0.60 % less_minus_one_simps(1)
% 0.30/0.60 thf(fact_6088_less__minus__one__simps_I1_J,axiom,
% 0.30/0.60 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.30/0.60
% 0.30/0.60 % less_minus_one_simps(1)
% 0.30/0.60 thf(fact_6089_less__minus__one__simps_I1_J,axiom,
% 0.30/0.60 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.30/0.60
% 0.30/0.60 % less_minus_one_simps(1)
% 0.30/0.60 thf(fact_6090_less__minus__one__simps_I3_J,axiom,
% 0.30/0.60 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_minus_one_simps(3)
% 0.30/0.60 thf(fact_6091_less__minus__one__simps_I3_J,axiom,
% 0.30/0.60 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_minus_one_simps(3)
% 0.30/0.60 thf(fact_6092_less__minus__one__simps_I3_J,axiom,
% 0.30/0.60 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_minus_one_simps(3)
% 0.30/0.60 thf(fact_6093_less__minus__one__simps_I3_J,axiom,
% 0.30/0.60 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_minus_one_simps(3)
% 0.30/0.60 thf(fact_6094_neg__numeral__le__one,axiom,
% 0.30/0.60 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_one
% 0.30/0.60 thf(fact_6095_neg__numeral__le__one,axiom,
% 0.30/0.60 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_one
% 0.30/0.60 thf(fact_6096_neg__numeral__le__one,axiom,
% 0.30/0.60 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_one
% 0.30/0.60 thf(fact_6097_neg__numeral__le__one,axiom,
% 0.30/0.60 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_one
% 0.30/0.60 thf(fact_6098_neg__one__le__numeral,axiom,
% 0.30/0.60 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_le_numeral
% 0.30/0.60 thf(fact_6099_neg__one__le__numeral,axiom,
% 0.30/0.60 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_le_numeral
% 0.30/0.60 thf(fact_6100_neg__one__le__numeral,axiom,
% 0.30/0.60 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_le_numeral
% 0.30/0.60 thf(fact_6101_neg__one__le__numeral,axiom,
% 0.30/0.60 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_le_numeral
% 0.30/0.60 thf(fact_6102_neg__numeral__le__neg__one,axiom,
% 0.30/0.60 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_neg_one
% 0.30/0.60 thf(fact_6103_neg__numeral__le__neg__one,axiom,
% 0.30/0.60 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_neg_one
% 0.30/0.60 thf(fact_6104_neg__numeral__le__neg__one,axiom,
% 0.30/0.60 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_neg_one
% 0.30/0.60 thf(fact_6105_neg__numeral__le__neg__one,axiom,
% 0.30/0.60 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_le_neg_one
% 0.30/0.60 thf(fact_6106_not__numeral__le__neg__one,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_numeral_le_neg_one
% 0.30/0.60 thf(fact_6107_not__numeral__le__neg__one,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_numeral_le_neg_one
% 0.30/0.60 thf(fact_6108_not__numeral__le__neg__one,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_numeral_le_neg_one
% 0.30/0.60 thf(fact_6109_not__numeral__le__neg__one,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_numeral_le_neg_one
% 0.30/0.60 thf(fact_6110_not__one__le__neg__numeral,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_one_le_neg_numeral
% 0.30/0.60 thf(fact_6111_not__one__le__neg__numeral,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_one_le_neg_numeral
% 0.30/0.60 thf(fact_6112_not__one__le__neg__numeral,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_one_le_neg_numeral
% 0.30/0.60 thf(fact_6113_not__one__le__neg__numeral,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_one_le_neg_numeral
% 0.30/0.60 thf(fact_6114_not__neg__one__less__neg__numeral,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_neg_one_less_neg_numeral
% 0.30/0.60 thf(fact_6115_not__neg__one__less__neg__numeral,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_neg_one_less_neg_numeral
% 0.30/0.60 thf(fact_6116_not__neg__one__less__neg__numeral,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_neg_one_less_neg_numeral
% 0.30/0.60 thf(fact_6117_not__neg__one__less__neg__numeral,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_neg_one_less_neg_numeral
% 0.30/0.60 thf(fact_6118_not__one__less__neg__numeral,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_one_less_neg_numeral
% 0.30/0.60 thf(fact_6119_not__one__less__neg__numeral,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_one_less_neg_numeral
% 0.30/0.60 thf(fact_6120_not__one__less__neg__numeral,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_one_less_neg_numeral
% 0.30/0.60 thf(fact_6121_not__one__less__neg__numeral,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_one_less_neg_numeral
% 0.30/0.60 thf(fact_6122_not__numeral__less__neg__one,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_numeral_less_neg_one
% 0.30/0.60 thf(fact_6123_not__numeral__less__neg__one,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_numeral_less_neg_one
% 0.30/0.60 thf(fact_6124_not__numeral__less__neg__one,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_numeral_less_neg_one
% 0.30/0.60 thf(fact_6125_not__numeral__less__neg__one,axiom,
% 0.30/0.60 ! [M: num] :
% 0.30/0.60 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % not_numeral_less_neg_one
% 0.30/0.60 thf(fact_6126_neg__one__less__numeral,axiom,
% 0.30/0.60 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_less_numeral
% 0.30/0.60 thf(fact_6127_neg__one__less__numeral,axiom,
% 0.30/0.60 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_less_numeral
% 0.30/0.60 thf(fact_6128_neg__one__less__numeral,axiom,
% 0.30/0.60 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_less_numeral
% 0.30/0.60 thf(fact_6129_neg__one__less__numeral,axiom,
% 0.30/0.60 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_one_less_numeral
% 0.30/0.60 thf(fact_6130_neg__numeral__less__one,axiom,
% 0.30/0.60 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_one
% 0.30/0.60 thf(fact_6131_neg__numeral__less__one,axiom,
% 0.30/0.60 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_one
% 0.30/0.60 thf(fact_6132_neg__numeral__less__one,axiom,
% 0.30/0.60 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_one
% 0.30/0.60 thf(fact_6133_neg__numeral__less__one,axiom,
% 0.30/0.60 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.30/0.60
% 0.30/0.60 % neg_numeral_less_one
% 0.30/0.60 thf(fact_6134_nonzero__neg__divide__eq__eq2,axiom,
% 0.30/0.60 ! [B: real,C: real,A: real] :
% 0.30/0.60 ( ( B != zero_zero_real )
% 0.30/0.60 => ( ( C
% 0.30/0.60 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
% 0.30/0.60 = ( ( times_times_real @ C @ B )
% 0.30/0.60 = ( uminus_uminus_real @ A ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % nonzero_neg_divide_eq_eq2
% 0.30/0.60 thf(fact_6135_nonzero__neg__divide__eq__eq2,axiom,
% 0.30/0.60 ! [B: complex,C: complex,A: complex] :
% 0.30/0.60 ( ( B != zero_zero_complex )
% 0.30/0.60 => ( ( C
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.30/0.60 = ( ( times_times_complex @ C @ B )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % nonzero_neg_divide_eq_eq2
% 0.30/0.60 thf(fact_6136_nonzero__neg__divide__eq__eq2,axiom,
% 0.30/0.60 ! [B: rat,C: rat,A: rat] :
% 0.30/0.60 ( ( B != zero_zero_rat )
% 0.30/0.60 => ( ( C
% 0.30/0.60 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
% 0.30/0.60 = ( ( times_times_rat @ C @ B )
% 0.30/0.60 = ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % nonzero_neg_divide_eq_eq2
% 0.30/0.60 thf(fact_6137_nonzero__neg__divide__eq__eq,axiom,
% 0.30/0.60 ! [B: real,A: real,C: real] :
% 0.30/0.60 ( ( B != zero_zero_real )
% 0.30/0.60 => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.60 = C )
% 0.30/0.60 = ( ( uminus_uminus_real @ A )
% 0.30/0.60 = ( times_times_real @ C @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % nonzero_neg_divide_eq_eq
% 0.30/0.60 thf(fact_6138_nonzero__neg__divide__eq__eq,axiom,
% 0.30/0.60 ! [B: complex,A: complex,C: complex] :
% 0.30/0.60 ( ( B != zero_zero_complex )
% 0.30/0.60 => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.30/0.60 = C )
% 0.30/0.60 = ( ( uminus1482373934393186551omplex @ A )
% 0.30/0.60 = ( times_times_complex @ C @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % nonzero_neg_divide_eq_eq
% 0.30/0.60 thf(fact_6139_nonzero__neg__divide__eq__eq,axiom,
% 0.30/0.60 ! [B: rat,A: rat,C: rat] :
% 0.30/0.60 ( ( B != zero_zero_rat )
% 0.30/0.60 => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.30/0.60 = C )
% 0.30/0.60 = ( ( uminus_uminus_rat @ A )
% 0.30/0.60 = ( times_times_rat @ C @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % nonzero_neg_divide_eq_eq
% 0.30/0.60 thf(fact_6140_minus__divide__eq__eq,axiom,
% 0.30/0.60 ! [B: real,C: real,A: real] :
% 0.30/0.60 ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
% 0.30/0.60 = A )
% 0.30/0.60 = ( ( ( C != zero_zero_real )
% 0.30/0.60 => ( ( uminus_uminus_real @ B )
% 0.30/0.60 = ( times_times_real @ A @ C ) ) )
% 0.30/0.60 & ( ( C = zero_zero_real )
% 0.30/0.60 => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_eq_eq
% 0.30/0.60 thf(fact_6141_minus__divide__eq__eq,axiom,
% 0.30/0.60 ! [B: complex,C: complex,A: complex] :
% 0.30/0.60 ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.30/0.60 = A )
% 0.30/0.60 = ( ( ( C != zero_zero_complex )
% 0.30/0.60 => ( ( uminus1482373934393186551omplex @ B )
% 0.30/0.60 = ( times_times_complex @ A @ C ) ) )
% 0.30/0.60 & ( ( C = zero_zero_complex )
% 0.30/0.60 => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_eq_eq
% 0.30/0.60 thf(fact_6142_minus__divide__eq__eq,axiom,
% 0.30/0.60 ! [B: rat,C: rat,A: rat] :
% 0.30/0.60 ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.60 = A )
% 0.30/0.60 = ( ( ( C != zero_zero_rat )
% 0.30/0.60 => ( ( uminus_uminus_rat @ B )
% 0.30/0.60 = ( times_times_rat @ A @ C ) ) )
% 0.30/0.60 & ( ( C = zero_zero_rat )
% 0.30/0.60 => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_eq_eq
% 0.30/0.60 thf(fact_6143_eq__minus__divide__eq,axiom,
% 0.30/0.60 ! [A: real,B: real,C: real] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.30/0.60 = ( ( ( C != zero_zero_real )
% 0.30/0.60 => ( ( times_times_real @ A @ C )
% 0.30/0.60 = ( uminus_uminus_real @ B ) ) )
% 0.30/0.60 & ( ( C = zero_zero_real )
% 0.30/0.60 => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % eq_minus_divide_eq
% 0.30/0.60 thf(fact_6144_eq__minus__divide__eq,axiom,
% 0.30/0.60 ! [A: complex,B: complex,C: complex] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
% 0.30/0.60 = ( ( ( C != zero_zero_complex )
% 0.30/0.60 => ( ( times_times_complex @ A @ C )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ B ) ) )
% 0.30/0.60 & ( ( C = zero_zero_complex )
% 0.30/0.60 => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % eq_minus_divide_eq
% 0.30/0.60 thf(fact_6145_eq__minus__divide__eq,axiom,
% 0.30/0.60 ! [A: rat,B: rat,C: rat] :
% 0.30/0.60 ( ( A
% 0.30/0.60 = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.30/0.60 = ( ( ( C != zero_zero_rat )
% 0.30/0.60 => ( ( times_times_rat @ A @ C )
% 0.30/0.60 = ( uminus_uminus_rat @ B ) ) )
% 0.30/0.60 & ( ( C = zero_zero_rat )
% 0.30/0.60 => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % eq_minus_divide_eq
% 0.30/0.60 thf(fact_6146_divide__eq__minus__1__iff,axiom,
% 0.30/0.60 ! [A: real,B: real] :
% 0.30/0.60 ( ( ( divide_divide_real @ A @ B )
% 0.30/0.60 = ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.60 = ( ( B != zero_zero_real )
% 0.30/0.60 & ( A
% 0.30/0.60 = ( uminus_uminus_real @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % divide_eq_minus_1_iff
% 0.30/0.60 thf(fact_6147_divide__eq__minus__1__iff,axiom,
% 0.30/0.60 ! [A: complex,B: complex] :
% 0.30/0.60 ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.60 = ( ( B != zero_zero_complex )
% 0.30/0.60 & ( A
% 0.30/0.60 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % divide_eq_minus_1_iff
% 0.30/0.60 thf(fact_6148_divide__eq__minus__1__iff,axiom,
% 0.30/0.60 ! [A: rat,B: rat] :
% 0.30/0.60 ( ( ( divide_divide_rat @ A @ B )
% 0.30/0.60 = ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.60 = ( ( B != zero_zero_rat )
% 0.30/0.60 & ( A
% 0.30/0.60 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % divide_eq_minus_1_iff
% 0.30/0.60 thf(fact_6149_mult__1s__ring__1_I1_J,axiom,
% 0.30/0.60 ! [B: int] :
% 0.30/0.60 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
% 0.30/0.60 = ( uminus_uminus_int @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_1s_ring_1(1)
% 0.30/0.60 thf(fact_6150_mult__1s__ring__1_I1_J,axiom,
% 0.30/0.60 ! [B: real] :
% 0.30/0.60 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
% 0.30/0.60 = ( uminus_uminus_real @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_1s_ring_1(1)
% 0.30/0.60 thf(fact_6151_mult__1s__ring__1_I1_J,axiom,
% 0.30/0.60 ! [B: complex] :
% 0.30/0.60 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_1s_ring_1(1)
% 0.30/0.60 thf(fact_6152_mult__1s__ring__1_I1_J,axiom,
% 0.30/0.60 ! [B: code_integer] :
% 0.30/0.60 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_1s_ring_1(1)
% 0.30/0.60 thf(fact_6153_mult__1s__ring__1_I1_J,axiom,
% 0.30/0.60 ! [B: rat] :
% 0.30/0.60 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
% 0.30/0.60 = ( uminus_uminus_rat @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_1s_ring_1(1)
% 0.30/0.60 thf(fact_6154_mult__1s__ring__1_I2_J,axiom,
% 0.30/0.60 ! [B: int] :
% 0.30/0.60 ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
% 0.30/0.60 = ( uminus_uminus_int @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_1s_ring_1(2)
% 0.30/0.60 thf(fact_6155_mult__1s__ring__1_I2_J,axiom,
% 0.30/0.60 ! [B: real] :
% 0.30/0.60 ( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
% 0.30/0.60 = ( uminus_uminus_real @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_1s_ring_1(2)
% 0.30/0.60 thf(fact_6156_mult__1s__ring__1_I2_J,axiom,
% 0.30/0.60 ! [B: complex] :
% 0.30/0.60 ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_1s_ring_1(2)
% 0.30/0.60 thf(fact_6157_mult__1s__ring__1_I2_J,axiom,
% 0.30/0.60 ! [B: code_integer] :
% 0.30/0.60 ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_1s_ring_1(2)
% 0.30/0.60 thf(fact_6158_mult__1s__ring__1_I2_J,axiom,
% 0.30/0.60 ! [B: rat] :
% 0.30/0.60 ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
% 0.30/0.60 = ( uminus_uminus_rat @ B ) ) ).
% 0.30/0.60
% 0.30/0.60 % mult_1s_ring_1(2)
% 0.30/0.60 thf(fact_6159_uminus__numeral__One,axiom,
% 0.30/0.60 ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
% 0.30/0.60 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.60
% 0.30/0.60 % uminus_numeral_One
% 0.30/0.60 thf(fact_6160_uminus__numeral__One,axiom,
% 0.30/0.60 ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
% 0.30/0.60 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.60
% 0.30/0.60 % uminus_numeral_One
% 0.30/0.60 thf(fact_6161_uminus__numeral__One,axiom,
% 0.30/0.60 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.60
% 0.30/0.60 % uminus_numeral_One
% 0.30/0.60 thf(fact_6162_uminus__numeral__One,axiom,
% 0.30/0.60 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.60
% 0.30/0.60 % uminus_numeral_One
% 0.30/0.60 thf(fact_6163_uminus__numeral__One,axiom,
% 0.30/0.60 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
% 0.30/0.60 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.60
% 0.30/0.60 % uminus_numeral_One
% 0.30/0.60 thf(fact_6164_power__minus,axiom,
% 0.30/0.60 ! [A: int,N: nat] :
% 0.30/0.60 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.30/0.60 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus
% 0.30/0.60 thf(fact_6165_power__minus,axiom,
% 0.30/0.60 ! [A: real,N: nat] :
% 0.30/0.60 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.30/0.60 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus
% 0.30/0.60 thf(fact_6166_power__minus,axiom,
% 0.30/0.60 ! [A: complex,N: nat] :
% 0.30/0.60 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.30/0.60 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus
% 0.30/0.60 thf(fact_6167_power__minus,axiom,
% 0.30/0.60 ! [A: code_integer,N: nat] :
% 0.30/0.60 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.30/0.60 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus
% 0.30/0.60 thf(fact_6168_power__minus,axiom,
% 0.30/0.60 ! [A: rat,N: nat] :
% 0.30/0.60 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.30/0.60 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus
% 0.30/0.60 thf(fact_6169_power__minus__Bit0,axiom,
% 0.30/0.60 ! [X2: int,K: num] :
% 0.30/0.60 ( ( power_power_int @ ( uminus_uminus_int @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.30/0.60 = ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus_Bit0
% 0.30/0.60 thf(fact_6170_power__minus__Bit0,axiom,
% 0.30/0.60 ! [X2: real,K: num] :
% 0.30/0.60 ( ( power_power_real @ ( uminus_uminus_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.30/0.60 = ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus_Bit0
% 0.30/0.60 thf(fact_6171_power__minus__Bit0,axiom,
% 0.30/0.60 ! [X2: complex,K: num] :
% 0.30/0.60 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.30/0.60 = ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus_Bit0
% 0.30/0.60 thf(fact_6172_power__minus__Bit0,axiom,
% 0.30/0.60 ! [X2: code_integer,K: num] :
% 0.30/0.60 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.30/0.60 = ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus_Bit0
% 0.30/0.60 thf(fact_6173_power__minus__Bit0,axiom,
% 0.30/0.60 ! [X2: rat,K: num] :
% 0.30/0.60 ( ( power_power_rat @ ( uminus_uminus_rat @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.30/0.60 = ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus_Bit0
% 0.30/0.60 thf(fact_6174_power__minus__Bit1,axiom,
% 0.30/0.60 ! [X2: int,K: num] :
% 0.30/0.60 ( ( power_power_int @ ( uminus_uminus_int @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.30/0.60 = ( uminus_uminus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus_Bit1
% 0.30/0.60 thf(fact_6175_power__minus__Bit1,axiom,
% 0.30/0.60 ! [X2: real,K: num] :
% 0.30/0.60 ( ( power_power_real @ ( uminus_uminus_real @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.30/0.60 = ( uminus_uminus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus_Bit1
% 0.30/0.60 thf(fact_6176_power__minus__Bit1,axiom,
% 0.30/0.60 ! [X2: complex,K: num] :
% 0.30/0.60 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus_Bit1
% 0.30/0.60 thf(fact_6177_power__minus__Bit1,axiom,
% 0.30/0.60 ! [X2: code_integer,K: num] :
% 0.30/0.60 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.30/0.60 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus_Bit1
% 0.30/0.60 thf(fact_6178_power__minus__Bit1,axiom,
% 0.30/0.60 ! [X2: rat,K: num] :
% 0.30/0.60 ( ( power_power_rat @ ( uminus_uminus_rat @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.30/0.60 = ( uminus_uminus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % power_minus_Bit1
% 0.30/0.60 thf(fact_6179_real__0__less__add__iff,axiom,
% 0.30/0.60 ! [X2: real,Y3: real] :
% 0.30/0.60 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 0.30/0.60 = ( ord_less_real @ ( uminus_uminus_real @ X2 ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % real_0_less_add_iff
% 0.30/0.60 thf(fact_6180_real__add__less__0__iff,axiom,
% 0.30/0.60 ! [X2: real,Y3: real] :
% 0.30/0.60 ( ( ord_less_real @ ( plus_plus_real @ X2 @ Y3 ) @ zero_zero_real )
% 0.30/0.60 = ( ord_less_real @ Y3 @ ( uminus_uminus_real @ X2 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % real_add_less_0_iff
% 0.30/0.60 thf(fact_6181_real__0__le__add__iff,axiom,
% 0.30/0.60 ! [X2: real,Y3: real] :
% 0.30/0.60 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 0.30/0.60 = ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ Y3 ) ) ).
% 0.30/0.60
% 0.30/0.60 % real_0_le_add_iff
% 0.30/0.60 thf(fact_6182_real__add__le__0__iff,axiom,
% 0.30/0.60 ! [X2: real,Y3: real] :
% 0.30/0.60 ( ( ord_less_eq_real @ ( plus_plus_real @ X2 @ Y3 ) @ zero_zero_real )
% 0.30/0.60 = ( ord_less_eq_real @ Y3 @ ( uminus_uminus_real @ X2 ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % real_add_le_0_iff
% 0.30/0.60 thf(fact_6183_zmod__zminus1__eq__if,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( ( ( modulo_modulo_int @ A @ B )
% 0.30/0.60 = zero_zero_int )
% 0.30/0.60 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.60 = zero_zero_int ) )
% 0.30/0.60 & ( ( ( modulo_modulo_int @ A @ B )
% 0.30/0.60 != zero_zero_int )
% 0.30/0.60 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.60 = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % zmod_zminus1_eq_if
% 0.30/0.60 thf(fact_6184_zmod__zminus2__eq__if,axiom,
% 0.30/0.60 ! [A: int,B: int] :
% 0.30/0.60 ( ( ( ( modulo_modulo_int @ A @ B )
% 0.30/0.60 = zero_zero_int )
% 0.30/0.60 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 0.30/0.60 = zero_zero_int ) )
% 0.30/0.60 & ( ( ( modulo_modulo_int @ A @ B )
% 0.30/0.60 != zero_zero_int )
% 0.30/0.60 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 0.30/0.60 = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % zmod_zminus2_eq_if
% 0.30/0.60 thf(fact_6185_pred__numeral__def,axiom,
% 0.30/0.60 ( pred_numeral
% 0.30/0.60 = ( ^ [K2: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K2 ) @ one_one_nat ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % pred_numeral_def
% 0.30/0.60 thf(fact_6186_less__minus__divide__eq,axiom,
% 0.30/0.60 ! [A: real,B: real,C: real] :
% 0.30/0.60 ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.30/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.30/0.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.30/0.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_minus_divide_eq
% 0.30/0.60 thf(fact_6187_less__minus__divide__eq,axiom,
% 0.30/0.60 ! [A: rat,B: rat,C: rat] :
% 0.30/0.60 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.30/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.30/0.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % less_minus_divide_eq
% 0.30/0.60 thf(fact_6188_minus__divide__less__eq,axiom,
% 0.30/0.60 ! [B: real,C: real,A: real] :
% 0.30/0.60 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.30/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.30/0.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.30/0.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_less_eq
% 0.30/0.60 thf(fact_6189_minus__divide__less__eq,axiom,
% 0.30/0.60 ! [B: rat,C: rat,A: rat] :
% 0.30/0.60 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.30/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.30/0.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_less_eq
% 0.30/0.60 thf(fact_6190_neg__less__minus__divide__eq,axiom,
% 0.30/0.60 ! [C: real,A: real,B: real] :
% 0.30/0.60 ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.30/0.60 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_less_minus_divide_eq
% 0.30/0.60 thf(fact_6191_neg__less__minus__divide__eq,axiom,
% 0.30/0.60 ! [C: rat,A: rat,B: rat] :
% 0.30/0.60 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.30/0.60 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_less_minus_divide_eq
% 0.30/0.60 thf(fact_6192_neg__minus__divide__less__eq,axiom,
% 0.30/0.60 ! [C: real,B: real,A: real] :
% 0.30/0.60 ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.30/0.60 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_minus_divide_less_eq
% 0.30/0.60 thf(fact_6193_neg__minus__divide__less__eq,axiom,
% 0.30/0.60 ! [C: rat,B: rat,A: rat] :
% 0.30/0.60 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.30/0.60 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % neg_minus_divide_less_eq
% 0.30/0.60 thf(fact_6194_pos__less__minus__divide__eq,axiom,
% 0.30/0.60 ! [C: real,A: real,B: real] :
% 0.30/0.60 ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.30/0.60 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % pos_less_minus_divide_eq
% 0.30/0.60 thf(fact_6195_pos__less__minus__divide__eq,axiom,
% 0.30/0.60 ! [C: rat,A: rat,B: rat] :
% 0.30/0.60 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.30/0.60 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % pos_less_minus_divide_eq
% 0.30/0.60 thf(fact_6196_pos__minus__divide__less__eq,axiom,
% 0.30/0.60 ! [C: real,B: real,A: real] :
% 0.30/0.60 ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.30/0.60 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % pos_minus_divide_less_eq
% 0.30/0.60 thf(fact_6197_pos__minus__divide__less__eq,axiom,
% 0.30/0.60 ! [C: rat,B: rat,A: rat] :
% 0.30/0.60 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.30/0.60 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % pos_minus_divide_less_eq
% 0.30/0.60 thf(fact_6198_eq__divide__eq__numeral_I2_J,axiom,
% 0.30/0.60 ! [W: num,B: real,C: real] :
% 0.30/0.60 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.30/0.60 = ( divide_divide_real @ B @ C ) )
% 0.30/0.60 = ( ( ( C != zero_zero_real )
% 0.30/0.60 => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
% 0.30/0.60 = B ) )
% 0.30/0.60 & ( ( C = zero_zero_real )
% 0.30/0.60 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.30/0.60 = zero_zero_real ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % eq_divide_eq_numeral(2)
% 0.30/0.60 thf(fact_6199_eq__divide__eq__numeral_I2_J,axiom,
% 0.30/0.60 ! [W: num,B: complex,C: complex] :
% 0.30/0.60 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.60 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.30/0.60 = ( ( ( C != zero_zero_complex )
% 0.30/0.60 => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
% 0.30/0.60 = B ) )
% 0.30/0.60 & ( ( C = zero_zero_complex )
% 0.30/0.60 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.60 = zero_zero_complex ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % eq_divide_eq_numeral(2)
% 0.30/0.60 thf(fact_6200_eq__divide__eq__numeral_I2_J,axiom,
% 0.30/0.60 ! [W: num,B: rat,C: rat] :
% 0.30/0.60 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.30/0.60 = ( divide_divide_rat @ B @ C ) )
% 0.30/0.60 = ( ( ( C != zero_zero_rat )
% 0.30/0.60 => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
% 0.30/0.60 = B ) )
% 0.30/0.60 & ( ( C = zero_zero_rat )
% 0.30/0.60 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.30/0.60 = zero_zero_rat ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % eq_divide_eq_numeral(2)
% 0.30/0.60 thf(fact_6201_divide__eq__eq__numeral_I2_J,axiom,
% 0.30/0.60 ! [B: real,C: real,W: num] :
% 0.30/0.60 ( ( ( divide_divide_real @ B @ C )
% 0.30/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.60 = ( ( ( C != zero_zero_real )
% 0.30/0.60 => ( B
% 0.30/0.60 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.30/0.60 & ( ( C = zero_zero_real )
% 0.30/0.60 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.30/0.60 = zero_zero_real ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % divide_eq_eq_numeral(2)
% 0.30/0.60 thf(fact_6202_divide__eq__eq__numeral_I2_J,axiom,
% 0.30/0.60 ! [B: complex,C: complex,W: num] :
% 0.30/0.60 ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.60 = ( ( ( C != zero_zero_complex )
% 0.30/0.60 => ( B
% 0.30/0.60 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
% 0.30/0.60 & ( ( C = zero_zero_complex )
% 0.30/0.60 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.60 = zero_zero_complex ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % divide_eq_eq_numeral(2)
% 0.30/0.60 thf(fact_6203_divide__eq__eq__numeral_I2_J,axiom,
% 0.30/0.60 ! [B: rat,C: rat,W: num] :
% 0.30/0.60 ( ( ( divide_divide_rat @ B @ C )
% 0.30/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.30/0.60 = ( ( ( C != zero_zero_rat )
% 0.30/0.60 => ( B
% 0.30/0.60 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.30/0.60 & ( ( C = zero_zero_rat )
% 0.30/0.60 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.30/0.60 = zero_zero_rat ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % divide_eq_eq_numeral(2)
% 0.30/0.60 thf(fact_6204_add__divide__eq__if__simps_I3_J,axiom,
% 0.30/0.60 ! [Z: real,A: real,B: real] :
% 0.30/0.60 ( ( ( Z = zero_zero_real )
% 0.30/0.60 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.30/0.60 = B ) )
% 0.30/0.60 & ( ( Z != zero_zero_real )
% 0.30/0.60 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.30/0.60 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add_divide_eq_if_simps(3)
% 0.30/0.60 thf(fact_6205_add__divide__eq__if__simps_I3_J,axiom,
% 0.30/0.60 ! [Z: complex,A: complex,B: complex] :
% 0.30/0.60 ( ( ( Z = zero_zero_complex )
% 0.30/0.60 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.30/0.60 = B ) )
% 0.30/0.60 & ( ( Z != zero_zero_complex )
% 0.30/0.60 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.30/0.60 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add_divide_eq_if_simps(3)
% 0.30/0.60 thf(fact_6206_add__divide__eq__if__simps_I3_J,axiom,
% 0.30/0.60 ! [Z: rat,A: rat,B: rat] :
% 0.30/0.60 ( ( ( Z = zero_zero_rat )
% 0.30/0.60 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.30/0.60 = B ) )
% 0.30/0.60 & ( ( Z != zero_zero_rat )
% 0.30/0.60 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.30/0.60 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add_divide_eq_if_simps(3)
% 0.30/0.60 thf(fact_6207_minus__divide__add__eq__iff,axiom,
% 0.30/0.60 ! [Z: real,X2: real,Y3: real] :
% 0.30/0.60 ( ( Z != zero_zero_real )
% 0.30/0.60 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X2 @ Z ) ) @ Y3 )
% 0.30/0.60 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_add_eq_iff
% 0.30/0.60 thf(fact_6208_minus__divide__add__eq__iff,axiom,
% 0.30/0.60 ! [Z: complex,X2: complex,Y3: complex] :
% 0.30/0.60 ( ( Z != zero_zero_complex )
% 0.30/0.60 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X2 @ Z ) ) @ Y3 )
% 0.30/0.60 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_add_eq_iff
% 0.30/0.60 thf(fact_6209_minus__divide__add__eq__iff,axiom,
% 0.30/0.60 ! [Z: rat,X2: rat,Y3: rat] :
% 0.30/0.60 ( ( Z != zero_zero_rat )
% 0.30/0.60 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X2 @ Z ) ) @ Y3 )
% 0.30/0.60 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X2 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_add_eq_iff
% 0.30/0.60 thf(fact_6210_minus__divide__diff__eq__iff,axiom,
% 0.30/0.60 ! [Z: real,X2: real,Y3: real] :
% 0.30/0.60 ( ( Z != zero_zero_real )
% 0.30/0.60 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X2 @ Z ) ) @ Y3 )
% 0.30/0.60 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_diff_eq_iff
% 0.30/0.60 thf(fact_6211_minus__divide__diff__eq__iff,axiom,
% 0.30/0.60 ! [Z: complex,X2: complex,Y3: complex] :
% 0.30/0.60 ( ( Z != zero_zero_complex )
% 0.30/0.60 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X2 @ Z ) ) @ Y3 )
% 0.30/0.60 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_diff_eq_iff
% 0.30/0.60 thf(fact_6212_minus__divide__diff__eq__iff,axiom,
% 0.30/0.60 ! [Z: rat,X2: rat,Y3: rat] :
% 0.30/0.60 ( ( Z != zero_zero_rat )
% 0.30/0.60 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X2 @ Z ) ) @ Y3 )
% 0.30/0.60 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X2 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % minus_divide_diff_eq_iff
% 0.30/0.60 thf(fact_6213_add__divide__eq__if__simps_I5_J,axiom,
% 0.30/0.60 ! [Z: real,A: real,B: real] :
% 0.30/0.60 ( ( ( Z = zero_zero_real )
% 0.30/0.60 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.30/0.60 = ( uminus_uminus_real @ B ) ) )
% 0.30/0.60 & ( ( Z != zero_zero_real )
% 0.30/0.60 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.30/0.60 = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.60
% 0.30/0.60 % add_divide_eq_if_simps(5)
% 0.30/0.60 thf(fact_6214_add__divide__eq__if__simps_I5_J,axiom,
% 0.30/0.60 ! [Z: complex,A: complex,B: complex] :
% 0.30/0.60 ( ( ( Z = zero_zero_complex )
% 0.30/0.60 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.30/0.60 = ( uminus1482373934393186551omplex @ B ) ) )
% 0.30/0.60 & ( ( Z != zero_zero_complex )
% 0.30/0.60 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.30/0.61 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % add_divide_eq_if_simps(5)
% 0.30/0.61 thf(fact_6215_add__divide__eq__if__simps_I5_J,axiom,
% 0.30/0.61 ! [Z: rat,A: rat,B: rat] :
% 0.30/0.61 ( ( ( Z = zero_zero_rat )
% 0.30/0.61 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.30/0.61 = ( uminus_uminus_rat @ B ) ) )
% 0.30/0.61 & ( ( Z != zero_zero_rat )
% 0.30/0.61 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.30/0.61 = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % add_divide_eq_if_simps(5)
% 0.30/0.61 thf(fact_6216_add__divide__eq__if__simps_I6_J,axiom,
% 0.30/0.61 ! [Z: real,A: real,B: real] :
% 0.30/0.61 ( ( ( Z = zero_zero_real )
% 0.30/0.61 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.30/0.61 = ( uminus_uminus_real @ B ) ) )
% 0.30/0.61 & ( ( Z != zero_zero_real )
% 0.30/0.61 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.30/0.61 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % add_divide_eq_if_simps(6)
% 0.30/0.61 thf(fact_6217_add__divide__eq__if__simps_I6_J,axiom,
% 0.30/0.61 ! [Z: complex,A: complex,B: complex] :
% 0.30/0.61 ( ( ( Z = zero_zero_complex )
% 0.30/0.61 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.30/0.61 = ( uminus1482373934393186551omplex @ B ) ) )
% 0.30/0.61 & ( ( Z != zero_zero_complex )
% 0.30/0.61 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.30/0.61 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % add_divide_eq_if_simps(6)
% 0.30/0.61 thf(fact_6218_add__divide__eq__if__simps_I6_J,axiom,
% 0.30/0.61 ! [Z: rat,A: rat,B: rat] :
% 0.30/0.61 ( ( ( Z = zero_zero_rat )
% 0.30/0.61 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.30/0.61 = ( uminus_uminus_rat @ B ) ) )
% 0.30/0.61 & ( ( Z != zero_zero_rat )
% 0.30/0.61 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.30/0.61 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % add_divide_eq_if_simps(6)
% 0.30/0.61 thf(fact_6219_even__minus,axiom,
% 0.30/0.61 ! [A: int] :
% 0.30/0.61 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
% 0.30/0.61 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.30/0.61
% 0.30/0.61 % even_minus
% 0.30/0.61 thf(fact_6220_even__minus,axiom,
% 0.30/0.61 ! [A: code_integer] :
% 0.30/0.61 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.61 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.30/0.61
% 0.30/0.61 % even_minus
% 0.30/0.61 thf(fact_6221_power2__eq__iff,axiom,
% 0.30/0.61 ! [X2: int,Y3: int] :
% 0.30/0.61 ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.61 = ( ( X2 = Y3 )
% 0.30/0.61 | ( X2
% 0.30/0.61 = ( uminus_uminus_int @ Y3 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % power2_eq_iff
% 0.30/0.61 thf(fact_6222_power2__eq__iff,axiom,
% 0.30/0.61 ! [X2: real,Y3: real] :
% 0.30/0.61 ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.61 = ( ( X2 = Y3 )
% 0.30/0.61 | ( X2
% 0.30/0.61 = ( uminus_uminus_real @ Y3 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % power2_eq_iff
% 0.30/0.61 thf(fact_6223_power2__eq__iff,axiom,
% 0.30/0.61 ! [X2: complex,Y3: complex] :
% 0.30/0.61 ( ( ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61 = ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.61 = ( ( X2 = Y3 )
% 0.30/0.61 | ( X2
% 0.30/0.61 = ( uminus1482373934393186551omplex @ Y3 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % power2_eq_iff
% 0.30/0.61 thf(fact_6224_power2__eq__iff,axiom,
% 0.30/0.61 ! [X2: code_integer,Y3: code_integer] :
% 0.30/0.61 ( ( ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61 = ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.61 = ( ( X2 = Y3 )
% 0.30/0.61 | ( X2
% 0.30/0.61 = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % power2_eq_iff
% 0.30/0.61 thf(fact_6225_power2__eq__iff,axiom,
% 0.30/0.61 ! [X2: rat,Y3: rat] :
% 0.30/0.61 ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.61 = ( ( X2 = Y3 )
% 0.30/0.61 | ( X2
% 0.30/0.61 = ( uminus_uminus_rat @ Y3 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % power2_eq_iff
% 0.30/0.61 thf(fact_6226_verit__less__mono__div__int2,axiom,
% 0.30/0.61 ! [A2: int,B3: int,N: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.30/0.61 => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
% 0.30/0.61 => ( ord_less_eq_int @ ( divide_divide_int @ B3 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % verit_less_mono_div_int2
% 0.30/0.61 thf(fact_6227_div__eq__minus1,axiom,
% 0.30/0.61 ! [B: int] :
% 0.30/0.61 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.61 => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 0.30/0.61 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % div_eq_minus1
% 0.30/0.61 thf(fact_6228_pos__minus__divide__le__eq,axiom,
% 0.30/0.61 ! [C: real,B: real,A: real] :
% 0.30/0.61 ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.30/0.61 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % pos_minus_divide_le_eq
% 0.30/0.61 thf(fact_6229_pos__minus__divide__le__eq,axiom,
% 0.30/0.61 ! [C: rat,B: rat,A: rat] :
% 0.30/0.61 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.30/0.61 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % pos_minus_divide_le_eq
% 0.30/0.61 thf(fact_6230_pos__le__minus__divide__eq,axiom,
% 0.30/0.61 ! [C: real,A: real,B: real] :
% 0.30/0.61 ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.30/0.61 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % pos_le_minus_divide_eq
% 0.30/0.61 thf(fact_6231_pos__le__minus__divide__eq,axiom,
% 0.30/0.61 ! [C: rat,A: rat,B: rat] :
% 0.30/0.61 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.30/0.61 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % pos_le_minus_divide_eq
% 0.30/0.61 thf(fact_6232_neg__minus__divide__le__eq,axiom,
% 0.30/0.61 ! [C: real,B: real,A: real] :
% 0.30/0.61 ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.30/0.61 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_minus_divide_le_eq
% 0.30/0.61 thf(fact_6233_neg__minus__divide__le__eq,axiom,
% 0.30/0.61 ! [C: rat,B: rat,A: rat] :
% 0.30/0.61 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.30/0.61 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_minus_divide_le_eq
% 0.30/0.61 thf(fact_6234_neg__le__minus__divide__eq,axiom,
% 0.30/0.61 ! [C: real,A: real,B: real] :
% 0.30/0.61 ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.30/0.61 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_le_minus_divide_eq
% 0.30/0.61 thf(fact_6235_neg__le__minus__divide__eq,axiom,
% 0.30/0.61 ! [C: rat,A: rat,B: rat] :
% 0.30/0.61 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.30/0.61 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_le_minus_divide_eq
% 0.30/0.61 thf(fact_6236_minus__divide__le__eq,axiom,
% 0.30/0.61 ! [B: real,C: real,A: real] :
% 0.30/0.61 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.30/0.61 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.30/0.61 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.30/0.61 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % minus_divide_le_eq
% 0.30/0.61 thf(fact_6237_minus__divide__le__eq,axiom,
% 0.30/0.61 ! [B: rat,C: rat,A: rat] :
% 0.30/0.61 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.30/0.61 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.61 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.30/0.61 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % minus_divide_le_eq
% 0.30/0.61 thf(fact_6238_le__minus__divide__eq,axiom,
% 0.30/0.61 ! [A: real,B: real,C: real] :
% 0.30/0.61 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.30/0.61 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.30/0.61 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.30/0.61 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % le_minus_divide_eq
% 0.30/0.61 thf(fact_6239_le__minus__divide__eq,axiom,
% 0.30/0.61 ! [A: rat,B: rat,C: rat] :
% 0.30/0.61 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.30/0.61 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.30/0.61 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.61 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % le_minus_divide_eq
% 0.30/0.61 thf(fact_6240_less__divide__eq__numeral_I2_J,axiom,
% 0.30/0.61 ! [W: num,B: real,C: real] :
% 0.30/0.61 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 0.30/0.61 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.30/0.61 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.30/0.61 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61 => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % less_divide_eq_numeral(2)
% 0.30/0.61 thf(fact_6241_less__divide__eq__numeral_I2_J,axiom,
% 0.30/0.61 ! [W: num,B: rat,C: rat] :
% 0.30/0.61 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.61 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.30/0.61 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.30/0.61 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61 => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % less_divide_eq_numeral(2)
% 0.30/0.61 thf(fact_6242_divide__less__eq__numeral_I2_J,axiom,
% 0.30/0.61 ! [B: real,C: real,W: num] :
% 0.30/0.61 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.61 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.30/0.61 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.30/0.61 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61 => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % divide_less_eq_numeral(2)
% 0.30/0.61 thf(fact_6243_divide__less__eq__numeral_I2_J,axiom,
% 0.30/0.61 ! [B: rat,C: rat,W: num] :
% 0.30/0.61 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.30/0.61 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.30/0.61 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.30/0.61 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61 => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % divide_less_eq_numeral(2)
% 0.30/0.61 thf(fact_6244_power2__eq__1__iff,axiom,
% 0.30/0.61 ! [A: int] :
% 0.30/0.61 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61 = one_one_int )
% 0.30/0.61 = ( ( A = one_one_int )
% 0.30/0.61 | ( A
% 0.30/0.61 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % power2_eq_1_iff
% 0.30/0.61 thf(fact_6245_power2__eq__1__iff,axiom,
% 0.30/0.61 ! [A: real] :
% 0.30/0.61 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61 = one_one_real )
% 0.30/0.61 = ( ( A = one_one_real )
% 0.30/0.61 | ( A
% 0.30/0.61 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % power2_eq_1_iff
% 0.30/0.61 thf(fact_6246_power2__eq__1__iff,axiom,
% 0.30/0.61 ! [A: complex] :
% 0.30/0.61 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61 = one_one_complex )
% 0.30/0.61 = ( ( A = one_one_complex )
% 0.30/0.61 | ( A
% 0.30/0.61 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % power2_eq_1_iff
% 0.30/0.61 thf(fact_6247_power2__eq__1__iff,axiom,
% 0.30/0.61 ! [A: code_integer] :
% 0.30/0.61 ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61 = one_one_Code_integer )
% 0.30/0.61 = ( ( A = one_one_Code_integer )
% 0.30/0.61 | ( A
% 0.30/0.61 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % power2_eq_1_iff
% 0.30/0.61 thf(fact_6248_power2__eq__1__iff,axiom,
% 0.30/0.61 ! [A: rat] :
% 0.30/0.61 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61 = one_one_rat )
% 0.30/0.61 = ( ( A = one_one_rat )
% 0.30/0.61 | ( A
% 0.30/0.61 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % power2_eq_1_iff
% 0.30/0.61 thf(fact_6249_uminus__power__if,axiom,
% 0.30/0.61 ! [N: nat,A: int] :
% 0.30/0.61 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.30/0.61 = ( power_power_int @ A @ N ) ) )
% 0.30/0.61 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.30/0.61 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % uminus_power_if
% 0.30/0.61 thf(fact_6250_uminus__power__if,axiom,
% 0.30/0.61 ! [N: nat,A: real] :
% 0.30/0.61 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.30/0.61 = ( power_power_real @ A @ N ) ) )
% 0.30/0.61 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.30/0.61 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % uminus_power_if
% 0.30/0.61 thf(fact_6251_uminus__power__if,axiom,
% 0.30/0.61 ! [N: nat,A: complex] :
% 0.30/0.61 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.30/0.61 = ( power_power_complex @ A @ N ) ) )
% 0.30/0.61 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.30/0.61 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % uminus_power_if
% 0.30/0.61 thf(fact_6252_uminus__power__if,axiom,
% 0.30/0.61 ! [N: nat,A: code_integer] :
% 0.30/0.61 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.30/0.61 = ( power_8256067586552552935nteger @ A @ N ) ) )
% 0.30/0.61 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.30/0.61 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % uminus_power_if
% 0.30/0.61 thf(fact_6253_uminus__power__if,axiom,
% 0.30/0.61 ! [N: nat,A: rat] :
% 0.30/0.61 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.30/0.61 = ( power_power_rat @ A @ N ) ) )
% 0.30/0.61 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.30/0.61 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % uminus_power_if
% 0.30/0.61 thf(fact_6254_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.30/0.61 ! [K: nat,N: nat] :
% 0.30/0.61 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.61 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
% 0.30/0.61 = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_one_power_add_eq_neg_one_power_diff
% 0.30/0.61 thf(fact_6255_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.30/0.61 ! [K: nat,N: nat] :
% 0.30/0.61 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.61 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
% 0.30/0.61 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_one_power_add_eq_neg_one_power_diff
% 0.30/0.61 thf(fact_6256_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.30/0.61 ! [K: nat,N: nat] :
% 0.30/0.61 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.61 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
% 0.30/0.61 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_one_power_add_eq_neg_one_power_diff
% 0.30/0.61 thf(fact_6257_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.30/0.61 ! [K: nat,N: nat] :
% 0.30/0.61 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.61 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
% 0.30/0.61 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_one_power_add_eq_neg_one_power_diff
% 0.30/0.61 thf(fact_6258_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.30/0.61 ! [K: nat,N: nat] :
% 0.30/0.61 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.61 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
% 0.30/0.61 = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_one_power_add_eq_neg_one_power_diff
% 0.30/0.61 thf(fact_6259_realpow__square__minus__le,axiom,
% 0.30/0.61 ! [U: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % realpow_square_minus_le
% 0.30/0.61 thf(fact_6260_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 0.30/0.61 ! [N: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ).
% 0.30/0.61
% 0.30/0.61 % signed_take_bit_int_greater_eq_minus_exp
% 0.30/0.61 thf(fact_6261_signed__take__bit__int__less__eq__self__iff,axiom,
% 0.30/0.61 ! [N: nat,K: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 0.30/0.61 = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% 0.30/0.61
% 0.30/0.61 % signed_take_bit_int_less_eq_self_iff
% 0.30/0.61 thf(fact_6262_signed__take__bit__int__greater__self__iff,axiom,
% 0.30/0.61 ! [K: int,N: nat] :
% 0.30/0.61 ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.30/0.61 = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % signed_take_bit_int_greater_self_iff
% 0.30/0.61 thf(fact_6263_minus__mod__int__eq,axiom,
% 0.30/0.61 ! [L: int,K: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ zero_zero_int @ L )
% 0.30/0.61 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
% 0.30/0.61 = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % minus_mod_int_eq
% 0.30/0.61 thf(fact_6264_zmod__minus1,axiom,
% 0.30/0.61 ! [B: int] :
% 0.30/0.61 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.61 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 0.30/0.61 = ( minus_minus_int @ B @ one_one_int ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % zmod_minus1
% 0.30/0.61 thf(fact_6265_zdiv__zminus2__eq__if,axiom,
% 0.30/0.61 ! [B: int,A: int] :
% 0.30/0.61 ( ( B != zero_zero_int )
% 0.30/0.61 => ( ( ( ( modulo_modulo_int @ A @ B )
% 0.30/0.61 = zero_zero_int )
% 0.30/0.61 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.30/0.61 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 0.30/0.61 & ( ( ( modulo_modulo_int @ A @ B )
% 0.30/0.61 != zero_zero_int )
% 0.30/0.61 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.30/0.61 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % zdiv_zminus2_eq_if
% 0.30/0.61 thf(fact_6266_zdiv__zminus1__eq__if,axiom,
% 0.30/0.61 ! [B: int,A: int] :
% 0.30/0.61 ( ( B != zero_zero_int )
% 0.30/0.61 => ( ( ( ( modulo_modulo_int @ A @ B )
% 0.30/0.61 = zero_zero_int )
% 0.30/0.61 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.61 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 0.30/0.61 & ( ( ( modulo_modulo_int @ A @ B )
% 0.30/0.61 != zero_zero_int )
% 0.30/0.61 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.61 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % zdiv_zminus1_eq_if
% 0.30/0.61 thf(fact_6267_zminus1__lemma,axiom,
% 0.30/0.61 ! [A: int,B: int,Q3: int,R2: int] :
% 0.30/0.61 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.30/0.61 => ( ( B != zero_zero_int )
% 0.30/0.61 => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q3 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q3 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R2 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % zminus1_lemma
% 0.30/0.61 thf(fact_6268_le__divide__eq__numeral_I2_J,axiom,
% 0.30/0.61 ! [W: num,B: real,C: real] :
% 0.30/0.61 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 0.30/0.61 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.30/0.61 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.30/0.61 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % le_divide_eq_numeral(2)
% 0.30/0.61 thf(fact_6269_le__divide__eq__numeral_I2_J,axiom,
% 0.30/0.61 ! [W: num,B: rat,C: rat] :
% 0.30/0.61 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.61 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.30/0.61 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.30/0.61 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % le_divide_eq_numeral(2)
% 0.30/0.61 thf(fact_6270_divide__le__eq__numeral_I2_J,axiom,
% 0.30/0.61 ! [B: real,C: real,W: num] :
% 0.30/0.61 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.61 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.30/0.61 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.30/0.61 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % divide_le_eq_numeral(2)
% 0.30/0.61 thf(fact_6271_divide__le__eq__numeral_I2_J,axiom,
% 0.30/0.61 ! [B: rat,C: rat,W: num] :
% 0.30/0.61 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.30/0.61 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.30/0.61 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.30/0.61 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % divide_le_eq_numeral(2)
% 0.30/0.61 thf(fact_6272_square__le__1,axiom,
% 0.30/0.61 ! [X2: real] :
% 0.30/0.61 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.61 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.30/0.61 => ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % square_le_1
% 0.30/0.61 thf(fact_6273_square__le__1,axiom,
% 0.30/0.61 ! [X2: code_integer] :
% 0.30/0.61 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X2 )
% 0.30/0.61 => ( ( ord_le3102999989581377725nteger @ X2 @ one_one_Code_integer )
% 0.30/0.61 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % square_le_1
% 0.30/0.61 thf(fact_6274_square__le__1,axiom,
% 0.30/0.61 ! [X2: rat] :
% 0.30/0.61 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X2 )
% 0.30/0.61 => ( ( ord_less_eq_rat @ X2 @ one_one_rat )
% 0.30/0.61 => ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % square_le_1
% 0.30/0.61 thf(fact_6275_square__le__1,axiom,
% 0.30/0.61 ! [X2: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X2 )
% 0.30/0.61 => ( ( ord_less_eq_int @ X2 @ one_one_int )
% 0.30/0.61 => ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % square_le_1
% 0.30/0.61 thf(fact_6276_minus__power__mult__self,axiom,
% 0.30/0.61 ! [A: int,N: nat] :
% 0.30/0.61 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 0.30/0.61 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % minus_power_mult_self
% 0.30/0.61 thf(fact_6277_minus__power__mult__self,axiom,
% 0.30/0.61 ! [A: real,N: nat] :
% 0.30/0.61 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 0.30/0.61 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % minus_power_mult_self
% 0.30/0.61 thf(fact_6278_minus__power__mult__self,axiom,
% 0.30/0.61 ! [A: complex,N: nat] :
% 0.30/0.61 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
% 0.30/0.61 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % minus_power_mult_self
% 0.30/0.61 thf(fact_6279_minus__power__mult__self,axiom,
% 0.30/0.61 ! [A: code_integer,N: nat] :
% 0.30/0.61 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 0.30/0.61 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % minus_power_mult_self
% 0.30/0.61 thf(fact_6280_minus__power__mult__self,axiom,
% 0.30/0.61 ! [A: rat,N: nat] :
% 0.30/0.61 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 0.30/0.61 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % minus_power_mult_self
% 0.30/0.61 thf(fact_6281_minus__one__power__iff,axiom,
% 0.30/0.61 ! [N: nat] :
% 0.30/0.61 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.30/0.61 = one_one_int ) )
% 0.30/0.61 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.30/0.61 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % minus_one_power_iff
% 0.30/0.61 thf(fact_6282_minus__one__power__iff,axiom,
% 0.30/0.61 ! [N: nat] :
% 0.30/0.61 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.30/0.61 = one_one_real ) )
% 0.30/0.61 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.30/0.61 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % minus_one_power_iff
% 0.30/0.61 thf(fact_6283_minus__one__power__iff,axiom,
% 0.30/0.61 ! [N: nat] :
% 0.30/0.61 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.30/0.61 = one_one_complex ) )
% 0.30/0.61 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.30/0.61 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % minus_one_power_iff
% 0.30/0.61 thf(fact_6284_minus__one__power__iff,axiom,
% 0.30/0.61 ! [N: nat] :
% 0.30/0.61 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.30/0.61 = one_one_Code_integer ) )
% 0.30/0.61 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.30/0.61 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % minus_one_power_iff
% 0.30/0.61 thf(fact_6285_minus__one__power__iff,axiom,
% 0.30/0.61 ! [N: nat] :
% 0.30/0.61 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.30/0.61 = one_one_rat ) )
% 0.30/0.61 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.30/0.61 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % minus_one_power_iff
% 0.30/0.61 thf(fact_6286_signed__take__bit__int__eq__self__iff,axiom,
% 0.30/0.61 ! [N: nat,K: int] :
% 0.30/0.61 ( ( ( bit_ri631733984087533419it_int @ N @ K )
% 0.30/0.61 = K )
% 0.30/0.61 = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 0.30/0.61 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % signed_take_bit_int_eq_self_iff
% 0.30/0.61 thf(fact_6287_signed__take__bit__int__eq__self,axiom,
% 0.30/0.61 ! [N: nat,K: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 0.30/0.61 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61 => ( ( bit_ri631733984087533419it_int @ N @ K )
% 0.30/0.61 = K ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % signed_take_bit_int_eq_self
% 0.30/0.61 thf(fact_6288_minus__1__div__exp__eq__int,axiom,
% 0.30/0.61 ! [N: nat] :
% 0.30/0.61 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % minus_1_div_exp_eq_int
% 0.30/0.61 thf(fact_6289_div__pos__neg__trivial,axiom,
% 0.30/0.61 ! [K: int,L: int] :
% 0.30/0.61 ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.61 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 0.30/0.61 => ( ( divide_divide_int @ K @ L )
% 0.30/0.61 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % div_pos_neg_trivial
% 0.30/0.61 thf(fact_6290_add__0__iff,axiom,
% 0.30/0.61 ! [B: complex,A: complex] :
% 0.30/0.61 ( ( B
% 0.30/0.61 = ( plus_plus_complex @ B @ A ) )
% 0.30/0.61 = ( A = zero_zero_complex ) ) ).
% 0.30/0.61
% 0.30/0.61 % add_0_iff
% 0.30/0.61 thf(fact_6291_add__0__iff,axiom,
% 0.30/0.61 ! [B: real,A: real] :
% 0.30/0.61 ( ( B
% 0.30/0.61 = ( plus_plus_real @ B @ A ) )
% 0.30/0.61 = ( A = zero_zero_real ) ) ).
% 0.30/0.61
% 0.30/0.61 % add_0_iff
% 0.30/0.61 thf(fact_6292_add__0__iff,axiom,
% 0.30/0.61 ! [B: rat,A: rat] :
% 0.30/0.61 ( ( B
% 0.30/0.61 = ( plus_plus_rat @ B @ A ) )
% 0.30/0.61 = ( A = zero_zero_rat ) ) ).
% 0.30/0.61
% 0.30/0.61 % add_0_iff
% 0.30/0.61 thf(fact_6293_add__0__iff,axiom,
% 0.30/0.61 ! [B: nat,A: nat] :
% 0.30/0.61 ( ( B
% 0.30/0.61 = ( plus_plus_nat @ B @ A ) )
% 0.30/0.61 = ( A = zero_zero_nat ) ) ).
% 0.30/0.61
% 0.30/0.61 % add_0_iff
% 0.30/0.61 thf(fact_6294_add__0__iff,axiom,
% 0.30/0.61 ! [B: int,A: int] :
% 0.30/0.61 ( ( B
% 0.30/0.61 = ( plus_plus_int @ B @ A ) )
% 0.30/0.61 = ( A = zero_zero_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % add_0_iff
% 0.30/0.61 thf(fact_6295_crossproduct__eq,axiom,
% 0.30/0.61 ! [W: real,Y3: real,X2: real,Z: real] :
% 0.30/0.61 ( ( ( plus_plus_real @ ( times_times_real @ W @ Y3 ) @ ( times_times_real @ X2 @ Z ) )
% 0.30/0.61 = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X2 @ Y3 ) ) )
% 0.30/0.61 = ( ( W = X2 )
% 0.30/0.61 | ( Y3 = Z ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % crossproduct_eq
% 0.30/0.61 thf(fact_6296_crossproduct__eq,axiom,
% 0.30/0.61 ! [W: rat,Y3: rat,X2: rat,Z: rat] :
% 0.30/0.61 ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y3 ) @ ( times_times_rat @ X2 @ Z ) )
% 0.30/0.61 = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X2 @ Y3 ) ) )
% 0.30/0.61 = ( ( W = X2 )
% 0.30/0.61 | ( Y3 = Z ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % crossproduct_eq
% 0.30/0.61 thf(fact_6297_crossproduct__eq,axiom,
% 0.30/0.61 ! [W: nat,Y3: nat,X2: nat,Z: nat] :
% 0.30/0.61 ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y3 ) @ ( times_times_nat @ X2 @ Z ) )
% 0.30/0.61 = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X2 @ Y3 ) ) )
% 0.30/0.61 = ( ( W = X2 )
% 0.30/0.61 | ( Y3 = Z ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % crossproduct_eq
% 0.30/0.61 thf(fact_6298_crossproduct__eq,axiom,
% 0.30/0.61 ! [W: int,Y3: int,X2: int,Z: int] :
% 0.30/0.61 ( ( ( plus_plus_int @ ( times_times_int @ W @ Y3 ) @ ( times_times_int @ X2 @ Z ) )
% 0.30/0.61 = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X2 @ Y3 ) ) )
% 0.30/0.61 = ( ( W = X2 )
% 0.30/0.61 | ( Y3 = Z ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % crossproduct_eq
% 0.30/0.61 thf(fact_6299_crossproduct__noteq,axiom,
% 0.30/0.61 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.61 ( ( ( A != B )
% 0.30/0.61 & ( C != D ) )
% 0.30/0.61 = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
% 0.30/0.61 != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % crossproduct_noteq
% 0.30/0.61 thf(fact_6300_crossproduct__noteq,axiom,
% 0.30/0.61 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.61 ( ( ( A != B )
% 0.30/0.61 & ( C != D ) )
% 0.30/0.61 = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
% 0.30/0.61 != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % crossproduct_noteq
% 0.30/0.61 thf(fact_6301_crossproduct__noteq,axiom,
% 0.30/0.61 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.61 ( ( ( A != B )
% 0.30/0.61 & ( C != D ) )
% 0.30/0.61 = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
% 0.30/0.61 != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % crossproduct_noteq
% 0.30/0.61 thf(fact_6302_crossproduct__noteq,axiom,
% 0.30/0.61 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.61 ( ( ( A != B )
% 0.30/0.61 & ( C != D ) )
% 0.30/0.61 = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
% 0.30/0.61 != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % crossproduct_noteq
% 0.30/0.61 thf(fact_6303_divmod__nat__def,axiom,
% 0.30/0.61 ( divmod_nat
% 0.30/0.61 = ( ^ [M3: nat,N3: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M3 @ N3 ) @ ( modulo_modulo_nat @ M3 @ N3 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % divmod_nat_def
% 0.30/0.61 thf(fact_6304_power__minus1__odd,axiom,
% 0.30/0.61 ! [N: nat] :
% 0.30/0.61 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.61 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % power_minus1_odd
% 0.30/0.61 thf(fact_6305_power__minus1__odd,axiom,
% 0.30/0.61 ! [N: nat] :
% 0.30/0.61 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.61 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.61
% 0.30/0.61 % power_minus1_odd
% 0.30/0.61 thf(fact_6306_power__minus1__odd,axiom,
% 0.30/0.61 ! [N: nat] :
% 0.30/0.61 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.61 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.61
% 0.30/0.61 % power_minus1_odd
% 0.30/0.61 thf(fact_6307_power__minus1__odd,axiom,
% 0.30/0.61 ! [N: nat] :
% 0.30/0.61 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.61 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.61
% 0.30/0.61 % power_minus1_odd
% 0.30/0.61 thf(fact_6308_power__minus1__odd,axiom,
% 0.30/0.61 ! [N: nat] :
% 0.30/0.61 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.61 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.61
% 0.30/0.61 % power_minus1_odd
% 0.30/0.61 thf(fact_6309_int__bit__induct,axiom,
% 0.30/0.61 ! [P: int > $o,K: int] :
% 0.30/0.61 ( ( P @ zero_zero_int )
% 0.30/0.61 => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.61 => ( ! [K3: int] :
% 0.30/0.61 ( ( P @ K3 )
% 0.30/0.61 => ( ( K3 != zero_zero_int )
% 0.30/0.61 => ( P @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.61 => ( ! [K3: int] :
% 0.30/0.61 ( ( P @ K3 )
% 0.30/0.61 => ( ( K3
% 0.30/0.61 != ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.61 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
% 0.30/0.61 => ( P @ K ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % int_bit_induct
% 0.30/0.61 thf(fact_6310_signed__take__bit__int__greater__eq,axiom,
% 0.30/0.61 ! [K: int,N: nat] :
% 0.30/0.61 ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.61 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % signed_take_bit_int_greater_eq
% 0.30/0.61 thf(fact_6311_one__div__minus__numeral,axiom,
% 0.30/0.61 ! [N: num] :
% 0.30/0.61 ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.61 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % one_div_minus_numeral
% 0.30/0.61 thf(fact_6312_minus__one__div__numeral,axiom,
% 0.30/0.61 ! [N: num] :
% 0.30/0.61 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.61 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % minus_one_div_numeral
% 0.30/0.61 thf(fact_6313_numeral__div__minus__numeral,axiom,
% 0.30/0.61 ! [M: num,N: num] :
% 0.30/0.61 ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.61 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % numeral_div_minus_numeral
% 0.30/0.61 thf(fact_6314_minus__numeral__div__numeral,axiom,
% 0.30/0.61 ! [M: num,N: num] :
% 0.30/0.61 ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.61 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % minus_numeral_div_numeral
% 0.30/0.61 thf(fact_6315_dbl__dec__simps_I4_J,axiom,
% 0.30/0.61 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.61 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(4)
% 0.30/0.61 thf(fact_6316_dbl__dec__simps_I4_J,axiom,
% 0.30/0.61 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.61 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(4)
% 0.30/0.61 thf(fact_6317_dbl__dec__simps_I4_J,axiom,
% 0.30/0.61 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.61 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(4)
% 0.30/0.61 thf(fact_6318_dbl__dec__simps_I4_J,axiom,
% 0.30/0.61 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.61 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(4)
% 0.30/0.61 thf(fact_6319_dbl__dec__simps_I4_J,axiom,
% 0.30/0.61 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.61 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(4)
% 0.30/0.61 thf(fact_6320_of__int__code__if,axiom,
% 0.30/0.61 ( ring_1_of_int_int
% 0.30/0.61 = ( ^ [K2: int] :
% 0.30/0.61 ( if_int @ ( K2 = zero_zero_int ) @ zero_zero_int
% 0.30/0.61 @ ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K2 ) ) )
% 0.30/0.61 @ ( if_int
% 0.30/0.61 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.61 = zero_zero_int )
% 0.30/0.61 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.30/0.61 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_code_if
% 0.30/0.61 thf(fact_6321_of__int__code__if,axiom,
% 0.30/0.61 ( ring_1_of_int_real
% 0.30/0.61 = ( ^ [K2: int] :
% 0.30/0.61 ( if_real @ ( K2 = zero_zero_int ) @ zero_zero_real
% 0.30/0.61 @ ( if_real @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K2 ) ) )
% 0.30/0.61 @ ( if_real
% 0.30/0.61 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.61 = zero_zero_int )
% 0.30/0.61 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.30/0.61 @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_code_if
% 0.30/0.61 thf(fact_6322_of__int__code__if,axiom,
% 0.30/0.61 ( ring_17405671764205052669omplex
% 0.30/0.61 = ( ^ [K2: int] :
% 0.30/0.61 ( if_complex @ ( K2 = zero_zero_int ) @ zero_zero_complex
% 0.30/0.61 @ ( if_complex @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K2 ) ) )
% 0.30/0.61 @ ( if_complex
% 0.30/0.61 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.61 = zero_zero_int )
% 0.30/0.61 @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.30/0.61 @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_code_if
% 0.30/0.61 thf(fact_6323_of__int__code__if,axiom,
% 0.30/0.61 ( ring_18347121197199848620nteger
% 0.30/0.61 = ( ^ [K2: int] :
% 0.30/0.61 ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 0.30/0.61 @ ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K2 ) ) )
% 0.30/0.61 @ ( if_Code_integer
% 0.30/0.61 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.61 = zero_zero_int )
% 0.30/0.61 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.30/0.61 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_code_if
% 0.30/0.61 thf(fact_6324_of__int__code__if,axiom,
% 0.30/0.61 ( ring_1_of_int_rat
% 0.30/0.61 = ( ^ [K2: int] :
% 0.30/0.61 ( if_rat @ ( K2 = zero_zero_int ) @ zero_zero_rat
% 0.30/0.61 @ ( if_rat @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K2 ) ) )
% 0.30/0.61 @ ( if_rat
% 0.30/0.61 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.61 = zero_zero_int )
% 0.30/0.61 @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.30/0.61 @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_code_if
% 0.30/0.61 thf(fact_6325_int__ge__less__than__def,axiom,
% 0.30/0.61 ( int_ge_less_than
% 0.30/0.61 = ( ^ [D2: int] :
% 0.30/0.61 ( collec213857154873943460nt_int
% 0.30/0.61 @ ( produc4947309494688390418_int_o
% 0.30/0.61 @ ^ [Z6: int,Z3: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ D2 @ Z6 )
% 0.30/0.61 & ( ord_less_int @ Z6 @ Z3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % int_ge_less_than_def
% 0.30/0.61 thf(fact_6326_dbl__dec__simps_I3_J,axiom,
% 0.30/0.61 ( ( neg_nu6511756317524482435omplex @ one_one_complex )
% 0.30/0.61 = one_one_complex ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(3)
% 0.30/0.61 thf(fact_6327_dbl__dec__simps_I3_J,axiom,
% 0.30/0.61 ( ( neg_nu6075765906172075777c_real @ one_one_real )
% 0.30/0.61 = one_one_real ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(3)
% 0.30/0.61 thf(fact_6328_dbl__dec__simps_I3_J,axiom,
% 0.30/0.61 ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
% 0.30/0.61 = one_one_rat ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(3)
% 0.30/0.61 thf(fact_6329_dbl__dec__simps_I3_J,axiom,
% 0.30/0.61 ( ( neg_nu3811975205180677377ec_int @ one_one_int )
% 0.30/0.61 = one_one_int ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(3)
% 0.30/0.61 thf(fact_6330_of__int__numeral,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
% 0.30/0.61 = ( numera6690914467698888265omplex @ K ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_numeral
% 0.30/0.61 thf(fact_6331_of__int__numeral,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 0.30/0.61 = ( numeral_numeral_real @ K ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_numeral
% 0.30/0.61 thf(fact_6332_of__int__numeral,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
% 0.30/0.61 = ( numeral_numeral_rat @ K ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_numeral
% 0.30/0.61 thf(fact_6333_of__int__numeral,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 0.30/0.61 = ( numeral_numeral_int @ K ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_numeral
% 0.30/0.61 thf(fact_6334_of__int__eq__numeral__iff,axiom,
% 0.30/0.61 ! [Z: int,N: num] :
% 0.30/0.61 ( ( ( ring_17405671764205052669omplex @ Z )
% 0.30/0.61 = ( numera6690914467698888265omplex @ N ) )
% 0.30/0.61 = ( Z
% 0.30/0.61 = ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_numeral_iff
% 0.30/0.61 thf(fact_6335_of__int__eq__numeral__iff,axiom,
% 0.30/0.61 ! [Z: int,N: num] :
% 0.30/0.61 ( ( ( ring_1_of_int_real @ Z )
% 0.30/0.61 = ( numeral_numeral_real @ N ) )
% 0.30/0.61 = ( Z
% 0.30/0.61 = ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_numeral_iff
% 0.30/0.61 thf(fact_6336_of__int__eq__numeral__iff,axiom,
% 0.30/0.61 ! [Z: int,N: num] :
% 0.30/0.61 ( ( ( ring_1_of_int_rat @ Z )
% 0.30/0.61 = ( numeral_numeral_rat @ N ) )
% 0.30/0.61 = ( Z
% 0.30/0.61 = ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_numeral_iff
% 0.30/0.61 thf(fact_6337_of__int__eq__numeral__iff,axiom,
% 0.30/0.61 ! [Z: int,N: num] :
% 0.30/0.61 ( ( ( ring_1_of_int_int @ Z )
% 0.30/0.61 = ( numeral_numeral_int @ N ) )
% 0.30/0.61 = ( Z
% 0.30/0.61 = ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_numeral_iff
% 0.30/0.61 thf(fact_6338_of__int__le__iff,axiom,
% 0.30/0.61 ! [W: int,Z: int] :
% 0.30/0.61 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.61 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_iff
% 0.30/0.61 thf(fact_6339_of__int__le__iff,axiom,
% 0.30/0.61 ! [W: int,Z: int] :
% 0.30/0.61 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 0.30/0.61 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_iff
% 0.30/0.61 thf(fact_6340_of__int__le__iff,axiom,
% 0.30/0.61 ! [W: int,Z: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.30/0.61 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_iff
% 0.30/0.61 thf(fact_6341_of__int__less__iff,axiom,
% 0.30/0.61 ! [W: int,Z: int] :
% 0.30/0.61 ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.61 = ( ord_less_int @ W @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_iff
% 0.30/0.61 thf(fact_6342_of__int__less__iff,axiom,
% 0.30/0.61 ! [W: int,Z: int] :
% 0.30/0.61 ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 0.30/0.61 = ( ord_less_int @ W @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_iff
% 0.30/0.61 thf(fact_6343_of__int__less__iff,axiom,
% 0.30/0.61 ! [W: int,Z: int] :
% 0.30/0.61 ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.30/0.61 = ( ord_less_int @ W @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_iff
% 0.30/0.61 thf(fact_6344_of__int__eq__1__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ( ring_17405671764205052669omplex @ Z )
% 0.30/0.61 = one_one_complex )
% 0.30/0.61 = ( Z = one_one_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_1_iff
% 0.30/0.61 thf(fact_6345_of__int__eq__1__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ( ring_1_of_int_int @ Z )
% 0.30/0.61 = one_one_int )
% 0.30/0.61 = ( Z = one_one_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_1_iff
% 0.30/0.61 thf(fact_6346_of__int__eq__1__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ( ring_1_of_int_real @ Z )
% 0.30/0.61 = one_one_real )
% 0.30/0.61 = ( Z = one_one_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_1_iff
% 0.30/0.61 thf(fact_6347_of__int__eq__1__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ( ring_1_of_int_rat @ Z )
% 0.30/0.61 = one_one_rat )
% 0.30/0.61 = ( Z = one_one_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_1_iff
% 0.30/0.61 thf(fact_6348_of__int__1,axiom,
% 0.30/0.61 ( ( ring_17405671764205052669omplex @ one_one_int )
% 0.30/0.61 = one_one_complex ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_1
% 0.30/0.61 thf(fact_6349_of__int__1,axiom,
% 0.30/0.61 ( ( ring_1_of_int_int @ one_one_int )
% 0.30/0.61 = one_one_int ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_1
% 0.30/0.61 thf(fact_6350_of__int__1,axiom,
% 0.30/0.61 ( ( ring_1_of_int_real @ one_one_int )
% 0.30/0.61 = one_one_real ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_1
% 0.30/0.61 thf(fact_6351_of__int__1,axiom,
% 0.30/0.61 ( ( ring_1_of_int_rat @ one_one_int )
% 0.30/0.61 = one_one_rat ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_1
% 0.30/0.61 thf(fact_6352_of__int__mult,axiom,
% 0.30/0.61 ! [W: int,Z: int] :
% 0.30/0.61 ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
% 0.30/0.61 = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_mult
% 0.30/0.61 thf(fact_6353_of__int__mult,axiom,
% 0.30/0.61 ! [W: int,Z: int] :
% 0.30/0.61 ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
% 0.30/0.61 = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_mult
% 0.30/0.61 thf(fact_6354_of__int__mult,axiom,
% 0.30/0.61 ! [W: int,Z: int] :
% 0.30/0.61 ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
% 0.30/0.61 = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_mult
% 0.30/0.61 thf(fact_6355_of__int__add,axiom,
% 0.30/0.61 ! [W: int,Z: int] :
% 0.30/0.61 ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
% 0.30/0.61 = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_add
% 0.30/0.61 thf(fact_6356_of__int__add,axiom,
% 0.30/0.61 ! [W: int,Z: int] :
% 0.30/0.61 ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
% 0.30/0.61 = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_add
% 0.30/0.61 thf(fact_6357_of__int__add,axiom,
% 0.30/0.61 ! [W: int,Z: int] :
% 0.30/0.61 ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
% 0.30/0.61 = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_add
% 0.30/0.61 thf(fact_6358_of__int__diff,axiom,
% 0.30/0.61 ! [W: int,Z: int] :
% 0.30/0.61 ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
% 0.30/0.61 = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_diff
% 0.30/0.61 thf(fact_6359_of__int__diff,axiom,
% 0.30/0.61 ! [W: int,Z: int] :
% 0.30/0.61 ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
% 0.30/0.61 = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_diff
% 0.30/0.61 thf(fact_6360_of__int__diff,axiom,
% 0.30/0.61 ! [W: int,Z: int] :
% 0.30/0.61 ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
% 0.30/0.61 = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_diff
% 0.30/0.61 thf(fact_6361_of__int__power,axiom,
% 0.30/0.61 ! [Z: int,N: nat] :
% 0.30/0.61 ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N ) )
% 0.30/0.61 = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_power
% 0.30/0.61 thf(fact_6362_of__int__power,axiom,
% 0.30/0.61 ! [Z: int,N: nat] :
% 0.30/0.61 ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
% 0.30/0.61 = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_power
% 0.30/0.61 thf(fact_6363_of__int__power,axiom,
% 0.30/0.61 ! [Z: int,N: nat] :
% 0.30/0.61 ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
% 0.30/0.61 = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_power
% 0.30/0.61 thf(fact_6364_of__int__power,axiom,
% 0.30/0.61 ! [Z: int,N: nat] :
% 0.30/0.61 ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N ) )
% 0.30/0.61 = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_power
% 0.30/0.61 thf(fact_6365_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.30/0.61 ! [B: int,W: nat,X2: int] :
% 0.30/0.61 ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
% 0.30/0.61 = ( ring_1_of_int_rat @ X2 ) )
% 0.30/0.61 = ( ( power_power_int @ B @ W )
% 0.30/0.61 = X2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_of_int_power_cancel_iff
% 0.30/0.61 thf(fact_6366_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.30/0.61 ! [B: int,W: nat,X2: int] :
% 0.30/0.61 ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
% 0.30/0.61 = ( ring_1_of_int_real @ X2 ) )
% 0.30/0.61 = ( ( power_power_int @ B @ W )
% 0.30/0.61 = X2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_of_int_power_cancel_iff
% 0.30/0.61 thf(fact_6367_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.30/0.61 ! [B: int,W: nat,X2: int] :
% 0.30/0.61 ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
% 0.30/0.61 = ( ring_1_of_int_int @ X2 ) )
% 0.30/0.61 = ( ( power_power_int @ B @ W )
% 0.30/0.61 = X2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_of_int_power_cancel_iff
% 0.30/0.61 thf(fact_6368_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.30/0.61 ! [B: int,W: nat,X2: int] :
% 0.30/0.61 ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
% 0.30/0.61 = ( ring_17405671764205052669omplex @ X2 ) )
% 0.30/0.61 = ( ( power_power_int @ B @ W )
% 0.30/0.61 = X2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_of_int_power_cancel_iff
% 0.30/0.61 thf(fact_6369_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: int,B: int,W: nat] :
% 0.30/0.61 ( ( ( ring_1_of_int_rat @ X2 )
% 0.30/0.61 = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 0.30/0.61 = ( X2
% 0.30/0.61 = ( power_power_int @ B @ W ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_power_eq_of_int_cancel_iff
% 0.30/0.61 thf(fact_6370_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: int,B: int,W: nat] :
% 0.30/0.61 ( ( ( ring_1_of_int_real @ X2 )
% 0.30/0.61 = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.30/0.61 = ( X2
% 0.30/0.61 = ( power_power_int @ B @ W ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_power_eq_of_int_cancel_iff
% 0.30/0.61 thf(fact_6371_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: int,B: int,W: nat] :
% 0.30/0.61 ( ( ( ring_1_of_int_int @ X2 )
% 0.30/0.61 = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.30/0.61 = ( X2
% 0.30/0.61 = ( power_power_int @ B @ W ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_power_eq_of_int_cancel_iff
% 0.30/0.61 thf(fact_6372_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: int,B: int,W: nat] :
% 0.30/0.61 ( ( ( ring_17405671764205052669omplex @ X2 )
% 0.30/0.61 = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
% 0.30/0.61 = ( X2
% 0.30/0.61 = ( power_power_int @ B @ W ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_power_eq_of_int_cancel_iff
% 0.30/0.61 thf(fact_6373_dbl__dec__simps_I5_J,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.30/0.61 = ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(5)
% 0.30/0.61 thf(fact_6374_dbl__dec__simps_I5_J,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) )
% 0.30/0.61 = ( numeral_numeral_real @ ( bitM @ K ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(5)
% 0.30/0.61 thf(fact_6375_dbl__dec__simps_I5_J,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) )
% 0.30/0.61 = ( numeral_numeral_rat @ ( bitM @ K ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(5)
% 0.30/0.61 thf(fact_6376_dbl__dec__simps_I5_J,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
% 0.30/0.61 = ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(5)
% 0.30/0.61 thf(fact_6377_dbl__dec__simps_I2_J,axiom,
% 0.30/0.61 ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 0.30/0.61 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(2)
% 0.30/0.61 thf(fact_6378_dbl__dec__simps_I2_J,axiom,
% 0.30/0.61 ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 0.30/0.61 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(2)
% 0.30/0.61 thf(fact_6379_dbl__dec__simps_I2_J,axiom,
% 0.30/0.61 ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 0.30/0.61 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(2)
% 0.30/0.61 thf(fact_6380_dbl__dec__simps_I2_J,axiom,
% 0.30/0.61 ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 0.30/0.61 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(2)
% 0.30/0.61 thf(fact_6381_dbl__dec__simps_I2_J,axiom,
% 0.30/0.61 ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 0.30/0.61 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(2)
% 0.30/0.61 thf(fact_6382_dbl__dec__simps_I1_J,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.61 = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(1)
% 0.30/0.61 thf(fact_6383_dbl__dec__simps_I1_J,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.30/0.61 = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(1)
% 0.30/0.61 thf(fact_6384_dbl__dec__simps_I1_J,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.30/0.61 = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(1)
% 0.30/0.61 thf(fact_6385_dbl__dec__simps_I1_J,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.30/0.61 = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(1)
% 0.30/0.61 thf(fact_6386_dbl__dec__simps_I1_J,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.30/0.61 = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_simps(1)
% 0.30/0.61 thf(fact_6387_dbl__inc__simps_I1_J,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.61 = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_inc_simps(1)
% 0.30/0.61 thf(fact_6388_dbl__inc__simps_I1_J,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.30/0.61 = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_inc_simps(1)
% 0.30/0.61 thf(fact_6389_dbl__inc__simps_I1_J,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.30/0.61 = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_inc_simps(1)
% 0.30/0.61 thf(fact_6390_dbl__inc__simps_I1_J,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.30/0.61 = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_inc_simps(1)
% 0.30/0.61 thf(fact_6391_dbl__inc__simps_I1_J,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.30/0.61 = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_inc_simps(1)
% 0.30/0.61 thf(fact_6392_of__int__le__0__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.30/0.61 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_0_iff
% 0.30/0.61 thf(fact_6393_of__int__le__0__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 0.30/0.61 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_0_iff
% 0.30/0.61 thf(fact_6394_of__int__le__0__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.30/0.61 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_0_iff
% 0.30/0.61 thf(fact_6395_of__int__0__le__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.61 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_0_le_iff
% 0.30/0.61 thf(fact_6396_of__int__0__le__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.30/0.61 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_0_le_iff
% 0.30/0.61 thf(fact_6397_of__int__0__le__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.30/0.61 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_0_le_iff
% 0.30/0.61 thf(fact_6398_of__int__less__0__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.30/0.61 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_0_iff
% 0.30/0.61 thf(fact_6399_of__int__less__0__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 0.30/0.61 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_0_iff
% 0.30/0.61 thf(fact_6400_of__int__less__0__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.30/0.61 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_0_iff
% 0.30/0.61 thf(fact_6401_of__int__0__less__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.61 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_0_less_iff
% 0.30/0.61 thf(fact_6402_of__int__0__less__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.30/0.61 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_0_less_iff
% 0.30/0.61 thf(fact_6403_of__int__0__less__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.30/0.61 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_0_less_iff
% 0.30/0.61 thf(fact_6404_of__int__le__numeral__iff,axiom,
% 0.30/0.61 ! [Z: int,N: num] :
% 0.30/0.61 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 0.30/0.61 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_numeral_iff
% 0.30/0.61 thf(fact_6405_of__int__le__numeral__iff,axiom,
% 0.30/0.61 ! [Z: int,N: num] :
% 0.30/0.61 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 0.30/0.61 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_numeral_iff
% 0.30/0.61 thf(fact_6406_of__int__le__numeral__iff,axiom,
% 0.30/0.61 ! [Z: int,N: num] :
% 0.30/0.61 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.61 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_numeral_iff
% 0.30/0.61 thf(fact_6407_of__int__numeral__le__iff,axiom,
% 0.30/0.61 ! [N: num,Z: int] :
% 0.30/0.61 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.61 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_numeral_le_iff
% 0.30/0.61 thf(fact_6408_of__int__numeral__le__iff,axiom,
% 0.30/0.61 ! [N: num,Z: int] :
% 0.30/0.61 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 0.30/0.61 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_numeral_le_iff
% 0.30/0.61 thf(fact_6409_of__int__numeral__le__iff,axiom,
% 0.30/0.61 ! [N: num,Z: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 0.30/0.61 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_numeral_le_iff
% 0.30/0.61 thf(fact_6410_of__int__less__numeral__iff,axiom,
% 0.30/0.61 ! [Z: int,N: num] :
% 0.30/0.61 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 0.30/0.61 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_numeral_iff
% 0.30/0.61 thf(fact_6411_of__int__less__numeral__iff,axiom,
% 0.30/0.61 ! [Z: int,N: num] :
% 0.30/0.61 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 0.30/0.61 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_numeral_iff
% 0.30/0.61 thf(fact_6412_of__int__less__numeral__iff,axiom,
% 0.30/0.61 ! [Z: int,N: num] :
% 0.30/0.61 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.61 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_numeral_iff
% 0.30/0.61 thf(fact_6413_of__int__numeral__less__iff,axiom,
% 0.30/0.61 ! [N: num,Z: int] :
% 0.30/0.61 ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.61 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_numeral_less_iff
% 0.30/0.61 thf(fact_6414_of__int__numeral__less__iff,axiom,
% 0.30/0.61 ! [N: num,Z: int] :
% 0.30/0.61 ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 0.30/0.61 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_numeral_less_iff
% 0.30/0.61 thf(fact_6415_of__int__numeral__less__iff,axiom,
% 0.30/0.61 ! [N: num,Z: int] :
% 0.30/0.61 ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 0.30/0.61 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_numeral_less_iff
% 0.30/0.61 thf(fact_6416_of__int__le__1__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.30/0.61 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_1_iff
% 0.30/0.61 thf(fact_6417_of__int__le__1__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 0.30/0.61 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_1_iff
% 0.30/0.61 thf(fact_6418_of__int__le__1__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.30/0.61 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_1_iff
% 0.30/0.61 thf(fact_6419_of__int__1__le__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.61 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_1_le_iff
% 0.30/0.61 thf(fact_6420_of__int__1__le__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.30/0.61 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_1_le_iff
% 0.30/0.61 thf(fact_6421_of__int__1__le__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.30/0.61 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_1_le_iff
% 0.30/0.61 thf(fact_6422_of__int__1__less__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.61 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_1_less_iff
% 0.30/0.61 thf(fact_6423_of__int__1__less__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.30/0.61 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_1_less_iff
% 0.30/0.61 thf(fact_6424_of__int__1__less__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.30/0.61 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_1_less_iff
% 0.30/0.61 thf(fact_6425_of__int__less__1__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.30/0.61 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_1_iff
% 0.30/0.61 thf(fact_6426_of__int__less__1__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 0.30/0.61 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_1_iff
% 0.30/0.61 thf(fact_6427_of__int__less__1__iff,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.30/0.61 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_1_iff
% 0.30/0.61 thf(fact_6428_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [Y3: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ( ring_17405671764205052669omplex @ Y3 )
% 0.30/0.61 = ( power_power_complex @ ( numera6690914467698888265omplex @ X2 ) @ N ) )
% 0.30/0.61 = ( Y3
% 0.30/0.61 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6429_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [Y3: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ( ring_1_of_int_real @ Y3 )
% 0.30/0.61 = ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
% 0.30/0.61 = ( Y3
% 0.30/0.61 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6430_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [Y3: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ( ring_1_of_int_rat @ Y3 )
% 0.30/0.61 = ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) )
% 0.30/0.61 = ( Y3
% 0.30/0.61 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6431_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [Y3: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ( ring_1_of_int_int @ Y3 )
% 0.30/0.61 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) )
% 0.30/0.61 = ( Y3
% 0.30/0.61 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6432_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,Y3: int] :
% 0.30/0.61 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X2 ) @ N )
% 0.30/0.61 = ( ring_17405671764205052669omplex @ Y3 ) )
% 0.30/0.61 = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N )
% 0.30/0.61 = Y3 ) ) ).
% 0.30/0.61
% 0.30/0.61 % numeral_power_eq_of_int_cancel_iff
% 0.30/0.61 thf(fact_6433_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,Y3: int] :
% 0.30/0.61 ( ( ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N )
% 0.30/0.61 = ( ring_1_of_int_real @ Y3 ) )
% 0.30/0.61 = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N )
% 0.30/0.61 = Y3 ) ) ).
% 0.30/0.61
% 0.30/0.61 % numeral_power_eq_of_int_cancel_iff
% 0.30/0.61 thf(fact_6434_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,Y3: int] :
% 0.30/0.61 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N )
% 0.30/0.61 = ( ring_1_of_int_rat @ Y3 ) )
% 0.30/0.61 = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N )
% 0.30/0.61 = Y3 ) ) ).
% 0.30/0.61
% 0.30/0.61 % numeral_power_eq_of_int_cancel_iff
% 0.30/0.61 thf(fact_6435_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,Y3: int] :
% 0.30/0.61 ( ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N )
% 0.30/0.61 = ( ring_1_of_int_int @ Y3 ) )
% 0.30/0.61 = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N )
% 0.30/0.61 = Y3 ) ) ).
% 0.30/0.61
% 0.30/0.61 % numeral_power_eq_of_int_cancel_iff
% 0.30/0.61 thf(fact_6436_of__int__power__le__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: int,B: int,W: nat] :
% 0.30/0.61 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X2 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.30/0.61 = ( ord_less_eq_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_power_le_of_int_cancel_iff
% 0.30/0.61 thf(fact_6437_of__int__power__le__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: int,B: int,W: nat] :
% 0.30/0.61 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X2 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 0.30/0.61 = ( ord_less_eq_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_power_le_of_int_cancel_iff
% 0.30/0.61 thf(fact_6438_of__int__power__le__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: int,B: int,W: nat] :
% 0.30/0.61 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X2 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.30/0.61 = ( ord_less_eq_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_power_le_of_int_cancel_iff
% 0.30/0.61 thf(fact_6439_of__int__le__of__int__power__cancel__iff,axiom,
% 0.30/0.61 ! [B: int,W: nat,X2: int] :
% 0.30/0.61 ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X2 ) )
% 0.30/0.61 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_of_int_power_cancel_iff
% 0.30/0.61 thf(fact_6440_of__int__le__of__int__power__cancel__iff,axiom,
% 0.30/0.61 ! [B: int,W: nat,X2: int] :
% 0.30/0.61 ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X2 ) )
% 0.30/0.61 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_of_int_power_cancel_iff
% 0.30/0.61 thf(fact_6441_of__int__le__of__int__power__cancel__iff,axiom,
% 0.30/0.61 ! [B: int,W: nat,X2: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X2 ) )
% 0.30/0.61 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_of_int_power_cancel_iff
% 0.30/0.61 thf(fact_6442_of__int__power__less__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: int,B: int,W: nat] :
% 0.30/0.61 ( ( ord_less_real @ ( ring_1_of_int_real @ X2 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.30/0.61 = ( ord_less_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_power_less_of_int_cancel_iff
% 0.30/0.61 thf(fact_6443_of__int__power__less__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: int,B: int,W: nat] :
% 0.30/0.61 ( ( ord_less_rat @ ( ring_1_of_int_rat @ X2 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 0.30/0.61 = ( ord_less_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_power_less_of_int_cancel_iff
% 0.30/0.61 thf(fact_6444_of__int__power__less__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: int,B: int,W: nat] :
% 0.30/0.61 ( ( ord_less_int @ ( ring_1_of_int_int @ X2 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.30/0.61 = ( ord_less_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_power_less_of_int_cancel_iff
% 0.30/0.61 thf(fact_6445_of__int__less__of__int__power__cancel__iff,axiom,
% 0.30/0.61 ! [B: int,W: nat,X2: int] :
% 0.30/0.61 ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X2 ) )
% 0.30/0.61 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_of_int_power_cancel_iff
% 0.30/0.61 thf(fact_6446_of__int__less__of__int__power__cancel__iff,axiom,
% 0.30/0.61 ! [B: int,W: nat,X2: int] :
% 0.30/0.61 ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X2 ) )
% 0.30/0.61 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_of_int_power_cancel_iff
% 0.30/0.61 thf(fact_6447_of__int__less__of__int__power__cancel__iff,axiom,
% 0.30/0.61 ! [B: int,W: nat,X2: int] :
% 0.30/0.61 ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X2 ) )
% 0.30/0.61 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_of_int_power_cancel_iff
% 0.30/0.61 thf(fact_6448_numeral__power__le__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,A: int] :
% 0.30/0.61 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.30/0.61 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A ) ) ).
% 0.30/0.61
% 0.30/0.61 % numeral_power_le_of_int_cancel_iff
% 0.30/0.61 thf(fact_6449_numeral__power__le__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,A: int] :
% 0.30/0.61 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.30/0.61 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A ) ) ).
% 0.30/0.61
% 0.30/0.61 % numeral_power_le_of_int_cancel_iff
% 0.30/0.61 thf(fact_6450_numeral__power__le__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,A: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.30/0.61 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A ) ) ).
% 0.30/0.61
% 0.30/0.61 % numeral_power_le_of_int_cancel_iff
% 0.30/0.61 thf(fact_6451_of__int__le__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [A: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
% 0.30/0.61 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6452_of__int__le__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [A: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) )
% 0.30/0.61 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6453_of__int__le__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [A: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) )
% 0.30/0.61 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6454_numeral__power__less__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,A: int] :
% 0.30/0.61 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.30/0.61 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A ) ) ).
% 0.30/0.61
% 0.30/0.61 % numeral_power_less_of_int_cancel_iff
% 0.30/0.61 thf(fact_6455_numeral__power__less__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,A: int] :
% 0.30/0.61 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.30/0.61 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A ) ) ).
% 0.30/0.61
% 0.30/0.61 % numeral_power_less_of_int_cancel_iff
% 0.30/0.61 thf(fact_6456_numeral__power__less__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,A: int] :
% 0.30/0.61 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.30/0.61 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A ) ) ).
% 0.30/0.61
% 0.30/0.61 % numeral_power_less_of_int_cancel_iff
% 0.30/0.61 thf(fact_6457_of__int__less__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [A: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
% 0.30/0.61 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6458_of__int__less__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [A: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) )
% 0.30/0.61 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6459_of__int__less__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [A: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) )
% 0.30/0.61 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6460_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [Y3: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ( ring_1_of_int_int @ Y3 )
% 0.30/0.61 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) )
% 0.30/0.61 = ( Y3
% 0.30/0.61 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_neg_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6461_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [Y3: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ( ring_1_of_int_real @ Y3 )
% 0.30/0.61 = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) )
% 0.30/0.61 = ( Y3
% 0.30/0.61 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_neg_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6462_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [Y3: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ( ring_17405671764205052669omplex @ Y3 )
% 0.30/0.61 = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X2 ) ) @ N ) )
% 0.30/0.61 = ( Y3
% 0.30/0.61 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_neg_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6463_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [Y3: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ( ring_18347121197199848620nteger @ Y3 )
% 0.30/0.61 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) )
% 0.30/0.61 = ( Y3
% 0.30/0.61 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_neg_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6464_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [Y3: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ( ring_1_of_int_rat @ Y3 )
% 0.30/0.61 = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) )
% 0.30/0.61 = ( Y3
% 0.30/0.61 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_eq_neg_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6465_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,Y3: int] :
% 0.30/0.61 ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N )
% 0.30/0.61 = ( ring_1_of_int_int @ Y3 ) )
% 0.30/0.61 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N )
% 0.30/0.61 = Y3 ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_numeral_power_eq_of_int_cancel_iff
% 0.30/0.61 thf(fact_6466_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,Y3: int] :
% 0.30/0.61 ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N )
% 0.30/0.61 = ( ring_1_of_int_real @ Y3 ) )
% 0.30/0.61 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N )
% 0.30/0.61 = Y3 ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_numeral_power_eq_of_int_cancel_iff
% 0.30/0.61 thf(fact_6467_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,Y3: int] :
% 0.30/0.61 ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X2 ) ) @ N )
% 0.30/0.61 = ( ring_17405671764205052669omplex @ Y3 ) )
% 0.30/0.61 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N )
% 0.30/0.61 = Y3 ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_numeral_power_eq_of_int_cancel_iff
% 0.30/0.61 thf(fact_6468_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,Y3: int] :
% 0.30/0.61 ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N )
% 0.30/0.61 = ( ring_18347121197199848620nteger @ Y3 ) )
% 0.30/0.61 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N )
% 0.30/0.61 = Y3 ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_numeral_power_eq_of_int_cancel_iff
% 0.30/0.61 thf(fact_6469_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,Y3: int] :
% 0.30/0.61 ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N )
% 0.30/0.61 = ( ring_1_of_int_rat @ Y3 ) )
% 0.30/0.61 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N )
% 0.30/0.61 = Y3 ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_numeral_power_eq_of_int_cancel_iff
% 0.30/0.61 thf(fact_6470_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [A: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) )
% 0.30/0.61 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_neg_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6471_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [A: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) )
% 0.30/0.61 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_neg_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6472_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [A: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) )
% 0.30/0.61 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_neg_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6473_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [A: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) )
% 0.30/0.61 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_le_neg_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6474_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,A: int] :
% 0.30/0.61 ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.30/0.61 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_numeral_power_le_of_int_cancel_iff
% 0.30/0.61 thf(fact_6475_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,A: int] :
% 0.30/0.61 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.30/0.61 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_numeral_power_le_of_int_cancel_iff
% 0.30/0.61 thf(fact_6476_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,A: int] :
% 0.30/0.61 ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.30/0.61 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_numeral_power_le_of_int_cancel_iff
% 0.30/0.61 thf(fact_6477_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,A: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.30/0.61 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_numeral_power_le_of_int_cancel_iff
% 0.30/0.61 thf(fact_6478_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [A: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) )
% 0.30/0.61 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_neg_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6479_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [A: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) )
% 0.30/0.61 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_neg_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6480_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [A: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) )
% 0.30/0.61 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_neg_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6481_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.61 ! [A: int,X2: num,N: nat] :
% 0.30/0.61 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) )
% 0.30/0.61 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_less_neg_numeral_power_cancel_iff
% 0.30/0.61 thf(fact_6482_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,A: int] :
% 0.30/0.61 ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.30/0.61 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_numeral_power_less_of_int_cancel_iff
% 0.30/0.61 thf(fact_6483_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,A: int] :
% 0.30/0.61 ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.30/0.61 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_numeral_power_less_of_int_cancel_iff
% 0.30/0.61 thf(fact_6484_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,A: int] :
% 0.30/0.61 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.30/0.61 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_numeral_power_less_of_int_cancel_iff
% 0.30/0.61 thf(fact_6485_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.30/0.61 ! [X2: num,N: nat,A: int] :
% 0.30/0.61 ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.30/0.61 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A ) ) ).
% 0.30/0.61
% 0.30/0.61 % neg_numeral_power_less_of_int_cancel_iff
% 0.30/0.61 thf(fact_6486_Collect__case__prod__mono,axiom,
% 0.30/0.61 ! [A2: int > int > $o,B3: int > int > $o] :
% 0.30/0.61 ( ( ord_le6741204236512500942_int_o @ A2 @ B3 )
% 0.30/0.61 => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A2 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B3 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Collect_case_prod_mono
% 0.30/0.61 thf(fact_6487_mult__of__int__commute,axiom,
% 0.30/0.61 ! [X2: int,Y3: real] :
% 0.30/0.61 ( ( times_times_real @ ( ring_1_of_int_real @ X2 ) @ Y3 )
% 0.30/0.61 = ( times_times_real @ Y3 @ ( ring_1_of_int_real @ X2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % mult_of_int_commute
% 0.30/0.61 thf(fact_6488_mult__of__int__commute,axiom,
% 0.30/0.61 ! [X2: int,Y3: rat] :
% 0.30/0.61 ( ( times_times_rat @ ( ring_1_of_int_rat @ X2 ) @ Y3 )
% 0.30/0.61 = ( times_times_rat @ Y3 @ ( ring_1_of_int_rat @ X2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % mult_of_int_commute
% 0.30/0.61 thf(fact_6489_mult__of__int__commute,axiom,
% 0.30/0.61 ! [X2: int,Y3: int] :
% 0.30/0.61 ( ( times_times_int @ ( ring_1_of_int_int @ X2 ) @ Y3 )
% 0.30/0.61 = ( times_times_int @ Y3 @ ( ring_1_of_int_int @ X2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % mult_of_int_commute
% 0.30/0.61 thf(fact_6490_real__of__int__div4,axiom,
% 0.30/0.61 ! [N: int,X2: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X2 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % real_of_int_div4
% 0.30/0.61 thf(fact_6491_real__of__int__div,axiom,
% 0.30/0.61 ! [D: int,N: int] :
% 0.30/0.61 ( ( dvd_dvd_int @ D @ N )
% 0.30/0.61 => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
% 0.30/0.61 = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % real_of_int_div
% 0.30/0.61 thf(fact_6492_of__int__nonneg,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_nonneg
% 0.30/0.61 thf(fact_6493_of__int__nonneg,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_nonneg
% 0.30/0.61 thf(fact_6494_of__int__nonneg,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.61 => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_nonneg
% 0.30/0.61 thf(fact_6495_of__int__pos,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.30/0.61 => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_pos
% 0.30/0.61 thf(fact_6496_of__int__pos,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.30/0.61 => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_pos
% 0.30/0.61 thf(fact_6497_of__int__pos,axiom,
% 0.30/0.61 ! [Z: int] :
% 0.30/0.61 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.30/0.61 => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_pos
% 0.30/0.61 thf(fact_6498_of__int__neg__numeral,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.61 = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_neg_numeral
% 0.30/0.61 thf(fact_6499_of__int__neg__numeral,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.61 = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_neg_numeral
% 0.30/0.61 thf(fact_6500_of__int__neg__numeral,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.61 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_neg_numeral
% 0.30/0.61 thf(fact_6501_of__int__neg__numeral,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.61 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_neg_numeral
% 0.30/0.61 thf(fact_6502_of__int__neg__numeral,axiom,
% 0.30/0.61 ! [K: num] :
% 0.30/0.61 ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.61 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % of_int_neg_numeral
% 0.30/0.61 thf(fact_6503_int__le__real__less,axiom,
% 0.30/0.61 ( ord_less_eq_int
% 0.30/0.61 = ( ^ [N3: int,M3: int] : ( ord_less_real @ ( ring_1_of_int_real @ N3 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M3 ) @ one_one_real ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % int_le_real_less
% 0.30/0.61 thf(fact_6504_int__less__real__le,axiom,
% 0.30/0.61 ( ord_less_int
% 0.30/0.61 = ( ^ [N3: int,M3: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) @ ( ring_1_of_int_real @ M3 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % int_less_real_le
% 0.30/0.61 thf(fact_6505_real__of__int__div__aux,axiom,
% 0.30/0.61 ! [X2: int,D: int] :
% 0.30/0.61 ( ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ D ) )
% 0.30/0.61 = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X2 @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X2 @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % real_of_int_div_aux
% 0.30/0.61 thf(fact_6506_real__of__int__div2,axiom,
% 0.30/0.61 ! [N: int,X2: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X2 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X2 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % real_of_int_div2
% 0.30/0.61 thf(fact_6507_real__of__int__div3,axiom,
% 0.30/0.61 ! [N: int,X2: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X2 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X2 ) ) ) @ one_one_real ) ).
% 0.30/0.61
% 0.30/0.61 % real_of_int_div3
% 0.30/0.61 thf(fact_6508_even__of__int__iff,axiom,
% 0.30/0.61 ! [K: int] :
% 0.30/0.61 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
% 0.30/0.61 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 0.30/0.61
% 0.30/0.61 % even_of_int_iff
% 0.30/0.61 thf(fact_6509_even__of__int__iff,axiom,
% 0.30/0.61 ! [K: int] :
% 0.30/0.61 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
% 0.30/0.61 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 0.30/0.61
% 0.30/0.61 % even_of_int_iff
% 0.30/0.61 thf(fact_6510_dbl__dec__def,axiom,
% 0.30/0.61 ( neg_nu6511756317524482435omplex
% 0.30/0.61 = ( ^ [X: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_def
% 0.30/0.61 thf(fact_6511_dbl__dec__def,axiom,
% 0.30/0.61 ( neg_nu6075765906172075777c_real
% 0.30/0.61 = ( ^ [X: real] : ( minus_minus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_def
% 0.30/0.61 thf(fact_6512_dbl__dec__def,axiom,
% 0.30/0.61 ( neg_nu3179335615603231917ec_rat
% 0.30/0.61 = ( ^ [X: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_def
% 0.30/0.61 thf(fact_6513_dbl__dec__def,axiom,
% 0.30/0.61 ( neg_nu3811975205180677377ec_int
% 0.30/0.61 = ( ^ [X: int] : ( minus_minus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % dbl_dec_def
% 0.30/0.61 thf(fact_6514_floor__exists,axiom,
% 0.30/0.61 ! [X2: real] :
% 0.30/0.61 ? [Z2: int] :
% 0.30/0.61 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X2 )
% 0.30/0.61 & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % floor_exists
% 0.30/0.61 thf(fact_6515_floor__exists,axiom,
% 0.30/0.61 ! [X2: rat] :
% 0.30/0.61 ? [Z2: int] :
% 0.30/0.61 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X2 )
% 0.30/0.61 & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % floor_exists
% 0.30/0.61 thf(fact_6516_floor__exists1,axiom,
% 0.30/0.61 ! [X2: real] :
% 0.30/0.61 ? [X5: int] :
% 0.30/0.61 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X5 ) @ X2 )
% 0.30/0.61 & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ X5 @ one_one_int ) ) )
% 0.30/0.61 & ! [Y5: int] :
% 0.30/0.61 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X2 )
% 0.30/0.61 & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 0.30/0.61 => ( Y5 = X5 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % floor_exists1
% 0.30/0.61 thf(fact_6517_floor__exists1,axiom,
% 0.30/0.61 ! [X2: rat] :
% 0.30/0.61 ? [X5: int] :
% 0.30/0.61 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X5 ) @ X2 )
% 0.30/0.61 & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ X5 @ one_one_int ) ) )
% 0.30/0.61 & ! [Y5: int] :
% 0.30/0.61 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X2 )
% 0.30/0.61 & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 0.30/0.61 => ( Y5 = X5 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % floor_exists1
% 0.30/0.61 thf(fact_6518_int__ge__less__than2__def,axiom,
% 0.30/0.61 ( int_ge_less_than2
% 0.30/0.61 = ( ^ [D2: int] :
% 0.30/0.61 ( collec213857154873943460nt_int
% 0.30/0.61 @ ( produc4947309494688390418_int_o
% 0.30/0.61 @ ^ [Z6: int,Z3: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ D2 @ Z3 )
% 0.30/0.61 & ( ord_less_int @ Z6 @ Z3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % int_ge_less_than2_def
% 0.30/0.61 thf(fact_6519_ln__one__minus__pos__lower__bound,axiom,
% 0.30/0.61 ! [X2: real] :
% 0.30/0.61 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.61 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.61 => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ln_one_minus_pos_lower_bound
% 0.30/0.61 thf(fact_6520_artanh__def,axiom,
% 0.30/0.61 ( artanh_real
% 0.30/0.61 = ( ^ [X: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X ) @ ( minus_minus_real @ one_one_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % artanh_def
% 0.30/0.61 thf(fact_6521_Sum__Icc__int,axiom,
% 0.30/0.61 ! [M: int,N: int] :
% 0.30/0.61 ( ( ord_less_eq_int @ M @ N )
% 0.30/0.61 => ( ( groups4538972089207619220nt_int
% 0.30/0.61 @ ^ [X: int] : X
% 0.30/0.61 @ ( set_or1266510415728281911st_int @ M @ N ) )
% 0.30/0.61 = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % Sum_Icc_int
% 0.30/0.61 thf(fact_6522_ln__one,axiom,
% 0.30/0.61 ( ( ln_ln_real @ one_one_real )
% 0.30/0.61 = zero_zero_real ) ).
% 0.30/0.61
% 0.30/0.61 % ln_one
% 0.30/0.61 thf(fact_6523_ln__le__cancel__iff,axiom,
% 0.30/0.61 ! [X2: real,Y3: real] :
% 0.30/0.61 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.61 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.61 => ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) )
% 0.30/0.61 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ln_le_cancel_iff
% 0.30/0.61 thf(fact_6524_ln__eq__zero__iff,axiom,
% 0.30/0.61 ! [X2: real] :
% 0.30/0.61 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.61 => ( ( ( ln_ln_real @ X2 )
% 0.30/0.61 = zero_zero_real )
% 0.30/0.61 = ( X2 = one_one_real ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ln_eq_zero_iff
% 0.30/0.61 thf(fact_6525_ln__gt__zero__iff,axiom,
% 0.30/0.61 ! [X2: real] :
% 0.30/0.61 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.61 => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
% 0.30/0.61 = ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ln_gt_zero_iff
% 0.30/0.61 thf(fact_6526_ln__less__zero__iff,axiom,
% 0.30/0.61 ! [X2: real] :
% 0.30/0.61 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.61 => ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
% 0.30/0.61 = ( ord_less_real @ X2 @ one_one_real ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ln_less_zero_iff
% 0.30/0.61 thf(fact_6527_ln__le__zero__iff,axiom,
% 0.30/0.61 ! [X2: real] :
% 0.30/0.61 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.61 => ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
% 0.30/0.61 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ln_le_zero_iff
% 0.30/0.61 thf(fact_6528_ln__ge__zero__iff,axiom,
% 0.30/0.61 ! [X2: real] :
% 0.30/0.61 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.61 => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
% 0.30/0.61 = ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ln_ge_zero_iff
% 0.30/0.61 thf(fact_6529_sum__mono,axiom,
% 0.30/0.61 ! [K5: set_complex,F: complex > rat,G: complex > rat] :
% 0.30/0.61 ( ! [I3: complex] :
% 0.30/0.61 ( ( member_complex @ I3 @ K5 )
% 0.30/0.61 => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ K5 ) @ ( groups5058264527183730370ex_rat @ G @ K5 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono
% 0.30/0.61 thf(fact_6530_sum__mono,axiom,
% 0.30/0.61 ! [K5: set_real,F: real > rat,G: real > rat] :
% 0.30/0.61 ( ! [I3: real] :
% 0.30/0.61 ( ( member_real @ I3 @ K5 )
% 0.30/0.61 => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono
% 0.30/0.61 thf(fact_6531_sum__mono,axiom,
% 0.30/0.61 ! [K5: set_nat,F: nat > rat,G: nat > rat] :
% 0.30/0.61 ( ! [I3: nat] :
% 0.30/0.61 ( ( member_nat @ I3 @ K5 )
% 0.30/0.61 => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono
% 0.30/0.61 thf(fact_6532_sum__mono,axiom,
% 0.30/0.61 ! [K5: set_int,F: int > rat,G: int > rat] :
% 0.30/0.61 ( ! [I3: int] :
% 0.30/0.61 ( ( member_int @ I3 @ K5 )
% 0.30/0.61 => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono
% 0.30/0.61 thf(fact_6533_sum__mono,axiom,
% 0.30/0.61 ! [K5: set_complex,F: complex > nat,G: complex > nat] :
% 0.30/0.61 ( ! [I3: complex] :
% 0.30/0.61 ( ( member_complex @ I3 @ K5 )
% 0.30/0.61 => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K5 ) @ ( groups5693394587270226106ex_nat @ G @ K5 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono
% 0.30/0.61 thf(fact_6534_sum__mono,axiom,
% 0.30/0.61 ! [K5: set_real,F: real > nat,G: real > nat] :
% 0.30/0.61 ( ! [I3: real] :
% 0.30/0.61 ( ( member_real @ I3 @ K5 )
% 0.30/0.61 => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono
% 0.30/0.61 thf(fact_6535_sum__mono,axiom,
% 0.30/0.61 ! [K5: set_int,F: int > nat,G: int > nat] :
% 0.30/0.61 ( ! [I3: int] :
% 0.30/0.61 ( ( member_int @ I3 @ K5 )
% 0.30/0.61 => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono
% 0.30/0.61 thf(fact_6536_sum__mono,axiom,
% 0.30/0.61 ! [K5: set_complex,F: complex > int,G: complex > int] :
% 0.30/0.61 ( ! [I3: complex] :
% 0.30/0.61 ( ( member_complex @ I3 @ K5 )
% 0.30/0.61 => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K5 ) @ ( groups5690904116761175830ex_int @ G @ K5 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono
% 0.30/0.61 thf(fact_6537_sum__mono,axiom,
% 0.30/0.61 ! [K5: set_real,F: real > int,G: real > int] :
% 0.30/0.61 ( ! [I3: real] :
% 0.30/0.61 ( ( member_real @ I3 @ K5 )
% 0.30/0.61 => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono
% 0.30/0.61 thf(fact_6538_sum__mono,axiom,
% 0.30/0.61 ! [K5: set_nat,F: nat > int,G: nat > int] :
% 0.30/0.61 ( ! [I3: nat] :
% 0.30/0.61 ( ( member_nat @ I3 @ K5 )
% 0.30/0.61 => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono
% 0.30/0.61 thf(fact_6539_sum__product,axiom,
% 0.30/0.61 ! [F: int > int,A2: set_int,G: int > int,B3: set_int] :
% 0.30/0.61 ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ B3 ) )
% 0.30/0.61 = ( groups4538972089207619220nt_int
% 0.30/0.61 @ ^ [I4: int] :
% 0.30/0.61 ( groups4538972089207619220nt_int
% 0.30/0.61 @ ^ [J3: int] : ( times_times_int @ ( F @ I4 ) @ ( G @ J3 ) )
% 0.30/0.61 @ B3 )
% 0.30/0.61 @ A2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_product
% 0.30/0.61 thf(fact_6540_sum__product,axiom,
% 0.30/0.61 ! [F: complex > complex,A2: set_complex,G: complex > complex,B3: set_complex] :
% 0.30/0.61 ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ B3 ) )
% 0.30/0.61 = ( groups7754918857620584856omplex
% 0.30/0.61 @ ^ [I4: complex] :
% 0.30/0.61 ( groups7754918857620584856omplex
% 0.30/0.61 @ ^ [J3: complex] : ( times_times_complex @ ( F @ I4 ) @ ( G @ J3 ) )
% 0.30/0.61 @ B3 )
% 0.30/0.61 @ A2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_product
% 0.30/0.61 thf(fact_6541_sum__product,axiom,
% 0.30/0.61 ! [F: nat > nat,A2: set_nat,G: nat > nat,B3: set_nat] :
% 0.30/0.61 ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B3 ) )
% 0.30/0.61 = ( groups3542108847815614940at_nat
% 0.30/0.61 @ ^ [I4: nat] :
% 0.30/0.61 ( groups3542108847815614940at_nat
% 0.30/0.61 @ ^ [J3: nat] : ( times_times_nat @ ( F @ I4 ) @ ( G @ J3 ) )
% 0.30/0.61 @ B3 )
% 0.30/0.61 @ A2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_product
% 0.30/0.61 thf(fact_6542_sum__product,axiom,
% 0.30/0.61 ! [F: nat > real,A2: set_nat,G: nat > real,B3: set_nat] :
% 0.30/0.61 ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B3 ) )
% 0.30/0.61 = ( groups6591440286371151544t_real
% 0.30/0.61 @ ^ [I4: nat] :
% 0.30/0.61 ( groups6591440286371151544t_real
% 0.30/0.61 @ ^ [J3: nat] : ( times_times_real @ ( F @ I4 ) @ ( G @ J3 ) )
% 0.30/0.61 @ B3 )
% 0.30/0.61 @ A2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_product
% 0.30/0.61 thf(fact_6543_sum__distrib__right,axiom,
% 0.30/0.61 ! [F: int > int,A2: set_int,R2: int] :
% 0.30/0.61 ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ R2 )
% 0.30/0.61 = ( groups4538972089207619220nt_int
% 0.30/0.61 @ ^ [N3: int] : ( times_times_int @ ( F @ N3 ) @ R2 )
% 0.30/0.61 @ A2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_distrib_right
% 0.30/0.61 thf(fact_6544_sum__distrib__right,axiom,
% 0.30/0.61 ! [F: complex > complex,A2: set_complex,R2: complex] :
% 0.30/0.61 ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R2 )
% 0.30/0.61 = ( groups7754918857620584856omplex
% 0.30/0.61 @ ^ [N3: complex] : ( times_times_complex @ ( F @ N3 ) @ R2 )
% 0.30/0.61 @ A2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_distrib_right
% 0.30/0.61 thf(fact_6545_sum__distrib__right,axiom,
% 0.30/0.61 ! [F: nat > nat,A2: set_nat,R2: nat] :
% 0.30/0.61 ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R2 )
% 0.30/0.61 = ( groups3542108847815614940at_nat
% 0.30/0.61 @ ^ [N3: nat] : ( times_times_nat @ ( F @ N3 ) @ R2 )
% 0.30/0.61 @ A2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_distrib_right
% 0.30/0.61 thf(fact_6546_sum__distrib__right,axiom,
% 0.30/0.61 ! [F: nat > real,A2: set_nat,R2: real] :
% 0.30/0.61 ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R2 )
% 0.30/0.61 = ( groups6591440286371151544t_real
% 0.30/0.61 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ R2 )
% 0.30/0.61 @ A2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_distrib_right
% 0.30/0.61 thf(fact_6547_sum__distrib__left,axiom,
% 0.30/0.61 ! [R2: int,F: int > int,A2: set_int] :
% 0.30/0.61 ( ( times_times_int @ R2 @ ( groups4538972089207619220nt_int @ F @ A2 ) )
% 0.30/0.61 = ( groups4538972089207619220nt_int
% 0.30/0.61 @ ^ [N3: int] : ( times_times_int @ R2 @ ( F @ N3 ) )
% 0.30/0.61 @ A2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_distrib_left
% 0.30/0.61 thf(fact_6548_sum__distrib__left,axiom,
% 0.30/0.61 ! [R2: complex,F: complex > complex,A2: set_complex] :
% 0.30/0.61 ( ( times_times_complex @ R2 @ ( groups7754918857620584856omplex @ F @ A2 ) )
% 0.30/0.61 = ( groups7754918857620584856omplex
% 0.30/0.61 @ ^ [N3: complex] : ( times_times_complex @ R2 @ ( F @ N3 ) )
% 0.30/0.61 @ A2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_distrib_left
% 0.30/0.61 thf(fact_6549_sum__distrib__left,axiom,
% 0.30/0.61 ! [R2: nat,F: nat > nat,A2: set_nat] :
% 0.30/0.61 ( ( times_times_nat @ R2 @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.30/0.61 = ( groups3542108847815614940at_nat
% 0.30/0.61 @ ^ [N3: nat] : ( times_times_nat @ R2 @ ( F @ N3 ) )
% 0.30/0.61 @ A2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_distrib_left
% 0.30/0.61 thf(fact_6550_sum__distrib__left,axiom,
% 0.30/0.61 ! [R2: real,F: nat > real,A2: set_nat] :
% 0.30/0.61 ( ( times_times_real @ R2 @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 0.30/0.61 = ( groups6591440286371151544t_real
% 0.30/0.61 @ ^ [N3: nat] : ( times_times_real @ R2 @ ( F @ N3 ) )
% 0.30/0.61 @ A2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_distrib_left
% 0.30/0.61 thf(fact_6551_sum_Odistrib,axiom,
% 0.30/0.61 ! [G: int > int,H2: int > int,A2: set_int] :
% 0.30/0.61 ( ( groups4538972089207619220nt_int
% 0.30/0.61 @ ^ [X: int] : ( plus_plus_int @ ( G @ X ) @ ( H2 @ X ) )
% 0.30/0.61 @ A2 )
% 0.30/0.61 = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H2 @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.distrib
% 0.30/0.61 thf(fact_6552_sum_Odistrib,axiom,
% 0.30/0.61 ! [G: complex > complex,H2: complex > complex,A2: set_complex] :
% 0.30/0.61 ( ( groups7754918857620584856omplex
% 0.30/0.61 @ ^ [X: complex] : ( plus_plus_complex @ ( G @ X ) @ ( H2 @ X ) )
% 0.30/0.61 @ A2 )
% 0.30/0.61 = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.distrib
% 0.30/0.61 thf(fact_6553_sum_Odistrib,axiom,
% 0.30/0.61 ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
% 0.30/0.61 ( ( groups3542108847815614940at_nat
% 0.30/0.61 @ ^ [X: nat] : ( plus_plus_nat @ ( G @ X ) @ ( H2 @ X ) )
% 0.30/0.61 @ A2 )
% 0.30/0.61 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.distrib
% 0.30/0.61 thf(fact_6554_sum_Odistrib,axiom,
% 0.30/0.61 ! [G: nat > real,H2: nat > real,A2: set_nat] :
% 0.30/0.61 ( ( groups6591440286371151544t_real
% 0.30/0.61 @ ^ [X: nat] : ( plus_plus_real @ ( G @ X ) @ ( H2 @ X ) )
% 0.30/0.61 @ A2 )
% 0.30/0.61 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.distrib
% 0.30/0.61 thf(fact_6555_sum__subtractf,axiom,
% 0.30/0.61 ! [F: int > int,G: int > int,A2: set_int] :
% 0.30/0.61 ( ( groups4538972089207619220nt_int
% 0.30/0.61 @ ^ [X: int] : ( minus_minus_int @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.61 @ A2 )
% 0.30/0.61 = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_subtractf
% 0.30/0.61 thf(fact_6556_sum__subtractf,axiom,
% 0.30/0.61 ! [F: complex > complex,G: complex > complex,A2: set_complex] :
% 0.30/0.61 ( ( groups7754918857620584856omplex
% 0.30/0.61 @ ^ [X: complex] : ( minus_minus_complex @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.61 @ A2 )
% 0.30/0.61 = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_subtractf
% 0.30/0.61 thf(fact_6557_sum__subtractf,axiom,
% 0.30/0.61 ! [F: nat > real,G: nat > real,A2: set_nat] :
% 0.30/0.61 ( ( groups6591440286371151544t_real
% 0.30/0.61 @ ^ [X: nat] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.61 @ A2 )
% 0.30/0.61 = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_subtractf
% 0.30/0.61 thf(fact_6558_sum__divide__distrib,axiom,
% 0.30/0.61 ! [F: complex > complex,A2: set_complex,R2: complex] :
% 0.30/0.61 ( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R2 )
% 0.30/0.61 = ( groups7754918857620584856omplex
% 0.30/0.61 @ ^ [N3: complex] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ R2 )
% 0.30/0.61 @ A2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_divide_distrib
% 0.30/0.61 thf(fact_6559_sum__divide__distrib,axiom,
% 0.30/0.61 ! [F: nat > real,A2: set_nat,R2: real] :
% 0.30/0.61 ( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R2 )
% 0.30/0.61 = ( groups6591440286371151544t_real
% 0.30/0.61 @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ R2 )
% 0.30/0.61 @ A2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_divide_distrib
% 0.30/0.61 thf(fact_6560_mod__sum__eq,axiom,
% 0.30/0.61 ! [F: int > int,A: int,A2: set_int] :
% 0.30/0.61 ( ( modulo_modulo_int
% 0.30/0.61 @ ( groups4538972089207619220nt_int
% 0.30/0.61 @ ^ [I4: int] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
% 0.30/0.61 @ A2 )
% 0.30/0.61 @ A )
% 0.30/0.61 = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ A ) ) ).
% 0.30/0.61
% 0.30/0.61 % mod_sum_eq
% 0.30/0.61 thf(fact_6561_mod__sum__eq,axiom,
% 0.30/0.61 ! [F: nat > nat,A: nat,A2: set_nat] :
% 0.30/0.61 ( ( modulo_modulo_nat
% 0.30/0.61 @ ( groups3542108847815614940at_nat
% 0.30/0.61 @ ^ [I4: nat] : ( modulo_modulo_nat @ ( F @ I4 ) @ A )
% 0.30/0.61 @ A2 )
% 0.30/0.61 @ A )
% 0.30/0.61 = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ A ) ) ).
% 0.30/0.61
% 0.30/0.61 % mod_sum_eq
% 0.30/0.61 thf(fact_6562_sum__nonneg,axiom,
% 0.30/0.61 ! [A2: set_complex,F: complex > real] :
% 0.30/0.61 ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg
% 0.30/0.61 thf(fact_6563_sum__nonneg,axiom,
% 0.30/0.61 ! [A2: set_real,F: real > real] :
% 0.30/0.61 ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg
% 0.30/0.61 thf(fact_6564_sum__nonneg,axiom,
% 0.30/0.61 ! [A2: set_int,F: int > real] :
% 0.30/0.61 ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg
% 0.30/0.61 thf(fact_6565_sum__nonneg,axiom,
% 0.30/0.61 ! [A2: set_complex,F: complex > rat] :
% 0.30/0.61 ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg
% 0.30/0.61 thf(fact_6566_sum__nonneg,axiom,
% 0.30/0.61 ! [A2: set_real,F: real > rat] :
% 0.30/0.61 ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg
% 0.30/0.61 thf(fact_6567_sum__nonneg,axiom,
% 0.30/0.61 ! [A2: set_nat,F: nat > rat] :
% 0.30/0.61 ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg
% 0.30/0.61 thf(fact_6568_sum__nonneg,axiom,
% 0.30/0.61 ! [A2: set_int,F: int > rat] :
% 0.30/0.61 ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg
% 0.30/0.61 thf(fact_6569_sum__nonneg,axiom,
% 0.30/0.61 ! [A2: set_complex,F: complex > nat] :
% 0.30/0.61 ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg
% 0.30/0.61 thf(fact_6570_sum__nonneg,axiom,
% 0.30/0.61 ! [A2: set_real,F: real > nat] :
% 0.30/0.61 ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg
% 0.30/0.61 thf(fact_6571_sum__nonneg,axiom,
% 0.30/0.61 ! [A2: set_int,F: int > nat] :
% 0.30/0.61 ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg
% 0.30/0.61 thf(fact_6572_sum__nonpos,axiom,
% 0.30/0.61 ! [A2: set_complex,F: complex > real] :
% 0.30/0.61 ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
% 0.30/0.61 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonpos
% 0.30/0.61 thf(fact_6573_sum__nonpos,axiom,
% 0.30/0.61 ! [A2: set_real,F: real > real] :
% 0.30/0.61 ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
% 0.30/0.61 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonpos
% 0.30/0.61 thf(fact_6574_sum__nonpos,axiom,
% 0.30/0.61 ! [A2: set_int,F: int > real] :
% 0.30/0.61 ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
% 0.30/0.61 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonpos
% 0.30/0.61 thf(fact_6575_sum__nonpos,axiom,
% 0.30/0.61 ! [A2: set_complex,F: complex > rat] :
% 0.30/0.61 ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 0.30/0.61 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonpos
% 0.30/0.61 thf(fact_6576_sum__nonpos,axiom,
% 0.30/0.61 ! [A2: set_real,F: real > rat] :
% 0.30/0.61 ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 0.30/0.61 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonpos
% 0.30/0.61 thf(fact_6577_sum__nonpos,axiom,
% 0.30/0.61 ! [A2: set_nat,F: nat > rat] :
% 0.30/0.61 ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 0.30/0.61 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonpos
% 0.30/0.61 thf(fact_6578_sum__nonpos,axiom,
% 0.30/0.61 ! [A2: set_int,F: int > rat] :
% 0.30/0.61 ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 0.30/0.61 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonpos
% 0.30/0.61 thf(fact_6579_sum__nonpos,axiom,
% 0.30/0.61 ! [A2: set_complex,F: complex > nat] :
% 0.30/0.61 ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
% 0.30/0.61 => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonpos
% 0.30/0.61 thf(fact_6580_sum__nonpos,axiom,
% 0.30/0.61 ! [A2: set_real,F: real > nat] :
% 0.30/0.61 ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
% 0.30/0.61 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonpos
% 0.30/0.61 thf(fact_6581_sum__nonpos,axiom,
% 0.30/0.61 ! [A2: set_int,F: int > nat] :
% 0.30/0.61 ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
% 0.30/0.61 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonpos
% 0.30/0.61 thf(fact_6582_sum__mono__inv,axiom,
% 0.30/0.61 ! [F: real > rat,I5: set_real,G: real > rat,I2: real] :
% 0.30/0.61 ( ( ( groups1300246762558778688al_rat @ F @ I5 )
% 0.30/0.61 = ( groups1300246762558778688al_rat @ G @ I5 ) )
% 0.30/0.61 => ( ! [I3: real] :
% 0.30/0.61 ( ( member_real @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ( member_real @ I2 @ I5 )
% 0.30/0.61 => ( ( finite_finite_real @ I5 )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = ( G @ I2 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono_inv
% 0.30/0.61 thf(fact_6583_sum__mono__inv,axiom,
% 0.30/0.61 ! [F: nat > rat,I5: set_nat,G: nat > rat,I2: nat] :
% 0.30/0.61 ( ( ( groups2906978787729119204at_rat @ F @ I5 )
% 0.30/0.61 = ( groups2906978787729119204at_rat @ G @ I5 ) )
% 0.30/0.61 => ( ! [I3: nat] :
% 0.30/0.61 ( ( member_nat @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ( member_nat @ I2 @ I5 )
% 0.30/0.61 => ( ( finite_finite_nat @ I5 )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = ( G @ I2 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono_inv
% 0.30/0.61 thf(fact_6584_sum__mono__inv,axiom,
% 0.30/0.61 ! [F: int > rat,I5: set_int,G: int > rat,I2: int] :
% 0.30/0.61 ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
% 0.30/0.61 = ( groups3906332499630173760nt_rat @ G @ I5 ) )
% 0.30/0.61 => ( ! [I3: int] :
% 0.30/0.61 ( ( member_int @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ( member_int @ I2 @ I5 )
% 0.30/0.61 => ( ( finite_finite_int @ I5 )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = ( G @ I2 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono_inv
% 0.30/0.61 thf(fact_6585_sum__mono__inv,axiom,
% 0.30/0.61 ! [F: complex > rat,I5: set_complex,G: complex > rat,I2: complex] :
% 0.30/0.61 ( ( ( groups5058264527183730370ex_rat @ F @ I5 )
% 0.30/0.61 = ( groups5058264527183730370ex_rat @ G @ I5 ) )
% 0.30/0.61 => ( ! [I3: complex] :
% 0.30/0.61 ( ( member_complex @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ( member_complex @ I2 @ I5 )
% 0.30/0.61 => ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = ( G @ I2 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono_inv
% 0.30/0.61 thf(fact_6586_sum__mono__inv,axiom,
% 0.30/0.61 ! [F: real > nat,I5: set_real,G: real > nat,I2: real] :
% 0.30/0.61 ( ( ( groups1935376822645274424al_nat @ F @ I5 )
% 0.30/0.61 = ( groups1935376822645274424al_nat @ G @ I5 ) )
% 0.30/0.61 => ( ! [I3: real] :
% 0.30/0.61 ( ( member_real @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ( member_real @ I2 @ I5 )
% 0.30/0.61 => ( ( finite_finite_real @ I5 )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = ( G @ I2 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono_inv
% 0.30/0.61 thf(fact_6587_sum__mono__inv,axiom,
% 0.30/0.61 ! [F: int > nat,I5: set_int,G: int > nat,I2: int] :
% 0.30/0.61 ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
% 0.30/0.61 = ( groups4541462559716669496nt_nat @ G @ I5 ) )
% 0.30/0.61 => ( ! [I3: int] :
% 0.30/0.61 ( ( member_int @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ( member_int @ I2 @ I5 )
% 0.30/0.61 => ( ( finite_finite_int @ I5 )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = ( G @ I2 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono_inv
% 0.30/0.61 thf(fact_6588_sum__mono__inv,axiom,
% 0.30/0.61 ! [F: complex > nat,I5: set_complex,G: complex > nat,I2: complex] :
% 0.30/0.61 ( ( ( groups5693394587270226106ex_nat @ F @ I5 )
% 0.30/0.61 = ( groups5693394587270226106ex_nat @ G @ I5 ) )
% 0.30/0.61 => ( ! [I3: complex] :
% 0.30/0.61 ( ( member_complex @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ( member_complex @ I2 @ I5 )
% 0.30/0.61 => ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = ( G @ I2 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono_inv
% 0.30/0.61 thf(fact_6589_sum__mono__inv,axiom,
% 0.30/0.61 ! [F: real > int,I5: set_real,G: real > int,I2: real] :
% 0.30/0.61 ( ( ( groups1932886352136224148al_int @ F @ I5 )
% 0.30/0.61 = ( groups1932886352136224148al_int @ G @ I5 ) )
% 0.30/0.61 => ( ! [I3: real] :
% 0.30/0.61 ( ( member_real @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ( member_real @ I2 @ I5 )
% 0.30/0.61 => ( ( finite_finite_real @ I5 )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = ( G @ I2 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono_inv
% 0.30/0.61 thf(fact_6590_sum__mono__inv,axiom,
% 0.30/0.61 ! [F: nat > int,I5: set_nat,G: nat > int,I2: nat] :
% 0.30/0.61 ( ( ( groups3539618377306564664at_int @ F @ I5 )
% 0.30/0.61 = ( groups3539618377306564664at_int @ G @ I5 ) )
% 0.30/0.61 => ( ! [I3: nat] :
% 0.30/0.61 ( ( member_nat @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ( member_nat @ I2 @ I5 )
% 0.30/0.61 => ( ( finite_finite_nat @ I5 )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = ( G @ I2 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono_inv
% 0.30/0.61 thf(fact_6591_sum__mono__inv,axiom,
% 0.30/0.61 ! [F: complex > int,I5: set_complex,G: complex > int,I2: complex] :
% 0.30/0.61 ( ( ( groups5690904116761175830ex_int @ F @ I5 )
% 0.30/0.61 = ( groups5690904116761175830ex_int @ G @ I5 ) )
% 0.30/0.61 => ( ! [I3: complex] :
% 0.30/0.61 ( ( member_complex @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.61 => ( ( member_complex @ I2 @ I5 )
% 0.30/0.61 => ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = ( G @ I2 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_mono_inv
% 0.30/0.61 thf(fact_6592_sum__le__included,axiom,
% 0.30/0.61 ! [S: set_int,T: set_int,G: int > real,I2: int > int,F: int > real] :
% 0.30/0.61 ( ( finite_finite_int @ S )
% 0.30/0.61 => ( ( finite_finite_int @ T )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ T )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ S )
% 0.30/0.61 => ? [Xa: int] :
% 0.30/0.61 ( ( member_int @ Xa @ T )
% 0.30/0.61 & ( ( I2 @ Xa )
% 0.30/0.61 = X5 )
% 0.30/0.61 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 0.30/0.61 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_le_included
% 0.30/0.61 thf(fact_6593_sum__le__included,axiom,
% 0.30/0.61 ! [S: set_int,T: set_complex,G: complex > real,I2: complex > int,F: int > real] :
% 0.30/0.61 ( ( finite_finite_int @ S )
% 0.30/0.61 => ( ( finite3207457112153483333omplex @ T )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ T )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ S )
% 0.30/0.61 => ? [Xa: complex] :
% 0.30/0.61 ( ( member_complex @ Xa @ T )
% 0.30/0.61 & ( ( I2 @ Xa )
% 0.30/0.61 = X5 )
% 0.30/0.61 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 0.30/0.61 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_le_included
% 0.30/0.61 thf(fact_6594_sum__le__included,axiom,
% 0.30/0.61 ! [S: set_complex,T: set_int,G: int > real,I2: int > complex,F: complex > real] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ S )
% 0.30/0.61 => ( ( finite_finite_int @ T )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ T )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ S )
% 0.30/0.61 => ? [Xa: int] :
% 0.30/0.61 ( ( member_int @ Xa @ T )
% 0.30/0.61 & ( ( I2 @ Xa )
% 0.30/0.61 = X5 )
% 0.30/0.61 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 0.30/0.61 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_le_included
% 0.30/0.61 thf(fact_6595_sum__le__included,axiom,
% 0.30/0.61 ! [S: set_complex,T: set_complex,G: complex > real,I2: complex > complex,F: complex > real] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ S )
% 0.30/0.61 => ( ( finite3207457112153483333omplex @ T )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ T )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ S )
% 0.30/0.61 => ? [Xa: complex] :
% 0.30/0.61 ( ( member_complex @ Xa @ T )
% 0.30/0.61 & ( ( I2 @ Xa )
% 0.30/0.61 = X5 )
% 0.30/0.61 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 0.30/0.61 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_le_included
% 0.30/0.61 thf(fact_6596_sum__le__included,axiom,
% 0.30/0.61 ! [S: set_nat,T: set_nat,G: nat > rat,I2: nat > nat,F: nat > rat] :
% 0.30/0.61 ( ( finite_finite_nat @ S )
% 0.30/0.61 => ( ( finite_finite_nat @ T )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ T )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ S )
% 0.30/0.61 => ? [Xa: nat] :
% 0.30/0.61 ( ( member_nat @ Xa @ T )
% 0.30/0.61 & ( ( I2 @ Xa )
% 0.30/0.61 = X5 )
% 0.30/0.61 & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 0.30/0.61 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_le_included
% 0.30/0.61 thf(fact_6597_sum__le__included,axiom,
% 0.30/0.61 ! [S: set_nat,T: set_int,G: int > rat,I2: int > nat,F: nat > rat] :
% 0.30/0.61 ( ( finite_finite_nat @ S )
% 0.30/0.61 => ( ( finite_finite_int @ T )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ T )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ S )
% 0.30/0.61 => ? [Xa: int] :
% 0.30/0.61 ( ( member_int @ Xa @ T )
% 0.30/0.61 & ( ( I2 @ Xa )
% 0.30/0.61 = X5 )
% 0.30/0.61 & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 0.30/0.61 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_le_included
% 0.30/0.61 thf(fact_6598_sum__le__included,axiom,
% 0.30/0.61 ! [S: set_nat,T: set_complex,G: complex > rat,I2: complex > nat,F: nat > rat] :
% 0.30/0.61 ( ( finite_finite_nat @ S )
% 0.30/0.61 => ( ( finite3207457112153483333omplex @ T )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ T )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ S )
% 0.30/0.61 => ? [Xa: complex] :
% 0.30/0.61 ( ( member_complex @ Xa @ T )
% 0.30/0.61 & ( ( I2 @ Xa )
% 0.30/0.61 = X5 )
% 0.30/0.61 & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 0.30/0.61 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_le_included
% 0.30/0.61 thf(fact_6599_sum__le__included,axiom,
% 0.30/0.61 ! [S: set_int,T: set_nat,G: nat > rat,I2: nat > int,F: int > rat] :
% 0.30/0.61 ( ( finite_finite_int @ S )
% 0.30/0.61 => ( ( finite_finite_nat @ T )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ T )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ S )
% 0.30/0.61 => ? [Xa: nat] :
% 0.30/0.61 ( ( member_nat @ Xa @ T )
% 0.30/0.61 & ( ( I2 @ Xa )
% 0.30/0.61 = X5 )
% 0.30/0.61 & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 0.30/0.61 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_le_included
% 0.30/0.61 thf(fact_6600_sum__le__included,axiom,
% 0.30/0.61 ! [S: set_int,T: set_int,G: int > rat,I2: int > int,F: int > rat] :
% 0.30/0.61 ( ( finite_finite_int @ S )
% 0.30/0.61 => ( ( finite_finite_int @ T )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ T )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ S )
% 0.30/0.61 => ? [Xa: int] :
% 0.30/0.61 ( ( member_int @ Xa @ T )
% 0.30/0.61 & ( ( I2 @ Xa )
% 0.30/0.61 = X5 )
% 0.30/0.61 & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 0.30/0.61 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_le_included
% 0.30/0.61 thf(fact_6601_sum__le__included,axiom,
% 0.30/0.61 ! [S: set_int,T: set_complex,G: complex > rat,I2: complex > int,F: int > rat] :
% 0.30/0.61 ( ( finite_finite_int @ S )
% 0.30/0.61 => ( ( finite3207457112153483333omplex @ T )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ T )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ S )
% 0.30/0.61 => ? [Xa: complex] :
% 0.30/0.61 ( ( member_complex @ Xa @ T )
% 0.30/0.61 & ( ( I2 @ Xa )
% 0.30/0.61 = X5 )
% 0.30/0.61 & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 0.30/0.61 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_le_included
% 0.30/0.61 thf(fact_6602_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.61 ! [A2: set_real,F: real > real] :
% 0.30/0.61 ( ( finite_finite_real @ A2 )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ( ( groups8097168146408367636l_real @ F @ A2 )
% 0.30/0.61 = zero_zero_real )
% 0.30/0.61 = ( ! [X: real] :
% 0.30/0.61 ( ( member_real @ X @ A2 )
% 0.30/0.61 => ( ( F @ X )
% 0.30/0.61 = zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_eq_0_iff
% 0.30/0.61 thf(fact_6603_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.61 ! [A2: set_int,F: int > real] :
% 0.30/0.61 ( ( finite_finite_int @ A2 )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ( ( groups8778361861064173332t_real @ F @ A2 )
% 0.30/0.61 = zero_zero_real )
% 0.30/0.61 = ( ! [X: int] :
% 0.30/0.61 ( ( member_int @ X @ A2 )
% 0.30/0.61 => ( ( F @ X )
% 0.30/0.61 = zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_eq_0_iff
% 0.30/0.61 thf(fact_6604_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.61 ! [A2: set_complex,F: complex > real] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ( ( groups5808333547571424918x_real @ F @ A2 )
% 0.30/0.61 = zero_zero_real )
% 0.30/0.61 = ( ! [X: complex] :
% 0.30/0.61 ( ( member_complex @ X @ A2 )
% 0.30/0.61 => ( ( F @ X )
% 0.30/0.61 = zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_eq_0_iff
% 0.30/0.61 thf(fact_6605_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.61 ! [A2: set_real,F: real > rat] :
% 0.30/0.61 ( ( finite_finite_real @ A2 )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ( ( groups1300246762558778688al_rat @ F @ A2 )
% 0.30/0.61 = zero_zero_rat )
% 0.30/0.61 = ( ! [X: real] :
% 0.30/0.61 ( ( member_real @ X @ A2 )
% 0.30/0.61 => ( ( F @ X )
% 0.30/0.61 = zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_eq_0_iff
% 0.30/0.61 thf(fact_6606_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.61 ! [A2: set_nat,F: nat > rat] :
% 0.30/0.61 ( ( finite_finite_nat @ A2 )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ( ( groups2906978787729119204at_rat @ F @ A2 )
% 0.30/0.61 = zero_zero_rat )
% 0.30/0.61 = ( ! [X: nat] :
% 0.30/0.61 ( ( member_nat @ X @ A2 )
% 0.30/0.61 => ( ( F @ X )
% 0.30/0.61 = zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_eq_0_iff
% 0.30/0.61 thf(fact_6607_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.61 ! [A2: set_int,F: int > rat] :
% 0.30/0.61 ( ( finite_finite_int @ A2 )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
% 0.30/0.61 = zero_zero_rat )
% 0.30/0.61 = ( ! [X: int] :
% 0.30/0.61 ( ( member_int @ X @ A2 )
% 0.30/0.61 => ( ( F @ X )
% 0.30/0.61 = zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_eq_0_iff
% 0.30/0.61 thf(fact_6608_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.61 ! [A2: set_complex,F: complex > rat] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
% 0.30/0.61 = zero_zero_rat )
% 0.30/0.61 = ( ! [X: complex] :
% 0.30/0.61 ( ( member_complex @ X @ A2 )
% 0.30/0.61 => ( ( F @ X )
% 0.30/0.61 = zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_eq_0_iff
% 0.30/0.61 thf(fact_6609_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.61 ! [A2: set_real,F: real > nat] :
% 0.30/0.61 ( ( finite_finite_real @ A2 )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ( ( groups1935376822645274424al_nat @ F @ A2 )
% 0.30/0.61 = zero_zero_nat )
% 0.30/0.61 = ( ! [X: real] :
% 0.30/0.61 ( ( member_real @ X @ A2 )
% 0.30/0.61 => ( ( F @ X )
% 0.30/0.61 = zero_zero_nat ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_eq_0_iff
% 0.30/0.61 thf(fact_6610_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.61 ! [A2: set_int,F: int > nat] :
% 0.30/0.61 ( ( finite_finite_int @ A2 )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 0.30/0.61 = zero_zero_nat )
% 0.30/0.61 = ( ! [X: int] :
% 0.30/0.61 ( ( member_int @ X @ A2 )
% 0.30/0.61 => ( ( F @ X )
% 0.30/0.61 = zero_zero_nat ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_eq_0_iff
% 0.30/0.61 thf(fact_6611_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.61 ! [A2: set_complex,F: complex > nat] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.30/0.61 => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 0.30/0.61 = zero_zero_nat )
% 0.30/0.61 = ( ! [X: complex] :
% 0.30/0.61 ( ( member_complex @ X @ A2 )
% 0.30/0.61 => ( ( F @ X )
% 0.30/0.61 = zero_zero_nat ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_eq_0_iff
% 0.30/0.61 thf(fact_6612_sum__strict__mono__ex1,axiom,
% 0.30/0.61 ! [A2: set_int,F: int > real,G: int > real] :
% 0.30/0.61 ( ( finite_finite_int @ A2 )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ? [X3: int] :
% 0.30/0.61 ( ( member_int @ X3 @ A2 )
% 0.30/0.61 & ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.61 => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono_ex1
% 0.30/0.61 thf(fact_6613_sum__strict__mono__ex1,axiom,
% 0.30/0.61 ! [A2: set_complex,F: complex > real,G: complex > real] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ? [X3: complex] :
% 0.30/0.61 ( ( member_complex @ X3 @ A2 )
% 0.30/0.61 & ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.61 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono_ex1
% 0.30/0.61 thf(fact_6614_sum__strict__mono__ex1,axiom,
% 0.30/0.61 ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 0.30/0.61 ( ( finite_finite_nat @ A2 )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ? [X3: nat] :
% 0.30/0.61 ( ( member_nat @ X3 @ A2 )
% 0.30/0.61 & ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.61 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono_ex1
% 0.30/0.61 thf(fact_6615_sum__strict__mono__ex1,axiom,
% 0.30/0.61 ! [A2: set_int,F: int > rat,G: int > rat] :
% 0.30/0.61 ( ( finite_finite_int @ A2 )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ? [X3: int] :
% 0.30/0.61 ( ( member_int @ X3 @ A2 )
% 0.30/0.61 & ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.61 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono_ex1
% 0.30/0.61 thf(fact_6616_sum__strict__mono__ex1,axiom,
% 0.30/0.61 ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ? [X3: complex] :
% 0.30/0.61 ( ( member_complex @ X3 @ A2 )
% 0.30/0.61 & ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.61 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono_ex1
% 0.30/0.61 thf(fact_6617_sum__strict__mono__ex1,axiom,
% 0.30/0.61 ! [A2: set_int,F: int > nat,G: int > nat] :
% 0.30/0.61 ( ( finite_finite_int @ A2 )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ? [X3: int] :
% 0.30/0.61 ( ( member_int @ X3 @ A2 )
% 0.30/0.61 & ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.61 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono_ex1
% 0.30/0.61 thf(fact_6618_sum__strict__mono__ex1,axiom,
% 0.30/0.61 ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ? [X3: complex] :
% 0.30/0.61 ( ( member_complex @ X3 @ A2 )
% 0.30/0.61 & ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.61 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono_ex1
% 0.30/0.61 thf(fact_6619_sum__strict__mono__ex1,axiom,
% 0.30/0.61 ! [A2: set_nat,F: nat > int,G: nat > int] :
% 0.30/0.61 ( ( finite_finite_nat @ A2 )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ? [X3: nat] :
% 0.30/0.61 ( ( member_nat @ X3 @ A2 )
% 0.30/0.61 & ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.61 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono_ex1
% 0.30/0.61 thf(fact_6620_sum__strict__mono__ex1,axiom,
% 0.30/0.61 ! [A2: set_complex,F: complex > int,G: complex > int] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ? [X3: complex] :
% 0.30/0.61 ( ( member_complex @ X3 @ A2 )
% 0.30/0.61 & ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.61 => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono_ex1
% 0.30/0.61 thf(fact_6621_sum__strict__mono__ex1,axiom,
% 0.30/0.61 ! [A2: set_int,F: int > int,G: int > int] :
% 0.30/0.61 ( ( finite_finite_int @ A2 )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ? [X3: int] :
% 0.30/0.61 ( ( member_int @ X3 @ A2 )
% 0.30/0.61 & ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.30/0.61 => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono_ex1
% 0.30/0.61 thf(fact_6622_sum_Orelated,axiom,
% 0.30/0.61 ! [R: complex > complex > $o,S3: set_nat,H2: nat > complex,G: nat > complex] :
% 0.30/0.61 ( ( R @ zero_zero_complex @ zero_zero_complex )
% 0.30/0.61 => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
% 0.30/0.61 ( ( ( R @ X15 @ X23 )
% 0.30/0.61 & ( R @ Y15 @ Y23 ) )
% 0.30/0.61 => ( R @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
% 0.30/0.61 => ( ( finite_finite_nat @ S3 )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ S3 )
% 0.30/0.61 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( R @ ( groups2073611262835488442omplex @ H2 @ S3 ) @ ( groups2073611262835488442omplex @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.related
% 0.30/0.61 thf(fact_6623_sum_Orelated,axiom,
% 0.30/0.61 ! [R: complex > complex > $o,S3: set_int,H2: int > complex,G: int > complex] :
% 0.30/0.61 ( ( R @ zero_zero_complex @ zero_zero_complex )
% 0.30/0.61 => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
% 0.30/0.61 ( ( ( R @ X15 @ X23 )
% 0.30/0.61 & ( R @ Y15 @ Y23 ) )
% 0.30/0.61 => ( R @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
% 0.30/0.61 => ( ( finite_finite_int @ S3 )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ S3 )
% 0.30/0.61 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( R @ ( groups3049146728041665814omplex @ H2 @ S3 ) @ ( groups3049146728041665814omplex @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.related
% 0.30/0.61 thf(fact_6624_sum_Orelated,axiom,
% 0.30/0.61 ! [R: real > real > $o,S3: set_int,H2: int > real,G: int > real] :
% 0.30/0.61 ( ( R @ zero_zero_real @ zero_zero_real )
% 0.30/0.61 => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
% 0.30/0.61 ( ( ( R @ X15 @ X23 )
% 0.30/0.61 & ( R @ Y15 @ Y23 ) )
% 0.30/0.61 => ( R @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
% 0.30/0.61 => ( ( finite_finite_int @ S3 )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ S3 )
% 0.30/0.61 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( R @ ( groups8778361861064173332t_real @ H2 @ S3 ) @ ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.related
% 0.30/0.61 thf(fact_6625_sum_Orelated,axiom,
% 0.30/0.61 ! [R: real > real > $o,S3: set_complex,H2: complex > real,G: complex > real] :
% 0.30/0.61 ( ( R @ zero_zero_real @ zero_zero_real )
% 0.30/0.61 => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
% 0.30/0.61 ( ( ( R @ X15 @ X23 )
% 0.30/0.61 & ( R @ Y15 @ Y23 ) )
% 0.30/0.61 => ( R @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
% 0.30/0.61 => ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ S3 )
% 0.30/0.61 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( R @ ( groups5808333547571424918x_real @ H2 @ S3 ) @ ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.related
% 0.30/0.61 thf(fact_6626_sum_Orelated,axiom,
% 0.30/0.61 ! [R: rat > rat > $o,S3: set_nat,H2: nat > rat,G: nat > rat] :
% 0.30/0.61 ( ( R @ zero_zero_rat @ zero_zero_rat )
% 0.30/0.61 => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
% 0.30/0.61 ( ( ( R @ X15 @ X23 )
% 0.30/0.61 & ( R @ Y15 @ Y23 ) )
% 0.30/0.61 => ( R @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 0.30/0.61 => ( ( finite_finite_nat @ S3 )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ S3 )
% 0.30/0.61 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( R @ ( groups2906978787729119204at_rat @ H2 @ S3 ) @ ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.related
% 0.30/0.61 thf(fact_6627_sum_Orelated,axiom,
% 0.30/0.61 ! [R: rat > rat > $o,S3: set_int,H2: int > rat,G: int > rat] :
% 0.30/0.61 ( ( R @ zero_zero_rat @ zero_zero_rat )
% 0.30/0.61 => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
% 0.30/0.61 ( ( ( R @ X15 @ X23 )
% 0.30/0.61 & ( R @ Y15 @ Y23 ) )
% 0.30/0.61 => ( R @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 0.30/0.61 => ( ( finite_finite_int @ S3 )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ S3 )
% 0.30/0.61 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( R @ ( groups3906332499630173760nt_rat @ H2 @ S3 ) @ ( groups3906332499630173760nt_rat @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.related
% 0.30/0.61 thf(fact_6628_sum_Orelated,axiom,
% 0.30/0.61 ! [R: rat > rat > $o,S3: set_complex,H2: complex > rat,G: complex > rat] :
% 0.30/0.61 ( ( R @ zero_zero_rat @ zero_zero_rat )
% 0.30/0.61 => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
% 0.30/0.61 ( ( ( R @ X15 @ X23 )
% 0.30/0.61 & ( R @ Y15 @ Y23 ) )
% 0.30/0.61 => ( R @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 0.30/0.61 => ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ S3 )
% 0.30/0.61 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( R @ ( groups5058264527183730370ex_rat @ H2 @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.related
% 0.30/0.61 thf(fact_6629_sum_Orelated,axiom,
% 0.30/0.61 ! [R: nat > nat > $o,S3: set_int,H2: int > nat,G: int > nat] :
% 0.30/0.61 ( ( R @ zero_zero_nat @ zero_zero_nat )
% 0.30/0.61 => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
% 0.30/0.61 ( ( ( R @ X15 @ X23 )
% 0.30/0.61 & ( R @ Y15 @ Y23 ) )
% 0.30/0.61 => ( R @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
% 0.30/0.61 => ( ( finite_finite_int @ S3 )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ S3 )
% 0.30/0.61 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( R @ ( groups4541462559716669496nt_nat @ H2 @ S3 ) @ ( groups4541462559716669496nt_nat @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.related
% 0.30/0.61 thf(fact_6630_sum_Orelated,axiom,
% 0.30/0.61 ! [R: nat > nat > $o,S3: set_complex,H2: complex > nat,G: complex > nat] :
% 0.30/0.61 ( ( R @ zero_zero_nat @ zero_zero_nat )
% 0.30/0.61 => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
% 0.30/0.61 ( ( ( R @ X15 @ X23 )
% 0.30/0.61 & ( R @ Y15 @ Y23 ) )
% 0.30/0.61 => ( R @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
% 0.30/0.61 => ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ S3 )
% 0.30/0.61 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( R @ ( groups5693394587270226106ex_nat @ H2 @ S3 ) @ ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.related
% 0.30/0.61 thf(fact_6631_sum_Orelated,axiom,
% 0.30/0.61 ! [R: int > int > $o,S3: set_nat,H2: nat > int,G: nat > int] :
% 0.30/0.61 ( ( R @ zero_zero_int @ zero_zero_int )
% 0.30/0.61 => ( ! [X15: int,Y15: int,X23: int,Y23: int] :
% 0.30/0.61 ( ( ( R @ X15 @ X23 )
% 0.30/0.61 & ( R @ Y15 @ Y23 ) )
% 0.30/0.61 => ( R @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
% 0.30/0.61 => ( ( finite_finite_nat @ S3 )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ S3 )
% 0.30/0.61 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( R @ ( groups3539618377306564664at_int @ H2 @ S3 ) @ ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.related
% 0.30/0.61 thf(fact_6632_sum__strict__mono,axiom,
% 0.30/0.61 ! [A2: set_complex,F: complex > real,G: complex > real] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.61 => ( ( A2 != bot_bot_set_complex )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono
% 0.30/0.61 thf(fact_6633_sum__strict__mono,axiom,
% 0.30/0.61 ! [A2: set_int,F: int > real,G: int > real] :
% 0.30/0.61 ( ( finite_finite_int @ A2 )
% 0.30/0.61 => ( ( A2 != bot_bot_set_int )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono
% 0.30/0.61 thf(fact_6634_sum__strict__mono,axiom,
% 0.30/0.61 ! [A2: set_real,F: real > real,G: real > real] :
% 0.30/0.61 ( ( finite_finite_real @ A2 )
% 0.30/0.61 => ( ( A2 != bot_bot_set_real )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono
% 0.30/0.61 thf(fact_6635_sum__strict__mono,axiom,
% 0.30/0.61 ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.61 => ( ( A2 != bot_bot_set_complex )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono
% 0.30/0.61 thf(fact_6636_sum__strict__mono,axiom,
% 0.30/0.61 ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 0.30/0.61 ( ( finite_finite_nat @ A2 )
% 0.30/0.61 => ( ( A2 != bot_bot_set_nat )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono
% 0.30/0.61 thf(fact_6637_sum__strict__mono,axiom,
% 0.30/0.61 ! [A2: set_int,F: int > rat,G: int > rat] :
% 0.30/0.61 ( ( finite_finite_int @ A2 )
% 0.30/0.61 => ( ( A2 != bot_bot_set_int )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono
% 0.30/0.61 thf(fact_6638_sum__strict__mono,axiom,
% 0.30/0.61 ! [A2: set_real,F: real > rat,G: real > rat] :
% 0.30/0.61 ( ( finite_finite_real @ A2 )
% 0.30/0.61 => ( ( A2 != bot_bot_set_real )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono
% 0.30/0.61 thf(fact_6639_sum__strict__mono,axiom,
% 0.30/0.61 ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.61 => ( ( A2 != bot_bot_set_complex )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono
% 0.30/0.61 thf(fact_6640_sum__strict__mono,axiom,
% 0.30/0.61 ! [A2: set_int,F: int > nat,G: int > nat] :
% 0.30/0.61 ( ( finite_finite_int @ A2 )
% 0.30/0.61 => ( ( A2 != bot_bot_set_int )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono
% 0.30/0.61 thf(fact_6641_sum__strict__mono,axiom,
% 0.30/0.61 ! [A2: set_real,F: real > nat,G: real > nat] :
% 0.30/0.61 ( ( finite_finite_real @ A2 )
% 0.30/0.61 => ( ( A2 != bot_bot_set_real )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ A2 )
% 0.30/0.61 => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.61 => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_strict_mono
% 0.30/0.61 thf(fact_6642_ln__bound,axiom,
% 0.30/0.61 ! [X2: real] :
% 0.30/0.61 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.61 => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).
% 0.30/0.61
% 0.30/0.61 % ln_bound
% 0.30/0.61 thf(fact_6643_ln__gt__zero,axiom,
% 0.30/0.61 ! [X2: real] :
% 0.30/0.61 ( ( ord_less_real @ one_one_real @ X2 )
% 0.30/0.61 => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ln_gt_zero
% 0.30/0.61 thf(fact_6644_ln__less__zero,axiom,
% 0.30/0.61 ! [X2: real] :
% 0.30/0.61 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.61 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.30/0.61 => ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ln_less_zero
% 0.30/0.61 thf(fact_6645_ln__gt__zero__imp__gt__one,axiom,
% 0.30/0.61 ! [X2: real] :
% 0.30/0.61 ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
% 0.30/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.61 => ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ln_gt_zero_imp_gt_one
% 0.30/0.61 thf(fact_6646_ln__ge__zero,axiom,
% 0.30/0.61 ! [X2: real] :
% 0.30/0.61 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ln_ge_zero
% 0.30/0.61 thf(fact_6647_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.61 ! [S4: set_real,T4: set_real,S3: set_real,I2: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
% 0.30/0.61 ( ( finite_finite_real @ S4 )
% 0.30/0.61 => ( ( finite_finite_real @ T4 )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 0.30/0.61 => ( ( I2 @ ( J @ A4 ) )
% 0.30/0.61 = A4 ) )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 0.30/0.61 => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.30/0.61 => ( ! [B4: real] :
% 0.30/0.61 ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.30/0.61 => ( ( J @ ( I2 @ B4 ) )
% 0.30/0.61 = B4 ) )
% 0.30/0.61 => ( ! [B4: real] :
% 0.30/0.61 ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.30/0.61 => ( member_real @ ( I2 @ B4 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ S4 )
% 0.30/0.61 => ( ( G @ A4 )
% 0.30/0.61 = zero_zero_complex ) )
% 0.30/0.61 => ( ! [B4: real] :
% 0.30/0.61 ( ( member_real @ B4 @ T4 )
% 0.30/0.61 => ( ( H2 @ B4 )
% 0.30/0.61 = zero_zero_complex ) )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ S3 )
% 0.30/0.61 => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.61 = ( G @ A4 ) ) )
% 0.30/0.61 => ( ( groups5754745047067104278omplex @ G @ S3 )
% 0.30/0.61 = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.reindex_bij_witness_not_neutral
% 0.30/0.61 thf(fact_6648_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.61 ! [S4: set_real,T4: set_int,S3: set_real,I2: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
% 0.30/0.61 ( ( finite_finite_real @ S4 )
% 0.30/0.61 => ( ( finite_finite_int @ T4 )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 0.30/0.61 => ( ( I2 @ ( J @ A4 ) )
% 0.30/0.61 = A4 ) )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 0.30/0.61 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.30/0.61 => ( ! [B4: int] :
% 0.30/0.61 ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.30/0.61 => ( ( J @ ( I2 @ B4 ) )
% 0.30/0.61 = B4 ) )
% 0.30/0.61 => ( ! [B4: int] :
% 0.30/0.61 ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.30/0.61 => ( member_real @ ( I2 @ B4 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ S4 )
% 0.30/0.61 => ( ( G @ A4 )
% 0.30/0.61 = zero_zero_complex ) )
% 0.30/0.61 => ( ! [B4: int] :
% 0.30/0.61 ( ( member_int @ B4 @ T4 )
% 0.30/0.61 => ( ( H2 @ B4 )
% 0.30/0.61 = zero_zero_complex ) )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ S3 )
% 0.30/0.61 => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.61 = ( G @ A4 ) ) )
% 0.30/0.61 => ( ( groups5754745047067104278omplex @ G @ S3 )
% 0.30/0.61 = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.reindex_bij_witness_not_neutral
% 0.30/0.61 thf(fact_6649_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.61 ! [S4: set_int,T4: set_real,S3: set_int,I2: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
% 0.30/0.61 ( ( finite_finite_int @ S4 )
% 0.30/0.61 => ( ( finite_finite_real @ T4 )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 0.30/0.61 => ( ( I2 @ ( J @ A4 ) )
% 0.30/0.61 = A4 ) )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 0.30/0.61 => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.30/0.61 => ( ! [B4: real] :
% 0.30/0.61 ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.30/0.61 => ( ( J @ ( I2 @ B4 ) )
% 0.30/0.61 = B4 ) )
% 0.30/0.61 => ( ! [B4: real] :
% 0.30/0.61 ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.30/0.61 => ( member_int @ ( I2 @ B4 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ S4 )
% 0.30/0.61 => ( ( G @ A4 )
% 0.30/0.61 = zero_zero_complex ) )
% 0.30/0.61 => ( ! [B4: real] :
% 0.30/0.61 ( ( member_real @ B4 @ T4 )
% 0.30/0.61 => ( ( H2 @ B4 )
% 0.30/0.61 = zero_zero_complex ) )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ S3 )
% 0.30/0.61 => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.61 = ( G @ A4 ) ) )
% 0.30/0.61 => ( ( groups3049146728041665814omplex @ G @ S3 )
% 0.30/0.61 = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.reindex_bij_witness_not_neutral
% 0.30/0.61 thf(fact_6650_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.61 ! [S4: set_int,T4: set_int,S3: set_int,I2: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
% 0.30/0.61 ( ( finite_finite_int @ S4 )
% 0.30/0.61 => ( ( finite_finite_int @ T4 )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 0.30/0.61 => ( ( I2 @ ( J @ A4 ) )
% 0.30/0.61 = A4 ) )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 0.30/0.61 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.30/0.61 => ( ! [B4: int] :
% 0.30/0.61 ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.30/0.61 => ( ( J @ ( I2 @ B4 ) )
% 0.30/0.61 = B4 ) )
% 0.30/0.61 => ( ! [B4: int] :
% 0.30/0.61 ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.30/0.61 => ( member_int @ ( I2 @ B4 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ S4 )
% 0.30/0.61 => ( ( G @ A4 )
% 0.30/0.61 = zero_zero_complex ) )
% 0.30/0.61 => ( ! [B4: int] :
% 0.30/0.61 ( ( member_int @ B4 @ T4 )
% 0.30/0.61 => ( ( H2 @ B4 )
% 0.30/0.61 = zero_zero_complex ) )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ S3 )
% 0.30/0.61 => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.61 = ( G @ A4 ) ) )
% 0.30/0.61 => ( ( groups3049146728041665814omplex @ G @ S3 )
% 0.30/0.61 = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.reindex_bij_witness_not_neutral
% 0.30/0.61 thf(fact_6651_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.61 ! [S4: set_real,T4: set_real,S3: set_real,I2: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
% 0.30/0.61 ( ( finite_finite_real @ S4 )
% 0.30/0.61 => ( ( finite_finite_real @ T4 )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 0.30/0.61 => ( ( I2 @ ( J @ A4 ) )
% 0.30/0.61 = A4 ) )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 0.30/0.61 => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.30/0.61 => ( ! [B4: real] :
% 0.30/0.61 ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.30/0.61 => ( ( J @ ( I2 @ B4 ) )
% 0.30/0.61 = B4 ) )
% 0.30/0.61 => ( ! [B4: real] :
% 0.30/0.61 ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.30/0.61 => ( member_real @ ( I2 @ B4 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ S4 )
% 0.30/0.61 => ( ( G @ A4 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ! [B4: real] :
% 0.30/0.61 ( ( member_real @ B4 @ T4 )
% 0.30/0.61 => ( ( H2 @ B4 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ S3 )
% 0.30/0.61 => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.61 = ( G @ A4 ) ) )
% 0.30/0.61 => ( ( groups8097168146408367636l_real @ G @ S3 )
% 0.30/0.61 = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.reindex_bij_witness_not_neutral
% 0.30/0.61 thf(fact_6652_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.61 ! [S4: set_real,T4: set_int,S3: set_real,I2: int > real,J: real > int,T3: set_int,G: real > real,H2: int > real] :
% 0.30/0.61 ( ( finite_finite_real @ S4 )
% 0.30/0.61 => ( ( finite_finite_int @ T4 )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 0.30/0.61 => ( ( I2 @ ( J @ A4 ) )
% 0.30/0.61 = A4 ) )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 0.30/0.61 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.30/0.61 => ( ! [B4: int] :
% 0.30/0.61 ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.30/0.61 => ( ( J @ ( I2 @ B4 ) )
% 0.30/0.61 = B4 ) )
% 0.30/0.61 => ( ! [B4: int] :
% 0.30/0.61 ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.30/0.61 => ( member_real @ ( I2 @ B4 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ S4 )
% 0.30/0.61 => ( ( G @ A4 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ! [B4: int] :
% 0.30/0.61 ( ( member_int @ B4 @ T4 )
% 0.30/0.61 => ( ( H2 @ B4 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ S3 )
% 0.30/0.61 => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.61 = ( G @ A4 ) ) )
% 0.30/0.61 => ( ( groups8097168146408367636l_real @ G @ S3 )
% 0.30/0.61 = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.reindex_bij_witness_not_neutral
% 0.30/0.61 thf(fact_6653_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.61 ! [S4: set_real,T4: set_complex,S3: set_real,I2: complex > real,J: real > complex,T3: set_complex,G: real > real,H2: complex > real] :
% 0.30/0.61 ( ( finite_finite_real @ S4 )
% 0.30/0.61 => ( ( finite3207457112153483333omplex @ T4 )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 0.30/0.61 => ( ( I2 @ ( J @ A4 ) )
% 0.30/0.61 = A4 ) )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 0.30/0.61 => ( member_complex @ ( J @ A4 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 0.30/0.61 => ( ! [B4: complex] :
% 0.30/0.61 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.30/0.61 => ( ( J @ ( I2 @ B4 ) )
% 0.30/0.61 = B4 ) )
% 0.30/0.61 => ( ! [B4: complex] :
% 0.30/0.61 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.30/0.61 => ( member_real @ ( I2 @ B4 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ S4 )
% 0.30/0.61 => ( ( G @ A4 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ! [B4: complex] :
% 0.30/0.61 ( ( member_complex @ B4 @ T4 )
% 0.30/0.61 => ( ( H2 @ B4 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ S3 )
% 0.30/0.61 => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.61 = ( G @ A4 ) ) )
% 0.30/0.61 => ( ( groups8097168146408367636l_real @ G @ S3 )
% 0.30/0.61 = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.reindex_bij_witness_not_neutral
% 0.30/0.61 thf(fact_6654_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.61 ! [S4: set_int,T4: set_real,S3: set_int,I2: real > int,J: int > real,T3: set_real,G: int > real,H2: real > real] :
% 0.30/0.61 ( ( finite_finite_int @ S4 )
% 0.30/0.61 => ( ( finite_finite_real @ T4 )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 0.30/0.61 => ( ( I2 @ ( J @ A4 ) )
% 0.30/0.61 = A4 ) )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 0.30/0.61 => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.30/0.61 => ( ! [B4: real] :
% 0.30/0.61 ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.30/0.61 => ( ( J @ ( I2 @ B4 ) )
% 0.30/0.61 = B4 ) )
% 0.30/0.61 => ( ! [B4: real] :
% 0.30/0.61 ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.30/0.61 => ( member_int @ ( I2 @ B4 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ S4 )
% 0.30/0.61 => ( ( G @ A4 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ! [B4: real] :
% 0.30/0.61 ( ( member_real @ B4 @ T4 )
% 0.30/0.61 => ( ( H2 @ B4 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ S3 )
% 0.30/0.61 => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.61 = ( G @ A4 ) ) )
% 0.30/0.61 => ( ( groups8778361861064173332t_real @ G @ S3 )
% 0.30/0.61 = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.reindex_bij_witness_not_neutral
% 0.30/0.61 thf(fact_6655_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.61 ! [S4: set_int,T4: set_int,S3: set_int,I2: int > int,J: int > int,T3: set_int,G: int > real,H2: int > real] :
% 0.30/0.61 ( ( finite_finite_int @ S4 )
% 0.30/0.61 => ( ( finite_finite_int @ T4 )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 0.30/0.61 => ( ( I2 @ ( J @ A4 ) )
% 0.30/0.61 = A4 ) )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 0.30/0.61 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.30/0.61 => ( ! [B4: int] :
% 0.30/0.61 ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.30/0.61 => ( ( J @ ( I2 @ B4 ) )
% 0.30/0.61 = B4 ) )
% 0.30/0.61 => ( ! [B4: int] :
% 0.30/0.61 ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.30/0.61 => ( member_int @ ( I2 @ B4 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ S4 )
% 0.30/0.61 => ( ( G @ A4 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ! [B4: int] :
% 0.30/0.61 ( ( member_int @ B4 @ T4 )
% 0.30/0.61 => ( ( H2 @ B4 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ S3 )
% 0.30/0.61 => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.61 = ( G @ A4 ) ) )
% 0.30/0.61 => ( ( groups8778361861064173332t_real @ G @ S3 )
% 0.30/0.61 = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.reindex_bij_witness_not_neutral
% 0.30/0.61 thf(fact_6656_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.61 ! [S4: set_int,T4: set_complex,S3: set_int,I2: complex > int,J: int > complex,T3: set_complex,G: int > real,H2: complex > real] :
% 0.30/0.61 ( ( finite_finite_int @ S4 )
% 0.30/0.61 => ( ( finite3207457112153483333omplex @ T4 )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 0.30/0.61 => ( ( I2 @ ( J @ A4 ) )
% 0.30/0.61 = A4 ) )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 0.30/0.61 => ( member_complex @ ( J @ A4 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 0.30/0.61 => ( ! [B4: complex] :
% 0.30/0.61 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.30/0.61 => ( ( J @ ( I2 @ B4 ) )
% 0.30/0.61 = B4 ) )
% 0.30/0.61 => ( ! [B4: complex] :
% 0.30/0.61 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.30/0.61 => ( member_int @ ( I2 @ B4 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ S4 )
% 0.30/0.61 => ( ( G @ A4 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ! [B4: complex] :
% 0.30/0.61 ( ( member_complex @ B4 @ T4 )
% 0.30/0.61 => ( ( H2 @ B4 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ! [A4: int] :
% 0.30/0.61 ( ( member_int @ A4 @ S3 )
% 0.30/0.61 => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.61 = ( G @ A4 ) ) )
% 0.30/0.61 => ( ( groups8778361861064173332t_real @ G @ S3 )
% 0.30/0.61 = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.reindex_bij_witness_not_neutral
% 0.30/0.61 thf(fact_6657_sum__nonneg__leq__bound,axiom,
% 0.30/0.61 ! [S: set_real,F: real > real,B3: real,I2: real] :
% 0.30/0.61 ( ( finite_finite_real @ S )
% 0.30/0.61 => ( ! [I3: real] :
% 0.30/0.61 ( ( member_real @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups8097168146408367636l_real @ F @ S )
% 0.30/0.61 = B3 )
% 0.30/0.61 => ( ( member_real @ I2 @ S )
% 0.30/0.61 => ( ord_less_eq_real @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_leq_bound
% 0.30/0.61 thf(fact_6658_sum__nonneg__leq__bound,axiom,
% 0.30/0.61 ! [S: set_int,F: int > real,B3: real,I2: int] :
% 0.30/0.61 ( ( finite_finite_int @ S )
% 0.30/0.61 => ( ! [I3: int] :
% 0.30/0.61 ( ( member_int @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups8778361861064173332t_real @ F @ S )
% 0.30/0.61 = B3 )
% 0.30/0.61 => ( ( member_int @ I2 @ S )
% 0.30/0.61 => ( ord_less_eq_real @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_leq_bound
% 0.30/0.61 thf(fact_6659_sum__nonneg__leq__bound,axiom,
% 0.30/0.61 ! [S: set_complex,F: complex > real,B3: real,I2: complex] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ S )
% 0.30/0.61 => ( ! [I3: complex] :
% 0.30/0.61 ( ( member_complex @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups5808333547571424918x_real @ F @ S )
% 0.30/0.61 = B3 )
% 0.30/0.61 => ( ( member_complex @ I2 @ S )
% 0.30/0.61 => ( ord_less_eq_real @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_leq_bound
% 0.30/0.61 thf(fact_6660_sum__nonneg__leq__bound,axiom,
% 0.30/0.61 ! [S: set_real,F: real > rat,B3: rat,I2: real] :
% 0.30/0.61 ( ( finite_finite_real @ S )
% 0.30/0.61 => ( ! [I3: real] :
% 0.30/0.61 ( ( member_real @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups1300246762558778688al_rat @ F @ S )
% 0.30/0.61 = B3 )
% 0.30/0.61 => ( ( member_real @ I2 @ S )
% 0.30/0.61 => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_leq_bound
% 0.30/0.61 thf(fact_6661_sum__nonneg__leq__bound,axiom,
% 0.30/0.61 ! [S: set_nat,F: nat > rat,B3: rat,I2: nat] :
% 0.30/0.61 ( ( finite_finite_nat @ S )
% 0.30/0.61 => ( ! [I3: nat] :
% 0.30/0.61 ( ( member_nat @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups2906978787729119204at_rat @ F @ S )
% 0.30/0.61 = B3 )
% 0.30/0.61 => ( ( member_nat @ I2 @ S )
% 0.30/0.61 => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_leq_bound
% 0.30/0.61 thf(fact_6662_sum__nonneg__leq__bound,axiom,
% 0.30/0.61 ! [S: set_int,F: int > rat,B3: rat,I2: int] :
% 0.30/0.61 ( ( finite_finite_int @ S )
% 0.30/0.61 => ( ! [I3: int] :
% 0.30/0.61 ( ( member_int @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups3906332499630173760nt_rat @ F @ S )
% 0.30/0.61 = B3 )
% 0.30/0.61 => ( ( member_int @ I2 @ S )
% 0.30/0.61 => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_leq_bound
% 0.30/0.61 thf(fact_6663_sum__nonneg__leq__bound,axiom,
% 0.30/0.61 ! [S: set_complex,F: complex > rat,B3: rat,I2: complex] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ S )
% 0.30/0.61 => ( ! [I3: complex] :
% 0.30/0.61 ( ( member_complex @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups5058264527183730370ex_rat @ F @ S )
% 0.30/0.61 = B3 )
% 0.30/0.61 => ( ( member_complex @ I2 @ S )
% 0.30/0.61 => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_leq_bound
% 0.30/0.61 thf(fact_6664_sum__nonneg__leq__bound,axiom,
% 0.30/0.61 ! [S: set_real,F: real > nat,B3: nat,I2: real] :
% 0.30/0.61 ( ( finite_finite_real @ S )
% 0.30/0.61 => ( ! [I3: real] :
% 0.30/0.61 ( ( member_real @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups1935376822645274424al_nat @ F @ S )
% 0.30/0.61 = B3 )
% 0.30/0.61 => ( ( member_real @ I2 @ S )
% 0.30/0.61 => ( ord_less_eq_nat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_leq_bound
% 0.30/0.61 thf(fact_6665_sum__nonneg__leq__bound,axiom,
% 0.30/0.61 ! [S: set_int,F: int > nat,B3: nat,I2: int] :
% 0.30/0.61 ( ( finite_finite_int @ S )
% 0.30/0.61 => ( ! [I3: int] :
% 0.30/0.61 ( ( member_int @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups4541462559716669496nt_nat @ F @ S )
% 0.30/0.61 = B3 )
% 0.30/0.61 => ( ( member_int @ I2 @ S )
% 0.30/0.61 => ( ord_less_eq_nat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_leq_bound
% 0.30/0.61 thf(fact_6666_sum__nonneg__leq__bound,axiom,
% 0.30/0.61 ! [S: set_complex,F: complex > nat,B3: nat,I2: complex] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ S )
% 0.30/0.61 => ( ! [I3: complex] :
% 0.30/0.61 ( ( member_complex @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups5693394587270226106ex_nat @ F @ S )
% 0.30/0.61 = B3 )
% 0.30/0.61 => ( ( member_complex @ I2 @ S )
% 0.30/0.61 => ( ord_less_eq_nat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_leq_bound
% 0.30/0.61 thf(fact_6667_sum__nonneg__0,axiom,
% 0.30/0.61 ! [S: set_real,F: real > real,I2: real] :
% 0.30/0.61 ( ( finite_finite_real @ S )
% 0.30/0.61 => ( ! [I3: real] :
% 0.30/0.61 ( ( member_real @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups8097168146408367636l_real @ F @ S )
% 0.30/0.61 = zero_zero_real )
% 0.30/0.61 => ( ( member_real @ I2 @ S )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = zero_zero_real ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_0
% 0.30/0.61 thf(fact_6668_sum__nonneg__0,axiom,
% 0.30/0.61 ! [S: set_int,F: int > real,I2: int] :
% 0.30/0.61 ( ( finite_finite_int @ S )
% 0.30/0.61 => ( ! [I3: int] :
% 0.30/0.61 ( ( member_int @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups8778361861064173332t_real @ F @ S )
% 0.30/0.61 = zero_zero_real )
% 0.30/0.61 => ( ( member_int @ I2 @ S )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = zero_zero_real ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_0
% 0.30/0.61 thf(fact_6669_sum__nonneg__0,axiom,
% 0.30/0.61 ! [S: set_complex,F: complex > real,I2: complex] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ S )
% 0.30/0.61 => ( ! [I3: complex] :
% 0.30/0.61 ( ( member_complex @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups5808333547571424918x_real @ F @ S )
% 0.30/0.61 = zero_zero_real )
% 0.30/0.61 => ( ( member_complex @ I2 @ S )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = zero_zero_real ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_0
% 0.30/0.61 thf(fact_6670_sum__nonneg__0,axiom,
% 0.30/0.61 ! [S: set_real,F: real > rat,I2: real] :
% 0.30/0.61 ( ( finite_finite_real @ S )
% 0.30/0.61 => ( ! [I3: real] :
% 0.30/0.61 ( ( member_real @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups1300246762558778688al_rat @ F @ S )
% 0.30/0.61 = zero_zero_rat )
% 0.30/0.61 => ( ( member_real @ I2 @ S )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_0
% 0.30/0.61 thf(fact_6671_sum__nonneg__0,axiom,
% 0.30/0.61 ! [S: set_nat,F: nat > rat,I2: nat] :
% 0.30/0.61 ( ( finite_finite_nat @ S )
% 0.30/0.61 => ( ! [I3: nat] :
% 0.30/0.61 ( ( member_nat @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups2906978787729119204at_rat @ F @ S )
% 0.30/0.61 = zero_zero_rat )
% 0.30/0.61 => ( ( member_nat @ I2 @ S )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_0
% 0.30/0.61 thf(fact_6672_sum__nonneg__0,axiom,
% 0.30/0.61 ! [S: set_int,F: int > rat,I2: int] :
% 0.30/0.61 ( ( finite_finite_int @ S )
% 0.30/0.61 => ( ! [I3: int] :
% 0.30/0.61 ( ( member_int @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups3906332499630173760nt_rat @ F @ S )
% 0.30/0.61 = zero_zero_rat )
% 0.30/0.61 => ( ( member_int @ I2 @ S )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_0
% 0.30/0.61 thf(fact_6673_sum__nonneg__0,axiom,
% 0.30/0.61 ! [S: set_complex,F: complex > rat,I2: complex] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ S )
% 0.30/0.61 => ( ! [I3: complex] :
% 0.30/0.61 ( ( member_complex @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups5058264527183730370ex_rat @ F @ S )
% 0.30/0.61 = zero_zero_rat )
% 0.30/0.61 => ( ( member_complex @ I2 @ S )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_0
% 0.30/0.61 thf(fact_6674_sum__nonneg__0,axiom,
% 0.30/0.61 ! [S: set_real,F: real > nat,I2: real] :
% 0.30/0.61 ( ( finite_finite_real @ S )
% 0.30/0.61 => ( ! [I3: real] :
% 0.30/0.61 ( ( member_real @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups1935376822645274424al_nat @ F @ S )
% 0.30/0.61 = zero_zero_nat )
% 0.30/0.61 => ( ( member_real @ I2 @ S )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = zero_zero_nat ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_0
% 0.30/0.61 thf(fact_6675_sum__nonneg__0,axiom,
% 0.30/0.61 ! [S: set_int,F: int > nat,I2: int] :
% 0.30/0.61 ( ( finite_finite_int @ S )
% 0.30/0.61 => ( ! [I3: int] :
% 0.30/0.61 ( ( member_int @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups4541462559716669496nt_nat @ F @ S )
% 0.30/0.61 = zero_zero_nat )
% 0.30/0.61 => ( ( member_int @ I2 @ S )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = zero_zero_nat ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_0
% 0.30/0.61 thf(fact_6676_sum__nonneg__0,axiom,
% 0.30/0.61 ! [S: set_complex,F: complex > nat,I2: complex] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ S )
% 0.30/0.61 => ( ! [I3: complex] :
% 0.30/0.61 ( ( member_complex @ I3 @ S )
% 0.30/0.61 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ( ( groups5693394587270226106ex_nat @ F @ S )
% 0.30/0.61 = zero_zero_nat )
% 0.30/0.61 => ( ( member_complex @ I2 @ S )
% 0.30/0.61 => ( ( F @ I2 )
% 0.30/0.61 = zero_zero_nat ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_nonneg_0
% 0.30/0.61 thf(fact_6677_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.61 ! [A2: set_real,G: real > complex] :
% 0.30/0.61 ( ( finite_finite_real @ A2 )
% 0.30/0.61 => ( ( groups5754745047067104278omplex @ G
% 0.30/0.61 @ ( minus_minus_set_real @ A2
% 0.30/0.61 @ ( collect_real
% 0.30/0.61 @ ^ [X: real] :
% 0.30/0.61 ( ( G @ X )
% 0.30/0.61 = zero_zero_complex ) ) ) )
% 0.30/0.61 = ( groups5754745047067104278omplex @ G @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.setdiff_irrelevant
% 0.30/0.61 thf(fact_6678_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.61 ! [A2: set_int,G: int > complex] :
% 0.30/0.61 ( ( finite_finite_int @ A2 )
% 0.30/0.61 => ( ( groups3049146728041665814omplex @ G
% 0.30/0.61 @ ( minus_minus_set_int @ A2
% 0.30/0.61 @ ( collect_int
% 0.30/0.61 @ ^ [X: int] :
% 0.30/0.61 ( ( G @ X )
% 0.30/0.61 = zero_zero_complex ) ) ) )
% 0.30/0.61 = ( groups3049146728041665814omplex @ G @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.setdiff_irrelevant
% 0.30/0.61 thf(fact_6679_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.61 ! [A2: set_real,G: real > real] :
% 0.30/0.61 ( ( finite_finite_real @ A2 )
% 0.30/0.61 => ( ( groups8097168146408367636l_real @ G
% 0.30/0.61 @ ( minus_minus_set_real @ A2
% 0.30/0.61 @ ( collect_real
% 0.30/0.61 @ ^ [X: real] :
% 0.30/0.61 ( ( G @ X )
% 0.30/0.61 = zero_zero_real ) ) ) )
% 0.30/0.61 = ( groups8097168146408367636l_real @ G @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.setdiff_irrelevant
% 0.30/0.61 thf(fact_6680_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.61 ! [A2: set_int,G: int > real] :
% 0.30/0.61 ( ( finite_finite_int @ A2 )
% 0.30/0.61 => ( ( groups8778361861064173332t_real @ G
% 0.30/0.61 @ ( minus_minus_set_int @ A2
% 0.30/0.61 @ ( collect_int
% 0.30/0.61 @ ^ [X: int] :
% 0.30/0.61 ( ( G @ X )
% 0.30/0.61 = zero_zero_real ) ) ) )
% 0.30/0.61 = ( groups8778361861064173332t_real @ G @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.setdiff_irrelevant
% 0.30/0.61 thf(fact_6681_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.61 ! [A2: set_complex,G: complex > real] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.61 => ( ( groups5808333547571424918x_real @ G
% 0.30/0.61 @ ( minus_811609699411566653omplex @ A2
% 0.30/0.61 @ ( collect_complex
% 0.30/0.61 @ ^ [X: complex] :
% 0.30/0.61 ( ( G @ X )
% 0.30/0.61 = zero_zero_real ) ) ) )
% 0.30/0.61 = ( groups5808333547571424918x_real @ G @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.setdiff_irrelevant
% 0.30/0.61 thf(fact_6682_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.61 ! [A2: set_real,G: real > rat] :
% 0.30/0.61 ( ( finite_finite_real @ A2 )
% 0.30/0.61 => ( ( groups1300246762558778688al_rat @ G
% 0.30/0.61 @ ( minus_minus_set_real @ A2
% 0.30/0.61 @ ( collect_real
% 0.30/0.61 @ ^ [X: real] :
% 0.30/0.61 ( ( G @ X )
% 0.30/0.61 = zero_zero_rat ) ) ) )
% 0.30/0.61 = ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.setdiff_irrelevant
% 0.30/0.61 thf(fact_6683_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.61 ! [A2: set_int,G: int > rat] :
% 0.30/0.61 ( ( finite_finite_int @ A2 )
% 0.30/0.61 => ( ( groups3906332499630173760nt_rat @ G
% 0.30/0.61 @ ( minus_minus_set_int @ A2
% 0.30/0.61 @ ( collect_int
% 0.30/0.61 @ ^ [X: int] :
% 0.30/0.61 ( ( G @ X )
% 0.30/0.61 = zero_zero_rat ) ) ) )
% 0.30/0.61 = ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.setdiff_irrelevant
% 0.30/0.61 thf(fact_6684_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.61 ! [A2: set_complex,G: complex > rat] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.61 => ( ( groups5058264527183730370ex_rat @ G
% 0.30/0.61 @ ( minus_811609699411566653omplex @ A2
% 0.30/0.61 @ ( collect_complex
% 0.30/0.61 @ ^ [X: complex] :
% 0.30/0.61 ( ( G @ X )
% 0.30/0.61 = zero_zero_rat ) ) ) )
% 0.30/0.61 = ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.setdiff_irrelevant
% 0.30/0.61 thf(fact_6685_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.61 ! [A2: set_real,G: real > nat] :
% 0.30/0.61 ( ( finite_finite_real @ A2 )
% 0.30/0.61 => ( ( groups1935376822645274424al_nat @ G
% 0.30/0.61 @ ( minus_minus_set_real @ A2
% 0.30/0.61 @ ( collect_real
% 0.30/0.61 @ ^ [X: real] :
% 0.30/0.61 ( ( G @ X )
% 0.30/0.61 = zero_zero_nat ) ) ) )
% 0.30/0.61 = ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.setdiff_irrelevant
% 0.30/0.61 thf(fact_6686_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.61 ! [A2: set_int,G: int > nat] :
% 0.30/0.61 ( ( finite_finite_int @ A2 )
% 0.30/0.61 => ( ( groups4541462559716669496nt_nat @ G
% 0.30/0.61 @ ( minus_minus_set_int @ A2
% 0.30/0.61 @ ( collect_int
% 0.30/0.61 @ ^ [X: int] :
% 0.30/0.61 ( ( G @ X )
% 0.30/0.61 = zero_zero_nat ) ) ) )
% 0.30/0.61 = ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.setdiff_irrelevant
% 0.30/0.61 thf(fact_6687_sum__pos2,axiom,
% 0.30/0.61 ! [I5: set_real,I2: real,F: real > real] :
% 0.30/0.61 ( ( finite_finite_real @ I5 )
% 0.30/0.61 => ( ( member_real @ I2 @ I5 )
% 0.30/0.61 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 0.30/0.61 => ( ! [I3: real] :
% 0.30/0.61 ( ( member_real @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos2
% 0.30/0.61 thf(fact_6688_sum__pos2,axiom,
% 0.30/0.61 ! [I5: set_int,I2: int,F: int > real] :
% 0.30/0.61 ( ( finite_finite_int @ I5 )
% 0.30/0.61 => ( ( member_int @ I2 @ I5 )
% 0.30/0.61 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 0.30/0.61 => ( ! [I3: int] :
% 0.30/0.61 ( ( member_int @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos2
% 0.30/0.61 thf(fact_6689_sum__pos2,axiom,
% 0.30/0.61 ! [I5: set_complex,I2: complex,F: complex > real] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.61 => ( ( member_complex @ I2 @ I5 )
% 0.30/0.61 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 0.30/0.61 => ( ! [I3: complex] :
% 0.30/0.61 ( ( member_complex @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos2
% 0.30/0.61 thf(fact_6690_sum__pos2,axiom,
% 0.30/0.61 ! [I5: set_real,I2: real,F: real > rat] :
% 0.30/0.61 ( ( finite_finite_real @ I5 )
% 0.30/0.61 => ( ( member_real @ I2 @ I5 )
% 0.30/0.61 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
% 0.30/0.61 => ( ! [I3: real] :
% 0.30/0.61 ( ( member_real @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos2
% 0.30/0.61 thf(fact_6691_sum__pos2,axiom,
% 0.30/0.61 ! [I5: set_nat,I2: nat,F: nat > rat] :
% 0.30/0.61 ( ( finite_finite_nat @ I5 )
% 0.30/0.61 => ( ( member_nat @ I2 @ I5 )
% 0.30/0.61 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
% 0.30/0.61 => ( ! [I3: nat] :
% 0.30/0.61 ( ( member_nat @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos2
% 0.30/0.61 thf(fact_6692_sum__pos2,axiom,
% 0.30/0.61 ! [I5: set_int,I2: int,F: int > rat] :
% 0.30/0.61 ( ( finite_finite_int @ I5 )
% 0.30/0.61 => ( ( member_int @ I2 @ I5 )
% 0.30/0.61 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
% 0.30/0.61 => ( ! [I3: int] :
% 0.30/0.61 ( ( member_int @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos2
% 0.30/0.61 thf(fact_6693_sum__pos2,axiom,
% 0.30/0.61 ! [I5: set_complex,I2: complex,F: complex > rat] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.61 => ( ( member_complex @ I2 @ I5 )
% 0.30/0.61 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
% 0.30/0.61 => ( ! [I3: complex] :
% 0.30/0.61 ( ( member_complex @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos2
% 0.30/0.61 thf(fact_6694_sum__pos2,axiom,
% 0.30/0.61 ! [I5: set_real,I2: real,F: real > nat] :
% 0.30/0.61 ( ( finite_finite_real @ I5 )
% 0.30/0.61 => ( ( member_real @ I2 @ I5 )
% 0.30/0.61 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 0.30/0.61 => ( ! [I3: real] :
% 0.30/0.61 ( ( member_real @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos2
% 0.30/0.61 thf(fact_6695_sum__pos2,axiom,
% 0.30/0.61 ! [I5: set_int,I2: int,F: int > nat] :
% 0.30/0.61 ( ( finite_finite_int @ I5 )
% 0.30/0.61 => ( ( member_int @ I2 @ I5 )
% 0.30/0.61 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 0.30/0.61 => ( ! [I3: int] :
% 0.30/0.61 ( ( member_int @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos2
% 0.30/0.61 thf(fact_6696_sum__pos2,axiom,
% 0.30/0.61 ! [I5: set_complex,I2: complex,F: complex > nat] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.61 => ( ( member_complex @ I2 @ I5 )
% 0.30/0.61 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 0.30/0.61 => ( ! [I3: complex] :
% 0.30/0.61 ( ( member_complex @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos2
% 0.30/0.61 thf(fact_6697_sum__pos,axiom,
% 0.30/0.61 ! [I5: set_complex,F: complex > real] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.61 => ( ( I5 != bot_bot_set_complex )
% 0.30/0.61 => ( ! [I3: complex] :
% 0.30/0.61 ( ( member_complex @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos
% 0.30/0.61 thf(fact_6698_sum__pos,axiom,
% 0.30/0.61 ! [I5: set_int,F: int > real] :
% 0.30/0.61 ( ( finite_finite_int @ I5 )
% 0.30/0.61 => ( ( I5 != bot_bot_set_int )
% 0.30/0.61 => ( ! [I3: int] :
% 0.30/0.61 ( ( member_int @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos
% 0.30/0.61 thf(fact_6699_sum__pos,axiom,
% 0.30/0.61 ! [I5: set_real,F: real > real] :
% 0.30/0.61 ( ( finite_finite_real @ I5 )
% 0.30/0.61 => ( ( I5 != bot_bot_set_real )
% 0.30/0.61 => ( ! [I3: real] :
% 0.30/0.61 ( ( member_real @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos
% 0.30/0.61 thf(fact_6700_sum__pos,axiom,
% 0.30/0.61 ! [I5: set_complex,F: complex > rat] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.61 => ( ( I5 != bot_bot_set_complex )
% 0.30/0.61 => ( ! [I3: complex] :
% 0.30/0.61 ( ( member_complex @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos
% 0.30/0.61 thf(fact_6701_sum__pos,axiom,
% 0.30/0.61 ! [I5: set_nat,F: nat > rat] :
% 0.30/0.61 ( ( finite_finite_nat @ I5 )
% 0.30/0.61 => ( ( I5 != bot_bot_set_nat )
% 0.30/0.61 => ( ! [I3: nat] :
% 0.30/0.61 ( ( member_nat @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos
% 0.30/0.61 thf(fact_6702_sum__pos,axiom,
% 0.30/0.61 ! [I5: set_int,F: int > rat] :
% 0.30/0.61 ( ( finite_finite_int @ I5 )
% 0.30/0.61 => ( ( I5 != bot_bot_set_int )
% 0.30/0.61 => ( ! [I3: int] :
% 0.30/0.61 ( ( member_int @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos
% 0.30/0.61 thf(fact_6703_sum__pos,axiom,
% 0.30/0.61 ! [I5: set_real,F: real > rat] :
% 0.30/0.61 ( ( finite_finite_real @ I5 )
% 0.30/0.61 => ( ( I5 != bot_bot_set_real )
% 0.30/0.61 => ( ! [I3: real] :
% 0.30/0.61 ( ( member_real @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos
% 0.30/0.61 thf(fact_6704_sum__pos,axiom,
% 0.30/0.61 ! [I5: set_complex,F: complex > nat] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.61 => ( ( I5 != bot_bot_set_complex )
% 0.30/0.61 => ( ! [I3: complex] :
% 0.30/0.61 ( ( member_complex @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos
% 0.30/0.61 thf(fact_6705_sum__pos,axiom,
% 0.30/0.61 ! [I5: set_int,F: int > nat] :
% 0.30/0.61 ( ( finite_finite_int @ I5 )
% 0.30/0.61 => ( ( I5 != bot_bot_set_int )
% 0.30/0.61 => ( ! [I3: int] :
% 0.30/0.61 ( ( member_int @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos
% 0.30/0.61 thf(fact_6706_sum__pos,axiom,
% 0.30/0.61 ! [I5: set_real,F: real > nat] :
% 0.30/0.61 ( ( finite_finite_real @ I5 )
% 0.30/0.61 => ( ( I5 != bot_bot_set_real )
% 0.30/0.61 => ( ! [I3: real] :
% 0.30/0.61 ( ( member_real @ I3 @ I5 )
% 0.30/0.61 => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.61 => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum_pos
% 0.30/0.61 thf(fact_6707_ln__ge__zero__imp__ge__one,axiom,
% 0.30/0.61 ! [X2: real] :
% 0.30/0.61 ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
% 0.30/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.61 => ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % ln_ge_zero_imp_ge_one
% 0.30/0.61 thf(fact_6708_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.61 ! [T3: set_real,S3: set_real,G: real > complex,H2: real > complex] :
% 0.30/0.61 ( ( finite_finite_real @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_complex ) )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups5754745047067104278omplex @ G @ T3 )
% 0.30/0.61 = ( groups5754745047067104278omplex @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_right
% 0.30/0.61 thf(fact_6709_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.61 ! [T3: set_real,S3: set_real,G: real > real,H2: real > real] :
% 0.30/0.61 ( ( finite_finite_real @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups8097168146408367636l_real @ G @ T3 )
% 0.30/0.61 = ( groups8097168146408367636l_real @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_right
% 0.30/0.61 thf(fact_6710_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.61 ! [T3: set_complex,S3: set_complex,G: complex > real,H2: complex > real] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.61 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups5808333547571424918x_real @ G @ T3 )
% 0.30/0.61 = ( groups5808333547571424918x_real @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_right
% 0.30/0.61 thf(fact_6711_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.61 ! [T3: set_real,S3: set_real,G: real > rat,H2: real > rat] :
% 0.30/0.61 ( ( finite_finite_real @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_rat ) )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups1300246762558778688al_rat @ G @ T3 )
% 0.30/0.61 = ( groups1300246762558778688al_rat @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_right
% 0.30/0.61 thf(fact_6712_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.61 ! [T3: set_complex,S3: set_complex,G: complex > rat,H2: complex > rat] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.61 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_rat ) )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 0.30/0.61 = ( groups5058264527183730370ex_rat @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_right
% 0.30/0.61 thf(fact_6713_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.61 ! [T3: set_real,S3: set_real,G: real > nat,H2: real > nat] :
% 0.30/0.61 ( ( finite_finite_real @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_nat ) )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups1935376822645274424al_nat @ G @ T3 )
% 0.30/0.61 = ( groups1935376822645274424al_nat @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_right
% 0.30/0.61 thf(fact_6714_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.61 ! [T3: set_complex,S3: set_complex,G: complex > nat,H2: complex > nat] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.61 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_nat ) )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 0.30/0.61 = ( groups5693394587270226106ex_nat @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_right
% 0.30/0.61 thf(fact_6715_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.61 ! [T3: set_real,S3: set_real,G: real > int,H2: real > int] :
% 0.30/0.61 ( ( finite_finite_real @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_int ) )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups1932886352136224148al_int @ G @ T3 )
% 0.30/0.61 = ( groups1932886352136224148al_int @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_right
% 0.30/0.61 thf(fact_6716_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.61 ! [T3: set_complex,S3: set_complex,G: complex > int,H2: complex > int] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.61 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_int ) )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 0.30/0.61 = ( groups5690904116761175830ex_int @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_right
% 0.30/0.61 thf(fact_6717_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.61 ! [T3: set_nat,S3: set_nat,G: nat > complex,H2: nat > complex] :
% 0.30/0.61 ( ( finite_finite_nat @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_complex ) )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups2073611262835488442omplex @ G @ T3 )
% 0.30/0.61 = ( groups2073611262835488442omplex @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_right
% 0.30/0.61 thf(fact_6718_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.61 ! [T3: set_real,S3: set_real,H2: real > complex,G: real > complex] :
% 0.30/0.61 ( ( finite_finite_real @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.61 => ( ( H2 @ X5 )
% 0.30/0.61 = zero_zero_complex ) )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups5754745047067104278omplex @ G @ S3 )
% 0.30/0.61 = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_left
% 0.30/0.61 thf(fact_6719_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.61 ! [T3: set_real,S3: set_real,H2: real > real,G: real > real] :
% 0.30/0.61 ( ( finite_finite_real @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.61 => ( ( H2 @ X5 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups8097168146408367636l_real @ G @ S3 )
% 0.30/0.61 = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_left
% 0.30/0.61 thf(fact_6720_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.61 ! [T3: set_complex,S3: set_complex,H2: complex > real,G: complex > real] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.61 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.61 => ( ( H2 @ X5 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups5808333547571424918x_real @ G @ S3 )
% 0.30/0.61 = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_left
% 0.30/0.61 thf(fact_6721_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.61 ! [T3: set_real,S3: set_real,H2: real > rat,G: real > rat] :
% 0.30/0.61 ( ( finite_finite_real @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.61 => ( ( H2 @ X5 )
% 0.30/0.61 = zero_zero_rat ) )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups1300246762558778688al_rat @ G @ S3 )
% 0.30/0.61 = ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_left
% 0.30/0.61 thf(fact_6722_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.61 ! [T3: set_complex,S3: set_complex,H2: complex > rat,G: complex > rat] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.61 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.61 => ( ( H2 @ X5 )
% 0.30/0.61 = zero_zero_rat ) )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups5058264527183730370ex_rat @ G @ S3 )
% 0.30/0.61 = ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_left
% 0.30/0.61 thf(fact_6723_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.61 ! [T3: set_real,S3: set_real,H2: real > nat,G: real > nat] :
% 0.30/0.61 ( ( finite_finite_real @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.61 => ( ( H2 @ X5 )
% 0.30/0.61 = zero_zero_nat ) )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups1935376822645274424al_nat @ G @ S3 )
% 0.30/0.61 = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_left
% 0.30/0.61 thf(fact_6724_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.61 ! [T3: set_complex,S3: set_complex,H2: complex > nat,G: complex > nat] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.61 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.61 => ( ( H2 @ X5 )
% 0.30/0.61 = zero_zero_nat ) )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups5693394587270226106ex_nat @ G @ S3 )
% 0.30/0.61 = ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_left
% 0.30/0.61 thf(fact_6725_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.61 ! [T3: set_real,S3: set_real,H2: real > int,G: real > int] :
% 0.30/0.61 ( ( finite_finite_real @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.61 => ( ( H2 @ X5 )
% 0.30/0.61 = zero_zero_int ) )
% 0.30/0.61 => ( ! [X5: real] :
% 0.30/0.61 ( ( member_real @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups1932886352136224148al_int @ G @ S3 )
% 0.30/0.61 = ( groups1932886352136224148al_int @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_left
% 0.30/0.61 thf(fact_6726_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.61 ! [T3: set_complex,S3: set_complex,H2: complex > int,G: complex > int] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.61 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.61 => ( ( H2 @ X5 )
% 0.30/0.61 = zero_zero_int ) )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups5690904116761175830ex_int @ G @ S3 )
% 0.30/0.61 = ( groups5690904116761175830ex_int @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_left
% 0.30/0.61 thf(fact_6727_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.61 ! [T3: set_nat,S3: set_nat,H2: nat > complex,G: nat > complex] :
% 0.30/0.61 ( ( finite_finite_nat @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 0.30/0.61 => ( ( H2 @ X5 )
% 0.30/0.61 = zero_zero_complex ) )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ S3 )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = ( H2 @ X5 ) ) )
% 0.30/0.61 => ( ( groups2073611262835488442omplex @ G @ S3 )
% 0.30/0.61 = ( groups2073611262835488442omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_cong_left
% 0.30/0.61 thf(fact_6728_sum_Omono__neutral__right,axiom,
% 0.30/0.61 ! [T3: set_complex,S3: set_complex,G: complex > real] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.61 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ( groups5808333547571424918x_real @ G @ T3 )
% 0.30/0.61 = ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_right
% 0.30/0.61 thf(fact_6729_sum_Omono__neutral__right,axiom,
% 0.30/0.61 ! [T3: set_complex,S3: set_complex,G: complex > rat] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.61 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_rat ) )
% 0.30/0.61 => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 0.30/0.61 = ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_right
% 0.30/0.61 thf(fact_6730_sum_Omono__neutral__right,axiom,
% 0.30/0.61 ! [T3: set_complex,S3: set_complex,G: complex > nat] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.61 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_nat ) )
% 0.30/0.61 => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 0.30/0.61 = ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_right
% 0.30/0.61 thf(fact_6731_sum_Omono__neutral__right,axiom,
% 0.30/0.61 ! [T3: set_complex,S3: set_complex,G: complex > int] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.61 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_int ) )
% 0.30/0.61 => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 0.30/0.61 = ( groups5690904116761175830ex_int @ G @ S3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_right
% 0.30/0.61 thf(fact_6732_sum_Omono__neutral__right,axiom,
% 0.30/0.61 ! [T3: set_nat,S3: set_nat,G: nat > complex] :
% 0.30/0.61 ( ( finite_finite_nat @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_complex ) )
% 0.30/0.61 => ( ( groups2073611262835488442omplex @ G @ T3 )
% 0.30/0.61 = ( groups2073611262835488442omplex @ G @ S3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_right
% 0.30/0.61 thf(fact_6733_sum_Omono__neutral__right,axiom,
% 0.30/0.61 ! [T3: set_nat,S3: set_nat,G: nat > rat] :
% 0.30/0.61 ( ( finite_finite_nat @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_rat ) )
% 0.30/0.61 => ( ( groups2906978787729119204at_rat @ G @ T3 )
% 0.30/0.61 = ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_right
% 0.30/0.61 thf(fact_6734_sum_Omono__neutral__right,axiom,
% 0.30/0.61 ! [T3: set_nat,S3: set_nat,G: nat > int] :
% 0.30/0.61 ( ( finite_finite_nat @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_int ) )
% 0.30/0.61 => ( ( groups3539618377306564664at_int @ G @ T3 )
% 0.30/0.61 = ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_right
% 0.30/0.61 thf(fact_6735_sum_Omono__neutral__right,axiom,
% 0.30/0.61 ! [T3: set_int,S3: set_int,G: int > complex] :
% 0.30/0.61 ( ( finite_finite_int @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_complex ) )
% 0.30/0.61 => ( ( groups3049146728041665814omplex @ G @ T3 )
% 0.30/0.61 = ( groups3049146728041665814omplex @ G @ S3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_right
% 0.30/0.61 thf(fact_6736_sum_Omono__neutral__right,axiom,
% 0.30/0.61 ! [T3: set_int,S3: set_int,G: int > real] :
% 0.30/0.61 ( ( finite_finite_int @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ( groups8778361861064173332t_real @ G @ T3 )
% 0.30/0.61 = ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_right
% 0.30/0.61 thf(fact_6737_sum_Omono__neutral__right,axiom,
% 0.30/0.61 ! [T3: set_int,S3: set_int,G: int > rat] :
% 0.30/0.61 ( ( finite_finite_int @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_rat ) )
% 0.30/0.61 => ( ( groups3906332499630173760nt_rat @ G @ T3 )
% 0.30/0.61 = ( groups3906332499630173760nt_rat @ G @ S3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_right
% 0.30/0.61 thf(fact_6738_sum_Omono__neutral__left,axiom,
% 0.30/0.61 ! [T3: set_complex,S3: set_complex,G: complex > real] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.61 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ( groups5808333547571424918x_real @ G @ S3 )
% 0.30/0.61 = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_left
% 0.30/0.61 thf(fact_6739_sum_Omono__neutral__left,axiom,
% 0.30/0.61 ! [T3: set_complex,S3: set_complex,G: complex > rat] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.61 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_rat ) )
% 0.30/0.61 => ( ( groups5058264527183730370ex_rat @ G @ S3 )
% 0.30/0.61 = ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_left
% 0.30/0.61 thf(fact_6740_sum_Omono__neutral__left,axiom,
% 0.30/0.61 ! [T3: set_complex,S3: set_complex,G: complex > nat] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.61 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_nat ) )
% 0.30/0.61 => ( ( groups5693394587270226106ex_nat @ G @ S3 )
% 0.30/0.61 = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_left
% 0.30/0.61 thf(fact_6741_sum_Omono__neutral__left,axiom,
% 0.30/0.61 ! [T3: set_complex,S3: set_complex,G: complex > int] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.61 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: complex] :
% 0.30/0.61 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_int ) )
% 0.30/0.61 => ( ( groups5690904116761175830ex_int @ G @ S3 )
% 0.30/0.61 = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_left
% 0.30/0.61 thf(fact_6742_sum_Omono__neutral__left,axiom,
% 0.30/0.61 ! [T3: set_nat,S3: set_nat,G: nat > complex] :
% 0.30/0.61 ( ( finite_finite_nat @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_complex ) )
% 0.30/0.61 => ( ( groups2073611262835488442omplex @ G @ S3 )
% 0.30/0.61 = ( groups2073611262835488442omplex @ G @ T3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_left
% 0.30/0.61 thf(fact_6743_sum_Omono__neutral__left,axiom,
% 0.30/0.61 ! [T3: set_nat,S3: set_nat,G: nat > rat] :
% 0.30/0.61 ( ( finite_finite_nat @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_rat ) )
% 0.30/0.61 => ( ( groups2906978787729119204at_rat @ G @ S3 )
% 0.30/0.61 = ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_left
% 0.30/0.61 thf(fact_6744_sum_Omono__neutral__left,axiom,
% 0.30/0.61 ! [T3: set_nat,S3: set_nat,G: nat > int] :
% 0.30/0.61 ( ( finite_finite_nat @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: nat] :
% 0.30/0.61 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_int ) )
% 0.30/0.61 => ( ( groups3539618377306564664at_int @ G @ S3 )
% 0.30/0.61 = ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_left
% 0.30/0.61 thf(fact_6745_sum_Omono__neutral__left,axiom,
% 0.30/0.61 ! [T3: set_int,S3: set_int,G: int > complex] :
% 0.30/0.61 ( ( finite_finite_int @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_complex ) )
% 0.30/0.61 => ( ( groups3049146728041665814omplex @ G @ S3 )
% 0.30/0.61 = ( groups3049146728041665814omplex @ G @ T3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_left
% 0.30/0.61 thf(fact_6746_sum_Omono__neutral__left,axiom,
% 0.30/0.61 ! [T3: set_int,S3: set_int,G: int > real] :
% 0.30/0.61 ( ( finite_finite_int @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ( groups8778361861064173332t_real @ G @ S3 )
% 0.30/0.61 = ( groups8778361861064173332t_real @ G @ T3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_left
% 0.30/0.61 thf(fact_6747_sum_Omono__neutral__left,axiom,
% 0.30/0.61 ! [T3: set_int,S3: set_int,G: int > rat] :
% 0.30/0.61 ( ( finite_finite_int @ T3 )
% 0.30/0.61 => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 0.30/0.61 => ( ! [X5: int] :
% 0.30/0.61 ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 0.30/0.61 => ( ( G @ X5 )
% 0.30/0.61 = zero_zero_rat ) )
% 0.30/0.61 => ( ( groups3906332499630173760nt_rat @ G @ S3 )
% 0.30/0.61 = ( groups3906332499630173760nt_rat @ G @ T3 ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.mono_neutral_left
% 0.30/0.61 thf(fact_6748_sum_Osame__carrierI,axiom,
% 0.30/0.61 ! [C4: set_real,A2: set_real,B3: set_real,G: real > complex,H2: real > complex] :
% 0.30/0.61 ( ( finite_finite_real @ C4 )
% 0.30/0.61 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.61 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.61 => ( ( G @ A4 )
% 0.30/0.61 = zero_zero_complex ) )
% 0.30/0.61 => ( ! [B4: real] :
% 0.30/0.61 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 0.30/0.61 => ( ( H2 @ B4 )
% 0.30/0.61 = zero_zero_complex ) )
% 0.30/0.61 => ( ( ( groups5754745047067104278omplex @ G @ C4 )
% 0.30/0.61 = ( groups5754745047067104278omplex @ H2 @ C4 ) )
% 0.30/0.61 => ( ( groups5754745047067104278omplex @ G @ A2 )
% 0.30/0.61 = ( groups5754745047067104278omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.same_carrierI
% 0.30/0.61 thf(fact_6749_sum_Osame__carrierI,axiom,
% 0.30/0.61 ! [C4: set_real,A2: set_real,B3: set_real,G: real > real,H2: real > real] :
% 0.30/0.61 ( ( finite_finite_real @ C4 )
% 0.30/0.61 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.61 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 0.30/0.61 => ( ! [A4: real] :
% 0.30/0.61 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.61 => ( ( G @ A4 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ! [B4: real] :
% 0.30/0.61 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 0.30/0.61 => ( ( H2 @ B4 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ( ( groups8097168146408367636l_real @ G @ C4 )
% 0.30/0.61 = ( groups8097168146408367636l_real @ H2 @ C4 ) )
% 0.30/0.61 => ( ( groups8097168146408367636l_real @ G @ A2 )
% 0.30/0.61 = ( groups8097168146408367636l_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.same_carrierI
% 0.30/0.61 thf(fact_6750_sum_Osame__carrierI,axiom,
% 0.30/0.61 ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
% 0.30/0.61 ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.61 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.30/0.61 => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
% 0.30/0.61 => ( ! [A4: complex] :
% 0.30/0.61 ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.30/0.61 => ( ( G @ A4 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ! [B4: complex] :
% 0.30/0.61 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
% 0.30/0.61 => ( ( H2 @ B4 )
% 0.30/0.61 = zero_zero_real ) )
% 0.30/0.61 => ( ( ( groups5808333547571424918x_real @ G @ C4 )
% 0.30/0.61 = ( groups5808333547571424918x_real @ H2 @ C4 ) )
% 0.30/0.61 => ( ( groups5808333547571424918x_real @ G @ A2 )
% 0.30/0.61 = ( groups5808333547571424918x_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.30/0.61
% 0.30/0.61 % sum.same_carrierI
% 0.30/0.61 thf(fact_6751_sum_Osame__carrierI,axiom,
% 0.30/0.61 ! [C4: set_real,A2: set_real,B3: set_real,G: real > rat,H2: real > rat] :
% 0.30/0.61 ( ( finite_finite_real @ C4 )
% 0.30/0.61 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.61 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 0.30/0.62 => ( ! [A4: real] :
% 0.30/0.62 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.62 => ( ( G @ A4 )
% 0.30/0.62 = zero_zero_rat ) )
% 0.30/0.62 => ( ! [B4: real] :
% 0.30/0.62 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 0.30/0.62 => ( ( H2 @ B4 )
% 0.30/0.62 = zero_zero_rat ) )
% 0.30/0.62 => ( ( ( groups1300246762558778688al_rat @ G @ C4 )
% 0.30/0.62 = ( groups1300246762558778688al_rat @ H2 @ C4 ) )
% 0.30/0.62 => ( ( groups1300246762558778688al_rat @ G @ A2 )
% 0.30/0.62 = ( groups1300246762558778688al_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.same_carrierI
% 0.30/0.62 thf(fact_6752_sum_Osame__carrierI,axiom,
% 0.30/0.62 ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > rat,H2: complex > rat] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
% 0.30/0.62 => ( ! [A4: complex] :
% 0.30/0.62 ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.30/0.62 => ( ( G @ A4 )
% 0.30/0.62 = zero_zero_rat ) )
% 0.30/0.62 => ( ! [B4: complex] :
% 0.30/0.62 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
% 0.30/0.62 => ( ( H2 @ B4 )
% 0.30/0.62 = zero_zero_rat ) )
% 0.30/0.62 => ( ( ( groups5058264527183730370ex_rat @ G @ C4 )
% 0.30/0.62 = ( groups5058264527183730370ex_rat @ H2 @ C4 ) )
% 0.30/0.62 => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 0.30/0.62 = ( groups5058264527183730370ex_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.same_carrierI
% 0.30/0.62 thf(fact_6753_sum_Osame__carrierI,axiom,
% 0.30/0.62 ! [C4: set_real,A2: set_real,B3: set_real,G: real > nat,H2: real > nat] :
% 0.30/0.62 ( ( finite_finite_real @ C4 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 0.30/0.62 => ( ! [A4: real] :
% 0.30/0.62 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.62 => ( ( G @ A4 )
% 0.30/0.62 = zero_zero_nat ) )
% 0.30/0.62 => ( ! [B4: real] :
% 0.30/0.62 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 0.30/0.62 => ( ( H2 @ B4 )
% 0.30/0.62 = zero_zero_nat ) )
% 0.30/0.62 => ( ( ( groups1935376822645274424al_nat @ G @ C4 )
% 0.30/0.62 = ( groups1935376822645274424al_nat @ H2 @ C4 ) )
% 0.30/0.62 => ( ( groups1935376822645274424al_nat @ G @ A2 )
% 0.30/0.62 = ( groups1935376822645274424al_nat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.same_carrierI
% 0.30/0.62 thf(fact_6754_sum_Osame__carrierI,axiom,
% 0.30/0.62 ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > nat,H2: complex > nat] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
% 0.30/0.62 => ( ! [A4: complex] :
% 0.30/0.62 ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.30/0.62 => ( ( G @ A4 )
% 0.30/0.62 = zero_zero_nat ) )
% 0.30/0.62 => ( ! [B4: complex] :
% 0.30/0.62 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
% 0.30/0.62 => ( ( H2 @ B4 )
% 0.30/0.62 = zero_zero_nat ) )
% 0.30/0.62 => ( ( ( groups5693394587270226106ex_nat @ G @ C4 )
% 0.30/0.62 = ( groups5693394587270226106ex_nat @ H2 @ C4 ) )
% 0.30/0.62 => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 0.30/0.62 = ( groups5693394587270226106ex_nat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.same_carrierI
% 0.30/0.62 thf(fact_6755_sum_Osame__carrierI,axiom,
% 0.30/0.62 ! [C4: set_real,A2: set_real,B3: set_real,G: real > int,H2: real > int] :
% 0.30/0.62 ( ( finite_finite_real @ C4 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 0.30/0.62 => ( ! [A4: real] :
% 0.30/0.62 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.62 => ( ( G @ A4 )
% 0.30/0.62 = zero_zero_int ) )
% 0.30/0.62 => ( ! [B4: real] :
% 0.30/0.62 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 0.30/0.62 => ( ( H2 @ B4 )
% 0.30/0.62 = zero_zero_int ) )
% 0.30/0.62 => ( ( ( groups1932886352136224148al_int @ G @ C4 )
% 0.30/0.62 = ( groups1932886352136224148al_int @ H2 @ C4 ) )
% 0.30/0.62 => ( ( groups1932886352136224148al_int @ G @ A2 )
% 0.30/0.62 = ( groups1932886352136224148al_int @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.same_carrierI
% 0.30/0.62 thf(fact_6756_sum_Osame__carrierI,axiom,
% 0.30/0.62 ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > int,H2: complex > int] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
% 0.30/0.62 => ( ! [A4: complex] :
% 0.30/0.62 ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.30/0.62 => ( ( G @ A4 )
% 0.30/0.62 = zero_zero_int ) )
% 0.30/0.62 => ( ! [B4: complex] :
% 0.30/0.62 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
% 0.30/0.62 => ( ( H2 @ B4 )
% 0.30/0.62 = zero_zero_int ) )
% 0.30/0.62 => ( ( ( groups5690904116761175830ex_int @ G @ C4 )
% 0.30/0.62 = ( groups5690904116761175830ex_int @ H2 @ C4 ) )
% 0.30/0.62 => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 0.30/0.62 = ( groups5690904116761175830ex_int @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.same_carrierI
% 0.30/0.62 thf(fact_6757_sum_Osame__carrierI,axiom,
% 0.30/0.62 ! [C4: set_nat,A2: set_nat,B3: set_nat,G: nat > complex,H2: nat > complex] :
% 0.30/0.62 ( ( finite_finite_nat @ C4 )
% 0.30/0.62 => ( ( ord_less_eq_set_nat @ A2 @ C4 )
% 0.30/0.62 => ( ( ord_less_eq_set_nat @ B3 @ C4 )
% 0.30/0.62 => ( ! [A4: nat] :
% 0.30/0.62 ( ( member_nat @ A4 @ ( minus_minus_set_nat @ C4 @ A2 ) )
% 0.30/0.62 => ( ( G @ A4 )
% 0.30/0.62 = zero_zero_complex ) )
% 0.30/0.62 => ( ! [B4: nat] :
% 0.30/0.62 ( ( member_nat @ B4 @ ( minus_minus_set_nat @ C4 @ B3 ) )
% 0.30/0.62 => ( ( H2 @ B4 )
% 0.30/0.62 = zero_zero_complex ) )
% 0.30/0.62 => ( ( ( groups2073611262835488442omplex @ G @ C4 )
% 0.30/0.62 = ( groups2073611262835488442omplex @ H2 @ C4 ) )
% 0.30/0.62 => ( ( groups2073611262835488442omplex @ G @ A2 )
% 0.30/0.62 = ( groups2073611262835488442omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.same_carrierI
% 0.30/0.62 thf(fact_6758_sum_Osame__carrier,axiom,
% 0.30/0.62 ! [C4: set_real,A2: set_real,B3: set_real,G: real > complex,H2: real > complex] :
% 0.30/0.62 ( ( finite_finite_real @ C4 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 0.30/0.62 => ( ! [A4: real] :
% 0.30/0.62 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.62 => ( ( G @ A4 )
% 0.30/0.62 = zero_zero_complex ) )
% 0.30/0.62 => ( ! [B4: real] :
% 0.30/0.62 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 0.30/0.62 => ( ( H2 @ B4 )
% 0.30/0.62 = zero_zero_complex ) )
% 0.30/0.62 => ( ( ( groups5754745047067104278omplex @ G @ A2 )
% 0.30/0.62 = ( groups5754745047067104278omplex @ H2 @ B3 ) )
% 0.30/0.62 = ( ( groups5754745047067104278omplex @ G @ C4 )
% 0.30/0.62 = ( groups5754745047067104278omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.same_carrier
% 0.30/0.62 thf(fact_6759_sum_Osame__carrier,axiom,
% 0.30/0.62 ! [C4: set_real,A2: set_real,B3: set_real,G: real > real,H2: real > real] :
% 0.30/0.62 ( ( finite_finite_real @ C4 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 0.30/0.62 => ( ! [A4: real] :
% 0.30/0.62 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.62 => ( ( G @ A4 )
% 0.30/0.62 = zero_zero_real ) )
% 0.30/0.62 => ( ! [B4: real] :
% 0.30/0.62 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 0.30/0.62 => ( ( H2 @ B4 )
% 0.30/0.62 = zero_zero_real ) )
% 0.30/0.62 => ( ( ( groups8097168146408367636l_real @ G @ A2 )
% 0.30/0.62 = ( groups8097168146408367636l_real @ H2 @ B3 ) )
% 0.30/0.62 = ( ( groups8097168146408367636l_real @ G @ C4 )
% 0.30/0.62 = ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.same_carrier
% 0.30/0.62 thf(fact_6760_sum_Osame__carrier,axiom,
% 0.30/0.62 ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
% 0.30/0.62 => ( ! [A4: complex] :
% 0.30/0.62 ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.30/0.62 => ( ( G @ A4 )
% 0.30/0.62 = zero_zero_real ) )
% 0.30/0.62 => ( ! [B4: complex] :
% 0.30/0.62 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
% 0.30/0.62 => ( ( H2 @ B4 )
% 0.30/0.62 = zero_zero_real ) )
% 0.30/0.62 => ( ( ( groups5808333547571424918x_real @ G @ A2 )
% 0.30/0.62 = ( groups5808333547571424918x_real @ H2 @ B3 ) )
% 0.30/0.62 = ( ( groups5808333547571424918x_real @ G @ C4 )
% 0.30/0.62 = ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.same_carrier
% 0.30/0.62 thf(fact_6761_sum_Osame__carrier,axiom,
% 0.30/0.62 ! [C4: set_real,A2: set_real,B3: set_real,G: real > rat,H2: real > rat] :
% 0.30/0.62 ( ( finite_finite_real @ C4 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 0.30/0.62 => ( ! [A4: real] :
% 0.30/0.62 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.62 => ( ( G @ A4 )
% 0.30/0.62 = zero_zero_rat ) )
% 0.30/0.62 => ( ! [B4: real] :
% 0.30/0.62 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 0.30/0.62 => ( ( H2 @ B4 )
% 0.30/0.62 = zero_zero_rat ) )
% 0.30/0.62 => ( ( ( groups1300246762558778688al_rat @ G @ A2 )
% 0.30/0.62 = ( groups1300246762558778688al_rat @ H2 @ B3 ) )
% 0.30/0.62 = ( ( groups1300246762558778688al_rat @ G @ C4 )
% 0.30/0.62 = ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.same_carrier
% 0.30/0.62 thf(fact_6762_sum_Osame__carrier,axiom,
% 0.30/0.62 ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > rat,H2: complex > rat] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
% 0.30/0.62 => ( ! [A4: complex] :
% 0.30/0.62 ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.30/0.62 => ( ( G @ A4 )
% 0.30/0.62 = zero_zero_rat ) )
% 0.30/0.62 => ( ! [B4: complex] :
% 0.30/0.62 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
% 0.30/0.62 => ( ( H2 @ B4 )
% 0.30/0.62 = zero_zero_rat ) )
% 0.30/0.62 => ( ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 0.30/0.62 = ( groups5058264527183730370ex_rat @ H2 @ B3 ) )
% 0.30/0.62 = ( ( groups5058264527183730370ex_rat @ G @ C4 )
% 0.30/0.62 = ( groups5058264527183730370ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.same_carrier
% 0.30/0.62 thf(fact_6763_sum_Osame__carrier,axiom,
% 0.30/0.62 ! [C4: set_real,A2: set_real,B3: set_real,G: real > nat,H2: real > nat] :
% 0.30/0.62 ( ( finite_finite_real @ C4 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 0.30/0.62 => ( ! [A4: real] :
% 0.30/0.62 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.62 => ( ( G @ A4 )
% 0.30/0.62 = zero_zero_nat ) )
% 0.30/0.62 => ( ! [B4: real] :
% 0.30/0.62 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 0.30/0.62 => ( ( H2 @ B4 )
% 0.30/0.62 = zero_zero_nat ) )
% 0.30/0.62 => ( ( ( groups1935376822645274424al_nat @ G @ A2 )
% 0.30/0.62 = ( groups1935376822645274424al_nat @ H2 @ B3 ) )
% 0.30/0.62 = ( ( groups1935376822645274424al_nat @ G @ C4 )
% 0.30/0.62 = ( groups1935376822645274424al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.same_carrier
% 0.30/0.62 thf(fact_6764_sum_Osame__carrier,axiom,
% 0.30/0.62 ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > nat,H2: complex > nat] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
% 0.30/0.62 => ( ! [A4: complex] :
% 0.30/0.62 ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.30/0.62 => ( ( G @ A4 )
% 0.30/0.62 = zero_zero_nat ) )
% 0.30/0.62 => ( ! [B4: complex] :
% 0.30/0.62 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
% 0.30/0.62 => ( ( H2 @ B4 )
% 0.30/0.62 = zero_zero_nat ) )
% 0.30/0.62 => ( ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 0.30/0.62 = ( groups5693394587270226106ex_nat @ H2 @ B3 ) )
% 0.30/0.62 = ( ( groups5693394587270226106ex_nat @ G @ C4 )
% 0.30/0.62 = ( groups5693394587270226106ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.same_carrier
% 0.30/0.62 thf(fact_6765_sum_Osame__carrier,axiom,
% 0.30/0.62 ! [C4: set_real,A2: set_real,B3: set_real,G: real > int,H2: real > int] :
% 0.30/0.62 ( ( finite_finite_real @ C4 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 0.30/0.62 => ( ! [A4: real] :
% 0.30/0.62 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.62 => ( ( G @ A4 )
% 0.30/0.62 = zero_zero_int ) )
% 0.30/0.62 => ( ! [B4: real] :
% 0.30/0.62 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 0.30/0.62 => ( ( H2 @ B4 )
% 0.30/0.62 = zero_zero_int ) )
% 0.30/0.62 => ( ( ( groups1932886352136224148al_int @ G @ A2 )
% 0.30/0.62 = ( groups1932886352136224148al_int @ H2 @ B3 ) )
% 0.30/0.62 = ( ( groups1932886352136224148al_int @ G @ C4 )
% 0.30/0.62 = ( groups1932886352136224148al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.same_carrier
% 0.30/0.62 thf(fact_6766_sum_Osame__carrier,axiom,
% 0.30/0.62 ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > int,H2: complex > int] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
% 0.30/0.62 => ( ! [A4: complex] :
% 0.30/0.62 ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.30/0.62 => ( ( G @ A4 )
% 0.30/0.62 = zero_zero_int ) )
% 0.30/0.62 => ( ! [B4: complex] :
% 0.30/0.62 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
% 0.30/0.62 => ( ( H2 @ B4 )
% 0.30/0.62 = zero_zero_int ) )
% 0.30/0.62 => ( ( ( groups5690904116761175830ex_int @ G @ A2 )
% 0.30/0.62 = ( groups5690904116761175830ex_int @ H2 @ B3 ) )
% 0.30/0.62 = ( ( groups5690904116761175830ex_int @ G @ C4 )
% 0.30/0.62 = ( groups5690904116761175830ex_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.same_carrier
% 0.30/0.62 thf(fact_6767_sum_Osame__carrier,axiom,
% 0.30/0.62 ! [C4: set_nat,A2: set_nat,B3: set_nat,G: nat > complex,H2: nat > complex] :
% 0.30/0.62 ( ( finite_finite_nat @ C4 )
% 0.30/0.62 => ( ( ord_less_eq_set_nat @ A2 @ C4 )
% 0.30/0.62 => ( ( ord_less_eq_set_nat @ B3 @ C4 )
% 0.30/0.62 => ( ! [A4: nat] :
% 0.30/0.62 ( ( member_nat @ A4 @ ( minus_minus_set_nat @ C4 @ A2 ) )
% 0.30/0.62 => ( ( G @ A4 )
% 0.30/0.62 = zero_zero_complex ) )
% 0.30/0.62 => ( ! [B4: nat] :
% 0.30/0.62 ( ( member_nat @ B4 @ ( minus_minus_set_nat @ C4 @ B3 ) )
% 0.30/0.62 => ( ( H2 @ B4 )
% 0.30/0.62 = zero_zero_complex ) )
% 0.30/0.62 => ( ( ( groups2073611262835488442omplex @ G @ A2 )
% 0.30/0.62 = ( groups2073611262835488442omplex @ H2 @ B3 ) )
% 0.30/0.62 = ( ( groups2073611262835488442omplex @ G @ C4 )
% 0.30/0.62 = ( groups2073611262835488442omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.same_carrier
% 0.30/0.62 thf(fact_6768_sum_Osubset__diff,axiom,
% 0.30/0.62 ! [B3: set_complex,A2: set_complex,G: complex > real] :
% 0.30/0.62 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.30/0.62 => ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62 => ( ( groups5808333547571424918x_real @ G @ A2 )
% 0.30/0.62 = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5808333547571424918x_real @ G @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.subset_diff
% 0.30/0.62 thf(fact_6769_sum_Osubset__diff,axiom,
% 0.30/0.62 ! [B3: set_complex,A2: set_complex,G: complex > rat] :
% 0.30/0.62 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.30/0.62 => ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62 => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 0.30/0.62 = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5058264527183730370ex_rat @ G @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.subset_diff
% 0.30/0.62 thf(fact_6770_sum_Osubset__diff,axiom,
% 0.30/0.62 ! [B3: set_complex,A2: set_complex,G: complex > nat] :
% 0.30/0.62 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.30/0.62 => ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62 => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 0.30/0.62 = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5693394587270226106ex_nat @ G @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.subset_diff
% 0.30/0.62 thf(fact_6771_sum_Osubset__diff,axiom,
% 0.30/0.62 ! [B3: set_complex,A2: set_complex,G: complex > int] :
% 0.30/0.62 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.30/0.62 => ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62 => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 0.30/0.62 = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5690904116761175830ex_int @ G @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.subset_diff
% 0.30/0.62 thf(fact_6772_sum_Osubset__diff,axiom,
% 0.30/0.62 ! [B3: set_nat,A2: set_nat,G: nat > rat] :
% 0.30/0.62 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.30/0.62 => ( ( finite_finite_nat @ A2 )
% 0.30/0.62 => ( ( groups2906978787729119204at_rat @ G @ A2 )
% 0.30/0.62 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups2906978787729119204at_rat @ G @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.subset_diff
% 0.30/0.62 thf(fact_6773_sum_Osubset__diff,axiom,
% 0.30/0.62 ! [B3: set_nat,A2: set_nat,G: nat > int] :
% 0.30/0.62 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.30/0.62 => ( ( finite_finite_nat @ A2 )
% 0.30/0.62 => ( ( groups3539618377306564664at_int @ G @ A2 )
% 0.30/0.62 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups3539618377306564664at_int @ G @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.subset_diff
% 0.30/0.62 thf(fact_6774_sum_Osubset__diff,axiom,
% 0.30/0.62 ! [B3: set_int,A2: set_int,G: int > real] :
% 0.30/0.62 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.30/0.62 => ( ( finite_finite_int @ A2 )
% 0.30/0.62 => ( ( groups8778361861064173332t_real @ G @ A2 )
% 0.30/0.62 = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups8778361861064173332t_real @ G @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.subset_diff
% 0.30/0.62 thf(fact_6775_sum_Osubset__diff,axiom,
% 0.30/0.62 ! [B3: set_int,A2: set_int,G: int > rat] :
% 0.30/0.62 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.30/0.62 => ( ( finite_finite_int @ A2 )
% 0.30/0.62 => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 0.30/0.62 = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups3906332499630173760nt_rat @ G @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.subset_diff
% 0.30/0.62 thf(fact_6776_sum_Osubset__diff,axiom,
% 0.30/0.62 ! [B3: set_int,A2: set_int,G: int > nat] :
% 0.30/0.62 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.30/0.62 => ( ( finite_finite_int @ A2 )
% 0.30/0.62 => ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 0.30/0.62 = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups4541462559716669496nt_nat @ G @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.subset_diff
% 0.30/0.62 thf(fact_6777_sum_Osubset__diff,axiom,
% 0.30/0.62 ! [B3: set_int,A2: set_int,G: int > int] :
% 0.30/0.62 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.30/0.62 => ( ( finite_finite_int @ A2 )
% 0.30/0.62 => ( ( groups4538972089207619220nt_int @ G @ A2 )
% 0.30/0.62 = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups4538972089207619220nt_int @ G @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.subset_diff
% 0.30/0.62 thf(fact_6778_sum__diff,axiom,
% 0.30/0.62 ! [A2: set_complex,B3: set_complex,F: complex > real] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.30/0.62 => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.30/0.62 = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_diff
% 0.30/0.62 thf(fact_6779_sum__diff,axiom,
% 0.30/0.62 ! [A2: set_complex,B3: set_complex,F: complex > rat] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.30/0.62 => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.30/0.62 = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_diff
% 0.30/0.62 thf(fact_6780_sum__diff,axiom,
% 0.30/0.62 ! [A2: set_complex,B3: set_complex,F: complex > int] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.30/0.62 => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.30/0.62 = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_diff
% 0.30/0.62 thf(fact_6781_sum__diff,axiom,
% 0.30/0.62 ! [A2: set_nat,B3: set_nat,F: nat > rat] :
% 0.30/0.62 ( ( finite_finite_nat @ A2 )
% 0.30/0.62 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.30/0.62 => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 0.30/0.62 = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_diff
% 0.30/0.62 thf(fact_6782_sum__diff,axiom,
% 0.30/0.62 ! [A2: set_nat,B3: set_nat,F: nat > int] :
% 0.30/0.62 ( ( finite_finite_nat @ A2 )
% 0.30/0.62 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.30/0.62 => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 0.30/0.62 = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_diff
% 0.30/0.62 thf(fact_6783_sum__diff,axiom,
% 0.30/0.62 ! [A2: set_int,B3: set_int,F: int > real] :
% 0.30/0.62 ( ( finite_finite_int @ A2 )
% 0.30/0.62 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.30/0.62 => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
% 0.30/0.62 = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_diff
% 0.30/0.62 thf(fact_6784_sum__diff,axiom,
% 0.30/0.62 ! [A2: set_int,B3: set_int,F: int > rat] :
% 0.30/0.62 ( ( finite_finite_int @ A2 )
% 0.30/0.62 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.30/0.62 => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
% 0.30/0.62 = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_diff
% 0.30/0.62 thf(fact_6785_sum__diff,axiom,
% 0.30/0.62 ! [A2: set_int,B3: set_int,F: int > int] :
% 0.30/0.62 ( ( finite_finite_int @ A2 )
% 0.30/0.62 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.30/0.62 => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
% 0.30/0.62 = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_diff
% 0.30/0.62 thf(fact_6786_sum__diff,axiom,
% 0.30/0.62 ! [A2: set_complex,B3: set_complex,F: complex > complex] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.30/0.62 => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.30/0.62 = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_diff
% 0.30/0.62 thf(fact_6787_sum__diff,axiom,
% 0.30/0.62 ! [A2: set_nat,B3: set_nat,F: nat > real] :
% 0.30/0.62 ( ( finite_finite_nat @ A2 )
% 0.30/0.62 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.30/0.62 => ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 0.30/0.62 = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_diff
% 0.30/0.62 thf(fact_6788_ln__add__one__self__le__self,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.62 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).
% 0.30/0.62
% 0.30/0.62 % ln_add_one_self_le_self
% 0.30/0.62 thf(fact_6789_ln__mult,axiom,
% 0.30/0.62 ! [X2: real,Y3: real] :
% 0.30/0.62 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.62 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.62 => ( ( ln_ln_real @ ( times_times_real @ X2 @ Y3 ) )
% 0.30/0.62 = ( plus_plus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % ln_mult
% 0.30/0.62 thf(fact_6790_ln__eq__minus__one,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.62 => ( ( ( ln_ln_real @ X2 )
% 0.30/0.62 = ( minus_minus_real @ X2 @ one_one_real ) )
% 0.30/0.62 => ( X2 = one_one_real ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % ln_eq_minus_one
% 0.30/0.62 thf(fact_6791_ln__div,axiom,
% 0.30/0.62 ! [X2: real,Y3: real] :
% 0.30/0.62 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.62 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.62 => ( ( ln_ln_real @ ( divide_divide_real @ X2 @ Y3 ) )
% 0.30/0.62 = ( minus_minus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % ln_div
% 0.30/0.62 thf(fact_6792_sum__mono2,axiom,
% 0.30/0.62 ! [B3: set_real,A2: set_real,F: real > real] :
% 0.30/0.62 ( ( finite_finite_real @ B3 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.30/0.62 => ( ! [B4: real] :
% 0.30/0.62 ( ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A2 ) )
% 0.30/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
% 0.30/0.62 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_mono2
% 0.30/0.62 thf(fact_6793_sum__mono2,axiom,
% 0.30/0.62 ! [B3: set_complex,A2: set_complex,F: complex > real] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ B3 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.30/0.62 => ( ! [B4: complex] :
% 0.30/0.62 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 0.30/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
% 0.30/0.62 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_mono2
% 0.30/0.62 thf(fact_6794_sum__mono2,axiom,
% 0.30/0.62 ! [B3: set_real,A2: set_real,F: real > rat] :
% 0.30/0.62 ( ( finite_finite_real @ B3 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.30/0.62 => ( ! [B4: real] :
% 0.30/0.62 ( ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A2 ) )
% 0.30/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
% 0.30/0.62 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_mono2
% 0.30/0.62 thf(fact_6795_sum__mono2,axiom,
% 0.30/0.62 ! [B3: set_complex,A2: set_complex,F: complex > rat] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ B3 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.30/0.62 => ( ! [B4: complex] :
% 0.30/0.62 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 0.30/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
% 0.30/0.62 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_mono2
% 0.30/0.62 thf(fact_6796_sum__mono2,axiom,
% 0.30/0.62 ! [B3: set_nat,A2: set_nat,F: nat > rat] :
% 0.30/0.62 ( ( finite_finite_nat @ B3 )
% 0.30/0.62 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.30/0.62 => ( ! [B4: nat] :
% 0.30/0.62 ( ( member_nat @ B4 @ ( minus_minus_set_nat @ B3 @ A2 ) )
% 0.30/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
% 0.30/0.62 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_mono2
% 0.30/0.62 thf(fact_6797_sum__mono2,axiom,
% 0.30/0.62 ! [B3: set_real,A2: set_real,F: real > nat] :
% 0.30/0.62 ( ( finite_finite_real @ B3 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.30/0.62 => ( ! [B4: real] :
% 0.30/0.62 ( ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A2 ) )
% 0.30/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B4 ) ) )
% 0.30/0.62 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_mono2
% 0.30/0.62 thf(fact_6798_sum__mono2,axiom,
% 0.30/0.62 ! [B3: set_complex,A2: set_complex,F: complex > nat] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ B3 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.30/0.62 => ( ! [B4: complex] :
% 0.30/0.62 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 0.30/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B4 ) ) )
% 0.30/0.62 => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_mono2
% 0.30/0.62 thf(fact_6799_sum__mono2,axiom,
% 0.30/0.62 ! [B3: set_real,A2: set_real,F: real > int] :
% 0.30/0.62 ( ( finite_finite_real @ B3 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.30/0.62 => ( ! [B4: real] :
% 0.30/0.62 ( ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A2 ) )
% 0.30/0.62 => ( ord_less_eq_int @ zero_zero_int @ ( F @ B4 ) ) )
% 0.30/0.62 => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_mono2
% 0.30/0.62 thf(fact_6800_sum__mono2,axiom,
% 0.30/0.62 ! [B3: set_complex,A2: set_complex,F: complex > int] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ B3 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.30/0.62 => ( ! [B4: complex] :
% 0.30/0.62 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 0.30/0.62 => ( ord_less_eq_int @ zero_zero_int @ ( F @ B4 ) ) )
% 0.30/0.62 => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_mono2
% 0.30/0.62 thf(fact_6801_sum__mono2,axiom,
% 0.30/0.62 ! [B3: set_nat,A2: set_nat,F: nat > int] :
% 0.30/0.62 ( ( finite_finite_nat @ B3 )
% 0.30/0.62 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.30/0.62 => ( ! [B4: nat] :
% 0.30/0.62 ( ( member_nat @ B4 @ ( minus_minus_set_nat @ B3 @ A2 ) )
% 0.30/0.62 => ( ord_less_eq_int @ zero_zero_int @ ( F @ B4 ) ) )
% 0.30/0.62 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_mono2
% 0.30/0.62 thf(fact_6802_ln__2__less__1,axiom,
% 0.30/0.62 ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 0.30/0.62
% 0.30/0.62 % ln_2_less_1
% 0.30/0.62 thf(fact_6803_ln__le__minus__one,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.62 => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % ln_le_minus_one
% 0.30/0.62 thf(fact_6804_ln__diff__le,axiom,
% 0.30/0.62 ! [X2: real,Y3: real] :
% 0.30/0.62 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.62 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.62 => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X2 @ Y3 ) @ Y3 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % ln_diff_le
% 0.30/0.62 thf(fact_6805_ln__add__one__self__le__self2,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.62 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).
% 0.30/0.62
% 0.30/0.62 % ln_add_one_self_le_self2
% 0.30/0.62 thf(fact_6806_sum__strict__mono2,axiom,
% 0.30/0.62 ! [B3: set_real,A2: set_real,B: real,F: real > real] :
% 0.30/0.62 ( ( finite_finite_real @ B3 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.30/0.62 => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
% 0.30/0.62 => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 0.30/0.62 => ( ! [X5: real] :
% 0.30/0.62 ( ( member_real @ X5 @ B3 )
% 0.30/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.30/0.62 => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B3 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_strict_mono2
% 0.30/0.62 thf(fact_6807_sum__strict__mono2,axiom,
% 0.30/0.62 ! [B3: set_complex,A2: set_complex,B: complex,F: complex > real] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ B3 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.30/0.62 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 0.30/0.62 => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 0.30/0.62 => ( ! [X5: complex] :
% 0.30/0.62 ( ( member_complex @ X5 @ B3 )
% 0.30/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.30/0.62 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_strict_mono2
% 0.30/0.62 thf(fact_6808_sum__strict__mono2,axiom,
% 0.30/0.62 ! [B3: set_real,A2: set_real,B: real,F: real > rat] :
% 0.30/0.62 ( ( finite_finite_real @ B3 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.30/0.62 => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
% 0.30/0.62 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 0.30/0.62 => ( ! [X5: real] :
% 0.30/0.62 ( ( member_real @ X5 @ B3 )
% 0.30/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.62 => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B3 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_strict_mono2
% 0.30/0.62 thf(fact_6809_sum__strict__mono2,axiom,
% 0.30/0.62 ! [B3: set_complex,A2: set_complex,B: complex,F: complex > rat] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ B3 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.30/0.62 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 0.30/0.62 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 0.30/0.62 => ( ! [X5: complex] :
% 0.30/0.62 ( ( member_complex @ X5 @ B3 )
% 0.30/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.62 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_strict_mono2
% 0.30/0.62 thf(fact_6810_sum__strict__mono2,axiom,
% 0.30/0.62 ! [B3: set_nat,A2: set_nat,B: nat,F: nat > rat] :
% 0.30/0.62 ( ( finite_finite_nat @ B3 )
% 0.30/0.62 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.30/0.62 => ( ( member_nat @ B @ ( minus_minus_set_nat @ B3 @ A2 ) )
% 0.30/0.62 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 0.30/0.62 => ( ! [X5: nat] :
% 0.30/0.62 ( ( member_nat @ X5 @ B3 )
% 0.30/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.62 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B3 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_strict_mono2
% 0.30/0.62 thf(fact_6811_sum__strict__mono2,axiom,
% 0.30/0.62 ! [B3: set_real,A2: set_real,B: real,F: real > nat] :
% 0.30/0.62 ( ( finite_finite_real @ B3 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.30/0.62 => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
% 0.30/0.62 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 0.30/0.62 => ( ! [X5: real] :
% 0.30/0.62 ( ( member_real @ X5 @ B3 )
% 0.30/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.30/0.62 => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B3 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_strict_mono2
% 0.30/0.62 thf(fact_6812_sum__strict__mono2,axiom,
% 0.30/0.62 ! [B3: set_complex,A2: set_complex,B: complex,F: complex > nat] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ B3 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.30/0.62 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 0.30/0.62 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 0.30/0.62 => ( ! [X5: complex] :
% 0.30/0.62 ( ( member_complex @ X5 @ B3 )
% 0.30/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.30/0.62 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B3 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_strict_mono2
% 0.30/0.62 thf(fact_6813_sum__strict__mono2,axiom,
% 0.30/0.62 ! [B3: set_real,A2: set_real,B: real,F: real > int] :
% 0.30/0.62 ( ( finite_finite_real @ B3 )
% 0.30/0.62 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.30/0.62 => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
% 0.30/0.62 => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 0.30/0.62 => ( ! [X5: real] :
% 0.30/0.62 ( ( member_real @ X5 @ B3 )
% 0.30/0.62 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
% 0.30/0.62 => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B3 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_strict_mono2
% 0.30/0.62 thf(fact_6814_sum__strict__mono2,axiom,
% 0.30/0.62 ! [B3: set_complex,A2: set_complex,B: complex,F: complex > int] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ B3 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.30/0.62 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 0.30/0.62 => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 0.30/0.62 => ( ! [X5: complex] :
% 0.30/0.62 ( ( member_complex @ X5 @ B3 )
% 0.30/0.62 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
% 0.30/0.62 => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B3 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_strict_mono2
% 0.30/0.62 thf(fact_6815_sum__strict__mono2,axiom,
% 0.30/0.62 ! [B3: set_nat,A2: set_nat,B: nat,F: nat > int] :
% 0.30/0.62 ( ( finite_finite_nat @ B3 )
% 0.30/0.62 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.30/0.62 => ( ( member_nat @ B @ ( minus_minus_set_nat @ B3 @ A2 ) )
% 0.30/0.62 => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 0.30/0.62 => ( ! [X5: nat] :
% 0.30/0.62 ( ( member_nat @ X5 @ B3 )
% 0.30/0.62 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
% 0.30/0.62 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B3 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_strict_mono2
% 0.30/0.62 thf(fact_6816_ln__one__minus__pos__upper__bound,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.62 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.30/0.62 => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X2 ) ) @ ( uminus_uminus_real @ X2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % ln_one_minus_pos_upper_bound
% 0.30/0.62 thf(fact_6817_exists__least__lemma,axiom,
% 0.30/0.62 ! [P: nat > $o] :
% 0.30/0.62 ( ~ ( P @ zero_zero_nat )
% 0.30/0.62 => ( ? [X_1: nat] : ( P @ X_1 )
% 0.30/0.62 => ? [N2: nat] :
% 0.30/0.62 ( ~ ( P @ N2 )
% 0.30/0.62 & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % exists_least_lemma
% 0.30/0.62 thf(fact_6818_ex__le__of__int,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ? [Z2: int] : ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z2 ) ) ).
% 0.30/0.62
% 0.30/0.62 % ex_le_of_int
% 0.30/0.62 thf(fact_6819_ex__le__of__int,axiom,
% 0.30/0.62 ! [X2: rat] :
% 0.30/0.62 ? [Z2: int] : ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z2 ) ) ).
% 0.30/0.62
% 0.30/0.62 % ex_le_of_int
% 0.30/0.62 thf(fact_6820_ex__of__int__less,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ? [Z2: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X2 ) ).
% 0.30/0.62
% 0.30/0.62 % ex_of_int_less
% 0.30/0.62 thf(fact_6821_ex__of__int__less,axiom,
% 0.30/0.62 ! [X2: rat] :
% 0.30/0.62 ? [Z2: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X2 ) ).
% 0.30/0.62
% 0.30/0.62 % ex_of_int_less
% 0.30/0.62 thf(fact_6822_ex__less__of__int,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ? [Z2: int] : ( ord_less_real @ X2 @ ( ring_1_of_int_real @ Z2 ) ) ).
% 0.30/0.62
% 0.30/0.62 % ex_less_of_int
% 0.30/0.62 thf(fact_6823_ex__less__of__int,axiom,
% 0.30/0.62 ! [X2: rat] :
% 0.30/0.62 ? [Z2: int] : ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ Z2 ) ) ).
% 0.30/0.62
% 0.30/0.62 % ex_less_of_int
% 0.30/0.62 thf(fact_6824_ln__one__plus__pos__lower__bound,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.62 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.30/0.62 => ( ord_less_eq_real @ ( minus_minus_real @ X2 @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % ln_one_plus_pos_lower_bound
% 0.30/0.62 thf(fact_6825_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.30/0.62 => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.30/0.62 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_ln_one_plus_x_minus_x_bound_nonpos
% 0.30/0.62 thf(fact_6826_round__unique,axiom,
% 0.30/0.62 ! [X2: real,Y3: int] :
% 0.30/0.62 ( ( ord_less_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y3 ) )
% 0.30/0.62 => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y3 ) @ ( plus_plus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.62 => ( ( archim8280529875227126926d_real @ X2 )
% 0.30/0.62 = Y3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % round_unique
% 0.30/0.62 thf(fact_6827_round__unique,axiom,
% 0.30/0.62 ! [X2: rat,Y3: int] :
% 0.30/0.62 ( ( ord_less_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y3 ) )
% 0.30/0.62 => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y3 ) @ ( plus_plus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.62 => ( ( archim7778729529865785530nd_rat @ X2 )
% 0.30/0.62 = Y3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % round_unique
% 0.30/0.62 thf(fact_6828_tanh__ln__real,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.62 => ( ( tanh_real @ ( ln_ln_real @ X2 ) )
% 0.30/0.62 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % tanh_ln_real
% 0.30/0.62 thf(fact_6829_pred__subset__eq2,axiom,
% 0.30/0.62 ! [R: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
% 0.30/0.62 ( ( ord_le2162486998276636481er_o_o
% 0.30/0.62 @ ^ [X: code_integer,Y: $o] : ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X @ Y ) @ R )
% 0.30/0.62 @ ^ [X: code_integer,Y: $o] : ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X @ Y ) @ S3 ) )
% 0.30/0.62 = ( ord_le8980329558974975238eger_o @ R @ S3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % pred_subset_eq2
% 0.30/0.62 thf(fact_6830_pred__subset__eq2,axiom,
% 0.30/0.62 ! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
% 0.30/0.62 ( ( ord_le6124364862034508274_num_o
% 0.30/0.62 @ ^ [X: num,Y: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X @ Y ) @ R )
% 0.30/0.62 @ ^ [X: num,Y: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X @ Y ) @ S3 ) )
% 0.30/0.62 = ( ord_le880128212290418581um_num @ R @ S3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % pred_subset_eq2
% 0.30/0.62 thf(fact_6831_pred__subset__eq2,axiom,
% 0.30/0.62 ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
% 0.30/0.62 ( ( ord_le3404735783095501756_num_o
% 0.30/0.62 @ ^ [X: nat,Y: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ R )
% 0.30/0.62 @ ^ [X: nat,Y: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ S3 ) )
% 0.30/0.62 = ( ord_le8085105155179020875at_num @ R @ S3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % pred_subset_eq2
% 0.30/0.62 thf(fact_6832_pred__subset__eq2,axiom,
% 0.30/0.62 ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
% 0.30/0.62 ( ( ord_le2646555220125990790_nat_o
% 0.30/0.62 @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
% 0.30/0.62 @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S3 ) )
% 0.30/0.62 = ( ord_le3146513528884898305at_nat @ R @ S3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % pred_subset_eq2
% 0.30/0.62 thf(fact_6833_pred__subset__eq2,axiom,
% 0.30/0.62 ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
% 0.30/0.62 ( ( ord_le6741204236512500942_int_o
% 0.30/0.62 @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
% 0.30/0.62 @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S3 ) )
% 0.30/0.62 = ( ord_le2843351958646193337nt_int @ R @ S3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % pred_subset_eq2
% 0.30/0.62 thf(fact_6834_abs__ln__one__plus__x__minus__x__bound,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.62 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_ln_one_plus_x_minus_x_bound
% 0.30/0.62 thf(fact_6835_accp__subset,axiom,
% 0.30/0.62 ! [R1: product_prod_num_num > product_prod_num_num > $o,R22: product_prod_num_num > product_prod_num_num > $o] :
% 0.30/0.62 ( ( ord_le2556027599737686990_num_o @ R1 @ R22 )
% 0.30/0.62 => ( ord_le2239182809043710856_num_o @ ( accp_P3113834385874906142um_num @ R22 ) @ ( accp_P3113834385874906142um_num @ R1 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % accp_subset
% 0.30/0.62 thf(fact_6836_accp__subset,axiom,
% 0.30/0.62 ! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 0.30/0.62 ( ( ord_le5604493270027003598_nat_o @ R1 @ R22 )
% 0.30/0.62 => ( ord_le704812498762024988_nat_o @ ( accp_P4275260045618599050at_nat @ R22 ) @ ( accp_P4275260045618599050at_nat @ R1 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % accp_subset
% 0.30/0.62 thf(fact_6837_accp__subset,axiom,
% 0.30/0.62 ! [R1: product_prod_int_int > product_prod_int_int > $o,R22: product_prod_int_int > product_prod_int_int > $o] :
% 0.30/0.62 ( ( ord_le1598226405681992910_int_o @ R1 @ R22 )
% 0.30/0.62 => ( ord_le8369615600986905444_int_o @ ( accp_P1096762738010456898nt_int @ R22 ) @ ( accp_P1096762738010456898nt_int @ R1 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % accp_subset
% 0.30/0.62 thf(fact_6838_accp__subset,axiom,
% 0.30/0.62 ! [R1: list_nat > list_nat > $o,R22: list_nat > list_nat > $o] :
% 0.30/0.62 ( ( ord_le6558929396352911974_nat_o @ R1 @ R22 )
% 0.30/0.62 => ( ord_le1520216061033275535_nat_o @ ( accp_list_nat @ R22 ) @ ( accp_list_nat @ R1 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % accp_subset
% 0.30/0.62 thf(fact_6839_accp__subset,axiom,
% 0.30/0.62 ! [R1: nat > nat > $o,R22: nat > nat > $o] :
% 0.30/0.62 ( ( ord_le2646555220125990790_nat_o @ R1 @ R22 )
% 0.30/0.62 => ( ord_less_eq_nat_o @ ( accp_nat @ R22 ) @ ( accp_nat @ R1 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % accp_subset
% 0.30/0.62 thf(fact_6840_abs__idempotent,axiom,
% 0.30/0.62 ! [A: real] :
% 0.30/0.62 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.30/0.62 = ( abs_abs_real @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_idempotent
% 0.30/0.62 thf(fact_6841_abs__idempotent,axiom,
% 0.30/0.62 ! [A: int] :
% 0.30/0.62 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 0.30/0.62 = ( abs_abs_int @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_idempotent
% 0.30/0.62 thf(fact_6842_abs__idempotent,axiom,
% 0.30/0.62 ! [A: code_integer] :
% 0.30/0.62 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.30/0.62 = ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_idempotent
% 0.30/0.62 thf(fact_6843_abs__abs,axiom,
% 0.30/0.62 ! [A: real] :
% 0.30/0.62 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.30/0.62 = ( abs_abs_real @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_abs
% 0.30/0.62 thf(fact_6844_abs__abs,axiom,
% 0.30/0.62 ! [A: int] :
% 0.30/0.62 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 0.30/0.62 = ( abs_abs_int @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_abs
% 0.30/0.62 thf(fact_6845_abs__abs,axiom,
% 0.30/0.62 ! [A: code_integer] :
% 0.30/0.62 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.30/0.62 = ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_abs
% 0.30/0.62 thf(fact_6846_abs__zero,axiom,
% 0.30/0.62 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.30/0.62 = zero_z3403309356797280102nteger ) ).
% 0.30/0.62
% 0.30/0.62 % abs_zero
% 0.30/0.62 thf(fact_6847_abs__zero,axiom,
% 0.30/0.62 ( ( abs_abs_real @ zero_zero_real )
% 0.30/0.62 = zero_zero_real ) ).
% 0.30/0.62
% 0.30/0.62 % abs_zero
% 0.30/0.62 thf(fact_6848_abs__zero,axiom,
% 0.30/0.62 ( ( abs_abs_rat @ zero_zero_rat )
% 0.30/0.62 = zero_zero_rat ) ).
% 0.30/0.62
% 0.30/0.62 % abs_zero
% 0.30/0.62 thf(fact_6849_abs__zero,axiom,
% 0.30/0.62 ( ( abs_abs_int @ zero_zero_int )
% 0.30/0.62 = zero_zero_int ) ).
% 0.30/0.62
% 0.30/0.62 % abs_zero
% 0.30/0.62 thf(fact_6850_abs__eq__0,axiom,
% 0.30/0.62 ! [A: code_integer] :
% 0.30/0.62 ( ( ( abs_abs_Code_integer @ A )
% 0.30/0.62 = zero_z3403309356797280102nteger )
% 0.30/0.62 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_0
% 0.30/0.62 thf(fact_6851_abs__eq__0,axiom,
% 0.30/0.62 ! [A: real] :
% 0.30/0.62 ( ( ( abs_abs_real @ A )
% 0.30/0.62 = zero_zero_real )
% 0.30/0.62 = ( A = zero_zero_real ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_0
% 0.30/0.62 thf(fact_6852_abs__eq__0,axiom,
% 0.30/0.62 ! [A: rat] :
% 0.30/0.62 ( ( ( abs_abs_rat @ A )
% 0.30/0.62 = zero_zero_rat )
% 0.30/0.62 = ( A = zero_zero_rat ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_0
% 0.30/0.62 thf(fact_6853_abs__eq__0,axiom,
% 0.30/0.62 ! [A: int] :
% 0.30/0.62 ( ( ( abs_abs_int @ A )
% 0.30/0.62 = zero_zero_int )
% 0.30/0.62 = ( A = zero_zero_int ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_0
% 0.30/0.62 thf(fact_6854_abs__0__eq,axiom,
% 0.30/0.62 ! [A: code_integer] :
% 0.30/0.62 ( ( zero_z3403309356797280102nteger
% 0.30/0.62 = ( abs_abs_Code_integer @ A ) )
% 0.30/0.62 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_0_eq
% 0.30/0.62 thf(fact_6855_abs__0__eq,axiom,
% 0.30/0.62 ! [A: real] :
% 0.30/0.62 ( ( zero_zero_real
% 0.30/0.62 = ( abs_abs_real @ A ) )
% 0.30/0.62 = ( A = zero_zero_real ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_0_eq
% 0.30/0.62 thf(fact_6856_abs__0__eq,axiom,
% 0.30/0.62 ! [A: rat] :
% 0.30/0.62 ( ( zero_zero_rat
% 0.30/0.62 = ( abs_abs_rat @ A ) )
% 0.30/0.62 = ( A = zero_zero_rat ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_0_eq
% 0.30/0.62 thf(fact_6857_abs__0__eq,axiom,
% 0.30/0.62 ! [A: int] :
% 0.30/0.62 ( ( zero_zero_int
% 0.30/0.62 = ( abs_abs_int @ A ) )
% 0.30/0.62 = ( A = zero_zero_int ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_0_eq
% 0.30/0.62 thf(fact_6858_abs__0,axiom,
% 0.30/0.62 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.30/0.62 = zero_z3403309356797280102nteger ) ).
% 0.30/0.62
% 0.30/0.62 % abs_0
% 0.30/0.62 thf(fact_6859_abs__0,axiom,
% 0.30/0.62 ( ( abs_abs_complex @ zero_zero_complex )
% 0.30/0.62 = zero_zero_complex ) ).
% 0.30/0.62
% 0.30/0.62 % abs_0
% 0.30/0.62 thf(fact_6860_abs__0,axiom,
% 0.30/0.62 ( ( abs_abs_real @ zero_zero_real )
% 0.30/0.62 = zero_zero_real ) ).
% 0.30/0.62
% 0.30/0.62 % abs_0
% 0.30/0.62 thf(fact_6861_abs__0,axiom,
% 0.30/0.62 ( ( abs_abs_rat @ zero_zero_rat )
% 0.30/0.62 = zero_zero_rat ) ).
% 0.30/0.62
% 0.30/0.62 % abs_0
% 0.30/0.62 thf(fact_6862_abs__0,axiom,
% 0.30/0.62 ( ( abs_abs_int @ zero_zero_int )
% 0.30/0.62 = zero_zero_int ) ).
% 0.30/0.62
% 0.30/0.62 % abs_0
% 0.30/0.62 thf(fact_6863_abs__numeral,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
% 0.30/0.62 = ( numera6620942414471956472nteger @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_numeral
% 0.30/0.62 thf(fact_6864_abs__numeral,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
% 0.30/0.62 = ( numeral_numeral_real @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_numeral
% 0.30/0.62 thf(fact_6865_abs__numeral,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
% 0.30/0.62 = ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_numeral
% 0.30/0.62 thf(fact_6866_abs__numeral,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
% 0.30/0.62 = ( numeral_numeral_int @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_numeral
% 0.30/0.62 thf(fact_6867_abs__mult__self__eq,axiom,
% 0.30/0.62 ! [A: code_integer] :
% 0.30/0.62 ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 0.30/0.62 = ( times_3573771949741848930nteger @ A @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_mult_self_eq
% 0.30/0.62 thf(fact_6868_abs__mult__self__eq,axiom,
% 0.30/0.62 ! [A: real] :
% 0.30/0.62 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
% 0.30/0.62 = ( times_times_real @ A @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_mult_self_eq
% 0.30/0.62 thf(fact_6869_abs__mult__self__eq,axiom,
% 0.30/0.62 ! [A: rat] :
% 0.30/0.62 ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
% 0.30/0.62 = ( times_times_rat @ A @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_mult_self_eq
% 0.30/0.62 thf(fact_6870_abs__mult__self__eq,axiom,
% 0.30/0.62 ! [A: int] :
% 0.30/0.62 ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
% 0.30/0.62 = ( times_times_int @ A @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_mult_self_eq
% 0.30/0.62 thf(fact_6871_abs__1,axiom,
% 0.30/0.62 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 0.30/0.62 = one_one_Code_integer ) ).
% 0.30/0.62
% 0.30/0.62 % abs_1
% 0.30/0.62 thf(fact_6872_abs__1,axiom,
% 0.30/0.62 ( ( abs_abs_complex @ one_one_complex )
% 0.30/0.62 = one_one_complex ) ).
% 0.30/0.62
% 0.30/0.62 % abs_1
% 0.30/0.62 thf(fact_6873_abs__1,axiom,
% 0.30/0.62 ( ( abs_abs_real @ one_one_real )
% 0.30/0.62 = one_one_real ) ).
% 0.30/0.62
% 0.30/0.62 % abs_1
% 0.30/0.62 thf(fact_6874_abs__1,axiom,
% 0.30/0.62 ( ( abs_abs_rat @ one_one_rat )
% 0.30/0.62 = one_one_rat ) ).
% 0.30/0.62
% 0.30/0.62 % abs_1
% 0.30/0.62 thf(fact_6875_abs__1,axiom,
% 0.30/0.62 ( ( abs_abs_int @ one_one_int )
% 0.30/0.62 = one_one_int ) ).
% 0.30/0.62
% 0.30/0.62 % abs_1
% 0.30/0.62 thf(fact_6876_abs__add__abs,axiom,
% 0.30/0.62 ! [A: code_integer,B: code_integer] :
% 0.30/0.62 ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
% 0.30/0.62 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_add_abs
% 0.30/0.62 thf(fact_6877_abs__add__abs,axiom,
% 0.30/0.62 ! [A: real,B: real] :
% 0.30/0.62 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 0.30/0.62 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_add_abs
% 0.30/0.62 thf(fact_6878_abs__add__abs,axiom,
% 0.30/0.62 ! [A: rat,B: rat] :
% 0.30/0.62 ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
% 0.30/0.62 = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_add_abs
% 0.30/0.62 thf(fact_6879_abs__add__abs,axiom,
% 0.30/0.62 ! [A: int,B: int] :
% 0.30/0.62 ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
% 0.30/0.62 = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_add_abs
% 0.30/0.62 thf(fact_6880_abs__divide,axiom,
% 0.30/0.62 ! [A: complex,B: complex] :
% 0.30/0.62 ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.30/0.62 = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_divide
% 0.30/0.62 thf(fact_6881_abs__divide,axiom,
% 0.30/0.62 ! [A: real,B: real] :
% 0.30/0.62 ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.62 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_divide
% 0.30/0.62 thf(fact_6882_abs__divide,axiom,
% 0.30/0.62 ! [A: rat,B: rat] :
% 0.30/0.62 ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 0.30/0.62 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_divide
% 0.30/0.62 thf(fact_6883_abs__minus__cancel,axiom,
% 0.30/0.62 ! [A: int] :
% 0.30/0.62 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 0.30/0.62 = ( abs_abs_int @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_minus_cancel
% 0.30/0.62 thf(fact_6884_abs__minus__cancel,axiom,
% 0.30/0.62 ! [A: real] :
% 0.30/0.62 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.30/0.62 = ( abs_abs_real @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_minus_cancel
% 0.30/0.62 thf(fact_6885_abs__minus__cancel,axiom,
% 0.30/0.62 ! [A: code_integer] :
% 0.30/0.62 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.62 = ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_minus_cancel
% 0.30/0.62 thf(fact_6886_abs__minus__cancel,axiom,
% 0.30/0.62 ! [A: rat] :
% 0.30/0.62 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 0.30/0.62 = ( abs_abs_rat @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_minus_cancel
% 0.30/0.62 thf(fact_6887_abs__minus,axiom,
% 0.30/0.62 ! [A: int] :
% 0.30/0.62 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 0.30/0.62 = ( abs_abs_int @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_minus
% 0.30/0.62 thf(fact_6888_abs__minus,axiom,
% 0.30/0.62 ! [A: real] :
% 0.30/0.62 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.30/0.62 = ( abs_abs_real @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_minus
% 0.30/0.62 thf(fact_6889_abs__minus,axiom,
% 0.30/0.62 ! [A: complex] :
% 0.30/0.62 ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
% 0.30/0.62 = ( abs_abs_complex @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_minus
% 0.30/0.62 thf(fact_6890_abs__minus,axiom,
% 0.30/0.62 ! [A: code_integer] :
% 0.30/0.62 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.62 = ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_minus
% 0.30/0.62 thf(fact_6891_abs__minus,axiom,
% 0.30/0.62 ! [A: rat] :
% 0.30/0.62 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 0.30/0.62 = ( abs_abs_rat @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_minus
% 0.30/0.62 thf(fact_6892_dvd__abs__iff,axiom,
% 0.30/0.62 ! [M: real,K: real] :
% 0.30/0.62 ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
% 0.30/0.62 = ( dvd_dvd_real @ M @ K ) ) ).
% 0.30/0.62
% 0.30/0.62 % dvd_abs_iff
% 0.30/0.62 thf(fact_6893_dvd__abs__iff,axiom,
% 0.30/0.62 ! [M: int,K: int] :
% 0.30/0.62 ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
% 0.30/0.62 = ( dvd_dvd_int @ M @ K ) ) ).
% 0.30/0.62
% 0.30/0.62 % dvd_abs_iff
% 0.30/0.62 thf(fact_6894_dvd__abs__iff,axiom,
% 0.30/0.62 ! [M: code_integer,K: code_integer] :
% 0.30/0.62 ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
% 0.30/0.62 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 0.30/0.62
% 0.30/0.62 % dvd_abs_iff
% 0.30/0.62 thf(fact_6895_abs__dvd__iff,axiom,
% 0.30/0.62 ! [M: real,K: real] :
% 0.30/0.62 ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
% 0.30/0.62 = ( dvd_dvd_real @ M @ K ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_dvd_iff
% 0.30/0.62 thf(fact_6896_abs__dvd__iff,axiom,
% 0.30/0.62 ! [M: int,K: int] :
% 0.30/0.62 ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
% 0.30/0.62 = ( dvd_dvd_int @ M @ K ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_dvd_iff
% 0.30/0.62 thf(fact_6897_abs__dvd__iff,axiom,
% 0.30/0.62 ! [M: code_integer,K: code_integer] :
% 0.30/0.62 ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
% 0.30/0.62 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_dvd_iff
% 0.30/0.62 thf(fact_6898_abs__bool__eq,axiom,
% 0.30/0.62 ! [P: $o] :
% 0.30/0.62 ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
% 0.30/0.62 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_bool_eq
% 0.30/0.62 thf(fact_6899_abs__bool__eq,axiom,
% 0.30/0.62 ! [P: $o] :
% 0.30/0.62 ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.30/0.62 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_bool_eq
% 0.30/0.62 thf(fact_6900_abs__bool__eq,axiom,
% 0.30/0.62 ! [P: $o] :
% 0.30/0.62 ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
% 0.30/0.62 = ( zero_n356916108424825756nteger @ P ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_bool_eq
% 0.30/0.62 thf(fact_6901_tanh__real__le__iff,axiom,
% 0.30/0.62 ! [X2: real,Y3: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ ( tanh_real @ X2 ) @ ( tanh_real @ Y3 ) )
% 0.30/0.62 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % tanh_real_le_iff
% 0.30/0.62 thf(fact_6902_abs__of__nonneg,axiom,
% 0.30/0.62 ! [A: code_integer] :
% 0.30/0.62 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.62 => ( ( abs_abs_Code_integer @ A )
% 0.30/0.62 = A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_nonneg
% 0.30/0.62 thf(fact_6903_abs__of__nonneg,axiom,
% 0.30/0.62 ! [A: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.62 => ( ( abs_abs_real @ A )
% 0.30/0.62 = A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_nonneg
% 0.30/0.62 thf(fact_6904_abs__of__nonneg,axiom,
% 0.30/0.62 ! [A: rat] :
% 0.30/0.62 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.62 => ( ( abs_abs_rat @ A )
% 0.30/0.62 = A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_nonneg
% 0.30/0.62 thf(fact_6905_abs__of__nonneg,axiom,
% 0.30/0.62 ! [A: int] :
% 0.30/0.62 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.62 => ( ( abs_abs_int @ A )
% 0.30/0.62 = A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_nonneg
% 0.30/0.62 thf(fact_6906_abs__le__self__iff,axiom,
% 0.30/0.62 ! [A: code_integer] :
% 0.30/0.62 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 0.30/0.62 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_self_iff
% 0.30/0.62 thf(fact_6907_abs__le__self__iff,axiom,
% 0.30/0.62 ! [A: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 0.30/0.62 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_self_iff
% 0.30/0.62 thf(fact_6908_abs__le__self__iff,axiom,
% 0.30/0.62 ! [A: rat] :
% 0.30/0.62 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 0.30/0.62 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_self_iff
% 0.30/0.62 thf(fact_6909_abs__le__self__iff,axiom,
% 0.30/0.62 ! [A: int] :
% 0.30/0.62 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 0.30/0.62 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_self_iff
% 0.30/0.62 thf(fact_6910_abs__le__zero__iff,axiom,
% 0.30/0.62 ! [A: code_integer] :
% 0.30/0.62 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 0.30/0.62 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_zero_iff
% 0.30/0.62 thf(fact_6911_abs__le__zero__iff,axiom,
% 0.30/0.62 ! [A: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 0.30/0.62 = ( A = zero_zero_real ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_zero_iff
% 0.30/0.62 thf(fact_6912_abs__le__zero__iff,axiom,
% 0.30/0.62 ! [A: rat] :
% 0.30/0.62 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 0.30/0.62 = ( A = zero_zero_rat ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_zero_iff
% 0.30/0.62 thf(fact_6913_abs__le__zero__iff,axiom,
% 0.30/0.62 ! [A: int] :
% 0.30/0.62 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 0.30/0.62 = ( A = zero_zero_int ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_zero_iff
% 0.30/0.62 thf(fact_6914_zero__less__abs__iff,axiom,
% 0.30/0.62 ! [A: code_integer] :
% 0.30/0.62 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 0.30/0.62 = ( A != zero_z3403309356797280102nteger ) ) ).
% 0.30/0.62
% 0.30/0.62 % zero_less_abs_iff
% 0.30/0.62 thf(fact_6915_zero__less__abs__iff,axiom,
% 0.30/0.62 ! [A: real] :
% 0.30/0.62 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 0.30/0.62 = ( A != zero_zero_real ) ) ).
% 0.30/0.62
% 0.30/0.62 % zero_less_abs_iff
% 0.30/0.62 thf(fact_6916_zero__less__abs__iff,axiom,
% 0.30/0.62 ! [A: rat] :
% 0.30/0.62 ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 0.30/0.62 = ( A != zero_zero_rat ) ) ).
% 0.30/0.62
% 0.30/0.62 % zero_less_abs_iff
% 0.30/0.62 thf(fact_6917_zero__less__abs__iff,axiom,
% 0.30/0.62 ! [A: int] :
% 0.30/0.62 ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 0.30/0.62 = ( A != zero_zero_int ) ) ).
% 0.30/0.62
% 0.30/0.62 % zero_less_abs_iff
% 0.30/0.62 thf(fact_6918_abs__neg__numeral,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.62 = ( numeral_numeral_int @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_neg_numeral
% 0.30/0.62 thf(fact_6919_abs__neg__numeral,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.62 = ( numeral_numeral_real @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_neg_numeral
% 0.30/0.62 thf(fact_6920_abs__neg__numeral,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.62 = ( numera6620942414471956472nteger @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_neg_numeral
% 0.30/0.62 thf(fact_6921_abs__neg__numeral,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.62 = ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_neg_numeral
% 0.30/0.62 thf(fact_6922_abs__neg__one,axiom,
% 0.30/0.62 ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.62 = one_one_int ) ).
% 0.30/0.62
% 0.30/0.62 % abs_neg_one
% 0.30/0.62 thf(fact_6923_abs__neg__one,axiom,
% 0.30/0.62 ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.62 = one_one_real ) ).
% 0.30/0.62
% 0.30/0.62 % abs_neg_one
% 0.30/0.62 thf(fact_6924_abs__neg__one,axiom,
% 0.30/0.62 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.62 = one_one_Code_integer ) ).
% 0.30/0.62
% 0.30/0.62 % abs_neg_one
% 0.30/0.62 thf(fact_6925_abs__neg__one,axiom,
% 0.30/0.62 ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.62 = one_one_rat ) ).
% 0.30/0.62
% 0.30/0.62 % abs_neg_one
% 0.30/0.62 thf(fact_6926_abs__power__minus,axiom,
% 0.30/0.62 ! [A: int,N: nat] :
% 0.30/0.62 ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 0.30/0.62 = ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_power_minus
% 0.30/0.62 thf(fact_6927_abs__power__minus,axiom,
% 0.30/0.62 ! [A: real,N: nat] :
% 0.30/0.62 ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 0.30/0.62 = ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_power_minus
% 0.30/0.62 thf(fact_6928_abs__power__minus,axiom,
% 0.30/0.62 ! [A: code_integer,N: nat] :
% 0.30/0.62 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 0.30/0.62 = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_power_minus
% 0.30/0.62 thf(fact_6929_abs__power__minus,axiom,
% 0.30/0.62 ! [A: rat,N: nat] :
% 0.30/0.62 ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 0.30/0.62 = ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_power_minus
% 0.30/0.62 thf(fact_6930_tanh__real__nonneg__iff,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X2 ) )
% 0.30/0.62 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 0.30/0.62
% 0.30/0.62 % tanh_real_nonneg_iff
% 0.30/0.62 thf(fact_6931_tanh__real__nonpos__iff,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ ( tanh_real @ X2 ) @ zero_zero_real )
% 0.30/0.62 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 0.30/0.62
% 0.30/0.62 % tanh_real_nonpos_iff
% 0.30/0.62 thf(fact_6932_round__numeral,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
% 0.30/0.62 = ( numeral_numeral_int @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % round_numeral
% 0.30/0.62 thf(fact_6933_round__numeral,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
% 0.30/0.62 = ( numeral_numeral_int @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % round_numeral
% 0.30/0.62 thf(fact_6934_round__1,axiom,
% 0.30/0.62 ( ( archim8280529875227126926d_real @ one_one_real )
% 0.30/0.62 = one_one_int ) ).
% 0.30/0.62
% 0.30/0.62 % round_1
% 0.30/0.62 thf(fact_6935_round__1,axiom,
% 0.30/0.62 ( ( archim7778729529865785530nd_rat @ one_one_rat )
% 0.30/0.62 = one_one_int ) ).
% 0.30/0.62
% 0.30/0.62 % round_1
% 0.30/0.62 thf(fact_6936_sum__abs,axiom,
% 0.30/0.62 ! [F: int > int,A2: set_int] :
% 0.30/0.62 ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
% 0.30/0.62 @ ( groups4538972089207619220nt_int
% 0.30/0.62 @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
% 0.30/0.62 @ A2 ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_abs
% 0.30/0.62 thf(fact_6937_sum__abs,axiom,
% 0.30/0.62 ! [F: nat > real,A2: set_nat] :
% 0.30/0.62 ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 0.30/0.62 @ ( groups6591440286371151544t_real
% 0.30/0.62 @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 0.30/0.62 @ A2 ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_abs
% 0.30/0.62 thf(fact_6938_divide__le__0__abs__iff,axiom,
% 0.30/0.62 ! [A: real,B: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
% 0.30/0.62 = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.62 | ( B = zero_zero_real ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % divide_le_0_abs_iff
% 0.30/0.62 thf(fact_6939_divide__le__0__abs__iff,axiom,
% 0.30/0.62 ! [A: rat,B: rat] :
% 0.30/0.62 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
% 0.30/0.62 = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.62 | ( B = zero_zero_rat ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % divide_le_0_abs_iff
% 0.30/0.62 thf(fact_6940_zero__le__divide__abs__iff,axiom,
% 0.30/0.62 ! [A: real,B: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
% 0.30/0.62 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.62 | ( B = zero_zero_real ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % zero_le_divide_abs_iff
% 0.30/0.62 thf(fact_6941_zero__le__divide__abs__iff,axiom,
% 0.30/0.62 ! [A: rat,B: rat] :
% 0.30/0.62 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
% 0.30/0.62 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.62 | ( B = zero_zero_rat ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % zero_le_divide_abs_iff
% 0.30/0.62 thf(fact_6942_abs__of__nonpos,axiom,
% 0.30/0.62 ! [A: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.62 => ( ( abs_abs_real @ A )
% 0.30/0.62 = ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_nonpos
% 0.30/0.62 thf(fact_6943_abs__of__nonpos,axiom,
% 0.30/0.62 ! [A: code_integer] :
% 0.30/0.62 ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
% 0.30/0.62 => ( ( abs_abs_Code_integer @ A )
% 0.30/0.62 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_nonpos
% 0.30/0.62 thf(fact_6944_abs__of__nonpos,axiom,
% 0.30/0.62 ! [A: rat] :
% 0.30/0.62 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.62 => ( ( abs_abs_rat @ A )
% 0.30/0.62 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_nonpos
% 0.30/0.62 thf(fact_6945_abs__of__nonpos,axiom,
% 0.30/0.62 ! [A: int] :
% 0.30/0.62 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.62 => ( ( abs_abs_int @ A )
% 0.30/0.62 = ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_nonpos
% 0.30/0.62 thf(fact_6946_artanh__minus__real,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.30/0.62 => ( ( artanh_real @ ( uminus_uminus_real @ X2 ) )
% 0.30/0.62 = ( uminus_uminus_real @ ( artanh_real @ X2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % artanh_minus_real
% 0.30/0.62 thf(fact_6947_sum__abs__ge__zero,axiom,
% 0.30/0.62 ! [F: int > int,A2: set_int] :
% 0.30/0.62 ( ord_less_eq_int @ zero_zero_int
% 0.30/0.62 @ ( groups4538972089207619220nt_int
% 0.30/0.62 @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
% 0.30/0.62 @ A2 ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_abs_ge_zero
% 0.30/0.62 thf(fact_6948_sum__abs__ge__zero,axiom,
% 0.30/0.62 ! [F: nat > real,A2: set_nat] :
% 0.30/0.62 ( ord_less_eq_real @ zero_zero_real
% 0.30/0.62 @ ( groups6591440286371151544t_real
% 0.30/0.62 @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 0.30/0.62 @ A2 ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_abs_ge_zero
% 0.30/0.62 thf(fact_6949_zero__less__power__abs__iff,axiom,
% 0.30/0.62 ! [A: code_integer,N: nat] :
% 0.30/0.62 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
% 0.30/0.62 = ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.62 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % zero_less_power_abs_iff
% 0.30/0.62 thf(fact_6950_zero__less__power__abs__iff,axiom,
% 0.30/0.62 ! [A: real,N: nat] :
% 0.30/0.62 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 0.30/0.62 = ( ( A != zero_zero_real )
% 0.30/0.62 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % zero_less_power_abs_iff
% 0.30/0.62 thf(fact_6951_zero__less__power__abs__iff,axiom,
% 0.30/0.62 ! [A: rat,N: nat] :
% 0.30/0.62 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
% 0.30/0.62 = ( ( A != zero_zero_rat )
% 0.30/0.62 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % zero_less_power_abs_iff
% 0.30/0.62 thf(fact_6952_zero__less__power__abs__iff,axiom,
% 0.30/0.62 ! [A: int,N: nat] :
% 0.30/0.62 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
% 0.30/0.62 = ( ( A != zero_zero_int )
% 0.30/0.62 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % zero_less_power_abs_iff
% 0.30/0.62 thf(fact_6953_abs__power2,axiom,
% 0.30/0.62 ! [A: code_integer] :
% 0.30/0.62 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.62 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_power2
% 0.30/0.62 thf(fact_6954_abs__power2,axiom,
% 0.30/0.62 ! [A: real] :
% 0.30/0.62 ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.62 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_power2
% 0.30/0.62 thf(fact_6955_abs__power2,axiom,
% 0.30/0.62 ! [A: int] :
% 0.30/0.62 ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.62 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_power2
% 0.30/0.62 thf(fact_6956_power2__abs,axiom,
% 0.30/0.62 ! [A: code_integer] :
% 0.30/0.62 ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.62 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % power2_abs
% 0.30/0.62 thf(fact_6957_power2__abs,axiom,
% 0.30/0.62 ! [A: real] :
% 0.30/0.62 ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.62 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % power2_abs
% 0.30/0.62 thf(fact_6958_power2__abs,axiom,
% 0.30/0.62 ! [A: int] :
% 0.30/0.62 ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.62 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % power2_abs
% 0.30/0.62 thf(fact_6959_sum_Ocl__ivl__Suc,axiom,
% 0.30/0.62 ! [N: nat,M: nat,G: nat > complex] :
% 0.30/0.62 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62 => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62 = zero_zero_complex ) )
% 0.30/0.62 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62 => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62 = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.cl_ivl_Suc
% 0.30/0.62 thf(fact_6960_sum_Ocl__ivl__Suc,axiom,
% 0.30/0.62 ! [N: nat,M: nat,G: nat > rat] :
% 0.30/0.62 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62 = zero_zero_rat ) )
% 0.30/0.62 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.cl_ivl_Suc
% 0.30/0.62 thf(fact_6961_sum_Ocl__ivl__Suc,axiom,
% 0.30/0.62 ! [N: nat,M: nat,G: nat > int] :
% 0.30/0.62 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62 = zero_zero_int ) )
% 0.30/0.62 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.cl_ivl_Suc
% 0.30/0.62 thf(fact_6962_sum_Ocl__ivl__Suc,axiom,
% 0.30/0.62 ! [N: nat,M: nat,G: nat > nat] :
% 0.30/0.62 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62 = zero_zero_nat ) )
% 0.30/0.62 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.cl_ivl_Suc
% 0.30/0.62 thf(fact_6963_sum_Ocl__ivl__Suc,axiom,
% 0.30/0.62 ! [N: nat,M: nat,G: nat > real] :
% 0.30/0.62 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62 = zero_zero_real ) )
% 0.30/0.62 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.cl_ivl_Suc
% 0.30/0.62 thf(fact_6964_round__neg__numeral,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.62 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % round_neg_numeral
% 0.30/0.62 thf(fact_6965_round__neg__numeral,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.62 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % round_neg_numeral
% 0.30/0.62 thf(fact_6966_sum__zero__power,axiom,
% 0.30/0.62 ! [A2: set_nat,C: nat > complex] :
% 0.30/0.62 ( ( ( ( finite_finite_nat @ A2 )
% 0.30/0.62 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62 => ( ( groups2073611262835488442omplex
% 0.30/0.62 @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 0.30/0.62 @ A2 )
% 0.30/0.62 = ( C @ zero_zero_nat ) ) )
% 0.30/0.62 & ( ~ ( ( finite_finite_nat @ A2 )
% 0.30/0.62 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62 => ( ( groups2073611262835488442omplex
% 0.30/0.62 @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 0.30/0.62 @ A2 )
% 0.30/0.62 = zero_zero_complex ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_zero_power
% 0.30/0.62 thf(fact_6967_sum__zero__power,axiom,
% 0.30/0.62 ! [A2: set_nat,C: nat > rat] :
% 0.30/0.62 ( ( ( ( finite_finite_nat @ A2 )
% 0.30/0.62 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62 => ( ( groups2906978787729119204at_rat
% 0.30/0.62 @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 0.30/0.62 @ A2 )
% 0.30/0.62 = ( C @ zero_zero_nat ) ) )
% 0.30/0.62 & ( ~ ( ( finite_finite_nat @ A2 )
% 0.30/0.62 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62 => ( ( groups2906978787729119204at_rat
% 0.30/0.62 @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 0.30/0.62 @ A2 )
% 0.30/0.62 = zero_zero_rat ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_zero_power
% 0.30/0.62 thf(fact_6968_sum__zero__power,axiom,
% 0.30/0.62 ! [A2: set_nat,C: nat > real] :
% 0.30/0.62 ( ( ( ( finite_finite_nat @ A2 )
% 0.30/0.62 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62 => ( ( groups6591440286371151544t_real
% 0.30/0.62 @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 0.30/0.62 @ A2 )
% 0.30/0.62 = ( C @ zero_zero_nat ) ) )
% 0.30/0.62 & ( ~ ( ( finite_finite_nat @ A2 )
% 0.30/0.62 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62 => ( ( groups6591440286371151544t_real
% 0.30/0.62 @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 0.30/0.62 @ A2 )
% 0.30/0.62 = zero_zero_real ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_zero_power
% 0.30/0.62 thf(fact_6969_power__even__abs__numeral,axiom,
% 0.30/0.62 ! [W: num,A: code_integer] :
% 0.30/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.62 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.62 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % power_even_abs_numeral
% 0.30/0.62 thf(fact_6970_power__even__abs__numeral,axiom,
% 0.30/0.62 ! [W: num,A: real] :
% 0.30/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.62 => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.62 = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % power_even_abs_numeral
% 0.30/0.62 thf(fact_6971_power__even__abs__numeral,axiom,
% 0.30/0.62 ! [W: num,A: int] :
% 0.30/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.62 => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.62 = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % power_even_abs_numeral
% 0.30/0.62 thf(fact_6972_sum__zero__power_H,axiom,
% 0.30/0.62 ! [A2: set_nat,C: nat > complex,D: nat > complex] :
% 0.30/0.62 ( ( ( ( finite_finite_nat @ A2 )
% 0.30/0.62 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62 => ( ( groups2073611262835488442omplex
% 0.30/0.62 @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 0.30/0.62 @ A2 )
% 0.30/0.62 = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.30/0.62 & ( ~ ( ( finite_finite_nat @ A2 )
% 0.30/0.62 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62 => ( ( groups2073611262835488442omplex
% 0.30/0.62 @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 0.30/0.62 @ A2 )
% 0.30/0.62 = zero_zero_complex ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_zero_power'
% 0.30/0.62 thf(fact_6973_sum__zero__power_H,axiom,
% 0.30/0.62 ! [A2: set_nat,C: nat > rat,D: nat > rat] :
% 0.30/0.62 ( ( ( ( finite_finite_nat @ A2 )
% 0.30/0.62 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62 => ( ( groups2906978787729119204at_rat
% 0.30/0.62 @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 0.30/0.62 @ A2 )
% 0.30/0.62 = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.30/0.62 & ( ~ ( ( finite_finite_nat @ A2 )
% 0.30/0.62 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62 => ( ( groups2906978787729119204at_rat
% 0.30/0.62 @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 0.30/0.62 @ A2 )
% 0.30/0.62 = zero_zero_rat ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_zero_power'
% 0.30/0.62 thf(fact_6974_sum__zero__power_H,axiom,
% 0.30/0.62 ! [A2: set_nat,C: nat > real,D: nat > real] :
% 0.30/0.62 ( ( ( ( finite_finite_nat @ A2 )
% 0.30/0.62 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62 => ( ( groups6591440286371151544t_real
% 0.30/0.62 @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 0.30/0.62 @ A2 )
% 0.30/0.62 = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.30/0.62 & ( ~ ( ( finite_finite_nat @ A2 )
% 0.30/0.62 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62 => ( ( groups6591440286371151544t_real
% 0.30/0.62 @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 0.30/0.62 @ A2 )
% 0.30/0.62 = zero_zero_real ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_zero_power'
% 0.30/0.62 thf(fact_6975_abs__ge__self,axiom,
% 0.30/0.62 ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_ge_self
% 0.30/0.62 thf(fact_6976_abs__ge__self,axiom,
% 0.30/0.62 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_ge_self
% 0.30/0.62 thf(fact_6977_abs__ge__self,axiom,
% 0.30/0.62 ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_ge_self
% 0.30/0.62 thf(fact_6978_abs__ge__self,axiom,
% 0.30/0.62 ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_ge_self
% 0.30/0.62 thf(fact_6979_abs__le__D1,axiom,
% 0.30/0.62 ! [A: real,B: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.30/0.62 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_D1
% 0.30/0.62 thf(fact_6980_abs__le__D1,axiom,
% 0.30/0.62 ! [A: code_integer,B: code_integer] :
% 0.30/0.62 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.30/0.62 => ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_D1
% 0.30/0.62 thf(fact_6981_abs__le__D1,axiom,
% 0.30/0.62 ! [A: rat,B: rat] :
% 0.30/0.62 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.30/0.62 => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_D1
% 0.30/0.62 thf(fact_6982_abs__le__D1,axiom,
% 0.30/0.62 ! [A: int,B: int] :
% 0.30/0.62 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.30/0.62 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_D1
% 0.30/0.62 thf(fact_6983_abs__eq__0__iff,axiom,
% 0.30/0.62 ! [A: code_integer] :
% 0.30/0.62 ( ( ( abs_abs_Code_integer @ A )
% 0.30/0.62 = zero_z3403309356797280102nteger )
% 0.30/0.62 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_0_iff
% 0.30/0.62 thf(fact_6984_abs__eq__0__iff,axiom,
% 0.30/0.62 ! [A: complex] :
% 0.30/0.62 ( ( ( abs_abs_complex @ A )
% 0.30/0.62 = zero_zero_complex )
% 0.30/0.62 = ( A = zero_zero_complex ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_0_iff
% 0.30/0.62 thf(fact_6985_abs__eq__0__iff,axiom,
% 0.30/0.62 ! [A: real] :
% 0.30/0.62 ( ( ( abs_abs_real @ A )
% 0.30/0.62 = zero_zero_real )
% 0.30/0.62 = ( A = zero_zero_real ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_0_iff
% 0.30/0.62 thf(fact_6986_abs__eq__0__iff,axiom,
% 0.30/0.62 ! [A: rat] :
% 0.30/0.62 ( ( ( abs_abs_rat @ A )
% 0.30/0.62 = zero_zero_rat )
% 0.30/0.62 = ( A = zero_zero_rat ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_0_iff
% 0.30/0.62 thf(fact_6987_abs__eq__0__iff,axiom,
% 0.30/0.62 ! [A: int] :
% 0.30/0.62 ( ( ( abs_abs_int @ A )
% 0.30/0.62 = zero_zero_int )
% 0.30/0.62 = ( A = zero_zero_int ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_0_iff
% 0.30/0.62 thf(fact_6988_abs__mult,axiom,
% 0.30/0.62 ! [A: code_integer,B: code_integer] :
% 0.30/0.62 ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.30/0.62 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_mult
% 0.30/0.62 thf(fact_6989_abs__mult,axiom,
% 0.30/0.62 ! [A: real,B: real] :
% 0.30/0.62 ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 0.30/0.62 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_mult
% 0.30/0.62 thf(fact_6990_abs__mult,axiom,
% 0.30/0.62 ! [A: rat,B: rat] :
% 0.30/0.62 ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 0.30/0.62 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_mult
% 0.30/0.62 thf(fact_6991_abs__mult,axiom,
% 0.30/0.62 ! [A: int,B: int] :
% 0.30/0.62 ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 0.30/0.62 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_mult
% 0.30/0.62 thf(fact_6992_abs__one,axiom,
% 0.30/0.62 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 0.30/0.62 = one_one_Code_integer ) ).
% 0.30/0.62
% 0.30/0.62 % abs_one
% 0.30/0.62 thf(fact_6993_abs__one,axiom,
% 0.30/0.62 ( ( abs_abs_real @ one_one_real )
% 0.30/0.62 = one_one_real ) ).
% 0.30/0.62
% 0.30/0.62 % abs_one
% 0.30/0.62 thf(fact_6994_abs__one,axiom,
% 0.30/0.62 ( ( abs_abs_rat @ one_one_rat )
% 0.30/0.62 = one_one_rat ) ).
% 0.30/0.62
% 0.30/0.62 % abs_one
% 0.30/0.62 thf(fact_6995_abs__one,axiom,
% 0.30/0.62 ( ( abs_abs_int @ one_one_int )
% 0.30/0.62 = one_one_int ) ).
% 0.30/0.62
% 0.30/0.62 % abs_one
% 0.30/0.62 thf(fact_6996_abs__minus__commute,axiom,
% 0.30/0.62 ! [A: code_integer,B: code_integer] :
% 0.30/0.62 ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.30/0.62 = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_minus_commute
% 0.30/0.62 thf(fact_6997_abs__minus__commute,axiom,
% 0.30/0.62 ! [A: real,B: real] :
% 0.30/0.62 ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 0.30/0.62 = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_minus_commute
% 0.30/0.62 thf(fact_6998_abs__minus__commute,axiom,
% 0.30/0.62 ! [A: rat,B: rat] :
% 0.30/0.62 ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
% 0.30/0.62 = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_minus_commute
% 0.30/0.62 thf(fact_6999_abs__minus__commute,axiom,
% 0.30/0.62 ! [A: int,B: int] :
% 0.30/0.62 ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
% 0.30/0.62 = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_minus_commute
% 0.30/0.62 thf(fact_7000_abs__eq__iff,axiom,
% 0.30/0.62 ! [X2: int,Y3: int] :
% 0.30/0.62 ( ( ( abs_abs_int @ X2 )
% 0.30/0.62 = ( abs_abs_int @ Y3 ) )
% 0.30/0.62 = ( ( X2 = Y3 )
% 0.30/0.62 | ( X2
% 0.30/0.62 = ( uminus_uminus_int @ Y3 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_iff
% 0.30/0.62 thf(fact_7001_abs__eq__iff,axiom,
% 0.30/0.62 ! [X2: real,Y3: real] :
% 0.30/0.62 ( ( ( abs_abs_real @ X2 )
% 0.30/0.62 = ( abs_abs_real @ Y3 ) )
% 0.30/0.62 = ( ( X2 = Y3 )
% 0.30/0.62 | ( X2
% 0.30/0.62 = ( uminus_uminus_real @ Y3 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_iff
% 0.30/0.62 thf(fact_7002_abs__eq__iff,axiom,
% 0.30/0.62 ! [X2: code_integer,Y3: code_integer] :
% 0.30/0.62 ( ( ( abs_abs_Code_integer @ X2 )
% 0.30/0.62 = ( abs_abs_Code_integer @ Y3 ) )
% 0.30/0.62 = ( ( X2 = Y3 )
% 0.30/0.62 | ( X2
% 0.30/0.62 = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_iff
% 0.30/0.62 thf(fact_7003_abs__eq__iff,axiom,
% 0.30/0.62 ! [X2: rat,Y3: rat] :
% 0.30/0.62 ( ( ( abs_abs_rat @ X2 )
% 0.30/0.62 = ( abs_abs_rat @ Y3 ) )
% 0.30/0.62 = ( ( X2 = Y3 )
% 0.30/0.62 | ( X2
% 0.30/0.62 = ( uminus_uminus_rat @ Y3 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_iff
% 0.30/0.62 thf(fact_7004_power__abs,axiom,
% 0.30/0.62 ! [A: code_integer,N: nat] :
% 0.30/0.62 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
% 0.30/0.62 = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % power_abs
% 0.30/0.62 thf(fact_7005_power__abs,axiom,
% 0.30/0.62 ! [A: real,N: nat] :
% 0.30/0.62 ( ( abs_abs_real @ ( power_power_real @ A @ N ) )
% 0.30/0.62 = ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % power_abs
% 0.30/0.62 thf(fact_7006_power__abs,axiom,
% 0.30/0.62 ! [A: int,N: nat] :
% 0.30/0.62 ( ( abs_abs_int @ ( power_power_int @ A @ N ) )
% 0.30/0.62 = ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % power_abs
% 0.30/0.62 thf(fact_7007_dvd__if__abs__eq,axiom,
% 0.30/0.62 ! [L: real,K: real] :
% 0.30/0.62 ( ( ( abs_abs_real @ L )
% 0.30/0.62 = ( abs_abs_real @ K ) )
% 0.30/0.62 => ( dvd_dvd_real @ L @ K ) ) ).
% 0.30/0.62
% 0.30/0.62 % dvd_if_abs_eq
% 0.30/0.62 thf(fact_7008_dvd__if__abs__eq,axiom,
% 0.30/0.62 ! [L: int,K: int] :
% 0.30/0.62 ( ( ( abs_abs_int @ L )
% 0.30/0.62 = ( abs_abs_int @ K ) )
% 0.30/0.62 => ( dvd_dvd_int @ L @ K ) ) ).
% 0.30/0.62
% 0.30/0.62 % dvd_if_abs_eq
% 0.30/0.62 thf(fact_7009_dvd__if__abs__eq,axiom,
% 0.30/0.62 ! [L: code_integer,K: code_integer] :
% 0.30/0.62 ( ( ( abs_abs_Code_integer @ L )
% 0.30/0.62 = ( abs_abs_Code_integer @ K ) )
% 0.30/0.62 => ( dvd_dvd_Code_integer @ L @ K ) ) ).
% 0.30/0.62
% 0.30/0.62 % dvd_if_abs_eq
% 0.30/0.62 thf(fact_7010_round__diff__minimal,axiom,
% 0.30/0.62 ! [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 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % round_diff_minimal
% 0.30/0.62 thf(fact_7011_round__diff__minimal,axiom,
% 0.30/0.62 ! [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 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % round_diff_minimal
% 0.30/0.62 thf(fact_7012_abs__ge__zero,axiom,
% 0.30/0.62 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_ge_zero
% 0.30/0.62 thf(fact_7013_abs__ge__zero,axiom,
% 0.30/0.62 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_ge_zero
% 0.30/0.62 thf(fact_7014_abs__ge__zero,axiom,
% 0.30/0.62 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_ge_zero
% 0.30/0.62 thf(fact_7015_abs__ge__zero,axiom,
% 0.30/0.62 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_ge_zero
% 0.30/0.62 thf(fact_7016_abs__of__pos,axiom,
% 0.30/0.62 ! [A: code_integer] :
% 0.30/0.62 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.62 => ( ( abs_abs_Code_integer @ A )
% 0.30/0.62 = A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_pos
% 0.30/0.62 thf(fact_7017_abs__of__pos,axiom,
% 0.30/0.62 ! [A: real] :
% 0.30/0.62 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.62 => ( ( abs_abs_real @ A )
% 0.30/0.62 = A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_pos
% 0.30/0.62 thf(fact_7018_abs__of__pos,axiom,
% 0.30/0.62 ! [A: rat] :
% 0.30/0.62 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.62 => ( ( abs_abs_rat @ A )
% 0.30/0.62 = A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_pos
% 0.30/0.62 thf(fact_7019_abs__of__pos,axiom,
% 0.30/0.62 ! [A: int] :
% 0.30/0.62 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.62 => ( ( abs_abs_int @ A )
% 0.30/0.62 = A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_pos
% 0.30/0.62 thf(fact_7020_abs__not__less__zero,axiom,
% 0.30/0.62 ! [A: code_integer] :
% 0.30/0.62 ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 0.30/0.62
% 0.30/0.62 % abs_not_less_zero
% 0.30/0.62 thf(fact_7021_abs__not__less__zero,axiom,
% 0.30/0.62 ! [A: real] :
% 0.30/0.62 ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 0.30/0.62
% 0.30/0.62 % abs_not_less_zero
% 0.30/0.62 thf(fact_7022_abs__not__less__zero,axiom,
% 0.30/0.62 ! [A: rat] :
% 0.30/0.62 ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 0.30/0.62
% 0.30/0.62 % abs_not_less_zero
% 0.30/0.62 thf(fact_7023_abs__not__less__zero,axiom,
% 0.30/0.62 ! [A: int] :
% 0.30/0.62 ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 0.30/0.62
% 0.30/0.62 % abs_not_less_zero
% 0.30/0.62 thf(fact_7024_abs__triangle__ineq,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq
% 0.30/0.62 thf(fact_7025_abs__triangle__ineq,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq
% 0.30/0.62 thf(fact_7026_abs__triangle__ineq,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq
% 0.30/0.62 thf(fact_7027_abs__triangle__ineq,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq
% 0.30/0.62 thf(fact_7028_abs__mult__less,axiom,
% 0.30/0.62 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 0.30/0.62 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 0.30/0.62 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
% 0.30/0.62 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_mult_less
% 0.30/0.62 thf(fact_7029_abs__mult__less,axiom,
% 0.30/0.62 ! [A: real,C: real,B: real,D: real] :
% 0.30/0.62 ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 0.30/0.62 => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
% 0.30/0.62 => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_mult_less
% 0.30/0.62 thf(fact_7030_abs__mult__less,axiom,
% 0.30/0.62 ! [A: rat,C: rat,B: rat,D: rat] :
% 0.30/0.62 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 0.30/0.62 => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
% 0.30/0.62 => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_mult_less
% 0.30/0.62 thf(fact_7031_abs__mult__less,axiom,
% 0.30/0.62 ! [A: int,C: int,B: int,D: int] :
% 0.30/0.62 ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 0.30/0.62 => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
% 0.30/0.62 => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_mult_less
% 0.30/0.62 thf(fact_7032_abs__triangle__ineq2,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq2
% 0.30/0.62 thf(fact_7033_abs__triangle__ineq2,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq2
% 0.30/0.62 thf(fact_7034_abs__triangle__ineq2,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq2
% 0.30/0.62 thf(fact_7035_abs__triangle__ineq2,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq2
% 0.30/0.62 thf(fact_7036_abs__triangle__ineq3,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq3
% 0.30/0.62 thf(fact_7037_abs__triangle__ineq3,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq3
% 0.30/0.62 thf(fact_7038_abs__triangle__ineq3,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq3
% 0.30/0.62 thf(fact_7039_abs__triangle__ineq3,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq3
% 0.30/0.62 thf(fact_7040_abs__triangle__ineq2__sym,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq2_sym
% 0.30/0.62 thf(fact_7041_abs__triangle__ineq2__sym,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq2_sym
% 0.30/0.62 thf(fact_7042_abs__triangle__ineq2__sym,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq2_sym
% 0.30/0.62 thf(fact_7043_abs__triangle__ineq2__sym,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq2_sym
% 0.30/0.62 thf(fact_7044_nonzero__abs__divide,axiom,
% 0.30/0.62 ! [B: real,A: real] :
% 0.30/0.62 ( ( B != zero_zero_real )
% 0.30/0.62 => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.62 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % nonzero_abs_divide
% 0.30/0.62 thf(fact_7045_nonzero__abs__divide,axiom,
% 0.30/0.62 ! [B: rat,A: rat] :
% 0.30/0.62 ( ( B != zero_zero_rat )
% 0.30/0.62 => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 0.30/0.62 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % nonzero_abs_divide
% 0.30/0.62 thf(fact_7046_abs__ge__minus__self,axiom,
% 0.30/0.62 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_ge_minus_self
% 0.30/0.62 thf(fact_7047_abs__ge__minus__self,axiom,
% 0.30/0.62 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_ge_minus_self
% 0.30/0.62 thf(fact_7048_abs__ge__minus__self,axiom,
% 0.30/0.62 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_ge_minus_self
% 0.30/0.62 thf(fact_7049_abs__ge__minus__self,axiom,
% 0.30/0.62 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_ge_minus_self
% 0.30/0.62 thf(fact_7050_abs__le__iff,axiom,
% 0.30/0.62 ! [A: real,B: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.30/0.62 = ( ( ord_less_eq_real @ A @ B )
% 0.30/0.62 & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_iff
% 0.30/0.62 thf(fact_7051_abs__le__iff,axiom,
% 0.30/0.62 ! [A: code_integer,B: code_integer] :
% 0.30/0.62 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.30/0.62 = ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.30/0.62 & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_iff
% 0.30/0.62 thf(fact_7052_abs__le__iff,axiom,
% 0.30/0.62 ! [A: rat,B: rat] :
% 0.30/0.62 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.30/0.62 = ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.62 & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_iff
% 0.30/0.62 thf(fact_7053_abs__le__iff,axiom,
% 0.30/0.62 ! [A: int,B: int] :
% 0.30/0.62 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.30/0.62 = ( ( ord_less_eq_int @ A @ B )
% 0.30/0.62 & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_iff
% 0.30/0.62 thf(fact_7054_abs__le__D2,axiom,
% 0.30/0.62 ! [A: real,B: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.30/0.62 => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_D2
% 0.30/0.62 thf(fact_7055_abs__le__D2,axiom,
% 0.30/0.62 ! [A: code_integer,B: code_integer] :
% 0.30/0.62 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.30/0.62 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_D2
% 0.30/0.62 thf(fact_7056_abs__le__D2,axiom,
% 0.30/0.62 ! [A: rat,B: rat] :
% 0.30/0.62 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.30/0.62 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_D2
% 0.30/0.62 thf(fact_7057_abs__le__D2,axiom,
% 0.30/0.62 ! [A: int,B: int] :
% 0.30/0.62 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.30/0.62 => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_D2
% 0.30/0.62 thf(fact_7058_abs__leI,axiom,
% 0.30/0.62 ! [A: real,B: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.62 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.30/0.62 => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_leI
% 0.30/0.62 thf(fact_7059_abs__leI,axiom,
% 0.30/0.62 ! [A: code_integer,B: code_integer] :
% 0.30/0.62 ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.30/0.62 => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.30/0.62 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_leI
% 0.30/0.62 thf(fact_7060_abs__leI,axiom,
% 0.30/0.62 ! [A: rat,B: rat] :
% 0.30/0.62 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.62 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.30/0.62 => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_leI
% 0.30/0.62 thf(fact_7061_abs__leI,axiom,
% 0.30/0.62 ! [A: int,B: int] :
% 0.30/0.62 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.62 => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.62 => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_leI
% 0.30/0.62 thf(fact_7062_abs__less__iff,axiom,
% 0.30/0.62 ! [A: int,B: int] :
% 0.30/0.62 ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
% 0.30/0.62 = ( ( ord_less_int @ A @ B )
% 0.30/0.62 & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_less_iff
% 0.30/0.62 thf(fact_7063_abs__less__iff,axiom,
% 0.30/0.62 ! [A: real,B: real] :
% 0.30/0.62 ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
% 0.30/0.62 = ( ( ord_less_real @ A @ B )
% 0.30/0.62 & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_less_iff
% 0.30/0.62 thf(fact_7064_abs__less__iff,axiom,
% 0.30/0.62 ! [A: code_integer,B: code_integer] :
% 0.30/0.62 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.30/0.62 = ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.30/0.62 & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_less_iff
% 0.30/0.62 thf(fact_7065_abs__less__iff,axiom,
% 0.30/0.62 ! [A: rat,B: rat] :
% 0.30/0.62 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
% 0.30/0.62 = ( ( ord_less_rat @ A @ B )
% 0.30/0.62 & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_less_iff
% 0.30/0.62 thf(fact_7066_sum__cong__Suc,axiom,
% 0.30/0.62 ! [A2: set_nat,F: nat > nat,G: nat > nat] :
% 0.30/0.62 ( ~ ( member_nat @ zero_zero_nat @ A2 )
% 0.30/0.62 => ( ! [X5: nat] :
% 0.30/0.62 ( ( member_nat @ ( suc @ X5 ) @ A2 )
% 0.30/0.62 => ( ( F @ ( suc @ X5 ) )
% 0.30/0.62 = ( G @ ( suc @ X5 ) ) ) )
% 0.30/0.62 => ( ( groups3542108847815614940at_nat @ F @ A2 )
% 0.30/0.62 = ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_cong_Suc
% 0.30/0.62 thf(fact_7067_sum__cong__Suc,axiom,
% 0.30/0.62 ! [A2: set_nat,F: nat > real,G: nat > real] :
% 0.30/0.62 ( ~ ( member_nat @ zero_zero_nat @ A2 )
% 0.30/0.62 => ( ! [X5: nat] :
% 0.30/0.62 ( ( member_nat @ ( suc @ X5 ) @ A2 )
% 0.30/0.62 => ( ( F @ ( suc @ X5 ) )
% 0.30/0.62 = ( G @ ( suc @ X5 ) ) ) )
% 0.30/0.62 => ( ( groups6591440286371151544t_real @ F @ A2 )
% 0.30/0.62 = ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_cong_Suc
% 0.30/0.62 thf(fact_7068_sum__subtractf__nat,axiom,
% 0.30/0.62 ! [A2: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,F: product_prod_nat_nat > nat] :
% 0.30/0.62 ( ! [X5: product_prod_nat_nat] :
% 0.30/0.62 ( ( member8440522571783428010at_nat @ X5 @ A2 )
% 0.30/0.62 => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
% 0.30/0.62 => ( ( groups977919841031483927at_nat
% 0.30/0.62 @ ^ [X: product_prod_nat_nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.62 @ A2 )
% 0.30/0.62 = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( groups977919841031483927at_nat @ G @ A2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_subtractf_nat
% 0.30/0.62 thf(fact_7069_sum__subtractf__nat,axiom,
% 0.30/0.62 ! [A2: set_complex,G: complex > nat,F: complex > nat] :
% 0.30/0.62 ( ! [X5: complex] :
% 0.30/0.62 ( ( member_complex @ X5 @ A2 )
% 0.30/0.62 => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
% 0.30/0.62 => ( ( groups5693394587270226106ex_nat
% 0.30/0.62 @ ^ [X: complex] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.62 @ A2 )
% 0.30/0.62 = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_subtractf_nat
% 0.30/0.62 thf(fact_7070_sum__subtractf__nat,axiom,
% 0.30/0.62 ! [A2: set_real,G: real > nat,F: real > nat] :
% 0.30/0.62 ( ! [X5: real] :
% 0.30/0.62 ( ( member_real @ X5 @ A2 )
% 0.30/0.62 => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
% 0.30/0.62 => ( ( groups1935376822645274424al_nat
% 0.30/0.62 @ ^ [X: real] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.62 @ A2 )
% 0.30/0.62 = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_subtractf_nat
% 0.30/0.62 thf(fact_7071_sum__subtractf__nat,axiom,
% 0.30/0.62 ! [A2: set_int,G: int > nat,F: int > nat] :
% 0.30/0.62 ( ! [X5: int] :
% 0.30/0.62 ( ( member_int @ X5 @ A2 )
% 0.30/0.62 => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
% 0.30/0.62 => ( ( groups4541462559716669496nt_nat
% 0.30/0.62 @ ^ [X: int] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.62 @ A2 )
% 0.30/0.62 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_subtractf_nat
% 0.30/0.62 thf(fact_7072_sum__subtractf__nat,axiom,
% 0.30/0.62 ! [A2: set_nat,G: nat > nat,F: nat > nat] :
% 0.30/0.62 ( ! [X5: nat] :
% 0.30/0.62 ( ( member_nat @ X5 @ A2 )
% 0.30/0.62 => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
% 0.30/0.62 => ( ( groups3542108847815614940at_nat
% 0.30/0.62 @ ^ [X: nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.62 @ A2 )
% 0.30/0.62 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_subtractf_nat
% 0.30/0.62 thf(fact_7073_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 0.30/0.62 ! [G: nat > nat,M: nat,N: nat] :
% 0.30/0.62 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.62 = ( groups3542108847815614940at_nat
% 0.30/0.62 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.shift_bounds_cl_Suc_ivl
% 0.30/0.62 thf(fact_7074_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 0.30/0.62 ! [G: nat > real,M: nat,N: nat] :
% 0.30/0.62 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.62 = ( groups6591440286371151544t_real
% 0.30/0.62 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.shift_bounds_cl_Suc_ivl
% 0.30/0.62 thf(fact_7075_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 0.30/0.62 ! [G: nat > nat,M: nat,K: nat,N: nat] :
% 0.30/0.62 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 0.30/0.62 = ( groups3542108847815614940at_nat
% 0.30/0.62 @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.shift_bounds_cl_nat_ivl
% 0.30/0.62 thf(fact_7076_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 0.30/0.62 ! [G: nat > real,M: nat,K: nat,N: nat] :
% 0.30/0.62 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 0.30/0.62 = ( groups6591440286371151544t_real
% 0.30/0.62 @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.shift_bounds_cl_nat_ivl
% 0.30/0.62 thf(fact_7077_tanh__real__lt__1,axiom,
% 0.30/0.62 ! [X2: real] : ( ord_less_real @ ( tanh_real @ X2 ) @ one_one_real ) ).
% 0.30/0.62
% 0.30/0.62 % tanh_real_lt_1
% 0.30/0.62 thf(fact_7078_dense__eq0__I,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ( ! [E2: real] :
% 0.30/0.62 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.30/0.62 => ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ E2 ) )
% 0.30/0.62 => ( X2 = zero_zero_real ) ) ).
% 0.30/0.62
% 0.30/0.62 % dense_eq0_I
% 0.30/0.62 thf(fact_7079_dense__eq0__I,axiom,
% 0.30/0.62 ! [X2: rat] :
% 0.30/0.62 ( ! [E2: rat] :
% 0.30/0.62 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.30/0.62 => ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ E2 ) )
% 0.30/0.62 => ( X2 = zero_zero_rat ) ) ).
% 0.30/0.62
% 0.30/0.62 % dense_eq0_I
% 0.30/0.62 thf(fact_7080_abs__mult__pos,axiom,
% 0.30/0.62 ! [X2: code_integer,Y3: code_integer] :
% 0.30/0.62 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X2 )
% 0.30/0.62 => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y3 ) @ X2 )
% 0.30/0.62 = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y3 @ X2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_mult_pos
% 0.30/0.62 thf(fact_7081_abs__mult__pos,axiom,
% 0.30/0.62 ! [X2: real,Y3: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.62 => ( ( times_times_real @ ( abs_abs_real @ Y3 ) @ X2 )
% 0.30/0.62 = ( abs_abs_real @ ( times_times_real @ Y3 @ X2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_mult_pos
% 0.30/0.62 thf(fact_7082_abs__mult__pos,axiom,
% 0.30/0.62 ! [X2: rat,Y3: rat] :
% 0.30/0.62 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.30/0.62 => ( ( times_times_rat @ ( abs_abs_rat @ Y3 ) @ X2 )
% 0.30/0.62 = ( abs_abs_rat @ ( times_times_rat @ Y3 @ X2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_mult_pos
% 0.30/0.62 thf(fact_7083_abs__mult__pos,axiom,
% 0.30/0.62 ! [X2: int,Y3: int] :
% 0.30/0.62 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.62 => ( ( times_times_int @ ( abs_abs_int @ Y3 ) @ X2 )
% 0.30/0.62 = ( abs_abs_int @ ( times_times_int @ Y3 @ X2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_mult_pos
% 0.30/0.62 thf(fact_7084_abs__eq__mult,axiom,
% 0.30/0.62 ! [A: code_integer,B: code_integer] :
% 0.30/0.62 ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.62 | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 0.30/0.62 & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.62 | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
% 0.30/0.62 => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.30/0.62 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_mult
% 0.30/0.62 thf(fact_7085_abs__eq__mult,axiom,
% 0.30/0.62 ! [A: real,B: real] :
% 0.30/0.62 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.62 | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.30/0.62 & ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.62 | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.30/0.62 => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 0.30/0.62 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_mult
% 0.30/0.62 thf(fact_7086_abs__eq__mult,axiom,
% 0.30/0.62 ! [A: rat,B: rat] :
% 0.30/0.62 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.62 | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.30/0.62 & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.62 | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 0.30/0.62 => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 0.30/0.62 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_mult
% 0.30/0.62 thf(fact_7087_abs__eq__mult,axiom,
% 0.30/0.62 ! [A: int,B: int] :
% 0.30/0.62 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.62 | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.30/0.62 & ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.62 | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 0.30/0.62 => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 0.30/0.62 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_mult
% 0.30/0.62 thf(fact_7088_abs__minus__le__zero,axiom,
% 0.30/0.62 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 0.30/0.62
% 0.30/0.62 % abs_minus_le_zero
% 0.30/0.62 thf(fact_7089_abs__minus__le__zero,axiom,
% 0.30/0.62 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 0.30/0.62
% 0.30/0.62 % abs_minus_le_zero
% 0.30/0.62 thf(fact_7090_abs__minus__le__zero,axiom,
% 0.30/0.62 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 0.30/0.62
% 0.30/0.62 % abs_minus_le_zero
% 0.30/0.62 thf(fact_7091_abs__minus__le__zero,axiom,
% 0.30/0.62 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 0.30/0.62
% 0.30/0.62 % abs_minus_le_zero
% 0.30/0.62 thf(fact_7092_eq__abs__iff_H,axiom,
% 0.30/0.62 ! [A: real,B: real] :
% 0.30/0.62 ( ( A
% 0.30/0.62 = ( abs_abs_real @ B ) )
% 0.30/0.62 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.62 & ( ( B = A )
% 0.30/0.62 | ( B
% 0.30/0.62 = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % eq_abs_iff'
% 0.30/0.62 thf(fact_7093_eq__abs__iff_H,axiom,
% 0.30/0.62 ! [A: code_integer,B: code_integer] :
% 0.30/0.62 ( ( A
% 0.30/0.62 = ( abs_abs_Code_integer @ B ) )
% 0.30/0.62 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.62 & ( ( B = A )
% 0.30/0.62 | ( B
% 0.30/0.62 = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % eq_abs_iff'
% 0.30/0.62 thf(fact_7094_eq__abs__iff_H,axiom,
% 0.30/0.62 ! [A: rat,B: rat] :
% 0.30/0.62 ( ( A
% 0.30/0.62 = ( abs_abs_rat @ B ) )
% 0.30/0.62 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.62 & ( ( B = A )
% 0.30/0.62 | ( B
% 0.30/0.62 = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % eq_abs_iff'
% 0.30/0.62 thf(fact_7095_eq__abs__iff_H,axiom,
% 0.30/0.62 ! [A: int,B: int] :
% 0.30/0.62 ( ( A
% 0.30/0.62 = ( abs_abs_int @ B ) )
% 0.30/0.62 = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.62 & ( ( B = A )
% 0.30/0.62 | ( B
% 0.30/0.62 = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % eq_abs_iff'
% 0.30/0.62 thf(fact_7096_abs__eq__iff_H,axiom,
% 0.30/0.62 ! [A: real,B: real] :
% 0.30/0.62 ( ( ( abs_abs_real @ A )
% 0.30/0.62 = B )
% 0.30/0.62 = ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.62 & ( ( A = B )
% 0.30/0.62 | ( A
% 0.30/0.62 = ( uminus_uminus_real @ B ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_iff'
% 0.30/0.62 thf(fact_7097_abs__eq__iff_H,axiom,
% 0.30/0.62 ! [A: code_integer,B: code_integer] :
% 0.30/0.62 ( ( ( abs_abs_Code_integer @ A )
% 0.30/0.62 = B )
% 0.30/0.62 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.62 & ( ( A = B )
% 0.30/0.62 | ( A
% 0.30/0.62 = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_iff'
% 0.30/0.62 thf(fact_7098_abs__eq__iff_H,axiom,
% 0.30/0.62 ! [A: rat,B: rat] :
% 0.30/0.62 ( ( ( abs_abs_rat @ A )
% 0.30/0.62 = B )
% 0.30/0.62 = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.62 & ( ( A = B )
% 0.30/0.62 | ( A
% 0.30/0.62 = ( uminus_uminus_rat @ B ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_iff'
% 0.30/0.62 thf(fact_7099_abs__eq__iff_H,axiom,
% 0.30/0.62 ! [A: int,B: int] :
% 0.30/0.62 ( ( ( abs_abs_int @ A )
% 0.30/0.62 = B )
% 0.30/0.62 = ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.62 & ( ( A = B )
% 0.30/0.62 | ( A
% 0.30/0.62 = ( uminus_uminus_int @ B ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_eq_iff'
% 0.30/0.62 thf(fact_7100_abs__div__pos,axiom,
% 0.30/0.62 ! [Y3: real,X2: real] :
% 0.30/0.62 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.62 => ( ( divide_divide_real @ ( abs_abs_real @ X2 ) @ Y3 )
% 0.30/0.62 = ( abs_abs_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_div_pos
% 0.30/0.62 thf(fact_7101_abs__div__pos,axiom,
% 0.30/0.62 ! [Y3: rat,X2: rat] :
% 0.30/0.62 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.30/0.62 => ( ( divide_divide_rat @ ( abs_abs_rat @ X2 ) @ Y3 )
% 0.30/0.62 = ( abs_abs_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_div_pos
% 0.30/0.62 thf(fact_7102_zero__le__power__abs,axiom,
% 0.30/0.62 ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % zero_le_power_abs
% 0.30/0.62 thf(fact_7103_zero__le__power__abs,axiom,
% 0.30/0.62 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % zero_le_power_abs
% 0.30/0.62 thf(fact_7104_zero__le__power__abs,axiom,
% 0.30/0.62 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % zero_le_power_abs
% 0.30/0.62 thf(fact_7105_zero__le__power__abs,axiom,
% 0.30/0.62 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % zero_le_power_abs
% 0.30/0.62 thf(fact_7106_abs__if__raw,axiom,
% 0.30/0.62 ( abs_abs_int
% 0.30/0.62 = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_if_raw
% 0.30/0.62 thf(fact_7107_abs__if__raw,axiom,
% 0.30/0.62 ( abs_abs_real
% 0.30/0.62 = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_if_raw
% 0.30/0.62 thf(fact_7108_abs__if__raw,axiom,
% 0.30/0.62 ( abs_abs_Code_integer
% 0.30/0.62 = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_if_raw
% 0.30/0.62 thf(fact_7109_abs__if__raw,axiom,
% 0.30/0.62 ( abs_abs_rat
% 0.30/0.62 = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_if_raw
% 0.30/0.62 thf(fact_7110_abs__if,axiom,
% 0.30/0.62 ( abs_abs_int
% 0.30/0.62 = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_if
% 0.30/0.62 thf(fact_7111_abs__if,axiom,
% 0.30/0.62 ( abs_abs_real
% 0.30/0.62 = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_if
% 0.30/0.62 thf(fact_7112_abs__if,axiom,
% 0.30/0.62 ( abs_abs_Code_integer
% 0.30/0.62 = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_if
% 0.30/0.62 thf(fact_7113_abs__if,axiom,
% 0.30/0.62 ( abs_abs_rat
% 0.30/0.62 = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_if
% 0.30/0.62 thf(fact_7114_abs__of__neg,axiom,
% 0.30/0.62 ! [A: int] :
% 0.30/0.62 ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.62 => ( ( abs_abs_int @ A )
% 0.30/0.62 = ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_neg
% 0.30/0.62 thf(fact_7115_abs__of__neg,axiom,
% 0.30/0.62 ! [A: real] :
% 0.30/0.62 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.62 => ( ( abs_abs_real @ A )
% 0.30/0.62 = ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_neg
% 0.30/0.62 thf(fact_7116_abs__of__neg,axiom,
% 0.30/0.62 ! [A: code_integer] :
% 0.30/0.62 ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 0.30/0.62 => ( ( abs_abs_Code_integer @ A )
% 0.30/0.62 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_neg
% 0.30/0.62 thf(fact_7117_abs__of__neg,axiom,
% 0.30/0.62 ! [A: rat] :
% 0.30/0.62 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.62 => ( ( abs_abs_rat @ A )
% 0.30/0.62 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_neg
% 0.30/0.62 thf(fact_7118_abs__diff__le__iff,axiom,
% 0.30/0.62 ! [X2: code_integer,A: code_integer,R2: code_integer] :
% 0.30/0.62 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X2 @ A ) ) @ R2 )
% 0.30/0.62 = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X2 )
% 0.30/0.62 & ( ord_le3102999989581377725nteger @ X2 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_diff_le_iff
% 0.30/0.62 thf(fact_7119_abs__diff__le__iff,axiom,
% 0.30/0.62 ! [X2: real,A: real,R2: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ A ) ) @ R2 )
% 0.30/0.62 = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X2 )
% 0.30/0.62 & ( ord_less_eq_real @ X2 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_diff_le_iff
% 0.30/0.62 thf(fact_7120_abs__diff__le__iff,axiom,
% 0.30/0.62 ! [X2: rat,A: rat,R2: rat] :
% 0.30/0.62 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ A ) ) @ R2 )
% 0.30/0.62 = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X2 )
% 0.30/0.62 & ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_diff_le_iff
% 0.30/0.62 thf(fact_7121_abs__diff__le__iff,axiom,
% 0.30/0.62 ! [X2: int,A: int,R2: int] :
% 0.30/0.62 ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ A ) ) @ R2 )
% 0.30/0.62 = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X2 )
% 0.30/0.62 & ( ord_less_eq_int @ X2 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_diff_le_iff
% 0.30/0.62 thf(fact_7122_abs__diff__triangle__ineq,axiom,
% 0.30/0.62 ! [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 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_diff_triangle_ineq
% 0.30/0.62 thf(fact_7123_abs__diff__triangle__ineq,axiom,
% 0.30/0.62 ! [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 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_diff_triangle_ineq
% 0.30/0.62 thf(fact_7124_abs__diff__triangle__ineq,axiom,
% 0.30/0.62 ! [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 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_diff_triangle_ineq
% 0.30/0.62 thf(fact_7125_abs__diff__triangle__ineq,axiom,
% 0.30/0.62 ! [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 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_diff_triangle_ineq
% 0.30/0.62 thf(fact_7126_abs__triangle__ineq4,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq4
% 0.30/0.62 thf(fact_7127_abs__triangle__ineq4,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq4
% 0.30/0.62 thf(fact_7128_abs__triangle__ineq4,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq4
% 0.30/0.62 thf(fact_7129_abs__triangle__ineq4,axiom,
% 0.30/0.62 ! [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 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_triangle_ineq4
% 0.30/0.62 thf(fact_7130_abs__diff__less__iff,axiom,
% 0.30/0.62 ! [X2: code_integer,A: code_integer,R2: code_integer] :
% 0.30/0.62 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X2 @ A ) ) @ R2 )
% 0.30/0.62 = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X2 )
% 0.30/0.62 & ( ord_le6747313008572928689nteger @ X2 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_diff_less_iff
% 0.30/0.62 thf(fact_7131_abs__diff__less__iff,axiom,
% 0.30/0.62 ! [X2: real,A: real,R2: real] :
% 0.30/0.62 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ A ) ) @ R2 )
% 0.30/0.62 = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X2 )
% 0.30/0.62 & ( ord_less_real @ X2 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_diff_less_iff
% 0.30/0.62 thf(fact_7132_abs__diff__less__iff,axiom,
% 0.30/0.62 ! [X2: rat,A: rat,R2: rat] :
% 0.30/0.62 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ A ) ) @ R2 )
% 0.30/0.62 = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X2 )
% 0.30/0.62 & ( ord_less_rat @ X2 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_diff_less_iff
% 0.30/0.62 thf(fact_7133_abs__diff__less__iff,axiom,
% 0.30/0.62 ! [X2: int,A: int,R2: int] :
% 0.30/0.62 ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ A ) ) @ R2 )
% 0.30/0.62 = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X2 )
% 0.30/0.62 & ( ord_less_int @ X2 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_diff_less_iff
% 0.30/0.62 thf(fact_7134_sum__eq__Suc0__iff,axiom,
% 0.30/0.62 ! [A2: set_int,F: int > nat] :
% 0.30/0.62 ( ( finite_finite_int @ A2 )
% 0.30/0.62 => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 0.30/0.62 = ( suc @ zero_zero_nat ) )
% 0.30/0.62 = ( ? [X: int] :
% 0.30/0.62 ( ( member_int @ X @ A2 )
% 0.30/0.62 & ( ( F @ X )
% 0.30/0.62 = ( suc @ zero_zero_nat ) )
% 0.30/0.62 & ! [Y: int] :
% 0.30/0.62 ( ( member_int @ Y @ A2 )
% 0.30/0.62 => ( ( X != Y )
% 0.30/0.62 => ( ( F @ Y )
% 0.30/0.62 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_eq_Suc0_iff
% 0.30/0.62 thf(fact_7135_sum__eq__Suc0__iff,axiom,
% 0.30/0.62 ! [A2: set_complex,F: complex > nat] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62 => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 0.30/0.62 = ( suc @ zero_zero_nat ) )
% 0.30/0.62 = ( ? [X: complex] :
% 0.30/0.62 ( ( member_complex @ X @ A2 )
% 0.30/0.62 & ( ( F @ X )
% 0.30/0.62 = ( suc @ zero_zero_nat ) )
% 0.30/0.62 & ! [Y: complex] :
% 0.30/0.62 ( ( member_complex @ Y @ A2 )
% 0.30/0.62 => ( ( X != Y )
% 0.30/0.62 => ( ( F @ Y )
% 0.30/0.62 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_eq_Suc0_iff
% 0.30/0.62 thf(fact_7136_sum__eq__Suc0__iff,axiom,
% 0.30/0.62 ! [A2: set_nat,F: nat > nat] :
% 0.30/0.62 ( ( finite_finite_nat @ A2 )
% 0.30/0.62 => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 0.30/0.62 = ( suc @ zero_zero_nat ) )
% 0.30/0.62 = ( ? [X: nat] :
% 0.30/0.62 ( ( member_nat @ X @ A2 )
% 0.30/0.62 & ( ( F @ X )
% 0.30/0.62 = ( suc @ zero_zero_nat ) )
% 0.30/0.62 & ! [Y: nat] :
% 0.30/0.62 ( ( member_nat @ Y @ A2 )
% 0.30/0.62 => ( ( X != Y )
% 0.30/0.62 => ( ( F @ Y )
% 0.30/0.62 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_eq_Suc0_iff
% 0.30/0.62 thf(fact_7137_sum__SucD,axiom,
% 0.30/0.62 ! [F: nat > nat,A2: set_nat,N: nat] :
% 0.30/0.62 ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 0.30/0.62 = ( suc @ N ) )
% 0.30/0.62 => ? [X5: nat] :
% 0.30/0.62 ( ( member_nat @ X5 @ A2 )
% 0.30/0.62 & ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_SucD
% 0.30/0.62 thf(fact_7138_sum__eq__1__iff,axiom,
% 0.30/0.62 ! [A2: set_int,F: int > nat] :
% 0.30/0.62 ( ( finite_finite_int @ A2 )
% 0.30/0.62 => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 0.30/0.62 = one_one_nat )
% 0.30/0.62 = ( ? [X: int] :
% 0.30/0.62 ( ( member_int @ X @ A2 )
% 0.30/0.62 & ( ( F @ X )
% 0.30/0.62 = one_one_nat )
% 0.30/0.62 & ! [Y: int] :
% 0.30/0.62 ( ( member_int @ Y @ A2 )
% 0.30/0.62 => ( ( X != Y )
% 0.30/0.62 => ( ( F @ Y )
% 0.30/0.62 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_eq_1_iff
% 0.30/0.62 thf(fact_7139_sum__eq__1__iff,axiom,
% 0.30/0.62 ! [A2: set_complex,F: complex > nat] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62 => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 0.30/0.62 = one_one_nat )
% 0.30/0.62 = ( ? [X: complex] :
% 0.30/0.62 ( ( member_complex @ X @ A2 )
% 0.30/0.62 & ( ( F @ X )
% 0.30/0.62 = one_one_nat )
% 0.30/0.62 & ! [Y: complex] :
% 0.30/0.62 ( ( member_complex @ Y @ A2 )
% 0.30/0.62 => ( ( X != Y )
% 0.30/0.62 => ( ( F @ Y )
% 0.30/0.62 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_eq_1_iff
% 0.30/0.62 thf(fact_7140_sum__eq__1__iff,axiom,
% 0.30/0.62 ! [A2: set_nat,F: nat > nat] :
% 0.30/0.62 ( ( finite_finite_nat @ A2 )
% 0.30/0.62 => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 0.30/0.62 = one_one_nat )
% 0.30/0.62 = ( ? [X: nat] :
% 0.30/0.62 ( ( member_nat @ X @ A2 )
% 0.30/0.62 & ( ( F @ X )
% 0.30/0.62 = one_one_nat )
% 0.30/0.62 & ! [Y: nat] :
% 0.30/0.62 ( ( member_nat @ Y @ A2 )
% 0.30/0.62 => ( ( X != Y )
% 0.30/0.62 => ( ( F @ Y )
% 0.30/0.62 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_eq_1_iff
% 0.30/0.62 thf(fact_7141_round__mono,axiom,
% 0.30/0.62 ! [X2: rat,Y3: rat] :
% 0.30/0.62 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.30/0.62 => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X2 ) @ ( archim7778729529865785530nd_rat @ Y3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % round_mono
% 0.30/0.62 thf(fact_7142_abs__real__def,axiom,
% 0.30/0.62 ( abs_abs_real
% 0.30/0.62 = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_real_def
% 0.30/0.62 thf(fact_7143_lemma__interval__lt,axiom,
% 0.30/0.62 ! [A: real,X2: real,B: real] :
% 0.30/0.62 ( ( ord_less_real @ A @ X2 )
% 0.30/0.62 => ( ( ord_less_real @ X2 @ B )
% 0.30/0.62 => ? [D3: real] :
% 0.30/0.62 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.30/0.62 & ! [Y5: real] :
% 0.30/0.62 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y5 ) ) @ D3 )
% 0.30/0.62 => ( ( ord_less_real @ A @ Y5 )
% 0.30/0.62 & ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lemma_interval_lt
% 0.30/0.62 thf(fact_7144_sin__bound__lemma,axiom,
% 0.30/0.62 ! [X2: real,Y3: real,U: real,V: real] :
% 0.30/0.62 ( ( X2 = Y3 )
% 0.30/0.62 => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 0.30/0.62 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X2 @ U ) @ Y3 ) ) @ V ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sin_bound_lemma
% 0.30/0.62 thf(fact_7145_sum__power__add,axiom,
% 0.30/0.62 ! [X2: complex,M: nat,I5: set_nat] :
% 0.30/0.62 ( ( groups2073611262835488442omplex
% 0.30/0.62 @ ^ [I4: nat] : ( power_power_complex @ X2 @ ( plus_plus_nat @ M @ I4 ) )
% 0.30/0.62 @ I5 )
% 0.30/0.62 = ( times_times_complex @ ( power_power_complex @ X2 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ I5 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_power_add
% 0.30/0.62 thf(fact_7146_sum__power__add,axiom,
% 0.30/0.62 ! [X2: rat,M: nat,I5: set_nat] :
% 0.30/0.62 ( ( groups2906978787729119204at_rat
% 0.30/0.62 @ ^ [I4: nat] : ( power_power_rat @ X2 @ ( plus_plus_nat @ M @ I4 ) )
% 0.30/0.62 @ I5 )
% 0.30/0.62 = ( times_times_rat @ ( power_power_rat @ X2 @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ I5 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_power_add
% 0.30/0.62 thf(fact_7147_sum__power__add,axiom,
% 0.30/0.62 ! [X2: int,M: nat,I5: set_nat] :
% 0.30/0.62 ( ( groups3539618377306564664at_int
% 0.30/0.62 @ ^ [I4: nat] : ( power_power_int @ X2 @ ( plus_plus_nat @ M @ I4 ) )
% 0.30/0.62 @ I5 )
% 0.30/0.62 = ( times_times_int @ ( power_power_int @ X2 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ I5 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_power_add
% 0.30/0.62 thf(fact_7148_sum__power__add,axiom,
% 0.30/0.62 ! [X2: real,M: nat,I5: set_nat] :
% 0.30/0.62 ( ( groups6591440286371151544t_real
% 0.30/0.62 @ ^ [I4: nat] : ( power_power_real @ X2 @ ( plus_plus_nat @ M @ I4 ) )
% 0.30/0.62 @ I5 )
% 0.30/0.62 = ( times_times_real @ ( power_power_real @ X2 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ I5 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_power_add
% 0.30/0.62 thf(fact_7149_sum_OatLeastAtMost__rev,axiom,
% 0.30/0.62 ! [G: nat > nat,N: nat,M: nat] :
% 0.30/0.62 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 0.30/0.62 = ( groups3542108847815614940at_nat
% 0.30/0.62 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.atLeastAtMost_rev
% 0.30/0.62 thf(fact_7150_sum_OatLeastAtMost__rev,axiom,
% 0.30/0.62 ! [G: nat > real,N: nat,M: nat] :
% 0.30/0.62 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 0.30/0.62 = ( groups6591440286371151544t_real
% 0.30/0.62 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.atLeastAtMost_rev
% 0.30/0.62 thf(fact_7151_sum__nth__roots,axiom,
% 0.30/0.62 ! [N: nat,C: complex] :
% 0.30/0.62 ( ( ord_less_nat @ one_one_nat @ N )
% 0.30/0.62 => ( ( groups7754918857620584856omplex
% 0.30/0.62 @ ^ [X: complex] : X
% 0.30/0.62 @ ( collect_complex
% 0.30/0.62 @ ^ [Z3: complex] :
% 0.30/0.62 ( ( power_power_complex @ Z3 @ N )
% 0.30/0.62 = C ) ) )
% 0.30/0.62 = zero_zero_complex ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_nth_roots
% 0.30/0.62 thf(fact_7152_sum__roots__unity,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( ord_less_nat @ one_one_nat @ N )
% 0.30/0.62 => ( ( groups7754918857620584856omplex
% 0.30/0.62 @ ^ [X: complex] : X
% 0.30/0.62 @ ( collect_complex
% 0.30/0.62 @ ^ [Z3: complex] :
% 0.30/0.62 ( ( power_power_complex @ Z3 @ N )
% 0.30/0.62 = one_one_complex ) ) )
% 0.30/0.62 = zero_zero_complex ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_roots_unity
% 0.30/0.62 thf(fact_7153_tanh__real__gt__neg1,axiom,
% 0.30/0.62 ! [X2: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X2 ) ) ).
% 0.30/0.62
% 0.30/0.62 % tanh_real_gt_neg1
% 0.30/0.62 thf(fact_7154_abs__add__one__gt__zero,axiom,
% 0.30/0.62 ! [X2: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_add_one_gt_zero
% 0.30/0.62 thf(fact_7155_abs__add__one__gt__zero,axiom,
% 0.30/0.62 ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_add_one_gt_zero
% 0.30/0.62 thf(fact_7156_abs__add__one__gt__zero,axiom,
% 0.30/0.62 ! [X2: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_add_one_gt_zero
% 0.30/0.62 thf(fact_7157_abs__add__one__gt__zero,axiom,
% 0.30/0.62 ! [X2: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_add_one_gt_zero
% 0.30/0.62 thf(fact_7158_of__int__leD,axiom,
% 0.30/0.62 ! [N: int,X2: code_integer] :
% 0.30/0.62 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X2 )
% 0.30/0.62 => ( ( N = zero_zero_int )
% 0.30/0.62 | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_int_leD
% 0.30/0.62 thf(fact_7159_of__int__leD,axiom,
% 0.30/0.62 ! [N: int,X2: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X2 )
% 0.30/0.62 => ( ( N = zero_zero_int )
% 0.30/0.62 | ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_int_leD
% 0.30/0.62 thf(fact_7160_of__int__leD,axiom,
% 0.30/0.62 ! [N: int,X2: rat] :
% 0.30/0.62 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X2 )
% 0.30/0.62 => ( ( N = zero_zero_int )
% 0.30/0.62 | ( ord_less_eq_rat @ one_one_rat @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_int_leD
% 0.30/0.62 thf(fact_7161_of__int__leD,axiom,
% 0.30/0.62 ! [N: int,X2: int] :
% 0.30/0.62 ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X2 )
% 0.30/0.62 => ( ( N = zero_zero_int )
% 0.30/0.62 | ( ord_less_eq_int @ one_one_int @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_int_leD
% 0.30/0.62 thf(fact_7162_of__int__lessD,axiom,
% 0.30/0.62 ! [N: int,X2: code_integer] :
% 0.30/0.62 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X2 )
% 0.30/0.62 => ( ( N = zero_zero_int )
% 0.30/0.62 | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_int_lessD
% 0.30/0.62 thf(fact_7163_of__int__lessD,axiom,
% 0.30/0.62 ! [N: int,X2: real] :
% 0.30/0.62 ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X2 )
% 0.30/0.62 => ( ( N = zero_zero_int )
% 0.30/0.62 | ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_int_lessD
% 0.30/0.62 thf(fact_7164_of__int__lessD,axiom,
% 0.30/0.62 ! [N: int,X2: rat] :
% 0.30/0.62 ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X2 )
% 0.30/0.62 => ( ( N = zero_zero_int )
% 0.30/0.62 | ( ord_less_rat @ one_one_rat @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_int_lessD
% 0.30/0.62 thf(fact_7165_of__int__lessD,axiom,
% 0.30/0.62 ! [N: int,X2: int] :
% 0.30/0.62 ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X2 )
% 0.30/0.62 => ( ( N = zero_zero_int )
% 0.30/0.62 | ( ord_less_int @ one_one_int @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_int_lessD
% 0.30/0.62 thf(fact_7166_sum__diff__nat,axiom,
% 0.30/0.62 ! [B3: set_complex,A2: set_complex,F: complex > nat] :
% 0.30/0.62 ( ( finite3207457112153483333omplex @ B3 )
% 0.30/0.62 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.30/0.62 => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.30/0.62 = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_diff_nat
% 0.30/0.62 thf(fact_7167_sum__diff__nat,axiom,
% 0.30/0.62 ! [B3: set_int,A2: set_int,F: int > nat] :
% 0.30/0.62 ( ( finite_finite_int @ B3 )
% 0.30/0.62 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.30/0.62 => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
% 0.30/0.62 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_diff_nat
% 0.30/0.62 thf(fact_7168_sum__diff__nat,axiom,
% 0.30/0.62 ! [B3: set_nat,A2: set_nat,F: nat > nat] :
% 0.30/0.62 ( ( finite_finite_nat @ B3 )
% 0.30/0.62 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.30/0.62 => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 0.30/0.62 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B3 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_diff_nat
% 0.30/0.62 thf(fact_7169_sum__shift__lb__Suc0__0,axiom,
% 0.30/0.62 ! [F: nat > complex,K: nat] :
% 0.30/0.62 ( ( ( F @ zero_zero_nat )
% 0.30/0.62 = zero_zero_complex )
% 0.30/0.62 => ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.30/0.62 = ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_shift_lb_Suc0_0
% 0.30/0.62 thf(fact_7170_sum__shift__lb__Suc0__0,axiom,
% 0.30/0.62 ! [F: nat > rat,K: nat] :
% 0.30/0.62 ( ( ( F @ zero_zero_nat )
% 0.30/0.62 = zero_zero_rat )
% 0.30/0.62 => ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.30/0.62 = ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_shift_lb_Suc0_0
% 0.30/0.62 thf(fact_7171_sum__shift__lb__Suc0__0,axiom,
% 0.30/0.62 ! [F: nat > int,K: nat] :
% 0.30/0.62 ( ( ( F @ zero_zero_nat )
% 0.30/0.62 = zero_zero_int )
% 0.30/0.62 => ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.30/0.62 = ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_shift_lb_Suc0_0
% 0.30/0.62 thf(fact_7172_sum__shift__lb__Suc0__0,axiom,
% 0.30/0.62 ! [F: nat > nat,K: nat] :
% 0.30/0.62 ( ( ( F @ zero_zero_nat )
% 0.30/0.62 = zero_zero_nat )
% 0.30/0.62 => ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.30/0.62 = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_shift_lb_Suc0_0
% 0.30/0.62 thf(fact_7173_sum__shift__lb__Suc0__0,axiom,
% 0.30/0.62 ! [F: nat > real,K: nat] :
% 0.30/0.62 ( ( ( F @ zero_zero_nat )
% 0.30/0.62 = zero_zero_real )
% 0.30/0.62 => ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.30/0.62 = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_shift_lb_Suc0_0
% 0.30/0.62 thf(fact_7174_sum_OatLeast0__atMost__Suc,axiom,
% 0.30/0.62 ! [G: nat > rat,N: nat] :
% 0.30/0.62 ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.30/0.62 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.atLeast0_atMost_Suc
% 0.30/0.62 thf(fact_7175_sum_OatLeast0__atMost__Suc,axiom,
% 0.30/0.62 ! [G: nat > int,N: nat] :
% 0.30/0.62 ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.30/0.62 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.atLeast0_atMost_Suc
% 0.30/0.62 thf(fact_7176_sum_OatLeast0__atMost__Suc,axiom,
% 0.30/0.62 ! [G: nat > nat,N: nat] :
% 0.30/0.62 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.30/0.62 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.atLeast0_atMost_Suc
% 0.30/0.62 thf(fact_7177_sum_OatLeast0__atMost__Suc,axiom,
% 0.30/0.62 ! [G: nat > real,N: nat] :
% 0.30/0.62 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.30/0.62 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.atLeast0_atMost_Suc
% 0.30/0.62 thf(fact_7178_sum_OatLeast__Suc__atMost,axiom,
% 0.30/0.62 ! [M: nat,N: nat,G: nat > rat] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.atLeast_Suc_atMost
% 0.30/0.62 thf(fact_7179_sum_OatLeast__Suc__atMost,axiom,
% 0.30/0.62 ! [M: nat,N: nat,G: nat > int] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.atLeast_Suc_atMost
% 0.30/0.62 thf(fact_7180_sum_OatLeast__Suc__atMost,axiom,
% 0.30/0.62 ! [M: nat,N: nat,G: nat > nat] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.atLeast_Suc_atMost
% 0.30/0.62 thf(fact_7181_sum_OatLeast__Suc__atMost,axiom,
% 0.30/0.62 ! [M: nat,N: nat,G: nat > real] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.atLeast_Suc_atMost
% 0.30/0.62 thf(fact_7182_sum_Onat__ivl__Suc_H,axiom,
% 0.30/0.62 ! [M: nat,N: nat,G: nat > rat] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.62 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62 = ( plus_plus_rat @ ( G @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.nat_ivl_Suc'
% 0.30/0.62 thf(fact_7183_sum_Onat__ivl__Suc_H,axiom,
% 0.30/0.62 ! [M: nat,N: nat,G: nat > int] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.62 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62 = ( plus_plus_int @ ( G @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.nat_ivl_Suc'
% 0.30/0.62 thf(fact_7184_sum_Onat__ivl__Suc_H,axiom,
% 0.30/0.62 ! [M: nat,N: nat,G: nat > nat] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.62 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62 = ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.nat_ivl_Suc'
% 0.30/0.62 thf(fact_7185_sum_Onat__ivl__Suc_H,axiom,
% 0.30/0.62 ! [M: nat,N: nat,G: nat > real] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.62 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62 = ( plus_plus_real @ ( G @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.nat_ivl_Suc'
% 0.30/0.62 thf(fact_7186_lemma__interval,axiom,
% 0.30/0.62 ! [A: real,X2: real,B: real] :
% 0.30/0.62 ( ( ord_less_real @ A @ X2 )
% 0.30/0.62 => ( ( ord_less_real @ X2 @ B )
% 0.30/0.62 => ? [D3: real] :
% 0.30/0.62 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.30/0.62 & ! [Y5: real] :
% 0.30/0.62 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y5 ) ) @ D3 )
% 0.30/0.62 => ( ( ord_less_eq_real @ A @ Y5 )
% 0.30/0.62 & ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % lemma_interval
% 0.30/0.62 thf(fact_7187_sum_OSuc__reindex__ivl,axiom,
% 0.30/0.62 ! [M: nat,N: nat,G: nat > rat] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.30/0.62 = ( plus_plus_rat @ ( G @ M )
% 0.30/0.62 @ ( groups2906978787729119204at_rat
% 0.30/0.62 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.Suc_reindex_ivl
% 0.30/0.62 thf(fact_7188_sum_OSuc__reindex__ivl,axiom,
% 0.30/0.62 ! [M: nat,N: nat,G: nat > int] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.30/0.62 = ( plus_plus_int @ ( G @ M )
% 0.30/0.62 @ ( groups3539618377306564664at_int
% 0.30/0.62 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.Suc_reindex_ivl
% 0.30/0.62 thf(fact_7189_sum_OSuc__reindex__ivl,axiom,
% 0.30/0.62 ! [M: nat,N: nat,G: nat > nat] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.30/0.62 = ( plus_plus_nat @ ( G @ M )
% 0.30/0.62 @ ( groups3542108847815614940at_nat
% 0.30/0.62 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.Suc_reindex_ivl
% 0.30/0.62 thf(fact_7190_sum_OSuc__reindex__ivl,axiom,
% 0.30/0.62 ! [M: nat,N: nat,G: nat > real] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.30/0.62 = ( plus_plus_real @ ( G @ M )
% 0.30/0.62 @ ( groups6591440286371151544t_real
% 0.30/0.62 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.Suc_reindex_ivl
% 0.30/0.62 thf(fact_7191_sum__Suc__diff,axiom,
% 0.30/0.62 ! [M: nat,N: nat,F: nat > rat] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.62 => ( ( groups2906978787729119204at_rat
% 0.30/0.62 @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_Suc_diff
% 0.30/0.62 thf(fact_7192_sum__Suc__diff,axiom,
% 0.30/0.62 ! [M: nat,N: nat,F: nat > int] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.62 => ( ( groups3539618377306564664at_int
% 0.30/0.62 @ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_Suc_diff
% 0.30/0.62 thf(fact_7193_sum__Suc__diff,axiom,
% 0.30/0.62 ! [M: nat,N: nat,F: nat > real] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.62 => ( ( groups6591440286371151544t_real
% 0.30/0.62 @ ^ [I4: nat] : ( minus_minus_real @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_Suc_diff
% 0.30/0.62 thf(fact_7194_of__int__round__abs__le,axiom,
% 0.30/0.62 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) @ X2 ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_int_round_abs_le
% 0.30/0.62 thf(fact_7195_of__int__round__abs__le,axiom,
% 0.30/0.62 ! [X2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) @ X2 ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_int_round_abs_le
% 0.30/0.62 thf(fact_7196_round__unique_H,axiom,
% 0.30/0.62 ! [X2: real,N: int] :
% 0.30/0.62 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.62 => ( ( archim8280529875227126926d_real @ X2 )
% 0.30/0.62 = N ) ) ).
% 0.30/0.62
% 0.30/0.62 % round_unique'
% 0.30/0.62 thf(fact_7197_round__unique_H,axiom,
% 0.30/0.62 ! [X2: rat,N: int] :
% 0.30/0.62 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.30/0.62 => ( ( archim7778729529865785530nd_rat @ X2 )
% 0.30/0.62 = N ) ) ).
% 0.30/0.62
% 0.30/0.62 % round_unique'
% 0.30/0.62 thf(fact_7198_pred__equals__eq2,axiom,
% 0.30/0.62 ! [R: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
% 0.30/0.62 ( ( ( ^ [X: code_integer,Y: $o] : ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X @ Y ) @ R ) )
% 0.30/0.62 = ( ^ [X: code_integer,Y: $o] : ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X @ Y ) @ S3 ) ) )
% 0.30/0.62 = ( R = S3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % pred_equals_eq2
% 0.30/0.62 thf(fact_7199_pred__equals__eq2,axiom,
% 0.30/0.62 ! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
% 0.30/0.62 ( ( ( ^ [X: num,Y: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X @ Y ) @ R ) )
% 0.30/0.62 = ( ^ [X: num,Y: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X @ Y ) @ S3 ) ) )
% 0.30/0.62 = ( R = S3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % pred_equals_eq2
% 0.30/0.62 thf(fact_7200_pred__equals__eq2,axiom,
% 0.30/0.62 ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
% 0.30/0.62 ( ( ( ^ [X: nat,Y: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ R ) )
% 0.30/0.62 = ( ^ [X: nat,Y: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ S3 ) ) )
% 0.30/0.62 = ( R = S3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % pred_equals_eq2
% 0.30/0.62 thf(fact_7201_pred__equals__eq2,axiom,
% 0.30/0.62 ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
% 0.30/0.62 ( ( ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) )
% 0.30/0.62 = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S3 ) ) )
% 0.30/0.62 = ( R = S3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % pred_equals_eq2
% 0.30/0.62 thf(fact_7202_pred__equals__eq2,axiom,
% 0.30/0.62 ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
% 0.30/0.62 ( ( ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R ) )
% 0.30/0.62 = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S3 ) ) )
% 0.30/0.62 = ( R = S3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % pred_equals_eq2
% 0.30/0.62 thf(fact_7203_bot__empty__eq2,axiom,
% 0.30/0.62 ( bot_bo4731626569425807221er_o_o
% 0.30/0.62 = ( ^ [X: code_integer,Y: $o] : ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X @ Y ) @ bot_bo5379713665208646970eger_o ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bot_empty_eq2
% 0.30/0.62 thf(fact_7204_bot__empty__eq2,axiom,
% 0.30/0.62 ( bot_bot_num_num_o
% 0.30/0.62 = ( ^ [X: num,Y: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X @ Y ) @ bot_bo9056780473022590049um_num ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bot_empty_eq2
% 0.30/0.62 thf(fact_7205_bot__empty__eq2,axiom,
% 0.30/0.62 ( bot_bot_nat_num_o
% 0.30/0.62 = ( ^ [X: nat,Y: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ bot_bo7038385379056416535at_num ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bot_empty_eq2
% 0.30/0.62 thf(fact_7206_bot__empty__eq2,axiom,
% 0.30/0.62 ( bot_bot_nat_nat_o
% 0.30/0.62 = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ bot_bo2099793752762293965at_nat ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bot_empty_eq2
% 0.30/0.62 thf(fact_7207_bot__empty__eq2,axiom,
% 0.30/0.62 ( bot_bot_int_int_o
% 0.30/0.62 = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ bot_bo1796632182523588997nt_int ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % bot_empty_eq2
% 0.30/0.62 thf(fact_7208_abs__le__square__iff,axiom,
% 0.30/0.62 ! [X2: code_integer,Y3: code_integer] :
% 0.30/0.62 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ ( abs_abs_Code_integer @ Y3 ) )
% 0.30/0.62 = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_square_iff
% 0.30/0.62 thf(fact_7209_abs__le__square__iff,axiom,
% 0.30/0.62 ! [X2: real,Y3: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ Y3 ) )
% 0.30/0.62 = ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_square_iff
% 0.30/0.62 thf(fact_7210_abs__le__square__iff,axiom,
% 0.30/0.62 ! [X2: rat,Y3: rat] :
% 0.30/0.62 ( ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ ( abs_abs_rat @ Y3 ) )
% 0.30/0.62 = ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_square_iff
% 0.30/0.62 thf(fact_7211_abs__le__square__iff,axiom,
% 0.30/0.62 ! [X2: int,Y3: int] :
% 0.30/0.62 ( ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ ( abs_abs_int @ Y3 ) )
% 0.30/0.62 = ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_le_square_iff
% 0.30/0.62 thf(fact_7212_abs__square__eq__1,axiom,
% 0.30/0.62 ! [X2: code_integer] :
% 0.30/0.62 ( ( ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.62 = one_one_Code_integer )
% 0.30/0.62 = ( ( abs_abs_Code_integer @ X2 )
% 0.30/0.62 = one_one_Code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_square_eq_1
% 0.30/0.62 thf(fact_7213_abs__square__eq__1,axiom,
% 0.30/0.62 ! [X2: rat] :
% 0.30/0.62 ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.62 = one_one_rat )
% 0.30/0.62 = ( ( abs_abs_rat @ X2 )
% 0.30/0.62 = one_one_rat ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_square_eq_1
% 0.30/0.62 thf(fact_7214_abs__square__eq__1,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.62 = one_one_real )
% 0.30/0.62 = ( ( abs_abs_real @ X2 )
% 0.30/0.62 = one_one_real ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_square_eq_1
% 0.30/0.62 thf(fact_7215_abs__square__eq__1,axiom,
% 0.30/0.62 ! [X2: int] :
% 0.30/0.62 ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.62 = one_one_int )
% 0.30/0.62 = ( ( abs_abs_int @ X2 )
% 0.30/0.62 = one_one_int ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_square_eq_1
% 0.30/0.62 thf(fact_7216_power__even__abs,axiom,
% 0.30/0.62 ! [N: nat,A: code_integer] :
% 0.30/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.62 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
% 0.30/0.62 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % power_even_abs
% 0.30/0.62 thf(fact_7217_power__even__abs,axiom,
% 0.30/0.62 ! [N: nat,A: real] :
% 0.30/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.62 => ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
% 0.30/0.62 = ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % power_even_abs
% 0.30/0.62 thf(fact_7218_power__even__abs,axiom,
% 0.30/0.62 ! [N: nat,A: int] :
% 0.30/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.62 => ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
% 0.30/0.62 = ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % power_even_abs
% 0.30/0.62 thf(fact_7219_sum_Oub__add__nat,axiom,
% 0.30/0.62 ! [M: nat,N: nat,G: nat > rat,P6: nat] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.30/0.62 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 0.30/0.62 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.ub_add_nat
% 0.30/0.62 thf(fact_7220_sum_Oub__add__nat,axiom,
% 0.30/0.62 ! [M: nat,N: nat,G: nat > int,P6: nat] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.30/0.62 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 0.30/0.62 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.ub_add_nat
% 0.30/0.62 thf(fact_7221_sum_Oub__add__nat,axiom,
% 0.30/0.62 ! [M: nat,N: nat,G: nat > nat,P6: nat] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.30/0.62 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 0.30/0.62 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.ub_add_nat
% 0.30/0.62 thf(fact_7222_sum_Oub__add__nat,axiom,
% 0.30/0.62 ! [M: nat,N: nat,G: nat > real,P6: nat] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.30/0.62 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 0.30/0.62 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.ub_add_nat
% 0.30/0.62 thf(fact_7223_abs__sqrt__wlog,axiom,
% 0.30/0.62 ! [P: code_integer > code_integer > $o,X2: code_integer] :
% 0.30/0.62 ( ! [X5: code_integer] :
% 0.30/0.62 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X5 )
% 0.30/0.62 => ( P @ X5 @ ( power_8256067586552552935nteger @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.62 => ( P @ ( abs_abs_Code_integer @ X2 ) @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_sqrt_wlog
% 0.30/0.62 thf(fact_7224_abs__sqrt__wlog,axiom,
% 0.30/0.62 ! [P: real > real > $o,X2: real] :
% 0.30/0.62 ( ! [X5: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 0.30/0.62 => ( P @ X5 @ ( power_power_real @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.62 => ( P @ ( abs_abs_real @ X2 ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_sqrt_wlog
% 0.30/0.62 thf(fact_7225_abs__sqrt__wlog,axiom,
% 0.30/0.62 ! [P: rat > rat > $o,X2: rat] :
% 0.30/0.62 ( ! [X5: rat] :
% 0.30/0.62 ( ( ord_less_eq_rat @ zero_zero_rat @ X5 )
% 0.30/0.62 => ( P @ X5 @ ( power_power_rat @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.62 => ( P @ ( abs_abs_rat @ X2 ) @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_sqrt_wlog
% 0.30/0.62 thf(fact_7226_abs__sqrt__wlog,axiom,
% 0.30/0.62 ! [P: int > int > $o,X2: int] :
% 0.30/0.62 ( ! [X5: int] :
% 0.30/0.62 ( ( ord_less_eq_int @ zero_zero_int @ X5 )
% 0.30/0.62 => ( P @ X5 @ ( power_power_int @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.62 => ( P @ ( abs_abs_int @ X2 ) @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_sqrt_wlog
% 0.30/0.62 thf(fact_7227_power2__le__iff__abs__le,axiom,
% 0.30/0.62 ! [Y3: code_integer,X2: code_integer] :
% 0.30/0.62 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
% 0.30/0.62 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.62 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ Y3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % power2_le_iff_abs_le
% 0.30/0.62 thf(fact_7228_power2__le__iff__abs__le,axiom,
% 0.30/0.62 ! [Y3: real,X2: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.62 => ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.62 = ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ Y3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % power2_le_iff_abs_le
% 0.30/0.62 thf(fact_7229_power2__le__iff__abs__le,axiom,
% 0.30/0.62 ! [Y3: rat,X2: rat] :
% 0.30/0.62 ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.62 => ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.62 = ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ Y3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % power2_le_iff_abs_le
% 0.30/0.62 thf(fact_7230_power2__le__iff__abs__le,axiom,
% 0.30/0.62 ! [Y3: int,X2: int] :
% 0.30/0.62 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.62 => ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.62 = ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ Y3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % power2_le_iff_abs_le
% 0.30/0.62 thf(fact_7231_abs__square__le__1,axiom,
% 0.30/0.62 ! [X2: code_integer] :
% 0.30/0.62 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.30/0.62 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ one_one_Code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_square_le_1
% 0.30/0.62 thf(fact_7232_abs__square__le__1,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.30/0.62 = ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_square_le_1
% 0.30/0.62 thf(fact_7233_abs__square__le__1,axiom,
% 0.30/0.62 ! [X2: rat] :
% 0.30/0.62 ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.30/0.62 = ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ one_one_rat ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_square_le_1
% 0.30/0.62 thf(fact_7234_abs__square__le__1,axiom,
% 0.30/0.62 ! [X2: int] :
% 0.30/0.62 ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.30/0.62 = ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ one_one_int ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_square_le_1
% 0.30/0.62 thf(fact_7235_abs__square__less__1,axiom,
% 0.30/0.62 ! [X2: code_integer] :
% 0.30/0.62 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.30/0.62 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X2 ) @ one_one_Code_integer ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_square_less_1
% 0.30/0.62 thf(fact_7236_abs__square__less__1,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.30/0.62 = ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_square_less_1
% 0.30/0.62 thf(fact_7237_abs__square__less__1,axiom,
% 0.30/0.62 ! [X2: rat] :
% 0.30/0.62 ( ( ord_less_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.30/0.62 = ( ord_less_rat @ ( abs_abs_rat @ X2 ) @ one_one_rat ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_square_less_1
% 0.30/0.62 thf(fact_7238_abs__square__less__1,axiom,
% 0.30/0.62 ! [X2: int] :
% 0.30/0.62 ( ( ord_less_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.30/0.62 = ( ord_less_int @ ( abs_abs_int @ X2 ) @ one_one_int ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_square_less_1
% 0.30/0.62 thf(fact_7239_power__mono__even,axiom,
% 0.30/0.62 ! [N: nat,A: code_integer,B: code_integer] :
% 0.30/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.62 => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
% 0.30/0.62 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % power_mono_even
% 0.30/0.62 thf(fact_7240_power__mono__even,axiom,
% 0.30/0.62 ! [N: nat,A: real,B: real] :
% 0.30/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.62 => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
% 0.30/0.62 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % power_mono_even
% 0.30/0.62 thf(fact_7241_power__mono__even,axiom,
% 0.30/0.62 ! [N: nat,A: rat,B: rat] :
% 0.30/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.62 => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
% 0.30/0.62 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % power_mono_even
% 0.30/0.62 thf(fact_7242_power__mono__even,axiom,
% 0.30/0.62 ! [N: nat,A: int,B: int] :
% 0.30/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.62 => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
% 0.30/0.62 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % power_mono_even
% 0.30/0.62 thf(fact_7243_convex__sum__bound__le,axiom,
% 0.30/0.62 ! [I5: set_complex,X2: complex > code_integer,A: complex > code_integer,B: code_integer,Delta: code_integer] :
% 0.30/0.62 ( ! [I3: complex] :
% 0.30/0.62 ( ( member_complex @ I3 @ I5 )
% 0.30/0.62 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I3 ) ) )
% 0.30/0.62 => ( ( ( groups6621422865394947399nteger @ X2 @ I5 )
% 0.30/0.62 = one_one_Code_integer )
% 0.30/0.62 => ( ! [I3: complex] :
% 0.30/0.62 ( ( member_complex @ I3 @ I5 )
% 0.30/0.62 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62 => ( ord_le3102999989581377725nteger
% 0.30/0.62 @ ( abs_abs_Code_integer
% 0.30/0.62 @ ( minus_8373710615458151222nteger
% 0.30/0.62 @ ( groups6621422865394947399nteger
% 0.30/0.62 @ ^ [I4: complex] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 0.30/0.62 @ I5 )
% 0.30/0.62 @ B ) )
% 0.30/0.62 @ Delta ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % convex_sum_bound_le
% 0.30/0.62 thf(fact_7244_convex__sum__bound__le,axiom,
% 0.30/0.62 ! [I5: set_real,X2: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
% 0.30/0.62 ( ! [I3: real] :
% 0.30/0.62 ( ( member_real @ I3 @ I5 )
% 0.30/0.62 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I3 ) ) )
% 0.30/0.62 => ( ( ( groups7713935264441627589nteger @ X2 @ I5 )
% 0.30/0.62 = one_one_Code_integer )
% 0.30/0.62 => ( ! [I3: real] :
% 0.30/0.62 ( ( member_real @ I3 @ I5 )
% 0.30/0.62 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62 => ( ord_le3102999989581377725nteger
% 0.30/0.62 @ ( abs_abs_Code_integer
% 0.30/0.62 @ ( minus_8373710615458151222nteger
% 0.30/0.62 @ ( groups7713935264441627589nteger
% 0.30/0.62 @ ^ [I4: real] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 0.30/0.62 @ I5 )
% 0.30/0.62 @ B ) )
% 0.30/0.62 @ Delta ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % convex_sum_bound_le
% 0.30/0.62 thf(fact_7245_convex__sum__bound__le,axiom,
% 0.30/0.62 ! [I5: set_nat,X2: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
% 0.30/0.62 ( ! [I3: nat] :
% 0.30/0.62 ( ( member_nat @ I3 @ I5 )
% 0.30/0.62 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I3 ) ) )
% 0.30/0.62 => ( ( ( groups7501900531339628137nteger @ X2 @ I5 )
% 0.30/0.62 = one_one_Code_integer )
% 0.30/0.62 => ( ! [I3: nat] :
% 0.30/0.62 ( ( member_nat @ I3 @ I5 )
% 0.30/0.62 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62 => ( ord_le3102999989581377725nteger
% 0.30/0.62 @ ( abs_abs_Code_integer
% 0.30/0.62 @ ( minus_8373710615458151222nteger
% 0.30/0.62 @ ( groups7501900531339628137nteger
% 0.30/0.62 @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 0.30/0.62 @ I5 )
% 0.30/0.62 @ B ) )
% 0.30/0.62 @ Delta ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % convex_sum_bound_le
% 0.30/0.62 thf(fact_7246_convex__sum__bound__le,axiom,
% 0.30/0.62 ! [I5: set_int,X2: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
% 0.30/0.62 ( ! [I3: int] :
% 0.30/0.62 ( ( member_int @ I3 @ I5 )
% 0.30/0.62 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I3 ) ) )
% 0.30/0.62 => ( ( ( groups7873554091576472773nteger @ X2 @ I5 )
% 0.30/0.62 = one_one_Code_integer )
% 0.30/0.62 => ( ! [I3: int] :
% 0.30/0.62 ( ( member_int @ I3 @ I5 )
% 0.30/0.62 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62 => ( ord_le3102999989581377725nteger
% 0.30/0.62 @ ( abs_abs_Code_integer
% 0.30/0.62 @ ( minus_8373710615458151222nteger
% 0.30/0.62 @ ( groups7873554091576472773nteger
% 0.30/0.62 @ ^ [I4: int] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 0.30/0.62 @ I5 )
% 0.30/0.62 @ B ) )
% 0.30/0.62 @ Delta ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % convex_sum_bound_le
% 0.30/0.62 thf(fact_7247_convex__sum__bound__le,axiom,
% 0.30/0.62 ! [I5: set_complex,X2: complex > real,A: complex > real,B: real,Delta: real] :
% 0.30/0.62 ( ! [I3: complex] :
% 0.30/0.62 ( ( member_complex @ I3 @ I5 )
% 0.30/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I3 ) ) )
% 0.30/0.62 => ( ( ( groups5808333547571424918x_real @ X2 @ I5 )
% 0.30/0.62 = one_one_real )
% 0.30/0.62 => ( ! [I3: complex] :
% 0.30/0.62 ( ( member_complex @ I3 @ I5 )
% 0.30/0.62 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62 => ( ord_less_eq_real
% 0.30/0.62 @ ( abs_abs_real
% 0.30/0.62 @ ( minus_minus_real
% 0.30/0.62 @ ( groups5808333547571424918x_real
% 0.30/0.62 @ ^ [I4: complex] : ( times_times_real @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 0.30/0.62 @ I5 )
% 0.30/0.62 @ B ) )
% 0.30/0.62 @ Delta ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % convex_sum_bound_le
% 0.30/0.62 thf(fact_7248_convex__sum__bound__le,axiom,
% 0.30/0.62 ! [I5: set_real,X2: real > real,A: real > real,B: real,Delta: real] :
% 0.30/0.62 ( ! [I3: real] :
% 0.30/0.62 ( ( member_real @ I3 @ I5 )
% 0.30/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I3 ) ) )
% 0.30/0.62 => ( ( ( groups8097168146408367636l_real @ X2 @ I5 )
% 0.30/0.62 = one_one_real )
% 0.30/0.62 => ( ! [I3: real] :
% 0.30/0.62 ( ( member_real @ I3 @ I5 )
% 0.30/0.62 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62 => ( ord_less_eq_real
% 0.30/0.62 @ ( abs_abs_real
% 0.30/0.62 @ ( minus_minus_real
% 0.30/0.62 @ ( groups8097168146408367636l_real
% 0.30/0.62 @ ^ [I4: real] : ( times_times_real @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 0.30/0.62 @ I5 )
% 0.30/0.62 @ B ) )
% 0.30/0.62 @ Delta ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % convex_sum_bound_le
% 0.30/0.62 thf(fact_7249_convex__sum__bound__le,axiom,
% 0.30/0.62 ! [I5: set_int,X2: int > real,A: int > real,B: real,Delta: real] :
% 0.30/0.62 ( ! [I3: int] :
% 0.30/0.62 ( ( member_int @ I3 @ I5 )
% 0.30/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I3 ) ) )
% 0.30/0.62 => ( ( ( groups8778361861064173332t_real @ X2 @ I5 )
% 0.30/0.62 = one_one_real )
% 0.30/0.62 => ( ! [I3: int] :
% 0.30/0.62 ( ( member_int @ I3 @ I5 )
% 0.30/0.62 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62 => ( ord_less_eq_real
% 0.30/0.62 @ ( abs_abs_real
% 0.30/0.62 @ ( minus_minus_real
% 0.30/0.62 @ ( groups8778361861064173332t_real
% 0.30/0.62 @ ^ [I4: int] : ( times_times_real @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 0.30/0.62 @ I5 )
% 0.30/0.62 @ B ) )
% 0.30/0.62 @ Delta ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % convex_sum_bound_le
% 0.30/0.62 thf(fact_7250_convex__sum__bound__le,axiom,
% 0.30/0.62 ! [I5: set_complex,X2: complex > rat,A: complex > rat,B: rat,Delta: rat] :
% 0.30/0.62 ( ! [I3: complex] :
% 0.30/0.62 ( ( member_complex @ I3 @ I5 )
% 0.30/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( X2 @ I3 ) ) )
% 0.30/0.62 => ( ( ( groups5058264527183730370ex_rat @ X2 @ I5 )
% 0.30/0.62 = one_one_rat )
% 0.30/0.62 => ( ! [I3: complex] :
% 0.30/0.62 ( ( member_complex @ I3 @ I5 )
% 0.30/0.62 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62 => ( ord_less_eq_rat
% 0.30/0.62 @ ( abs_abs_rat
% 0.30/0.62 @ ( minus_minus_rat
% 0.30/0.62 @ ( groups5058264527183730370ex_rat
% 0.30/0.62 @ ^ [I4: complex] : ( times_times_rat @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 0.30/0.62 @ I5 )
% 0.30/0.62 @ B ) )
% 0.30/0.62 @ Delta ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % convex_sum_bound_le
% 0.30/0.62 thf(fact_7251_convex__sum__bound__le,axiom,
% 0.30/0.62 ! [I5: set_real,X2: real > rat,A: real > rat,B: rat,Delta: rat] :
% 0.30/0.62 ( ! [I3: real] :
% 0.30/0.62 ( ( member_real @ I3 @ I5 )
% 0.30/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( X2 @ I3 ) ) )
% 0.30/0.62 => ( ( ( groups1300246762558778688al_rat @ X2 @ I5 )
% 0.30/0.62 = one_one_rat )
% 0.30/0.62 => ( ! [I3: real] :
% 0.30/0.62 ( ( member_real @ I3 @ I5 )
% 0.30/0.62 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62 => ( ord_less_eq_rat
% 0.30/0.62 @ ( abs_abs_rat
% 0.30/0.62 @ ( minus_minus_rat
% 0.30/0.62 @ ( groups1300246762558778688al_rat
% 0.30/0.62 @ ^ [I4: real] : ( times_times_rat @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 0.30/0.62 @ I5 )
% 0.30/0.62 @ B ) )
% 0.30/0.62 @ Delta ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % convex_sum_bound_le
% 0.30/0.62 thf(fact_7252_convex__sum__bound__le,axiom,
% 0.30/0.62 ! [I5: set_nat,X2: nat > rat,A: nat > rat,B: rat,Delta: rat] :
% 0.30/0.62 ( ! [I3: nat] :
% 0.30/0.62 ( ( member_nat @ I3 @ I5 )
% 0.30/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( X2 @ I3 ) ) )
% 0.30/0.62 => ( ( ( groups2906978787729119204at_rat @ X2 @ I5 )
% 0.30/0.62 = one_one_rat )
% 0.30/0.62 => ( ! [I3: nat] :
% 0.30/0.62 ( ( member_nat @ I3 @ I5 )
% 0.30/0.62 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62 => ( ord_less_eq_rat
% 0.30/0.62 @ ( abs_abs_rat
% 0.30/0.62 @ ( minus_minus_rat
% 0.30/0.62 @ ( groups2906978787729119204at_rat
% 0.30/0.62 @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 0.30/0.62 @ I5 )
% 0.30/0.62 @ B ) )
% 0.30/0.62 @ Delta ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % convex_sum_bound_le
% 0.30/0.62 thf(fact_7253_sum__natinterval__diff,axiom,
% 0.30/0.62 ! [M: nat,N: nat,F: nat > complex] :
% 0.30/0.62 ( ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( groups2073611262835488442omplex
% 0.30/0.62 @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 0.30/0.62 & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( groups2073611262835488442omplex
% 0.30/0.62 @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = zero_zero_complex ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_natinterval_diff
% 0.30/0.62 thf(fact_7254_sum__natinterval__diff,axiom,
% 0.30/0.62 ! [M: nat,N: nat,F: nat > rat] :
% 0.30/0.62 ( ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( groups2906978787729119204at_rat
% 0.30/0.62 @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 0.30/0.62 & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( groups2906978787729119204at_rat
% 0.30/0.62 @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = zero_zero_rat ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_natinterval_diff
% 0.30/0.62 thf(fact_7255_sum__natinterval__diff,axiom,
% 0.30/0.62 ! [M: nat,N: nat,F: nat > int] :
% 0.30/0.62 ( ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( groups3539618377306564664at_int
% 0.30/0.62 @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 0.30/0.62 & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( groups3539618377306564664at_int
% 0.30/0.62 @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = zero_zero_int ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_natinterval_diff
% 0.30/0.62 thf(fact_7256_sum__natinterval__diff,axiom,
% 0.30/0.62 ! [M: nat,N: nat,F: nat > real] :
% 0.30/0.62 ( ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( groups6591440286371151544t_real
% 0.30/0.62 @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 0.30/0.62 & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( groups6591440286371151544t_real
% 0.30/0.62 @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = zero_zero_real ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_natinterval_diff
% 0.30/0.62 thf(fact_7257_sum__telescope_H_H,axiom,
% 0.30/0.62 ! [M: nat,N: nat,F: nat > rat] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( groups2906978787729119204at_rat
% 0.30/0.62 @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.30/0.62 = ( minus_minus_rat @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_telescope''
% 0.30/0.62 thf(fact_7258_sum__telescope_H_H,axiom,
% 0.30/0.62 ! [M: nat,N: nat,F: nat > int] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( groups3539618377306564664at_int
% 0.30/0.62 @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.30/0.62 = ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_telescope''
% 0.30/0.62 thf(fact_7259_sum__telescope_H_H,axiom,
% 0.30/0.62 ! [M: nat,N: nat,F: nat > real] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( groups6591440286371151544t_real
% 0.30/0.62 @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.30/0.62 = ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_telescope''
% 0.30/0.62 thf(fact_7260_mask__eq__sum__exp,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
% 0.30/0.62 = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.62 @ ( collect_nat
% 0.30/0.62 @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % mask_eq_sum_exp
% 0.30/0.62 thf(fact_7261_mask__eq__sum__exp,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
% 0.30/0.62 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.62 @ ( collect_nat
% 0.30/0.62 @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % mask_eq_sum_exp
% 0.30/0.62 thf(fact_7262_sum__gp__multiplied,axiom,
% 0.30/0.62 ! [M: nat,N: nat,X2: complex] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.30/0.62 = ( minus_minus_complex @ ( power_power_complex @ X2 @ M ) @ ( power_power_complex @ X2 @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_gp_multiplied
% 0.30/0.62 thf(fact_7263_sum__gp__multiplied,axiom,
% 0.30/0.62 ! [M: nat,N: nat,X2: rat] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.30/0.62 = ( minus_minus_rat @ ( power_power_rat @ X2 @ M ) @ ( power_power_rat @ X2 @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_gp_multiplied
% 0.30/0.62 thf(fact_7264_sum__gp__multiplied,axiom,
% 0.30/0.62 ! [M: nat,N: nat,X2: int] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.30/0.62 = ( minus_minus_int @ ( power_power_int @ X2 @ M ) @ ( power_power_int @ X2 @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_gp_multiplied
% 0.30/0.62 thf(fact_7265_sum__gp__multiplied,axiom,
% 0.30/0.62 ! [M: nat,N: nat,X2: real] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.30/0.62 = ( minus_minus_real @ ( power_power_real @ X2 @ M ) @ ( power_power_real @ X2 @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_gp_multiplied
% 0.30/0.62 thf(fact_7266_sum_Oin__pairs,axiom,
% 0.30/0.62 ! [G: nat > rat,M: nat,N: nat] :
% 0.30/0.62 ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.62 = ( groups2906978787729119204at_rat
% 0.30/0.62 @ ^ [I4: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.in_pairs
% 0.30/0.62 thf(fact_7267_sum_Oin__pairs,axiom,
% 0.30/0.62 ! [G: nat > int,M: nat,N: nat] :
% 0.30/0.62 ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.62 = ( groups3539618377306564664at_int
% 0.30/0.62 @ ^ [I4: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.in_pairs
% 0.30/0.62 thf(fact_7268_sum_Oin__pairs,axiom,
% 0.30/0.62 ! [G: nat > nat,M: nat,N: nat] :
% 0.30/0.62 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.62 = ( groups3542108847815614940at_nat
% 0.30/0.62 @ ^ [I4: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.in_pairs
% 0.30/0.62 thf(fact_7269_sum_Oin__pairs,axiom,
% 0.30/0.62 ! [G: nat > real,M: nat,N: nat] :
% 0.30/0.62 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.62 = ( groups6591440286371151544t_real
% 0.30/0.62 @ ^ [I4: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum.in_pairs
% 0.30/0.62 thf(fact_7270_subrelI,axiom,
% 0.30/0.62 ! [R2: set_Pr448751882837621926eger_o,S: set_Pr448751882837621926eger_o] :
% 0.30/0.62 ( ! [X5: code_integer,Y4: $o] :
% 0.30/0.62 ( ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X5 @ Y4 ) @ R2 )
% 0.30/0.62 => ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X5 @ Y4 ) @ S ) )
% 0.30/0.62 => ( ord_le8980329558974975238eger_o @ R2 @ S ) ) ).
% 0.30/0.62
% 0.30/0.62 % subrelI
% 0.30/0.62 thf(fact_7271_subrelI,axiom,
% 0.30/0.62 ! [R2: set_Pr8218934625190621173um_num,S: set_Pr8218934625190621173um_num] :
% 0.30/0.62 ( ! [X5: num,Y4: num] :
% 0.30/0.62 ( ( member7279096912039735102um_num @ ( product_Pair_num_num @ X5 @ Y4 ) @ R2 )
% 0.30/0.62 => ( member7279096912039735102um_num @ ( product_Pair_num_num @ X5 @ Y4 ) @ S ) )
% 0.30/0.62 => ( ord_le880128212290418581um_num @ R2 @ S ) ) ).
% 0.30/0.62
% 0.30/0.62 % subrelI
% 0.30/0.62 thf(fact_7272_subrelI,axiom,
% 0.30/0.62 ! [R2: set_Pr6200539531224447659at_num,S: set_Pr6200539531224447659at_num] :
% 0.30/0.62 ( ! [X5: nat,Y4: num] :
% 0.30/0.62 ( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X5 @ Y4 ) @ R2 )
% 0.30/0.62 => ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X5 @ Y4 ) @ S ) )
% 0.30/0.62 => ( ord_le8085105155179020875at_num @ R2 @ S ) ) ).
% 0.30/0.62
% 0.30/0.62 % subrelI
% 0.30/0.62 thf(fact_7273_subrelI,axiom,
% 0.30/0.62 ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
% 0.30/0.62 ( ! [X5: nat,Y4: nat] :
% 0.30/0.62 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ R2 )
% 0.30/0.62 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ S ) )
% 0.30/0.62 => ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).
% 0.30/0.62
% 0.30/0.62 % subrelI
% 0.30/0.62 thf(fact_7274_subrelI,axiom,
% 0.30/0.62 ! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
% 0.30/0.62 ( ! [X5: int,Y4: int] :
% 0.30/0.62 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ R2 )
% 0.30/0.62 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ S ) )
% 0.30/0.62 => ( ord_le2843351958646193337nt_int @ R2 @ S ) ) ).
% 0.30/0.62
% 0.30/0.62 % subrelI
% 0.30/0.62 thf(fact_7275_accp__subset__induct,axiom,
% 0.30/0.62 ! [D4: product_prod_num_num > $o,R: product_prod_num_num > product_prod_num_num > $o,X2: product_prod_num_num,P: product_prod_num_num > $o] :
% 0.30/0.62 ( ( ord_le2239182809043710856_num_o @ D4 @ ( accp_P3113834385874906142um_num @ R ) )
% 0.30/0.62 => ( ! [X5: product_prod_num_num,Z2: product_prod_num_num] :
% 0.30/0.62 ( ( D4 @ X5 )
% 0.30/0.62 => ( ( R @ Z2 @ X5 )
% 0.30/0.62 => ( D4 @ Z2 ) ) )
% 0.30/0.62 => ( ( D4 @ X2 )
% 0.30/0.62 => ( ! [X5: product_prod_num_num] :
% 0.30/0.62 ( ( D4 @ X5 )
% 0.30/0.62 => ( ! [Z4: product_prod_num_num] :
% 0.30/0.62 ( ( R @ Z4 @ X5 )
% 0.30/0.62 => ( P @ Z4 ) )
% 0.30/0.62 => ( P @ X5 ) ) )
% 0.30/0.62 => ( P @ X2 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % accp_subset_induct
% 0.30/0.62 thf(fact_7276_accp__subset__induct,axiom,
% 0.30/0.62 ! [D4: product_prod_nat_nat > $o,R: product_prod_nat_nat > product_prod_nat_nat > $o,X2: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
% 0.30/0.62 ( ( ord_le704812498762024988_nat_o @ D4 @ ( accp_P4275260045618599050at_nat @ R ) )
% 0.30/0.62 => ( ! [X5: product_prod_nat_nat,Z2: product_prod_nat_nat] :
% 0.30/0.62 ( ( D4 @ X5 )
% 0.30/0.62 => ( ( R @ Z2 @ X5 )
% 0.30/0.62 => ( D4 @ Z2 ) ) )
% 0.30/0.62 => ( ( D4 @ X2 )
% 0.30/0.62 => ( ! [X5: product_prod_nat_nat] :
% 0.30/0.62 ( ( D4 @ X5 )
% 0.30/0.62 => ( ! [Z4: product_prod_nat_nat] :
% 0.30/0.62 ( ( R @ Z4 @ X5 )
% 0.30/0.62 => ( P @ Z4 ) )
% 0.30/0.62 => ( P @ X5 ) ) )
% 0.30/0.62 => ( P @ X2 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % accp_subset_induct
% 0.30/0.62 thf(fact_7277_accp__subset__induct,axiom,
% 0.30/0.62 ! [D4: product_prod_int_int > $o,R: product_prod_int_int > product_prod_int_int > $o,X2: product_prod_int_int,P: product_prod_int_int > $o] :
% 0.30/0.62 ( ( ord_le8369615600986905444_int_o @ D4 @ ( accp_P1096762738010456898nt_int @ R ) )
% 0.30/0.62 => ( ! [X5: product_prod_int_int,Z2: product_prod_int_int] :
% 0.30/0.62 ( ( D4 @ X5 )
% 0.30/0.62 => ( ( R @ Z2 @ X5 )
% 0.30/0.62 => ( D4 @ Z2 ) ) )
% 0.30/0.62 => ( ( D4 @ X2 )
% 0.30/0.62 => ( ! [X5: product_prod_int_int] :
% 0.30/0.62 ( ( D4 @ X5 )
% 0.30/0.62 => ( ! [Z4: product_prod_int_int] :
% 0.30/0.62 ( ( R @ Z4 @ X5 )
% 0.30/0.62 => ( P @ Z4 ) )
% 0.30/0.62 => ( P @ X5 ) ) )
% 0.30/0.62 => ( P @ X2 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % accp_subset_induct
% 0.30/0.62 thf(fact_7278_accp__subset__induct,axiom,
% 0.30/0.62 ! [D4: list_nat > $o,R: list_nat > list_nat > $o,X2: list_nat,P: list_nat > $o] :
% 0.30/0.62 ( ( ord_le1520216061033275535_nat_o @ D4 @ ( accp_list_nat @ R ) )
% 0.30/0.62 => ( ! [X5: list_nat,Z2: list_nat] :
% 0.30/0.62 ( ( D4 @ X5 )
% 0.30/0.62 => ( ( R @ Z2 @ X5 )
% 0.30/0.62 => ( D4 @ Z2 ) ) )
% 0.30/0.62 => ( ( D4 @ X2 )
% 0.30/0.62 => ( ! [X5: list_nat] :
% 0.30/0.62 ( ( D4 @ X5 )
% 0.30/0.62 => ( ! [Z4: list_nat] :
% 0.30/0.62 ( ( R @ Z4 @ X5 )
% 0.30/0.62 => ( P @ Z4 ) )
% 0.30/0.62 => ( P @ X5 ) ) )
% 0.30/0.62 => ( P @ X2 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % accp_subset_induct
% 0.30/0.62 thf(fact_7279_accp__subset__induct,axiom,
% 0.30/0.62 ! [D4: nat > $o,R: nat > nat > $o,X2: nat,P: nat > $o] :
% 0.30/0.62 ( ( ord_less_eq_nat_o @ D4 @ ( accp_nat @ R ) )
% 0.30/0.62 => ( ! [X5: nat,Z2: nat] :
% 0.30/0.62 ( ( D4 @ X5 )
% 0.30/0.62 => ( ( R @ Z2 @ X5 )
% 0.30/0.62 => ( D4 @ Z2 ) ) )
% 0.30/0.62 => ( ( D4 @ X2 )
% 0.30/0.62 => ( ! [X5: nat] :
% 0.30/0.62 ( ( D4 @ X5 )
% 0.30/0.62 => ( ! [Z4: nat] :
% 0.30/0.62 ( ( R @ Z4 @ X5 )
% 0.30/0.62 => ( P @ Z4 ) )
% 0.30/0.62 => ( P @ X5 ) ) )
% 0.30/0.62 => ( P @ X2 ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % accp_subset_induct
% 0.30/0.62 thf(fact_7280_mask__eq__sum__exp__nat,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.62 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.62 @ ( collect_nat
% 0.30/0.62 @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % mask_eq_sum_exp_nat
% 0.30/0.62 thf(fact_7281_gauss__sum__nat,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( groups3542108847815614940at_nat
% 0.30/0.62 @ ^ [X: nat] : X
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.62 = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % gauss_sum_nat
% 0.30/0.62 thf(fact_7282_arith__series__nat,axiom,
% 0.30/0.62 ! [A: nat,D: nat,N: nat] :
% 0.30/0.62 ( ( groups3542108847815614940at_nat
% 0.30/0.62 @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I4 @ D ) )
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.62 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % arith_series_nat
% 0.30/0.62 thf(fact_7283_Sum__Icc__nat,axiom,
% 0.30/0.62 ! [M: nat,N: nat] :
% 0.30/0.62 ( ( groups3542108847815614940at_nat
% 0.30/0.62 @ ^ [X: nat] : X
% 0.30/0.62 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % Sum_Icc_nat
% 0.30/0.62 thf(fact_7284_pred__subset__eq,axiom,
% 0.30/0.62 ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
% 0.30/0.62 ( ( ord_le704812498762024988_nat_o
% 0.30/0.62 @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ R )
% 0.30/0.62 @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ S3 ) )
% 0.30/0.62 = ( ord_le3146513528884898305at_nat @ R @ S3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % pred_subset_eq
% 0.30/0.62 thf(fact_7285_pred__subset__eq,axiom,
% 0.30/0.62 ! [R: set_complex,S3: set_complex] :
% 0.30/0.62 ( ( ord_le4573692005234683329plex_o
% 0.30/0.62 @ ^ [X: complex] : ( member_complex @ X @ R )
% 0.30/0.62 @ ^ [X: complex] : ( member_complex @ X @ S3 ) )
% 0.30/0.62 = ( ord_le211207098394363844omplex @ R @ S3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % pred_subset_eq
% 0.30/0.62 thf(fact_7286_pred__subset__eq,axiom,
% 0.30/0.62 ! [R: set_real,S3: set_real] :
% 0.30/0.62 ( ( ord_less_eq_real_o
% 0.30/0.62 @ ^ [X: real] : ( member_real @ X @ R )
% 0.30/0.62 @ ^ [X: real] : ( member_real @ X @ S3 ) )
% 0.30/0.62 = ( ord_less_eq_set_real @ R @ S3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % pred_subset_eq
% 0.30/0.62 thf(fact_7287_pred__subset__eq,axiom,
% 0.30/0.62 ! [R: set_nat,S3: set_nat] :
% 0.30/0.62 ( ( ord_less_eq_nat_o
% 0.30/0.62 @ ^ [X: nat] : ( member_nat @ X @ R )
% 0.30/0.62 @ ^ [X: nat] : ( member_nat @ X @ S3 ) )
% 0.30/0.62 = ( ord_less_eq_set_nat @ R @ S3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % pred_subset_eq
% 0.30/0.62 thf(fact_7288_pred__subset__eq,axiom,
% 0.30/0.62 ! [R: set_int,S3: set_int] :
% 0.30/0.62 ( ( ord_less_eq_int_o
% 0.30/0.62 @ ^ [X: int] : ( member_int @ X @ R )
% 0.30/0.62 @ ^ [X: int] : ( member_int @ X @ S3 ) )
% 0.30/0.62 = ( ord_less_eq_set_int @ R @ S3 ) ) ).
% 0.30/0.62
% 0.30/0.62 % pred_subset_eq
% 0.30/0.62 thf(fact_7289_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.62 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.30/0.62 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_ln_one_plus_x_minus_x_bound_nonneg
% 0.30/0.62 thf(fact_7290_of__int__round__le,axiom,
% 0.30/0.62 ! [X2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) @ ( plus_plus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_int_round_le
% 0.30/0.62 thf(fact_7291_of__int__round__le,axiom,
% 0.30/0.62 ! [X2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) @ ( plus_plus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_int_round_le
% 0.30/0.62 thf(fact_7292_of__int__round__ge,axiom,
% 0.30/0.62 ! [X2: real] : ( ord_less_eq_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_int_round_ge
% 0.30/0.62 thf(fact_7293_of__int__round__ge,axiom,
% 0.30/0.62 ! [X2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_int_round_ge
% 0.30/0.62 thf(fact_7294_of__int__round__gt,axiom,
% 0.30/0.62 ! [X2: real] : ( ord_less_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_int_round_gt
% 0.30/0.62 thf(fact_7295_of__int__round__gt,axiom,
% 0.30/0.62 ! [X2: rat] : ( ord_less_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_int_round_gt
% 0.30/0.62 thf(fact_7296_vebt__maxt_Opelims,axiom,
% 0.30/0.62 ! [X2: vEBT_VEBT,Y3: option_nat] :
% 0.30/0.62 ( ( ( vEBT_vebt_maxt @ X2 )
% 0.30/0.62 = Y3 )
% 0.30/0.62 => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X2 )
% 0.30/0.62 => ( ! [A4: $o,B4: $o] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.62 => ( ( ( B4
% 0.30/0.62 => ( Y3
% 0.30/0.62 = ( some_nat @ one_one_nat ) ) )
% 0.30/0.62 & ( ~ B4
% 0.30/0.62 => ( ( A4
% 0.30/0.62 => ( Y3
% 0.30/0.62 = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.62 & ( ~ A4
% 0.30/0.62 => ( Y3 = none_nat ) ) ) ) )
% 0.30/0.62 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
% 0.30/0.62 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.62 => ( ( Y3 = none_nat )
% 0.30/0.62 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 0.30/0.62 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.30/0.62 => ( ( Y3
% 0.30/0.62 = ( some_nat @ Ma2 ) )
% 0.30/0.62 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % vebt_maxt.pelims
% 0.30/0.62 thf(fact_7297_vebt__mint_Opelims,axiom,
% 0.30/0.62 ! [X2: vEBT_VEBT,Y3: option_nat] :
% 0.30/0.62 ( ( ( vEBT_vebt_mint @ X2 )
% 0.30/0.62 = Y3 )
% 0.30/0.62 => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X2 )
% 0.30/0.62 => ( ! [A4: $o,B4: $o] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.62 => ( ( ( A4
% 0.30/0.62 => ( Y3
% 0.30/0.62 = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.62 & ( ~ A4
% 0.30/0.62 => ( ( B4
% 0.30/0.62 => ( Y3
% 0.30/0.62 = ( some_nat @ one_one_nat ) ) )
% 0.30/0.62 & ( ~ B4
% 0.30/0.62 => ( Y3 = none_nat ) ) ) ) )
% 0.30/0.62 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
% 0.30/0.62 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.62 => ( ( Y3 = none_nat )
% 0.30/0.62 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 0.30/0.62 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.30/0.62 => ( ( Y3
% 0.30/0.62 = ( some_nat @ Mi2 ) )
% 0.30/0.62 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % vebt_mint.pelims
% 0.30/0.62 thf(fact_7298_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
% 0.30/0.62 ! [X2: vEBT_VEBT,Y3: nat] :
% 0.30/0.62 ( ( ( vEBT_T_m_i_n_t @ X2 )
% 0.30/0.62 = Y3 )
% 0.30/0.62 => ( ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ X2 )
% 0.30/0.62 => ( ! [A4: $o,B4: $o] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.62 => ( ( Y3
% 0.30/0.62 = ( plus_plus_nat @ one_one_nat @ ( if_nat @ A4 @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
% 0.30/0.62 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
% 0.30/0.62 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.62 => ( ( Y3 = one_one_nat )
% 0.30/0.62 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 0.30/0.62 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.30/0.62 => ( ( Y3 = one_one_nat )
% 0.30/0.62 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
% 0.30/0.62 thf(fact_7299_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
% 0.30/0.62 ! [X2: vEBT_VEBT,Y3: nat] :
% 0.30/0.62 ( ( ( vEBT_T_m_a_x_t @ X2 )
% 0.30/0.62 = Y3 )
% 0.30/0.62 => ( ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ X2 )
% 0.30/0.62 => ( ! [A4: $o,B4: $o] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.62 => ( ( Y3
% 0.30/0.62 = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B4 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
% 0.30/0.62 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
% 0.30/0.62 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.62 => ( ( Y3 = one_one_nat )
% 0.30/0.62 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 0.30/0.62 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.30/0.62 => ( ( Y3 = one_one_nat )
% 0.30/0.62 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
% 0.30/0.62 thf(fact_7300_fold__atLeastAtMost__nat_Opinduct,axiom,
% 0.30/0.62 ! [A0: nat > num > num,A12: nat,A23: nat,A32: num,P: ( nat > num > num ) > nat > nat > num > $o] :
% 0.30/0.62 ( ( accp_P4916641582247091100at_num @ set_fo256927282339908995el_num @ ( produc851828971589881931at_num @ A0 @ ( produc1195630363706982562at_num @ A12 @ ( product_Pair_nat_num @ A23 @ A32 ) ) ) )
% 0.30/0.62 => ( ! [F2: nat > num > num,A4: nat,B4: nat,Acc: num] :
% 0.30/0.62 ( ( accp_P4916641582247091100at_num @ set_fo256927282339908995el_num @ ( produc851828971589881931at_num @ F2 @ ( produc1195630363706982562at_num @ A4 @ ( product_Pair_nat_num @ B4 @ Acc ) ) ) )
% 0.30/0.62 => ( ( ~ ( ord_less_nat @ B4 @ A4 )
% 0.30/0.62 => ( P @ F2 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B4 @ ( F2 @ A4 @ Acc ) ) )
% 0.30/0.62 => ( P @ F2 @ A4 @ B4 @ Acc ) ) )
% 0.30/0.62 => ( P @ A0 @ A12 @ A23 @ A32 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % fold_atLeastAtMost_nat.pinduct
% 0.30/0.62 thf(fact_7301_fold__atLeastAtMost__nat_Opinduct,axiom,
% 0.30/0.62 ! [A0: nat > nat > nat,A12: nat,A23: nat,A32: nat,P: ( nat > nat > nat ) > nat > nat > nat > $o] :
% 0.30/0.62 ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ A0 @ ( produc487386426758144856at_nat @ A12 @ ( product_Pair_nat_nat @ A23 @ A32 ) ) ) )
% 0.30/0.62 => ( ! [F2: nat > nat > nat,A4: nat,B4: nat,Acc: nat] :
% 0.30/0.62 ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A4 @ ( product_Pair_nat_nat @ B4 @ Acc ) ) ) )
% 0.30/0.62 => ( ( ~ ( ord_less_nat @ B4 @ A4 )
% 0.30/0.62 => ( P @ F2 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B4 @ ( F2 @ A4 @ Acc ) ) )
% 0.30/0.62 => ( P @ F2 @ A4 @ B4 @ Acc ) ) )
% 0.30/0.62 => ( P @ A0 @ A12 @ A23 @ A32 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % fold_atLeastAtMost_nat.pinduct
% 0.30/0.62 thf(fact_7302_VEBT__internal_Ooption__shift_Opelims,axiom,
% 0.30/0.62 ! [X2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
% 0.30/0.62 ( ( ( vEBT_V1502963449132264192at_nat @ X2 @ Xa2 @ Xb )
% 0.30/0.62 = Y3 )
% 0.30/0.62 => ( ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ Xa2 @ Xb ) ) )
% 0.30/0.62 => ( ( ( Xa2 = none_P5556105721700978146at_nat )
% 0.30/0.62 => ( ( Y3 = none_P5556105721700978146at_nat )
% 0.30/0.62 => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Xb ) ) ) ) )
% 0.30/0.62 => ( ! [V2: product_prod_nat_nat] :
% 0.30/0.62 ( ( Xa2
% 0.30/0.62 = ( some_P7363390416028606310at_nat @ V2 ) )
% 0.30/0.62 => ( ( Xb = none_P5556105721700978146at_nat )
% 0.30/0.62 => ( ( Y3 = none_P5556105721700978146at_nat )
% 0.30/0.62 => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) ) ) ) )
% 0.30/0.62 => ~ ! [A4: product_prod_nat_nat] :
% 0.30/0.62 ( ( Xa2
% 0.30/0.62 = ( some_P7363390416028606310at_nat @ A4 ) )
% 0.30/0.62 => ! [B4: product_prod_nat_nat] :
% 0.30/0.62 ( ( Xb
% 0.30/0.62 = ( some_P7363390416028606310at_nat @ B4 ) )
% 0.30/0.62 => ( ( Y3
% 0.30/0.62 = ( some_P7363390416028606310at_nat @ ( X2 @ A4 @ B4 ) ) )
% 0.30/0.62 => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A4 ) @ ( some_P7363390416028606310at_nat @ B4 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % VEBT_internal.option_shift.pelims
% 0.30/0.62 thf(fact_7303_VEBT__internal_Ooption__shift_Opelims,axiom,
% 0.30/0.62 ! [X2: num > num > num,Xa2: option_num,Xb: option_num,Y3: option_num] :
% 0.30/0.62 ( ( ( vEBT_V819420779217536731ft_num @ X2 @ Xa2 @ Xb )
% 0.30/0.62 = Y3 )
% 0.30/0.62 => ( ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ Xa2 @ Xb ) ) )
% 0.30/0.62 => ( ( ( Xa2 = none_num )
% 0.30/0.62 => ( ( Y3 = none_num )
% 0.30/0.62 => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ none_num @ Xb ) ) ) ) )
% 0.30/0.62 => ( ! [V2: num] :
% 0.30/0.62 ( ( Xa2
% 0.30/0.62 = ( some_num @ V2 ) )
% 0.30/0.62 => ( ( Xb = none_num )
% 0.30/0.62 => ( ( Y3 = none_num )
% 0.30/0.62 => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) ) ) ) )
% 0.30/0.62 => ~ ! [A4: num] :
% 0.30/0.62 ( ( Xa2
% 0.30/0.62 = ( some_num @ A4 ) )
% 0.30/0.62 => ! [B4: num] :
% 0.30/0.62 ( ( Xb
% 0.30/0.62 = ( some_num @ B4 ) )
% 0.30/0.62 => ( ( Y3
% 0.30/0.62 = ( some_num @ ( X2 @ A4 @ B4 ) ) )
% 0.30/0.62 => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ ( some_num @ A4 ) @ ( some_num @ B4 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % VEBT_internal.option_shift.pelims
% 0.30/0.62 thf(fact_7304_VEBT__internal_Ooption__shift_Opelims,axiom,
% 0.30/0.62 ! [X2: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y3: option_nat] :
% 0.30/0.62 ( ( ( vEBT_V4262088993061758097ft_nat @ X2 @ Xa2 @ Xb )
% 0.30/0.62 = Y3 )
% 0.30/0.62 => ( ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ Xa2 @ Xb ) ) )
% 0.30/0.62 => ( ( ( Xa2 = none_nat )
% 0.30/0.62 => ( ( Y3 = none_nat )
% 0.30/0.62 => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ none_nat @ Xb ) ) ) ) )
% 0.30/0.62 => ( ! [V2: nat] :
% 0.30/0.62 ( ( Xa2
% 0.30/0.62 = ( some_nat @ V2 ) )
% 0.30/0.62 => ( ( Xb = none_nat )
% 0.30/0.62 => ( ( Y3 = none_nat )
% 0.30/0.62 => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) ) ) ) )
% 0.30/0.62 => ~ ! [A4: nat] :
% 0.30/0.62 ( ( Xa2
% 0.30/0.62 = ( some_nat @ A4 ) )
% 0.30/0.62 => ! [B4: nat] :
% 0.30/0.62 ( ( Xb
% 0.30/0.62 = ( some_nat @ B4 ) )
% 0.30/0.62 => ( ( Y3
% 0.30/0.62 = ( some_nat @ ( X2 @ A4 @ B4 ) ) )
% 0.30/0.62 => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A4 ) @ ( some_nat @ B4 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % VEBT_internal.option_shift.pelims
% 0.30/0.62 thf(fact_7305_zdvd1__eq,axiom,
% 0.30/0.62 ! [X2: int] :
% 0.30/0.62 ( ( dvd_dvd_int @ X2 @ one_one_int )
% 0.30/0.62 = ( ( abs_abs_int @ X2 )
% 0.30/0.62 = one_one_int ) ) ).
% 0.30/0.62
% 0.30/0.62 % zdvd1_eq
% 0.30/0.62 thf(fact_7306_zabs__less__one__iff,axiom,
% 0.30/0.62 ! [Z: int] :
% 0.30/0.62 ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 0.30/0.62 = ( Z = zero_zero_int ) ) ).
% 0.30/0.62
% 0.30/0.62 % zabs_less_one_iff
% 0.30/0.62 thf(fact_7307_abs__zmult__eq__1,axiom,
% 0.30/0.62 ! [M: int,N: int] :
% 0.30/0.62 ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
% 0.30/0.62 = one_one_int )
% 0.30/0.62 => ( ( abs_abs_int @ M )
% 0.30/0.62 = one_one_int ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_zmult_eq_1
% 0.30/0.62 thf(fact_7308_dvd__imp__le__int,axiom,
% 0.30/0.62 ! [I2: int,D: int] :
% 0.30/0.62 ( ( I2 != zero_zero_int )
% 0.30/0.62 => ( ( dvd_dvd_int @ D @ I2 )
% 0.30/0.62 => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I2 ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % dvd_imp_le_int
% 0.30/0.62 thf(fact_7309_abs__mod__less,axiom,
% 0.30/0.62 ! [L: int,K: int] :
% 0.30/0.62 ( ( L != zero_zero_int )
% 0.30/0.62 => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_mod_less
% 0.30/0.62 thf(fact_7310_zdvd__mult__cancel1,axiom,
% 0.30/0.62 ! [M: int,N: int] :
% 0.30/0.62 ( ( M != zero_zero_int )
% 0.30/0.62 => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
% 0.30/0.62 = ( ( abs_abs_int @ N )
% 0.30/0.62 = one_one_int ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % zdvd_mult_cancel1
% 0.30/0.62 thf(fact_7311_even__add__abs__iff,axiom,
% 0.30/0.62 ! [K: int,L: int] :
% 0.30/0.62 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L ) ) )
% 0.30/0.62 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % even_add_abs_iff
% 0.30/0.62 thf(fact_7312_even__abs__add__iff,axiom,
% 0.30/0.62 ! [K: int,L: int] :
% 0.30/0.62 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L ) )
% 0.30/0.62 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % even_abs_add_iff
% 0.30/0.62 thf(fact_7313_nat__intermed__int__val,axiom,
% 0.30/0.62 ! [M: nat,N: nat,F: nat > int,K: int] :
% 0.30/0.62 ( ! [I3: nat] :
% 0.30/0.62 ( ( ( ord_less_eq_nat @ M @ I3 )
% 0.30/0.62 & ( ord_less_nat @ I3 @ N ) )
% 0.30/0.62 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 0.30/0.62 => ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.62 => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 0.30/0.62 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.30/0.62 => ? [I3: nat] :
% 0.30/0.62 ( ( ord_less_eq_nat @ M @ I3 )
% 0.30/0.62 & ( ord_less_eq_nat @ I3 @ N )
% 0.30/0.62 & ( ( F @ I3 )
% 0.30/0.62 = K ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % nat_intermed_int_val
% 0.30/0.62 thf(fact_7314_decr__lemma,axiom,
% 0.30/0.62 ! [D: int,X2: int,Z: int] :
% 0.30/0.62 ( ( ord_less_int @ zero_zero_int @ D )
% 0.30/0.62 => ( ord_less_int @ ( minus_minus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% 0.30/0.62
% 0.30/0.62 % decr_lemma
% 0.30/0.62 thf(fact_7315_incr__lemma,axiom,
% 0.30/0.62 ! [D: int,Z: int,X2: int] :
% 0.30/0.62 ( ( ord_less_int @ zero_zero_int @ D )
% 0.30/0.62 => ( ord_less_int @ Z @ ( plus_plus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % incr_lemma
% 0.30/0.62 thf(fact_7316_nat__ivt__aux,axiom,
% 0.30/0.62 ! [N: nat,F: nat > int,K: int] :
% 0.30/0.62 ( ! [I3: nat] :
% 0.30/0.62 ( ( ord_less_nat @ I3 @ N )
% 0.30/0.62 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 0.30/0.62 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.30/0.62 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.30/0.62 => ? [I3: nat] :
% 0.30/0.62 ( ( ord_less_eq_nat @ I3 @ N )
% 0.30/0.62 & ( ( F @ I3 )
% 0.30/0.62 = K ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % nat_ivt_aux
% 0.30/0.62 thf(fact_7317_nat0__intermed__int__val,axiom,
% 0.30/0.62 ! [N: nat,F: nat > int,K: int] :
% 0.30/0.62 ( ! [I3: nat] :
% 0.30/0.62 ( ( ord_less_nat @ I3 @ N )
% 0.30/0.62 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 0.30/0.62 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.30/0.62 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.30/0.62 => ? [I3: nat] :
% 0.30/0.62 ( ( ord_less_eq_nat @ I3 @ N )
% 0.30/0.62 & ( ( F @ I3 )
% 0.30/0.62 = K ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % nat0_intermed_int_val
% 0.30/0.62 thf(fact_7318_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
% 0.30/0.62 ! [X2: vEBT_VEBT,Y3: nat] :
% 0.30/0.62 ( ( ( vEBT_T_m_i_n_N_u_l_l @ X2 )
% 0.30/0.62 = Y3 )
% 0.30/0.62 => ( ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ X2 )
% 0.30/0.62 => ( ( ( X2
% 0.30/0.62 = ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.62 => ( ( Y3 = one_one_nat )
% 0.30/0.62 => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
% 0.30/0.62 => ( ! [Uv2: $o] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.30/0.62 => ( ( Y3 = one_one_nat )
% 0.30/0.62 => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
% 0.30/0.62 => ( ! [Uu2: $o] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.30/0.62 => ( ( Y3 = one_one_nat )
% 0.30/0.62 => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
% 0.30/0.62 => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.30/0.62 => ( ( Y3 = one_one_nat )
% 0.30/0.62 => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
% 0.30/0.62 => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.62 ( ( X2
% 0.30/0.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
% 0.30/0.62 => ( ( Y3 = one_one_nat )
% 0.30/0.62 => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
% 0.30/0.62 thf(fact_7319_fold__atLeastAtMost__nat_Opelims,axiom,
% 0.30/0.62 ! [X2: nat > num > num,Xa2: nat,Xb: nat,Xc: num,Y3: num] :
% 0.30/0.62 ( ( ( set_fo8365102181078989356at_num @ X2 @ Xa2 @ Xb @ Xc )
% 0.30/0.62 = Y3 )
% 0.30/0.62 => ( ( accp_P4916641582247091100at_num @ set_fo256927282339908995el_num @ ( produc851828971589881931at_num @ X2 @ ( produc1195630363706982562at_num @ Xa2 @ ( product_Pair_nat_num @ Xb @ Xc ) ) ) )
% 0.30/0.62 => ~ ( ( ( ( ord_less_nat @ Xb @ Xa2 )
% 0.30/0.62 => ( Y3 = Xc ) )
% 0.30/0.62 & ( ~ ( ord_less_nat @ Xb @ Xa2 )
% 0.30/0.62 => ( Y3
% 0.30/0.62 = ( set_fo8365102181078989356at_num @ X2 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X2 @ Xa2 @ Xc ) ) ) ) )
% 0.30/0.62 => ~ ( accp_P4916641582247091100at_num @ set_fo256927282339908995el_num @ ( produc851828971589881931at_num @ X2 @ ( produc1195630363706982562at_num @ Xa2 @ ( product_Pair_nat_num @ Xb @ Xc ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % fold_atLeastAtMost_nat.pelims
% 0.30/0.62 thf(fact_7320_fold__atLeastAtMost__nat_Opelims,axiom,
% 0.30/0.62 ! [X2: nat > nat > nat,Xa2: nat,Xb: nat,Xc: nat,Y3: nat] :
% 0.30/0.62 ( ( ( set_fo2584398358068434914at_nat @ X2 @ Xa2 @ Xb @ Xc )
% 0.30/0.62 = Y3 )
% 0.30/0.62 => ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X2 @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) )
% 0.30/0.62 => ~ ( ( ( ( ord_less_nat @ Xb @ Xa2 )
% 0.30/0.62 => ( Y3 = Xc ) )
% 0.30/0.62 & ( ~ ( ord_less_nat @ Xb @ Xa2 )
% 0.30/0.62 => ( Y3
% 0.30/0.62 = ( set_fo2584398358068434914at_nat @ X2 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X2 @ Xa2 @ Xc ) ) ) ) )
% 0.30/0.62 => ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X2 @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % fold_atLeastAtMost_nat.pelims
% 0.30/0.62 thf(fact_7321_fold__atLeastAtMost__nat_Opsimps,axiom,
% 0.30/0.62 ! [F: nat > num > num,A: nat,B: nat,Acc2: num] :
% 0.30/0.62 ( ( accp_P4916641582247091100at_num @ set_fo256927282339908995el_num @ ( produc851828971589881931at_num @ F @ ( produc1195630363706982562at_num @ A @ ( product_Pair_nat_num @ B @ Acc2 ) ) ) )
% 0.30/0.62 => ( ( ( ord_less_nat @ B @ A )
% 0.30/0.62 => ( ( set_fo8365102181078989356at_num @ F @ A @ B @ Acc2 )
% 0.30/0.62 = Acc2 ) )
% 0.30/0.62 & ( ~ ( ord_less_nat @ B @ A )
% 0.30/0.62 => ( ( set_fo8365102181078989356at_num @ F @ A @ B @ Acc2 )
% 0.30/0.62 = ( set_fo8365102181078989356at_num @ F @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F @ A @ Acc2 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % fold_atLeastAtMost_nat.psimps
% 0.30/0.62 thf(fact_7322_fold__atLeastAtMost__nat_Opsimps,axiom,
% 0.30/0.62 ! [F: nat > nat > nat,A: nat,B: nat,Acc2: nat] :
% 0.30/0.62 ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc2 ) ) ) )
% 0.30/0.62 => ( ( ( ord_less_nat @ B @ A )
% 0.30/0.62 => ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc2 )
% 0.30/0.62 = Acc2 ) )
% 0.30/0.62 & ( ~ ( ord_less_nat @ B @ A )
% 0.30/0.62 => ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc2 )
% 0.30/0.62 = ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F @ A @ Acc2 ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % fold_atLeastAtMost_nat.psimps
% 0.30/0.62 thf(fact_7323_in__measure,axiom,
% 0.30/0.62 ! [X2: num,Y3: num,F: num > nat] :
% 0.30/0.62 ( ( member7279096912039735102um_num @ ( product_Pair_num_num @ X2 @ Y3 ) @ ( measure_num @ F ) )
% 0.30/0.62 = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % in_measure
% 0.30/0.62 thf(fact_7324_in__measure,axiom,
% 0.30/0.62 ! [X2: nat,Y3: nat,F: nat > nat] :
% 0.30/0.62 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( measure_nat @ F ) )
% 0.30/0.62 = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % in_measure
% 0.30/0.62 thf(fact_7325_in__measure,axiom,
% 0.30/0.62 ! [X2: int,Y3: int,F: int > nat] :
% 0.30/0.62 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ ( measure_int @ F ) )
% 0.30/0.62 = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % in_measure
% 0.30/0.62 thf(fact_7326_sum__gp,axiom,
% 0.30/0.62 ! [N: nat,M: nat,X2: complex] :
% 0.30/0.62 ( ( ( ord_less_nat @ N @ M )
% 0.30/0.62 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = zero_zero_complex ) )
% 0.30/0.62 & ( ~ ( ord_less_nat @ N @ M )
% 0.30/0.62 => ( ( ( X2 = one_one_complex )
% 0.30/0.62 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 0.30/0.62 & ( ( X2 != one_one_complex )
% 0.30/0.62 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X2 @ M ) @ ( power_power_complex @ X2 @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_gp
% 0.30/0.62 thf(fact_7327_sum__gp,axiom,
% 0.30/0.62 ! [N: nat,M: nat,X2: rat] :
% 0.30/0.62 ( ( ( ord_less_nat @ N @ M )
% 0.30/0.62 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = zero_zero_rat ) )
% 0.30/0.62 & ( ~ ( ord_less_nat @ N @ M )
% 0.30/0.62 => ( ( ( X2 = one_one_rat )
% 0.30/0.62 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 0.30/0.62 & ( ( X2 != one_one_rat )
% 0.30/0.62 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X2 @ M ) @ ( power_power_rat @ X2 @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_gp
% 0.30/0.62 thf(fact_7328_sum__gp,axiom,
% 0.30/0.62 ! [N: nat,M: nat,X2: real] :
% 0.30/0.62 ( ( ( ord_less_nat @ N @ M )
% 0.30/0.62 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = zero_zero_real ) )
% 0.30/0.62 & ( ~ ( ord_less_nat @ N @ M )
% 0.30/0.62 => ( ( ( X2 = one_one_real )
% 0.30/0.62 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 0.30/0.62 & ( ( X2 != one_one_real )
% 0.30/0.62 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.62 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ M ) @ ( power_power_real @ X2 @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % sum_gp
% 0.30/0.62 thf(fact_7329_arctan__double,axiom,
% 0.30/0.62 ! [X2: real] :
% 0.30/0.62 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.30/0.62 => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X2 ) )
% 0.30/0.62 = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % arctan_double
% 0.30/0.62 thf(fact_7330_of__nat__eq__iff,axiom,
% 0.30/0.62 ! [M: nat,N: nat] :
% 0.30/0.62 ( ( ( semiri1314217659103216013at_int @ M )
% 0.30/0.62 = ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.62 = ( M = N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_eq_iff
% 0.30/0.62 thf(fact_7331_of__nat__eq__iff,axiom,
% 0.30/0.62 ! [M: nat,N: nat] :
% 0.30/0.62 ( ( ( semiri5074537144036343181t_real @ M )
% 0.30/0.62 = ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.62 = ( M = N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_eq_iff
% 0.30/0.62 thf(fact_7332_of__nat__eq__iff,axiom,
% 0.30/0.62 ! [M: nat,N: nat] :
% 0.30/0.62 ( ( ( semiri1316708129612266289at_nat @ M )
% 0.30/0.62 = ( semiri1316708129612266289at_nat @ N ) )
% 0.30/0.62 = ( M = N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_eq_iff
% 0.30/0.62 thf(fact_7333_of__nat__eq__iff,axiom,
% 0.30/0.62 ! [M: nat,N: nat] :
% 0.30/0.62 ( ( ( semiri681578069525770553at_rat @ M )
% 0.30/0.62 = ( semiri681578069525770553at_rat @ N ) )
% 0.30/0.62 = ( M = N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_eq_iff
% 0.30/0.62 thf(fact_7334_int__eq__iff__numeral,axiom,
% 0.30/0.62 ! [M: nat,V: num] :
% 0.30/0.62 ( ( ( semiri1314217659103216013at_int @ M )
% 0.30/0.62 = ( numeral_numeral_int @ V ) )
% 0.30/0.62 = ( M
% 0.30/0.62 = ( numeral_numeral_nat @ V ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % int_eq_iff_numeral
% 0.30/0.62 thf(fact_7335_abs__of__nat,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 0.30/0.62 = ( semiri4939895301339042750nteger @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_nat
% 0.30/0.62 thf(fact_7336_abs__of__nat,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.62 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_nat
% 0.30/0.62 thf(fact_7337_abs__of__nat,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.62 = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_nat
% 0.30/0.62 thf(fact_7338_abs__of__nat,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.30/0.62 = ( semiri681578069525770553at_rat @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % abs_of_nat
% 0.30/0.62 thf(fact_7339_negative__zle,axiom,
% 0.30/0.62 ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 0.30/0.62
% 0.30/0.62 % negative_zle
% 0.30/0.62 thf(fact_7340_of__nat__eq__0__iff,axiom,
% 0.30/0.62 ! [M: nat] :
% 0.30/0.62 ( ( ( semiri8010041392384452111omplex @ M )
% 0.30/0.62 = zero_zero_complex )
% 0.30/0.62 = ( M = zero_zero_nat ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_eq_0_iff
% 0.30/0.62 thf(fact_7341_of__nat__eq__0__iff,axiom,
% 0.30/0.62 ! [M: nat] :
% 0.30/0.62 ( ( ( semiri1314217659103216013at_int @ M )
% 0.30/0.62 = zero_zero_int )
% 0.30/0.62 = ( M = zero_zero_nat ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_eq_0_iff
% 0.30/0.62 thf(fact_7342_of__nat__eq__0__iff,axiom,
% 0.30/0.62 ! [M: nat] :
% 0.30/0.62 ( ( ( semiri5074537144036343181t_real @ M )
% 0.30/0.62 = zero_zero_real )
% 0.30/0.62 = ( M = zero_zero_nat ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_eq_0_iff
% 0.30/0.62 thf(fact_7343_of__nat__eq__0__iff,axiom,
% 0.30/0.62 ! [M: nat] :
% 0.30/0.62 ( ( ( semiri1316708129612266289at_nat @ M )
% 0.30/0.62 = zero_zero_nat )
% 0.30/0.62 = ( M = zero_zero_nat ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_eq_0_iff
% 0.30/0.62 thf(fact_7344_of__nat__eq__0__iff,axiom,
% 0.30/0.62 ! [M: nat] :
% 0.30/0.62 ( ( ( semiri681578069525770553at_rat @ M )
% 0.30/0.62 = zero_zero_rat )
% 0.30/0.62 = ( M = zero_zero_nat ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_eq_0_iff
% 0.30/0.62 thf(fact_7345_of__nat__0__eq__iff,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( zero_zero_complex
% 0.30/0.62 = ( semiri8010041392384452111omplex @ N ) )
% 0.30/0.62 = ( zero_zero_nat = N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_0_eq_iff
% 0.30/0.62 thf(fact_7346_of__nat__0__eq__iff,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( zero_zero_int
% 0.30/0.62 = ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.62 = ( zero_zero_nat = N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_0_eq_iff
% 0.30/0.62 thf(fact_7347_of__nat__0__eq__iff,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( zero_zero_real
% 0.30/0.62 = ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.62 = ( zero_zero_nat = N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_0_eq_iff
% 0.30/0.62 thf(fact_7348_of__nat__0__eq__iff,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( zero_zero_nat
% 0.30/0.62 = ( semiri1316708129612266289at_nat @ N ) )
% 0.30/0.62 = ( zero_zero_nat = N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_0_eq_iff
% 0.30/0.62 thf(fact_7349_of__nat__0__eq__iff,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( zero_zero_rat
% 0.30/0.62 = ( semiri681578069525770553at_rat @ N ) )
% 0.30/0.62 = ( zero_zero_nat = N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_0_eq_iff
% 0.30/0.62 thf(fact_7350_of__nat__0,axiom,
% 0.30/0.62 ( ( semiri8010041392384452111omplex @ zero_zero_nat )
% 0.30/0.62 = zero_zero_complex ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_0
% 0.30/0.62 thf(fact_7351_of__nat__0,axiom,
% 0.30/0.62 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 0.30/0.62 = zero_zero_int ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_0
% 0.30/0.62 thf(fact_7352_of__nat__0,axiom,
% 0.30/0.62 ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 0.30/0.62 = zero_zero_real ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_0
% 0.30/0.62 thf(fact_7353_of__nat__0,axiom,
% 0.30/0.62 ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 0.30/0.62 = zero_zero_nat ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_0
% 0.30/0.62 thf(fact_7354_of__nat__0,axiom,
% 0.30/0.62 ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 0.30/0.62 = zero_zero_rat ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_0
% 0.30/0.62 thf(fact_7355_of__nat__less__iff,axiom,
% 0.30/0.62 ! [M: nat,N: nat] :
% 0.30/0.62 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.62 = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_less_iff
% 0.30/0.62 thf(fact_7356_of__nat__less__iff,axiom,
% 0.30/0.62 ! [M: nat,N: nat] :
% 0.30/0.62 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.62 = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_less_iff
% 0.30/0.62 thf(fact_7357_of__nat__less__iff,axiom,
% 0.30/0.62 ! [M: nat,N: nat] :
% 0.30/0.62 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.30/0.62 = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_less_iff
% 0.30/0.62 thf(fact_7358_of__nat__less__iff,axiom,
% 0.30/0.62 ! [M: nat,N: nat] :
% 0.30/0.62 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.30/0.62 = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_less_iff
% 0.30/0.62 thf(fact_7359_of__nat__numeral,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
% 0.30/0.62 = ( numera6690914467698888265omplex @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_numeral
% 0.30/0.62 thf(fact_7360_of__nat__numeral,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 0.30/0.62 = ( numeral_numeral_int @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_numeral
% 0.30/0.62 thf(fact_7361_of__nat__numeral,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
% 0.30/0.62 = ( numeral_numeral_real @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_numeral
% 0.30/0.62 thf(fact_7362_of__nat__numeral,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
% 0.30/0.62 = ( numeral_numeral_nat @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_numeral
% 0.30/0.62 thf(fact_7363_of__nat__numeral,axiom,
% 0.30/0.62 ! [N: num] :
% 0.30/0.62 ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
% 0.30/0.62 = ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_numeral
% 0.30/0.62 thf(fact_7364_of__nat__le__iff,axiom,
% 0.30/0.62 ! [M: nat,N: nat] :
% 0.30/0.62 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.62 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_le_iff
% 0.30/0.62 thf(fact_7365_of__nat__le__iff,axiom,
% 0.30/0.62 ! [M: nat,N: nat] :
% 0.30/0.62 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.30/0.62 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_le_iff
% 0.30/0.62 thf(fact_7366_of__nat__le__iff,axiom,
% 0.30/0.62 ! [M: nat,N: nat] :
% 0.30/0.62 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.30/0.62 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_le_iff
% 0.30/0.62 thf(fact_7367_of__nat__le__iff,axiom,
% 0.30/0.62 ! [M: nat,N: nat] :
% 0.30/0.62 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.62 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_le_iff
% 0.30/0.62 thf(fact_7368_of__nat__add,axiom,
% 0.30/0.62 ! [M: nat,N: nat] :
% 0.30/0.62 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.62 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_add
% 0.30/0.62 thf(fact_7369_of__nat__add,axiom,
% 0.30/0.62 ! [M: nat,N: nat] :
% 0.30/0.62 ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.62 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_add
% 0.30/0.62 thf(fact_7370_of__nat__add,axiom,
% 0.30/0.62 ! [M: nat,N: nat] :
% 0.30/0.62 ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.62 = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_add
% 0.30/0.62 thf(fact_7371_of__nat__add,axiom,
% 0.30/0.62 ! [M: nat,N: nat] :
% 0.30/0.62 ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.62 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_add
% 0.30/0.62 thf(fact_7372_of__nat__eq__1__iff,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( ( semiri8010041392384452111omplex @ N )
% 0.30/0.62 = one_one_complex )
% 0.30/0.62 = ( N = one_one_nat ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_eq_1_iff
% 0.30/0.62 thf(fact_7373_of__nat__eq__1__iff,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( ( semiri1314217659103216013at_int @ N )
% 0.30/0.62 = one_one_int )
% 0.30/0.62 = ( N = one_one_nat ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_eq_1_iff
% 0.30/0.62 thf(fact_7374_of__nat__eq__1__iff,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( ( semiri5074537144036343181t_real @ N )
% 0.30/0.62 = one_one_real )
% 0.30/0.62 = ( N = one_one_nat ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_eq_1_iff
% 0.30/0.62 thf(fact_7375_of__nat__eq__1__iff,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( ( semiri1316708129612266289at_nat @ N )
% 0.30/0.62 = one_one_nat )
% 0.30/0.62 = ( N = one_one_nat ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_eq_1_iff
% 0.30/0.62 thf(fact_7376_of__nat__eq__1__iff,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( ( semiri681578069525770553at_rat @ N )
% 0.30/0.62 = one_one_rat )
% 0.30/0.62 = ( N = one_one_nat ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_eq_1_iff
% 0.30/0.62 thf(fact_7377_of__nat__1__eq__iff,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( one_one_complex
% 0.30/0.62 = ( semiri8010041392384452111omplex @ N ) )
% 0.30/0.62 = ( N = one_one_nat ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_1_eq_iff
% 0.30/0.62 thf(fact_7378_of__nat__1__eq__iff,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( one_one_int
% 0.30/0.62 = ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.62 = ( N = one_one_nat ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_1_eq_iff
% 0.30/0.62 thf(fact_7379_of__nat__1__eq__iff,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( one_one_real
% 0.30/0.62 = ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.62 = ( N = one_one_nat ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_1_eq_iff
% 0.30/0.62 thf(fact_7380_of__nat__1__eq__iff,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( one_one_nat
% 0.30/0.62 = ( semiri1316708129612266289at_nat @ N ) )
% 0.30/0.62 = ( N = one_one_nat ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_1_eq_iff
% 0.30/0.62 thf(fact_7381_of__nat__1__eq__iff,axiom,
% 0.30/0.62 ! [N: nat] :
% 0.30/0.62 ( ( one_one_rat
% 0.30/0.62 = ( semiri681578069525770553at_rat @ N ) )
% 0.30/0.62 = ( N = one_one_nat ) ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_1_eq_iff
% 0.30/0.62 thf(fact_7382_of__nat__1,axiom,
% 0.30/0.62 ( ( semiri8010041392384452111omplex @ one_one_nat )
% 0.30/0.62 = one_one_complex ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_1
% 0.30/0.62 thf(fact_7383_of__nat__1,axiom,
% 0.30/0.62 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 0.30/0.62 = one_one_int ) ).
% 0.30/0.62
% 0.30/0.62 % of_nat_1
% 0.30/0.62 thf(fact_7384_of__nat__1,axiom,
% 0.30/0.62 ( ( semiri5074537144036343181t_real @ one_one_nat )
% 0.30/0.62 = one_one_real ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_1
% 0.30/0.63 thf(fact_7385_of__nat__1,axiom,
% 0.30/0.63 ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 0.30/0.63 = one_one_nat ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_1
% 0.30/0.63 thf(fact_7386_of__nat__1,axiom,
% 0.30/0.63 ( ( semiri681578069525770553at_rat @ one_one_nat )
% 0.30/0.63 = one_one_rat ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_1
% 0.30/0.63 thf(fact_7387_of__nat__mult,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
% 0.30/0.63 = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_mult
% 0.30/0.63 thf(fact_7388_of__nat__mult,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
% 0.30/0.63 = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_mult
% 0.30/0.63 thf(fact_7389_of__nat__mult,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
% 0.30/0.63 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_mult
% 0.30/0.63 thf(fact_7390_of__nat__mult,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
% 0.30/0.63 = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_mult
% 0.30/0.63 thf(fact_7391_of__nat__power,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
% 0.30/0.63 = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_power
% 0.30/0.63 thf(fact_7392_of__nat__power,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
% 0.30/0.63 = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_power
% 0.30/0.63 thf(fact_7393_of__nat__power,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
% 0.30/0.63 = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_power
% 0.30/0.63 thf(fact_7394_of__nat__power,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
% 0.30/0.63 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_power
% 0.30/0.63 thf(fact_7395_of__nat__power,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N ) )
% 0.30/0.63 = ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_power
% 0.30/0.63 thf(fact_7396_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.30/0.63 ! [B: nat,W: nat,X2: nat] :
% 0.30/0.63 ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
% 0.30/0.63 = ( semiri8010041392384452111omplex @ X2 ) )
% 0.30/0.63 = ( ( power_power_nat @ B @ W )
% 0.30/0.63 = X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_eq_of_nat_power_cancel_iff
% 0.30/0.63 thf(fact_7397_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.30/0.63 ! [B: nat,W: nat,X2: nat] :
% 0.30/0.63 ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
% 0.30/0.63 = ( semiri1314217659103216013at_int @ X2 ) )
% 0.30/0.63 = ( ( power_power_nat @ B @ W )
% 0.30/0.63 = X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_eq_of_nat_power_cancel_iff
% 0.30/0.63 thf(fact_7398_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.30/0.63 ! [B: nat,W: nat,X2: nat] :
% 0.30/0.63 ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
% 0.30/0.63 = ( semiri5074537144036343181t_real @ X2 ) )
% 0.30/0.63 = ( ( power_power_nat @ B @ W )
% 0.30/0.63 = X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_eq_of_nat_power_cancel_iff
% 0.30/0.63 thf(fact_7399_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.30/0.63 ! [B: nat,W: nat,X2: nat] :
% 0.30/0.63 ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
% 0.30/0.63 = ( semiri1316708129612266289at_nat @ X2 ) )
% 0.30/0.63 = ( ( power_power_nat @ B @ W )
% 0.30/0.63 = X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_eq_of_nat_power_cancel_iff
% 0.30/0.63 thf(fact_7400_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.30/0.63 ! [B: nat,W: nat,X2: nat] :
% 0.30/0.63 ( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W )
% 0.30/0.63 = ( semiri681578069525770553at_rat @ X2 ) )
% 0.30/0.63 = ( ( power_power_nat @ B @ W )
% 0.30/0.63 = X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_eq_of_nat_power_cancel_iff
% 0.30/0.63 thf(fact_7401_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,B: nat,W: nat] :
% 0.30/0.63 ( ( ( semiri8010041392384452111omplex @ X2 )
% 0.30/0.63 = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
% 0.30/0.63 = ( X2
% 0.30/0.63 = ( power_power_nat @ B @ W ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_power_eq_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7402_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,B: nat,W: nat] :
% 0.30/0.63 ( ( ( semiri1314217659103216013at_int @ X2 )
% 0.30/0.63 = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 0.30/0.63 = ( X2
% 0.30/0.63 = ( power_power_nat @ B @ W ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_power_eq_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7403_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,B: nat,W: nat] :
% 0.30/0.63 ( ( ( semiri5074537144036343181t_real @ X2 )
% 0.30/0.63 = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 0.30/0.63 = ( X2
% 0.30/0.63 = ( power_power_nat @ B @ W ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_power_eq_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7404_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,B: nat,W: nat] :
% 0.30/0.63 ( ( ( semiri1316708129612266289at_nat @ X2 )
% 0.30/0.63 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 0.30/0.63 = ( X2
% 0.30/0.63 = ( power_power_nat @ B @ W ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_power_eq_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7405_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,B: nat,W: nat] :
% 0.30/0.63 ( ( ( semiri681578069525770553at_rat @ X2 )
% 0.30/0.63 = ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 0.30/0.63 = ( X2
% 0.30/0.63 = ( power_power_nat @ B @ W ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_power_eq_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7406_negative__zless,axiom,
% 0.30/0.63 ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 0.30/0.63
% 0.30/0.63 % negative_zless
% 0.30/0.63 thf(fact_7407_arctan__le__zero__iff,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( arctan @ X2 ) @ zero_zero_real )
% 0.30/0.63 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 0.30/0.63
% 0.30/0.63 % arctan_le_zero_iff
% 0.30/0.63 thf(fact_7408_zero__le__arctan__iff,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X2 ) )
% 0.30/0.63 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % zero_le_arctan_iff
% 0.30/0.63 thf(fact_7409_of__nat__of__bool,axiom,
% 0.30/0.63 ! [P: $o] :
% 0.30/0.63 ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.30/0.63 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_of_bool
% 0.30/0.63 thf(fact_7410_of__nat__of__bool,axiom,
% 0.30/0.63 ! [P: $o] :
% 0.30/0.63 ( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.30/0.63 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_of_bool
% 0.30/0.63 thf(fact_7411_of__nat__of__bool,axiom,
% 0.30/0.63 ! [P: $o] :
% 0.30/0.63 ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.30/0.63 = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_of_bool
% 0.30/0.63 thf(fact_7412_of__nat__of__bool,axiom,
% 0.30/0.63 ! [P: $o] :
% 0.30/0.63 ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.30/0.63 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_of_bool
% 0.30/0.63 thf(fact_7413_of__nat__of__bool,axiom,
% 0.30/0.63 ! [P: $o] :
% 0.30/0.63 ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.30/0.63 = ( zero_n356916108424825756nteger @ P ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_of_bool
% 0.30/0.63 thf(fact_7414_of__nat__le__0__iff,axiom,
% 0.30/0.63 ! [M: nat] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 0.30/0.63 = ( M = zero_zero_nat ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_le_0_iff
% 0.30/0.63 thf(fact_7415_of__nat__le__0__iff,axiom,
% 0.30/0.63 ! [M: nat] :
% 0.30/0.63 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 0.30/0.63 = ( M = zero_zero_nat ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_le_0_iff
% 0.30/0.63 thf(fact_7416_of__nat__le__0__iff,axiom,
% 0.30/0.63 ! [M: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 0.30/0.63 = ( M = zero_zero_nat ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_le_0_iff
% 0.30/0.63 thf(fact_7417_of__nat__le__0__iff,axiom,
% 0.30/0.63 ! [M: nat] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 0.30/0.63 = ( M = zero_zero_nat ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_le_0_iff
% 0.30/0.63 thf(fact_7418_of__nat__Suc,axiom,
% 0.30/0.63 ! [M: nat] :
% 0.30/0.63 ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 0.30/0.63 = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_Suc
% 0.30/0.63 thf(fact_7419_of__nat__Suc,axiom,
% 0.30/0.63 ! [M: nat] :
% 0.30/0.63 ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 0.30/0.63 = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_Suc
% 0.30/0.63 thf(fact_7420_of__nat__Suc,axiom,
% 0.30/0.63 ! [M: nat] :
% 0.30/0.63 ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 0.30/0.63 = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_Suc
% 0.30/0.63 thf(fact_7421_of__nat__Suc,axiom,
% 0.30/0.63 ! [M: nat] :
% 0.30/0.63 ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 0.30/0.63 = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_Suc
% 0.30/0.63 thf(fact_7422_of__nat__Suc,axiom,
% 0.30/0.63 ! [M: nat] :
% 0.30/0.63 ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 0.30/0.63 = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_Suc
% 0.30/0.63 thf(fact_7423_numeral__less__real__of__nat__iff,axiom,
% 0.30/0.63 ! [W: num,N: nat] :
% 0.30/0.63 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.63 = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % numeral_less_real_of_nat_iff
% 0.30/0.63 thf(fact_7424_real__of__nat__less__numeral__iff,axiom,
% 0.30/0.63 ! [N: nat,W: num] :
% 0.30/0.63 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
% 0.30/0.63 = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % real_of_nat_less_numeral_iff
% 0.30/0.63 thf(fact_7425_numeral__le__real__of__nat__iff,axiom,
% 0.30/0.63 ! [N: num,M: nat] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
% 0.30/0.63 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 0.30/0.63
% 0.30/0.63 % numeral_le_real_of_nat_iff
% 0.30/0.63 thf(fact_7426_of__nat__0__less__iff,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.63 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_0_less_iff
% 0.30/0.63 thf(fact_7427_of__nat__0__less__iff,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.63 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_0_less_iff
% 0.30/0.63 thf(fact_7428_of__nat__0__less__iff,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
% 0.30/0.63 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_0_less_iff
% 0.30/0.63 thf(fact_7429_of__nat__0__less__iff,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.30/0.63 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_0_less_iff
% 0.30/0.63 thf(fact_7430_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.30/0.63 ! [B: nat,W: nat,X2: nat] :
% 0.30/0.63 ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X2 ) )
% 0.30/0.63 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_of_nat_power_cancel_iff
% 0.30/0.63 thf(fact_7431_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.30/0.63 ! [B: nat,W: nat,X2: nat] :
% 0.30/0.63 ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X2 ) )
% 0.30/0.63 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_of_nat_power_cancel_iff
% 0.30/0.63 thf(fact_7432_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.30/0.63 ! [B: nat,W: nat,X2: nat] :
% 0.30/0.63 ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X2 ) )
% 0.30/0.63 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_of_nat_power_cancel_iff
% 0.30/0.63 thf(fact_7433_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.30/0.63 ! [B: nat,W: nat,X2: nat] :
% 0.30/0.63 ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X2 ) )
% 0.30/0.63 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_of_nat_power_cancel_iff
% 0.30/0.63 thf(fact_7434_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,B: nat,W: nat] :
% 0.30/0.63 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 0.30/0.63 = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_power_less_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7435_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,B: nat,W: nat] :
% 0.30/0.63 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 0.30/0.63 = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_power_less_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7436_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,B: nat,W: nat] :
% 0.30/0.63 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 0.30/0.63 = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_power_less_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7437_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,B: nat,W: nat] :
% 0.30/0.63 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 0.30/0.63 = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_power_less_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7438_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [X2: num,N: nat,Y3: nat] :
% 0.30/0.63 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X2 ) @ N )
% 0.30/0.63 = ( semiri8010041392384452111omplex @ Y3 ) )
% 0.30/0.63 = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
% 0.30/0.63 = Y3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % numeral_power_eq_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7439_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [X2: num,N: nat,Y3: nat] :
% 0.30/0.63 ( ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N )
% 0.30/0.63 = ( semiri1314217659103216013at_int @ Y3 ) )
% 0.30/0.63 = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
% 0.30/0.63 = Y3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % numeral_power_eq_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7440_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [X2: num,N: nat,Y3: nat] :
% 0.30/0.63 ( ( ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N )
% 0.30/0.63 = ( semiri5074537144036343181t_real @ Y3 ) )
% 0.30/0.63 = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
% 0.30/0.63 = Y3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % numeral_power_eq_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7441_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [X2: num,N: nat,Y3: nat] :
% 0.30/0.63 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
% 0.30/0.63 = ( semiri1316708129612266289at_nat @ Y3 ) )
% 0.30/0.63 = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
% 0.30/0.63 = Y3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % numeral_power_eq_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7442_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [X2: num,N: nat,Y3: nat] :
% 0.30/0.63 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N )
% 0.30/0.63 = ( semiri681578069525770553at_rat @ Y3 ) )
% 0.30/0.63 = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
% 0.30/0.63 = Y3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % numeral_power_eq_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7443_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.63 ! [Y3: nat,X2: num,N: nat] :
% 0.30/0.63 ( ( ( semiri8010041392384452111omplex @ Y3 )
% 0.30/0.63 = ( power_power_complex @ ( numera6690914467698888265omplex @ X2 ) @ N ) )
% 0.30/0.63 = ( Y3
% 0.30/0.63 = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % real_of_nat_eq_numeral_power_cancel_iff
% 0.30/0.63 thf(fact_7444_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.63 ! [Y3: nat,X2: num,N: nat] :
% 0.30/0.63 ( ( ( semiri1314217659103216013at_int @ Y3 )
% 0.30/0.63 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) )
% 0.30/0.63 = ( Y3
% 0.30/0.63 = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % real_of_nat_eq_numeral_power_cancel_iff
% 0.30/0.63 thf(fact_7445_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.63 ! [Y3: nat,X2: num,N: nat] :
% 0.30/0.63 ( ( ( semiri5074537144036343181t_real @ Y3 )
% 0.30/0.63 = ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
% 0.30/0.63 = ( Y3
% 0.30/0.63 = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % real_of_nat_eq_numeral_power_cancel_iff
% 0.30/0.63 thf(fact_7446_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.63 ! [Y3: nat,X2: num,N: nat] :
% 0.30/0.63 ( ( ( semiri1316708129612266289at_nat @ Y3 )
% 0.30/0.63 = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) )
% 0.30/0.63 = ( Y3
% 0.30/0.63 = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % real_of_nat_eq_numeral_power_cancel_iff
% 0.30/0.63 thf(fact_7447_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.63 ! [Y3: nat,X2: num,N: nat] :
% 0.30/0.63 ( ( ( semiri681578069525770553at_rat @ Y3 )
% 0.30/0.63 = ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) )
% 0.30/0.63 = ( Y3
% 0.30/0.63 = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % real_of_nat_eq_numeral_power_cancel_iff
% 0.30/0.63 thf(fact_7448_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,B: nat,W: nat] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 0.30/0.63 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_power_le_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7449_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,B: nat,W: nat] :
% 0.30/0.63 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 0.30/0.63 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_power_le_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7450_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,B: nat,W: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 0.30/0.63 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_power_le_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7451_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,B: nat,W: nat] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 0.30/0.63 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_power_le_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7452_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.30/0.63 ! [B: nat,W: nat,X2: nat] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X2 ) )
% 0.30/0.63 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_le_of_nat_power_cancel_iff
% 0.30/0.63 thf(fact_7453_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.30/0.63 ! [B: nat,W: nat,X2: nat] :
% 0.30/0.63 ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X2 ) )
% 0.30/0.63 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_le_of_nat_power_cancel_iff
% 0.30/0.63 thf(fact_7454_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.30/0.63 ! [B: nat,W: nat,X2: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X2 ) )
% 0.30/0.63 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_le_of_nat_power_cancel_iff
% 0.30/0.63 thf(fact_7455_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.30/0.63 ! [B: nat,W: nat,X2: nat] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X2 ) )
% 0.30/0.63 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_le_of_nat_power_cancel_iff
% 0.30/0.63 thf(fact_7456_of__nat__zero__less__power__iff,axiom,
% 0.30/0.63 ! [X2: nat,N: nat] :
% 0.30/0.63 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X2 ) @ N ) )
% 0.30/0.63 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 0.30/0.63 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_zero_less_power_iff
% 0.30/0.63 thf(fact_7457_of__nat__zero__less__power__iff,axiom,
% 0.30/0.63 ! [X2: nat,N: nat] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X2 ) @ N ) )
% 0.30/0.63 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 0.30/0.63 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_zero_less_power_iff
% 0.30/0.63 thf(fact_7458_of__nat__zero__less__power__iff,axiom,
% 0.30/0.63 ! [X2: nat,N: nat] :
% 0.30/0.63 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ N ) )
% 0.30/0.63 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 0.30/0.63 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_zero_less_power_iff
% 0.30/0.63 thf(fact_7459_of__nat__zero__less__power__iff,axiom,
% 0.30/0.63 ! [X2: nat,N: nat] :
% 0.30/0.63 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X2 ) @ N ) )
% 0.30/0.63 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 0.30/0.63 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_zero_less_power_iff
% 0.30/0.63 thf(fact_7460_even__of__nat,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
% 0.30/0.63 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % even_of_nat
% 0.30/0.63 thf(fact_7461_even__of__nat,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.63 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % even_of_nat
% 0.30/0.63 thf(fact_7462_even__of__nat,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.30/0.63 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % even_of_nat
% 0.30/0.63 thf(fact_7463_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [I2: num,N: nat,X2: nat] :
% 0.30/0.63 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) @ ( semiri1314217659103216013at_int @ X2 ) )
% 0.30/0.63 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % numeral_power_less_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7464_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [I2: num,N: nat,X2: nat] :
% 0.30/0.63 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) @ ( semiri5074537144036343181t_real @ X2 ) )
% 0.30/0.63 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % numeral_power_less_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7465_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [I2: num,N: nat,X2: nat] :
% 0.30/0.63 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ ( semiri1316708129612266289at_nat @ X2 ) )
% 0.30/0.63 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % numeral_power_less_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7466_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [I2: num,N: nat,X2: nat] :
% 0.30/0.63 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) @ ( semiri681578069525770553at_rat @ X2 ) )
% 0.30/0.63 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % numeral_power_less_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7467_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,I2: num,N: nat] :
% 0.30/0.63 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) )
% 0.30/0.63 = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_numeral_power_cancel_iff
% 0.30/0.63 thf(fact_7468_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,I2: num,N: nat] :
% 0.30/0.63 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) )
% 0.30/0.63 = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_numeral_power_cancel_iff
% 0.30/0.63 thf(fact_7469_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,I2: num,N: nat] :
% 0.30/0.63 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) )
% 0.30/0.63 = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_numeral_power_cancel_iff
% 0.30/0.63 thf(fact_7470_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,I2: num,N: nat] :
% 0.30/0.63 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) )
% 0.30/0.63 = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_numeral_power_cancel_iff
% 0.30/0.63 thf(fact_7471_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [I2: num,N: nat,X2: nat] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) @ ( semiri5074537144036343181t_real @ X2 ) )
% 0.30/0.63 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % numeral_power_le_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7472_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [I2: num,N: nat,X2: nat] :
% 0.30/0.63 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) @ ( semiri681578069525770553at_rat @ X2 ) )
% 0.30/0.63 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % numeral_power_le_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7473_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [I2: num,N: nat,X2: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ ( semiri1316708129612266289at_nat @ X2 ) )
% 0.30/0.63 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % numeral_power_le_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7474_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63 ! [I2: num,N: nat,X2: nat] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) @ ( semiri1314217659103216013at_int @ X2 ) )
% 0.30/0.63 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % numeral_power_le_of_nat_cancel_iff
% 0.30/0.63 thf(fact_7475_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,I2: num,N: nat] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) )
% 0.30/0.63 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_le_numeral_power_cancel_iff
% 0.30/0.63 thf(fact_7476_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,I2: num,N: nat] :
% 0.30/0.63 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) )
% 0.30/0.63 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_le_numeral_power_cancel_iff
% 0.30/0.63 thf(fact_7477_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,I2: num,N: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) )
% 0.30/0.63 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_le_numeral_power_cancel_iff
% 0.30/0.63 thf(fact_7478_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.63 ! [X2: nat,I2: num,N: nat] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) )
% 0.30/0.63 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_le_numeral_power_cancel_iff
% 0.30/0.63 thf(fact_7479_real__arch__simple,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ? [N2: nat] : ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % real_arch_simple
% 0.30/0.63 thf(fact_7480_real__arch__simple,axiom,
% 0.30/0.63 ! [X2: rat] :
% 0.30/0.63 ? [N2: nat] : ( ord_less_eq_rat @ X2 @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % real_arch_simple
% 0.30/0.63 thf(fact_7481_reals__Archimedean2,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ? [N2: nat] : ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % reals_Archimedean2
% 0.30/0.63 thf(fact_7482_reals__Archimedean2,axiom,
% 0.30/0.63 ! [X2: rat] :
% 0.30/0.63 ? [N2: nat] : ( ord_less_rat @ X2 @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % reals_Archimedean2
% 0.30/0.63 thf(fact_7483_mult__of__nat__commute,axiom,
% 0.30/0.63 ! [X2: nat,Y3: int] :
% 0.30/0.63 ( ( times_times_int @ ( semiri1314217659103216013at_int @ X2 ) @ Y3 )
% 0.30/0.63 = ( times_times_int @ Y3 @ ( semiri1314217659103216013at_int @ X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % mult_of_nat_commute
% 0.30/0.63 thf(fact_7484_mult__of__nat__commute,axiom,
% 0.30/0.63 ! [X2: nat,Y3: real] :
% 0.30/0.63 ( ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ Y3 )
% 0.30/0.63 = ( times_times_real @ Y3 @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % mult_of_nat_commute
% 0.30/0.63 thf(fact_7485_mult__of__nat__commute,axiom,
% 0.30/0.63 ! [X2: nat,Y3: nat] :
% 0.30/0.63 ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ Y3 )
% 0.30/0.63 = ( times_times_nat @ Y3 @ ( semiri1316708129612266289at_nat @ X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % mult_of_nat_commute
% 0.30/0.63 thf(fact_7486_mult__of__nat__commute,axiom,
% 0.30/0.63 ! [X2: nat,Y3: rat] :
% 0.30/0.63 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X2 ) @ Y3 )
% 0.30/0.63 = ( times_times_rat @ Y3 @ ( semiri681578069525770553at_rat @ X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % mult_of_nat_commute
% 0.30/0.63 thf(fact_7487_arctan__monotone_H,axiom,
% 0.30/0.63 ! [X2: real,Y3: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.30/0.63 => ( ord_less_eq_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % arctan_monotone'
% 0.30/0.63 thf(fact_7488_arctan__le__iff,axiom,
% 0.30/0.63 ! [X2: real,Y3: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) )
% 0.30/0.63 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % arctan_le_iff
% 0.30/0.63 thf(fact_7489_int__diff__cases,axiom,
% 0.30/0.63 ! [Z: int] :
% 0.30/0.63 ~ ! [M4: nat,N2: nat] :
% 0.30/0.63 ( Z
% 0.30/0.63 != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % int_diff_cases
% 0.30/0.63 thf(fact_7490_of__nat__less__of__int__iff,axiom,
% 0.30/0.63 ! [N: nat,X2: int] :
% 0.30/0.63 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X2 ) )
% 0.30/0.63 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_of_int_iff
% 0.30/0.63 thf(fact_7491_of__nat__less__of__int__iff,axiom,
% 0.30/0.63 ! [N: nat,X2: int] :
% 0.30/0.63 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X2 ) )
% 0.30/0.63 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_of_int_iff
% 0.30/0.63 thf(fact_7492_of__nat__less__of__int__iff,axiom,
% 0.30/0.63 ! [N: nat,X2: int] :
% 0.30/0.63 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X2 ) )
% 0.30/0.63 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_of_int_iff
% 0.30/0.63 thf(fact_7493_of__nat__0__le__iff,axiom,
% 0.30/0.63 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_0_le_iff
% 0.30/0.63 thf(fact_7494_of__nat__0__le__iff,axiom,
% 0.30/0.63 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_0_le_iff
% 0.30/0.63 thf(fact_7495_of__nat__0__le__iff,axiom,
% 0.30/0.63 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_0_le_iff
% 0.30/0.63 thf(fact_7496_of__nat__0__le__iff,axiom,
% 0.30/0.63 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_0_le_iff
% 0.30/0.63 thf(fact_7497_of__nat__less__0__iff,axiom,
% 0.30/0.63 ! [M: nat] :
% 0.30/0.63 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_0_iff
% 0.30/0.63 thf(fact_7498_of__nat__less__0__iff,axiom,
% 0.30/0.63 ! [M: nat] :
% 0.30/0.63 ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_0_iff
% 0.30/0.63 thf(fact_7499_of__nat__less__0__iff,axiom,
% 0.30/0.63 ! [M: nat] :
% 0.30/0.63 ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_0_iff
% 0.30/0.63 thf(fact_7500_of__nat__less__0__iff,axiom,
% 0.30/0.63 ! [M: nat] :
% 0.30/0.63 ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_0_iff
% 0.30/0.63 thf(fact_7501_of__nat__neq__0,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
% 0.30/0.63 != zero_zero_complex ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_neq_0
% 0.30/0.63 thf(fact_7502_of__nat__neq__0,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 0.30/0.63 != zero_zero_int ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_neq_0
% 0.30/0.63 thf(fact_7503_of__nat__neq__0,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
% 0.30/0.63 != zero_zero_real ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_neq_0
% 0.30/0.63 thf(fact_7504_of__nat__neq__0,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
% 0.30/0.63 != zero_zero_nat ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_neq_0
% 0.30/0.63 thf(fact_7505_of__nat__neq__0,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
% 0.30/0.63 != zero_zero_rat ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_neq_0
% 0.30/0.63 thf(fact_7506_div__mult2__eq_H,axiom,
% 0.30/0.63 ! [A: int,M: nat,N: nat] :
% 0.30/0.63 ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.30/0.63 = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % div_mult2_eq'
% 0.30/0.63 thf(fact_7507_div__mult2__eq_H,axiom,
% 0.30/0.63 ! [A: nat,M: nat,N: nat] :
% 0.30/0.63 ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 0.30/0.63 = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % div_mult2_eq'
% 0.30/0.63 thf(fact_7508_less__imp__of__nat__less,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( ord_less_nat @ M @ N )
% 0.30/0.63 => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % less_imp_of_nat_less
% 0.30/0.63 thf(fact_7509_less__imp__of__nat__less,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( ord_less_nat @ M @ N )
% 0.30/0.63 => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % less_imp_of_nat_less
% 0.30/0.63 thf(fact_7510_less__imp__of__nat__less,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( ord_less_nat @ M @ N )
% 0.30/0.63 => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % less_imp_of_nat_less
% 0.30/0.63 thf(fact_7511_less__imp__of__nat__less,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( ord_less_nat @ M @ N )
% 0.30/0.63 => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % less_imp_of_nat_less
% 0.30/0.63 thf(fact_7512_of__nat__less__imp__less,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.63 => ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_imp_less
% 0.30/0.63 thf(fact_7513_of__nat__less__imp__less,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.63 => ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_imp_less
% 0.30/0.63 thf(fact_7514_of__nat__less__imp__less,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.30/0.63 => ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_imp_less
% 0.30/0.63 thf(fact_7515_of__nat__less__imp__less,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.30/0.63 => ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_imp_less
% 0.30/0.63 thf(fact_7516_of__nat__mono,axiom,
% 0.30/0.63 ! [I2: nat,J: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.63 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I2 ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_mono
% 0.30/0.63 thf(fact_7517_of__nat__mono,axiom,
% 0.30/0.63 ! [I2: nat,J: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.63 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I2 ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_mono
% 0.30/0.63 thf(fact_7518_of__nat__mono,axiom,
% 0.30/0.63 ! [I2: nat,J: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.63 => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_mono
% 0.30/0.63 thf(fact_7519_of__nat__mono,axiom,
% 0.30/0.63 ! [I2: nat,J: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.63 => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_mono
% 0.30/0.63 thf(fact_7520_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
% 0.30/0.63 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unique_euclidean_semiring_with_nat_class.of_nat_div
% 0.30/0.63 thf(fact_7521_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
% 0.30/0.63 = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % unique_euclidean_semiring_with_nat_class.of_nat_div
% 0.30/0.63 thf(fact_7522_of__nat__dvd__iff,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
% 0.30/0.63 = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_dvd_iff
% 0.30/0.63 thf(fact_7523_of__nat__dvd__iff,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.63 = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_dvd_iff
% 0.30/0.63 thf(fact_7524_of__nat__dvd__iff,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.30/0.63 = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_dvd_iff
% 0.30/0.63 thf(fact_7525_int__ops_I3_J,axiom,
% 0.30/0.63 ! [N: num] :
% 0.30/0.63 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 0.30/0.63 = ( numeral_numeral_int @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % int_ops(3)
% 0.30/0.63 thf(fact_7526_int__cases,axiom,
% 0.30/0.63 ! [Z: int] :
% 0.30/0.63 ( ! [N2: nat] :
% 0.30/0.63 ( Z
% 0.30/0.63 != ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63 => ~ ! [N2: nat] :
% 0.30/0.63 ( Z
% 0.30/0.63 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % int_cases
% 0.30/0.63 thf(fact_7527_int__of__nat__induct,axiom,
% 0.30/0.63 ! [P: int > $o,Z: int] :
% 0.30/0.63 ( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63 => ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
% 0.30/0.63 => ( P @ Z ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % int_of_nat_induct
% 0.30/0.63 thf(fact_7528_nat__int__comparison_I2_J,axiom,
% 0.30/0.63 ( ord_less_nat
% 0.30/0.63 = ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_int_comparison(2)
% 0.30/0.63 thf(fact_7529_nat__int__comparison_I3_J,axiom,
% 0.30/0.63 ( ord_less_eq_nat
% 0.30/0.63 = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_int_comparison(3)
% 0.30/0.63 thf(fact_7530_zle__int,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.63 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % zle_int
% 0.30/0.63 thf(fact_7531_zero__le__imp__eq__int,axiom,
% 0.30/0.63 ! [K: int] :
% 0.30/0.63 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.63 => ? [N2: nat] :
% 0.30/0.63 ( K
% 0.30/0.63 = ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % zero_le_imp_eq_int
% 0.30/0.63 thf(fact_7532_nonneg__int__cases,axiom,
% 0.30/0.63 ! [K: int] :
% 0.30/0.63 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.63 => ~ ! [N2: nat] :
% 0.30/0.63 ( K
% 0.30/0.63 != ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nonneg_int_cases
% 0.30/0.63 thf(fact_7533_of__nat__mod,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
% 0.30/0.63 = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_mod
% 0.30/0.63 thf(fact_7534_of__nat__mod,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
% 0.30/0.63 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_mod
% 0.30/0.63 thf(fact_7535_of__nat__mod,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
% 0.30/0.63 = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_mod
% 0.30/0.63 thf(fact_7536_int__ops_I2_J,axiom,
% 0.30/0.63 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 0.30/0.63 = one_one_int ) ).
% 0.30/0.63
% 0.30/0.63 % int_ops(2)
% 0.30/0.63 thf(fact_7537_int__plus,axiom,
% 0.30/0.63 ! [N: nat,M: nat] :
% 0.30/0.63 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
% 0.30/0.63 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % int_plus
% 0.30/0.63 thf(fact_7538_int__ops_I5_J,axiom,
% 0.30/0.63 ! [A: nat,B: nat] :
% 0.30/0.63 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
% 0.30/0.63 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % int_ops(5)
% 0.30/0.63 thf(fact_7539_zadd__int__left,axiom,
% 0.30/0.63 ! [M: nat,N: nat,Z: int] :
% 0.30/0.63 ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
% 0.30/0.63 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% 0.30/0.63
% 0.30/0.63 % zadd_int_left
% 0.30/0.63 thf(fact_7540_int__ops_I7_J,axiom,
% 0.30/0.63 ! [A: nat,B: nat] :
% 0.30/0.63 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
% 0.30/0.63 = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % int_ops(7)
% 0.30/0.63 thf(fact_7541_zle__iff__zadd,axiom,
% 0.30/0.63 ( ord_less_eq_int
% 0.30/0.63 = ( ^ [W3: int,Z3: int] :
% 0.30/0.63 ? [N3: nat] :
% 0.30/0.63 ( Z3
% 0.30/0.63 = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % zle_iff_zadd
% 0.30/0.63 thf(fact_7542_zdiv__int,axiom,
% 0.30/0.63 ! [A: nat,B: nat] :
% 0.30/0.63 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
% 0.30/0.63 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % zdiv_int
% 0.30/0.63 thf(fact_7543_of__nat__max,axiom,
% 0.30/0.63 ! [X2: nat,Y3: nat] :
% 0.30/0.63 ( ( semiri4216267220026989637d_enat @ ( ord_max_nat @ X2 @ Y3 ) )
% 0.30/0.63 = ( ord_ma741700101516333627d_enat @ ( semiri4216267220026989637d_enat @ X2 ) @ ( semiri4216267220026989637d_enat @ Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_max
% 0.30/0.63 thf(fact_7544_of__nat__max,axiom,
% 0.30/0.63 ! [X2: nat,Y3: nat] :
% 0.30/0.63 ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X2 @ Y3 ) )
% 0.30/0.63 = ( ord_max_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_max
% 0.30/0.63 thf(fact_7545_of__nat__max,axiom,
% 0.30/0.63 ! [X2: nat,Y3: nat] :
% 0.30/0.63 ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X2 @ Y3 ) )
% 0.30/0.63 = ( ord_max_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( semiri5074537144036343181t_real @ Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_max
% 0.30/0.63 thf(fact_7546_of__nat__max,axiom,
% 0.30/0.63 ! [X2: nat,Y3: nat] :
% 0.30/0.63 ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X2 @ Y3 ) )
% 0.30/0.63 = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( semiri1316708129612266289at_nat @ Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_max
% 0.30/0.63 thf(fact_7547_of__nat__max,axiom,
% 0.30/0.63 ! [X2: nat,Y3: nat] :
% 0.30/0.63 ( ( semiri681578069525770553at_rat @ ( ord_max_nat @ X2 @ Y3 ) )
% 0.30/0.63 = ( ord_max_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( semiri681578069525770553at_rat @ Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_max
% 0.30/0.63 thf(fact_7548_nat__less__as__int,axiom,
% 0.30/0.63 ( ord_less_nat
% 0.30/0.63 = ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_less_as_int
% 0.30/0.63 thf(fact_7549_nat__leq__as__int,axiom,
% 0.30/0.63 ( ord_less_eq_nat
% 0.30/0.63 = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_leq_as_int
% 0.30/0.63 thf(fact_7550_ex__less__of__nat__mult,axiom,
% 0.30/0.63 ! [X2: real,Y3: real] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ? [N2: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ex_less_of_nat_mult
% 0.30/0.63 thf(fact_7551_ex__less__of__nat__mult,axiom,
% 0.30/0.63 ! [X2: rat,Y3: rat] :
% 0.30/0.63 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 0.30/0.63 => ? [N2: nat] : ( ord_less_rat @ Y3 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ex_less_of_nat_mult
% 0.30/0.63 thf(fact_7552_of__nat__diff,axiom,
% 0.30/0.63 ! [N: nat,M: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.63 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.63 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_diff
% 0.30/0.63 thf(fact_7553_of__nat__diff,axiom,
% 0.30/0.63 ! [N: nat,M: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.63 => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.63 = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_diff
% 0.30/0.63 thf(fact_7554_of__nat__diff,axiom,
% 0.30/0.63 ! [N: nat,M: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.63 => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.63 = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_diff
% 0.30/0.63 thf(fact_7555_of__nat__diff,axiom,
% 0.30/0.63 ! [N: nat,M: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.63 => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.63 = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_diff
% 0.30/0.63 thf(fact_7556_reals__Archimedean3,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ! [Y5: real] :
% 0.30/0.63 ? [N2: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % reals_Archimedean3
% 0.30/0.63 thf(fact_7557_int__cases4,axiom,
% 0.30/0.63 ! [M: int] :
% 0.30/0.63 ( ! [N2: nat] :
% 0.30/0.63 ( M
% 0.30/0.63 != ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63 => ~ ! [N2: nat] :
% 0.30/0.63 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.63 => ( M
% 0.30/0.63 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % int_cases4
% 0.30/0.63 thf(fact_7558_real__of__nat__div4,axiom,
% 0.30/0.63 ! [N: nat,X2: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % real_of_nat_div4
% 0.30/0.63 thf(fact_7559_int__ops_I4_J,axiom,
% 0.30/0.63 ! [A: nat] :
% 0.30/0.63 ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
% 0.30/0.63 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% 0.30/0.63
% 0.30/0.63 % int_ops(4)
% 0.30/0.63 thf(fact_7560_int__Suc,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 0.30/0.63 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% 0.30/0.63
% 0.30/0.63 % int_Suc
% 0.30/0.63 thf(fact_7561_int__zle__neg,axiom,
% 0.30/0.63 ! [N: nat,M: nat] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
% 0.30/0.63 = ( ( N = zero_zero_nat )
% 0.30/0.63 & ( M = zero_zero_nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % int_zle_neg
% 0.30/0.63 thf(fact_7562_zless__iff__Suc__zadd,axiom,
% 0.30/0.63 ( ord_less_int
% 0.30/0.63 = ( ^ [W3: int,Z3: int] :
% 0.30/0.63 ? [N3: nat] :
% 0.30/0.63 ( Z3
% 0.30/0.63 = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % zless_iff_Suc_zadd
% 0.30/0.63 thf(fact_7563_nonpos__int__cases,axiom,
% 0.30/0.63 ! [K: int] :
% 0.30/0.63 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.30/0.63 => ~ ! [N2: nat] :
% 0.30/0.63 ( K
% 0.30/0.63 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nonpos_int_cases
% 0.30/0.63 thf(fact_7564_negative__zle__0,axiom,
% 0.30/0.63 ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% 0.30/0.63
% 0.30/0.63 % negative_zle_0
% 0.30/0.63 thf(fact_7565_real__of__nat__div,axiom,
% 0.30/0.63 ! [D: nat,N: nat] :
% 0.30/0.63 ( ( dvd_dvd_nat @ D @ N )
% 0.30/0.63 => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
% 0.30/0.63 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % real_of_nat_div
% 0.30/0.63 thf(fact_7566_mod__mult2__eq_H,axiom,
% 0.30/0.63 ! [A: code_integer,M: nat,N: nat] :
% 0.30/0.63 ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
% 0.30/0.63 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % mod_mult2_eq'
% 0.30/0.63 thf(fact_7567_mod__mult2__eq_H,axiom,
% 0.30/0.63 ! [A: int,M: nat,N: nat] :
% 0.30/0.63 ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.30/0.63 = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % mod_mult2_eq'
% 0.30/0.63 thf(fact_7568_mod__mult2__eq_H,axiom,
% 0.30/0.63 ! [A: nat,M: nat,N: nat] :
% 0.30/0.63 ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 0.30/0.63 = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % mod_mult2_eq'
% 0.30/0.63 thf(fact_7569_field__char__0__class_Oof__nat__div,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
% 0.30/0.63 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % field_char_0_class.of_nat_div
% 0.30/0.63 thf(fact_7570_field__char__0__class_Oof__nat__div,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
% 0.30/0.63 = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % field_char_0_class.of_nat_div
% 0.30/0.63 thf(fact_7571_field__char__0__class_Oof__nat__div,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
% 0.30/0.63 = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % field_char_0_class.of_nat_div
% 0.30/0.63 thf(fact_7572_zero__less__imp__eq__int,axiom,
% 0.30/0.63 ! [K: int] :
% 0.30/0.63 ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.63 => ? [N2: nat] :
% 0.30/0.63 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.63 & ( K
% 0.30/0.63 = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % zero_less_imp_eq_int
% 0.30/0.63 thf(fact_7573_pos__int__cases,axiom,
% 0.30/0.63 ! [K: int] :
% 0.30/0.63 ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.63 => ~ ! [N2: nat] :
% 0.30/0.63 ( ( K
% 0.30/0.63 = ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % pos_int_cases
% 0.30/0.63 thf(fact_7574_int__cases3,axiom,
% 0.30/0.63 ! [K: int] :
% 0.30/0.63 ( ( K != zero_zero_int )
% 0.30/0.63 => ( ! [N2: nat] :
% 0.30/0.63 ( ( K
% 0.30/0.63 = ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.63 => ~ ! [N2: nat] :
% 0.30/0.63 ( ( K
% 0.30/0.63 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.30/0.63 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % int_cases3
% 0.30/0.63 thf(fact_7575_nat__less__real__le,axiom,
% 0.30/0.63 ( ord_less_nat
% 0.30/0.63 = ( ^ [N3: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_less_real_le
% 0.30/0.63 thf(fact_7576_nat__le__real__less,axiom,
% 0.30/0.63 ( ord_less_eq_nat
% 0.30/0.63 = ( ^ [N3: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_le_real_less
% 0.30/0.63 thf(fact_7577_zmult__zless__mono2__lemma,axiom,
% 0.30/0.63 ! [I2: int,J: int,K: nat] :
% 0.30/0.63 ( ( ord_less_int @ I2 @ J )
% 0.30/0.63 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.63 => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % zmult_zless_mono2_lemma
% 0.30/0.63 thf(fact_7578_not__zle__0__negative,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % not_zle_0_negative
% 0.30/0.63 thf(fact_7579_negative__zless__0,axiom,
% 0.30/0.63 ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% 0.30/0.63
% 0.30/0.63 % negative_zless_0
% 0.30/0.63 thf(fact_7580_negD,axiom,
% 0.30/0.63 ! [X2: int] :
% 0.30/0.63 ( ( ord_less_int @ X2 @ zero_zero_int )
% 0.30/0.63 => ? [N2: nat] :
% 0.30/0.63 ( X2
% 0.30/0.63 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % negD
% 0.30/0.63 thf(fact_7581_int__ops_I6_J,axiom,
% 0.30/0.63 ! [A: nat,B: nat] :
% 0.30/0.63 ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 0.30/0.63 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 0.30/0.63 = zero_zero_int ) )
% 0.30/0.63 & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 0.30/0.63 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 0.30/0.63 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % int_ops(6)
% 0.30/0.63 thf(fact_7582_real__of__nat__div__aux,axiom,
% 0.30/0.63 ! [X2: nat,D: nat] :
% 0.30/0.63 ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( semiri5074537144036343181t_real @ D ) )
% 0.30/0.63 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X2 @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X2 @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % real_of_nat_div_aux
% 0.30/0.63 thf(fact_7583_nat__approx__posE,axiom,
% 0.30/0.63 ! [E: real] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ E )
% 0.30/0.63 => ~ ! [N2: nat] :
% 0.30/0.63 ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ E ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_approx_posE
% 0.30/0.63 thf(fact_7584_nat__approx__posE,axiom,
% 0.30/0.63 ! [E: rat] :
% 0.30/0.63 ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.30/0.63 => ~ ! [N2: nat] :
% 0.30/0.63 ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ E ) ) ).
% 0.30/0.63
% 0.30/0.63 % nat_approx_posE
% 0.30/0.63 thf(fact_7585_of__nat__less__two__power,axiom,
% 0.30/0.63 ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_two_power
% 0.30/0.63 thf(fact_7586_of__nat__less__two__power,axiom,
% 0.30/0.63 ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_two_power
% 0.30/0.63 thf(fact_7587_of__nat__less__two__power,axiom,
% 0.30/0.63 ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_less_two_power
% 0.30/0.63 thf(fact_7588_inverse__of__nat__le,axiom,
% 0.30/0.63 ! [N: nat,M: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.63 => ( ( N != zero_zero_nat )
% 0.30/0.63 => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % inverse_of_nat_le
% 0.30/0.63 thf(fact_7589_inverse__of__nat__le,axiom,
% 0.30/0.63 ! [N: nat,M: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.63 => ( ( N != zero_zero_nat )
% 0.30/0.63 => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % inverse_of_nat_le
% 0.30/0.63 thf(fact_7590_fold__atLeastAtMost__nat_Osimps,axiom,
% 0.30/0.63 ( set_fo2584398358068434914at_nat
% 0.30/0.63 = ( ^ [F3: nat > nat > nat,A3: nat,B2: nat,Acc3: nat] : ( if_nat @ ( ord_less_nat @ B2 @ A3 ) @ Acc3 @ ( set_fo2584398358068434914at_nat @ F3 @ ( plus_plus_nat @ A3 @ one_one_nat ) @ B2 @ ( F3 @ A3 @ Acc3 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % fold_atLeastAtMost_nat.simps
% 0.30/0.63 thf(fact_7591_fold__atLeastAtMost__nat_Oelims,axiom,
% 0.30/0.63 ! [X2: nat > nat > nat,Xa2: nat,Xb: nat,Xc: nat,Y3: nat] :
% 0.30/0.63 ( ( ( set_fo2584398358068434914at_nat @ X2 @ Xa2 @ Xb @ Xc )
% 0.30/0.63 = Y3 )
% 0.30/0.63 => ( ( ( ord_less_nat @ Xb @ Xa2 )
% 0.30/0.63 => ( Y3 = Xc ) )
% 0.30/0.63 & ( ~ ( ord_less_nat @ Xb @ Xa2 )
% 0.30/0.63 => ( Y3
% 0.30/0.63 = ( set_fo2584398358068434914at_nat @ X2 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X2 @ Xa2 @ Xc ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % fold_atLeastAtMost_nat.elims
% 0.30/0.63 thf(fact_7592_real__archimedian__rdiv__eq__0,axiom,
% 0.30/0.63 ! [X2: real,C: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.63 => ( ! [M4: nat] :
% 0.30/0.63 ( ( ord_less_nat @ zero_zero_nat @ M4 )
% 0.30/0.63 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X2 ) @ C ) )
% 0.30/0.63 => ( X2 = zero_zero_real ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % real_archimedian_rdiv_eq_0
% 0.30/0.63 thf(fact_7593_neg__int__cases,axiom,
% 0.30/0.63 ! [K: int] :
% 0.30/0.63 ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.63 => ~ ! [N2: nat] :
% 0.30/0.63 ( ( K
% 0.30/0.63 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.30/0.63 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % neg_int_cases
% 0.30/0.63 thf(fact_7594_zdiff__int__split,axiom,
% 0.30/0.63 ! [P: int > $o,X2: nat,Y3: nat] :
% 0.30/0.63 ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X2 @ Y3 ) ) )
% 0.30/0.63 = ( ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.30/0.63 => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) )
% 0.30/0.63 & ( ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.63 => ( P @ zero_zero_int ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % zdiff_int_split
% 0.30/0.63 thf(fact_7595_real__of__nat__div2,axiom,
% 0.30/0.63 ! [N: nat,X2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % real_of_nat_div2
% 0.30/0.63 thf(fact_7596_real__of__nat__div3,axiom,
% 0.30/0.63 ! [N: nat,X2: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) ) @ one_one_real ) ).
% 0.30/0.63
% 0.30/0.63 % real_of_nat_div3
% 0.30/0.63 thf(fact_7597_ln__realpow,axiom,
% 0.30/0.63 ! [X2: real,N: nat] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ( ln_ln_real @ ( power_power_real @ X2 @ N ) )
% 0.30/0.63 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ln_realpow
% 0.30/0.63 thf(fact_7598_linear__plus__1__le__power,axiom,
% 0.30/0.63 ! [X2: real,N: nat] :
% 0.30/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X2 @ one_one_real ) @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % linear_plus_1_le_power
% 0.30/0.63 thf(fact_7599_Bernoulli__inequality,axiom,
% 0.30/0.63 ! [X2: real,N: nat] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.63 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X2 ) @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Bernoulli_inequality
% 0.30/0.63 thf(fact_7600_sum__atLeastAtMost__code,axiom,
% 0.30/0.63 ! [F: nat > complex,A: nat,B: nat] :
% 0.30/0.63 ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.30/0.63 = ( set_fo1517530859248394432omplex
% 0.30/0.63 @ ^ [A3: nat] : ( plus_plus_complex @ ( F @ A3 ) )
% 0.30/0.63 @ A
% 0.30/0.63 @ B
% 0.30/0.63 @ zero_zero_complex ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_atLeastAtMost_code
% 0.30/0.63 thf(fact_7601_sum__atLeastAtMost__code,axiom,
% 0.30/0.63 ! [F: nat > rat,A: nat,B: nat] :
% 0.30/0.63 ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.30/0.63 = ( set_fo1949268297981939178at_rat
% 0.30/0.63 @ ^ [A3: nat] : ( plus_plus_rat @ ( F @ A3 ) )
% 0.30/0.63 @ A
% 0.30/0.63 @ B
% 0.30/0.63 @ zero_zero_rat ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_atLeastAtMost_code
% 0.30/0.63 thf(fact_7602_sum__atLeastAtMost__code,axiom,
% 0.30/0.63 ! [F: nat > int,A: nat,B: nat] :
% 0.30/0.63 ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.30/0.63 = ( set_fo2581907887559384638at_int
% 0.30/0.63 @ ^ [A3: nat] : ( plus_plus_int @ ( F @ A3 ) )
% 0.30/0.63 @ A
% 0.30/0.63 @ B
% 0.30/0.63 @ zero_zero_int ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_atLeastAtMost_code
% 0.30/0.63 thf(fact_7603_sum__atLeastAtMost__code,axiom,
% 0.30/0.63 ! [F: nat > nat,A: nat,B: nat] :
% 0.30/0.63 ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.30/0.63 = ( set_fo2584398358068434914at_nat
% 0.30/0.63 @ ^ [A3: nat] : ( plus_plus_nat @ ( F @ A3 ) )
% 0.30/0.63 @ A
% 0.30/0.63 @ B
% 0.30/0.63 @ zero_zero_nat ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_atLeastAtMost_code
% 0.30/0.63 thf(fact_7604_sum__atLeastAtMost__code,axiom,
% 0.30/0.63 ! [F: nat > real,A: nat,B: nat] :
% 0.30/0.63 ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.30/0.63 = ( set_fo3111899725591712190t_real
% 0.30/0.63 @ ^ [A3: nat] : ( plus_plus_real @ ( F @ A3 ) )
% 0.30/0.63 @ A
% 0.30/0.63 @ B
% 0.30/0.63 @ zero_zero_real ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_atLeastAtMost_code
% 0.30/0.63 thf(fact_7605_arctan__add,axiom,
% 0.30/0.63 ! [X2: real,Y3: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.30/0.63 => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.30/0.63 => ( ( plus_plus_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) )
% 0.30/0.63 = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % arctan_add
% 0.30/0.63 thf(fact_7606_double__gauss__sum,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % double_gauss_sum
% 0.30/0.63 thf(fact_7607_double__gauss__sum,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % double_gauss_sum
% 0.30/0.63 thf(fact_7608_double__gauss__sum,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % double_gauss_sum
% 0.30/0.63 thf(fact_7609_double__gauss__sum,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % double_gauss_sum
% 0.30/0.63 thf(fact_7610_double__gauss__sum,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % double_gauss_sum
% 0.30/0.63 thf(fact_7611_double__arith__series,axiom,
% 0.30/0.63 ! [A: complex,D: complex,N: nat] :
% 0.30/0.63 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 0.30/0.63 @ ( groups2073611262835488442omplex
% 0.30/0.63 @ ^ [I4: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I4 ) @ D ) )
% 0.30/0.63 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63 = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ D ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % double_arith_series
% 0.30/0.63 thf(fact_7612_double__arith__series,axiom,
% 0.30/0.63 ! [A: int,D: int,N: nat] :
% 0.30/0.63 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 0.30/0.63 @ ( groups3539618377306564664at_int
% 0.30/0.63 @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 0.30/0.63 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63 = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % double_arith_series
% 0.30/0.63 thf(fact_7613_double__arith__series,axiom,
% 0.30/0.63 ! [A: rat,D: rat,N: nat] :
% 0.30/0.63 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 0.30/0.63 @ ( groups2906978787729119204at_rat
% 0.30/0.63 @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I4 ) @ D ) )
% 0.30/0.63 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63 = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % double_arith_series
% 0.30/0.63 thf(fact_7614_double__arith__series,axiom,
% 0.30/0.63 ! [A: nat,D: nat,N: nat] :
% 0.30/0.63 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
% 0.30/0.63 @ ( groups3542108847815614940at_nat
% 0.30/0.63 @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 0.30/0.63 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63 = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % double_arith_series
% 0.30/0.63 thf(fact_7615_double__arith__series,axiom,
% 0.30/0.63 ! [A: real,D: real,N: nat] :
% 0.30/0.63 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.30/0.63 @ ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [I4: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I4 ) @ D ) )
% 0.30/0.63 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63 = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ D ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % double_arith_series
% 0.30/0.63 thf(fact_7616_Bernoulli__inequality__even,axiom,
% 0.30/0.63 ! [N: nat,X2: real] :
% 0.30/0.63 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.63 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X2 ) @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Bernoulli_inequality_even
% 0.30/0.63 thf(fact_7617_gauss__sum,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.63 = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % gauss_sum
% 0.30/0.63 thf(fact_7618_gauss__sum,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.63 = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % gauss_sum
% 0.30/0.63 thf(fact_7619_arith__series,axiom,
% 0.30/0.63 ! [A: int,D: int,N: nat] :
% 0.30/0.63 ( ( groups3539618377306564664at_int
% 0.30/0.63 @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 0.30/0.63 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.63 = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % arith_series
% 0.30/0.63 thf(fact_7620_arith__series,axiom,
% 0.30/0.63 ! [A: nat,D: nat,N: nat] :
% 0.30/0.63 ( ( groups3542108847815614940at_nat
% 0.30/0.63 @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 0.30/0.63 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.63 = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % arith_series
% 0.30/0.63 thf(fact_7621_double__gauss__sum__from__Suc__0,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.30/0.63 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % double_gauss_sum_from_Suc_0
% 0.30/0.63 thf(fact_7622_double__gauss__sum__from__Suc__0,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.30/0.63 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % double_gauss_sum_from_Suc_0
% 0.30/0.63 thf(fact_7623_double__gauss__sum__from__Suc__0,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.30/0.63 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % double_gauss_sum_from_Suc_0
% 0.30/0.63 thf(fact_7624_double__gauss__sum__from__Suc__0,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.30/0.63 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % double_gauss_sum_from_Suc_0
% 0.30/0.63 thf(fact_7625_double__gauss__sum__from__Suc__0,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.30/0.63 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % double_gauss_sum_from_Suc_0
% 0.30/0.63 thf(fact_7626_sum__gp__offset,axiom,
% 0.30/0.63 ! [X2: complex,M: nat,N: nat] :
% 0.30/0.63 ( ( ( X2 = one_one_complex )
% 0.30/0.63 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 0.30/0.63 = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) )
% 0.30/0.63 & ( ( X2 != one_one_complex )
% 0.30/0.63 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 0.30/0.63 = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X2 @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_gp_offset
% 0.30/0.63 thf(fact_7627_sum__gp__offset,axiom,
% 0.30/0.63 ! [X2: rat,M: nat,N: nat] :
% 0.30/0.63 ( ( ( X2 = one_one_rat )
% 0.30/0.63 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 0.30/0.63 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
% 0.30/0.63 & ( ( X2 != one_one_rat )
% 0.30/0.63 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 0.30/0.63 = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X2 @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_gp_offset
% 0.30/0.63 thf(fact_7628_sum__gp__offset,axiom,
% 0.30/0.63 ! [X2: real,M: nat,N: nat] :
% 0.30/0.63 ( ( ( X2 = one_one_real )
% 0.30/0.63 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 0.30/0.63 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) )
% 0.30/0.63 & ( ( X2 != one_one_real )
% 0.30/0.63 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 0.30/0.63 = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X2 @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( suc @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_gp_offset
% 0.30/0.63 thf(fact_7629_gauss__sum__from__Suc__0,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.30/0.63 = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % gauss_sum_from_Suc_0
% 0.30/0.63 thf(fact_7630_gauss__sum__from__Suc__0,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.30/0.63 = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % gauss_sum_from_Suc_0
% 0.30/0.63 thf(fact_7631_of__nat__code__if,axiom,
% 0.30/0.63 ( semiri8010041392384452111omplex
% 0.30/0.63 = ( ^ [N3: nat] :
% 0.30/0.63 ( if_complex @ ( N3 = zero_zero_nat ) @ zero_zero_complex
% 0.30/0.63 @ ( produc1917071388513777916omplex
% 0.30/0.63 @ ^ [M3: nat,Q4: nat] : ( if_complex @ ( Q4 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ one_one_complex ) )
% 0.30/0.63 @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_code_if
% 0.30/0.63 thf(fact_7632_of__nat__code__if,axiom,
% 0.30/0.63 ( semiri1314217659103216013at_int
% 0.30/0.63 = ( ^ [N3: nat] :
% 0.30/0.63 ( if_int @ ( N3 = zero_zero_nat ) @ zero_zero_int
% 0.30/0.63 @ ( produc6840382203811409530at_int
% 0.30/0.63 @ ^ [M3: nat,Q4: nat] : ( if_int @ ( Q4 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ one_one_int ) )
% 0.30/0.63 @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_code_if
% 0.30/0.63 thf(fact_7633_of__nat__code__if,axiom,
% 0.30/0.63 ( semiri5074537144036343181t_real
% 0.30/0.63 = ( ^ [N3: nat] :
% 0.30/0.63 ( if_real @ ( N3 = zero_zero_nat ) @ zero_zero_real
% 0.30/0.63 @ ( produc1703576794950452218t_real
% 0.30/0.63 @ ^ [M3: nat,Q4: nat] : ( if_real @ ( Q4 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ one_one_real ) )
% 0.30/0.63 @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_code_if
% 0.30/0.63 thf(fact_7634_of__nat__code__if,axiom,
% 0.30/0.63 ( semiri681578069525770553at_rat
% 0.30/0.63 = ( ^ [N3: nat] :
% 0.30/0.63 ( if_rat @ ( N3 = zero_zero_nat ) @ zero_zero_rat
% 0.30/0.63 @ ( produc6207742614233964070at_rat
% 0.30/0.63 @ ^ [M3: nat,Q4: nat] : ( if_rat @ ( Q4 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ one_one_rat ) )
% 0.30/0.63 @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_code_if
% 0.30/0.63 thf(fact_7635_of__nat__code__if,axiom,
% 0.30/0.63 ( semiri1316708129612266289at_nat
% 0.30/0.63 = ( ^ [N3: nat] :
% 0.30/0.63 ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat
% 0.30/0.63 @ ( produc6842872674320459806at_nat
% 0.30/0.63 @ ^ [M3: nat,Q4: nat] : ( if_nat @ ( Q4 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ one_one_nat ) )
% 0.30/0.63 @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_nat_code_if
% 0.30/0.63 thf(fact_7636_height__double__log__univ__size,axiom,
% 0.30/0.63 ! [U: real,Deg: nat,T: vEBT_VEBT] :
% 0.30/0.63 ( ( U
% 0.30/0.63 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Deg ) )
% 0.30/0.63 => ( ( vEBT_invar_vebt @ T @ Deg )
% 0.30/0.63 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_height @ T ) ) @ ( plus_plus_real @ one_one_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % height_double_log_univ_size
% 0.30/0.63 thf(fact_7637_monoseq__arctan__series,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.30/0.63 => ( topolo6980174941875973593q_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % monoseq_arctan_series
% 0.30/0.63 thf(fact_7638_lemma__termdiff3,axiom,
% 0.30/0.63 ! [H2: real,Z: real,K5: real,N: nat] :
% 0.30/0.63 ( ( H2 != zero_zero_real )
% 0.30/0.63 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
% 0.30/0.63 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K5 )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lemma_termdiff3
% 0.30/0.63 thf(fact_7639_lemma__termdiff3,axiom,
% 0.30/0.63 ! [H2: complex,Z: complex,K5: real,N: nat] :
% 0.30/0.63 ( ( H2 != zero_zero_complex )
% 0.30/0.63 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
% 0.30/0.63 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K5 )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lemma_termdiff3
% 0.30/0.63 thf(fact_7640_ln__series,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.63 => ( ( ln_ln_real @ X2 )
% 0.30/0.63 = ( suminf_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X2 @ one_one_real ) @ ( suc @ N3 ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ln_series
% 0.30/0.63 thf(fact_7641_lemma__termdiff2,axiom,
% 0.30/0.63 ! [H2: complex,Z: complex,N: nat] :
% 0.30/0.63 ( ( H2 != zero_zero_complex )
% 0.30/0.63 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.30/0.63 = ( times_times_complex @ H2
% 0.30/0.63 @ ( groups2073611262835488442omplex
% 0.30/0.63 @ ^ [P4: nat] :
% 0.30/0.63 ( groups2073611262835488442omplex
% 0.30/0.63 @ ^ [Q4: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q4 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P4 ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lemma_termdiff2
% 0.30/0.63 thf(fact_7642_lemma__termdiff2,axiom,
% 0.30/0.63 ! [H2: rat,Z: rat,N: nat] :
% 0.30/0.63 ( ( H2 != zero_zero_rat )
% 0.30/0.63 => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ N ) @ ( power_power_rat @ Z @ N ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.30/0.63 = ( times_times_rat @ H2
% 0.30/0.63 @ ( groups2906978787729119204at_rat
% 0.30/0.63 @ ^ [P4: nat] :
% 0.30/0.63 ( groups2906978787729119204at_rat
% 0.30/0.63 @ ^ [Q4: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ Q4 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P4 ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lemma_termdiff2
% 0.30/0.63 thf(fact_7643_lemma__termdiff2,axiom,
% 0.30/0.63 ! [H2: real,Z: real,N: nat] :
% 0.30/0.63 ( ( H2 != zero_zero_real )
% 0.30/0.63 => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.30/0.63 = ( times_times_real @ H2
% 0.30/0.63 @ ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [P4: nat] :
% 0.30/0.63 ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [Q4: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q4 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P4 ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lemma_termdiff2
% 0.30/0.63 thf(fact_7644_lessThan__iff,axiom,
% 0.30/0.63 ! [I2: rat,K: rat] :
% 0.30/0.63 ( ( member_rat @ I2 @ ( set_ord_lessThan_rat @ K ) )
% 0.30/0.63 = ( ord_less_rat @ I2 @ K ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_iff
% 0.30/0.63 thf(fact_7645_lessThan__iff,axiom,
% 0.30/0.63 ! [I2: num,K: num] :
% 0.30/0.63 ( ( member_num @ I2 @ ( set_ord_lessThan_num @ K ) )
% 0.30/0.63 = ( ord_less_num @ I2 @ K ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_iff
% 0.30/0.63 thf(fact_7646_lessThan__iff,axiom,
% 0.30/0.63 ! [I2: nat,K: nat] :
% 0.30/0.63 ( ( member_nat @ I2 @ ( set_ord_lessThan_nat @ K ) )
% 0.30/0.63 = ( ord_less_nat @ I2 @ K ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_iff
% 0.30/0.63 thf(fact_7647_lessThan__iff,axiom,
% 0.30/0.63 ! [I2: int,K: int] :
% 0.30/0.63 ( ( member_int @ I2 @ ( set_ord_lessThan_int @ K ) )
% 0.30/0.63 = ( ord_less_int @ I2 @ K ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_iff
% 0.30/0.63 thf(fact_7648_lessThan__iff,axiom,
% 0.30/0.63 ! [I2: real,K: real] :
% 0.30/0.63 ( ( member_real @ I2 @ ( set_or5984915006950818249n_real @ K ) )
% 0.30/0.63 = ( ord_less_real @ I2 @ K ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_iff
% 0.30/0.63 thf(fact_7649_lessThan__subset__iff,axiom,
% 0.30/0.63 ! [X2: rat,Y3: rat] :
% 0.30/0.63 ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X2 ) @ ( set_ord_lessThan_rat @ Y3 ) )
% 0.30/0.63 = ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_subset_iff
% 0.30/0.63 thf(fact_7650_lessThan__subset__iff,axiom,
% 0.30/0.63 ! [X2: num,Y3: num] :
% 0.30/0.63 ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X2 ) @ ( set_ord_lessThan_num @ Y3 ) )
% 0.30/0.63 = ( ord_less_eq_num @ X2 @ Y3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_subset_iff
% 0.30/0.63 thf(fact_7651_lessThan__subset__iff,axiom,
% 0.30/0.63 ! [X2: nat,Y3: nat] :
% 0.30/0.63 ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X2 ) @ ( set_ord_lessThan_nat @ Y3 ) )
% 0.30/0.63 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_subset_iff
% 0.30/0.63 thf(fact_7652_lessThan__subset__iff,axiom,
% 0.30/0.63 ! [X2: int,Y3: int] :
% 0.30/0.63 ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X2 ) @ ( set_ord_lessThan_int @ Y3 ) )
% 0.30/0.63 = ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_subset_iff
% 0.30/0.63 thf(fact_7653_lessThan__subset__iff,axiom,
% 0.30/0.63 ! [X2: real,Y3: real] :
% 0.30/0.63 ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X2 ) @ ( set_or5984915006950818249n_real @ Y3 ) )
% 0.30/0.63 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_subset_iff
% 0.30/0.63 thf(fact_7654_log__one,axiom,
% 0.30/0.63 ! [A: real] :
% 0.30/0.63 ( ( log @ A @ one_one_real )
% 0.30/0.63 = zero_zero_real ) ).
% 0.30/0.63
% 0.30/0.63 % log_one
% 0.30/0.63 thf(fact_7655_sum_OlessThan__Suc,axiom,
% 0.30/0.63 ! [G: nat > rat,N: nat] :
% 0.30/0.63 ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.63 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum.lessThan_Suc
% 0.30/0.63 thf(fact_7656_sum_OlessThan__Suc,axiom,
% 0.30/0.63 ! [G: nat > int,N: nat] :
% 0.30/0.63 ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.63 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum.lessThan_Suc
% 0.30/0.63 thf(fact_7657_sum_OlessThan__Suc,axiom,
% 0.30/0.63 ! [G: nat > nat,N: nat] :
% 0.30/0.63 ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.63 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum.lessThan_Suc
% 0.30/0.63 thf(fact_7658_sum_OlessThan__Suc,axiom,
% 0.30/0.63 ! [G: nat > real,N: nat] :
% 0.30/0.63 ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.63 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum.lessThan_Suc
% 0.30/0.63 thf(fact_7659_log__eq__one,axiom,
% 0.30/0.63 ! [A: real] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.63 => ( ( A != one_one_real )
% 0.30/0.63 => ( ( log @ A @ A )
% 0.30/0.63 = one_one_real ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log_eq_one
% 0.30/0.63 thf(fact_7660_log__less__cancel__iff,axiom,
% 0.30/0.63 ! [A: real,X2: real,Y3: real] :
% 0.30/0.63 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.63 => ( ( ord_less_real @ ( log @ A @ X2 ) @ ( log @ A @ Y3 ) )
% 0.30/0.63 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log_less_cancel_iff
% 0.30/0.63 thf(fact_7661_log__less__one__cancel__iff,axiom,
% 0.30/0.63 ! [A: real,X2: real] :
% 0.30/0.63 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ( ord_less_real @ ( log @ A @ X2 ) @ one_one_real )
% 0.30/0.63 = ( ord_less_real @ X2 @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log_less_one_cancel_iff
% 0.30/0.63 thf(fact_7662_one__less__log__cancel__iff,axiom,
% 0.30/0.63 ! [A: real,X2: real] :
% 0.30/0.63 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ( ord_less_real @ one_one_real @ ( log @ A @ X2 ) )
% 0.30/0.63 = ( ord_less_real @ A @ X2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % one_less_log_cancel_iff
% 0.30/0.63 thf(fact_7663_log__less__zero__cancel__iff,axiom,
% 0.30/0.63 ! [A: real,X2: real] :
% 0.30/0.63 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ( ord_less_real @ ( log @ A @ X2 ) @ zero_zero_real )
% 0.30/0.63 = ( ord_less_real @ X2 @ one_one_real ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log_less_zero_cancel_iff
% 0.30/0.63 thf(fact_7664_zero__less__log__cancel__iff,axiom,
% 0.30/0.63 ! [A: real,X2: real] :
% 0.30/0.63 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X2 ) )
% 0.30/0.63 = ( ord_less_real @ one_one_real @ X2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % zero_less_log_cancel_iff
% 0.30/0.63 thf(fact_7665_zero__le__log__cancel__iff,axiom,
% 0.30/0.63 ! [A: real,X2: real] :
% 0.30/0.63 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X2 ) )
% 0.30/0.63 = ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % zero_le_log_cancel_iff
% 0.30/0.63 thf(fact_7666_log__le__zero__cancel__iff,axiom,
% 0.30/0.63 ! [A: real,X2: real] :
% 0.30/0.63 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ( ord_less_eq_real @ ( log @ A @ X2 ) @ zero_zero_real )
% 0.30/0.63 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log_le_zero_cancel_iff
% 0.30/0.63 thf(fact_7667_one__le__log__cancel__iff,axiom,
% 0.30/0.63 ! [A: real,X2: real] :
% 0.30/0.63 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X2 ) )
% 0.30/0.63 = ( ord_less_eq_real @ A @ X2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % one_le_log_cancel_iff
% 0.30/0.63 thf(fact_7668_log__le__one__cancel__iff,axiom,
% 0.30/0.63 ! [A: real,X2: real] :
% 0.30/0.63 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ( ord_less_eq_real @ ( log @ A @ X2 ) @ one_one_real )
% 0.30/0.63 = ( ord_less_eq_real @ X2 @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log_le_one_cancel_iff
% 0.30/0.63 thf(fact_7669_log__le__cancel__iff,axiom,
% 0.30/0.63 ! [A: real,X2: real,Y3: real] :
% 0.30/0.63 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.63 => ( ( ord_less_eq_real @ ( log @ A @ X2 ) @ ( log @ A @ Y3 ) )
% 0.30/0.63 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log_le_cancel_iff
% 0.30/0.63 thf(fact_7670_powser__zero,axiom,
% 0.30/0.63 ! [F: nat > complex] :
% 0.30/0.63 ( ( suminf_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) )
% 0.30/0.63 = ( F @ zero_zero_nat ) ) ).
% 0.30/0.63
% 0.30/0.63 % powser_zero
% 0.30/0.63 thf(fact_7671_powser__zero,axiom,
% 0.30/0.63 ! [F: nat > real] :
% 0.30/0.63 ( ( suminf_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) )
% 0.30/0.63 = ( F @ zero_zero_nat ) ) ).
% 0.30/0.63
% 0.30/0.63 % powser_zero
% 0.30/0.63 thf(fact_7672_log__pow__cancel,axiom,
% 0.30/0.63 ! [A: real,B: nat] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.63 => ( ( A != one_one_real )
% 0.30/0.63 => ( ( log @ A @ ( power_power_real @ A @ B ) )
% 0.30/0.63 = ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log_pow_cancel
% 0.30/0.63 thf(fact_7673_lessThan__def,axiom,
% 0.30/0.63 ( set_ord_lessThan_rat
% 0.30/0.63 = ( ^ [U2: rat] :
% 0.30/0.63 ( collect_rat
% 0.30/0.63 @ ^ [X: rat] : ( ord_less_rat @ X @ U2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_def
% 0.30/0.63 thf(fact_7674_lessThan__def,axiom,
% 0.30/0.63 ( set_ord_lessThan_num
% 0.30/0.63 = ( ^ [U2: num] :
% 0.30/0.63 ( collect_num
% 0.30/0.63 @ ^ [X: num] : ( ord_less_num @ X @ U2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_def
% 0.30/0.63 thf(fact_7675_lessThan__def,axiom,
% 0.30/0.63 ( set_ord_lessThan_nat
% 0.30/0.63 = ( ^ [U2: nat] :
% 0.30/0.63 ( collect_nat
% 0.30/0.63 @ ^ [X: nat] : ( ord_less_nat @ X @ U2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_def
% 0.30/0.63 thf(fact_7676_lessThan__def,axiom,
% 0.30/0.63 ( set_ord_lessThan_int
% 0.30/0.63 = ( ^ [U2: int] :
% 0.30/0.63 ( collect_int
% 0.30/0.63 @ ^ [X: int] : ( ord_less_int @ X @ U2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_def
% 0.30/0.63 thf(fact_7677_lessThan__def,axiom,
% 0.30/0.63 ( set_or5984915006950818249n_real
% 0.30/0.63 = ( ^ [U2: real] :
% 0.30/0.63 ( collect_real
% 0.30/0.63 @ ^ [X: real] : ( ord_less_real @ X @ U2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_def
% 0.30/0.63 thf(fact_7678_lessThan__strict__subset__iff,axiom,
% 0.30/0.63 ! [M: rat,N: rat] :
% 0.30/0.63 ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N ) )
% 0.30/0.63 = ( ord_less_rat @ M @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_strict_subset_iff
% 0.30/0.63 thf(fact_7679_lessThan__strict__subset__iff,axiom,
% 0.30/0.63 ! [M: num,N: num] :
% 0.30/0.63 ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
% 0.30/0.63 = ( ord_less_num @ M @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_strict_subset_iff
% 0.30/0.63 thf(fact_7680_lessThan__strict__subset__iff,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.63 = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_strict_subset_iff
% 0.30/0.63 thf(fact_7681_lessThan__strict__subset__iff,axiom,
% 0.30/0.63 ! [M: int,N: int] :
% 0.30/0.63 ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
% 0.30/0.63 = ( ord_less_int @ M @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_strict_subset_iff
% 0.30/0.63 thf(fact_7682_lessThan__strict__subset__iff,axiom,
% 0.30/0.63 ! [M: real,N: real] :
% 0.30/0.63 ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
% 0.30/0.63 = ( ord_less_real @ M @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % lessThan_strict_subset_iff
% 0.30/0.63 thf(fact_7683_complex__mod__minus__le__complex__mod,axiom,
% 0.30/0.63 ! [X2: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X2 ) ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % complex_mod_minus_le_complex_mod
% 0.30/0.63 thf(fact_7684_complex__mod__triangle__ineq2,axiom,
% 0.30/0.63 ! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% 0.30/0.63
% 0.30/0.63 % complex_mod_triangle_ineq2
% 0.30/0.63 thf(fact_7685_log__of__power__eq,axiom,
% 0.30/0.63 ! [M: nat,B: real,N: nat] :
% 0.30/0.63 ( ( ( semiri5074537144036343181t_real @ M )
% 0.30/0.63 = ( power_power_real @ B @ N ) )
% 0.30/0.63 => ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.63 => ( ( semiri5074537144036343181t_real @ N )
% 0.30/0.63 = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log_of_power_eq
% 0.30/0.63 thf(fact_7686_less__log__of__power,axiom,
% 0.30/0.63 ! [B: real,N: nat,M: real] :
% 0.30/0.63 ( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
% 0.30/0.63 => ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.63 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % less_log_of_power
% 0.30/0.63 thf(fact_7687_sum_Onat__diff__reindex,axiom,
% 0.30/0.63 ! [G: nat > nat,N: nat] :
% 0.30/0.63 ( ( groups3542108847815614940at_nat
% 0.30/0.63 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.63 = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum.nat_diff_reindex
% 0.30/0.63 thf(fact_7688_sum_Onat__diff__reindex,axiom,
% 0.30/0.63 ! [G: nat > real,N: nat] :
% 0.30/0.63 ( ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.63 = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum.nat_diff_reindex
% 0.30/0.63 thf(fact_7689_sum__diff__distrib,axiom,
% 0.30/0.63 ! [Q: int > nat,P: int > nat,N: int] :
% 0.30/0.63 ( ! [X5: int] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
% 0.30/0.63 => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N ) ) )
% 0.30/0.63 = ( groups4541462559716669496nt_nat
% 0.30/0.63 @ ^ [X: int] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 0.30/0.63 @ ( set_ord_lessThan_int @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_diff_distrib
% 0.30/0.63 thf(fact_7690_sum__diff__distrib,axiom,
% 0.30/0.63 ! [Q: real > nat,P: real > nat,N: real] :
% 0.30/0.63 ( ! [X5: real] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
% 0.30/0.63 => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N ) ) )
% 0.30/0.63 = ( groups1935376822645274424al_nat
% 0.30/0.63 @ ^ [X: real] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 0.30/0.63 @ ( set_or5984915006950818249n_real @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_diff_distrib
% 0.30/0.63 thf(fact_7691_sum__diff__distrib,axiom,
% 0.30/0.63 ! [Q: nat > nat,P: nat > nat,N: nat] :
% 0.30/0.63 ( ! [X5: nat] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
% 0.30/0.63 => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
% 0.30/0.63 = ( groups3542108847815614940at_nat
% 0.30/0.63 @ ^ [X: nat] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_diff_distrib
% 0.30/0.63 thf(fact_7692_log__base__change,axiom,
% 0.30/0.63 ! [A: real,B: real,X2: real] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.63 => ( ( A != one_one_real )
% 0.30/0.63 => ( ( log @ B @ X2 )
% 0.30/0.63 = ( divide_divide_real @ ( log @ A @ X2 ) @ ( log @ A @ B ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log_base_change
% 0.30/0.63 thf(fact_7693_le__log__of__power,axiom,
% 0.30/0.63 ! [B: real,N: nat,M: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
% 0.30/0.63 => ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.63 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % le_log_of_power
% 0.30/0.63 thf(fact_7694_log__base__pow,axiom,
% 0.30/0.63 ! [A: real,N: nat,X2: real] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.63 => ( ( log @ ( power_power_real @ A @ N ) @ X2 )
% 0.30/0.63 = ( divide_divide_real @ ( log @ A @ X2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log_base_pow
% 0.30/0.63 thf(fact_7695_log__nat__power,axiom,
% 0.30/0.63 ! [X2: real,B: real,N: nat] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ( log @ B @ ( power_power_real @ X2 @ N ) )
% 0.30/0.63 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log_nat_power
% 0.30/0.63 thf(fact_7696_sum_OlessThan__Suc__shift,axiom,
% 0.30/0.63 ! [G: nat > rat,N: nat] :
% 0.30/0.63 ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.63 = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 0.30/0.63 @ ( groups2906978787729119204at_rat
% 0.30/0.63 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum.lessThan_Suc_shift
% 0.30/0.63 thf(fact_7697_sum_OlessThan__Suc__shift,axiom,
% 0.30/0.63 ! [G: nat > int,N: nat] :
% 0.30/0.63 ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.63 = ( plus_plus_int @ ( G @ zero_zero_nat )
% 0.30/0.63 @ ( groups3539618377306564664at_int
% 0.30/0.63 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum.lessThan_Suc_shift
% 0.30/0.63 thf(fact_7698_sum_OlessThan__Suc__shift,axiom,
% 0.30/0.63 ! [G: nat > nat,N: nat] :
% 0.30/0.63 ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.63 = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 0.30/0.63 @ ( groups3542108847815614940at_nat
% 0.30/0.63 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum.lessThan_Suc_shift
% 0.30/0.63 thf(fact_7699_sum_OlessThan__Suc__shift,axiom,
% 0.30/0.63 ! [G: nat > real,N: nat] :
% 0.30/0.63 ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.63 = ( plus_plus_real @ ( G @ zero_zero_nat )
% 0.30/0.63 @ ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum.lessThan_Suc_shift
% 0.30/0.63 thf(fact_7700_sum__lessThan__telescope,axiom,
% 0.30/0.63 ! [F: nat > rat,M: nat] :
% 0.30/0.63 ( ( groups2906978787729119204at_rat
% 0.30/0.63 @ ^ [N3: nat] : ( minus_minus_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63 = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_lessThan_telescope
% 0.30/0.63 thf(fact_7701_sum__lessThan__telescope,axiom,
% 0.30/0.63 ! [F: nat > int,M: nat] :
% 0.30/0.63 ( ( groups3539618377306564664at_int
% 0.30/0.63 @ ^ [N3: nat] : ( minus_minus_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63 = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_lessThan_telescope
% 0.30/0.63 thf(fact_7702_sum__lessThan__telescope,axiom,
% 0.30/0.63 ! [F: nat > real,M: nat] :
% 0.30/0.63 ( ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [N3: nat] : ( minus_minus_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63 = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_lessThan_telescope
% 0.30/0.63 thf(fact_7703_sum__lessThan__telescope_H,axiom,
% 0.30/0.63 ! [F: nat > rat,M: nat] :
% 0.30/0.63 ( ( groups2906978787729119204at_rat
% 0.30/0.63 @ ^ [N3: nat] : ( minus_minus_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63 = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_lessThan_telescope'
% 0.30/0.63 thf(fact_7704_sum__lessThan__telescope_H,axiom,
% 0.30/0.63 ! [F: nat > int,M: nat] :
% 0.30/0.63 ( ( groups3539618377306564664at_int
% 0.30/0.63 @ ^ [N3: nat] : ( minus_minus_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63 = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_lessThan_telescope'
% 0.30/0.63 thf(fact_7705_sum__lessThan__telescope_H,axiom,
% 0.30/0.63 ! [F: nat > real,M: nat] :
% 0.30/0.63 ( ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63 = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_lessThan_telescope'
% 0.30/0.63 thf(fact_7706_sumr__diff__mult__const2,axiom,
% 0.30/0.63 ! [F: nat > int,N: nat,R2: int] :
% 0.30/0.63 ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ R2 ) )
% 0.30/0.63 = ( groups3539618377306564664at_int
% 0.30/0.63 @ ^ [I4: nat] : ( minus_minus_int @ ( F @ I4 ) @ R2 )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sumr_diff_mult_const2
% 0.30/0.63 thf(fact_7707_sumr__diff__mult__const2,axiom,
% 0.30/0.63 ! [F: nat > rat,N: nat,R2: rat] :
% 0.30/0.63 ( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ R2 ) )
% 0.30/0.63 = ( groups2906978787729119204at_rat
% 0.30/0.63 @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ I4 ) @ R2 )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sumr_diff_mult_const2
% 0.30/0.63 thf(fact_7708_sumr__diff__mult__const2,axiom,
% 0.30/0.63 ! [F: nat > real,N: nat,R2: real] :
% 0.30/0.63 ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ R2 ) )
% 0.30/0.63 = ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [I4: nat] : ( minus_minus_real @ ( F @ I4 ) @ R2 )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sumr_diff_mult_const2
% 0.30/0.63 thf(fact_7709_sum_OatLeast1__atMost__eq,axiom,
% 0.30/0.63 ! [G: nat > nat,N: nat] :
% 0.30/0.63 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.30/0.63 = ( groups3542108847815614940at_nat
% 0.30/0.63 @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum.atLeast1_atMost_eq
% 0.30/0.63 thf(fact_7710_sum_OatLeast1__atMost__eq,axiom,
% 0.30/0.63 ! [G: nat > real,N: nat] :
% 0.30/0.63 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.30/0.63 = ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum.atLeast1_atMost_eq
% 0.30/0.63 thf(fact_7711_log2__of__power__eq,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( M
% 0.30/0.63 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.63 => ( ( semiri5074537144036343181t_real @ N )
% 0.30/0.63 = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log2_of_power_eq
% 0.30/0.63 thf(fact_7712_log__of__power__less,axiom,
% 0.30/0.63 ! [M: nat,B: real,N: nat] :
% 0.30/0.63 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 0.30/0.63 => ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.63 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.63 => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log_of_power_less
% 0.30/0.63 thf(fact_7713_log__mult,axiom,
% 0.30/0.63 ! [A: real,X2: real,Y3: real] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.63 => ( ( A != one_one_real )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.63 => ( ( log @ A @ ( times_times_real @ X2 @ Y3 ) )
% 0.30/0.63 = ( plus_plus_real @ ( log @ A @ X2 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log_mult
% 0.30/0.63 thf(fact_7714_log__divide,axiom,
% 0.30/0.63 ! [A: real,X2: real,Y3: real] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.63 => ( ( A != one_one_real )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.63 => ( ( log @ A @ ( divide_divide_real @ X2 @ Y3 ) )
% 0.30/0.63 = ( minus_minus_real @ ( log @ A @ X2 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log_divide
% 0.30/0.63 thf(fact_7715_one__diff__power__eq,axiom,
% 0.30/0.63 ! [X2: complex,N: nat] :
% 0.30/0.63 ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ N ) )
% 0.30/0.63 = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % one_diff_power_eq
% 0.30/0.63 thf(fact_7716_one__diff__power__eq,axiom,
% 0.30/0.63 ! [X2: rat,N: nat] :
% 0.30/0.63 ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N ) )
% 0.30/0.63 = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % one_diff_power_eq
% 0.30/0.63 thf(fact_7717_one__diff__power__eq,axiom,
% 0.30/0.63 ! [X2: int,N: nat] :
% 0.30/0.63 ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X2 @ N ) )
% 0.30/0.63 = ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % one_diff_power_eq
% 0.30/0.63 thf(fact_7718_one__diff__power__eq,axiom,
% 0.30/0.63 ! [X2: real,N: nat] :
% 0.30/0.63 ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N ) )
% 0.30/0.63 = ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % one_diff_power_eq
% 0.30/0.63 thf(fact_7719_power__diff__1__eq,axiom,
% 0.30/0.63 ! [X2: complex,N: nat] :
% 0.30/0.63 ( ( minus_minus_complex @ ( power_power_complex @ X2 @ N ) @ one_one_complex )
% 0.30/0.63 = ( times_times_complex @ ( minus_minus_complex @ X2 @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % power_diff_1_eq
% 0.30/0.63 thf(fact_7720_power__diff__1__eq,axiom,
% 0.30/0.63 ! [X2: rat,N: nat] :
% 0.30/0.63 ( ( minus_minus_rat @ ( power_power_rat @ X2 @ N ) @ one_one_rat )
% 0.30/0.63 = ( times_times_rat @ ( minus_minus_rat @ X2 @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % power_diff_1_eq
% 0.30/0.63 thf(fact_7721_power__diff__1__eq,axiom,
% 0.30/0.63 ! [X2: int,N: nat] :
% 0.30/0.63 ( ( minus_minus_int @ ( power_power_int @ X2 @ N ) @ one_one_int )
% 0.30/0.63 = ( times_times_int @ ( minus_minus_int @ X2 @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % power_diff_1_eq
% 0.30/0.63 thf(fact_7722_power__diff__1__eq,axiom,
% 0.30/0.63 ! [X2: real,N: nat] :
% 0.30/0.63 ( ( minus_minus_real @ ( power_power_real @ X2 @ N ) @ one_one_real )
% 0.30/0.63 = ( times_times_real @ ( minus_minus_real @ X2 @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % power_diff_1_eq
% 0.30/0.63 thf(fact_7723_geometric__sum,axiom,
% 0.30/0.63 ! [X2: complex,N: nat] :
% 0.30/0.63 ( ( X2 != one_one_complex )
% 0.30/0.63 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.63 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X2 @ N ) @ one_one_complex ) @ ( minus_minus_complex @ X2 @ one_one_complex ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % geometric_sum
% 0.30/0.63 thf(fact_7724_geometric__sum,axiom,
% 0.30/0.63 ! [X2: rat,N: nat] :
% 0.30/0.63 ( ( X2 != one_one_rat )
% 0.30/0.63 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.63 = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X2 @ N ) @ one_one_rat ) @ ( minus_minus_rat @ X2 @ one_one_rat ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % geometric_sum
% 0.30/0.63 thf(fact_7725_geometric__sum,axiom,
% 0.30/0.63 ! [X2: real,N: nat] :
% 0.30/0.63 ( ( X2 != one_one_real )
% 0.30/0.63 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.63 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ N ) @ one_one_real ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % geometric_sum
% 0.30/0.63 thf(fact_7726_log__of__power__le,axiom,
% 0.30/0.63 ! [M: nat,B: real,N: nat] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 0.30/0.63 => ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.63 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.63 => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log_of_power_le
% 0.30/0.63 thf(fact_7727_log__eq__div__ln__mult__log,axiom,
% 0.30/0.63 ! [A: real,B: real,X2: real] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.63 => ( ( A != one_one_real )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.63 => ( ( B != one_one_real )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ( log @ A @ X2 )
% 0.30/0.63 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X2 ) ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log_eq_div_ln_mult_log
% 0.30/0.63 thf(fact_7728_monoseq__realpow,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.30/0.63 => ( topolo6980174941875973593q_real @ ( power_power_real @ X2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % monoseq_realpow
% 0.30/0.63 thf(fact_7729_sum__gp__strict,axiom,
% 0.30/0.63 ! [X2: complex,N: nat] :
% 0.30/0.63 ( ( ( X2 = one_one_complex )
% 0.30/0.63 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.63 = ( semiri8010041392384452111omplex @ N ) ) )
% 0.30/0.63 & ( ( X2 != one_one_complex )
% 0.30/0.63 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.63 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ N ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_gp_strict
% 0.30/0.63 thf(fact_7730_sum__gp__strict,axiom,
% 0.30/0.63 ! [X2: rat,N: nat] :
% 0.30/0.63 ( ( ( X2 = one_one_rat )
% 0.30/0.63 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.63 = ( semiri681578069525770553at_rat @ N ) ) )
% 0.30/0.63 & ( ( X2 != one_one_rat )
% 0.30/0.63 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.63 = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_gp_strict
% 0.30/0.63 thf(fact_7731_sum__gp__strict,axiom,
% 0.30/0.63 ! [X2: real,N: nat] :
% 0.30/0.63 ( ( ( X2 = one_one_real )
% 0.30/0.63 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.63 = ( semiri5074537144036343181t_real @ N ) ) )
% 0.30/0.63 & ( ( X2 != one_one_real )
% 0.30/0.63 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.63 = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_gp_strict
% 0.30/0.63 thf(fact_7732_lemma__termdiff1,axiom,
% 0.30/0.63 ! [Z: complex,H2: complex,M: nat] :
% 0.30/0.63 ( ( groups2073611262835488442omplex
% 0.30/0.63 @ ^ [P4: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_complex @ Z @ P4 ) ) @ ( power_power_complex @ Z @ M ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63 = ( groups2073611262835488442omplex
% 0.30/0.63 @ ^ [P4: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P4 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lemma_termdiff1
% 0.30/0.63 thf(fact_7733_lemma__termdiff1,axiom,
% 0.30/0.63 ! [Z: rat,H2: rat,M: nat] :
% 0.30/0.63 ( ( groups2906978787729119204at_rat
% 0.30/0.63 @ ^ [P4: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_rat @ Z @ P4 ) ) @ ( power_power_rat @ Z @ M ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63 = ( groups2906978787729119204at_rat
% 0.30/0.63 @ ^ [P4: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P4 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lemma_termdiff1
% 0.30/0.63 thf(fact_7734_lemma__termdiff1,axiom,
% 0.30/0.63 ! [Z: int,H2: int,M: nat] :
% 0.30/0.63 ( ( groups3539618377306564664at_int
% 0.30/0.63 @ ^ [P4: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_int @ Z @ P4 ) ) @ ( power_power_int @ Z @ M ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63 = ( groups3539618377306564664at_int
% 0.30/0.63 @ ^ [P4: nat] : ( times_times_int @ ( power_power_int @ Z @ P4 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lemma_termdiff1
% 0.30/0.63 thf(fact_7735_lemma__termdiff1,axiom,
% 0.30/0.63 ! [Z: real,H2: real,M: nat] :
% 0.30/0.63 ( ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [P4: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_real @ Z @ P4 ) ) @ ( power_power_real @ Z @ M ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63 = ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [P4: nat] : ( times_times_real @ ( power_power_real @ Z @ P4 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % lemma_termdiff1
% 0.30/0.63 thf(fact_7736_diff__power__eq__sum,axiom,
% 0.30/0.63 ! [X2: complex,N: nat,Y3: complex] :
% 0.30/0.63 ( ( minus_minus_complex @ ( power_power_complex @ X2 @ ( suc @ N ) ) @ ( power_power_complex @ Y3 @ ( suc @ N ) ) )
% 0.30/0.63 = ( times_times_complex @ ( minus_minus_complex @ X2 @ Y3 )
% 0.30/0.63 @ ( groups2073611262835488442omplex
% 0.30/0.63 @ ^ [P4: nat] : ( times_times_complex @ ( power_power_complex @ X2 @ P4 ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ P4 ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % diff_power_eq_sum
% 0.30/0.63 thf(fact_7737_diff__power__eq__sum,axiom,
% 0.30/0.63 ! [X2: rat,N: nat,Y3: rat] :
% 0.30/0.63 ( ( minus_minus_rat @ ( power_power_rat @ X2 @ ( suc @ N ) ) @ ( power_power_rat @ Y3 @ ( suc @ N ) ) )
% 0.30/0.63 = ( times_times_rat @ ( minus_minus_rat @ X2 @ Y3 )
% 0.30/0.63 @ ( groups2906978787729119204at_rat
% 0.30/0.63 @ ^ [P4: nat] : ( times_times_rat @ ( power_power_rat @ X2 @ P4 ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ P4 ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % diff_power_eq_sum
% 0.30/0.63 thf(fact_7738_diff__power__eq__sum,axiom,
% 0.30/0.63 ! [X2: int,N: nat,Y3: int] :
% 0.30/0.63 ( ( minus_minus_int @ ( power_power_int @ X2 @ ( suc @ N ) ) @ ( power_power_int @ Y3 @ ( suc @ N ) ) )
% 0.30/0.63 = ( times_times_int @ ( minus_minus_int @ X2 @ Y3 )
% 0.30/0.63 @ ( groups3539618377306564664at_int
% 0.30/0.63 @ ^ [P4: nat] : ( times_times_int @ ( power_power_int @ X2 @ P4 ) @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ P4 ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % diff_power_eq_sum
% 0.30/0.63 thf(fact_7739_diff__power__eq__sum,axiom,
% 0.30/0.63 ! [X2: real,N: nat,Y3: real] :
% 0.30/0.63 ( ( minus_minus_real @ ( power_power_real @ X2 @ ( suc @ N ) ) @ ( power_power_real @ Y3 @ ( suc @ N ) ) )
% 0.30/0.63 = ( times_times_real @ ( minus_minus_real @ X2 @ Y3 )
% 0.30/0.63 @ ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [P4: nat] : ( times_times_real @ ( power_power_real @ X2 @ P4 ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ P4 ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % diff_power_eq_sum
% 0.30/0.63 thf(fact_7740_power__diff__sumr2,axiom,
% 0.30/0.63 ! [X2: complex,N: nat,Y3: complex] :
% 0.30/0.63 ( ( minus_minus_complex @ ( power_power_complex @ X2 @ N ) @ ( power_power_complex @ Y3 @ N ) )
% 0.30/0.63 = ( times_times_complex @ ( minus_minus_complex @ X2 @ Y3 )
% 0.30/0.63 @ ( groups2073611262835488442omplex
% 0.30/0.63 @ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_complex @ X2 @ I4 ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % power_diff_sumr2
% 0.30/0.63 thf(fact_7741_power__diff__sumr2,axiom,
% 0.30/0.63 ! [X2: rat,N: nat,Y3: rat] :
% 0.30/0.63 ( ( minus_minus_rat @ ( power_power_rat @ X2 @ N ) @ ( power_power_rat @ Y3 @ N ) )
% 0.30/0.63 = ( times_times_rat @ ( minus_minus_rat @ X2 @ Y3 )
% 0.30/0.63 @ ( groups2906978787729119204at_rat
% 0.30/0.63 @ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_rat @ X2 @ I4 ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % power_diff_sumr2
% 0.30/0.63 thf(fact_7742_power__diff__sumr2,axiom,
% 0.30/0.63 ! [X2: int,N: nat,Y3: int] :
% 0.30/0.63 ( ( minus_minus_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y3 @ N ) )
% 0.30/0.63 = ( times_times_int @ ( minus_minus_int @ X2 @ Y3 )
% 0.30/0.63 @ ( groups3539618377306564664at_int
% 0.30/0.63 @ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_int @ X2 @ I4 ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % power_diff_sumr2
% 0.30/0.63 thf(fact_7743_power__diff__sumr2,axiom,
% 0.30/0.63 ! [X2: real,N: nat,Y3: real] :
% 0.30/0.63 ( ( minus_minus_real @ ( power_power_real @ X2 @ N ) @ ( power_power_real @ Y3 @ N ) )
% 0.30/0.63 = ( times_times_real @ ( minus_minus_real @ X2 @ Y3 )
% 0.30/0.63 @ ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_real @ X2 @ I4 ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % power_diff_sumr2
% 0.30/0.63 thf(fact_7744_less__log2__of__power,axiom,
% 0.30/0.63 ! [N: nat,M: nat] :
% 0.30/0.63 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.30/0.63 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % less_log2_of_power
% 0.30/0.63 thf(fact_7745_le__log2__of__power,axiom,
% 0.30/0.63 ! [N: nat,M: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.30/0.63 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % le_log2_of_power
% 0.30/0.63 thf(fact_7746_real__sum__nat__ivl__bounded2,axiom,
% 0.30/0.63 ! [N: nat,F: nat > rat,K5: rat,K: nat] :
% 0.30/0.63 ( ! [P7: nat] :
% 0.30/0.63 ( ( ord_less_nat @ P7 @ N )
% 0.30/0.63 => ( ord_less_eq_rat @ ( F @ P7 ) @ K5 ) )
% 0.30/0.63 => ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
% 0.30/0.63 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K5 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % real_sum_nat_ivl_bounded2
% 0.30/0.63 thf(fact_7747_real__sum__nat__ivl__bounded2,axiom,
% 0.30/0.63 ! [N: nat,F: nat > int,K5: int,K: nat] :
% 0.30/0.63 ( ! [P7: nat] :
% 0.30/0.63 ( ( ord_less_nat @ P7 @ N )
% 0.30/0.63 => ( ord_less_eq_int @ ( F @ P7 ) @ K5 ) )
% 0.30/0.63 => ( ( ord_less_eq_int @ zero_zero_int @ K5 )
% 0.30/0.63 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K5 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % real_sum_nat_ivl_bounded2
% 0.30/0.63 thf(fact_7748_real__sum__nat__ivl__bounded2,axiom,
% 0.30/0.63 ! [N: nat,F: nat > nat,K5: nat,K: nat] :
% 0.30/0.63 ( ! [P7: nat] :
% 0.30/0.63 ( ( ord_less_nat @ P7 @ N )
% 0.30/0.63 => ( ord_less_eq_nat @ ( F @ P7 ) @ K5 ) )
% 0.30/0.63 => ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
% 0.30/0.63 => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K5 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % real_sum_nat_ivl_bounded2
% 0.30/0.63 thf(fact_7749_real__sum__nat__ivl__bounded2,axiom,
% 0.30/0.63 ! [N: nat,F: nat > real,K5: real,K: nat] :
% 0.30/0.63 ( ! [P7: nat] :
% 0.30/0.63 ( ( ord_less_nat @ P7 @ N )
% 0.30/0.63 => ( ord_less_eq_real @ ( F @ P7 ) @ K5 ) )
% 0.30/0.63 => ( ( ord_less_eq_real @ zero_zero_real @ K5 )
% 0.30/0.63 => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K5 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % real_sum_nat_ivl_bounded2
% 0.30/0.63 thf(fact_7750_one__diff__power__eq_H,axiom,
% 0.30/0.63 ! [X2: complex,N: nat] :
% 0.30/0.63 ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ N ) )
% 0.30/0.63 = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X2 )
% 0.30/0.63 @ ( groups2073611262835488442omplex
% 0.30/0.63 @ ^ [I4: nat] : ( power_power_complex @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % one_diff_power_eq'
% 0.30/0.63 thf(fact_7751_one__diff__power__eq_H,axiom,
% 0.30/0.63 ! [X2: rat,N: nat] :
% 0.30/0.63 ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N ) )
% 0.30/0.63 = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 )
% 0.30/0.63 @ ( groups2906978787729119204at_rat
% 0.30/0.63 @ ^ [I4: nat] : ( power_power_rat @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % one_diff_power_eq'
% 0.30/0.63 thf(fact_7752_one__diff__power__eq_H,axiom,
% 0.30/0.63 ! [X2: int,N: nat] :
% 0.30/0.63 ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X2 @ N ) )
% 0.30/0.63 = ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 )
% 0.30/0.63 @ ( groups3539618377306564664at_int
% 0.30/0.63 @ ^ [I4: nat] : ( power_power_int @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % one_diff_power_eq'
% 0.30/0.63 thf(fact_7753_one__diff__power__eq_H,axiom,
% 0.30/0.63 ! [X2: real,N: nat] :
% 0.30/0.63 ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N ) )
% 0.30/0.63 = ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 )
% 0.30/0.63 @ ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [I4: nat] : ( power_power_real @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % one_diff_power_eq'
% 0.30/0.63 thf(fact_7754_log2__of__power__less,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.63 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.63 => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log2_of_power_less
% 0.30/0.63 thf(fact_7755_sum__split__even__odd,axiom,
% 0.30/0.63 ! [F: nat > real,G: nat > real,N: nat] :
% 0.30/0.63 ( ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( F @ I4 ) @ ( G @ I4 ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.63 = ( plus_plus_real
% 0.30/0.63 @ ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [I4: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.63 @ ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ one_one_nat ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_split_even_odd
% 0.30/0.63 thf(fact_7756_log2__of__power__le,axiom,
% 0.30/0.63 ! [M: nat,N: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.63 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.63 => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log2_of_power_le
% 0.30/0.63 thf(fact_7757_insert__bound__size__univ,axiom,
% 0.30/0.63 ! [T: vEBT_VEBT,N: nat,U: real,X2: nat] :
% 0.30/0.63 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.63 => ( ( U
% 0.30/0.63 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.63 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_i_n_s_e_r_t @ T @ X2 ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % insert_bound_size_univ
% 0.30/0.63 thf(fact_7758_member__bound__size__univ,axiom,
% 0.30/0.63 ! [T: vEBT_VEBT,N: nat,U: real,X2: nat] :
% 0.30/0.63 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.63 => ( ( U
% 0.30/0.63 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.63 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_m_e_m_b_e_r @ T @ X2 ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % member_bound_size_univ
% 0.30/0.63 thf(fact_7759_arctan__series,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.30/0.63 => ( ( arctan @ X2 )
% 0.30/0.63 = ( suminf_real
% 0.30/0.63 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % arctan_series
% 0.30/0.63 thf(fact_7760_norm__divide__numeral,axiom,
% 0.30/0.63 ! [A: real,W: num] :
% 0.30/0.63 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.63 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_divide_numeral
% 0.30/0.63 thf(fact_7761_norm__divide__numeral,axiom,
% 0.30/0.63 ! [A: complex,W: num] :
% 0.30/0.63 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.63 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_divide_numeral
% 0.30/0.63 thf(fact_7762_norm__mult__numeral2,axiom,
% 0.30/0.63 ! [A: real,W: num] :
% 0.30/0.63 ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.63 = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_mult_numeral2
% 0.30/0.63 thf(fact_7763_norm__mult__numeral2,axiom,
% 0.30/0.63 ! [A: complex,W: num] :
% 0.30/0.63 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.63 = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_mult_numeral2
% 0.30/0.63 thf(fact_7764_norm__mult__numeral1,axiom,
% 0.30/0.63 ! [W: num,A: real] :
% 0.30/0.63 ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 0.30/0.63 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_mult_numeral1
% 0.30/0.63 thf(fact_7765_norm__mult__numeral1,axiom,
% 0.30/0.63 ! [W: num,A: complex] :
% 0.30/0.63 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 0.30/0.63 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_mult_numeral1
% 0.30/0.63 thf(fact_7766_norm__neg__numeral,axiom,
% 0.30/0.63 ! [W: num] :
% 0.30/0.63 ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.63 = ( numeral_numeral_real @ W ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_neg_numeral
% 0.30/0.63 thf(fact_7767_norm__neg__numeral,axiom,
% 0.30/0.63 ! [W: num] :
% 0.30/0.63 ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.63 = ( numeral_numeral_real @ W ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_neg_numeral
% 0.30/0.63 thf(fact_7768_norm__le__zero__iff,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X2 ) @ zero_zero_real )
% 0.30/0.63 = ( X2 = zero_zero_real ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_le_zero_iff
% 0.30/0.63 thf(fact_7769_norm__le__zero__iff,axiom,
% 0.30/0.63 ! [X2: complex] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ zero_zero_real )
% 0.30/0.63 = ( X2 = zero_zero_complex ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_le_zero_iff
% 0.30/0.63 thf(fact_7770_norm__one,axiom,
% 0.30/0.63 ( ( real_V7735802525324610683m_real @ one_one_real )
% 0.30/0.63 = one_one_real ) ).
% 0.30/0.63
% 0.30/0.63 % norm_one
% 0.30/0.63 thf(fact_7771_norm__one,axiom,
% 0.30/0.63 ( ( real_V1022390504157884413omplex @ one_one_complex )
% 0.30/0.63 = one_one_real ) ).
% 0.30/0.63
% 0.30/0.63 % norm_one
% 0.30/0.63 thf(fact_7772_norm__numeral,axiom,
% 0.30/0.63 ! [W: num] :
% 0.30/0.63 ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
% 0.30/0.63 = ( numeral_numeral_real @ W ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_numeral
% 0.30/0.63 thf(fact_7773_norm__numeral,axiom,
% 0.30/0.63 ! [W: num] :
% 0.30/0.63 ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.63 = ( numeral_numeral_real @ W ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_numeral
% 0.30/0.63 thf(fact_7774_norm__minus__commute,axiom,
% 0.30/0.63 ! [A: real,B: real] :
% 0.30/0.63 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) )
% 0.30/0.63 = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_minus_commute
% 0.30/0.63 thf(fact_7775_norm__minus__commute,axiom,
% 0.30/0.63 ! [A: complex,B: complex] :
% 0.30/0.63 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) )
% 0.30/0.63 = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ A ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_minus_commute
% 0.30/0.63 thf(fact_7776_norm__ge__zero,axiom,
% 0.30/0.63 ! [X2: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_ge_zero
% 0.30/0.63 thf(fact_7777_norm__mult,axiom,
% 0.30/0.63 ! [X2: real,Y3: real] :
% 0.30/0.63 ( ( real_V7735802525324610683m_real @ ( times_times_real @ X2 @ Y3 ) )
% 0.30/0.63 = ( times_times_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_mult
% 0.30/0.63 thf(fact_7778_norm__mult,axiom,
% 0.30/0.63 ! [X2: complex,Y3: complex] :
% 0.30/0.63 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X2 @ Y3 ) )
% 0.30/0.63 = ( times_times_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_mult
% 0.30/0.63 thf(fact_7779_norm__divide,axiom,
% 0.30/0.63 ! [A: real,B: real] :
% 0.30/0.63 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.63 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_divide
% 0.30/0.63 thf(fact_7780_norm__divide,axiom,
% 0.30/0.63 ! [A: complex,B: complex] :
% 0.30/0.63 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.30/0.63 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_divide
% 0.30/0.63 thf(fact_7781_sum__norm__le,axiom,
% 0.30/0.63 ! [S3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > complex,G: product_prod_nat_nat > real] :
% 0.30/0.63 ( ! [X5: product_prod_nat_nat] :
% 0.30/0.63 ( ( member8440522571783428010at_nat @ X5 @ S3 )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups6381953495645901045omplex @ F @ S3 ) ) @ ( groups4567486121110086003t_real @ G @ S3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_norm_le
% 0.30/0.63 thf(fact_7782_sum__norm__le,axiom,
% 0.30/0.63 ! [S3: set_real,F: real > complex,G: real > real] :
% 0.30/0.63 ( ! [X5: real] :
% 0.30/0.63 ( ( member_real @ X5 @ S3 )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S3 ) ) @ ( groups8097168146408367636l_real @ G @ S3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_norm_le
% 0.30/0.63 thf(fact_7783_sum__norm__le,axiom,
% 0.30/0.63 ! [S3: set_int,F: int > complex,G: int > real] :
% 0.30/0.63 ( ! [X5: int] :
% 0.30/0.63 ( ( member_int @ X5 @ S3 )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S3 ) ) @ ( groups8778361861064173332t_real @ G @ S3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_norm_le
% 0.30/0.63 thf(fact_7784_sum__norm__le,axiom,
% 0.30/0.63 ! [S3: set_nat,F: nat > complex,G: nat > real] :
% 0.30/0.63 ( ! [X5: nat] :
% 0.30/0.63 ( ( member_nat @ X5 @ S3 )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S3 ) ) @ ( groups6591440286371151544t_real @ G @ S3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_norm_le
% 0.30/0.63 thf(fact_7785_sum__norm__le,axiom,
% 0.30/0.63 ! [S3: set_complex,F: complex > complex,G: complex > real] :
% 0.30/0.63 ( ! [X5: complex] :
% 0.30/0.63 ( ( member_complex @ X5 @ S3 )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S3 ) ) @ ( groups5808333547571424918x_real @ G @ S3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_norm_le
% 0.30/0.63 thf(fact_7786_sum__norm__le,axiom,
% 0.30/0.63 ! [S3: set_nat,F: nat > real,G: nat > real] :
% 0.30/0.63 ( ! [X5: nat] :
% 0.30/0.63 ( ( member_nat @ X5 @ S3 )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S3 ) ) @ ( groups6591440286371151544t_real @ G @ S3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_norm_le
% 0.30/0.63 thf(fact_7787_norm__power,axiom,
% 0.30/0.63 ! [X2: real,N: nat] :
% 0.30/0.63 ( ( real_V7735802525324610683m_real @ ( power_power_real @ X2 @ N ) )
% 0.30/0.63 = ( power_power_real @ ( real_V7735802525324610683m_real @ X2 ) @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_power
% 0.30/0.63 thf(fact_7788_norm__power,axiom,
% 0.30/0.63 ! [X2: complex,N: nat] :
% 0.30/0.63 ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X2 @ N ) )
% 0.30/0.63 = ( power_power_real @ ( real_V1022390504157884413omplex @ X2 ) @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_power
% 0.30/0.63 thf(fact_7789_norm__sum,axiom,
% 0.30/0.63 ! [F: nat > complex,A2: set_nat] :
% 0.30/0.63 ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A2 ) )
% 0.30/0.63 @ ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [I4: nat] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
% 0.30/0.63 @ A2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_sum
% 0.30/0.63 thf(fact_7790_norm__sum,axiom,
% 0.30/0.63 ! [F: complex > complex,A2: set_complex] :
% 0.30/0.63 ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A2 ) )
% 0.30/0.63 @ ( groups5808333547571424918x_real
% 0.30/0.63 @ ^ [I4: complex] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
% 0.30/0.63 @ A2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_sum
% 0.30/0.63 thf(fact_7791_norm__sum,axiom,
% 0.30/0.63 ! [F: nat > real,A2: set_nat] :
% 0.30/0.63 ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 0.30/0.63 @ ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [I4: nat] : ( real_V7735802525324610683m_real @ ( F @ I4 ) )
% 0.30/0.63 @ A2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_sum
% 0.30/0.63 thf(fact_7792_norm__uminus__minus,axiom,
% 0.30/0.63 ! [X2: real,Y3: real] :
% 0.30/0.63 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X2 ) @ Y3 ) )
% 0.30/0.63 = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_uminus_minus
% 0.30/0.63 thf(fact_7793_norm__uminus__minus,axiom,
% 0.30/0.63 ! [X2: complex,Y3: complex] :
% 0.30/0.63 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X2 ) @ Y3 ) )
% 0.30/0.63 = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_uminus_minus
% 0.30/0.63 thf(fact_7794_nonzero__norm__divide,axiom,
% 0.30/0.63 ! [B: real,A: real] :
% 0.30/0.63 ( ( B != zero_zero_real )
% 0.30/0.63 => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.63 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nonzero_norm_divide
% 0.30/0.63 thf(fact_7795_nonzero__norm__divide,axiom,
% 0.30/0.63 ! [B: complex,A: complex] :
% 0.30/0.63 ( ( B != zero_zero_complex )
% 0.30/0.63 => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.30/0.63 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % nonzero_norm_divide
% 0.30/0.63 thf(fact_7796_power__eq__imp__eq__norm,axiom,
% 0.30/0.63 ! [W: real,N: nat,Z: real] :
% 0.30/0.63 ( ( ( power_power_real @ W @ N )
% 0.30/0.63 = ( power_power_real @ Z @ N ) )
% 0.30/0.63 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.63 => ( ( real_V7735802525324610683m_real @ W )
% 0.30/0.63 = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % power_eq_imp_eq_norm
% 0.30/0.63 thf(fact_7797_power__eq__imp__eq__norm,axiom,
% 0.30/0.63 ! [W: complex,N: nat,Z: complex] :
% 0.30/0.63 ( ( ( power_power_complex @ W @ N )
% 0.30/0.63 = ( power_power_complex @ Z @ N ) )
% 0.30/0.63 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.63 => ( ( real_V1022390504157884413omplex @ W )
% 0.30/0.63 = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % power_eq_imp_eq_norm
% 0.30/0.63 thf(fact_7798_norm__mult__less,axiom,
% 0.30/0.63 ! [X2: real,R2: real,Y3: real,S: real] :
% 0.30/0.63 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ R2 )
% 0.30/0.63 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
% 0.30/0.63 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X2 @ Y3 ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_mult_less
% 0.30/0.63 thf(fact_7799_norm__mult__less,axiom,
% 0.30/0.63 ! [X2: complex,R2: real,Y3: complex,S: real] :
% 0.30/0.63 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ R2 )
% 0.30/0.63 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
% 0.30/0.63 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X2 @ Y3 ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_mult_less
% 0.30/0.63 thf(fact_7800_norm__mult__ineq,axiom,
% 0.30/0.63 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X2 @ Y3 ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_mult_ineq
% 0.30/0.63 thf(fact_7801_norm__mult__ineq,axiom,
% 0.30/0.63 ! [X2: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X2 @ Y3 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_mult_ineq
% 0.30/0.63 thf(fact_7802_norm__triangle__lt,axiom,
% 0.30/0.63 ! [X2: real,Y3: real,E: real] :
% 0.30/0.63 ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 0.30/0.63 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ E ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_triangle_lt
% 0.30/0.63 thf(fact_7803_norm__triangle__lt,axiom,
% 0.30/0.63 ! [X2: complex,Y3: complex,E: real] :
% 0.30/0.63 ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 0.30/0.63 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ E ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_triangle_lt
% 0.30/0.63 thf(fact_7804_norm__add__less,axiom,
% 0.30/0.63 ! [X2: real,R2: real,Y3: real,S: real] :
% 0.30/0.63 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ R2 )
% 0.30/0.63 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
% 0.30/0.63 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_add_less
% 0.30/0.63 thf(fact_7805_norm__add__less,axiom,
% 0.30/0.63 ! [X2: complex,R2: real,Y3: complex,S: real] :
% 0.30/0.63 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ R2 )
% 0.30/0.63 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
% 0.30/0.63 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_add_less
% 0.30/0.63 thf(fact_7806_norm__add__leD,axiom,
% 0.30/0.63 ! [A: real,B: real,C: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_add_leD
% 0.30/0.63 thf(fact_7807_norm__add__leD,axiom,
% 0.30/0.63 ! [A: complex,B: complex,C: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_add_leD
% 0.30/0.63 thf(fact_7808_norm__triangle__le,axiom,
% 0.30/0.63 ! [X2: real,Y3: real,E: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ E ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_triangle_le
% 0.30/0.63 thf(fact_7809_norm__triangle__le,axiom,
% 0.30/0.63 ! [X2: complex,Y3: complex,E: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ E ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_triangle_le
% 0.30/0.63 thf(fact_7810_norm__triangle__ineq,axiom,
% 0.30/0.63 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_triangle_ineq
% 0.30/0.63 thf(fact_7811_norm__triangle__ineq,axiom,
% 0.30/0.63 ! [X2: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_triangle_ineq
% 0.30/0.63 thf(fact_7812_norm__triangle__mono,axiom,
% 0.30/0.63 ! [A: real,R2: real,B: real,S: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
% 0.30/0.63 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_triangle_mono
% 0.30/0.63 thf(fact_7813_norm__triangle__mono,axiom,
% 0.30/0.63 ! [A: complex,R2: real,B: complex,S: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
% 0.30/0.63 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_triangle_mono
% 0.30/0.63 thf(fact_7814_norm__power__ineq,axiom,
% 0.30/0.63 ! [X2: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X2 @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X2 ) @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_power_ineq
% 0.30/0.63 thf(fact_7815_norm__power__ineq,axiom,
% 0.30/0.63 ! [X2: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X2 @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X2 ) @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_power_ineq
% 0.30/0.63 thf(fact_7816_norm__diff__triangle__less,axiom,
% 0.30/0.63 ! [X2: real,Y3: real,E1: real,Z: real,E22: real] :
% 0.30/0.63 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ E1 )
% 0.30/0.63 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
% 0.30/0.63 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_diff_triangle_less
% 0.30/0.63 thf(fact_7817_norm__diff__triangle__less,axiom,
% 0.30/0.63 ! [X2: complex,Y3: complex,E1: real,Z: complex,E22: real] :
% 0.30/0.63 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) @ E1 )
% 0.30/0.63 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
% 0.30/0.63 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_diff_triangle_less
% 0.30/0.63 thf(fact_7818_norm__triangle__sub,axiom,
% 0.30/0.63 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y3 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_triangle_sub
% 0.30/0.63 thf(fact_7819_norm__triangle__sub,axiom,
% 0.30/0.63 ! [X2: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y3 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_triangle_sub
% 0.30/0.63 thf(fact_7820_norm__triangle__ineq4,axiom,
% 0.30/0.63 ! [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 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_triangle_ineq4
% 0.30/0.63 thf(fact_7821_norm__triangle__ineq4,axiom,
% 0.30/0.63 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_triangle_ineq4
% 0.30/0.63 thf(fact_7822_norm__diff__triangle__le,axiom,
% 0.30/0.63 ! [X2: real,Y3: real,E1: real,Z: real,E22: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ E1 )
% 0.30/0.63 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_diff_triangle_le
% 0.30/0.63 thf(fact_7823_norm__diff__triangle__le,axiom,
% 0.30/0.63 ! [X2: complex,Y3: complex,E1: real,Z: complex,E22: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) @ E1 )
% 0.30/0.63 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_diff_triangle_le
% 0.30/0.63 thf(fact_7824_norm__triangle__le__diff,axiom,
% 0.30/0.63 ! [X2: real,Y3: real,E: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ E ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_triangle_le_diff
% 0.30/0.63 thf(fact_7825_norm__triangle__le__diff,axiom,
% 0.30/0.63 ! [X2: complex,Y3: complex,E: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) @ E ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_triangle_le_diff
% 0.30/0.63 thf(fact_7826_norm__diff__ineq,axiom,
% 0.30/0.63 ! [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 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_diff_ineq
% 0.30/0.63 thf(fact_7827_norm__diff__ineq,axiom,
% 0.30/0.63 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_diff_ineq
% 0.30/0.63 thf(fact_7828_norm__triangle__ineq2,axiom,
% 0.30/0.63 ! [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 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_triangle_ineq2
% 0.30/0.63 thf(fact_7829_norm__triangle__ineq2,axiom,
% 0.30/0.63 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_triangle_ineq2
% 0.30/0.63 thf(fact_7830_power__eq__1__iff,axiom,
% 0.30/0.63 ! [W: real,N: nat] :
% 0.30/0.63 ( ( ( power_power_real @ W @ N )
% 0.30/0.63 = one_one_real )
% 0.30/0.63 => ( ( ( real_V7735802525324610683m_real @ W )
% 0.30/0.63 = one_one_real )
% 0.30/0.63 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % power_eq_1_iff
% 0.30/0.63 thf(fact_7831_power__eq__1__iff,axiom,
% 0.30/0.63 ! [W: complex,N: nat] :
% 0.30/0.63 ( ( ( power_power_complex @ W @ N )
% 0.30/0.63 = one_one_complex )
% 0.30/0.63 => ( ( ( real_V1022390504157884413omplex @ W )
% 0.30/0.63 = one_one_real )
% 0.30/0.63 | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % power_eq_1_iff
% 0.30/0.63 thf(fact_7832_norm__diff__triangle__ineq,axiom,
% 0.30/0.63 ! [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 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_diff_triangle_ineq
% 0.30/0.63 thf(fact_7833_norm__diff__triangle__ineq,axiom,
% 0.30/0.63 ! [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 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_diff_triangle_ineq
% 0.30/0.63 thf(fact_7834_norm__triangle__ineq3,axiom,
% 0.30/0.63 ! [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 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_triangle_ineq3
% 0.30/0.63 thf(fact_7835_norm__triangle__ineq3,axiom,
% 0.30/0.63 ! [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 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_triangle_ineq3
% 0.30/0.63 thf(fact_7836_square__norm__one,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.63 = one_one_real )
% 0.30/0.63 => ( ( real_V7735802525324610683m_real @ X2 )
% 0.30/0.63 = one_one_real ) ) ).
% 0.30/0.63
% 0.30/0.63 % square_norm_one
% 0.30/0.63 thf(fact_7837_square__norm__one,axiom,
% 0.30/0.63 ! [X2: complex] :
% 0.30/0.63 ( ( ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.63 = one_one_complex )
% 0.30/0.63 => ( ( real_V1022390504157884413omplex @ X2 )
% 0.30/0.63 = one_one_real ) ) ).
% 0.30/0.63
% 0.30/0.63 % square_norm_one
% 0.30/0.63 thf(fact_7838_norm__power__diff,axiom,
% 0.30/0.63 ! [Z: real,W: real,M: nat] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 0.30/0.63 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
% 0.30/0.63 => ( 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 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_power_diff
% 0.30/0.63 thf(fact_7839_norm__power__diff,axiom,
% 0.30/0.63 ! [Z: complex,W: complex,M: nat] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 0.30/0.63 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
% 0.30/0.63 => ( 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 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % norm_power_diff
% 0.30/0.63 thf(fact_7840_suminf__geometric,axiom,
% 0.30/0.63 ! [C: real] :
% 0.30/0.63 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 0.30/0.63 => ( ( suminf_real @ ( power_power_real @ C ) )
% 0.30/0.63 = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_geometric
% 0.30/0.63 thf(fact_7841_suminf__geometric,axiom,
% 0.30/0.63 ! [C: complex] :
% 0.30/0.63 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 0.30/0.63 => ( ( suminf_complex @ ( power_power_complex @ C ) )
% 0.30/0.63 = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_geometric
% 0.30/0.63 thf(fact_7842_sum__bounds__lt__plus1,axiom,
% 0.30/0.63 ! [F: nat > nat,Mm: nat] :
% 0.30/0.63 ( ( groups3542108847815614940at_nat
% 0.30/0.63 @ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ Mm ) )
% 0.30/0.63 = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_bounds_lt_plus1
% 0.30/0.63 thf(fact_7843_sum__bounds__lt__plus1,axiom,
% 0.30/0.63 ! [F: nat > real,Mm: nat] :
% 0.30/0.63 ( ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ Mm ) )
% 0.30/0.63 = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_bounds_lt_plus1
% 0.30/0.63 thf(fact_7844_heigt__uplog__rel,axiom,
% 0.30/0.63 ! [T: vEBT_VEBT,N: nat] :
% 0.30/0.63 ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.63 => ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ T ) )
% 0.30/0.63 = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % heigt_uplog_rel
% 0.30/0.63 thf(fact_7845_sumr__cos__zero__one,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( groups6591440286371151544t_real
% 0.30/0.63 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ zero_zero_real @ M3 ) )
% 0.30/0.63 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.63 = one_one_real ) ).
% 0.30/0.63
% 0.30/0.63 % sumr_cos_zero_one
% 0.30/0.63 thf(fact_7846_log__ceil__idem,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 0.30/0.63 => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) )
% 0.30/0.63 = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % log_ceil_idem
% 0.30/0.63 thf(fact_7847_of__int__ceiling__cancel,axiom,
% 0.30/0.63 ! [X2: rat] :
% 0.30/0.63 ( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.30/0.63 = X2 )
% 0.30/0.63 = ( ? [N3: int] :
% 0.30/0.63 ( X2
% 0.30/0.63 = ( ring_1_of_int_rat @ N3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_int_ceiling_cancel
% 0.30/0.63 thf(fact_7848_of__int__ceiling__cancel,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) )
% 0.30/0.63 = X2 )
% 0.30/0.63 = ( ? [N3: int] :
% 0.30/0.63 ( X2
% 0.30/0.63 = ( ring_1_of_int_real @ N3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_int_ceiling_cancel
% 0.30/0.63 thf(fact_7849_ceiling__numeral,axiom,
% 0.30/0.63 ! [V: num] :
% 0.30/0.63 ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
% 0.30/0.63 = ( numeral_numeral_int @ V ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_numeral
% 0.30/0.63 thf(fact_7850_ceiling__numeral,axiom,
% 0.30/0.63 ! [V: num] :
% 0.30/0.63 ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
% 0.30/0.63 = ( numeral_numeral_int @ V ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_numeral
% 0.30/0.63 thf(fact_7851_ceiling__one,axiom,
% 0.30/0.63 ( ( archim2889992004027027881ng_rat @ one_one_rat )
% 0.30/0.63 = one_one_int ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_one
% 0.30/0.63 thf(fact_7852_ceiling__one,axiom,
% 0.30/0.63 ( ( archim7802044766580827645g_real @ one_one_real )
% 0.30/0.63 = one_one_int ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_one
% 0.30/0.63 thf(fact_7853_cos__coeff__0,axiom,
% 0.30/0.63 ( ( cos_coeff @ zero_zero_nat )
% 0.30/0.63 = one_one_real ) ).
% 0.30/0.63
% 0.30/0.63 % cos_coeff_0
% 0.30/0.63 thf(fact_7854_ceiling__add__of__int,axiom,
% 0.30/0.63 ! [X2: rat,Z: int] :
% 0.30/0.63 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) )
% 0.30/0.63 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ Z ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_add_of_int
% 0.30/0.63 thf(fact_7855_ceiling__add__of__int,axiom,
% 0.30/0.63 ! [X2: real,Z: int] :
% 0.30/0.63 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X2 @ ( ring_1_of_int_real @ Z ) ) )
% 0.30/0.63 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X2 ) @ Z ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_add_of_int
% 0.30/0.63 thf(fact_7856_ceiling__diff__of__int,axiom,
% 0.30/0.63 ! [X2: rat,Z: int] :
% 0.30/0.63 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) )
% 0.30/0.63 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ Z ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_diff_of_int
% 0.30/0.63 thf(fact_7857_ceiling__diff__of__int,axiom,
% 0.30/0.63 ! [X2: real,Z: int] :
% 0.30/0.63 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X2 @ ( ring_1_of_int_real @ Z ) ) )
% 0.30/0.63 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X2 ) @ Z ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_diff_of_int
% 0.30/0.63 thf(fact_7858_ceiling__le__zero,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ zero_zero_int )
% 0.30/0.63 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_le_zero
% 0.30/0.63 thf(fact_7859_ceiling__le__zero,axiom,
% 0.30/0.63 ! [X2: rat] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ zero_zero_int )
% 0.30/0.63 = ( ord_less_eq_rat @ X2 @ zero_zero_rat ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_le_zero
% 0.30/0.63 thf(fact_7860_zero__less__ceiling,axiom,
% 0.30/0.63 ! [X2: rat] :
% 0.30/0.63 ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.30/0.63 = ( ord_less_rat @ zero_zero_rat @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % zero_less_ceiling
% 0.30/0.63 thf(fact_7861_zero__less__ceiling,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X2 ) )
% 0.30/0.63 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % zero_less_ceiling
% 0.30/0.63 thf(fact_7862_ceiling__le__numeral,axiom,
% 0.30/0.63 ! [X2: real,V: num] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.63 = ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ V ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_le_numeral
% 0.30/0.63 thf(fact_7863_ceiling__le__numeral,axiom,
% 0.30/0.63 ! [X2: rat,V: num] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.63 = ( ord_less_eq_rat @ X2 @ ( numeral_numeral_rat @ V ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_le_numeral
% 0.30/0.63 thf(fact_7864_ceiling__less__one,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ one_one_int )
% 0.30/0.63 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_less_one
% 0.30/0.63 thf(fact_7865_ceiling__less__one,axiom,
% 0.30/0.63 ! [X2: rat] :
% 0.30/0.63 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ one_one_int )
% 0.30/0.63 = ( ord_less_eq_rat @ X2 @ zero_zero_rat ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_less_one
% 0.30/0.63 thf(fact_7866_one__le__ceiling,axiom,
% 0.30/0.63 ! [X2: rat] :
% 0.30/0.63 ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.30/0.63 = ( ord_less_rat @ zero_zero_rat @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % one_le_ceiling
% 0.30/0.63 thf(fact_7867_one__le__ceiling,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X2 ) )
% 0.30/0.63 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % one_le_ceiling
% 0.30/0.63 thf(fact_7868_numeral__less__ceiling,axiom,
% 0.30/0.63 ! [V: num,X2: real] :
% 0.30/0.63 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X2 ) )
% 0.30/0.63 = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % numeral_less_ceiling
% 0.30/0.63 thf(fact_7869_numeral__less__ceiling,axiom,
% 0.30/0.63 ! [V: num,X2: rat] :
% 0.30/0.63 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.30/0.63 = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % numeral_less_ceiling
% 0.30/0.63 thf(fact_7870_ceiling__le__one,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ one_one_int )
% 0.30/0.63 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_le_one
% 0.30/0.63 thf(fact_7871_ceiling__le__one,axiom,
% 0.30/0.63 ! [X2: rat] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ one_one_int )
% 0.30/0.63 = ( ord_less_eq_rat @ X2 @ one_one_rat ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_le_one
% 0.30/0.63 thf(fact_7872_one__less__ceiling,axiom,
% 0.30/0.63 ! [X2: rat] :
% 0.30/0.63 ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.30/0.63 = ( ord_less_rat @ one_one_rat @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % one_less_ceiling
% 0.30/0.63 thf(fact_7873_one__less__ceiling,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X2 ) )
% 0.30/0.63 = ( ord_less_real @ one_one_real @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % one_less_ceiling
% 0.30/0.63 thf(fact_7874_ceiling__add__numeral,axiom,
% 0.30/0.63 ! [X2: real,V: num] :
% 0.30/0.63 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X2 @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.63 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X2 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_add_numeral
% 0.30/0.63 thf(fact_7875_ceiling__add__numeral,axiom,
% 0.30/0.63 ! [X2: rat,V: num] :
% 0.30/0.63 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X2 @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.63 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_add_numeral
% 0.30/0.63 thf(fact_7876_ceiling__neg__numeral,axiom,
% 0.30/0.63 ! [V: num] :
% 0.30/0.63 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.63 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_neg_numeral
% 0.30/0.63 thf(fact_7877_ceiling__neg__numeral,axiom,
% 0.30/0.63 ! [V: num] :
% 0.30/0.63 ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.63 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_neg_numeral
% 0.30/0.63 thf(fact_7878_ceiling__add__one,axiom,
% 0.30/0.63 ! [X2: rat] :
% 0.30/0.63 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X2 @ one_one_rat ) )
% 0.30/0.63 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ one_one_int ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_add_one
% 0.30/0.63 thf(fact_7879_ceiling__add__one,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X2 @ one_one_real ) )
% 0.30/0.63 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X2 ) @ one_one_int ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_add_one
% 0.30/0.63 thf(fact_7880_ceiling__diff__numeral,axiom,
% 0.30/0.63 ! [X2: real,V: num] :
% 0.30/0.63 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X2 @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.63 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X2 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_diff_numeral
% 0.30/0.63 thf(fact_7881_ceiling__diff__numeral,axiom,
% 0.30/0.63 ! [X2: rat,V: num] :
% 0.30/0.63 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X2 @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.63 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_diff_numeral
% 0.30/0.63 thf(fact_7882_ceiling__diff__one,axiom,
% 0.30/0.63 ! [X2: rat] :
% 0.30/0.63 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X2 @ one_one_rat ) )
% 0.30/0.63 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ one_one_int ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_diff_one
% 0.30/0.63 thf(fact_7883_ceiling__diff__one,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X2 @ one_one_real ) )
% 0.30/0.63 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X2 ) @ one_one_int ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_diff_one
% 0.30/0.63 thf(fact_7884_ceiling__numeral__power,axiom,
% 0.30/0.63 ! [X2: num,N: nat] :
% 0.30/0.63 ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
% 0.30/0.63 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_numeral_power
% 0.30/0.63 thf(fact_7885_ceiling__numeral__power,axiom,
% 0.30/0.63 ! [X2: num,N: nat] :
% 0.30/0.63 ( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) )
% 0.30/0.63 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_numeral_power
% 0.30/0.63 thf(fact_7886_ceiling__less__zero,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ zero_zero_int )
% 0.30/0.63 = ( ord_less_eq_real @ X2 @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_less_zero
% 0.30/0.63 thf(fact_7887_ceiling__less__zero,axiom,
% 0.30/0.63 ! [X2: rat] :
% 0.30/0.63 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ zero_zero_int )
% 0.30/0.63 = ( ord_less_eq_rat @ X2 @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_less_zero
% 0.30/0.63 thf(fact_7888_zero__le__ceiling,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X2 ) )
% 0.30/0.63 = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % zero_le_ceiling
% 0.30/0.63 thf(fact_7889_zero__le__ceiling,axiom,
% 0.30/0.63 ! [X2: rat] :
% 0.30/0.63 ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.30/0.63 = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % zero_le_ceiling
% 0.30/0.63 thf(fact_7890_ceiling__divide__eq__div__numeral,axiom,
% 0.30/0.63 ! [A: num,B: num] :
% 0.30/0.63 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 0.30/0.63 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_divide_eq_div_numeral
% 0.30/0.63 thf(fact_7891_ceiling__less__numeral,axiom,
% 0.30/0.63 ! [X2: real,V: num] :
% 0.30/0.63 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.63 = ( ord_less_eq_real @ X2 @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_less_numeral
% 0.30/0.63 thf(fact_7892_ceiling__less__numeral,axiom,
% 0.30/0.63 ! [X2: rat,V: num] :
% 0.30/0.63 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.63 = ( ord_less_eq_rat @ X2 @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_less_numeral
% 0.30/0.63 thf(fact_7893_numeral__le__ceiling,axiom,
% 0.30/0.63 ! [V: num,X2: real] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X2 ) )
% 0.30/0.63 = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % numeral_le_ceiling
% 0.30/0.63 thf(fact_7894_numeral__le__ceiling,axiom,
% 0.30/0.63 ! [V: num,X2: rat] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.30/0.63 = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % numeral_le_ceiling
% 0.30/0.63 thf(fact_7895_ceiling__le__neg__numeral,axiom,
% 0.30/0.63 ! [X2: real,V: num] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.63 = ( ord_less_eq_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_le_neg_numeral
% 0.30/0.63 thf(fact_7896_ceiling__le__neg__numeral,axiom,
% 0.30/0.63 ! [X2: rat,V: num] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.63 = ( ord_less_eq_rat @ X2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_le_neg_numeral
% 0.30/0.63 thf(fact_7897_neg__numeral__less__ceiling,axiom,
% 0.30/0.63 ! [V: num,X2: real] :
% 0.30/0.63 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X2 ) )
% 0.30/0.63 = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % neg_numeral_less_ceiling
% 0.30/0.63 thf(fact_7898_neg__numeral__less__ceiling,axiom,
% 0.30/0.63 ! [V: num,X2: rat] :
% 0.30/0.63 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.30/0.63 = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % neg_numeral_less_ceiling
% 0.30/0.63 thf(fact_7899_ceiling__minus__divide__eq__div__numeral,axiom,
% 0.30/0.63 ! [A: num,B: num] :
% 0.30/0.63 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 0.30/0.63 = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_minus_divide_eq_div_numeral
% 0.30/0.63 thf(fact_7900_ceiling__less__neg__numeral,axiom,
% 0.30/0.63 ! [X2: real,V: num] :
% 0.30/0.63 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.63 = ( ord_less_eq_real @ X2 @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_less_neg_numeral
% 0.30/0.63 thf(fact_7901_ceiling__less__neg__numeral,axiom,
% 0.30/0.63 ! [X2: rat,V: num] :
% 0.30/0.63 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.63 = ( ord_less_eq_rat @ X2 @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_less_neg_numeral
% 0.30/0.63 thf(fact_7902_neg__numeral__le__ceiling,axiom,
% 0.30/0.63 ! [V: num,X2: real] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X2 ) )
% 0.30/0.63 = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % neg_numeral_le_ceiling
% 0.30/0.63 thf(fact_7903_neg__numeral__le__ceiling,axiom,
% 0.30/0.63 ! [V: num,X2: rat] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.30/0.63 = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % neg_numeral_le_ceiling
% 0.30/0.63 thf(fact_7904_ceiling__mono,axiom,
% 0.30/0.63 ! [Y3: real,X2: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ Y3 @ X2 )
% 0.30/0.63 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y3 ) @ ( archim7802044766580827645g_real @ X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_mono
% 0.30/0.63 thf(fact_7905_ceiling__mono,axiom,
% 0.30/0.63 ! [Y3: rat,X2: rat] :
% 0.30/0.63 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 0.30/0.63 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y3 ) @ ( archim2889992004027027881ng_rat @ X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_mono
% 0.30/0.63 thf(fact_7906_le__of__int__ceiling,axiom,
% 0.30/0.63 ! [X2: real] : ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % le_of_int_ceiling
% 0.30/0.63 thf(fact_7907_le__of__int__ceiling,axiom,
% 0.30/0.63 ! [X2: rat] : ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % le_of_int_ceiling
% 0.30/0.63 thf(fact_7908_ceiling__less__cancel,axiom,
% 0.30/0.63 ! [X2: rat,Y3: rat] :
% 0.30/0.63 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( archim2889992004027027881ng_rat @ Y3 ) )
% 0.30/0.63 => ( ord_less_rat @ X2 @ Y3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_less_cancel
% 0.30/0.63 thf(fact_7909_ceiling__less__cancel,axiom,
% 0.30/0.63 ! [X2: real,Y3: real] :
% 0.30/0.63 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ ( archim7802044766580827645g_real @ Y3 ) )
% 0.30/0.63 => ( ord_less_real @ X2 @ Y3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_less_cancel
% 0.30/0.63 thf(fact_7910_ceiling__ge__round,axiom,
% 0.30/0.63 ! [X2: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X2 ) @ ( archim7802044766580827645g_real @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_ge_round
% 0.30/0.63 thf(fact_7911_ceiling__le__iff,axiom,
% 0.30/0.63 ! [X2: real,Z: int] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ Z )
% 0.30/0.63 = ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_le_iff
% 0.30/0.63 thf(fact_7912_ceiling__le__iff,axiom,
% 0.30/0.63 ! [X2: rat,Z: int] :
% 0.30/0.63 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ Z )
% 0.30/0.63 = ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_le_iff
% 0.30/0.63 thf(fact_7913_ceiling__le,axiom,
% 0.30/0.63 ! [X2: real,A: int] :
% 0.30/0.63 ( ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ A ) )
% 0.30/0.63 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ A ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_le
% 0.30/0.63 thf(fact_7914_ceiling__le,axiom,
% 0.30/0.63 ! [X2: rat,A: int] :
% 0.30/0.63 ( ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ A ) )
% 0.30/0.63 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ A ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_le
% 0.30/0.63 thf(fact_7915_less__ceiling__iff,axiom,
% 0.30/0.63 ! [Z: int,X2: rat] :
% 0.30/0.63 ( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.30/0.63 = ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % less_ceiling_iff
% 0.30/0.63 thf(fact_7916_less__ceiling__iff,axiom,
% 0.30/0.63 ! [Z: int,X2: real] :
% 0.30/0.63 ( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X2 ) )
% 0.30/0.63 = ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % less_ceiling_iff
% 0.30/0.63 thf(fact_7917_ceiling__add__le,axiom,
% 0.30/0.63 ! [X2: rat,Y3: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X2 @ Y3 ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( archim2889992004027027881ng_rat @ Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_add_le
% 0.30/0.63 thf(fact_7918_ceiling__add__le,axiom,
% 0.30/0.63 ! [X2: real,Y3: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X2 ) @ ( archim7802044766580827645g_real @ Y3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_add_le
% 0.30/0.63 thf(fact_7919_of__int__ceiling__le__add__one,axiom,
% 0.30/0.63 ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_int_ceiling_le_add_one
% 0.30/0.63 thf(fact_7920_of__int__ceiling__le__add__one,axiom,
% 0.30/0.63 ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).
% 0.30/0.63
% 0.30/0.63 % of_int_ceiling_le_add_one
% 0.30/0.63 thf(fact_7921_of__int__ceiling__diff__one__le,axiom,
% 0.30/0.63 ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).
% 0.30/0.63
% 0.30/0.63 % of_int_ceiling_diff_one_le
% 0.30/0.63 thf(fact_7922_of__int__ceiling__diff__one__le,axiom,
% 0.30/0.63 ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).
% 0.30/0.63
% 0.30/0.63 % of_int_ceiling_diff_one_le
% 0.30/0.63 thf(fact_7923_ceiling__divide__eq__div,axiom,
% 0.30/0.63 ! [A: int,B: int] :
% 0.30/0.63 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) )
% 0.30/0.63 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_divide_eq_div
% 0.30/0.63 thf(fact_7924_ceiling__divide__eq__div,axiom,
% 0.30/0.63 ! [A: int,B: int] :
% 0.30/0.63 ( ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) )
% 0.30/0.63 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_divide_eq_div
% 0.30/0.63 thf(fact_7925_ceiling__correct,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) @ one_one_real ) @ X2 )
% 0.30/0.63 & ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_correct
% 0.30/0.63 thf(fact_7926_ceiling__correct,axiom,
% 0.30/0.63 ! [X2: rat] :
% 0.30/0.63 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) ) @ one_one_rat ) @ X2 )
% 0.30/0.63 & ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_correct
% 0.30/0.63 thf(fact_7927_ceiling__unique,axiom,
% 0.30/0.63 ! [Z: int,X2: real] :
% 0.30/0.63 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X2 )
% 0.30/0.63 => ( ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.63 => ( ( archim7802044766580827645g_real @ X2 )
% 0.30/0.63 = Z ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_unique
% 0.30/0.63 thf(fact_7928_ceiling__unique,axiom,
% 0.30/0.63 ! [Z: int,X2: rat] :
% 0.30/0.63 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X2 )
% 0.30/0.63 => ( ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z ) )
% 0.30/0.63 => ( ( archim2889992004027027881ng_rat @ X2 )
% 0.30/0.63 = Z ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_unique
% 0.30/0.63 thf(fact_7929_ceiling__eq__iff,axiom,
% 0.30/0.63 ! [X2: real,A: int] :
% 0.30/0.63 ( ( ( archim7802044766580827645g_real @ X2 )
% 0.30/0.63 = A )
% 0.30/0.63 = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X2 )
% 0.30/0.63 & ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_eq_iff
% 0.30/0.63 thf(fact_7930_ceiling__eq__iff,axiom,
% 0.30/0.63 ! [X2: rat,A: int] :
% 0.30/0.63 ( ( ( archim2889992004027027881ng_rat @ X2 )
% 0.30/0.63 = A )
% 0.30/0.63 = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X2 )
% 0.30/0.63 & ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ A ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_eq_iff
% 0.30/0.63 thf(fact_7931_ceiling__split,axiom,
% 0.30/0.63 ! [P: int > $o,T: real] :
% 0.30/0.63 ( ( P @ ( archim7802044766580827645g_real @ T ) )
% 0.30/0.63 = ( ! [I4: int] :
% 0.30/0.63 ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) @ T )
% 0.30/0.63 & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I4 ) ) )
% 0.30/0.63 => ( P @ I4 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_split
% 0.30/0.63 thf(fact_7932_ceiling__split,axiom,
% 0.30/0.63 ! [P: int > $o,T: rat] :
% 0.30/0.63 ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
% 0.30/0.63 = ( ! [I4: int] :
% 0.30/0.63 ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) @ T )
% 0.30/0.63 & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I4 ) ) )
% 0.30/0.63 => ( P @ I4 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_split
% 0.30/0.63 thf(fact_7933_mult__ceiling__le,axiom,
% 0.30/0.63 ! [A: real,B: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.63 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.63 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % mult_ceiling_le
% 0.30/0.63 thf(fact_7934_mult__ceiling__le,axiom,
% 0.30/0.63 ! [A: rat,B: rat] :
% 0.30/0.63 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.63 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.63 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % mult_ceiling_le
% 0.30/0.63 thf(fact_7935_ceiling__less__iff,axiom,
% 0.30/0.63 ! [X2: real,Z: int] :
% 0.30/0.63 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ Z )
% 0.30/0.63 = ( ord_less_eq_real @ X2 @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_less_iff
% 0.30/0.63 thf(fact_7936_ceiling__less__iff,axiom,
% 0.30/0.63 ! [X2: rat,Z: int] :
% 0.30/0.63 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ Z )
% 0.30/0.63 = ( ord_less_eq_rat @ X2 @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_less_iff
% 0.30/0.63 thf(fact_7937_le__ceiling__iff,axiom,
% 0.30/0.63 ! [Z: int,X2: rat] :
% 0.30/0.63 ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.30/0.63 = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % le_ceiling_iff
% 0.30/0.63 thf(fact_7938_le__ceiling__iff,axiom,
% 0.30/0.63 ! [Z: int,X2: real] :
% 0.30/0.63 ( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X2 ) )
% 0.30/0.63 = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % le_ceiling_iff
% 0.30/0.63 thf(fact_7939_ceiling__divide__upper,axiom,
% 0.30/0.63 ! [Q3: real,P6: real] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ Q3 )
% 0.30/0.63 => ( ord_less_eq_real @ P6 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P6 @ Q3 ) ) ) @ Q3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_divide_upper
% 0.30/0.63 thf(fact_7940_ceiling__divide__upper,axiom,
% 0.30/0.63 ! [Q3: rat,P6: rat] :
% 0.30/0.63 ( ( ord_less_rat @ zero_zero_rat @ Q3 )
% 0.30/0.63 => ( ord_less_eq_rat @ P6 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P6 @ Q3 ) ) ) @ Q3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_divide_upper
% 0.30/0.63 thf(fact_7941_ceiling__divide__lower,axiom,
% 0.30/0.63 ! [Q3: real,P6: real] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ Q3 )
% 0.30/0.63 => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P6 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) @ P6 ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_divide_lower
% 0.30/0.63 thf(fact_7942_ceiling__divide__lower,axiom,
% 0.30/0.63 ! [Q3: rat,P6: rat] :
% 0.30/0.63 ( ( ord_less_rat @ zero_zero_rat @ Q3 )
% 0.30/0.63 => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P6 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) @ P6 ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_divide_lower
% 0.30/0.63 thf(fact_7943_ceiling__eq,axiom,
% 0.30/0.63 ! [N: int,X2: real] :
% 0.30/0.63 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X2 )
% 0.30/0.63 => ( ( ord_less_eq_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.30/0.63 => ( ( archim7802044766580827645g_real @ X2 )
% 0.30/0.63 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_eq
% 0.30/0.63 thf(fact_7944_ceiling__eq,axiom,
% 0.30/0.63 ! [N: int,X2: rat] :
% 0.30/0.63 ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X2 )
% 0.30/0.63 => ( ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
% 0.30/0.63 => ( ( archim2889992004027027881ng_rat @ X2 )
% 0.30/0.63 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_eq
% 0.30/0.63 thf(fact_7945_ceiling__log2__div2,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.63 => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.30/0.63 = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_log2_div2
% 0.30/0.63 thf(fact_7946_ceiling__log__nat__eq__if,axiom,
% 0.30/0.63 ! [B: nat,N: nat,K: nat] :
% 0.30/0.63 ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.30/0.63 => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 0.30/0.63 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.30/0.63 => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.30/0.63 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_log_nat_eq_if
% 0.30/0.63 thf(fact_7947_ceiling__log__nat__eq__powr__iff,axiom,
% 0.30/0.63 ! [B: nat,K: nat,N: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.30/0.63 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.63 => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.30/0.63 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
% 0.30/0.63 = ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.30/0.63 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_log_nat_eq_powr_iff
% 0.30/0.63 thf(fact_7948_pi__series,axiom,
% 0.30/0.63 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.63 = ( suminf_real
% 0.30/0.63 @ ^ [K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % pi_series
% 0.30/0.63 thf(fact_7949_summable__arctan__series,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.30/0.63 => ( summable_real
% 0.30/0.63 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_arctan_series
% 0.30/0.63 thf(fact_7950_finite__nat__bounded,axiom,
% 0.30/0.63 ! [S3: set_nat] :
% 0.30/0.63 ( ( finite_finite_nat @ S3 )
% 0.30/0.63 => ? [K3: nat] : ( ord_less_eq_set_nat @ S3 @ ( set_ord_lessThan_nat @ K3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % finite_nat_bounded
% 0.30/0.63 thf(fact_7951_finite__nat__iff__bounded,axiom,
% 0.30/0.63 ( finite_finite_nat
% 0.30/0.63 = ( ^ [S5: set_nat] :
% 0.30/0.63 ? [K2: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_lessThan_nat @ K2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % finite_nat_iff_bounded
% 0.30/0.63 thf(fact_7952_infinite__int__iff__unbounded__le,axiom,
% 0.30/0.63 ! [S3: set_int] :
% 0.30/0.63 ( ( ~ ( finite_finite_int @ S3 ) )
% 0.30/0.63 = ( ! [M3: int] :
% 0.30/0.63 ? [N3: int] :
% 0.30/0.63 ( ( ord_less_eq_int @ M3 @ ( abs_abs_int @ N3 ) )
% 0.30/0.63 & ( member_int @ N3 @ S3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % infinite_int_iff_unbounded_le
% 0.30/0.63 thf(fact_7953_summable__iff__shift,axiom,
% 0.30/0.63 ! [F: nat > real,K: nat] :
% 0.30/0.63 ( ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) )
% 0.30/0.63 = ( summable_real @ F ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_iff_shift
% 0.30/0.63 thf(fact_7954_summable__cmult__iff,axiom,
% 0.30/0.63 ! [C: complex,F: nat > complex] :
% 0.30/0.63 ( ( summable_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) ) )
% 0.30/0.63 = ( ( C = zero_zero_complex )
% 0.30/0.63 | ( summable_complex @ F ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_cmult_iff
% 0.30/0.63 thf(fact_7955_summable__cmult__iff,axiom,
% 0.30/0.63 ! [C: real,F: nat > real] :
% 0.30/0.63 ( ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) )
% 0.30/0.63 = ( ( C = zero_zero_real )
% 0.30/0.63 | ( summable_real @ F ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_cmult_iff
% 0.30/0.63 thf(fact_7956_summable__divide__iff,axiom,
% 0.30/0.63 ! [F: nat > complex,C: complex] :
% 0.30/0.63 ( ( summable_complex
% 0.30/0.63 @ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C ) )
% 0.30/0.63 = ( ( C = zero_zero_complex )
% 0.30/0.63 | ( summable_complex @ F ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_divide_iff
% 0.30/0.63 thf(fact_7957_summable__divide__iff,axiom,
% 0.30/0.63 ! [F: nat > real,C: real] :
% 0.30/0.63 ( ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C ) )
% 0.30/0.63 = ( ( C = zero_zero_real )
% 0.30/0.63 | ( summable_real @ F ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_divide_iff
% 0.30/0.63 thf(fact_7958_summable__geometric__iff,axiom,
% 0.30/0.63 ! [C: real] :
% 0.30/0.63 ( ( summable_real @ ( power_power_real @ C ) )
% 0.30/0.63 = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_geometric_iff
% 0.30/0.63 thf(fact_7959_summable__geometric__iff,axiom,
% 0.30/0.63 ! [C: complex] :
% 0.30/0.63 ( ( summable_complex @ ( power_power_complex @ C ) )
% 0.30/0.63 = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_geometric_iff
% 0.30/0.63 thf(fact_7960_summable__comparison__test,axiom,
% 0.30/0.63 ! [F: nat > real,G: nat > real] :
% 0.30/0.63 ( ? [N7: nat] :
% 0.30/0.63 ! [N2: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N7 @ N2 )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
% 0.30/0.63 => ( ( summable_real @ G )
% 0.30/0.63 => ( summable_real @ F ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_comparison_test
% 0.30/0.63 thf(fact_7961_summable__comparison__test,axiom,
% 0.30/0.63 ! [F: nat > complex,G: nat > real] :
% 0.30/0.63 ( ? [N7: nat] :
% 0.30/0.63 ! [N2: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N7 @ N2 )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
% 0.30/0.63 => ( ( summable_real @ G )
% 0.30/0.63 => ( summable_complex @ F ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_comparison_test
% 0.30/0.63 thf(fact_7962_summable__comparison__test_H,axiom,
% 0.30/0.63 ! [G: nat > real,N5: nat,F: nat > real] :
% 0.30/0.63 ( ( summable_real @ G )
% 0.30/0.63 => ( ! [N2: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N5 @ N2 )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
% 0.30/0.63 => ( summable_real @ F ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_comparison_test'
% 0.30/0.63 thf(fact_7963_summable__comparison__test_H,axiom,
% 0.30/0.63 ! [G: nat > real,N5: nat,F: nat > complex] :
% 0.30/0.63 ( ( summable_real @ G )
% 0.30/0.63 => ( ! [N2: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N5 @ N2 )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
% 0.30/0.63 => ( summable_complex @ F ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_comparison_test'
% 0.30/0.63 thf(fact_7964_summable__mult2,axiom,
% 0.30/0.63 ! [F: nat > real,C: real] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_mult2
% 0.30/0.63 thf(fact_7965_summable__mult,axiom,
% 0.30/0.63 ! [F: nat > real,C: real] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_mult
% 0.30/0.63 thf(fact_7966_summable__add,axiom,
% 0.30/0.63 ! [F: nat > real,G: nat > real] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ( summable_real @ G )
% 0.30/0.63 => ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( plus_plus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_add
% 0.30/0.63 thf(fact_7967_summable__add,axiom,
% 0.30/0.63 ! [F: nat > nat,G: nat > nat] :
% 0.30/0.63 ( ( summable_nat @ F )
% 0.30/0.63 => ( ( summable_nat @ G )
% 0.30/0.63 => ( summable_nat
% 0.30/0.63 @ ^ [N3: nat] : ( plus_plus_nat @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_add
% 0.30/0.63 thf(fact_7968_summable__add,axiom,
% 0.30/0.63 ! [F: nat > int,G: nat > int] :
% 0.30/0.63 ( ( summable_int @ F )
% 0.30/0.63 => ( ( summable_int @ G )
% 0.30/0.63 => ( summable_int
% 0.30/0.63 @ ^ [N3: nat] : ( plus_plus_int @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_add
% 0.30/0.63 thf(fact_7969_summable__diff,axiom,
% 0.30/0.63 ! [F: nat > real,G: nat > real] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ( summable_real @ G )
% 0.30/0.63 => ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_diff
% 0.30/0.63 thf(fact_7970_summable__divide,axiom,
% 0.30/0.63 ! [F: nat > complex,C: complex] :
% 0.30/0.63 ( ( summable_complex @ F )
% 0.30/0.63 => ( summable_complex
% 0.30/0.63 @ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_divide
% 0.30/0.63 thf(fact_7971_summable__divide,axiom,
% 0.30/0.63 ! [F: nat > real,C: real] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_divide
% 0.30/0.63 thf(fact_7972_summable__Suc__iff,axiom,
% 0.30/0.63 ! [F: nat > real] :
% 0.30/0.63 ( ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
% 0.30/0.63 = ( summable_real @ F ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_Suc_iff
% 0.30/0.63 thf(fact_7973_summable__ignore__initial__segment,axiom,
% 0.30/0.63 ! [F: nat > real,K: nat] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_ignore_initial_segment
% 0.30/0.63 thf(fact_7974_powser__insidea,axiom,
% 0.30/0.63 ! [F: nat > real,X2: real,Z: real] :
% 0.30/0.63 ( ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X2 @ N3 ) ) )
% 0.30/0.63 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X2 ) )
% 0.30/0.63 => ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % powser_insidea
% 0.30/0.63 thf(fact_7975_powser__insidea,axiom,
% 0.30/0.63 ! [F: nat > complex,X2: complex,Z: complex] :
% 0.30/0.63 ( ( summable_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ X2 @ N3 ) ) )
% 0.30/0.63 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X2 ) )
% 0.30/0.63 => ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % powser_insidea
% 0.30/0.63 thf(fact_7976_suminf__le,axiom,
% 0.30/0.63 ! [F: nat > real,G: nat > real] :
% 0.30/0.63 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.30/0.63 => ( ( summable_real @ F )
% 0.30/0.63 => ( ( summable_real @ G )
% 0.30/0.63 => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_le
% 0.30/0.63 thf(fact_7977_suminf__le,axiom,
% 0.30/0.63 ! [F: nat > nat,G: nat > nat] :
% 0.30/0.63 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.30/0.63 => ( ( summable_nat @ F )
% 0.30/0.63 => ( ( summable_nat @ G )
% 0.30/0.63 => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_le
% 0.30/0.63 thf(fact_7978_suminf__le,axiom,
% 0.30/0.63 ! [F: nat > int,G: nat > int] :
% 0.30/0.63 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.30/0.63 => ( ( summable_int @ F )
% 0.30/0.63 => ( ( summable_int @ G )
% 0.30/0.63 => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_le
% 0.30/0.63 thf(fact_7979_summable__mult__D,axiom,
% 0.30/0.63 ! [C: complex,F: nat > complex] :
% 0.30/0.63 ( ( summable_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) ) )
% 0.30/0.63 => ( ( C != zero_zero_complex )
% 0.30/0.63 => ( summable_complex @ F ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_mult_D
% 0.30/0.63 thf(fact_7980_summable__mult__D,axiom,
% 0.30/0.63 ! [C: real,F: nat > real] :
% 0.30/0.63 ( ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) )
% 0.30/0.63 => ( ( C != zero_zero_real )
% 0.30/0.63 => ( summable_real @ F ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_mult_D
% 0.30/0.63 thf(fact_7981_summable__zero__power,axiom,
% 0.30/0.63 summable_real @ ( power_power_real @ zero_zero_real ) ).
% 0.30/0.63
% 0.30/0.63 % summable_zero_power
% 0.30/0.63 thf(fact_7982_summable__zero__power,axiom,
% 0.30/0.63 summable_int @ ( power_power_int @ zero_zero_int ) ).
% 0.30/0.63
% 0.30/0.63 % summable_zero_power
% 0.30/0.63 thf(fact_7983_summable__zero__power,axiom,
% 0.30/0.63 summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% 0.30/0.63
% 0.30/0.63 % summable_zero_power
% 0.30/0.63 thf(fact_7984_pi__ge__zero,axiom,
% 0.30/0.63 ord_less_eq_real @ zero_zero_real @ pi ).
% 0.30/0.63
% 0.30/0.63 % pi_ge_zero
% 0.30/0.63 thf(fact_7985_suminf__mult,axiom,
% 0.30/0.63 ! [F: nat > real,C: real] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ( suminf_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) )
% 0.30/0.63 = ( times_times_real @ C @ ( suminf_real @ F ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_mult
% 0.30/0.63 thf(fact_7986_suminf__mult2,axiom,
% 0.30/0.63 ! [F: nat > real,C: real] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ( times_times_real @ ( suminf_real @ F ) @ C )
% 0.30/0.63 = ( suminf_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_mult2
% 0.30/0.63 thf(fact_7987_suminf__add,axiom,
% 0.30/0.63 ! [F: nat > real,G: nat > real] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ( summable_real @ G )
% 0.30/0.63 => ( ( plus_plus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
% 0.30/0.63 = ( suminf_real
% 0.30/0.63 @ ^ [N3: nat] : ( plus_plus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_add
% 0.30/0.63 thf(fact_7988_suminf__add,axiom,
% 0.30/0.63 ! [F: nat > nat,G: nat > nat] :
% 0.30/0.63 ( ( summable_nat @ F )
% 0.30/0.63 => ( ( summable_nat @ G )
% 0.30/0.63 => ( ( plus_plus_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) )
% 0.30/0.63 = ( suminf_nat
% 0.30/0.63 @ ^ [N3: nat] : ( plus_plus_nat @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_add
% 0.30/0.63 thf(fact_7989_suminf__add,axiom,
% 0.30/0.63 ! [F: nat > int,G: nat > int] :
% 0.30/0.63 ( ( summable_int @ F )
% 0.30/0.63 => ( ( summable_int @ G )
% 0.30/0.63 => ( ( plus_plus_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) )
% 0.30/0.63 = ( suminf_int
% 0.30/0.63 @ ^ [N3: nat] : ( plus_plus_int @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_add
% 0.30/0.63 thf(fact_7990_suminf__diff,axiom,
% 0.30/0.63 ! [F: nat > real,G: nat > real] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ( summable_real @ G )
% 0.30/0.63 => ( ( minus_minus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
% 0.30/0.63 = ( suminf_real
% 0.30/0.63 @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_diff
% 0.30/0.63 thf(fact_7991_suminf__divide,axiom,
% 0.30/0.63 ! [F: nat > complex,C: complex] :
% 0.30/0.63 ( ( summable_complex @ F )
% 0.30/0.63 => ( ( suminf_complex
% 0.30/0.63 @ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C ) )
% 0.30/0.63 = ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_divide
% 0.30/0.63 thf(fact_7992_suminf__divide,axiom,
% 0.30/0.63 ! [F: nat > real,C: real] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ( suminf_real
% 0.30/0.63 @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C ) )
% 0.30/0.63 = ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_divide
% 0.30/0.63 thf(fact_7993_suminf__eq__zero__iff,axiom,
% 0.30/0.63 ! [F: nat > real] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
% 0.30/0.63 => ( ( ( suminf_real @ F )
% 0.30/0.63 = zero_zero_real )
% 0.30/0.63 = ( ! [N3: nat] :
% 0.30/0.63 ( ( F @ N3 )
% 0.30/0.63 = zero_zero_real ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_eq_zero_iff
% 0.30/0.63 thf(fact_7994_suminf__eq__zero__iff,axiom,
% 0.30/0.63 ! [F: nat > nat] :
% 0.30/0.63 ( ( summable_nat @ F )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
% 0.30/0.63 => ( ( ( suminf_nat @ F )
% 0.30/0.63 = zero_zero_nat )
% 0.30/0.63 = ( ! [N3: nat] :
% 0.30/0.63 ( ( F @ N3 )
% 0.30/0.63 = zero_zero_nat ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_eq_zero_iff
% 0.30/0.63 thf(fact_7995_suminf__eq__zero__iff,axiom,
% 0.30/0.63 ! [F: nat > int] :
% 0.30/0.63 ( ( summable_int @ F )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
% 0.30/0.63 => ( ( ( suminf_int @ F )
% 0.30/0.63 = zero_zero_int )
% 0.30/0.63 = ( ! [N3: nat] :
% 0.30/0.63 ( ( F @ N3 )
% 0.30/0.63 = zero_zero_int ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_eq_zero_iff
% 0.30/0.63 thf(fact_7996_suminf__nonneg,axiom,
% 0.30/0.63 ! [F: nat > real] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
% 0.30/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_nonneg
% 0.30/0.63 thf(fact_7997_suminf__nonneg,axiom,
% 0.30/0.63 ! [F: nat > nat] :
% 0.30/0.63 ( ( summable_nat @ F )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
% 0.30/0.63 => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_nonneg
% 0.30/0.63 thf(fact_7998_suminf__nonneg,axiom,
% 0.30/0.63 ! [F: nat > int] :
% 0.30/0.63 ( ( summable_int @ F )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
% 0.30/0.63 => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_nonneg
% 0.30/0.63 thf(fact_7999_suminf__pos,axiom,
% 0.30/0.63 ! [F: nat > real] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N2 ) )
% 0.30/0.63 => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_pos
% 0.30/0.63 thf(fact_8000_suminf__pos,axiom,
% 0.30/0.63 ! [F: nat > nat] :
% 0.30/0.63 ( ( summable_nat @ F )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N2 ) )
% 0.30/0.63 => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_pos
% 0.30/0.63 thf(fact_8001_suminf__pos,axiom,
% 0.30/0.63 ! [F: nat > int] :
% 0.30/0.63 ( ( summable_int @ F )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N2 ) )
% 0.30/0.63 => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_pos
% 0.30/0.63 thf(fact_8002_summable__0__powser,axiom,
% 0.30/0.63 ! [F: nat > complex] :
% 0.30/0.63 ( summable_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_0_powser
% 0.30/0.63 thf(fact_8003_summable__0__powser,axiom,
% 0.30/0.63 ! [F: nat > real] :
% 0.30/0.63 ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_0_powser
% 0.30/0.63 thf(fact_8004_summable__zero__power_H,axiom,
% 0.30/0.63 ! [F: nat > complex] :
% 0.30/0.63 ( summable_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_zero_power'
% 0.30/0.63 thf(fact_8005_summable__zero__power_H,axiom,
% 0.30/0.63 ! [F: nat > real] :
% 0.30/0.63 ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_zero_power'
% 0.30/0.63 thf(fact_8006_summable__zero__power_H,axiom,
% 0.30/0.63 ! [F: nat > int] :
% 0.30/0.63 ( summable_int
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_int @ ( F @ N3 ) @ ( power_power_int @ zero_zero_int @ N3 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_zero_power'
% 0.30/0.63 thf(fact_8007_powser__split__head_I3_J,axiom,
% 0.30/0.63 ! [F: nat > complex,Z: complex] :
% 0.30/0.63 ( ( summable_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
% 0.30/0.63 => ( summable_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % powser_split_head(3)
% 0.30/0.63 thf(fact_8008_powser__split__head_I3_J,axiom,
% 0.30/0.63 ! [F: nat > real,Z: real] :
% 0.30/0.63 ( ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
% 0.30/0.63 => ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % powser_split_head(3)
% 0.30/0.63 thf(fact_8009_summable__powser__split__head,axiom,
% 0.30/0.63 ! [F: nat > complex,Z: complex] :
% 0.30/0.63 ( ( summable_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) )
% 0.30/0.63 = ( summable_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_powser_split_head
% 0.30/0.63 thf(fact_8010_summable__powser__split__head,axiom,
% 0.30/0.63 ! [F: nat > real,Z: real] :
% 0.30/0.63 ( ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) )
% 0.30/0.63 = ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_powser_split_head
% 0.30/0.63 thf(fact_8011_summable__powser__ignore__initial__segment,axiom,
% 0.30/0.63 ! [F: nat > complex,M: nat,Z: complex] :
% 0.30/0.63 ( ( summable_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N3 @ M ) ) @ ( power_power_complex @ Z @ N3 ) ) )
% 0.30/0.63 = ( summable_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_powser_ignore_initial_segment
% 0.30/0.63 thf(fact_8012_summable__powser__ignore__initial__segment,axiom,
% 0.30/0.63 ! [F: nat > real,M: nat,Z: real] :
% 0.30/0.63 ( ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N3 @ M ) ) @ ( power_power_real @ Z @ N3 ) ) )
% 0.30/0.63 = ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_powser_ignore_initial_segment
% 0.30/0.63 thf(fact_8013_summable__norm__comparison__test,axiom,
% 0.30/0.63 ! [F: nat > complex,G: nat > real] :
% 0.30/0.63 ( ? [N7: nat] :
% 0.30/0.63 ! [N2: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N7 @ N2 )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
% 0.30/0.63 => ( ( summable_real @ G )
% 0.30/0.63 => ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_norm_comparison_test
% 0.30/0.63 thf(fact_8014_summable__rabs__comparison__test,axiom,
% 0.30/0.63 ! [F: nat > real,G: nat > real] :
% 0.30/0.63 ( ? [N7: nat] :
% 0.30/0.63 ! [N2: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N7 @ N2 )
% 0.30/0.63 => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
% 0.30/0.63 => ( ( summable_real @ G )
% 0.30/0.63 => ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_rabs_comparison_test
% 0.30/0.63 thf(fact_8015_summable__rabs,axiom,
% 0.30/0.63 ! [F: nat > real] :
% 0.30/0.63 ( ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) )
% 0.30/0.63 => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
% 0.30/0.63 @ ( suminf_real
% 0.30/0.63 @ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_rabs
% 0.30/0.63 thf(fact_8016_suminf__pos__iff,axiom,
% 0.30/0.63 ! [F: nat > real] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
% 0.30/0.63 = ( ? [I4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_pos_iff
% 0.30/0.63 thf(fact_8017_suminf__pos__iff,axiom,
% 0.30/0.63 ! [F: nat > nat] :
% 0.30/0.63 ( ( summable_nat @ F )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
% 0.30/0.63 => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
% 0.30/0.63 = ( ? [I4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_pos_iff
% 0.30/0.63 thf(fact_8018_suminf__pos__iff,axiom,
% 0.30/0.63 ! [F: nat > int] :
% 0.30/0.63 ( ( summable_int @ F )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
% 0.30/0.63 => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
% 0.30/0.63 = ( ? [I4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I4 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_pos_iff
% 0.30/0.63 thf(fact_8019_suminf__pos2,axiom,
% 0.30/0.63 ! [F: nat > real,I2: nat] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 0.30/0.63 => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_pos2
% 0.30/0.63 thf(fact_8020_suminf__pos2,axiom,
% 0.30/0.63 ! [F: nat > nat,I2: nat] :
% 0.30/0.63 ( ( summable_nat @ F )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
% 0.30/0.63 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 0.30/0.63 => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_pos2
% 0.30/0.63 thf(fact_8021_suminf__pos2,axiom,
% 0.30/0.63 ! [F: nat > int,I2: nat] :
% 0.30/0.63 ( ( summable_int @ F )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
% 0.30/0.63 => ( ( ord_less_int @ zero_zero_int @ ( F @ I2 ) )
% 0.30/0.63 => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_pos2
% 0.30/0.63 thf(fact_8022_suminf__le__const,axiom,
% 0.30/0.63 ! [F: nat > int,X2: int] :
% 0.30/0.63 ( ( summable_int @ F )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X2 )
% 0.30/0.63 => ( ord_less_eq_int @ ( suminf_int @ F ) @ X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_le_const
% 0.30/0.63 thf(fact_8023_suminf__le__const,axiom,
% 0.30/0.63 ! [F: nat > nat,X2: nat] :
% 0.30/0.63 ( ( summable_nat @ F )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X2 )
% 0.30/0.63 => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_le_const
% 0.30/0.63 thf(fact_8024_suminf__le__const,axiom,
% 0.30/0.63 ! [F: nat > real,X2: real] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X2 )
% 0.30/0.63 => ( ord_less_eq_real @ ( suminf_real @ F ) @ X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_le_const
% 0.30/0.63 thf(fact_8025_powser__inside,axiom,
% 0.30/0.63 ! [F: nat > real,X2: real,Z: real] :
% 0.30/0.63 ( ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X2 @ N3 ) ) )
% 0.30/0.63 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X2 ) )
% 0.30/0.63 => ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % powser_inside
% 0.30/0.63 thf(fact_8026_powser__inside,axiom,
% 0.30/0.63 ! [F: nat > complex,X2: complex,Z: complex] :
% 0.30/0.63 ( ( summable_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ X2 @ N3 ) ) )
% 0.30/0.63 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X2 ) )
% 0.30/0.63 => ( summable_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % powser_inside
% 0.30/0.63 thf(fact_8027_summableI__nonneg__bounded,axiom,
% 0.30/0.63 ! [F: nat > int,X2: int] :
% 0.30/0.63 ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X2 )
% 0.30/0.63 => ( summable_int @ F ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summableI_nonneg_bounded
% 0.30/0.63 thf(fact_8028_summableI__nonneg__bounded,axiom,
% 0.30/0.63 ! [F: nat > nat,X2: nat] :
% 0.30/0.63 ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X2 )
% 0.30/0.63 => ( summable_nat @ F ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summableI_nonneg_bounded
% 0.30/0.63 thf(fact_8029_summableI__nonneg__bounded,axiom,
% 0.30/0.63 ! [F: nat > real,X2: real] :
% 0.30/0.63 ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X2 )
% 0.30/0.63 => ( summable_real @ F ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summableI_nonneg_bounded
% 0.30/0.63 thf(fact_8030_summable__geometric,axiom,
% 0.30/0.63 ! [C: real] :
% 0.30/0.63 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 0.30/0.63 => ( summable_real @ ( power_power_real @ C ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_geometric
% 0.30/0.63 thf(fact_8031_summable__geometric,axiom,
% 0.30/0.63 ! [C: complex] :
% 0.30/0.63 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 0.30/0.63 => ( summable_complex @ ( power_power_complex @ C ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_geometric
% 0.30/0.63 thf(fact_8032_complete__algebra__summable__geometric,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ one_one_real )
% 0.30/0.63 => ( summable_real @ ( power_power_real @ X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % complete_algebra_summable_geometric
% 0.30/0.63 thf(fact_8033_complete__algebra__summable__geometric,axiom,
% 0.30/0.63 ! [X2: complex] :
% 0.30/0.63 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ one_one_real )
% 0.30/0.63 => ( summable_complex @ ( power_power_complex @ X2 ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % complete_algebra_summable_geometric
% 0.30/0.63 thf(fact_8034_suminf__split__head,axiom,
% 0.30/0.63 ! [F: nat > real] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ( suminf_real
% 0.30/0.63 @ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
% 0.30/0.63 = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_split_head
% 0.30/0.63 thf(fact_8035_pi__less__4,axiom,
% 0.30/0.63 ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % pi_less_4
% 0.30/0.63 thf(fact_8036_pi__ge__two,axiom,
% 0.30/0.63 ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% 0.30/0.63
% 0.30/0.63 % pi_ge_two
% 0.30/0.63 thf(fact_8037_pi__half__neq__two,axiom,
% 0.30/0.63 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.63 != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % pi_half_neq_two
% 0.30/0.63 thf(fact_8038_summable__norm,axiom,
% 0.30/0.63 ! [F: nat > real] :
% 0.30/0.63 ( ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( real_V7735802525324610683m_real @ ( F @ N3 ) ) )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
% 0.30/0.63 @ ( suminf_real
% 0.30/0.63 @ ^ [N3: nat] : ( real_V7735802525324610683m_real @ ( F @ N3 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_norm
% 0.30/0.63 thf(fact_8039_summable__norm,axiom,
% 0.30/0.63 ! [F: nat > complex] :
% 0.30/0.63 ( ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( F @ N3 ) ) )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
% 0.30/0.63 @ ( suminf_real
% 0.30/0.63 @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_norm
% 0.30/0.63 thf(fact_8040_sum__le__suminf,axiom,
% 0.30/0.63 ! [F: nat > int,I5: set_nat] :
% 0.30/0.63 ( ( summable_int @ F )
% 0.30/0.63 => ( ( finite_finite_nat @ I5 )
% 0.30/0.63 => ( ! [N2: nat] :
% 0.30/0.63 ( ( member_nat @ N2 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 0.30/0.63 => ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) ) )
% 0.30/0.63 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_le_suminf
% 0.30/0.63 thf(fact_8041_sum__le__suminf,axiom,
% 0.30/0.63 ! [F: nat > nat,I5: set_nat] :
% 0.30/0.63 ( ( summable_nat @ F )
% 0.30/0.63 => ( ( finite_finite_nat @ I5 )
% 0.30/0.63 => ( ! [N2: nat] :
% 0.30/0.63 ( ( member_nat @ N2 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 0.30/0.63 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) ) )
% 0.30/0.63 => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_le_suminf
% 0.30/0.63 thf(fact_8042_sum__le__suminf,axiom,
% 0.30/0.63 ! [F: nat > real,I5: set_nat] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ( finite_finite_nat @ I5 )
% 0.30/0.63 => ( ! [N2: nat] :
% 0.30/0.63 ( ( member_nat @ N2 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 0.30/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) ) )
% 0.30/0.63 => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_le_suminf
% 0.30/0.63 thf(fact_8043_suminf__split__initial__segment,axiom,
% 0.30/0.63 ! [F: nat > real,K: nat] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ( suminf_real @ F )
% 0.30/0.63 = ( plus_plus_real
% 0.30/0.63 @ ( suminf_real
% 0.30/0.63 @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) )
% 0.30/0.63 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_split_initial_segment
% 0.30/0.63 thf(fact_8044_suminf__minus__initial__segment,axiom,
% 0.30/0.63 ! [F: nat > real,K: nat] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ( suminf_real
% 0.30/0.63 @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) )
% 0.30/0.63 = ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_minus_initial_segment
% 0.30/0.63 thf(fact_8045_pi__half__neq__zero,axiom,
% 0.30/0.63 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.63 != zero_zero_real ) ).
% 0.30/0.63
% 0.30/0.63 % pi_half_neq_zero
% 0.30/0.63 thf(fact_8046_pi__half__less__two,axiom,
% 0.30/0.63 ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.30/0.63
% 0.30/0.63 % pi_half_less_two
% 0.30/0.63 thf(fact_8047_pi__half__le__two,axiom,
% 0.30/0.63 ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.30/0.63
% 0.30/0.63 % pi_half_le_two
% 0.30/0.63 thf(fact_8048_sum__less__suminf,axiom,
% 0.30/0.63 ! [F: nat > int,N: nat] :
% 0.30/0.63 ( ( summable_int @ F )
% 0.30/0.63 => ( ! [M4: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N @ M4 )
% 0.30/0.63 => ( ord_less_int @ zero_zero_int @ ( F @ M4 ) ) )
% 0.30/0.63 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_less_suminf
% 0.30/0.63 thf(fact_8049_sum__less__suminf,axiom,
% 0.30/0.63 ! [F: nat > nat,N: nat] :
% 0.30/0.63 ( ( summable_nat @ F )
% 0.30/0.63 => ( ! [M4: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N @ M4 )
% 0.30/0.63 => ( ord_less_nat @ zero_zero_nat @ ( F @ M4 ) ) )
% 0.30/0.63 => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_less_suminf
% 0.30/0.63 thf(fact_8050_sum__less__suminf,axiom,
% 0.30/0.63 ! [F: nat > real,N: nat] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ! [M4: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N @ M4 )
% 0.30/0.63 => ( ord_less_real @ zero_zero_real @ ( F @ M4 ) ) )
% 0.30/0.63 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_less_suminf
% 0.30/0.63 thf(fact_8051_powser__split__head_I1_J,axiom,
% 0.30/0.63 ! [F: nat > complex,Z: complex] :
% 0.30/0.63 ( ( summable_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
% 0.30/0.63 => ( ( suminf_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
% 0.30/0.63 = ( plus_plus_complex @ ( F @ zero_zero_nat )
% 0.30/0.63 @ ( times_times_complex
% 0.30/0.63 @ ( suminf_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) )
% 0.30/0.63 @ Z ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % powser_split_head(1)
% 0.30/0.63 thf(fact_8052_powser__split__head_I1_J,axiom,
% 0.30/0.63 ! [F: nat > real,Z: real] :
% 0.30/0.63 ( ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
% 0.30/0.63 => ( ( suminf_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
% 0.30/0.63 = ( plus_plus_real @ ( F @ zero_zero_nat )
% 0.30/0.63 @ ( times_times_real
% 0.30/0.63 @ ( suminf_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) )
% 0.30/0.63 @ Z ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % powser_split_head(1)
% 0.30/0.63 thf(fact_8053_powser__split__head_I2_J,axiom,
% 0.30/0.63 ! [F: nat > complex,Z: complex] :
% 0.30/0.63 ( ( summable_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
% 0.30/0.63 => ( ( times_times_complex
% 0.30/0.63 @ ( suminf_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) )
% 0.30/0.63 @ Z )
% 0.30/0.63 = ( minus_minus_complex
% 0.30/0.63 @ ( suminf_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
% 0.30/0.63 @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % powser_split_head(2)
% 0.30/0.63 thf(fact_8054_powser__split__head_I2_J,axiom,
% 0.30/0.63 ! [F: nat > real,Z: real] :
% 0.30/0.63 ( ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
% 0.30/0.63 => ( ( times_times_real
% 0.30/0.63 @ ( suminf_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) )
% 0.30/0.63 @ Z )
% 0.30/0.63 = ( minus_minus_real
% 0.30/0.63 @ ( suminf_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
% 0.30/0.63 @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % powser_split_head(2)
% 0.30/0.63 thf(fact_8055_summable__partial__sum__bound,axiom,
% 0.30/0.63 ! [F: nat > complex,E: real] :
% 0.30/0.63 ( ( summable_complex @ F )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.30/0.63 => ~ ! [N8: nat] :
% 0.30/0.63 ~ ! [M2: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N8 @ M2 )
% 0.30/0.63 => ! [N9: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M2 @ N9 ) ) ) @ E ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_partial_sum_bound
% 0.30/0.63 thf(fact_8056_summable__partial__sum__bound,axiom,
% 0.30/0.63 ! [F: nat > real,E: real] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.30/0.63 => ~ ! [N8: nat] :
% 0.30/0.63 ~ ! [M2: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N8 @ M2 )
% 0.30/0.63 => ! [N9: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M2 @ N9 ) ) ) @ E ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_partial_sum_bound
% 0.30/0.63 thf(fact_8057_suminf__exist__split,axiom,
% 0.30/0.63 ! [R2: real,F: nat > real] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.30/0.63 => ( ( summable_real @ F )
% 0.30/0.63 => ? [N8: nat] :
% 0.30/0.63 ! [N9: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N8 @ N9 )
% 0.30/0.63 => ( ord_less_real
% 0.30/0.63 @ ( real_V7735802525324610683m_real
% 0.30/0.63 @ ( suminf_real
% 0.30/0.63 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N9 ) ) ) )
% 0.30/0.63 @ R2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_exist_split
% 0.30/0.63 thf(fact_8058_suminf__exist__split,axiom,
% 0.30/0.63 ! [R2: real,F: nat > complex] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.30/0.63 => ( ( summable_complex @ F )
% 0.30/0.63 => ? [N8: nat] :
% 0.30/0.63 ! [N9: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N8 @ N9 )
% 0.30/0.63 => ( ord_less_real
% 0.30/0.63 @ ( real_V1022390504157884413omplex
% 0.30/0.63 @ ( suminf_complex
% 0.30/0.63 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N9 ) ) ) )
% 0.30/0.63 @ R2 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % suminf_exist_split
% 0.30/0.63 thf(fact_8059_summable__power__series,axiom,
% 0.30/0.63 ! [F: nat > real,Z: real] :
% 0.30/0.63 ( ! [I3: nat] : ( ord_less_eq_real @ ( F @ I3 ) @ one_one_real )
% 0.30/0.63 => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 0.30/0.63 => ( ( ord_less_eq_real @ zero_zero_real @ Z )
% 0.30/0.63 => ( ( ord_less_real @ Z @ one_one_real )
% 0.30/0.63 => ( summable_real
% 0.30/0.63 @ ^ [I4: nat] : ( times_times_real @ ( F @ I4 ) @ ( power_power_real @ Z @ I4 ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_power_series
% 0.30/0.63 thf(fact_8060_Abel__lemma,axiom,
% 0.30/0.63 ! [R2: real,R0: real,A: nat > complex,M7: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 0.30/0.63 => ( ( ord_less_real @ R2 @ R0 )
% 0.30/0.63 => ( ! [N2: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N2 ) ) @ ( power_power_real @ R0 @ N2 ) ) @ M7 )
% 0.30/0.63 => ( summable_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N3 ) ) @ ( power_power_real @ R2 @ N3 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % Abel_lemma
% 0.30/0.63 thf(fact_8061_pi__half__gt__zero,axiom,
% 0.30/0.63 ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % pi_half_gt_zero
% 0.30/0.63 thf(fact_8062_pi__half__ge__zero,axiom,
% 0.30/0.63 ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % pi_half_ge_zero
% 0.30/0.63 thf(fact_8063_m2pi__less__pi,axiom,
% 0.30/0.63 ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 0.30/0.63
% 0.30/0.63 % m2pi_less_pi
% 0.30/0.63 thf(fact_8064_summable__ratio__test,axiom,
% 0.30/0.63 ! [C: real,N5: nat,F: nat > real] :
% 0.30/0.63 ( ( ord_less_real @ C @ one_one_real )
% 0.30/0.63 => ( ! [N2: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N5 @ N2 )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N2 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) ) ) )
% 0.30/0.63 => ( summable_real @ F ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_ratio_test
% 0.30/0.63 thf(fact_8065_summable__ratio__test,axiom,
% 0.30/0.63 ! [C: real,N5: nat,F: nat > complex] :
% 0.30/0.63 ( ( ord_less_real @ C @ one_one_real )
% 0.30/0.63 => ( ! [N2: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N5 @ N2 )
% 0.30/0.63 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N2 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) )
% 0.30/0.63 => ( summable_complex @ F ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % summable_ratio_test
% 0.30/0.63 thf(fact_8066_sum__less__suminf2,axiom,
% 0.30/0.63 ! [F: nat > int,N: nat,I2: nat] :
% 0.30/0.63 ( ( summable_int @ F )
% 0.30/0.63 => ( ! [M4: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N @ M4 )
% 0.30/0.63 => ( ord_less_eq_int @ zero_zero_int @ ( F @ M4 ) ) )
% 0.30/0.63 => ( ( ord_less_eq_nat @ N @ I2 )
% 0.30/0.63 => ( ( ord_less_int @ zero_zero_int @ ( F @ I2 ) )
% 0.30/0.63 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_less_suminf2
% 0.30/0.63 thf(fact_8067_sum__less__suminf2,axiom,
% 0.30/0.63 ! [F: nat > nat,N: nat,I2: nat] :
% 0.30/0.63 ( ( summable_nat @ F )
% 0.30/0.63 => ( ! [M4: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N @ M4 )
% 0.30/0.63 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M4 ) ) )
% 0.30/0.63 => ( ( ord_less_eq_nat @ N @ I2 )
% 0.30/0.63 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 0.30/0.63 => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_less_suminf2
% 0.30/0.63 thf(fact_8068_sum__less__suminf2,axiom,
% 0.30/0.63 ! [F: nat > real,N: nat,I2: nat] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ! [M4: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ N @ M4 )
% 0.30/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( F @ M4 ) ) )
% 0.30/0.63 => ( ( ord_less_eq_nat @ N @ I2 )
% 0.30/0.63 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 0.30/0.63 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_less_suminf2
% 0.30/0.63 thf(fact_8069_arctan__ubound,axiom,
% 0.30/0.63 ! [Y3: real] : ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % arctan_ubound
% 0.30/0.63 thf(fact_8070_arctan__one,axiom,
% 0.30/0.63 ( ( arctan @ one_one_real )
% 0.30/0.63 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % arctan_one
% 0.30/0.63 thf(fact_8071_minus__pi__half__less__zero,axiom,
% 0.30/0.63 ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 0.30/0.63
% 0.30/0.63 % minus_pi_half_less_zero
% 0.30/0.63 thf(fact_8072_arctan__bounded,axiom,
% 0.30/0.63 ! [Y3: real] :
% 0.30/0.63 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 0.30/0.63 & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % arctan_bounded
% 0.30/0.63 thf(fact_8073_arctan__lbound,axiom,
% 0.30/0.63 ! [Y3: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % arctan_lbound
% 0.30/0.63 thf(fact_8074_infinite__nat__iff__unbounded,axiom,
% 0.30/0.63 ! [S3: set_nat] :
% 0.30/0.63 ( ( ~ ( finite_finite_nat @ S3 ) )
% 0.30/0.63 = ( ! [M3: nat] :
% 0.30/0.63 ? [N3: nat] :
% 0.30/0.63 ( ( ord_less_nat @ M3 @ N3 )
% 0.30/0.63 & ( member_nat @ N3 @ S3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % infinite_nat_iff_unbounded
% 0.30/0.63 thf(fact_8075_unbounded__k__infinite,axiom,
% 0.30/0.63 ! [K: nat,S3: set_nat] :
% 0.30/0.63 ( ! [M4: nat] :
% 0.30/0.63 ( ( ord_less_nat @ K @ M4 )
% 0.30/0.63 => ? [N9: nat] :
% 0.30/0.63 ( ( ord_less_nat @ M4 @ N9 )
% 0.30/0.63 & ( member_nat @ N9 @ S3 ) ) )
% 0.30/0.63 => ~ ( finite_finite_nat @ S3 ) ) ).
% 0.30/0.63
% 0.30/0.63 % unbounded_k_infinite
% 0.30/0.63 thf(fact_8076_infinite__nat__iff__unbounded__le,axiom,
% 0.30/0.63 ! [S3: set_nat] :
% 0.30/0.63 ( ( ~ ( finite_finite_nat @ S3 ) )
% 0.30/0.63 = ( ! [M3: nat] :
% 0.30/0.63 ? [N3: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.63 & ( member_nat @ N3 @ S3 ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % infinite_nat_iff_unbounded_le
% 0.30/0.63 thf(fact_8077_machin__Euler,axiom,
% 0.30/0.63 ( ( 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 ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.63 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % machin_Euler
% 0.30/0.63 thf(fact_8078_machin,axiom,
% 0.30/0.63 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.63 = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % machin
% 0.30/0.63 thf(fact_8079_sum__pos__lt__pair,axiom,
% 0.30/0.63 ! [F: nat > real,K: nat] :
% 0.30/0.63 ( ( summable_real @ F )
% 0.30/0.63 => ( ! [D3: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) @ one_one_nat ) ) ) ) )
% 0.30/0.63 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % sum_pos_lt_pair
% 0.30/0.63 thf(fact_8080_sin__cos__npi,axiom,
% 0.30/0.63 ! [N: nat] :
% 0.30/0.63 ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.63 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.30/0.63
% 0.30/0.63 % sin_cos_npi
% 0.30/0.63 thf(fact_8081_cos__pi__eq__zero,axiom,
% 0.30/0.63 ! [M: nat] :
% 0.30/0.63 ( ( 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 ) ) ) )
% 0.30/0.63 = zero_zero_real ) ).
% 0.30/0.63
% 0.30/0.63 % cos_pi_eq_zero
% 0.30/0.63 thf(fact_8082_ceiling__log__eq__powr__iff,axiom,
% 0.30/0.63 ! [X2: real,B: real,K: nat] :
% 0.30/0.63 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.63 => ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.63 => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X2 ) )
% 0.30/0.63 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 0.30/0.63 = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X2 )
% 0.30/0.63 & ( ord_less_eq_real @ X2 @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % ceiling_log_eq_powr_iff
% 0.30/0.63 thf(fact_8083_geometric__deriv__sums,axiom,
% 0.30/0.63 ! [Z: real] :
% 0.30/0.63 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 0.30/0.63 => ( sums_real
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) )
% 0.30/0.63 @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % geometric_deriv_sums
% 0.30/0.63 thf(fact_8084_geometric__deriv__sums,axiom,
% 0.30/0.63 ! [Z: complex] :
% 0.30/0.63 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 0.30/0.63 => ( sums_complex
% 0.30/0.63 @ ^ [N3: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) )
% 0.30/0.63 @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % geometric_deriv_sums
% 0.30/0.63 thf(fact_8085_monoseq__def,axiom,
% 0.30/0.63 ( topolo6980174941875973593q_real
% 0.30/0.63 = ( ^ [X4: nat > real] :
% 0.30/0.63 ( ! [M3: nat,N3: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.63 => ( ord_less_eq_real @ ( X4 @ M3 ) @ ( X4 @ N3 ) ) )
% 0.30/0.63 | ! [M3: nat,N3: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.63 => ( ord_less_eq_real @ ( X4 @ N3 ) @ ( X4 @ M3 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % monoseq_def
% 0.30/0.63 thf(fact_8086_monoseq__def,axiom,
% 0.30/0.63 ( topolo3100542954746470799et_int
% 0.30/0.63 = ( ^ [X4: nat > set_int] :
% 0.30/0.63 ( ! [M3: nat,N3: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.63 => ( ord_less_eq_set_int @ ( X4 @ M3 ) @ ( X4 @ N3 ) ) )
% 0.30/0.63 | ! [M3: nat,N3: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.63 => ( ord_less_eq_set_int @ ( X4 @ N3 ) @ ( X4 @ M3 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % monoseq_def
% 0.30/0.63 thf(fact_8087_monoseq__def,axiom,
% 0.30/0.63 ( topolo4267028734544971653eq_rat
% 0.30/0.63 = ( ^ [X4: nat > rat] :
% 0.30/0.63 ( ! [M3: nat,N3: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.63 => ( ord_less_eq_rat @ ( X4 @ M3 ) @ ( X4 @ N3 ) ) )
% 0.30/0.63 | ! [M3: nat,N3: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.63 => ( ord_less_eq_rat @ ( X4 @ N3 ) @ ( X4 @ M3 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % monoseq_def
% 0.30/0.63 thf(fact_8088_monoseq__def,axiom,
% 0.30/0.63 ( topolo1459490580787246023eq_num
% 0.30/0.63 = ( ^ [X4: nat > num] :
% 0.30/0.63 ( ! [M3: nat,N3: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.63 => ( ord_less_eq_num @ ( X4 @ M3 ) @ ( X4 @ N3 ) ) )
% 0.30/0.63 | ! [M3: nat,N3: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.63 => ( ord_less_eq_num @ ( X4 @ N3 ) @ ( X4 @ M3 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % monoseq_def
% 0.30/0.63 thf(fact_8089_monoseq__def,axiom,
% 0.30/0.63 ( topolo4902158794631467389eq_nat
% 0.30/0.63 = ( ^ [X4: nat > nat] :
% 0.30/0.63 ( ! [M3: nat,N3: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.63 => ( ord_less_eq_nat @ ( X4 @ M3 ) @ ( X4 @ N3 ) ) )
% 0.30/0.63 | ! [M3: nat,N3: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.63 => ( ord_less_eq_nat @ ( X4 @ N3 ) @ ( X4 @ M3 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % monoseq_def
% 0.30/0.63 thf(fact_8090_monoseq__def,axiom,
% 0.30/0.63 ( topolo4899668324122417113eq_int
% 0.30/0.63 = ( ^ [X4: nat > int] :
% 0.30/0.63 ( ! [M3: nat,N3: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.63 => ( ord_less_eq_int @ ( X4 @ M3 ) @ ( X4 @ N3 ) ) )
% 0.30/0.63 | ! [M3: nat,N3: nat] :
% 0.30/0.63 ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.63 => ( ord_less_eq_int @ ( X4 @ N3 ) @ ( X4 @ M3 ) ) ) ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % monoseq_def
% 0.30/0.63 thf(fact_8091_powr__one__eq__one,axiom,
% 0.30/0.63 ! [A: real] :
% 0.30/0.63 ( ( powr_real @ one_one_real @ A )
% 0.30/0.63 = one_one_real ) ).
% 0.30/0.63
% 0.30/0.63 % powr_one_eq_one
% 0.30/0.63 thf(fact_8092_cos__zero,axiom,
% 0.30/0.63 ( ( cos_complex @ zero_zero_complex )
% 0.30/0.63 = one_one_complex ) ).
% 0.30/0.63
% 0.30/0.63 % cos_zero
% 0.30/0.63 thf(fact_8093_cos__zero,axiom,
% 0.30/0.63 ( ( cos_real @ zero_zero_real )
% 0.30/0.63 = one_one_real ) ).
% 0.30/0.63
% 0.30/0.63 % cos_zero
% 0.30/0.63 thf(fact_8094_powr__zero__eq__one,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ( X2 = zero_zero_real )
% 0.30/0.63 => ( ( powr_real @ X2 @ zero_zero_real )
% 0.30/0.63 = zero_zero_real ) )
% 0.30/0.63 & ( ( X2 != zero_zero_real )
% 0.30/0.63 => ( ( powr_real @ X2 @ zero_zero_real )
% 0.30/0.63 = one_one_real ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % powr_zero_eq_one
% 0.30/0.63 thf(fact_8095_powr__nonneg__iff,axiom,
% 0.30/0.63 ! [A: real,X2: real] :
% 0.30/0.63 ( ( ord_less_eq_real @ ( powr_real @ A @ X2 ) @ zero_zero_real )
% 0.30/0.63 = ( A = zero_zero_real ) ) ).
% 0.30/0.63
% 0.30/0.63 % powr_nonneg_iff
% 0.30/0.63 thf(fact_8096_powr__less__cancel__iff,axiom,
% 0.30/0.63 ! [X2: real,A: real,B: real] :
% 0.30/0.63 ( ( ord_less_real @ one_one_real @ X2 )
% 0.30/0.63 => ( ( ord_less_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) )
% 0.30/0.63 = ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % powr_less_cancel_iff
% 0.30/0.63 thf(fact_8097_sin__pi__minus,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( sin_real @ ( minus_minus_real @ pi @ X2 ) )
% 0.30/0.63 = ( sin_real @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % sin_pi_minus
% 0.30/0.63 thf(fact_8098_powr__eq__one__iff,axiom,
% 0.30/0.63 ! [A: real,X2: real] :
% 0.30/0.63 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63 => ( ( ( powr_real @ A @ X2 )
% 0.30/0.63 = one_one_real )
% 0.30/0.63 = ( X2 = zero_zero_real ) ) ) ).
% 0.30/0.63
% 0.30/0.63 % powr_eq_one_iff
% 0.30/0.63 thf(fact_8099_powr__one__gt__zero__iff,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.63 ( ( ( powr_real @ X2 @ one_one_real )
% 0.30/0.63 = X2 )
% 0.30/0.63 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 0.30/0.63
% 0.30/0.63 % powr_one_gt_zero_iff
% 0.30/0.63 thf(fact_8100_powr__one,axiom,
% 0.30/0.63 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( powr_real @ X2 @ one_one_real )
% 0.30/0.64 = X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_one
% 0.30/0.64 thf(fact_8101_powr__le__cancel__iff,axiom,
% 0.30/0.64 ! [X2: real,A: real,B: real] :
% 0.30/0.64 ( ( ord_less_real @ one_one_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) )
% 0.30/0.64 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_le_cancel_iff
% 0.30/0.64 thf(fact_8102_numeral__powr__numeral__real,axiom,
% 0.30/0.64 ! [M: num,N: num] :
% 0.30/0.64 ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.30/0.64 = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % numeral_powr_numeral_real
% 0.30/0.64 thf(fact_8103_cos__pi,axiom,
% 0.30/0.64 ( ( cos_real @ pi )
% 0.30/0.64 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_pi
% 0.30/0.64 thf(fact_8104_cos__minus__pi,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( cos_real @ ( minus_minus_real @ X2 @ pi ) )
% 0.30/0.64 = ( uminus_uminus_real @ ( cos_real @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_minus_pi
% 0.30/0.64 thf(fact_8105_cos__pi__minus,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( cos_real @ ( minus_minus_real @ pi @ X2 ) )
% 0.30/0.64 = ( uminus_uminus_real @ ( cos_real @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_pi_minus
% 0.30/0.64 thf(fact_8106_sin__minus__pi,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( sin_real @ ( minus_minus_real @ X2 @ pi ) )
% 0.30/0.64 = ( uminus_uminus_real @ ( sin_real @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_minus_pi
% 0.30/0.64 thf(fact_8107_sin__cos__squared__add3,axiom,
% 0.30/0.64 ! [X2: complex] :
% 0.30/0.64 ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X2 ) @ ( cos_complex @ X2 ) ) @ ( times_times_complex @ ( sin_complex @ X2 ) @ ( sin_complex @ X2 ) ) )
% 0.30/0.64 = one_one_complex ) ).
% 0.30/0.64
% 0.30/0.64 % sin_cos_squared_add3
% 0.30/0.64 thf(fact_8108_sin__cos__squared__add3,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ X2 ) ) @ ( times_times_real @ ( sin_real @ X2 ) @ ( sin_real @ X2 ) ) )
% 0.30/0.64 = one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % sin_cos_squared_add3
% 0.30/0.64 thf(fact_8109_log__powr__cancel,axiom,
% 0.30/0.64 ! [A: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.64 => ( ( A != one_one_real )
% 0.30/0.64 => ( ( log @ A @ ( powr_real @ A @ Y3 ) )
% 0.30/0.64 = Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % log_powr_cancel
% 0.30/0.64 thf(fact_8110_powr__log__cancel,axiom,
% 0.30/0.64 ! [A: real,X2: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.64 => ( ( A != one_one_real )
% 0.30/0.64 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( powr_real @ A @ ( log @ A @ X2 ) )
% 0.30/0.64 = X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_log_cancel
% 0.30/0.64 thf(fact_8111_sin__npi2,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.30/0.64 = zero_zero_real ) ).
% 0.30/0.64
% 0.30/0.64 % sin_npi2
% 0.30/0.64 thf(fact_8112_sin__npi,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.30/0.64 = zero_zero_real ) ).
% 0.30/0.64
% 0.30/0.64 % sin_npi
% 0.30/0.64 thf(fact_8113_sin__npi__int,axiom,
% 0.30/0.64 ! [N: int] :
% 0.30/0.64 ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.30/0.64 = zero_zero_real ) ).
% 0.30/0.64
% 0.30/0.64 % sin_npi_int
% 0.30/0.64 thf(fact_8114_powser__sums__zero__iff,axiom,
% 0.30/0.64 ! [A: nat > complex,X2: complex] :
% 0.30/0.64 ( ( sums_complex
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_complex @ ( A @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) )
% 0.30/0.64 @ X2 )
% 0.30/0.64 = ( ( A @ zero_zero_nat )
% 0.30/0.64 = X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % powser_sums_zero_iff
% 0.30/0.64 thf(fact_8115_powser__sums__zero__iff,axiom,
% 0.30/0.64 ! [A: nat > real,X2: real] :
% 0.30/0.64 ( ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_real @ ( A @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) )
% 0.30/0.64 @ X2 )
% 0.30/0.64 = ( ( A @ zero_zero_nat )
% 0.30/0.64 = X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % powser_sums_zero_iff
% 0.30/0.64 thf(fact_8116_powr__numeral,axiom,
% 0.30/0.64 ! [X2: real,N: num] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( powr_real @ X2 @ ( numeral_numeral_real @ N ) )
% 0.30/0.64 = ( power_power_real @ X2 @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_numeral
% 0.30/0.64 thf(fact_8117_cos__pi__half,axiom,
% 0.30/0.64 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 = zero_zero_real ) ).
% 0.30/0.64
% 0.30/0.64 % cos_pi_half
% 0.30/0.64 thf(fact_8118_sin__two__pi,axiom,
% 0.30/0.64 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.30/0.64 = zero_zero_real ) ).
% 0.30/0.64
% 0.30/0.64 % sin_two_pi
% 0.30/0.64 thf(fact_8119_sin__pi__half,axiom,
% 0.30/0.64 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 = one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % sin_pi_half
% 0.30/0.64 thf(fact_8120_cos__two__pi,axiom,
% 0.30/0.64 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.30/0.64 = one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % cos_two_pi
% 0.30/0.64 thf(fact_8121_cos__periodic,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( cos_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.30/0.64 = ( cos_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_periodic
% 0.30/0.64 thf(fact_8122_sin__periodic,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( sin_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.30/0.64 = ( sin_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_periodic
% 0.30/0.64 thf(fact_8123_cos__2pi__minus,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X2 ) )
% 0.30/0.64 = ( cos_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_2pi_minus
% 0.30/0.64 thf(fact_8124_cos__npi,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.30/0.64 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_npi
% 0.30/0.64 thf(fact_8125_cos__npi2,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.30/0.64 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_npi2
% 0.30/0.64 thf(fact_8126_sin__cos__squared__add2,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.64 = one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % sin_cos_squared_add2
% 0.30/0.64 thf(fact_8127_sin__cos__squared__add2,axiom,
% 0.30/0.64 ! [X2: complex] :
% 0.30/0.64 ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.64 = one_one_complex ) ).
% 0.30/0.64
% 0.30/0.64 % sin_cos_squared_add2
% 0.30/0.64 thf(fact_8128_sin__cos__squared__add,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.64 = one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % sin_cos_squared_add
% 0.30/0.64 thf(fact_8129_sin__cos__squared__add,axiom,
% 0.30/0.64 ! [X2: complex] :
% 0.30/0.64 ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.64 = one_one_complex ) ).
% 0.30/0.64
% 0.30/0.64 % sin_cos_squared_add
% 0.30/0.64 thf(fact_8130_sin__2npi,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 0.30/0.64 = zero_zero_real ) ).
% 0.30/0.64
% 0.30/0.64 % sin_2npi
% 0.30/0.64 thf(fact_8131_cos__2npi,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 0.30/0.64 = one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % cos_2npi
% 0.30/0.64 thf(fact_8132_sin__2pi__minus,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X2 ) )
% 0.30/0.64 = ( uminus_uminus_real @ ( sin_real @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_2pi_minus
% 0.30/0.64 thf(fact_8133_sin__int__2pin,axiom,
% 0.30/0.64 ! [N: int] :
% 0.30/0.64 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 0.30/0.64 = zero_zero_real ) ).
% 0.30/0.64
% 0.30/0.64 % sin_int_2pin
% 0.30/0.64 thf(fact_8134_cos__int__2pin,axiom,
% 0.30/0.64 ! [N: int] :
% 0.30/0.64 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 0.30/0.64 = one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % cos_int_2pin
% 0.30/0.64 thf(fact_8135_cos__3over2__pi,axiom,
% 0.30/0.64 ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.30/0.64 = zero_zero_real ) ).
% 0.30/0.64
% 0.30/0.64 % cos_3over2_pi
% 0.30/0.64 thf(fact_8136_square__powr__half,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( powr_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 = ( abs_abs_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % square_powr_half
% 0.30/0.64 thf(fact_8137_sin__3over2__pi,axiom,
% 0.30/0.64 ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.30/0.64 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_3over2_pi
% 0.30/0.64 thf(fact_8138_cos__npi__int,axiom,
% 0.30/0.64 ! [N: int] :
% 0.30/0.64 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.30/0.64 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.30/0.64 = one_one_real ) )
% 0.30/0.64 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.30/0.64 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.30/0.64 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_npi_int
% 0.30/0.64 thf(fact_8139_sin__add,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( sin_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X2 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_add
% 0.30/0.64 thf(fact_8140_polar__Ex,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ? [R3: real,A4: real] :
% 0.30/0.64 ( ( X2
% 0.30/0.64 = ( times_times_real @ R3 @ ( cos_real @ A4 ) ) )
% 0.30/0.64 & ( Y3
% 0.30/0.64 = ( times_times_real @ R3 @ ( sin_real @ A4 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % polar_Ex
% 0.30/0.64 thf(fact_8141_sin__diff,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( sin_real @ ( minus_minus_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X2 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_diff
% 0.30/0.64 thf(fact_8142_cos__one__sin__zero,axiom,
% 0.30/0.64 ! [X2: complex] :
% 0.30/0.64 ( ( ( cos_complex @ X2 )
% 0.30/0.64 = one_one_complex )
% 0.30/0.64 => ( ( sin_complex @ X2 )
% 0.30/0.64 = zero_zero_complex ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_one_sin_zero
% 0.30/0.64 thf(fact_8143_cos__one__sin__zero,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ( cos_real @ X2 )
% 0.30/0.64 = one_one_real )
% 0.30/0.64 => ( ( sin_real @ X2 )
% 0.30/0.64 = zero_zero_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_one_sin_zero
% 0.30/0.64 thf(fact_8144_powr__powr,axiom,
% 0.30/0.64 ! [X2: real,A: real,B: real] :
% 0.30/0.64 ( ( powr_real @ ( powr_real @ X2 @ A ) @ B )
% 0.30/0.64 = ( powr_real @ X2 @ ( times_times_real @ A @ B ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_powr
% 0.30/0.64 thf(fact_8145_sums__le,axiom,
% 0.30/0.64 ! [F: nat > real,G: nat > real,S: real,T: real] :
% 0.30/0.64 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.30/0.64 => ( ( sums_real @ F @ S )
% 0.30/0.64 => ( ( sums_real @ G @ T )
% 0.30/0.64 => ( ord_less_eq_real @ S @ T ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_le
% 0.30/0.64 thf(fact_8146_sums__le,axiom,
% 0.30/0.64 ! [F: nat > nat,G: nat > nat,S: nat,T: nat] :
% 0.30/0.64 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.30/0.64 => ( ( sums_nat @ F @ S )
% 0.30/0.64 => ( ( sums_nat @ G @ T )
% 0.30/0.64 => ( ord_less_eq_nat @ S @ T ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_le
% 0.30/0.64 thf(fact_8147_sums__le,axiom,
% 0.30/0.64 ! [F: nat > int,G: nat > int,S: int,T: int] :
% 0.30/0.64 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.30/0.64 => ( ( sums_int @ F @ S )
% 0.30/0.64 => ( ( sums_int @ G @ T )
% 0.30/0.64 => ( ord_less_eq_int @ S @ T ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_le
% 0.30/0.64 thf(fact_8148_sums__mult,axiom,
% 0.30/0.64 ! [F: nat > real,A: real,C: real] :
% 0.30/0.64 ( ( sums_real @ F @ A )
% 0.30/0.64 => ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) )
% 0.30/0.64 @ ( times_times_real @ C @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_mult
% 0.30/0.64 thf(fact_8149_sums__mult2,axiom,
% 0.30/0.64 ! [F: nat > real,A: real,C: real] :
% 0.30/0.64 ( ( sums_real @ F @ A )
% 0.30/0.64 => ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C )
% 0.30/0.64 @ ( times_times_real @ A @ C ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_mult2
% 0.30/0.64 thf(fact_8150_sums__add,axiom,
% 0.30/0.64 ! [F: nat > real,A: real,G: nat > real,B: real] :
% 0.30/0.64 ( ( sums_real @ F @ A )
% 0.30/0.64 => ( ( sums_real @ G @ B )
% 0.30/0.64 => ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( plus_plus_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.30/0.64 @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_add
% 0.30/0.64 thf(fact_8151_sums__add,axiom,
% 0.30/0.64 ! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
% 0.30/0.64 ( ( sums_nat @ F @ A )
% 0.30/0.64 => ( ( sums_nat @ G @ B )
% 0.30/0.64 => ( sums_nat
% 0.30/0.64 @ ^ [N3: nat] : ( plus_plus_nat @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.30/0.64 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_add
% 0.30/0.64 thf(fact_8152_sums__add,axiom,
% 0.30/0.64 ! [F: nat > int,A: int,G: nat > int,B: int] :
% 0.30/0.64 ( ( sums_int @ F @ A )
% 0.30/0.64 => ( ( sums_int @ G @ B )
% 0.30/0.64 => ( sums_int
% 0.30/0.64 @ ^ [N3: nat] : ( plus_plus_int @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.30/0.64 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_add
% 0.30/0.64 thf(fact_8153_sums__diff,axiom,
% 0.30/0.64 ! [F: nat > real,A: real,G: nat > real,B: real] :
% 0.30/0.64 ( ( sums_real @ F @ A )
% 0.30/0.64 => ( ( sums_real @ G @ B )
% 0.30/0.64 => ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.30/0.64 @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_diff
% 0.30/0.64 thf(fact_8154_sums__divide,axiom,
% 0.30/0.64 ! [F: nat > complex,A: complex,C: complex] :
% 0.30/0.64 ( ( sums_complex @ F @ A )
% 0.30/0.64 => ( sums_complex
% 0.30/0.64 @ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C )
% 0.30/0.64 @ ( divide1717551699836669952omplex @ A @ C ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_divide
% 0.30/0.64 thf(fact_8155_sums__divide,axiom,
% 0.30/0.64 ! [F: nat > real,A: real,C: real] :
% 0.30/0.64 ( ( sums_real @ F @ A )
% 0.30/0.64 => ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C )
% 0.30/0.64 @ ( divide_divide_real @ A @ C ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_divide
% 0.30/0.64 thf(fact_8156_cos__diff,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( cos_real @ ( minus_minus_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X2 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_diff
% 0.30/0.64 thf(fact_8157_cos__add,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( cos_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X2 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_add
% 0.30/0.64 thf(fact_8158_sin__zero__norm__cos__one,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ( sin_real @ X2 )
% 0.30/0.64 = zero_zero_real )
% 0.30/0.64 => ( ( real_V7735802525324610683m_real @ ( cos_real @ X2 ) )
% 0.30/0.64 = one_one_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_zero_norm_cos_one
% 0.30/0.64 thf(fact_8159_sin__zero__norm__cos__one,axiom,
% 0.30/0.64 ! [X2: complex] :
% 0.30/0.64 ( ( ( sin_complex @ X2 )
% 0.30/0.64 = zero_zero_complex )
% 0.30/0.64 => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X2 ) )
% 0.30/0.64 = one_one_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_zero_norm_cos_one
% 0.30/0.64 thf(fact_8160_sin__zero__abs__cos__one,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ( sin_real @ X2 )
% 0.30/0.64 = zero_zero_real )
% 0.30/0.64 => ( ( abs_abs_real @ ( cos_real @ X2 ) )
% 0.30/0.64 = one_one_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_zero_abs_cos_one
% 0.30/0.64 thf(fact_8161_sin__double,axiom,
% 0.30/0.64 ! [X2: complex] :
% 0.30/0.64 ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) )
% 0.30/0.64 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X2 ) ) @ ( cos_complex @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_double
% 0.30/0.64 thf(fact_8162_sin__double,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) )
% 0.30/0.64 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X2 ) ) @ ( cos_real @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_double
% 0.30/0.64 thf(fact_8163_sincos__principal__value,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ? [Y4: real] :
% 0.30/0.64 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
% 0.30/0.64 & ( ord_less_eq_real @ Y4 @ pi )
% 0.30/0.64 & ( ( sin_real @ Y4 )
% 0.30/0.64 = ( sin_real @ X2 ) )
% 0.30/0.64 & ( ( cos_real @ Y4 )
% 0.30/0.64 = ( cos_real @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sincos_principal_value
% 0.30/0.64 thf(fact_8164_powr__mono2,axiom,
% 0.30/0.64 ! [A: real,X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.30/0.64 => ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_mono2
% 0.30/0.64 thf(fact_8165_powr__ge__pzero,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X2 @ Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_ge_pzero
% 0.30/0.64 thf(fact_8166_powr__less__cancel,axiom,
% 0.30/0.64 ! [X2: real,A: real,B: real] :
% 0.30/0.64 ( ( ord_less_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) )
% 0.30/0.64 => ( ( ord_less_real @ one_one_real @ X2 )
% 0.30/0.64 => ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_less_cancel
% 0.30/0.64 thf(fact_8167_powr__less__mono,axiom,
% 0.30/0.64 ! [A: real,B: real,X2: real] :
% 0.30/0.64 ( ( ord_less_real @ A @ B )
% 0.30/0.64 => ( ( ord_less_real @ one_one_real @ X2 )
% 0.30/0.64 => ( ord_less_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_less_mono
% 0.30/0.64 thf(fact_8168_powr__mono,axiom,
% 0.30/0.64 ! [A: real,B: real,X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.64 => ( ( ord_less_eq_real @ one_one_real @ X2 )
% 0.30/0.64 => ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_mono
% 0.30/0.64 thf(fact_8169_sin__x__le__x,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ord_less_eq_real @ ( sin_real @ X2 ) @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_x_le_x
% 0.30/0.64 thf(fact_8170_sin__le__one,axiom,
% 0.30/0.64 ! [X2: real] : ( ord_less_eq_real @ ( sin_real @ X2 ) @ one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % sin_le_one
% 0.30/0.64 thf(fact_8171_cos__le__one,axiom,
% 0.30/0.64 ! [X2: real] : ( ord_less_eq_real @ ( cos_real @ X2 ) @ one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % cos_le_one
% 0.30/0.64 thf(fact_8172_abs__sin__x__le__abs__x,axiom,
% 0.30/0.64 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X2 ) ) @ ( abs_abs_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % abs_sin_x_le_abs_x
% 0.30/0.64 thf(fact_8173_sums__mult__iff,axiom,
% 0.30/0.64 ! [C: complex,F: nat > complex,D: complex] :
% 0.30/0.64 ( ( C != zero_zero_complex )
% 0.30/0.64 => ( ( sums_complex
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) )
% 0.30/0.64 @ ( times_times_complex @ C @ D ) )
% 0.30/0.64 = ( sums_complex @ F @ D ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_mult_iff
% 0.30/0.64 thf(fact_8174_sums__mult__iff,axiom,
% 0.30/0.64 ! [C: real,F: nat > real,D: real] :
% 0.30/0.64 ( ( C != zero_zero_real )
% 0.30/0.64 => ( ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) )
% 0.30/0.64 @ ( times_times_real @ C @ D ) )
% 0.30/0.64 = ( sums_real @ F @ D ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_mult_iff
% 0.30/0.64 thf(fact_8175_sums__mult2__iff,axiom,
% 0.30/0.64 ! [C: complex,F: nat > complex,D: complex] :
% 0.30/0.64 ( ( C != zero_zero_complex )
% 0.30/0.64 => ( ( sums_complex
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ C )
% 0.30/0.64 @ ( times_times_complex @ D @ C ) )
% 0.30/0.64 = ( sums_complex @ F @ D ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_mult2_iff
% 0.30/0.64 thf(fact_8176_sums__mult2__iff,axiom,
% 0.30/0.64 ! [C: real,F: nat > real,D: real] :
% 0.30/0.64 ( ( C != zero_zero_real )
% 0.30/0.64 => ( ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C )
% 0.30/0.64 @ ( times_times_real @ D @ C ) )
% 0.30/0.64 = ( sums_real @ F @ D ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_mult2_iff
% 0.30/0.64 thf(fact_8177_sin__cos__le1,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X2 ) @ ( sin_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) ) ) @ one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % sin_cos_le1
% 0.30/0.64 thf(fact_8178_sin__squared__eq,axiom,
% 0.30/0.64 ! [X2: complex] :
% 0.30/0.64 ( ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_squared_eq
% 0.30/0.64 thf(fact_8179_sin__squared__eq,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_squared_eq
% 0.30/0.64 thf(fact_8180_cos__squared__eq,axiom,
% 0.30/0.64 ! [X2: complex] :
% 0.30/0.64 ( ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_squared_eq
% 0.30/0.64 thf(fact_8181_cos__squared__eq,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_squared_eq
% 0.30/0.64 thf(fact_8182_powr__mono2_H,axiom,
% 0.30/0.64 ! [A: real,X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.64 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.30/0.64 => ( ord_less_eq_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X2 @ A ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_mono2'
% 0.30/0.64 thf(fact_8183_powr__less__mono2,axiom,
% 0.30/0.64 ! [A: real,X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.64 => ( ord_less_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_less_mono2
% 0.30/0.64 thf(fact_8184_powr__inj,axiom,
% 0.30/0.64 ! [A: real,X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.64 => ( ( A != one_one_real )
% 0.30/0.64 => ( ( ( powr_real @ A @ X2 )
% 0.30/0.64 = ( powr_real @ A @ Y3 ) )
% 0.30/0.64 = ( X2 = Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_inj
% 0.30/0.64 thf(fact_8185_gr__one__powr,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ one_one_real @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ( ord_less_real @ one_one_real @ ( powr_real @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % gr_one_powr
% 0.30/0.64 thf(fact_8186_ge__one__powr__ge__zero,axiom,
% 0.30/0.64 ! [X2: real,A: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.64 => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X2 @ A ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ge_one_powr_ge_zero
% 0.30/0.64 thf(fact_8187_powr__mono__both,axiom,
% 0.30/0.64 ! [A: real,B: real,X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.64 => ( ( ord_less_eq_real @ A @ B )
% 0.30/0.64 => ( ( ord_less_eq_real @ one_one_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.30/0.64 => ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y3 @ B ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_mono_both
% 0.30/0.64 thf(fact_8188_powr__le1,axiom,
% 0.30/0.64 ! [A: real,X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.30/0.64 => ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ one_one_real ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_le1
% 0.30/0.64 thf(fact_8189_powr__divide,axiom,
% 0.30/0.64 ! [X2: real,Y3: real,A: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ( ( powr_real @ ( divide_divide_real @ X2 @ Y3 ) @ A )
% 0.30/0.64 = ( divide_divide_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_divide
% 0.30/0.64 thf(fact_8190_powr__mult,axiom,
% 0.30/0.64 ! [X2: real,Y3: real,A: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ( ( powr_real @ ( times_times_real @ X2 @ Y3 ) @ A )
% 0.30/0.64 = ( times_times_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_mult
% 0.30/0.64 thf(fact_8191_sums__mult__D,axiom,
% 0.30/0.64 ! [C: complex,F: nat > complex,A: complex] :
% 0.30/0.64 ( ( sums_complex
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) )
% 0.30/0.64 @ A )
% 0.30/0.64 => ( ( C != zero_zero_complex )
% 0.30/0.64 => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_mult_D
% 0.30/0.64 thf(fact_8192_sums__mult__D,axiom,
% 0.30/0.64 ! [C: real,F: nat > real,A: real] :
% 0.30/0.64 ( ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) )
% 0.30/0.64 @ A )
% 0.30/0.64 => ( ( C != zero_zero_real )
% 0.30/0.64 => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_mult_D
% 0.30/0.64 thf(fact_8193_sums__Suc__imp,axiom,
% 0.30/0.64 ! [F: nat > complex,S: complex] :
% 0.30/0.64 ( ( ( F @ zero_zero_nat )
% 0.30/0.64 = zero_zero_complex )
% 0.30/0.64 => ( ( sums_complex
% 0.30/0.64 @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
% 0.30/0.64 @ S )
% 0.30/0.64 => ( sums_complex @ F @ S ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_Suc_imp
% 0.30/0.64 thf(fact_8194_sums__Suc__imp,axiom,
% 0.30/0.64 ! [F: nat > real,S: real] :
% 0.30/0.64 ( ( ( F @ zero_zero_nat )
% 0.30/0.64 = zero_zero_real )
% 0.30/0.64 => ( ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
% 0.30/0.64 @ S )
% 0.30/0.64 => ( sums_real @ F @ S ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_Suc_imp
% 0.30/0.64 thf(fact_8195_sums__Suc__iff,axiom,
% 0.30/0.64 ! [F: nat > real,S: real] :
% 0.30/0.64 ( ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
% 0.30/0.64 @ S )
% 0.30/0.64 = ( sums_real @ F @ ( plus_plus_real @ S @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_Suc_iff
% 0.30/0.64 thf(fact_8196_sums__Suc,axiom,
% 0.30/0.64 ! [F: nat > real,L: real] :
% 0.30/0.64 ( ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
% 0.30/0.64 @ L )
% 0.30/0.64 => ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_Suc
% 0.30/0.64 thf(fact_8197_sums__Suc,axiom,
% 0.30/0.64 ! [F: nat > nat,L: nat] :
% 0.30/0.64 ( ( sums_nat
% 0.30/0.64 @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
% 0.30/0.64 @ L )
% 0.30/0.64 => ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_Suc
% 0.30/0.64 thf(fact_8198_sums__Suc,axiom,
% 0.30/0.64 ! [F: nat > int,L: int] :
% 0.30/0.64 ( ( sums_int
% 0.30/0.64 @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
% 0.30/0.64 @ L )
% 0.30/0.64 => ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_Suc
% 0.30/0.64 thf(fact_8199_divide__powr__uminus,axiom,
% 0.30/0.64 ! [A: real,B: real,C: real] :
% 0.30/0.64 ( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
% 0.30/0.64 = ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % divide_powr_uminus
% 0.30/0.64 thf(fact_8200_sums__zero__iff__shift,axiom,
% 0.30/0.64 ! [N: nat,F: nat > complex,S: complex] :
% 0.30/0.64 ( ! [I3: nat] :
% 0.30/0.64 ( ( ord_less_nat @ I3 @ N )
% 0.30/0.64 => ( ( F @ I3 )
% 0.30/0.64 = zero_zero_complex ) )
% 0.30/0.64 => ( ( sums_complex
% 0.30/0.64 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 0.30/0.64 @ S )
% 0.30/0.64 = ( sums_complex @ F @ S ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_zero_iff_shift
% 0.30/0.64 thf(fact_8201_sums__zero__iff__shift,axiom,
% 0.30/0.64 ! [N: nat,F: nat > real,S: real] :
% 0.30/0.64 ( ! [I3: nat] :
% 0.30/0.64 ( ( ord_less_nat @ I3 @ N )
% 0.30/0.64 => ( ( F @ I3 )
% 0.30/0.64 = zero_zero_real ) )
% 0.30/0.64 => ( ( sums_real
% 0.30/0.64 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 0.30/0.64 @ S )
% 0.30/0.64 = ( sums_real @ F @ S ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_zero_iff_shift
% 0.30/0.64 thf(fact_8202_sin__x__ge__neg__x,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ ( sin_real @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_x_ge_neg_x
% 0.30/0.64 thf(fact_8203_sin__ge__zero,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ pi )
% 0.30/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_ge_zero
% 0.30/0.64 thf(fact_8204_log__powr,axiom,
% 0.30/0.64 ! [X2: real,B: real,Y3: real] :
% 0.30/0.64 ( ( X2 != zero_zero_real )
% 0.30/0.64 => ( ( log @ B @ ( powr_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( times_times_real @ Y3 @ ( log @ B @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % log_powr
% 0.30/0.64 thf(fact_8205_ln__powr,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( X2 != zero_zero_real )
% 0.30/0.64 => ( ( ln_ln_real @ ( powr_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( times_times_real @ Y3 @ ( ln_ln_real @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ln_powr
% 0.30/0.64 thf(fact_8206_sin__ge__minus__one,axiom,
% 0.30/0.64 ! [X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_ge_minus_one
% 0.30/0.64 thf(fact_8207_cos__monotone__0__pi__le,axiom,
% 0.30/0.64 ! [Y3: real,X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ pi )
% 0.30/0.64 => ( ord_less_eq_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_monotone_0_pi_le
% 0.30/0.64 thf(fact_8208_cos__mono__le__eq,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ pi )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ pi )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) )
% 0.30/0.64 = ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_mono_le_eq
% 0.30/0.64 thf(fact_8209_cos__inj__pi,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ pi )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ pi )
% 0.30/0.64 => ( ( ( cos_real @ X2 )
% 0.30/0.64 = ( cos_real @ Y3 ) )
% 0.30/0.64 => ( X2 = Y3 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_inj_pi
% 0.30/0.64 thf(fact_8210_cos__ge__minus__one,axiom,
% 0.30/0.64 ! [X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_ge_minus_one
% 0.30/0.64 thf(fact_8211_abs__sin__le__one,axiom,
% 0.30/0.64 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X2 ) ) @ one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % abs_sin_le_one
% 0.30/0.64 thf(fact_8212_abs__cos__le__one,axiom,
% 0.30/0.64 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X2 ) ) @ one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % abs_cos_le_one
% 0.30/0.64 thf(fact_8213_sin__times__sin,axiom,
% 0.30/0.64 ! [W: complex,Z: complex] :
% 0.30/0.64 ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.30/0.64 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_times_sin
% 0.30/0.64 thf(fact_8214_sin__times__sin,axiom,
% 0.30/0.64 ! [W: real,Z: real] :
% 0.30/0.64 ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.30/0.64 = ( 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 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_times_sin
% 0.30/0.64 thf(fact_8215_sin__times__cos,axiom,
% 0.30/0.64 ! [W: complex,Z: complex] :
% 0.30/0.64 ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
% 0.30/0.64 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_times_cos
% 0.30/0.64 thf(fact_8216_sin__times__cos,axiom,
% 0.30/0.64 ! [W: real,Z: real] :
% 0.30/0.64 ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
% 0.30/0.64 = ( 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 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_times_cos
% 0.30/0.64 thf(fact_8217_cos__times__sin,axiom,
% 0.30/0.64 ! [W: complex,Z: complex] :
% 0.30/0.64 ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
% 0.30/0.64 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_times_sin
% 0.30/0.64 thf(fact_8218_cos__times__sin,axiom,
% 0.30/0.64 ! [W: real,Z: real] :
% 0.30/0.64 ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
% 0.30/0.64 = ( 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 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_times_sin
% 0.30/0.64 thf(fact_8219_sin__plus__sin,axiom,
% 0.30/0.64 ! [W: complex,Z: complex] :
% 0.30/0.64 ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.30/0.64 = ( 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 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_plus_sin
% 0.30/0.64 thf(fact_8220_sin__plus__sin,axiom,
% 0.30/0.64 ! [W: real,Z: real] :
% 0.30/0.64 ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.30/0.64 = ( 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 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_plus_sin
% 0.30/0.64 thf(fact_8221_sin__diff__sin,axiom,
% 0.30/0.64 ! [W: complex,Z: complex] :
% 0.30/0.64 ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.30/0.64 = ( 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 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_diff_sin
% 0.30/0.64 thf(fact_8222_sin__diff__sin,axiom,
% 0.30/0.64 ! [W: real,Z: real] :
% 0.30/0.64 ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.30/0.64 = ( 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 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_diff_sin
% 0.30/0.64 thf(fact_8223_cos__diff__cos,axiom,
% 0.30/0.64 ! [W: complex,Z: complex] :
% 0.30/0.64 ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.30/0.64 = ( 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 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_diff_cos
% 0.30/0.64 thf(fact_8224_cos__diff__cos,axiom,
% 0.30/0.64 ! [W: real,Z: real] :
% 0.30/0.64 ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.30/0.64 = ( 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 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_diff_cos
% 0.30/0.64 thf(fact_8225_cos__double,axiom,
% 0.30/0.64 ! [X2: complex] :
% 0.30/0.64 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) )
% 0.30/0.64 = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_double
% 0.30/0.64 thf(fact_8226_cos__double,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) )
% 0.30/0.64 = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_double
% 0.30/0.64 thf(fact_8227_powr__add,axiom,
% 0.30/0.64 ! [X2: real,A: real,B: real] :
% 0.30/0.64 ( ( powr_real @ X2 @ ( plus_plus_real @ A @ B ) )
% 0.30/0.64 = ( times_times_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_add
% 0.30/0.64 thf(fact_8228_powr__diff,axiom,
% 0.30/0.64 ! [W: real,Z1: real,Z22: real] :
% 0.30/0.64 ( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z22 ) )
% 0.30/0.64 = ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z22 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_diff
% 0.30/0.64 thf(fact_8229_cos__double__sin,axiom,
% 0.30/0.64 ! [W: complex] :
% 0.30/0.64 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 0.30/0.64 = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_double_sin
% 0.30/0.64 thf(fact_8230_cos__double__sin,axiom,
% 0.30/0.64 ! [W: real] :
% 0.30/0.64 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 0.30/0.64 = ( 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 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_double_sin
% 0.30/0.64 thf(fact_8231_powser__sums__if,axiom,
% 0.30/0.64 ! [M: nat,Z: complex] :
% 0.30/0.64 ( sums_complex
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_complex @ ( if_complex @ ( N3 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N3 ) )
% 0.30/0.64 @ ( power_power_complex @ Z @ M ) ) ).
% 0.30/0.64
% 0.30/0.64 % powser_sums_if
% 0.30/0.64 thf(fact_8232_powser__sums__if,axiom,
% 0.30/0.64 ! [M: nat,Z: real] :
% 0.30/0.64 ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_real @ ( if_real @ ( N3 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N3 ) )
% 0.30/0.64 @ ( power_power_real @ Z @ M ) ) ).
% 0.30/0.64
% 0.30/0.64 % powser_sums_if
% 0.30/0.64 thf(fact_8233_powser__sums__if,axiom,
% 0.30/0.64 ! [M: nat,Z: int] :
% 0.30/0.64 ( sums_int
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_int @ ( if_int @ ( N3 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N3 ) )
% 0.30/0.64 @ ( power_power_int @ Z @ M ) ) ).
% 0.30/0.64
% 0.30/0.64 % powser_sums_if
% 0.30/0.64 thf(fact_8234_powser__sums__zero,axiom,
% 0.30/0.64 ! [A: nat > complex] :
% 0.30/0.64 ( sums_complex
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_complex @ ( A @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) )
% 0.30/0.64 @ ( A @ zero_zero_nat ) ) ).
% 0.30/0.64
% 0.30/0.64 % powser_sums_zero
% 0.30/0.64 thf(fact_8235_powser__sums__zero,axiom,
% 0.30/0.64 ! [A: nat > real] :
% 0.30/0.64 ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_real @ ( A @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) )
% 0.30/0.64 @ ( A @ zero_zero_nat ) ) ).
% 0.30/0.64
% 0.30/0.64 % powser_sums_zero
% 0.30/0.64 thf(fact_8236_cos__two__neq__zero,axiom,
% 0.30/0.64 ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.64 != zero_zero_real ) ).
% 0.30/0.64
% 0.30/0.64 % cos_two_neq_zero
% 0.30/0.64 thf(fact_8237_powr__realpow,axiom,
% 0.30/0.64 ! [X2: real,N: nat] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( powr_real @ X2 @ ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.64 = ( power_power_real @ X2 @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_realpow
% 0.30/0.64 thf(fact_8238_powr__less__iff,axiom,
% 0.30/0.64 ! [B: real,X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ ( powr_real @ B @ Y3 ) @ X2 )
% 0.30/0.64 = ( ord_less_real @ Y3 @ ( log @ B @ X2 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_less_iff
% 0.30/0.64 thf(fact_8239_less__powr__iff,axiom,
% 0.30/0.64 ! [B: real,X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ ( powr_real @ B @ Y3 ) )
% 0.30/0.64 = ( ord_less_real @ ( log @ B @ X2 ) @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % less_powr_iff
% 0.30/0.64 thf(fact_8240_log__less__iff,axiom,
% 0.30/0.64 ! [B: real,X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ ( log @ B @ X2 ) @ Y3 )
% 0.30/0.64 = ( ord_less_real @ X2 @ ( powr_real @ B @ Y3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % log_less_iff
% 0.30/0.64 thf(fact_8241_less__log__iff,axiom,
% 0.30/0.64 ! [B: real,X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ Y3 @ ( log @ B @ X2 ) )
% 0.30/0.64 = ( ord_less_real @ ( powr_real @ B @ Y3 ) @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % less_log_iff
% 0.30/0.64 thf(fact_8242_cos__mono__less__eq,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ pi )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ pi )
% 0.30/0.64 => ( ( ord_less_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) )
% 0.30/0.64 = ( ord_less_real @ Y3 @ X2 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_mono_less_eq
% 0.30/0.64 thf(fact_8243_cos__monotone__0__pi,axiom,
% 0.30/0.64 ! [Y3: real,X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ( ( ord_less_real @ Y3 @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ pi )
% 0.30/0.64 => ( ord_less_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_monotone_0_pi
% 0.30/0.64 thf(fact_8244_sums__iff__shift,axiom,
% 0.30/0.64 ! [F: nat > real,N: nat,S: real] :
% 0.30/0.64 ( ( sums_real
% 0.30/0.64 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 0.30/0.64 @ S )
% 0.30/0.64 = ( sums_real @ F @ ( plus_plus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_iff_shift
% 0.30/0.64 thf(fact_8245_sums__iff__shift_H,axiom,
% 0.30/0.64 ! [F: nat > real,N: nat,S: real] :
% 0.30/0.64 ( ( sums_real
% 0.30/0.64 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 0.30/0.64 @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) )
% 0.30/0.64 = ( sums_real @ F @ S ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_iff_shift'
% 0.30/0.64 thf(fact_8246_sums__split__initial__segment,axiom,
% 0.30/0.64 ! [F: nat > real,S: real,N: nat] :
% 0.30/0.64 ( ( sums_real @ F @ S )
% 0.30/0.64 => ( sums_real
% 0.30/0.64 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 0.30/0.64 @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_split_initial_segment
% 0.30/0.64 thf(fact_8247_sums__If__finite__set_H,axiom,
% 0.30/0.64 ! [G: nat > real,S3: real,A2: set_nat,S4: real,F: nat > real] :
% 0.30/0.64 ( ( sums_real @ G @ S3 )
% 0.30/0.64 => ( ( finite_finite_nat @ A2 )
% 0.30/0.64 => ( ( S4
% 0.30/0.64 = ( plus_plus_real @ S3
% 0.30/0.64 @ ( groups6591440286371151544t_real
% 0.30/0.64 @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.30/0.64 @ A2 ) ) )
% 0.30/0.64 => ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( if_real @ ( member_nat @ N3 @ A2 ) @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.30/0.64 @ S4 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_If_finite_set'
% 0.30/0.64 thf(fact_8248_cos__monotone__minus__pi__0_H,axiom,
% 0.30/0.64 ! [Y3: real,X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.30/0.64 => ( ord_less_eq_real @ ( cos_real @ Y3 ) @ ( cos_real @ X2 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_monotone_minus_pi_0'
% 0.30/0.64 thf(fact_8249_sin__zero__iff__int2,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ( sin_real @ X2 )
% 0.30/0.64 = zero_zero_real )
% 0.30/0.64 = ( ? [I4: int] :
% 0.30/0.64 ( X2
% 0.30/0.64 = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ pi ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_zero_iff_int2
% 0.30/0.64 thf(fact_8250_sincos__total__pi,axiom,
% 0.30/0.64 ! [Y3: real,X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.64 = one_one_real )
% 0.30/0.64 => ? [T5: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.30/0.64 & ( ord_less_eq_real @ T5 @ pi )
% 0.30/0.64 & ( X2
% 0.30/0.64 = ( cos_real @ T5 ) )
% 0.30/0.64 & ( Y3
% 0.30/0.64 = ( sin_real @ T5 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sincos_total_pi
% 0.30/0.64 thf(fact_8251_sin__expansion__lemma,axiom,
% 0.30/0.64 ! [X2: real,M: nat] :
% 0.30/0.64 ( ( sin_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64 = ( cos_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_expansion_lemma
% 0.30/0.64 thf(fact_8252_powr__minus__divide,axiom,
% 0.30/0.64 ! [X2: real,A: real] :
% 0.30/0.64 ( ( powr_real @ X2 @ ( uminus_uminus_real @ A ) )
% 0.30/0.64 = ( divide_divide_real @ one_one_real @ ( powr_real @ X2 @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_minus_divide
% 0.30/0.64 thf(fact_8253_cos__expansion__lemma,axiom,
% 0.30/0.64 ! [X2: real,M: nat] :
% 0.30/0.64 ( ( cos_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64 = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_expansion_lemma
% 0.30/0.64 thf(fact_8254_powr__neg__one,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( powr_real @ X2 @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.64 = ( divide_divide_real @ one_one_real @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_neg_one
% 0.30/0.64 thf(fact_8255_sin__gt__zero__02,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.64 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_gt_zero_02
% 0.30/0.64 thf(fact_8256_powr__mult__base,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( times_times_real @ X2 @ ( powr_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( powr_real @ X2 @ ( plus_plus_real @ one_one_real @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_mult_base
% 0.30/0.64 thf(fact_8257_le__log__iff,axiom,
% 0.30/0.64 ! [B: real,X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ ( log @ B @ X2 ) )
% 0.30/0.64 = ( ord_less_eq_real @ ( powr_real @ B @ Y3 ) @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % le_log_iff
% 0.30/0.64 thf(fact_8258_log__le__iff,axiom,
% 0.30/0.64 ! [B: real,X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( log @ B @ X2 ) @ Y3 )
% 0.30/0.64 = ( ord_less_eq_real @ X2 @ ( powr_real @ B @ Y3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % log_le_iff
% 0.30/0.64 thf(fact_8259_le__powr__iff,axiom,
% 0.30/0.64 ! [B: real,X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ ( powr_real @ B @ Y3 ) )
% 0.30/0.64 = ( ord_less_eq_real @ ( log @ B @ X2 ) @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % le_powr_iff
% 0.30/0.64 thf(fact_8260_powr__le__iff,axiom,
% 0.30/0.64 ! [B: real,X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( powr_real @ B @ Y3 ) @ X2 )
% 0.30/0.64 = ( ord_less_eq_real @ Y3 @ ( log @ B @ X2 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_le_iff
% 0.30/0.64 thf(fact_8261_cos__two__less__zero,axiom,
% 0.30/0.64 ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 0.30/0.64
% 0.30/0.64 % cos_two_less_zero
% 0.30/0.64 thf(fact_8262_cos__two__le__zero,axiom,
% 0.30/0.64 ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 0.30/0.64
% 0.30/0.64 % cos_two_le_zero
% 0.30/0.64 thf(fact_8263_cos__is__zero,axiom,
% 0.30/0.64 ? [X5: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 0.30/0.64 & ( ord_less_eq_real @ X5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.64 & ( ( cos_real @ X5 )
% 0.30/0.64 = zero_zero_real )
% 0.30/0.64 & ! [Y5: real] :
% 0.30/0.64 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 0.30/0.64 & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.64 & ( ( cos_real @ Y5 )
% 0.30/0.64 = zero_zero_real ) )
% 0.30/0.64 => ( Y5 = X5 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_is_zero
% 0.30/0.64 thf(fact_8264_cos__monotone__minus__pi__0,axiom,
% 0.30/0.64 ! [Y3: real,X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_real @ Y3 @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.30/0.64 => ( ord_less_real @ ( cos_real @ Y3 ) @ ( cos_real @ X2 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_monotone_minus_pi_0
% 0.30/0.64 thf(fact_8265_cos__total,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64 => ? [X5: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 0.30/0.64 & ( ord_less_eq_real @ X5 @ pi )
% 0.30/0.64 & ( ( cos_real @ X5 )
% 0.30/0.64 = Y3 )
% 0.30/0.64 & ! [Y5: real] :
% 0.30/0.64 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 0.30/0.64 & ( ord_less_eq_real @ Y5 @ pi )
% 0.30/0.64 & ( ( cos_real @ Y5 )
% 0.30/0.64 = Y3 ) )
% 0.30/0.64 => ( Y5 = X5 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_total
% 0.30/0.64 thf(fact_8266_sincos__total__pi__half,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.64 = one_one_real )
% 0.30/0.64 => ? [T5: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.30/0.64 & ( ord_less_eq_real @ T5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 & ( X2
% 0.30/0.64 = ( cos_real @ T5 ) )
% 0.30/0.64 & ( Y3
% 0.30/0.64 = ( sin_real @ T5 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sincos_total_pi_half
% 0.30/0.64 thf(fact_8267_sincos__total__2pi__le,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.64 = one_one_real )
% 0.30/0.64 => ? [T5: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.30/0.64 & ( ord_less_eq_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.30/0.64 & ( X2
% 0.30/0.64 = ( cos_real @ T5 ) )
% 0.30/0.64 & ( Y3
% 0.30/0.64 = ( sin_real @ T5 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sincos_total_2pi_le
% 0.30/0.64 thf(fact_8268_sincos__total__2pi,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.64 = one_one_real )
% 0.30/0.64 => ~ ! [T5: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.30/0.64 => ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.30/0.64 => ( ( X2
% 0.30/0.64 = ( cos_real @ T5 ) )
% 0.30/0.64 => ( Y3
% 0.30/0.64 != ( sin_real @ T5 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sincos_total_2pi
% 0.30/0.64 thf(fact_8269_ln__powr__bound,axiom,
% 0.30/0.64 ! [X2: real,A: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.64 => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( divide_divide_real @ ( powr_real @ X2 @ A ) @ A ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ln_powr_bound
% 0.30/0.64 thf(fact_8270_ln__powr__bound2,axiom,
% 0.30/0.64 ! [X2: real,A: real] :
% 0.30/0.64 ( ( ord_less_real @ one_one_real @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.64 => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X2 ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ln_powr_bound2
% 0.30/0.64 thf(fact_8271_log__add__eq__powr,axiom,
% 0.30/0.64 ! [B: real,X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.64 => ( ( B != one_one_real )
% 0.30/0.64 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( plus_plus_real @ ( log @ B @ X2 ) @ Y3 )
% 0.30/0.64 = ( log @ B @ ( times_times_real @ X2 @ ( powr_real @ B @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % log_add_eq_powr
% 0.30/0.64 thf(fact_8272_add__log__eq__powr,axiom,
% 0.30/0.64 ! [B: real,X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.64 => ( ( B != one_one_real )
% 0.30/0.64 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( plus_plus_real @ Y3 @ ( log @ B @ X2 ) )
% 0.30/0.64 = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y3 ) @ X2 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % add_log_eq_powr
% 0.30/0.64 thf(fact_8273_minus__log__eq__powr,axiom,
% 0.30/0.64 ! [B: real,X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.64 => ( ( B != one_one_real )
% 0.30/0.64 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( minus_minus_real @ Y3 @ ( log @ B @ X2 ) )
% 0.30/0.64 = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y3 ) @ X2 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % minus_log_eq_powr
% 0.30/0.64 thf(fact_8274_sin__pi__divide__n__ge__0,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( N != zero_zero_nat )
% 0.30/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_pi_divide_n_ge_0
% 0.30/0.64 thf(fact_8275_cos__times__cos,axiom,
% 0.30/0.64 ! [W: complex,Z: complex] :
% 0.30/0.64 ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.30/0.64 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_times_cos
% 0.30/0.64 thf(fact_8276_cos__times__cos,axiom,
% 0.30/0.64 ! [W: real,Z: real] :
% 0.30/0.64 ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.30/0.64 = ( 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 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_times_cos
% 0.30/0.64 thf(fact_8277_cos__plus__cos,axiom,
% 0.30/0.64 ! [W: complex,Z: complex] :
% 0.30/0.64 ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.30/0.64 = ( 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 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_plus_cos
% 0.30/0.64 thf(fact_8278_cos__plus__cos,axiom,
% 0.30/0.64 ! [W: real,Z: real] :
% 0.30/0.64 ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.30/0.64 = ( 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 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_plus_cos
% 0.30/0.64 thf(fact_8279_geometric__sums,axiom,
% 0.30/0.64 ! [C: real] :
% 0.30/0.64 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 0.30/0.64 => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % geometric_sums
% 0.30/0.64 thf(fact_8280_geometric__sums,axiom,
% 0.30/0.64 ! [C: complex] :
% 0.30/0.64 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 0.30/0.64 => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % geometric_sums
% 0.30/0.64 thf(fact_8281_power__half__series,axiom,
% 0.30/0.64 ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N3 ) )
% 0.30/0.64 @ one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % power_half_series
% 0.30/0.64 thf(fact_8282_sin__gt__zero2,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_gt_zero2
% 0.30/0.64 thf(fact_8283_sin__lt__zero,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ pi @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.30/0.64 => ( ord_less_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_lt_zero
% 0.30/0.64 thf(fact_8284_cos__double__less__one,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.64 => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_real ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_double_less_one
% 0.30/0.64 thf(fact_8285_sin__30,axiom,
% 0.30/0.64 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.30/0.64 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_30
% 0.30/0.64 thf(fact_8286_cos__gt__zero,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_gt_zero
% 0.30/0.64 thf(fact_8287_sin__inj__pi,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( ( sin_real @ X2 )
% 0.30/0.64 = ( sin_real @ Y3 ) )
% 0.30/0.64 => ( X2 = Y3 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_inj_pi
% 0.30/0.64 thf(fact_8288_sin__mono__le__eq,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( sin_real @ X2 ) @ ( sin_real @ Y3 ) )
% 0.30/0.64 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_mono_le_eq
% 0.30/0.64 thf(fact_8289_sin__monotone__2pi__le,axiom,
% 0.30/0.64 ! [Y3: real,X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ord_less_eq_real @ ( sin_real @ Y3 ) @ ( sin_real @ X2 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_monotone_2pi_le
% 0.30/0.64 thf(fact_8290_log__minus__eq__powr,axiom,
% 0.30/0.64 ! [B: real,X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.64 => ( ( B != one_one_real )
% 0.30/0.64 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( minus_minus_real @ ( log @ B @ X2 ) @ Y3 )
% 0.30/0.64 = ( log @ B @ ( times_times_real @ X2 @ ( powr_real @ B @ ( uminus_uminus_real @ Y3 ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % log_minus_eq_powr
% 0.30/0.64 thf(fact_8291_cos__60,axiom,
% 0.30/0.64 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.30/0.64 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_60
% 0.30/0.64 thf(fact_8292_cos__one__2pi__int,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ( cos_real @ X2 )
% 0.30/0.64 = one_one_real )
% 0.30/0.64 = ( ? [X: int] :
% 0.30/0.64 ( X2
% 0.30/0.64 = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_one_2pi_int
% 0.30/0.64 thf(fact_8293_cos__double__cos,axiom,
% 0.30/0.64 ! [W: complex] :
% 0.30/0.64 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 0.30/0.64 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_double_cos
% 0.30/0.64 thf(fact_8294_cos__double__cos,axiom,
% 0.30/0.64 ! [W: real] :
% 0.30/0.64 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 0.30/0.64 = ( 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 ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_double_cos
% 0.30/0.64 thf(fact_8295_cos__treble__cos,axiom,
% 0.30/0.64 ! [X2: complex] :
% 0.30/0.64 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X2 ) )
% 0.30/0.64 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_treble_cos
% 0.30/0.64 thf(fact_8296_cos__treble__cos,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X2 ) )
% 0.30/0.64 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_treble_cos
% 0.30/0.64 thf(fact_8297_sums__if_H,axiom,
% 0.30/0.64 ! [G: nat > real,X2: real] :
% 0.30/0.64 ( ( sums_real @ G @ X2 )
% 0.30/0.64 => ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64 @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_if'
% 0.30/0.64 thf(fact_8298_sums__if,axiom,
% 0.30/0.64 ! [G: nat > real,X2: real,F: nat > real,Y3: real] :
% 0.30/0.64 ( ( sums_real @ G @ X2 )
% 0.30/0.64 => ( ( sums_real @ F @ Y3 )
% 0.30/0.64 => ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( F @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64 @ ( plus_plus_real @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sums_if
% 0.30/0.64 thf(fact_8299_powr__neg__numeral,axiom,
% 0.30/0.64 ! [X2: real,N: num] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( powr_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.64 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_neg_numeral
% 0.30/0.64 thf(fact_8300_sin__le__zero,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ pi @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.30/0.64 => ( ord_less_eq_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_le_zero
% 0.30/0.64 thf(fact_8301_sin__less__zero,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.30/0.64 => ( ord_less_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_less_zero
% 0.30/0.64 thf(fact_8302_sin__mono__less__eq,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( ord_less_real @ ( sin_real @ X2 ) @ ( sin_real @ Y3 ) )
% 0.30/0.64 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_mono_less_eq
% 0.30/0.64 thf(fact_8303_sin__monotone__2pi,axiom,
% 0.30/0.64 ! [Y3: real,X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_real @ Y3 @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ord_less_real @ ( sin_real @ Y3 ) @ ( sin_real @ X2 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_monotone_2pi
% 0.30/0.64 thf(fact_8304_sin__total,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64 => ? [X5: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
% 0.30/0.64 & ( ord_less_eq_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 & ( ( sin_real @ X5 )
% 0.30/0.64 = Y3 )
% 0.30/0.64 & ! [Y5: real] :
% 0.30/0.64 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 0.30/0.64 & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 & ( ( sin_real @ Y5 )
% 0.30/0.64 = Y3 ) )
% 0.30/0.64 => ( Y5 = X5 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_total
% 0.30/0.64 thf(fact_8305_cos__gt__zero__pi,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_gt_zero_pi
% 0.30/0.64 thf(fact_8306_cos__ge__zero,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_ge_zero
% 0.30/0.64 thf(fact_8307_cos__one__2pi,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ( cos_real @ X2 )
% 0.30/0.64 = one_one_real )
% 0.30/0.64 = ( ? [X: nat] :
% 0.30/0.64 ( X2
% 0.30/0.64 = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.30/0.64 | ? [X: nat] :
% 0.30/0.64 ( X2
% 0.30/0.64 = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_one_2pi
% 0.30/0.64 thf(fact_8308_sin__pi__divide__n__gt__0,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.64 => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_pi_divide_n_gt_0
% 0.30/0.64 thf(fact_8309_sin__zero__iff__int,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ( sin_real @ X2 )
% 0.30/0.64 = zero_zero_real )
% 0.30/0.64 = ( ? [I4: int] :
% 0.30/0.64 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
% 0.30/0.64 & ( X2
% 0.30/0.64 = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_zero_iff_int
% 0.30/0.64 thf(fact_8310_cos__zero__iff__int,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ( cos_real @ X2 )
% 0.30/0.64 = zero_zero_real )
% 0.30/0.64 = ( ? [I4: int] :
% 0.30/0.64 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
% 0.30/0.64 & ( X2
% 0.30/0.64 = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_zero_iff_int
% 0.30/0.64 thf(fact_8311_sin__zero__lemma,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ( sin_real @ X2 )
% 0.30/0.64 = zero_zero_real )
% 0.30/0.64 => ? [N2: nat] :
% 0.30/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.64 & ( X2
% 0.30/0.64 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_zero_lemma
% 0.30/0.64 thf(fact_8312_sin__zero__iff,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ( sin_real @ X2 )
% 0.30/0.64 = zero_zero_real )
% 0.30/0.64 = ( ? [N3: nat] :
% 0.30/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 0.30/0.64 & ( X2
% 0.30/0.64 = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.64 | ? [N3: nat] :
% 0.30/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 0.30/0.64 & ( X2
% 0.30/0.64 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_zero_iff
% 0.30/0.64 thf(fact_8313_cos__zero__lemma,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ( cos_real @ X2 )
% 0.30/0.64 = zero_zero_real )
% 0.30/0.64 => ? [N2: nat] :
% 0.30/0.64 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.64 & ( X2
% 0.30/0.64 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_zero_lemma
% 0.30/0.64 thf(fact_8314_cos__zero__iff,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ( cos_real @ X2 )
% 0.30/0.64 = zero_zero_real )
% 0.30/0.64 = ( ? [N3: nat] :
% 0.30/0.64 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 0.30/0.64 & ( X2
% 0.30/0.64 = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.64 | ? [N3: nat] :
% 0.30/0.64 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 0.30/0.64 & ( X2
% 0.30/0.64 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_zero_iff
% 0.30/0.64 thf(fact_8315_mono__SucI1,axiom,
% 0.30/0.64 ! [X8: nat > real] :
% 0.30/0.64 ( ! [N2: nat] : ( ord_less_eq_real @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 0.30/0.64 => ( topolo6980174941875973593q_real @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % mono_SucI1
% 0.30/0.64 thf(fact_8316_mono__SucI1,axiom,
% 0.30/0.64 ! [X8: nat > set_int] :
% 0.30/0.64 ( ! [N2: nat] : ( ord_less_eq_set_int @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 0.30/0.64 => ( topolo3100542954746470799et_int @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % mono_SucI1
% 0.30/0.64 thf(fact_8317_mono__SucI1,axiom,
% 0.30/0.64 ! [X8: nat > rat] :
% 0.30/0.64 ( ! [N2: nat] : ( ord_less_eq_rat @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 0.30/0.64 => ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % mono_SucI1
% 0.30/0.64 thf(fact_8318_mono__SucI1,axiom,
% 0.30/0.64 ! [X8: nat > num] :
% 0.30/0.64 ( ! [N2: nat] : ( ord_less_eq_num @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 0.30/0.64 => ( topolo1459490580787246023eq_num @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % mono_SucI1
% 0.30/0.64 thf(fact_8319_mono__SucI1,axiom,
% 0.30/0.64 ! [X8: nat > nat] :
% 0.30/0.64 ( ! [N2: nat] : ( ord_less_eq_nat @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 0.30/0.64 => ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % mono_SucI1
% 0.30/0.64 thf(fact_8320_mono__SucI1,axiom,
% 0.30/0.64 ! [X8: nat > int] :
% 0.30/0.64 ( ! [N2: nat] : ( ord_less_eq_int @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 0.30/0.64 => ( topolo4899668324122417113eq_int @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % mono_SucI1
% 0.30/0.64 thf(fact_8321_mono__SucI2,axiom,
% 0.30/0.64 ! [X8: nat > real] :
% 0.30/0.64 ( ! [N2: nat] : ( ord_less_eq_real @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) )
% 0.30/0.64 => ( topolo6980174941875973593q_real @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % mono_SucI2
% 0.30/0.64 thf(fact_8322_mono__SucI2,axiom,
% 0.30/0.64 ! [X8: nat > set_int] :
% 0.30/0.64 ( ! [N2: nat] : ( ord_less_eq_set_int @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) )
% 0.30/0.64 => ( topolo3100542954746470799et_int @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % mono_SucI2
% 0.30/0.64 thf(fact_8323_mono__SucI2,axiom,
% 0.30/0.64 ! [X8: nat > rat] :
% 0.30/0.64 ( ! [N2: nat] : ( ord_less_eq_rat @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) )
% 0.30/0.64 => ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % mono_SucI2
% 0.30/0.64 thf(fact_8324_mono__SucI2,axiom,
% 0.30/0.64 ! [X8: nat > num] :
% 0.30/0.64 ( ! [N2: nat] : ( ord_less_eq_num @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) )
% 0.30/0.64 => ( topolo1459490580787246023eq_num @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % mono_SucI2
% 0.30/0.64 thf(fact_8325_mono__SucI2,axiom,
% 0.30/0.64 ! [X8: nat > nat] :
% 0.30/0.64 ( ! [N2: nat] : ( ord_less_eq_nat @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) )
% 0.30/0.64 => ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % mono_SucI2
% 0.30/0.64 thf(fact_8326_mono__SucI2,axiom,
% 0.30/0.64 ! [X8: nat > int] :
% 0.30/0.64 ( ! [N2: nat] : ( ord_less_eq_int @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) )
% 0.30/0.64 => ( topolo4899668324122417113eq_int @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % mono_SucI2
% 0.30/0.64 thf(fact_8327_monoseq__Suc,axiom,
% 0.30/0.64 ( topolo6980174941875973593q_real
% 0.30/0.64 = ( ^ [X4: nat > real] :
% 0.30/0.64 ( ! [N3: nat] : ( ord_less_eq_real @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
% 0.30/0.64 | ! [N3: nat] : ( ord_less_eq_real @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoseq_Suc
% 0.30/0.64 thf(fact_8328_monoseq__Suc,axiom,
% 0.30/0.64 ( topolo3100542954746470799et_int
% 0.30/0.64 = ( ^ [X4: nat > set_int] :
% 0.30/0.64 ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
% 0.30/0.64 | ! [N3: nat] : ( ord_less_eq_set_int @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoseq_Suc
% 0.30/0.64 thf(fact_8329_monoseq__Suc,axiom,
% 0.30/0.64 ( topolo4267028734544971653eq_rat
% 0.30/0.64 = ( ^ [X4: nat > rat] :
% 0.30/0.64 ( ! [N3: nat] : ( ord_less_eq_rat @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
% 0.30/0.64 | ! [N3: nat] : ( ord_less_eq_rat @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoseq_Suc
% 0.30/0.64 thf(fact_8330_monoseq__Suc,axiom,
% 0.30/0.64 ( topolo1459490580787246023eq_num
% 0.30/0.64 = ( ^ [X4: nat > num] :
% 0.30/0.64 ( ! [N3: nat] : ( ord_less_eq_num @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
% 0.30/0.64 | ! [N3: nat] : ( ord_less_eq_num @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoseq_Suc
% 0.30/0.64 thf(fact_8331_monoseq__Suc,axiom,
% 0.30/0.64 ( topolo4902158794631467389eq_nat
% 0.30/0.64 = ( ^ [X4: nat > nat] :
% 0.30/0.64 ( ! [N3: nat] : ( ord_less_eq_nat @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
% 0.30/0.64 | ! [N3: nat] : ( ord_less_eq_nat @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoseq_Suc
% 0.30/0.64 thf(fact_8332_monoseq__Suc,axiom,
% 0.30/0.64 ( topolo4899668324122417113eq_int
% 0.30/0.64 = ( ^ [X4: nat > int] :
% 0.30/0.64 ( ! [N3: nat] : ( ord_less_eq_int @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
% 0.30/0.64 | ! [N3: nat] : ( ord_less_eq_int @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoseq_Suc
% 0.30/0.64 thf(fact_8333_monoI1,axiom,
% 0.30/0.64 ! [X8: nat > real] :
% 0.30/0.64 ( ! [M4: nat,N2: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.30/0.64 => ( ord_less_eq_real @ ( X8 @ M4 ) @ ( X8 @ N2 ) ) )
% 0.30/0.64 => ( topolo6980174941875973593q_real @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoI1
% 0.30/0.64 thf(fact_8334_monoI1,axiom,
% 0.30/0.64 ! [X8: nat > set_int] :
% 0.30/0.64 ( ! [M4: nat,N2: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.30/0.64 => ( ord_less_eq_set_int @ ( X8 @ M4 ) @ ( X8 @ N2 ) ) )
% 0.30/0.64 => ( topolo3100542954746470799et_int @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoI1
% 0.30/0.64 thf(fact_8335_monoI1,axiom,
% 0.30/0.64 ! [X8: nat > rat] :
% 0.30/0.64 ( ! [M4: nat,N2: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.30/0.64 => ( ord_less_eq_rat @ ( X8 @ M4 ) @ ( X8 @ N2 ) ) )
% 0.30/0.64 => ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoI1
% 0.30/0.64 thf(fact_8336_monoI1,axiom,
% 0.30/0.64 ! [X8: nat > num] :
% 0.30/0.64 ( ! [M4: nat,N2: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.30/0.64 => ( ord_less_eq_num @ ( X8 @ M4 ) @ ( X8 @ N2 ) ) )
% 0.30/0.64 => ( topolo1459490580787246023eq_num @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoI1
% 0.30/0.64 thf(fact_8337_monoI1,axiom,
% 0.30/0.64 ! [X8: nat > nat] :
% 0.30/0.64 ( ! [M4: nat,N2: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.30/0.64 => ( ord_less_eq_nat @ ( X8 @ M4 ) @ ( X8 @ N2 ) ) )
% 0.30/0.64 => ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoI1
% 0.30/0.64 thf(fact_8338_monoI1,axiom,
% 0.30/0.64 ! [X8: nat > int] :
% 0.30/0.64 ( ! [M4: nat,N2: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.30/0.64 => ( ord_less_eq_int @ ( X8 @ M4 ) @ ( X8 @ N2 ) ) )
% 0.30/0.64 => ( topolo4899668324122417113eq_int @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoI1
% 0.30/0.64 thf(fact_8339_monoI2,axiom,
% 0.30/0.64 ! [X8: nat > real] :
% 0.30/0.64 ( ! [M4: nat,N2: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.30/0.64 => ( ord_less_eq_real @ ( X8 @ N2 ) @ ( X8 @ M4 ) ) )
% 0.30/0.64 => ( topolo6980174941875973593q_real @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoI2
% 0.30/0.64 thf(fact_8340_monoI2,axiom,
% 0.30/0.64 ! [X8: nat > set_int] :
% 0.30/0.64 ( ! [M4: nat,N2: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.30/0.64 => ( ord_less_eq_set_int @ ( X8 @ N2 ) @ ( X8 @ M4 ) ) )
% 0.30/0.64 => ( topolo3100542954746470799et_int @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoI2
% 0.30/0.64 thf(fact_8341_monoI2,axiom,
% 0.30/0.64 ! [X8: nat > rat] :
% 0.30/0.64 ( ! [M4: nat,N2: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.30/0.64 => ( ord_less_eq_rat @ ( X8 @ N2 ) @ ( X8 @ M4 ) ) )
% 0.30/0.64 => ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoI2
% 0.30/0.64 thf(fact_8342_monoI2,axiom,
% 0.30/0.64 ! [X8: nat > num] :
% 0.30/0.64 ( ! [M4: nat,N2: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.30/0.64 => ( ord_less_eq_num @ ( X8 @ N2 ) @ ( X8 @ M4 ) ) )
% 0.30/0.64 => ( topolo1459490580787246023eq_num @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoI2
% 0.30/0.64 thf(fact_8343_monoI2,axiom,
% 0.30/0.64 ! [X8: nat > nat] :
% 0.30/0.64 ( ! [M4: nat,N2: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.30/0.64 => ( ord_less_eq_nat @ ( X8 @ N2 ) @ ( X8 @ M4 ) ) )
% 0.30/0.64 => ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoI2
% 0.30/0.64 thf(fact_8344_monoI2,axiom,
% 0.30/0.64 ! [X8: nat > int] :
% 0.30/0.64 ( ! [M4: nat,N2: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.30/0.64 => ( ord_less_eq_int @ ( X8 @ N2 ) @ ( X8 @ M4 ) ) )
% 0.30/0.64 => ( topolo4899668324122417113eq_int @ X8 ) ) ).
% 0.30/0.64
% 0.30/0.64 % monoI2
% 0.30/0.64 thf(fact_8345_Maclaurin__cos__expansion2,axiom,
% 0.30/0.64 ! [X2: real,N: nat] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ? [T5: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ T5 )
% 0.30/0.64 & ( ord_less_real @ T5 @ X2 )
% 0.30/0.64 & ( ( cos_real @ X2 )
% 0.30/0.64 = ( plus_plus_real
% 0.30/0.64 @ ( groups6591440286371151544t_real
% 0.30/0.64 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
% 0.30/0.64 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Maclaurin_cos_expansion2
% 0.30/0.64 thf(fact_8346_Maclaurin__minus__cos__expansion,axiom,
% 0.30/0.64 ! [N: nat,X2: real] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.30/0.64 => ? [T5: real] :
% 0.30/0.64 ( ( ord_less_real @ X2 @ T5 )
% 0.30/0.64 & ( ord_less_real @ T5 @ zero_zero_real )
% 0.30/0.64 & ( ( cos_real @ X2 )
% 0.30/0.64 = ( plus_plus_real
% 0.30/0.64 @ ( groups6591440286371151544t_real
% 0.30/0.64 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
% 0.30/0.64 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Maclaurin_minus_cos_expansion
% 0.30/0.64 thf(fact_8347_Maclaurin__cos__expansion,axiom,
% 0.30/0.64 ! [X2: real,N: nat] :
% 0.30/0.64 ? [T5: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X2 ) )
% 0.30/0.64 & ( ( cos_real @ X2 )
% 0.30/0.64 = ( plus_plus_real
% 0.30/0.64 @ ( groups6591440286371151544t_real
% 0.30/0.64 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
% 0.30/0.64 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Maclaurin_cos_expansion
% 0.30/0.64 thf(fact_8348_diffs__equiv,axiom,
% 0.30/0.64 ! [C: nat > complex,X2: complex] :
% 0.30/0.64 ( ( summable_complex
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X2 @ N3 ) ) )
% 0.30/0.64 => ( sums_complex
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N3 ) @ ( C @ N3 ) ) @ ( power_power_complex @ X2 @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) )
% 0.30/0.64 @ ( suminf_complex
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X2 @ N3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % diffs_equiv
% 0.30/0.64 thf(fact_8349_diffs__equiv,axiom,
% 0.30/0.64 ! [C: nat > real,X2: real] :
% 0.30/0.64 ( ( summable_real
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X2 @ N3 ) ) )
% 0.30/0.64 => ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( C @ N3 ) ) @ ( power_power_real @ X2 @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) )
% 0.30/0.64 @ ( suminf_real
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X2 @ N3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % diffs_equiv
% 0.30/0.64 thf(fact_8350_tan__double,axiom,
% 0.30/0.64 ! [X2: complex] :
% 0.30/0.64 ( ( ( cos_complex @ X2 )
% 0.30/0.64 != zero_zero_complex )
% 0.30/0.64 => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) )
% 0.30/0.64 != zero_zero_complex )
% 0.30/0.64 => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) )
% 0.30/0.64 = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X2 ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_double
% 0.30/0.64 thf(fact_8351_tan__double,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ( cos_real @ X2 )
% 0.30/0.64 != zero_zero_real )
% 0.30/0.64 => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) )
% 0.30/0.64 != zero_zero_real )
% 0.30/0.64 => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) )
% 0.30/0.64 = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X2 ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_double
% 0.30/0.64 thf(fact_8352_fact__0,axiom,
% 0.30/0.64 ( ( semiri5044797733671781792omplex @ zero_zero_nat )
% 0.30/0.64 = one_one_complex ) ).
% 0.30/0.64
% 0.30/0.64 % fact_0
% 0.30/0.64 thf(fact_8353_fact__0,axiom,
% 0.30/0.64 ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
% 0.30/0.64 = one_one_rat ) ).
% 0.30/0.64
% 0.30/0.64 % fact_0
% 0.30/0.64 thf(fact_8354_fact__0,axiom,
% 0.30/0.64 ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
% 0.30/0.64 = one_one_int ) ).
% 0.30/0.64
% 0.30/0.64 % fact_0
% 0.30/0.64 thf(fact_8355_fact__0,axiom,
% 0.30/0.64 ( ( semiri2265585572941072030t_real @ zero_zero_nat )
% 0.30/0.64 = one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % fact_0
% 0.30/0.64 thf(fact_8356_fact__0,axiom,
% 0.30/0.64 ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
% 0.30/0.64 = one_one_nat ) ).
% 0.30/0.64
% 0.30/0.64 % fact_0
% 0.30/0.64 thf(fact_8357_fact__1,axiom,
% 0.30/0.64 ( ( semiri5044797733671781792omplex @ one_one_nat )
% 0.30/0.64 = one_one_complex ) ).
% 0.30/0.64
% 0.30/0.64 % fact_1
% 0.30/0.64 thf(fact_8358_fact__1,axiom,
% 0.30/0.64 ( ( semiri773545260158071498ct_rat @ one_one_nat )
% 0.30/0.64 = one_one_rat ) ).
% 0.30/0.64
% 0.30/0.64 % fact_1
% 0.30/0.64 thf(fact_8359_fact__1,axiom,
% 0.30/0.64 ( ( semiri1406184849735516958ct_int @ one_one_nat )
% 0.30/0.64 = one_one_int ) ).
% 0.30/0.64
% 0.30/0.64 % fact_1
% 0.30/0.64 thf(fact_8360_fact__1,axiom,
% 0.30/0.64 ( ( semiri2265585572941072030t_real @ one_one_nat )
% 0.30/0.64 = one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % fact_1
% 0.30/0.64 thf(fact_8361_fact__1,axiom,
% 0.30/0.64 ( ( semiri1408675320244567234ct_nat @ one_one_nat )
% 0.30/0.64 = one_one_nat ) ).
% 0.30/0.64
% 0.30/0.64 % fact_1
% 0.30/0.64 thf(fact_8362_fact__Suc__0,axiom,
% 0.30/0.64 ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
% 0.30/0.64 = one_one_complex ) ).
% 0.30/0.64
% 0.30/0.64 % fact_Suc_0
% 0.30/0.64 thf(fact_8363_fact__Suc__0,axiom,
% 0.30/0.64 ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
% 0.30/0.64 = one_one_rat ) ).
% 0.30/0.64
% 0.30/0.64 % fact_Suc_0
% 0.30/0.64 thf(fact_8364_fact__Suc__0,axiom,
% 0.30/0.64 ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
% 0.30/0.64 = one_one_int ) ).
% 0.30/0.64
% 0.30/0.64 % fact_Suc_0
% 0.30/0.64 thf(fact_8365_fact__Suc__0,axiom,
% 0.30/0.64 ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
% 0.30/0.64 = one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % fact_Suc_0
% 0.30/0.64 thf(fact_8366_fact__Suc__0,axiom,
% 0.30/0.64 ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
% 0.30/0.64 = one_one_nat ) ).
% 0.30/0.64
% 0.30/0.64 % fact_Suc_0
% 0.30/0.64 thf(fact_8367_fact__Suc,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
% 0.30/0.64 = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_Suc
% 0.30/0.64 thf(fact_8368_fact__Suc,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
% 0.30/0.64 = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_Suc
% 0.30/0.64 thf(fact_8369_fact__Suc,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
% 0.30/0.64 = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_Suc
% 0.30/0.64 thf(fact_8370_fact__Suc,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
% 0.30/0.64 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_Suc
% 0.30/0.64 thf(fact_8371_tan__npi,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.30/0.64 = zero_zero_real ) ).
% 0.30/0.64
% 0.30/0.64 % tan_npi
% 0.30/0.64 thf(fact_8372_tan__periodic__n,axiom,
% 0.30/0.64 ! [X2: real,N: num] :
% 0.30/0.64 ( ( tan_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
% 0.30/0.64 = ( tan_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_periodic_n
% 0.30/0.64 thf(fact_8373_tan__periodic__nat,axiom,
% 0.30/0.64 ! [X2: real,N: nat] :
% 0.30/0.64 ( ( tan_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
% 0.30/0.64 = ( tan_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_periodic_nat
% 0.30/0.64 thf(fact_8374_tan__periodic__int,axiom,
% 0.30/0.64 ! [X2: real,I2: int] :
% 0.30/0.64 ( ( tan_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ pi ) ) )
% 0.30/0.64 = ( tan_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_periodic_int
% 0.30/0.64 thf(fact_8375_fact__2,axiom,
% 0.30/0.64 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_2
% 0.30/0.64 thf(fact_8376_fact__2,axiom,
% 0.30/0.64 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_2
% 0.30/0.64 thf(fact_8377_fact__2,axiom,
% 0.30/0.64 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_2
% 0.30/0.64 thf(fact_8378_fact__2,axiom,
% 0.30/0.64 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_2
% 0.30/0.64 thf(fact_8379_fact__2,axiom,
% 0.30/0.64 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_2
% 0.30/0.64 thf(fact_8380_tan__periodic,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( tan_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.30/0.64 = ( tan_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_periodic
% 0.30/0.64 thf(fact_8381_fact__ge__zero,axiom,
% 0.30/0.64 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_ge_zero
% 0.30/0.64 thf(fact_8382_fact__ge__zero,axiom,
% 0.30/0.64 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_ge_zero
% 0.30/0.64 thf(fact_8383_fact__ge__zero,axiom,
% 0.30/0.64 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_ge_zero
% 0.30/0.64 thf(fact_8384_fact__ge__zero,axiom,
% 0.30/0.64 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_ge_zero
% 0.30/0.64 thf(fact_8385_fact__not__neg,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% 0.30/0.64
% 0.30/0.64 % fact_not_neg
% 0.30/0.64 thf(fact_8386_fact__not__neg,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% 0.30/0.64
% 0.30/0.64 % fact_not_neg
% 0.30/0.64 thf(fact_8387_fact__not__neg,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% 0.30/0.64
% 0.30/0.64 % fact_not_neg
% 0.30/0.64 thf(fact_8388_fact__not__neg,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% 0.30/0.64
% 0.30/0.64 % fact_not_neg
% 0.30/0.64 thf(fact_8389_fact__gt__zero,axiom,
% 0.30/0.64 ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_gt_zero
% 0.30/0.64 thf(fact_8390_fact__gt__zero,axiom,
% 0.30/0.64 ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_gt_zero
% 0.30/0.64 thf(fact_8391_fact__gt__zero,axiom,
% 0.30/0.64 ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_gt_zero
% 0.30/0.64 thf(fact_8392_fact__gt__zero,axiom,
% 0.30/0.64 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_gt_zero
% 0.30/0.64 thf(fact_8393_fact__ge__1,axiom,
% 0.30/0.64 ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_ge_1
% 0.30/0.64 thf(fact_8394_fact__ge__1,axiom,
% 0.30/0.64 ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_ge_1
% 0.30/0.64 thf(fact_8395_fact__ge__1,axiom,
% 0.30/0.64 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_ge_1
% 0.30/0.64 thf(fact_8396_fact__ge__1,axiom,
% 0.30/0.64 ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_ge_1
% 0.30/0.64 thf(fact_8397_fact__mono,axiom,
% 0.30/0.64 ! [M: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.64 => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_mono
% 0.30/0.64 thf(fact_8398_fact__mono,axiom,
% 0.30/0.64 ! [M: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.64 => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_mono
% 0.30/0.64 thf(fact_8399_fact__mono,axiom,
% 0.30/0.64 ! [M: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.64 => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_mono
% 0.30/0.64 thf(fact_8400_fact__mono,axiom,
% 0.30/0.64 ! [M: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.64 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_mono
% 0.30/0.64 thf(fact_8401_fact__dvd,axiom,
% 0.30/0.64 ! [N: nat,M: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.64 => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_dvd
% 0.30/0.64 thf(fact_8402_fact__dvd,axiom,
% 0.30/0.64 ! [N: nat,M: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.64 => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_dvd
% 0.30/0.64 thf(fact_8403_fact__dvd,axiom,
% 0.30/0.64 ! [N: nat,M: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.64 => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_dvd
% 0.30/0.64 thf(fact_8404_fact__dvd,axiom,
% 0.30/0.64 ! [N: nat,M: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.64 => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_dvd
% 0.30/0.64 thf(fact_8405_fact__less__mono,axiom,
% 0.30/0.64 ! [M: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.64 => ( ( ord_less_nat @ M @ N )
% 0.30/0.64 => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_less_mono
% 0.30/0.64 thf(fact_8406_fact__less__mono,axiom,
% 0.30/0.64 ! [M: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.64 => ( ( ord_less_nat @ M @ N )
% 0.30/0.64 => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_less_mono
% 0.30/0.64 thf(fact_8407_fact__less__mono,axiom,
% 0.30/0.64 ! [M: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.64 => ( ( ord_less_nat @ M @ N )
% 0.30/0.64 => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_less_mono
% 0.30/0.64 thf(fact_8408_fact__less__mono,axiom,
% 0.30/0.64 ! [M: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.64 => ( ( ord_less_nat @ M @ N )
% 0.30/0.64 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_less_mono
% 0.30/0.64 thf(fact_8409_fact__fact__dvd__fact,axiom,
% 0.30/0.64 ! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_fact_dvd_fact
% 0.30/0.64 thf(fact_8410_fact__fact__dvd__fact,axiom,
% 0.30/0.64 ! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_fact_dvd_fact
% 0.30/0.64 thf(fact_8411_fact__fact__dvd__fact,axiom,
% 0.30/0.64 ! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_fact_dvd_fact
% 0.30/0.64 thf(fact_8412_fact__fact__dvd__fact,axiom,
% 0.30/0.64 ! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_fact_dvd_fact
% 0.30/0.64 thf(fact_8413_fact__fact__dvd__fact,axiom,
% 0.30/0.64 ! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_fact_dvd_fact
% 0.30/0.64 thf(fact_8414_fact__mod,axiom,
% 0.30/0.64 ! [M: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.64 => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
% 0.30/0.64 = zero_zero_int ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_mod
% 0.30/0.64 thf(fact_8415_fact__mod,axiom,
% 0.30/0.64 ! [M: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.64 => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
% 0.30/0.64 = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_mod
% 0.30/0.64 thf(fact_8416_fact__mod,axiom,
% 0.30/0.64 ! [M: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.64 => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
% 0.30/0.64 = zero_zero_nat ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_mod
% 0.30/0.64 thf(fact_8417_fact__le__power,axiom,
% 0.30/0.64 ! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_le_power
% 0.30/0.64 thf(fact_8418_fact__le__power,axiom,
% 0.30/0.64 ! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_le_power
% 0.30/0.64 thf(fact_8419_fact__le__power,axiom,
% 0.30/0.64 ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_le_power
% 0.30/0.64 thf(fact_8420_fact__le__power,axiom,
% 0.30/0.64 ! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_le_power
% 0.30/0.64 thf(fact_8421_tan__def,axiom,
% 0.30/0.64 ( tan_complex
% 0.30/0.64 = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X ) @ ( cos_complex @ X ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_def
% 0.30/0.64 thf(fact_8422_tan__def,axiom,
% 0.30/0.64 ( tan_real
% 0.30/0.64 = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ X ) @ ( cos_real @ X ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_def
% 0.30/0.64 thf(fact_8423_choose__dvd,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % choose_dvd
% 0.30/0.64 thf(fact_8424_choose__dvd,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64 => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % choose_dvd
% 0.30/0.64 thf(fact_8425_choose__dvd,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64 => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % choose_dvd
% 0.30/0.64 thf(fact_8426_choose__dvd,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64 => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % choose_dvd
% 0.30/0.64 thf(fact_8427_choose__dvd,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64 => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % choose_dvd
% 0.30/0.64 thf(fact_8428_diffs__def,axiom,
% 0.30/0.64 ( diffs_int
% 0.30/0.64 = ( ^ [C2: nat > int,N3: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) @ ( C2 @ ( suc @ N3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % diffs_def
% 0.30/0.64 thf(fact_8429_diffs__def,axiom,
% 0.30/0.64 ( diffs_real
% 0.30/0.64 = ( ^ [C2: nat > real,N3: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) @ ( C2 @ ( suc @ N3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % diffs_def
% 0.30/0.64 thf(fact_8430_diffs__def,axiom,
% 0.30/0.64 ( diffs_rat
% 0.30/0.64 = ( ^ [C2: nat > rat,N3: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) @ ( C2 @ ( suc @ N3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % diffs_def
% 0.30/0.64 thf(fact_8431_fact__numeral,axiom,
% 0.30/0.64 ! [K: num] :
% 0.30/0.64 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
% 0.30/0.64 = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_numeral
% 0.30/0.64 thf(fact_8432_fact__numeral,axiom,
% 0.30/0.64 ! [K: num] :
% 0.30/0.64 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
% 0.30/0.64 = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_numeral
% 0.30/0.64 thf(fact_8433_fact__numeral,axiom,
% 0.30/0.64 ! [K: num] :
% 0.30/0.64 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
% 0.30/0.64 = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_numeral
% 0.30/0.64 thf(fact_8434_fact__numeral,axiom,
% 0.30/0.64 ! [K: num] :
% 0.30/0.64 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
% 0.30/0.64 = ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_numeral
% 0.30/0.64 thf(fact_8435_fact__numeral,axiom,
% 0.30/0.64 ! [K: num] :
% 0.30/0.64 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
% 0.30/0.64 = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_numeral
% 0.30/0.64 thf(fact_8436_termdiff__converges__all,axiom,
% 0.30/0.64 ! [C: nat > complex,X2: complex] :
% 0.30/0.64 ( ! [X5: complex] :
% 0.30/0.64 ( summable_complex
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_complex @ ( C @ N3 ) @ ( power_power_complex @ X5 @ N3 ) ) )
% 0.30/0.64 => ( summable_complex
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X2 @ N3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % termdiff_converges_all
% 0.30/0.64 thf(fact_8437_termdiff__converges__all,axiom,
% 0.30/0.64 ! [C: nat > real,X2: real] :
% 0.30/0.64 ( ! [X5: real] :
% 0.30/0.64 ( summable_real
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_real @ ( C @ N3 ) @ ( power_power_real @ X5 @ N3 ) ) )
% 0.30/0.64 => ( summable_real
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X2 @ N3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % termdiff_converges_all
% 0.30/0.64 thf(fact_8438_square__fact__le__2__fact,axiom,
% 0.30/0.64 ! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % square_fact_le_2_fact
% 0.30/0.64 thf(fact_8439_tan__45,axiom,
% 0.30/0.64 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64 = one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % tan_45
% 0.30/0.64 thf(fact_8440_fact__num__eq__if,axiom,
% 0.30/0.64 ( semiri5044797733671781792omplex
% 0.30/0.64 = ( ^ [M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M3 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_num_eq_if
% 0.30/0.64 thf(fact_8441_fact__num__eq__if,axiom,
% 0.30/0.64 ( semiri1406184849735516958ct_int
% 0.30/0.64 = ( ^ [M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_num_eq_if
% 0.30/0.64 thf(fact_8442_fact__num__eq__if,axiom,
% 0.30/0.64 ( semiri773545260158071498ct_rat
% 0.30/0.64 = ( ^ [M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M3 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_num_eq_if
% 0.30/0.64 thf(fact_8443_fact__num__eq__if,axiom,
% 0.30/0.64 ( semiri2265585572941072030t_real
% 0.30/0.64 = ( ^ [M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_num_eq_if
% 0.30/0.64 thf(fact_8444_fact__num__eq__if,axiom,
% 0.30/0.64 ( semiri1408675320244567234ct_nat
% 0.30/0.64 = ( ^ [M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_num_eq_if
% 0.30/0.64 thf(fact_8445_fact__code,axiom,
% 0.30/0.64 ( semiri1406184849735516958ct_int
% 0.30/0.64 = ( ^ [N3: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_code
% 0.30/0.64 thf(fact_8446_fact__code,axiom,
% 0.30/0.64 ( semiri773545260158071498ct_rat
% 0.30/0.64 = ( ^ [N3: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_code
% 0.30/0.64 thf(fact_8447_fact__code,axiom,
% 0.30/0.64 ( semiri2265585572941072030t_real
% 0.30/0.64 = ( ^ [N3: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_code
% 0.30/0.64 thf(fact_8448_fact__code,axiom,
% 0.30/0.64 ( semiri1408675320244567234ct_nat
% 0.30/0.64 = ( ^ [N3: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_code
% 0.30/0.64 thf(fact_8449_fact__reduce,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ( semiri1406184849735516958ct_int @ N )
% 0.30/0.64 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_reduce
% 0.30/0.64 thf(fact_8450_fact__reduce,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ( semiri773545260158071498ct_rat @ N )
% 0.30/0.64 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_reduce
% 0.30/0.64 thf(fact_8451_fact__reduce,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ( semiri2265585572941072030t_real @ N )
% 0.30/0.64 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_reduce
% 0.30/0.64 thf(fact_8452_fact__reduce,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ( semiri1408675320244567234ct_nat @ N )
% 0.30/0.64 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_reduce
% 0.30/0.64 thf(fact_8453_lemma__tan__total,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ? [X5: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ X5 )
% 0.30/0.64 & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 & ( ord_less_real @ Y3 @ ( tan_real @ X5 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % lemma_tan_total
% 0.30/0.64 thf(fact_8454_tan__gt__zero,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ord_less_real @ zero_zero_real @ ( tan_real @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_gt_zero
% 0.30/0.64 thf(fact_8455_lemma__tan__total1,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ? [X5: real] :
% 0.30/0.64 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
% 0.30/0.64 & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 & ( ( tan_real @ X5 )
% 0.30/0.64 = Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % lemma_tan_total1
% 0.30/0.64 thf(fact_8456_tan__mono__lt__eq,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( ord_less_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) )
% 0.30/0.64 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_mono_lt_eq
% 0.30/0.64 thf(fact_8457_tan__monotone_H,axiom,
% 0.30/0.64 ! [Y3: real,X2: real] :
% 0.30/0.64 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( ord_less_real @ Y3 @ X2 )
% 0.30/0.64 = ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_monotone'
% 0.30/0.64 thf(fact_8458_tan__monotone,axiom,
% 0.30/0.64 ! [Y3: real,X2: real] :
% 0.30/0.64 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_real @ Y3 @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X2 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_monotone
% 0.30/0.64 thf(fact_8459_tan__total,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ? [X5: real] :
% 0.30/0.64 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
% 0.30/0.64 & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 & ( ( tan_real @ X5 )
% 0.30/0.64 = Y3 )
% 0.30/0.64 & ! [Y5: real] :
% 0.30/0.64 ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 0.30/0.64 & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 & ( ( tan_real @ Y5 )
% 0.30/0.64 = Y3 ) )
% 0.30/0.64 => ( Y5 = X5 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_total
% 0.30/0.64 thf(fact_8460_tan__minus__45,axiom,
% 0.30/0.64 ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.64 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_minus_45
% 0.30/0.64 thf(fact_8461_tan__inverse,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y3 ) )
% 0.30/0.64 = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_inverse
% 0.30/0.64 thf(fact_8462_add__tan__eq,axiom,
% 0.30/0.64 ! [X2: complex,Y3: complex] :
% 0.30/0.64 ( ( ( cos_complex @ X2 )
% 0.30/0.64 != zero_zero_complex )
% 0.30/0.64 => ( ( ( cos_complex @ Y3 )
% 0.30/0.64 != zero_zero_complex )
% 0.30/0.64 => ( ( plus_plus_complex @ ( tan_complex @ X2 ) @ ( tan_complex @ Y3 ) )
% 0.30/0.64 = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X2 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % add_tan_eq
% 0.30/0.64 thf(fact_8463_add__tan__eq,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ( cos_real @ X2 )
% 0.30/0.64 != zero_zero_real )
% 0.30/0.64 => ( ( ( cos_real @ Y3 )
% 0.30/0.64 != zero_zero_real )
% 0.30/0.64 => ( ( plus_plus_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) )
% 0.30/0.64 = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % add_tan_eq
% 0.30/0.64 thf(fact_8464_termdiff__converges,axiom,
% 0.30/0.64 ! [X2: real,K5: real,C: nat > real] :
% 0.30/0.64 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ K5 )
% 0.30/0.64 => ( ! [X5: real] :
% 0.30/0.64 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X5 ) @ K5 )
% 0.30/0.64 => ( summable_real
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_real @ ( C @ N3 ) @ ( power_power_real @ X5 @ N3 ) ) ) )
% 0.30/0.64 => ( summable_real
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X2 @ N3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % termdiff_converges
% 0.30/0.64 thf(fact_8465_termdiff__converges,axiom,
% 0.30/0.64 ! [X2: complex,K5: real,C: nat > complex] :
% 0.30/0.64 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ K5 )
% 0.30/0.64 => ( ! [X5: complex] :
% 0.30/0.64 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X5 ) @ K5 )
% 0.30/0.64 => ( summable_complex
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_complex @ ( C @ N3 ) @ ( power_power_complex @ X5 @ N3 ) ) ) )
% 0.30/0.64 => ( summable_complex
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X2 @ N3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % termdiff_converges
% 0.30/0.64 thf(fact_8466_tan__pos__pi2__le,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_pos_pi2_le
% 0.30/0.64 thf(fact_8467_tan__total__pos,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ? [X5: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 0.30/0.64 & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 & ( ( tan_real @ X5 )
% 0.30/0.64 = Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_total_pos
% 0.30/0.64 thf(fact_8468_tan__less__zero,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.30/0.64 => ( ord_less_real @ ( tan_real @ X2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_less_zero
% 0.30/0.64 thf(fact_8469_tan__mono__le__eq,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) )
% 0.30/0.64 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_mono_le_eq
% 0.30/0.64 thf(fact_8470_tan__mono__le,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.30/0.64 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ord_less_eq_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_mono_le
% 0.30/0.64 thf(fact_8471_tan__bound__pi2,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64 => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X2 ) ) @ one_one_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_bound_pi2
% 0.30/0.64 thf(fact_8472_arctan__unique,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( ( tan_real @ X2 )
% 0.30/0.64 = Y3 )
% 0.30/0.64 => ( ( arctan @ Y3 )
% 0.30/0.64 = X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arctan_unique
% 0.30/0.64 thf(fact_8473_arctan__tan,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( arctan @ ( tan_real @ X2 ) )
% 0.30/0.64 = X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arctan_tan
% 0.30/0.64 thf(fact_8474_arctan,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 0.30/0.64 & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 & ( ( tan_real @ ( arctan @ Y3 ) )
% 0.30/0.64 = Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % arctan
% 0.30/0.64 thf(fact_8475_Maclaurin__zero,axiom,
% 0.30/0.64 ! [X2: real,N: nat,Diff: nat > complex > real] :
% 0.30/0.64 ( ( X2 = zero_zero_real )
% 0.30/0.64 => ( ( N != zero_zero_nat )
% 0.30/0.64 => ( ( groups6591440286371151544t_real
% 0.30/0.64 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
% 0.30/0.64 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64 = ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Maclaurin_zero
% 0.30/0.64 thf(fact_8476_Maclaurin__zero,axiom,
% 0.30/0.64 ! [X2: real,N: nat,Diff: nat > real > real] :
% 0.30/0.64 ( ( X2 = zero_zero_real )
% 0.30/0.64 => ( ( N != zero_zero_nat )
% 0.30/0.64 => ( ( groups6591440286371151544t_real
% 0.30/0.64 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
% 0.30/0.64 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64 = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Maclaurin_zero
% 0.30/0.64 thf(fact_8477_Maclaurin__zero,axiom,
% 0.30/0.64 ! [X2: real,N: nat,Diff: nat > rat > real] :
% 0.30/0.64 ( ( X2 = zero_zero_real )
% 0.30/0.64 => ( ( N != zero_zero_nat )
% 0.30/0.64 => ( ( groups6591440286371151544t_real
% 0.30/0.64 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
% 0.30/0.64 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64 = ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Maclaurin_zero
% 0.30/0.64 thf(fact_8478_Maclaurin__zero,axiom,
% 0.30/0.64 ! [X2: real,N: nat,Diff: nat > nat > real] :
% 0.30/0.64 ( ( X2 = zero_zero_real )
% 0.30/0.64 => ( ( N != zero_zero_nat )
% 0.30/0.64 => ( ( groups6591440286371151544t_real
% 0.30/0.64 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
% 0.30/0.64 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64 = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Maclaurin_zero
% 0.30/0.64 thf(fact_8479_Maclaurin__zero,axiom,
% 0.30/0.64 ! [X2: real,N: nat,Diff: nat > int > real] :
% 0.30/0.64 ( ( X2 = zero_zero_real )
% 0.30/0.64 => ( ( N != zero_zero_nat )
% 0.30/0.64 => ( ( groups6591440286371151544t_real
% 0.30/0.64 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
% 0.30/0.64 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64 = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Maclaurin_zero
% 0.30/0.64 thf(fact_8480_Maclaurin__lemma,axiom,
% 0.30/0.64 ! [H2: real,F: real > real,J: nat > real,N: nat] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.30/0.64 => ? [B8: real] :
% 0.30/0.64 ( ( F @ H2 )
% 0.30/0.64 = ( plus_plus_real
% 0.30/0.64 @ ( groups6591440286371151544t_real
% 0.30/0.64 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 0.30/0.64 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64 @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Maclaurin_lemma
% 0.30/0.64 thf(fact_8481_lemma__tan__add1,axiom,
% 0.30/0.64 ! [X2: complex,Y3: complex] :
% 0.30/0.64 ( ( ( cos_complex @ X2 )
% 0.30/0.64 != zero_zero_complex )
% 0.30/0.64 => ( ( ( cos_complex @ Y3 )
% 0.30/0.64 != zero_zero_complex )
% 0.30/0.64 => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X2 ) @ ( tan_complex @ Y3 ) ) )
% 0.30/0.64 = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X2 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % lemma_tan_add1
% 0.30/0.64 thf(fact_8482_lemma__tan__add1,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ( cos_real @ X2 )
% 0.30/0.64 != zero_zero_real )
% 0.30/0.64 => ( ( ( cos_real @ Y3 )
% 0.30/0.64 != zero_zero_real )
% 0.30/0.64 => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) ) )
% 0.30/0.64 = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % lemma_tan_add1
% 0.30/0.64 thf(fact_8483_tan__diff,axiom,
% 0.30/0.64 ! [X2: complex,Y3: complex] :
% 0.30/0.64 ( ( ( cos_complex @ X2 )
% 0.30/0.64 != zero_zero_complex )
% 0.30/0.64 => ( ( ( cos_complex @ Y3 )
% 0.30/0.64 != zero_zero_complex )
% 0.30/0.64 => ( ( ( cos_complex @ ( minus_minus_complex @ X2 @ Y3 ) )
% 0.30/0.64 != zero_zero_complex )
% 0.30/0.64 => ( ( tan_complex @ ( minus_minus_complex @ X2 @ Y3 ) )
% 0.30/0.64 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X2 ) @ ( tan_complex @ Y3 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X2 ) @ ( tan_complex @ Y3 ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_diff
% 0.30/0.64 thf(fact_8484_tan__diff,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ( cos_real @ X2 )
% 0.30/0.64 != zero_zero_real )
% 0.30/0.64 => ( ( ( cos_real @ Y3 )
% 0.30/0.64 != zero_zero_real )
% 0.30/0.64 => ( ( ( cos_real @ ( minus_minus_real @ X2 @ Y3 ) )
% 0.30/0.64 != zero_zero_real )
% 0.30/0.64 => ( ( tan_real @ ( minus_minus_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_diff
% 0.30/0.64 thf(fact_8485_tan__add,axiom,
% 0.30/0.64 ! [X2: complex,Y3: complex] :
% 0.30/0.64 ( ( ( cos_complex @ X2 )
% 0.30/0.64 != zero_zero_complex )
% 0.30/0.64 => ( ( ( cos_complex @ Y3 )
% 0.30/0.64 != zero_zero_complex )
% 0.30/0.64 => ( ( ( cos_complex @ ( plus_plus_complex @ X2 @ Y3 ) )
% 0.30/0.64 != zero_zero_complex )
% 0.30/0.64 => ( ( tan_complex @ ( plus_plus_complex @ X2 @ Y3 ) )
% 0.30/0.64 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X2 ) @ ( tan_complex @ Y3 ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X2 ) @ ( tan_complex @ Y3 ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_add
% 0.30/0.64 thf(fact_8486_tan__add,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ( cos_real @ X2 )
% 0.30/0.64 != zero_zero_real )
% 0.30/0.64 => ( ( ( cos_real @ Y3 )
% 0.30/0.64 != zero_zero_real )
% 0.30/0.64 => ( ( ( cos_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 0.30/0.64 != zero_zero_real )
% 0.30/0.64 => ( ( tan_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_add
% 0.30/0.64 thf(fact_8487_tan__total__pi4,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.30/0.64 => ? [Z2: real] :
% 0.30/0.64 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z2 )
% 0.30/0.64 & ( ord_less_real @ Z2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64 & ( ( tan_real @ Z2 )
% 0.30/0.64 = X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_total_pi4
% 0.30/0.64 thf(fact_8488_cos__paired,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_real @ X2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 0.30/0.64 @ ( cos_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_paired
% 0.30/0.64 thf(fact_8489_tan__half,axiom,
% 0.30/0.64 ( tan_complex
% 0.30/0.64 = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ one_one_complex ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_half
% 0.30/0.64 thf(fact_8490_tan__half,axiom,
% 0.30/0.64 ( tan_real
% 0.30/0.64 = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_half
% 0.30/0.64 thf(fact_8491_cos__coeff__def,axiom,
% 0.30/0.64 ( cos_coeff
% 0.30/0.64 = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ zero_zero_real ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_coeff_def
% 0.30/0.64 thf(fact_8492_sin__paired,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( sums_real
% 0.30/0.64 @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 0.30/0.64 @ ( sin_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_paired
% 0.30/0.64 thf(fact_8493_Maclaurin__sin__expansion3,axiom,
% 0.30/0.64 ! [N: nat,X2: real] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ? [T5: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ T5 )
% 0.30/0.64 & ( ord_less_real @ T5 @ X2 )
% 0.30/0.64 & ( ( sin_real @ X2 )
% 0.30/0.64 = ( plus_plus_real
% 0.30/0.64 @ ( groups6591440286371151544t_real
% 0.30/0.64 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
% 0.30/0.64 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Maclaurin_sin_expansion3
% 0.30/0.64 thf(fact_8494_Maclaurin__sin__expansion4,axiom,
% 0.30/0.64 ! [X2: real,N: nat] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ? [T5: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ T5 )
% 0.30/0.64 & ( ord_less_eq_real @ T5 @ X2 )
% 0.30/0.64 & ( ( sin_real @ X2 )
% 0.30/0.64 = ( plus_plus_real
% 0.30/0.64 @ ( groups6591440286371151544t_real
% 0.30/0.64 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
% 0.30/0.64 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Maclaurin_sin_expansion4
% 0.30/0.64 thf(fact_8495_Maclaurin__sin__expansion2,axiom,
% 0.30/0.64 ! [X2: real,N: nat] :
% 0.30/0.64 ? [T5: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X2 ) )
% 0.30/0.64 & ( ( sin_real @ X2 )
% 0.30/0.64 = ( plus_plus_real
% 0.30/0.64 @ ( groups6591440286371151544t_real
% 0.30/0.64 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
% 0.30/0.64 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Maclaurin_sin_expansion2
% 0.30/0.64 thf(fact_8496_Maclaurin__sin__expansion,axiom,
% 0.30/0.64 ! [X2: real,N: nat] :
% 0.30/0.64 ? [T5: real] :
% 0.30/0.64 ( ( sin_real @ X2 )
% 0.30/0.64 = ( plus_plus_real
% 0.30/0.64 @ ( groups6591440286371151544t_real
% 0.30/0.64 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
% 0.30/0.64 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Maclaurin_sin_expansion
% 0.30/0.64 thf(fact_8497_sin__coeff__def,axiom,
% 0.30/0.64 ( sin_coeff
% 0.30/0.64 = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_coeff_def
% 0.30/0.64 thf(fact_8498_fact__ge__self,axiom,
% 0.30/0.64 ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_ge_self
% 0.30/0.64 thf(fact_8499_fact__mono__nat,axiom,
% 0.30/0.64 ! [M: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.64 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_mono_nat
% 0.30/0.64 thf(fact_8500_fact__less__mono__nat,axiom,
% 0.30/0.64 ! [M: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.64 => ( ( ord_less_nat @ M @ N )
% 0.30/0.64 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_less_mono_nat
% 0.30/0.64 thf(fact_8501_fact__ge__Suc__0__nat,axiom,
% 0.30/0.64 ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_ge_Suc_0_nat
% 0.30/0.64 thf(fact_8502_dvd__fact,axiom,
% 0.30/0.64 ! [M: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.30/0.64 => ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.64 => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % dvd_fact
% 0.30/0.64 thf(fact_8503_fact__diff__Suc,axiom,
% 0.30/0.64 ! [N: nat,M: nat] :
% 0.30/0.64 ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.30/0.64 => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
% 0.30/0.64 = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_diff_Suc
% 0.30/0.64 thf(fact_8504_fact__div__fact__le__pow,axiom,
% 0.30/0.64 ! [R2: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ R2 @ N )
% 0.30/0.64 => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_div_fact_le_pow
% 0.30/0.64 thf(fact_8505_sin__coeff__Suc,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( sin_coeff @ ( suc @ N ) )
% 0.30/0.64 = ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_coeff_Suc
% 0.30/0.64 thf(fact_8506_cos__coeff__Suc,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( cos_coeff @ ( suc @ N ) )
% 0.30/0.64 = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_coeff_Suc
% 0.30/0.64 thf(fact_8507_sin__tan,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( sin_real @ X2 )
% 0.30/0.64 = ( divide_divide_real @ ( tan_real @ X2 ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_tan
% 0.30/0.64 thf(fact_8508_cos__tan,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( cos_real @ X2 )
% 0.30/0.64 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_tan
% 0.30/0.64 thf(fact_8509_arcosh__def,axiom,
% 0.30/0.64 ( arcosh_real
% 0.30/0.64 = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arcosh_def
% 0.30/0.64 thf(fact_8510_complex__unimodular__polar,axiom,
% 0.30/0.64 ! [Z: complex] :
% 0.30/0.64 ( ( ( real_V1022390504157884413omplex @ Z )
% 0.30/0.64 = one_one_real )
% 0.30/0.64 => ~ ! [T5: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.30/0.64 => ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.30/0.64 => ( Z
% 0.30/0.64 != ( complex2 @ ( cos_real @ T5 ) @ ( sin_real @ T5 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % complex_unimodular_polar
% 0.30/0.64 thf(fact_8511_Maclaurin__exp__lt,axiom,
% 0.30/0.64 ! [X2: real,N: nat] :
% 0.30/0.64 ( ( X2 != zero_zero_real )
% 0.30/0.64 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ? [T5: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
% 0.30/0.64 & ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X2 ) )
% 0.30/0.64 & ( ( exp_real @ X2 )
% 0.30/0.64 = ( plus_plus_real
% 0.30/0.64 @ ( groups6591440286371151544t_real
% 0.30/0.64 @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X2 @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
% 0.30/0.64 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Maclaurin_exp_lt
% 0.30/0.64 thf(fact_8512_real__sqrt__eq__iff,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ( sqrt @ X2 )
% 0.30/0.64 = ( sqrt @ Y3 ) )
% 0.30/0.64 = ( X2 = Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_eq_iff
% 0.30/0.64 thf(fact_8513_real__sqrt__eq__zero__cancel__iff,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ( sqrt @ X2 )
% 0.30/0.64 = zero_zero_real )
% 0.30/0.64 = ( X2 = zero_zero_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_eq_zero_cancel_iff
% 0.30/0.64 thf(fact_8514_real__sqrt__zero,axiom,
% 0.30/0.64 ( ( sqrt @ zero_zero_real )
% 0.30/0.64 = zero_zero_real ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_zero
% 0.30/0.64 thf(fact_8515_real__sqrt__less__iff,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) )
% 0.30/0.64 = ( ord_less_real @ X2 @ Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_less_iff
% 0.30/0.64 thf(fact_8516_real__sqrt__le__iff,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) )
% 0.30/0.64 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_le_iff
% 0.30/0.64 thf(fact_8517_real__sqrt__eq__1__iff,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ( sqrt @ X2 )
% 0.30/0.64 = one_one_real )
% 0.30/0.64 = ( X2 = one_one_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_eq_1_iff
% 0.30/0.64 thf(fact_8518_real__sqrt__one,axiom,
% 0.30/0.64 ( ( sqrt @ one_one_real )
% 0.30/0.64 = one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_one
% 0.30/0.64 thf(fact_8519_exp__le__cancel__iff,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) )
% 0.30/0.64 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_le_cancel_iff
% 0.30/0.64 thf(fact_8520_exp__zero,axiom,
% 0.30/0.64 ( ( exp_complex @ zero_zero_complex )
% 0.30/0.64 = one_one_complex ) ).
% 0.30/0.64
% 0.30/0.64 % exp_zero
% 0.30/0.64 thf(fact_8521_exp__zero,axiom,
% 0.30/0.64 ( ( exp_real @ zero_zero_real )
% 0.30/0.64 = one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % exp_zero
% 0.30/0.64 thf(fact_8522_of__real__eq__1__iff,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ( real_V1803761363581548252l_real @ X2 )
% 0.30/0.64 = one_one_real )
% 0.30/0.64 = ( X2 = one_one_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_real_eq_1_iff
% 0.30/0.64 thf(fact_8523_of__real__eq__1__iff,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ( real_V4546457046886955230omplex @ X2 )
% 0.30/0.64 = one_one_complex )
% 0.30/0.64 = ( X2 = one_one_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_real_eq_1_iff
% 0.30/0.64 thf(fact_8524_of__real__1,axiom,
% 0.30/0.64 ( ( real_V1803761363581548252l_real @ one_one_real )
% 0.30/0.64 = one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % of_real_1
% 0.30/0.64 thf(fact_8525_of__real__1,axiom,
% 0.30/0.64 ( ( real_V4546457046886955230omplex @ one_one_real )
% 0.30/0.64 = one_one_complex ) ).
% 0.30/0.64
% 0.30/0.64 % of_real_1
% 0.30/0.64 thf(fact_8526_of__real__numeral,axiom,
% 0.30/0.64 ! [W: num] :
% 0.30/0.64 ( ( real_V1803761363581548252l_real @ ( numeral_numeral_real @ W ) )
% 0.30/0.64 = ( numeral_numeral_real @ W ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_real_numeral
% 0.30/0.64 thf(fact_8527_of__real__numeral,axiom,
% 0.30/0.64 ! [W: num] :
% 0.30/0.64 ( ( real_V4546457046886955230omplex @ ( numeral_numeral_real @ W ) )
% 0.30/0.64 = ( numera6690914467698888265omplex @ W ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_real_numeral
% 0.30/0.64 thf(fact_8528_of__real__mult,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( real_V1803761363581548252l_real @ ( times_times_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( times_times_real @ ( real_V1803761363581548252l_real @ X2 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_real_mult
% 0.30/0.64 thf(fact_8529_of__real__mult,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( real_V4546457046886955230omplex @ ( times_times_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( times_times_complex @ ( real_V4546457046886955230omplex @ X2 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_real_mult
% 0.30/0.64 thf(fact_8530_real__sqrt__lt__0__iff,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ ( sqrt @ X2 ) @ zero_zero_real )
% 0.30/0.64 = ( ord_less_real @ X2 @ zero_zero_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_lt_0_iff
% 0.30/0.64 thf(fact_8531_real__sqrt__gt__0__iff,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 0.30/0.64 = ( ord_less_real @ zero_zero_real @ Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_gt_0_iff
% 0.30/0.64 thf(fact_8532_of__real__divide,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X2 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_real_divide
% 0.30/0.64 thf(fact_8533_of__real__divide,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X2 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_real_divide
% 0.30/0.64 thf(fact_8534_real__sqrt__ge__0__iff,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 0.30/0.64 = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_ge_0_iff
% 0.30/0.64 thf(fact_8535_real__sqrt__le__0__iff,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ zero_zero_real )
% 0.30/0.64 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_le_0_iff
% 0.30/0.64 thf(fact_8536_of__real__add,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( plus_plus_real @ ( real_V1803761363581548252l_real @ X2 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_real_add
% 0.30/0.64 thf(fact_8537_of__real__add,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X2 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_real_add
% 0.30/0.64 thf(fact_8538_of__real__power,axiom,
% 0.30/0.64 ! [X2: real,N: nat] :
% 0.30/0.64 ( ( real_V1803761363581548252l_real @ ( power_power_real @ X2 @ N ) )
% 0.30/0.64 = ( power_power_real @ ( real_V1803761363581548252l_real @ X2 ) @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_real_power
% 0.30/0.64 thf(fact_8539_of__real__power,axiom,
% 0.30/0.64 ! [X2: real,N: nat] :
% 0.30/0.64 ( ( real_V4546457046886955230omplex @ ( power_power_real @ X2 @ N ) )
% 0.30/0.64 = ( power_power_complex @ ( real_V4546457046886955230omplex @ X2 ) @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_real_power
% 0.30/0.64 thf(fact_8540_real__sqrt__gt__1__iff,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ one_one_real @ ( sqrt @ Y3 ) )
% 0.30/0.64 = ( ord_less_real @ one_one_real @ Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_gt_1_iff
% 0.30/0.64 thf(fact_8541_real__sqrt__lt__1__iff,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ ( sqrt @ X2 ) @ one_one_real )
% 0.30/0.64 = ( ord_less_real @ X2 @ one_one_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_lt_1_iff
% 0.30/0.64 thf(fact_8542_real__sqrt__ge__1__iff,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y3 ) )
% 0.30/0.64 = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_ge_1_iff
% 0.30/0.64 thf(fact_8543_real__sqrt__le__1__iff,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ one_one_real )
% 0.30/0.64 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_le_1_iff
% 0.30/0.64 thf(fact_8544_of__real__diff,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( real_V1803761363581548252l_real @ ( minus_minus_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( minus_minus_real @ ( real_V1803761363581548252l_real @ X2 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_real_diff
% 0.30/0.64 thf(fact_8545_of__real__diff,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( real_V4546457046886955230omplex @ ( minus_minus_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( minus_minus_complex @ ( real_V4546457046886955230omplex @ X2 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_real_diff
% 0.30/0.64 thf(fact_8546_exp__eq__one__iff,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ( exp_real @ X2 )
% 0.30/0.64 = one_one_real )
% 0.30/0.64 = ( X2 = zero_zero_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_eq_one_iff
% 0.30/0.64 thf(fact_8547_real__sqrt__abs2,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( sqrt @ ( times_times_real @ X2 @ X2 ) )
% 0.30/0.64 = ( abs_abs_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_abs2
% 0.30/0.64 thf(fact_8548_real__sqrt__mult__self,axiom,
% 0.30/0.64 ! [A: real] :
% 0.30/0.64 ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
% 0.30/0.64 = ( abs_abs_real @ A ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_mult_self
% 0.30/0.64 thf(fact_8549_real__sqrt__four,axiom,
% 0.30/0.64 ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.64 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_four
% 0.30/0.64 thf(fact_8550_one__less__exp__iff,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ one_one_real @ ( exp_real @ X2 ) )
% 0.30/0.64 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % one_less_exp_iff
% 0.30/0.64 thf(fact_8551_exp__less__one__iff,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ ( exp_real @ X2 ) @ one_one_real )
% 0.30/0.64 = ( ord_less_real @ X2 @ zero_zero_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_less_one_iff
% 0.30/0.64 thf(fact_8552_exp__le__one__iff,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( exp_real @ X2 ) @ one_one_real )
% 0.30/0.64 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_le_one_iff
% 0.30/0.64 thf(fact_8553_one__le__exp__iff,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X2 ) )
% 0.30/0.64 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % one_le_exp_iff
% 0.30/0.64 thf(fact_8554_of__real__neg__numeral,axiom,
% 0.30/0.64 ! [W: num] :
% 0.30/0.64 ( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.64 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_real_neg_numeral
% 0.30/0.64 thf(fact_8555_of__real__neg__numeral,axiom,
% 0.30/0.64 ! [W: num] :
% 0.30/0.64 ( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.64 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_real_neg_numeral
% 0.30/0.64 thf(fact_8556_cos__of__real__pi,axiom,
% 0.30/0.64 ( ( cos_real @ ( real_V1803761363581548252l_real @ pi ) )
% 0.30/0.64 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_of_real_pi
% 0.30/0.64 thf(fact_8557_cos__of__real__pi,axiom,
% 0.30/0.64 ( ( cos_complex @ ( real_V4546457046886955230omplex @ pi ) )
% 0.30/0.64 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_of_real_pi
% 0.30/0.64 thf(fact_8558_norm__cos__sin,axiom,
% 0.30/0.64 ! [T: real] :
% 0.30/0.64 ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
% 0.30/0.64 = one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % norm_cos_sin
% 0.30/0.64 thf(fact_8559_real__sqrt__abs,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( sqrt @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.64 = ( abs_abs_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_abs
% 0.30/0.64 thf(fact_8560_real__sqrt__pow2,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( power_power_real @ ( sqrt @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64 = X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_pow2
% 0.30/0.64 thf(fact_8561_real__sqrt__pow2__iff,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ( power_power_real @ ( sqrt @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64 = X2 )
% 0.30/0.64 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_pow2_iff
% 0.30/0.64 thf(fact_8562_real__sqrt__sum__squares__mult__squared__eq,axiom,
% 0.30/0.64 ! [X2: real,Y3: real,Xa2: real,Ya: real] :
% 0.30/0.64 ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64 = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_sum_squares_mult_squared_eq
% 0.30/0.64 thf(fact_8563_norm__of__real__add1,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X2 ) @ one_one_real ) )
% 0.30/0.64 = ( abs_abs_real @ ( plus_plus_real @ X2 @ one_one_real ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % norm_of_real_add1
% 0.30/0.64 thf(fact_8564_norm__of__real__add1,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X2 ) @ one_one_complex ) )
% 0.30/0.64 = ( abs_abs_real @ ( plus_plus_real @ X2 @ one_one_real ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % norm_of_real_add1
% 0.30/0.64 thf(fact_8565_norm__of__real__addn,axiom,
% 0.30/0.64 ! [X2: real,B: num] :
% 0.30/0.64 ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X2 ) @ ( numeral_numeral_real @ B ) ) )
% 0.30/0.64 = ( abs_abs_real @ ( plus_plus_real @ X2 @ ( numeral_numeral_real @ B ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % norm_of_real_addn
% 0.30/0.64 thf(fact_8566_norm__of__real__addn,axiom,
% 0.30/0.64 ! [X2: real,B: num] :
% 0.30/0.64 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X2 ) @ ( numera6690914467698888265omplex @ B ) ) )
% 0.30/0.64 = ( abs_abs_real @ ( plus_plus_real @ X2 @ ( numeral_numeral_real @ B ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % norm_of_real_addn
% 0.30/0.64 thf(fact_8567_cos__of__real__pi__half,axiom,
% 0.30/0.64 ( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 = zero_zero_real ) ).
% 0.30/0.64
% 0.30/0.64 % cos_of_real_pi_half
% 0.30/0.64 thf(fact_8568_cos__of__real__pi__half,axiom,
% 0.30/0.64 ( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 0.30/0.64 = zero_zero_complex ) ).
% 0.30/0.64
% 0.30/0.64 % cos_of_real_pi_half
% 0.30/0.64 thf(fact_8569_sin__of__real__pi__half,axiom,
% 0.30/0.64 ( ( sin_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 = one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % sin_of_real_pi_half
% 0.30/0.64 thf(fact_8570_sin__of__real__pi__half,axiom,
% 0.30/0.64 ( ( sin_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 0.30/0.64 = one_one_complex ) ).
% 0.30/0.64
% 0.30/0.64 % sin_of_real_pi_half
% 0.30/0.64 thf(fact_8571_real__sqrt__minus,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( sqrt @ ( uminus_uminus_real @ X2 ) )
% 0.30/0.64 = ( uminus_uminus_real @ ( sqrt @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_minus
% 0.30/0.64 thf(fact_8572_real__sqrt__less__mono,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.64 => ( ord_less_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_less_mono
% 0.30/0.64 thf(fact_8573_real__sqrt__le__mono,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.30/0.64 => ( ord_less_eq_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_le_mono
% 0.30/0.64 thf(fact_8574_complex__of__real__mult__Complex,axiom,
% 0.30/0.64 ! [R2: real,X2: real,Y3: real] :
% 0.30/0.64 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X2 @ Y3 ) )
% 0.30/0.64 = ( complex2 @ ( times_times_real @ R2 @ X2 ) @ ( times_times_real @ R2 @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % complex_of_real_mult_Complex
% 0.30/0.64 thf(fact_8575_Complex__mult__complex__of__real,axiom,
% 0.30/0.64 ! [X2: real,Y3: real,R2: real] :
% 0.30/0.64 ( ( times_times_complex @ ( complex2 @ X2 @ Y3 ) @ ( real_V4546457046886955230omplex @ R2 ) )
% 0.30/0.64 = ( complex2 @ ( times_times_real @ X2 @ R2 ) @ ( times_times_real @ Y3 @ R2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Complex_mult_complex_of_real
% 0.30/0.64 thf(fact_8576_real__sqrt__power,axiom,
% 0.30/0.64 ! [X2: real,K: nat] :
% 0.30/0.64 ( ( sqrt @ ( power_power_real @ X2 @ K ) )
% 0.30/0.64 = ( power_power_real @ ( sqrt @ X2 ) @ K ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_power
% 0.30/0.64 thf(fact_8577_real__sqrt__mult,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( sqrt @ ( times_times_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( times_times_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_mult
% 0.30/0.64 thf(fact_8578_real__sqrt__divide,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( sqrt @ ( divide_divide_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( divide_divide_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_divide
% 0.30/0.64 thf(fact_8579_norm__exp,axiom,
% 0.30/0.64 ! [X2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X2 ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % norm_exp
% 0.30/0.64 thf(fact_8580_norm__exp,axiom,
% 0.30/0.64 ! [X2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X2 ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % norm_exp
% 0.30/0.64 thf(fact_8581_exp__times__arg__commute,axiom,
% 0.30/0.64 ! [A2: complex] :
% 0.30/0.64 ( ( times_times_complex @ ( exp_complex @ A2 ) @ A2 )
% 0.30/0.64 = ( times_times_complex @ A2 @ ( exp_complex @ A2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_times_arg_commute
% 0.30/0.64 thf(fact_8582_exp__times__arg__commute,axiom,
% 0.30/0.64 ! [A2: real] :
% 0.30/0.64 ( ( times_times_real @ ( exp_real @ A2 ) @ A2 )
% 0.30/0.64 = ( times_times_real @ A2 @ ( exp_real @ A2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_times_arg_commute
% 0.30/0.64 thf(fact_8583_complex__diff,axiom,
% 0.30/0.64 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.64 ( ( minus_minus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 0.30/0.64 = ( complex2 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % complex_diff
% 0.30/0.64 thf(fact_8584_real__sqrt__gt__zero,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ord_less_real @ zero_zero_real @ ( sqrt @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_gt_zero
% 0.30/0.64 thf(fact_8585_real__sqrt__ge__zero,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_ge_zero
% 0.30/0.64 thf(fact_8586_real__sqrt__eq__zero__cancel,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ( sqrt @ X2 )
% 0.30/0.64 = zero_zero_real )
% 0.30/0.64 => ( X2 = zero_zero_real ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_eq_zero_cancel
% 0.30/0.64 thf(fact_8587_not__exp__le__zero,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ~ ( ord_less_eq_real @ ( exp_real @ X2 ) @ zero_zero_real ) ).
% 0.30/0.64
% 0.30/0.64 % not_exp_le_zero
% 0.30/0.64 thf(fact_8588_exp__ge__zero,axiom,
% 0.30/0.64 ! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_ge_zero
% 0.30/0.64 thf(fact_8589_real__sqrt__ge__one,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 0.30/0.64 => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_ge_one
% 0.30/0.64 thf(fact_8590_mult__exp__exp,axiom,
% 0.30/0.64 ! [X2: complex,Y3: complex] :
% 0.30/0.64 ( ( times_times_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ Y3 ) )
% 0.30/0.64 = ( exp_complex @ ( plus_plus_complex @ X2 @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % mult_exp_exp
% 0.30/0.64 thf(fact_8591_mult__exp__exp,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( times_times_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) )
% 0.30/0.64 = ( exp_real @ ( plus_plus_real @ X2 @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % mult_exp_exp
% 0.30/0.64 thf(fact_8592_exp__add__commuting,axiom,
% 0.30/0.64 ! [X2: complex,Y3: complex] :
% 0.30/0.64 ( ( ( times_times_complex @ X2 @ Y3 )
% 0.30/0.64 = ( times_times_complex @ Y3 @ X2 ) )
% 0.30/0.64 => ( ( exp_complex @ ( plus_plus_complex @ X2 @ Y3 ) )
% 0.30/0.64 = ( times_times_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_add_commuting
% 0.30/0.64 thf(fact_8593_exp__add__commuting,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ( times_times_real @ X2 @ Y3 )
% 0.30/0.64 = ( times_times_real @ Y3 @ X2 ) )
% 0.30/0.64 => ( ( exp_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( times_times_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_add_commuting
% 0.30/0.64 thf(fact_8594_exp__diff,axiom,
% 0.30/0.64 ! [X2: complex,Y3: complex] :
% 0.30/0.64 ( ( exp_complex @ ( minus_minus_complex @ X2 @ Y3 ) )
% 0.30/0.64 = ( divide1717551699836669952omplex @ ( exp_complex @ X2 ) @ ( exp_complex @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_diff
% 0.30/0.64 thf(fact_8595_exp__diff,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( exp_real @ ( minus_minus_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( divide_divide_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_diff
% 0.30/0.64 thf(fact_8596_Complex__eq__numeral,axiom,
% 0.30/0.64 ! [A: real,B: real,W: num] :
% 0.30/0.64 ( ( ( complex2 @ A @ B )
% 0.30/0.64 = ( numera6690914467698888265omplex @ W ) )
% 0.30/0.64 = ( ( A
% 0.30/0.64 = ( numeral_numeral_real @ W ) )
% 0.30/0.64 & ( B = zero_zero_real ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Complex_eq_numeral
% 0.30/0.64 thf(fact_8597_complex__norm,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( real_V1022390504157884413omplex @ ( complex2 @ X2 @ Y3 ) )
% 0.30/0.64 = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % complex_norm
% 0.30/0.64 thf(fact_8598_nonzero__of__real__divide,axiom,
% 0.30/0.64 ! [Y3: real,X2: real] :
% 0.30/0.64 ( ( Y3 != zero_zero_real )
% 0.30/0.64 => ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X2 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % nonzero_of_real_divide
% 0.30/0.64 thf(fact_8599_nonzero__of__real__divide,axiom,
% 0.30/0.64 ! [Y3: real,X2: real] :
% 0.30/0.64 ( ( Y3 != zero_zero_real )
% 0.30/0.64 => ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X2 @ Y3 ) )
% 0.30/0.64 = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X2 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % nonzero_of_real_divide
% 0.30/0.64 thf(fact_8600_exp__gt__one,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ord_less_real @ one_one_real @ ( exp_real @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_gt_one
% 0.30/0.64 thf(fact_8601_real__div__sqrt,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( divide_divide_real @ X2 @ ( sqrt @ X2 ) )
% 0.30/0.64 = ( sqrt @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_div_sqrt
% 0.30/0.64 thf(fact_8602_sqrt__add__le__add__sqrt,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sqrt_add_le_add_sqrt
% 0.30/0.64 thf(fact_8603_exp__ge__add__one__self,axiom,
% 0.30/0.64 ! [X2: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( exp_real @ X2 ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_ge_add_one_self
% 0.30/0.64 thf(fact_8604_le__real__sqrt__sumsq,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] : ( ord_less_eq_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % le_real_sqrt_sumsq
% 0.30/0.64 thf(fact_8605_exp__minus__inverse,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( times_times_real @ ( exp_real @ X2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) )
% 0.30/0.64 = one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % exp_minus_inverse
% 0.30/0.64 thf(fact_8606_exp__minus__inverse,axiom,
% 0.30/0.64 ! [X2: complex] :
% 0.30/0.64 ( ( times_times_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X2 ) ) )
% 0.30/0.64 = one_one_complex ) ).
% 0.30/0.64
% 0.30/0.64 % exp_minus_inverse
% 0.30/0.64 thf(fact_8607_exp__of__nat__mult,axiom,
% 0.30/0.64 ! [N: nat,X2: complex] :
% 0.30/0.64 ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ X2 ) )
% 0.30/0.64 = ( power_power_complex @ ( exp_complex @ X2 ) @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_of_nat_mult
% 0.30/0.64 thf(fact_8608_exp__of__nat__mult,axiom,
% 0.30/0.64 ! [N: nat,X2: real] :
% 0.30/0.64 ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) )
% 0.30/0.64 = ( power_power_real @ ( exp_real @ X2 ) @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_of_nat_mult
% 0.30/0.64 thf(fact_8609_exp__of__nat2__mult,axiom,
% 0.30/0.64 ! [X2: complex,N: nat] :
% 0.30/0.64 ( ( exp_complex @ ( times_times_complex @ X2 @ ( semiri8010041392384452111omplex @ N ) ) )
% 0.30/0.64 = ( power_power_complex @ ( exp_complex @ X2 ) @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_of_nat2_mult
% 0.30/0.64 thf(fact_8610_exp__of__nat2__mult,axiom,
% 0.30/0.64 ! [X2: real,N: nat] :
% 0.30/0.64 ( ( exp_real @ ( times_times_real @ X2 @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.30/0.64 = ( power_power_real @ ( exp_real @ X2 ) @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_of_nat2_mult
% 0.30/0.64 thf(fact_8611_Complex__eq__neg__numeral,axiom,
% 0.30/0.64 ! [A: real,B: real,W: num] :
% 0.30/0.64 ( ( ( complex2 @ A @ B )
% 0.30/0.64 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.64 = ( ( A
% 0.30/0.64 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.64 & ( B = zero_zero_real ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Complex_eq_neg_numeral
% 0.30/0.64 thf(fact_8612_log__ln,axiom,
% 0.30/0.64 ( ln_ln_real
% 0.30/0.64 = ( log @ ( exp_real @ one_one_real ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % log_ln
% 0.30/0.64 thf(fact_8613_complex__mult,axiom,
% 0.30/0.64 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.64 ( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 0.30/0.64 = ( 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 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % complex_mult
% 0.30/0.64 thf(fact_8614_one__complex_Ocode,axiom,
% 0.30/0.64 ( one_one_complex
% 0.30/0.64 = ( complex2 @ one_one_real @ zero_zero_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % one_complex.code
% 0.30/0.64 thf(fact_8615_Complex__eq__1,axiom,
% 0.30/0.64 ! [A: real,B: real] :
% 0.30/0.64 ( ( ( complex2 @ A @ B )
% 0.30/0.64 = one_one_complex )
% 0.30/0.64 = ( ( A = one_one_real )
% 0.30/0.64 & ( B = zero_zero_real ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Complex_eq_1
% 0.30/0.64 thf(fact_8616_norm__less__p1,axiom,
% 0.30/0.64 ! [X2: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X2 ) ) @ one_one_real ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % norm_less_p1
% 0.30/0.64 thf(fact_8617_norm__less__p1,axiom,
% 0.30/0.64 ! [X2: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X2 ) ) @ one_one_complex ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % norm_less_p1
% 0.30/0.64 thf(fact_8618_sqrt2__less__2,axiom,
% 0.30/0.64 ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.30/0.64
% 0.30/0.64 % sqrt2_less_2
% 0.30/0.64 thf(fact_8619_exp__ge__add__one__self__aux,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( exp_real @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_ge_add_one_self_aux
% 0.30/0.64 thf(fact_8620_lemma__exp__total,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ one_one_real @ Y3 )
% 0.30/0.64 => ? [X5: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 0.30/0.64 & ( ord_less_eq_real @ X5 @ ( minus_minus_real @ Y3 @ one_one_real ) )
% 0.30/0.64 & ( ( exp_real @ X5 )
% 0.30/0.64 = Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % lemma_exp_total
% 0.30/0.64 thf(fact_8621_ln__ge__iff,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ ( ln_ln_real @ X2 ) )
% 0.30/0.64 = ( ord_less_eq_real @ ( exp_real @ Y3 ) @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ln_ge_iff
% 0.30/0.64 thf(fact_8622_ln__x__over__x__mono,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.30/0.64 => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y3 ) @ Y3 ) @ ( divide_divide_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ln_x_over_x_mono
% 0.30/0.64 thf(fact_8623_powr__def,axiom,
% 0.30/0.64 ( powr_real
% 0.30/0.64 = ( ^ [X: real,A3: real] : ( if_real @ ( X = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A3 @ ( ln_ln_real @ X ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_def
% 0.30/0.64 thf(fact_8624_Complex__eq__neg__1,axiom,
% 0.30/0.64 ! [A: real,B: real] :
% 0.30/0.64 ( ( ( complex2 @ A @ B )
% 0.30/0.64 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.64 = ( ( A
% 0.30/0.64 = ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.64 & ( B = zero_zero_real ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Complex_eq_neg_1
% 0.30/0.64 thf(fact_8625_real__less__rsqrt,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 0.30/0.64 => ( ord_less_real @ X2 @ ( sqrt @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_less_rsqrt
% 0.30/0.64 thf(fact_8626_sqrt__le__D,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ Y3 )
% 0.30/0.64 => ( ord_less_eq_real @ X2 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sqrt_le_D
% 0.30/0.64 thf(fact_8627_real__le__rsqrt,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 0.30/0.64 => ( ord_less_eq_real @ X2 @ ( sqrt @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_le_rsqrt
% 0.30/0.64 thf(fact_8628_exp__le,axiom,
% 0.30/0.64 ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_le
% 0.30/0.64 thf(fact_8629_norm__of__real__diff,axiom,
% 0.30/0.64 ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( real_V1803761363581548252l_real @ B ) @ ( real_V1803761363581548252l_real @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % norm_of_real_diff
% 0.30/0.64 thf(fact_8630_norm__of__real__diff,axiom,
% 0.30/0.64 ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( real_V4546457046886955230omplex @ B ) @ ( real_V4546457046886955230omplex @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % norm_of_real_diff
% 0.30/0.64 thf(fact_8631_exp__divide__power__eq,axiom,
% 0.30/0.64 ! [N: nat,X2: complex] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X2 @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
% 0.30/0.64 = ( exp_complex @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_divide_power_eq
% 0.30/0.64 thf(fact_8632_exp__divide__power__eq,axiom,
% 0.30/0.64 ! [N: nat,X2: real] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
% 0.30/0.64 = ( exp_real @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_divide_power_eq
% 0.30/0.64 thf(fact_8633_tanh__altdef,axiom,
% 0.30/0.64 ( tanh_real
% 0.30/0.64 = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tanh_altdef
% 0.30/0.64 thf(fact_8634_tanh__altdef,axiom,
% 0.30/0.64 ( tanh_complex
% 0.30/0.64 = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tanh_altdef
% 0.30/0.64 thf(fact_8635_real__le__lsqrt,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ord_less_eq_real @ ( sqrt @ X2 ) @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_le_lsqrt
% 0.30/0.64 thf(fact_8636_real__sqrt__unique,axiom,
% 0.30/0.64 ! [Y3: real,X2: real] :
% 0.30/0.64 ( ( ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64 = X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ( ( sqrt @ X2 )
% 0.30/0.64 = Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_unique
% 0.30/0.64 thf(fact_8637_lemma__real__divide__sqrt__less,axiom,
% 0.30/0.64 ! [U: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ U )
% 0.30/0.64 => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 0.30/0.64
% 0.30/0.64 % lemma_real_divide_sqrt_less
% 0.30/0.64 thf(fact_8638_real__sqrt__sum__squares__eq__cancel,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64 = X2 )
% 0.30/0.64 => ( Y3 = zero_zero_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_sum_squares_eq_cancel
% 0.30/0.64 thf(fact_8639_real__sqrt__sum__squares__eq__cancel2,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64 = Y3 )
% 0.30/0.64 => ( X2 = zero_zero_real ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_sum_squares_eq_cancel2
% 0.30/0.64 thf(fact_8640_real__sqrt__sum__squares__ge1,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] : ( ord_less_eq_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_sum_squares_ge1
% 0.30/0.64 thf(fact_8641_real__sqrt__sum__squares__ge2,axiom,
% 0.30/0.64 ! [Y3: real,X2: real] : ( ord_less_eq_real @ Y3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_sum_squares_ge2
% 0.30/0.64 thf(fact_8642_real__sqrt__sum__squares__triangle__ineq,axiom,
% 0.30/0.64 ! [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 ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_sum_squares_triangle_ineq
% 0.30/0.64 thf(fact_8643_exp__half__le2,axiom,
% 0.30/0.64 ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_half_le2
% 0.30/0.64 thf(fact_8644_sqrt__ge__absD,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( sqrt @ Y3 ) )
% 0.30/0.64 => ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % sqrt_ge_absD
% 0.30/0.64 thf(fact_8645_cos__45,axiom,
% 0.30/0.64 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_45
% 0.30/0.64 thf(fact_8646_sin__45,axiom,
% 0.30/0.64 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_45
% 0.30/0.64 thf(fact_8647_tan__60,axiom,
% 0.30/0.64 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.30/0.64 = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_60
% 0.30/0.64 thf(fact_8648_cos__int__times__real,axiom,
% 0.30/0.64 ! [M: int,X2: real] :
% 0.30/0.64 ( ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X2 ) ) )
% 0.30/0.64 = ( real_V1803761363581548252l_real @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_int_times_real
% 0.30/0.64 thf(fact_8649_cos__int__times__real,axiom,
% 0.30/0.64 ! [M: int,X2: real] :
% 0.30/0.64 ( ( cos_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X2 ) ) )
% 0.30/0.64 = ( real_V4546457046886955230omplex @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_int_times_real
% 0.30/0.64 thf(fact_8650_sin__int__times__real,axiom,
% 0.30/0.64 ! [M: int,X2: real] :
% 0.30/0.64 ( ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X2 ) ) )
% 0.30/0.64 = ( real_V1803761363581548252l_real @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_int_times_real
% 0.30/0.64 thf(fact_8651_sin__int__times__real,axiom,
% 0.30/0.64 ! [M: int,X2: real] :
% 0.30/0.64 ( ( sin_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X2 ) ) )
% 0.30/0.64 = ( real_V4546457046886955230omplex @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_int_times_real
% 0.30/0.64 thf(fact_8652_exp__double,axiom,
% 0.30/0.64 ! [Z: complex] :
% 0.30/0.64 ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) )
% 0.30/0.64 = ( power_power_complex @ ( exp_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_double
% 0.30/0.64 thf(fact_8653_exp__double,axiom,
% 0.30/0.64 ! [Z: real] :
% 0.30/0.64 ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) )
% 0.30/0.64 = ( power_power_real @ ( exp_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_double
% 0.30/0.64 thf(fact_8654_real__less__lsqrt,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ord_less_real @ ( sqrt @ X2 ) @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_less_lsqrt
% 0.30/0.64 thf(fact_8655_sqrt__sum__squares__le__sum,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sqrt_sum_squares_le_sum
% 0.30/0.64 thf(fact_8656_sqrt__even__pow2,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.64 => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.64 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sqrt_even_pow2
% 0.30/0.64 thf(fact_8657_real__sqrt__ge__abs1,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_ge_abs1
% 0.30/0.64 thf(fact_8658_real__sqrt__ge__abs2,axiom,
% 0.30/0.64 ! [Y3: real,X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_ge_abs2
% 0.30/0.64 thf(fact_8659_sqrt__sum__squares__le__sum__abs,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sqrt_sum_squares_le_sum_abs
% 0.30/0.64 thf(fact_8660_ln__sqrt,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ln_ln_real @ ( sqrt @ X2 ) )
% 0.30/0.64 = ( divide_divide_real @ ( ln_ln_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % ln_sqrt
% 0.30/0.64 thf(fact_8661_cos__30,axiom,
% 0.30/0.64 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.30/0.64 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_30
% 0.30/0.64 thf(fact_8662_sin__60,axiom,
% 0.30/0.64 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.30/0.64 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_60
% 0.30/0.64 thf(fact_8663_exp__bound__half,axiom,
% 0.30/0.64 ! [Z: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_bound_half
% 0.30/0.64 thf(fact_8664_exp__bound__half,axiom,
% 0.30/0.64 ! [Z: complex] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_bound_half
% 0.30/0.64 thf(fact_8665_sin__cos__eq,axiom,
% 0.30/0.64 ( sin_real
% 0.30/0.64 = ( ^ [X: real] : ( cos_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_cos_eq
% 0.30/0.64 thf(fact_8666_sin__cos__eq,axiom,
% 0.30/0.64 ( sin_complex
% 0.30/0.64 = ( ^ [X: complex] : ( cos_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_cos_eq
% 0.30/0.64 thf(fact_8667_cos__sin__eq,axiom,
% 0.30/0.64 ( cos_real
% 0.30/0.64 = ( ^ [X: real] : ( sin_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_sin_eq
% 0.30/0.64 thf(fact_8668_cos__sin__eq,axiom,
% 0.30/0.64 ( cos_complex
% 0.30/0.64 = ( ^ [X: complex] : ( sin_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_sin_eq
% 0.30/0.64 thf(fact_8669_arsinh__real__aux,axiom,
% 0.30/0.64 ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arsinh_real_aux
% 0.30/0.64 thf(fact_8670_real__sqrt__power__even,axiom,
% 0.30/0.64 ! [N: nat,X2: real] :
% 0.30/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( power_power_real @ ( sqrt @ X2 ) @ N )
% 0.30/0.64 = ( power_power_real @ X2 @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_power_even
% 0.30/0.64 thf(fact_8671_exp__bound,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.30/0.64 => ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_bound
% 0.30/0.64 thf(fact_8672_real__sqrt__sum__squares__mult__ge__zero,axiom,
% 0.30/0.64 ! [X2: real,Y3: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_sum_squares_mult_ge_zero
% 0.30/0.64 thf(fact_8673_arith__geo__mean__sqrt,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X2 @ Y3 ) ) @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arith_geo_mean_sqrt
% 0.30/0.64 thf(fact_8674_powr__half__sqrt,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( powr_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 = ( sqrt @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % powr_half_sqrt
% 0.30/0.64 thf(fact_8675_tan__30,axiom,
% 0.30/0.64 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.30/0.64 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tan_30
% 0.30/0.64 thf(fact_8676_minus__sin__cos__eq,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( uminus_uminus_real @ ( sin_real @ X2 ) )
% 0.30/0.64 = ( cos_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % minus_sin_cos_eq
% 0.30/0.64 thf(fact_8677_minus__sin__cos__eq,axiom,
% 0.30/0.64 ! [X2: complex] :
% 0.30/0.64 ( ( uminus1482373934393186551omplex @ ( sin_complex @ X2 ) )
% 0.30/0.64 = ( cos_complex @ ( plus_plus_complex @ X2 @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % minus_sin_cos_eq
% 0.30/0.64 thf(fact_8678_real__exp__bound__lemma,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_exp_bound_lemma
% 0.30/0.64 thf(fact_8679_cos__x__y__le__one,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % cos_x_y_le_one
% 0.30/0.64 thf(fact_8680_real__sqrt__sum__squares__less,axiom,
% 0.30/0.64 ! [X2: real,U: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64 => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % real_sqrt_sum_squares_less
% 0.30/0.64 thf(fact_8681_arcosh__real__def,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 0.30/0.64 => ( ( arcosh_real @ X2 )
% 0.30/0.64 = ( ln_ln_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arcosh_real_def
% 0.30/0.64 thf(fact_8682_exp__ge__one__plus__x__over__n__power__n,axiom,
% 0.30/0.64 ! [N: nat,X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X2 )
% 0.30/0.64 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_ge_one_plus_x_over_n_power_n
% 0.30/0.64 thf(fact_8683_exp__ge__one__minus__x__over__n__power__n,axiom,
% 0.30/0.64 ! [X2: real,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.64 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_ge_one_minus_x_over_n_power_n
% 0.30/0.64 thf(fact_8684_cos__arctan,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( cos_real @ ( arctan @ X2 ) )
% 0.30/0.64 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_arctan
% 0.30/0.64 thf(fact_8685_sin__arctan,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( sin_real @ ( arctan @ X2 ) )
% 0.30/0.64 = ( divide_divide_real @ X2 @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_arctan
% 0.30/0.64 thf(fact_8686_exp__bound__lemma,axiom,
% 0.30/0.64 ! [Z: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( 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 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_bound_lemma
% 0.30/0.64 thf(fact_8687_exp__bound__lemma,axiom,
% 0.30/0.64 ! [Z: complex] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( 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 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_bound_lemma
% 0.30/0.64 thf(fact_8688_Maclaurin__exp__le,axiom,
% 0.30/0.64 ! [X2: real,N: nat] :
% 0.30/0.64 ? [T5: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X2 ) )
% 0.30/0.64 & ( ( exp_real @ X2 )
% 0.30/0.64 = ( plus_plus_real
% 0.30/0.64 @ ( groups6591440286371151544t_real
% 0.30/0.64 @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X2 @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
% 0.30/0.64 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Maclaurin_exp_le
% 0.30/0.64 thf(fact_8689_sqrt__sum__squares__half__less,axiom,
% 0.30/0.64 ! [X2: real,U: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ X2 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sqrt_sum_squares_half_less
% 0.30/0.64 thf(fact_8690_exp__lower__Taylor__quadratic,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( divide_divide_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % exp_lower_Taylor_quadratic
% 0.30/0.64 thf(fact_8691_log__base__10__eq2,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.30/0.64 = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % log_base_10_eq2
% 0.30/0.64 thf(fact_8692_sin__cos__sqrt,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X2 ) )
% 0.30/0.64 => ( ( sin_real @ X2 )
% 0.30/0.64 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_cos_sqrt
% 0.30/0.64 thf(fact_8693_arctan__half,axiom,
% 0.30/0.64 ( arctan
% 0.30/0.64 = ( ^ [X: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arctan_half
% 0.30/0.64 thf(fact_8694_tanh__real__altdef,axiom,
% 0.30/0.64 ( tanh_real
% 0.30/0.64 = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % tanh_real_altdef
% 0.30/0.64 thf(fact_8695_log__base__10__eq1,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.30/0.64 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % log_base_10_eq1
% 0.30/0.64 thf(fact_8696_arsinh__def,axiom,
% 0.30/0.64 ( arsinh_real
% 0.30/0.64 = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arsinh_def
% 0.30/0.64 thf(fact_8697_arsinh__real__def,axiom,
% 0.30/0.64 ( arsinh_real
% 0.30/0.64 = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arsinh_real_def
% 0.30/0.64 thf(fact_8698_binomial__code,axiom,
% 0.30/0.64 ( binomial
% 0.30/0.64 = ( ^ [N3: nat,K2: nat] : ( if_nat @ ( ord_less_nat @ N3 @ K2 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( binomial @ N3 @ ( minus_minus_nat @ N3 @ K2 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N3 @ K2 ) @ one_one_nat ) @ N3 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K2 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_code
% 0.30/0.64 thf(fact_8699_cos__arcsin,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.30/0.64 => ( ( cos_real @ ( arcsin @ X2 ) )
% 0.30/0.64 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_arcsin
% 0.30/0.64 thf(fact_8700_sin__arccos__abs,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.30/0.64 => ( ( sin_real @ ( arccos @ Y3 ) )
% 0.30/0.64 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_arccos_abs
% 0.30/0.64 thf(fact_8701_binomial__Suc__n,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( binomial @ ( suc @ N ) @ N )
% 0.30/0.64 = ( suc @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_Suc_n
% 0.30/0.64 thf(fact_8702_binomial__n__n,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( binomial @ N @ N )
% 0.30/0.64 = one_one_nat ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_n_n
% 0.30/0.64 thf(fact_8703_binomial__1,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
% 0.30/0.64 = N ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_1
% 0.30/0.64 thf(fact_8704_binomial__0__Suc,axiom,
% 0.30/0.64 ! [K: nat] :
% 0.30/0.64 ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
% 0.30/0.64 = zero_zero_nat ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_0_Suc
% 0.30/0.64 thf(fact_8705_binomial__eq__0__iff,axiom,
% 0.30/0.64 ! [N: nat,K: nat] :
% 0.30/0.64 ( ( ( binomial @ N @ K )
% 0.30/0.64 = zero_zero_nat )
% 0.30/0.64 = ( ord_less_nat @ N @ K ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_eq_0_iff
% 0.30/0.64 thf(fact_8706_binomial__Suc__Suc,axiom,
% 0.30/0.64 ! [N: nat,K: nat] :
% 0.30/0.64 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 0.30/0.64 = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_Suc_Suc
% 0.30/0.64 thf(fact_8707_binomial__n__0,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( binomial @ N @ zero_zero_nat )
% 0.30/0.64 = one_one_nat ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_n_0
% 0.30/0.64 thf(fact_8708_arccos__1,axiom,
% 0.30/0.64 ( ( arccos @ one_one_real )
% 0.30/0.64 = zero_zero_real ) ).
% 0.30/0.64
% 0.30/0.64 % arccos_1
% 0.30/0.64 thf(fact_8709_zero__less__binomial__iff,axiom,
% 0.30/0.64 ! [N: nat,K: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
% 0.30/0.64 = ( ord_less_eq_nat @ K @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % zero_less_binomial_iff
% 0.30/0.64 thf(fact_8710_arccos__minus__1,axiom,
% 0.30/0.64 ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.64 = pi ) ).
% 0.30/0.64
% 0.30/0.64 % arccos_minus_1
% 0.30/0.64 thf(fact_8711_cos__arccos,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64 => ( ( cos_real @ ( arccos @ Y3 ) )
% 0.30/0.64 = Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_arccos
% 0.30/0.64 thf(fact_8712_sin__arcsin,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64 => ( ( sin_real @ ( arcsin @ Y3 ) )
% 0.30/0.64 = Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_arcsin
% 0.30/0.64 thf(fact_8713_arccos__0,axiom,
% 0.30/0.64 ( ( arccos @ zero_zero_real )
% 0.30/0.64 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arccos_0
% 0.30/0.64 thf(fact_8714_arcsin__1,axiom,
% 0.30/0.64 ( ( arcsin @ one_one_real )
% 0.30/0.64 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arcsin_1
% 0.30/0.64 thf(fact_8715_arcsin__minus__1,axiom,
% 0.30/0.64 ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.64 = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arcsin_minus_1
% 0.30/0.64 thf(fact_8716_complex__exp__exists,axiom,
% 0.30/0.64 ! [Z: complex] :
% 0.30/0.64 ? [A4: complex,R3: real] :
% 0.30/0.64 ( Z
% 0.30/0.64 = ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( exp_complex @ A4 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % complex_exp_exists
% 0.30/0.64 thf(fact_8717_choose__one,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( binomial @ N @ one_one_nat )
% 0.30/0.64 = N ) ).
% 0.30/0.64
% 0.30/0.64 % choose_one
% 0.30/0.64 thf(fact_8718_binomial__eq__0,axiom,
% 0.30/0.64 ! [N: nat,K: nat] :
% 0.30/0.64 ( ( ord_less_nat @ N @ K )
% 0.30/0.64 => ( ( binomial @ N @ K )
% 0.30/0.64 = zero_zero_nat ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_eq_0
% 0.30/0.64 thf(fact_8719_Suc__times__binomial,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
% 0.30/0.64 = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Suc_times_binomial
% 0.30/0.64 thf(fact_8720_Suc__times__binomial__eq,axiom,
% 0.30/0.64 ! [N: nat,K: nat] :
% 0.30/0.64 ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
% 0.30/0.64 = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Suc_times_binomial_eq
% 0.30/0.64 thf(fact_8721_binomial__symmetric,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64 => ( ( binomial @ N @ K )
% 0.30/0.64 = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_symmetric
% 0.30/0.64 thf(fact_8722_choose__mult__lemma,axiom,
% 0.30/0.64 ! [M: nat,R2: nat,K: nat] :
% 0.30/0.64 ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
% 0.30/0.64 = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % choose_mult_lemma
% 0.30/0.64 thf(fact_8723_binomial__le__pow,axiom,
% 0.30/0.64 ! [R2: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ R2 @ N )
% 0.30/0.64 => ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_le_pow
% 0.30/0.64 thf(fact_8724_zero__less__binomial,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64 => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % zero_less_binomial
% 0.30/0.64 thf(fact_8725_Suc__times__binomial__add,axiom,
% 0.30/0.64 ! [A: nat,B: nat] :
% 0.30/0.64 ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
% 0.30/0.64 = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Suc_times_binomial_add
% 0.30/0.64 thf(fact_8726_binomial__Suc__Suc__eq__times,axiom,
% 0.30/0.64 ! [N: nat,K: nat] :
% 0.30/0.64 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 0.30/0.64 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_Suc_Suc_eq_times
% 0.30/0.64 thf(fact_8727_choose__mult,axiom,
% 0.30/0.64 ! [K: nat,M: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.64 => ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.64 => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 0.30/0.64 = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % choose_mult
% 0.30/0.64 thf(fact_8728_binomial__absorb__comp,axiom,
% 0.30/0.64 ! [N: nat,K: nat] :
% 0.30/0.64 ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
% 0.30/0.64 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_absorb_comp
% 0.30/0.64 thf(fact_8729_arccos__le__arccos,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( arccos @ X2 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arccos_le_arccos
% 0.30/0.64 thf(fact_8730_arccos__eq__iff,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.30/0.64 & ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real ) )
% 0.30/0.64 => ( ( ( arccos @ X2 )
% 0.30/0.64 = ( arccos @ Y3 ) )
% 0.30/0.64 = ( X2 = Y3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arccos_eq_iff
% 0.30/0.64 thf(fact_8731_arccos__le__mono,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( arccos @ X2 ) @ ( arccos @ Y3 ) )
% 0.30/0.64 = ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arccos_le_mono
% 0.30/0.64 thf(fact_8732_arcsin__le__arcsin,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64 => ( ord_less_eq_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arcsin_le_arcsin
% 0.30/0.64 thf(fact_8733_arcsin__minus,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.30/0.64 => ( ( arcsin @ ( uminus_uminus_real @ X2 ) )
% 0.30/0.64 = ( uminus_uminus_real @ ( arcsin @ X2 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arcsin_minus
% 0.30/0.64 thf(fact_8734_arcsin__le__mono,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) )
% 0.30/0.64 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arcsin_le_mono
% 0.30/0.64 thf(fact_8735_arcsin__eq__iff,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.30/0.64 => ( ( ( arcsin @ X2 )
% 0.30/0.64 = ( arcsin @ Y3 ) )
% 0.30/0.64 = ( X2 = Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arcsin_eq_iff
% 0.30/0.64 thf(fact_8736_binomial__absorption,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
% 0.30/0.64 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_absorption
% 0.30/0.64 thf(fact_8737_binomial__fact__lemma,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64 => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 0.30/0.64 = ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_fact_lemma
% 0.30/0.64 thf(fact_8738_binomial__ge__n__over__k__pow__k,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64 => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_ge_n_over_k_pow_k
% 0.30/0.64 thf(fact_8739_binomial__ge__n__over__k__pow__k,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64 => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_ge_n_over_k_pow_k
% 0.30/0.64 thf(fact_8740_binomial__maximum_H,axiom,
% 0.30/0.64 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_maximum'
% 0.30/0.64 thf(fact_8741_binomial__mono,axiom,
% 0.30/0.64 ! [K: nat,K6: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ K6 )
% 0.30/0.64 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 0.30/0.64 => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_mono
% 0.30/0.64 thf(fact_8742_binomial__antimono,axiom,
% 0.30/0.64 ! [K: nat,K6: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ K6 )
% 0.30/0.64 => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 0.30/0.64 => ( ( ord_less_eq_nat @ K6 @ N )
% 0.30/0.64 => ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_antimono
% 0.30/0.64 thf(fact_8743_binomial__maximum,axiom,
% 0.30/0.64 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_maximum
% 0.30/0.64 thf(fact_8744_binomial__le__pow2,axiom,
% 0.30/0.64 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_le_pow2
% 0.30/0.64 thf(fact_8745_choose__reduce__nat,axiom,
% 0.30/0.64 ! [N: nat,K: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.64 => ( ( binomial @ N @ K )
% 0.30/0.64 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % choose_reduce_nat
% 0.30/0.64 thf(fact_8746_arccos__lbound,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arccos_lbound
% 0.30/0.64 thf(fact_8747_arccos__less__arccos,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64 => ( ord_less_real @ ( arccos @ Y3 ) @ ( arccos @ X2 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arccos_less_arccos
% 0.30/0.64 thf(fact_8748_times__binomial__minus1__eq,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.64 => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
% 0.30/0.64 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % times_binomial_minus1_eq
% 0.30/0.64 thf(fact_8749_arccos__less__mono,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.30/0.64 => ( ( ord_less_real @ ( arccos @ X2 ) @ ( arccos @ Y3 ) )
% 0.30/0.64 = ( ord_less_real @ Y3 @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arccos_less_mono
% 0.30/0.64 thf(fact_8750_arccos__ubound,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arccos_ubound
% 0.30/0.64 thf(fact_8751_arccos__cos,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ pi )
% 0.30/0.64 => ( ( arccos @ ( cos_real @ X2 ) )
% 0.30/0.64 = X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arccos_cos
% 0.30/0.64 thf(fact_8752_arcsin__less__arcsin,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64 => ( ord_less_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arcsin_less_arcsin
% 0.30/0.64 thf(fact_8753_arcsin__less__mono,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.30/0.64 => ( ( ord_less_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) )
% 0.30/0.64 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arcsin_less_mono
% 0.30/0.64 thf(fact_8754_cos__arccos__abs,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.30/0.64 => ( ( cos_real @ ( arccos @ Y3 ) )
% 0.30/0.64 = Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_arccos_abs
% 0.30/0.64 thf(fact_8755_arccos__cos__eq__abs,axiom,
% 0.30/0.64 ! [Theta: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
% 0.30/0.64 => ( ( arccos @ ( cos_real @ Theta ) )
% 0.30/0.64 = ( abs_abs_real @ Theta ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arccos_cos_eq_abs
% 0.30/0.64 thf(fact_8756_binomial__altdef__nat,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64 => ( ( binomial @ N @ K )
% 0.30/0.64 = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_altdef_nat
% 0.30/0.64 thf(fact_8757_binomial__less__binomial__Suc,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_less_binomial_Suc
% 0.30/0.64 thf(fact_8758_binomial__strict__antimono,axiom,
% 0.30/0.64 ! [K: nat,K6: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ K @ K6 )
% 0.30/0.64 => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 0.30/0.64 => ( ( ord_less_eq_nat @ K6 @ N )
% 0.30/0.64 => ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_strict_antimono
% 0.30/0.64 thf(fact_8759_binomial__strict__mono,axiom,
% 0.30/0.64 ! [K: nat,K6: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ K @ K6 )
% 0.30/0.64 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 0.30/0.64 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_strict_mono
% 0.30/0.64 thf(fact_8760_central__binomial__odd,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.64 => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64 = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % central_binomial_odd
% 0.30/0.64 thf(fact_8761_binomial__addition__formula,axiom,
% 0.30/0.64 ! [N: nat,K: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ( binomial @ N @ ( suc @ K ) )
% 0.30/0.64 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_addition_formula
% 0.30/0.64 thf(fact_8762_binomial__fact,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64 => ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 0.30/0.64 = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_fact
% 0.30/0.64 thf(fact_8763_binomial__fact,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64 => ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 0.30/0.64 = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_fact
% 0.30/0.64 thf(fact_8764_binomial__fact,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64 => ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 0.30/0.64 = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % binomial_fact
% 0.30/0.64 thf(fact_8765_fact__binomial,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64 => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
% 0.30/0.64 = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_binomial
% 0.30/0.64 thf(fact_8766_fact__binomial,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64 => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
% 0.30/0.64 = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_binomial
% 0.30/0.64 thf(fact_8767_fact__binomial,axiom,
% 0.30/0.64 ! [K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64 => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
% 0.30/0.64 = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % fact_binomial
% 0.30/0.64 thf(fact_8768_arccos__lt__bounded,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_real @ Y3 @ one_one_real )
% 0.30/0.64 => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 0.30/0.64 & ( ord_less_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arccos_lt_bounded
% 0.30/0.64 thf(fact_8769_arccos__bounded,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 0.30/0.64 & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arccos_bounded
% 0.30/0.64 thf(fact_8770_sin__arccos__nonzero,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.30/0.64 => ( ( sin_real @ ( arccos @ X2 ) )
% 0.30/0.64 != zero_zero_real ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_arccos_nonzero
% 0.30/0.64 thf(fact_8771_arccos__cos2,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X2 )
% 0.30/0.64 => ( ( arccos @ ( cos_real @ X2 ) )
% 0.30/0.64 = ( uminus_uminus_real @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arccos_cos2
% 0.30/0.64 thf(fact_8772_arccos__minus,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.30/0.64 => ( ( arccos @ ( uminus_uminus_real @ X2 ) )
% 0.30/0.64 = ( minus_minus_real @ pi @ ( arccos @ X2 ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arccos_minus
% 0.30/0.64 thf(fact_8773_cos__arcsin__nonzero,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.64 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.30/0.64 => ( ( cos_real @ ( arcsin @ X2 ) )
% 0.30/0.64 != zero_zero_real ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % cos_arcsin_nonzero
% 0.30/0.64 thf(fact_8774_choose__two,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64 = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % choose_two
% 0.30/0.64 thf(fact_8775_arccos,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 0.30/0.64 & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi )
% 0.30/0.64 & ( ( cos_real @ ( arccos @ Y3 ) )
% 0.30/0.64 = Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arccos
% 0.30/0.64 thf(fact_8776_arccos__minus__abs,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.30/0.64 => ( ( arccos @ ( uminus_uminus_real @ X2 ) )
% 0.30/0.64 = ( minus_minus_real @ pi @ ( arccos @ X2 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arccos_minus_abs
% 0.30/0.64 thf(fact_8777_arccos__le__pi2,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arccos_le_pi2
% 0.30/0.64 thf(fact_8778_arcsin__lt__bounded,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_real @ Y3 @ one_one_real )
% 0.30/0.64 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.30/0.64 & ( ord_less_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arcsin_lt_bounded
% 0.30/0.64 thf(fact_8779_arcsin__lbound,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arcsin_lbound
% 0.30/0.64 thf(fact_8780_arcsin__ubound,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64 => ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arcsin_ubound
% 0.30/0.64 thf(fact_8781_arcsin__bounded,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.30/0.64 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arcsin_bounded
% 0.30/0.64 thf(fact_8782_arcsin__sin,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( arcsin @ ( sin_real @ X2 ) )
% 0.30/0.64 = X2 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arcsin_sin
% 0.30/0.64 thf(fact_8783_arcsin,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.30/0.64 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 & ( ( sin_real @ ( arcsin @ Y3 ) )
% 0.30/0.64 = Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arcsin
% 0.30/0.64 thf(fact_8784_arcsin__pi,axiom,
% 0.30/0.64 ! [Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.30/0.64 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ pi )
% 0.30/0.64 & ( ( sin_real @ ( arcsin @ Y3 ) )
% 0.30/0.64 = Y3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arcsin_pi
% 0.30/0.64 thf(fact_8785_arcsin__le__iff,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( arcsin @ X2 ) @ Y3 )
% 0.30/0.64 = ( ord_less_eq_real @ X2 @ ( sin_real @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arcsin_le_iff
% 0.30/0.64 thf(fact_8786_le__arcsin__iff,axiom,
% 0.30/0.64 ! [X2: real,Y3: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.30/0.64 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64 => ( ( ord_less_eq_real @ Y3 @ ( arcsin @ X2 ) )
% 0.30/0.64 = ( ord_less_eq_real @ ( sin_real @ Y3 ) @ X2 ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % le_arcsin_iff
% 0.30/0.64 thf(fact_8787_arccos__cos__eq__abs__2pi,axiom,
% 0.30/0.64 ! [Theta: real] :
% 0.30/0.64 ~ ! [K3: int] :
% 0.30/0.64 ( ( arccos @ ( cos_real @ Theta ) )
% 0.30/0.64 != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % arccos_cos_eq_abs_2pi
% 0.30/0.64 thf(fact_8788_central__binomial__lower__bound,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % central_binomial_lower_bound
% 0.30/0.64 thf(fact_8789_sin__arccos,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.64 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.30/0.64 => ( ( sin_real @ ( arccos @ X2 ) )
% 0.30/0.64 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % sin_arccos
% 0.30/0.64 thf(fact_8790_choose__odd__sum,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 0.30/0.64 @ ( groups2073611262835488442omplex
% 0.30/0.64 @ ^ [I4: nat] :
% 0.30/0.64 ( if_complex
% 0.30/0.64 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 0.30/0.64 @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) )
% 0.30/0.64 @ zero_zero_complex )
% 0.30/0.64 @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.64 = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % choose_odd_sum
% 0.30/0.64 thf(fact_8791_choose__odd__sum,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 0.30/0.64 @ ( groups3539618377306564664at_int
% 0.30/0.64 @ ^ [I4: nat] :
% 0.30/0.64 ( if_int
% 0.30/0.64 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 0.30/0.64 @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) )
% 0.30/0.64 @ zero_zero_int )
% 0.30/0.64 @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.64 = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % choose_odd_sum
% 0.30/0.64 thf(fact_8792_choose__odd__sum,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 0.30/0.64 @ ( groups2906978787729119204at_rat
% 0.30/0.64 @ ^ [I4: nat] :
% 0.30/0.64 ( if_rat
% 0.30/0.64 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 0.30/0.64 @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) )
% 0.30/0.64 @ zero_zero_rat )
% 0.30/0.64 @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.64 = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % choose_odd_sum
% 0.30/0.64 thf(fact_8793_choose__odd__sum,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.30/0.64 @ ( groups6591440286371151544t_real
% 0.30/0.64 @ ^ [I4: nat] :
% 0.30/0.64 ( if_real
% 0.30/0.64 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 0.30/0.64 @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) )
% 0.30/0.64 @ zero_zero_real )
% 0.30/0.64 @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.64 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % choose_odd_sum
% 0.30/0.64 thf(fact_8794_choose__even__sum,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 0.30/0.64 @ ( groups2073611262835488442omplex
% 0.30/0.64 @ ^ [I4: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) @ zero_zero_complex )
% 0.30/0.64 @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.64 = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % choose_even_sum
% 0.30/0.64 thf(fact_8795_choose__even__sum,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 0.30/0.64 @ ( groups3539618377306564664at_int
% 0.30/0.64 @ ^ [I4: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) @ zero_zero_int )
% 0.30/0.64 @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.64 = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % choose_even_sum
% 0.30/0.64 thf(fact_8796_choose__even__sum,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 0.30/0.64 @ ( groups2906978787729119204at_rat
% 0.30/0.64 @ ^ [I4: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) @ zero_zero_rat )
% 0.30/0.64 @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.64 = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % choose_even_sum
% 0.30/0.64 thf(fact_8797_choose__even__sum,axiom,
% 0.30/0.64 ! [N: nat] :
% 0.30/0.64 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64 => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.30/0.64 @ ( groups6591440286371151544t_real
% 0.30/0.64 @ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) @ zero_zero_real )
% 0.30/0.64 @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.64 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % choose_even_sum
% 0.30/0.64 thf(fact_8798_pochhammer__double,axiom,
% 0.30/0.64 ! [Z: complex,N: nat] :
% 0.30/0.64 ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.64 = ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % pochhammer_double
% 0.30/0.64 thf(fact_8799_pochhammer__double,axiom,
% 0.30/0.64 ! [Z: real,N: nat] :
% 0.30/0.64 ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.64 = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % pochhammer_double
% 0.30/0.64 thf(fact_8800_pochhammer__double,axiom,
% 0.30/0.64 ! [Z: rat,N: nat] :
% 0.30/0.64 ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.64 = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % pochhammer_double
% 0.30/0.64 thf(fact_8801_floor__log__nat__eq__powr__iff,axiom,
% 0.30/0.64 ! [B: nat,K: nat,N: nat] :
% 0.30/0.64 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.30/0.64 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.64 => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.30/0.64 = ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.64 = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.30/0.64 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % floor_log_nat_eq_powr_iff
% 0.30/0.64 thf(fact_8802_of__nat__code,axiom,
% 0.30/0.64 ( semiri8010041392384452111omplex
% 0.30/0.64 = ( ^ [N3: nat] :
% 0.30/0.64 ( semiri2816024913162550771omplex
% 0.30/0.64 @ ^ [I4: complex] : ( plus_plus_complex @ I4 @ one_one_complex )
% 0.30/0.64 @ N3
% 0.30/0.64 @ zero_zero_complex ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_nat_code
% 0.30/0.64 thf(fact_8803_of__nat__code,axiom,
% 0.30/0.64 ( semiri1314217659103216013at_int
% 0.30/0.64 = ( ^ [N3: nat] :
% 0.30/0.64 ( semiri8420488043553186161ux_int
% 0.30/0.64 @ ^ [I4: int] : ( plus_plus_int @ I4 @ one_one_int )
% 0.30/0.64 @ N3
% 0.30/0.64 @ zero_zero_int ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_nat_code
% 0.30/0.64 thf(fact_8804_of__nat__code,axiom,
% 0.30/0.64 ( semiri5074537144036343181t_real
% 0.30/0.64 = ( ^ [N3: nat] :
% 0.30/0.64 ( semiri7260567687927622513x_real
% 0.30/0.64 @ ^ [I4: real] : ( plus_plus_real @ I4 @ one_one_real )
% 0.30/0.64 @ N3
% 0.30/0.64 @ zero_zero_real ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_nat_code
% 0.30/0.64 thf(fact_8805_of__nat__code,axiom,
% 0.30/0.64 ( semiri1316708129612266289at_nat
% 0.30/0.64 = ( ^ [N3: nat] :
% 0.30/0.64 ( semiri8422978514062236437ux_nat
% 0.30/0.64 @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ one_one_nat )
% 0.30/0.64 @ N3
% 0.30/0.64 @ zero_zero_nat ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_nat_code
% 0.30/0.64 thf(fact_8806_of__nat__code,axiom,
% 0.30/0.64 ( semiri681578069525770553at_rat
% 0.30/0.64 = ( ^ [N3: nat] :
% 0.30/0.64 ( semiri7787848453975740701ux_rat
% 0.30/0.64 @ ^ [I4: rat] : ( plus_plus_rat @ I4 @ one_one_rat )
% 0.30/0.64 @ N3
% 0.30/0.64 @ zero_zero_rat ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_nat_code
% 0.30/0.64 thf(fact_8807_atMost__iff,axiom,
% 0.30/0.64 ! [I2: real,K: real] :
% 0.30/0.64 ( ( member_real @ I2 @ ( set_ord_atMost_real @ K ) )
% 0.30/0.64 = ( ord_less_eq_real @ I2 @ K ) ) ).
% 0.30/0.64
% 0.30/0.64 % atMost_iff
% 0.30/0.64 thf(fact_8808_atMost__iff,axiom,
% 0.30/0.64 ! [I2: set_int,K: set_int] :
% 0.30/0.64 ( ( member_set_int @ I2 @ ( set_or58775011639299419et_int @ K ) )
% 0.30/0.64 = ( ord_less_eq_set_int @ I2 @ K ) ) ).
% 0.30/0.64
% 0.30/0.64 % atMost_iff
% 0.30/0.64 thf(fact_8809_atMost__iff,axiom,
% 0.30/0.64 ! [I2: rat,K: rat] :
% 0.30/0.64 ( ( member_rat @ I2 @ ( set_ord_atMost_rat @ K ) )
% 0.30/0.64 = ( ord_less_eq_rat @ I2 @ K ) ) ).
% 0.30/0.64
% 0.30/0.64 % atMost_iff
% 0.30/0.64 thf(fact_8810_atMost__iff,axiom,
% 0.30/0.64 ! [I2: num,K: num] :
% 0.30/0.64 ( ( member_num @ I2 @ ( set_ord_atMost_num @ K ) )
% 0.30/0.64 = ( ord_less_eq_num @ I2 @ K ) ) ).
% 0.30/0.64
% 0.30/0.64 % atMost_iff
% 0.30/0.64 thf(fact_8811_atMost__iff,axiom,
% 0.30/0.64 ! [I2: nat,K: nat] :
% 0.30/0.64 ( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K ) )
% 0.30/0.64 = ( ord_less_eq_nat @ I2 @ K ) ) ).
% 0.30/0.64
% 0.30/0.64 % atMost_iff
% 0.30/0.64 thf(fact_8812_atMost__iff,axiom,
% 0.30/0.64 ! [I2: int,K: int] :
% 0.30/0.64 ( ( member_int @ I2 @ ( set_ord_atMost_int @ K ) )
% 0.30/0.64 = ( ord_less_eq_int @ I2 @ K ) ) ).
% 0.30/0.64
% 0.30/0.64 % atMost_iff
% 0.30/0.64 thf(fact_8813_of__int__floor__cancel,axiom,
% 0.30/0.64 ! [X2: real] :
% 0.30/0.64 ( ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X2 ) )
% 0.30/0.64 = X2 )
% 0.30/0.64 = ( ? [N3: int] :
% 0.30/0.64 ( X2
% 0.30/0.64 = ( ring_1_of_int_real @ N3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_int_floor_cancel
% 0.30/0.64 thf(fact_8814_of__int__floor__cancel,axiom,
% 0.30/0.64 ! [X2: rat] :
% 0.30/0.64 ( ( ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X2 ) )
% 0.30/0.64 = X2 )
% 0.30/0.64 = ( ? [N3: int] :
% 0.30/0.64 ( X2
% 0.30/0.64 = ( ring_1_of_int_rat @ N3 ) ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % of_int_floor_cancel
% 0.30/0.64 thf(fact_8815_atMost__subset__iff,axiom,
% 0.30/0.64 ! [X2: set_int,Y3: set_int] :
% 0.30/0.64 ( ( ord_le4403425263959731960et_int @ ( set_or58775011639299419et_int @ X2 ) @ ( set_or58775011639299419et_int @ Y3 ) )
% 0.30/0.64 = ( ord_less_eq_set_int @ X2 @ Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % atMost_subset_iff
% 0.30/0.64 thf(fact_8816_atMost__subset__iff,axiom,
% 0.30/0.64 ! [X2: rat,Y3: rat] :
% 0.30/0.64 ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X2 ) @ ( set_ord_atMost_rat @ Y3 ) )
% 0.30/0.64 = ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % atMost_subset_iff
% 0.30/0.64 thf(fact_8817_atMost__subset__iff,axiom,
% 0.30/0.64 ! [X2: num,Y3: num] :
% 0.30/0.64 ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X2 ) @ ( set_ord_atMost_num @ Y3 ) )
% 0.30/0.64 = ( ord_less_eq_num @ X2 @ Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % atMost_subset_iff
% 0.30/0.64 thf(fact_8818_atMost__subset__iff,axiom,
% 0.30/0.64 ! [X2: nat,Y3: nat] :
% 0.30/0.64 ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X2 ) @ ( set_ord_atMost_nat @ Y3 ) )
% 0.30/0.64 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % atMost_subset_iff
% 0.30/0.64 thf(fact_8819_atMost__subset__iff,axiom,
% 0.30/0.64 ! [X2: int,Y3: int] :
% 0.30/0.64 ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X2 ) @ ( set_ord_atMost_int @ Y3 ) )
% 0.30/0.64 = ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% 0.30/0.64
% 0.30/0.64 % atMost_subset_iff
% 0.30/0.64 thf(fact_8820_floor__numeral,axiom,
% 0.30/0.64 ! [V: num] :
% 0.30/0.64 ( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V ) )
% 0.30/0.64 = ( numeral_numeral_int @ V ) ) ).
% 0.30/0.64
% 0.30/0.64 % floor_numeral
% 0.30/0.64 thf(fact_8821_floor__numeral,axiom,
% 0.30/0.64 ! [V: num] :
% 0.30/0.64 ( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
% 0.30/0.64 = ( numeral_numeral_int @ V ) ) ).
% 0.30/0.64
% 0.30/0.64 % floor_numeral
% 0.30/0.64 thf(fact_8822_floor__one,axiom,
% 0.30/0.64 ( ( archim6058952711729229775r_real @ one_one_real )
% 0.30/0.64 = one_one_int ) ).
% 0.30/0.64
% 0.30/0.64 % floor_one
% 0.30/0.64 thf(fact_8823_floor__one,axiom,
% 0.30/0.64 ( ( archim3151403230148437115or_rat @ one_one_rat )
% 0.30/0.64 = one_one_int ) ).
% 0.30/0.64
% 0.30/0.64 % floor_one
% 0.30/0.64 thf(fact_8824_pochhammer__0,axiom,
% 0.30/0.64 ! [A: complex] :
% 0.30/0.64 ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
% 0.30/0.64 = one_one_complex ) ).
% 0.30/0.64
% 0.30/0.64 % pochhammer_0
% 0.30/0.64 thf(fact_8825_pochhammer__0,axiom,
% 0.30/0.64 ! [A: real] :
% 0.30/0.64 ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
% 0.30/0.64 = one_one_real ) ).
% 0.30/0.64
% 0.30/0.64 % pochhammer_0
% 0.30/0.64 thf(fact_8826_pochhammer__0,axiom,
% 0.30/0.64 ! [A: rat] :
% 0.30/0.64 ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
% 0.30/0.64 = one_one_rat ) ).
% 0.30/0.64
% 0.30/0.64 % pochhammer_0
% 0.30/0.64 thf(fact_8827_pochhammer__0,axiom,
% 0.30/0.64 ! [A: nat] :
% 0.30/0.64 ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
% 0.30/0.64 = one_one_nat ) ).
% 0.30/0.64
% 0.30/0.64 % pochhammer_0
% 0.30/0.64 thf(fact_8828_pochhammer__0,axiom,
% 0.30/0.64 ! [A: int] :
% 0.30/0.64 ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
% 0.30/0.64 = one_one_int ) ).
% 0.30/0.64
% 0.30/0.64 % pochhammer_0
% 0.30/0.64 thf(fact_8829_Icc__subset__Iic__iff,axiom,
% 0.30/0.64 ! [L: set_int,H2: set_int,H3: set_int] :
% 0.30/0.64 ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ L @ H2 ) @ ( set_or58775011639299419et_int @ H3 ) )
% 0.30/0.64 = ( ~ ( ord_less_eq_set_int @ L @ H2 )
% 0.30/0.64 | ( ord_less_eq_set_int @ H2 @ H3 ) ) ) ).
% 0.30/0.64
% 0.30/0.64 % Icc_subset_Iic_iff
% 0.30/0.64 thf(fact_8830_Icc__subset__Iic__iff,axiom,
% 0.30/0.64 ! [L: rat,H2: rat,H3: rat] :
% 0.30/0.64 ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H2 ) @ ( set_ord_atMost_rat @ H3 ) )
% 0.30/0.64 = ( ~ ( ord_less_eq_rat @ L @ H2 )
% 0.30/0.65 | ( ord_less_eq_rat @ H2 @ H3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Icc_subset_Iic_iff
% 0.30/0.65 thf(fact_8831_Icc__subset__Iic__iff,axiom,
% 0.30/0.65 ! [L: num,H2: num,H3: num] :
% 0.30/0.65 ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L @ H2 ) @ ( set_ord_atMost_num @ H3 ) )
% 0.30/0.65 = ( ~ ( ord_less_eq_num @ L @ H2 )
% 0.30/0.65 | ( ord_less_eq_num @ H2 @ H3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Icc_subset_Iic_iff
% 0.30/0.65 thf(fact_8832_Icc__subset__Iic__iff,axiom,
% 0.30/0.65 ! [L: nat,H2: nat,H3: nat] :
% 0.30/0.65 ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H2 ) @ ( set_ord_atMost_nat @ H3 ) )
% 0.30/0.65 = ( ~ ( ord_less_eq_nat @ L @ H2 )
% 0.30/0.65 | ( ord_less_eq_nat @ H2 @ H3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Icc_subset_Iic_iff
% 0.30/0.65 thf(fact_8833_Icc__subset__Iic__iff,axiom,
% 0.30/0.65 ! [L: int,H2: int,H3: int] :
% 0.30/0.65 ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H2 ) @ ( set_ord_atMost_int @ H3 ) )
% 0.30/0.65 = ( ~ ( ord_less_eq_int @ L @ H2 )
% 0.30/0.65 | ( ord_less_eq_int @ H2 @ H3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Icc_subset_Iic_iff
% 0.30/0.65 thf(fact_8834_Icc__subset__Iic__iff,axiom,
% 0.30/0.65 ! [L: real,H2: real,H3: real] :
% 0.30/0.65 ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H2 ) @ ( set_ord_atMost_real @ H3 ) )
% 0.30/0.65 = ( ~ ( ord_less_eq_real @ L @ H2 )
% 0.30/0.65 | ( ord_less_eq_real @ H2 @ H3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Icc_subset_Iic_iff
% 0.30/0.65 thf(fact_8835_sum_OatMost__Suc,axiom,
% 0.30/0.65 ! [G: nat > rat,N: nat] :
% 0.30/0.65 ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % sum.atMost_Suc
% 0.30/0.65 thf(fact_8836_sum_OatMost__Suc,axiom,
% 0.30/0.65 ! [G: nat > int,N: nat] :
% 0.30/0.65 ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % sum.atMost_Suc
% 0.30/0.65 thf(fact_8837_sum_OatMost__Suc,axiom,
% 0.30/0.65 ! [G: nat > nat,N: nat] :
% 0.30/0.65 ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % sum.atMost_Suc
% 0.30/0.65 thf(fact_8838_sum_OatMost__Suc,axiom,
% 0.30/0.65 ! [G: nat > real,N: nat] :
% 0.30/0.65 ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % sum.atMost_Suc
% 0.30/0.65 thf(fact_8839_floor__diff__of__int,axiom,
% 0.30/0.65 ! [X2: real,Z: int] :
% 0.30/0.65 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X2 @ ( ring_1_of_int_real @ Z ) ) )
% 0.30/0.65 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X2 ) @ Z ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_diff_of_int
% 0.30/0.65 thf(fact_8840_floor__diff__of__int,axiom,
% 0.30/0.65 ! [X2: rat,Z: int] :
% 0.30/0.65 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) )
% 0.30/0.65 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X2 ) @ Z ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_diff_of_int
% 0.30/0.65 thf(fact_8841_zero__le__floor,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % zero_le_floor
% 0.30/0.65 thf(fact_8842_zero__le__floor,axiom,
% 0.30/0.65 ! [X2: rat] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_rat @ zero_zero_rat @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % zero_le_floor
% 0.30/0.65 thf(fact_8843_floor__less__zero,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ zero_zero_int )
% 0.30/0.65 = ( ord_less_real @ X2 @ zero_zero_real ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_less_zero
% 0.30/0.65 thf(fact_8844_floor__less__zero,axiom,
% 0.30/0.65 ! [X2: rat] :
% 0.30/0.65 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X2 ) @ zero_zero_int )
% 0.30/0.65 = ( ord_less_rat @ X2 @ zero_zero_rat ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_less_zero
% 0.30/0.65 thf(fact_8845_numeral__le__floor,axiom,
% 0.30/0.65 ! [V: num,X2: real] :
% 0.30/0.65 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % numeral_le_floor
% 0.30/0.65 thf(fact_8846_numeral__le__floor,axiom,
% 0.30/0.65 ! [V: num,X2: rat] :
% 0.30/0.65 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % numeral_le_floor
% 0.30/0.65 thf(fact_8847_zero__less__floor,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_real @ one_one_real @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % zero_less_floor
% 0.30/0.65 thf(fact_8848_zero__less__floor,axiom,
% 0.30/0.65 ! [X2: rat] :
% 0.30/0.65 ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_rat @ one_one_rat @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % zero_less_floor
% 0.30/0.65 thf(fact_8849_floor__le__zero,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ zero_zero_int )
% 0.30/0.65 = ( ord_less_real @ X2 @ one_one_real ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_le_zero
% 0.30/0.65 thf(fact_8850_floor__le__zero,axiom,
% 0.30/0.65 ! [X2: rat] :
% 0.30/0.65 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ zero_zero_int )
% 0.30/0.65 = ( ord_less_rat @ X2 @ one_one_rat ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_le_zero
% 0.30/0.65 thf(fact_8851_floor__less__numeral,axiom,
% 0.30/0.65 ! [X2: real,V: num] :
% 0.30/0.65 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.65 = ( ord_less_real @ X2 @ ( numeral_numeral_real @ V ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_less_numeral
% 0.30/0.65 thf(fact_8852_floor__less__numeral,axiom,
% 0.30/0.65 ! [X2: rat,V: num] :
% 0.30/0.65 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.65 = ( ord_less_rat @ X2 @ ( numeral_numeral_rat @ V ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_less_numeral
% 0.30/0.65 thf(fact_8853_one__le__floor,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_real @ one_one_real @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % one_le_floor
% 0.30/0.65 thf(fact_8854_one__le__floor,axiom,
% 0.30/0.65 ! [X2: rat] :
% 0.30/0.65 ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_rat @ one_one_rat @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % one_le_floor
% 0.30/0.65 thf(fact_8855_floor__less__one,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ one_one_int )
% 0.30/0.65 = ( ord_less_real @ X2 @ one_one_real ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_less_one
% 0.30/0.65 thf(fact_8856_floor__less__one,axiom,
% 0.30/0.65 ! [X2: rat] :
% 0.30/0.65 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X2 ) @ one_one_int )
% 0.30/0.65 = ( ord_less_rat @ X2 @ one_one_rat ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_less_one
% 0.30/0.65 thf(fact_8857_floor__neg__numeral,axiom,
% 0.30/0.65 ! [V: num] :
% 0.30/0.65 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_neg_numeral
% 0.30/0.65 thf(fact_8858_floor__neg__numeral,axiom,
% 0.30/0.65 ! [V: num] :
% 0.30/0.65 ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_neg_numeral
% 0.30/0.65 thf(fact_8859_floor__diff__numeral,axiom,
% 0.30/0.65 ! [X2: real,V: num] :
% 0.30/0.65 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X2 @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.65 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X2 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_diff_numeral
% 0.30/0.65 thf(fact_8860_floor__diff__numeral,axiom,
% 0.30/0.65 ! [X2: rat,V: num] :
% 0.30/0.65 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X2 @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.65 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_diff_numeral
% 0.30/0.65 thf(fact_8861_floor__diff__one,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X2 @ one_one_real ) )
% 0.30/0.65 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X2 ) @ one_one_int ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_diff_one
% 0.30/0.65 thf(fact_8862_floor__diff__one,axiom,
% 0.30/0.65 ! [X2: rat] :
% 0.30/0.65 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X2 @ one_one_rat ) )
% 0.30/0.65 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X2 ) @ one_one_int ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_diff_one
% 0.30/0.65 thf(fact_8863_floor__numeral__power,axiom,
% 0.30/0.65 ! [X2: num,N: nat] :
% 0.30/0.65 ( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
% 0.30/0.65 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_numeral_power
% 0.30/0.65 thf(fact_8864_floor__numeral__power,axiom,
% 0.30/0.65 ! [X2: num,N: nat] :
% 0.30/0.65 ( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) )
% 0.30/0.65 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_numeral_power
% 0.30/0.65 thf(fact_8865_floor__divide__eq__div__numeral,axiom,
% 0.30/0.65 ! [A: num,B: num] :
% 0.30/0.65 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 0.30/0.65 = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_divide_eq_div_numeral
% 0.30/0.65 thf(fact_8866_numeral__less__floor,axiom,
% 0.30/0.65 ! [V: num,X2: real] :
% 0.30/0.65 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % numeral_less_floor
% 0.30/0.65 thf(fact_8867_numeral__less__floor,axiom,
% 0.30/0.65 ! [V: num,X2: rat] :
% 0.30/0.65 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % numeral_less_floor
% 0.30/0.65 thf(fact_8868_floor__le__numeral,axiom,
% 0.30/0.65 ! [X2: real,V: num] :
% 0.30/0.65 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.65 = ( ord_less_real @ X2 @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_le_numeral
% 0.30/0.65 thf(fact_8869_floor__le__numeral,axiom,
% 0.30/0.65 ! [X2: rat,V: num] :
% 0.30/0.65 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.65 = ( ord_less_rat @ X2 @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_le_numeral
% 0.30/0.65 thf(fact_8870_one__less__floor,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % one_less_floor
% 0.30/0.65 thf(fact_8871_one__less__floor,axiom,
% 0.30/0.65 ! [X2: rat] :
% 0.30/0.65 ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % one_less_floor
% 0.30/0.65 thf(fact_8872_floor__le__one,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ one_one_int )
% 0.30/0.65 = ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_le_one
% 0.30/0.65 thf(fact_8873_floor__le__one,axiom,
% 0.30/0.65 ! [X2: rat] :
% 0.30/0.65 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ one_one_int )
% 0.30/0.65 = ( ord_less_rat @ X2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_le_one
% 0.30/0.65 thf(fact_8874_neg__numeral__le__floor,axiom,
% 0.30/0.65 ! [V: num,X2: real] :
% 0.30/0.65 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % neg_numeral_le_floor
% 0.30/0.65 thf(fact_8875_neg__numeral__le__floor,axiom,
% 0.30/0.65 ! [V: num,X2: rat] :
% 0.30/0.65 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % neg_numeral_le_floor
% 0.30/0.65 thf(fact_8876_floor__less__neg__numeral,axiom,
% 0.30/0.65 ! [X2: real,V: num] :
% 0.30/0.65 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.65 = ( ord_less_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_less_neg_numeral
% 0.30/0.65 thf(fact_8877_floor__less__neg__numeral,axiom,
% 0.30/0.65 ! [X2: rat,V: num] :
% 0.30/0.65 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.65 = ( ord_less_rat @ X2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_less_neg_numeral
% 0.30/0.65 thf(fact_8878_floor__one__divide__eq__div__numeral,axiom,
% 0.30/0.65 ! [B: num] :
% 0.30/0.65 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
% 0.30/0.65 = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_one_divide_eq_div_numeral
% 0.30/0.65 thf(fact_8879_floor__minus__divide__eq__div__numeral,axiom,
% 0.30/0.65 ! [A: num,B: num] :
% 0.30/0.65 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 0.30/0.65 = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_minus_divide_eq_div_numeral
% 0.30/0.65 thf(fact_8880_neg__numeral__less__floor,axiom,
% 0.30/0.65 ! [V: num,X2: real] :
% 0.30/0.65 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % neg_numeral_less_floor
% 0.30/0.65 thf(fact_8881_neg__numeral__less__floor,axiom,
% 0.30/0.65 ! [V: num,X2: rat] :
% 0.30/0.65 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % neg_numeral_less_floor
% 0.30/0.65 thf(fact_8882_floor__le__neg__numeral,axiom,
% 0.30/0.65 ! [X2: real,V: num] :
% 0.30/0.65 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.65 = ( ord_less_real @ X2 @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_le_neg_numeral
% 0.30/0.65 thf(fact_8883_floor__le__neg__numeral,axiom,
% 0.30/0.65 ! [X2: rat,V: num] :
% 0.30/0.65 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.65 = ( ord_less_rat @ X2 @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_le_neg_numeral
% 0.30/0.65 thf(fact_8884_floor__minus__one__divide__eq__div__numeral,axiom,
% 0.30/0.65 ! [B: num] :
% 0.30/0.65 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
% 0.30/0.65 = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_minus_one_divide_eq_div_numeral
% 0.30/0.65 thf(fact_8885_atMost__def,axiom,
% 0.30/0.65 ( set_ord_atMost_real
% 0.30/0.65 = ( ^ [U2: real] :
% 0.30/0.65 ( collect_real
% 0.30/0.65 @ ^ [X: real] : ( ord_less_eq_real @ X @ U2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % atMost_def
% 0.30/0.65 thf(fact_8886_atMost__def,axiom,
% 0.30/0.65 ( set_or58775011639299419et_int
% 0.30/0.65 = ( ^ [U2: set_int] :
% 0.30/0.65 ( collect_set_int
% 0.30/0.65 @ ^ [X: set_int] : ( ord_less_eq_set_int @ X @ U2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % atMost_def
% 0.30/0.65 thf(fact_8887_atMost__def,axiom,
% 0.30/0.65 ( set_ord_atMost_rat
% 0.30/0.65 = ( ^ [U2: rat] :
% 0.30/0.65 ( collect_rat
% 0.30/0.65 @ ^ [X: rat] : ( ord_less_eq_rat @ X @ U2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % atMost_def
% 0.30/0.65 thf(fact_8888_atMost__def,axiom,
% 0.30/0.65 ( set_ord_atMost_num
% 0.30/0.65 = ( ^ [U2: num] :
% 0.30/0.65 ( collect_num
% 0.30/0.65 @ ^ [X: num] : ( ord_less_eq_num @ X @ U2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % atMost_def
% 0.30/0.65 thf(fact_8889_atMost__def,axiom,
% 0.30/0.65 ( set_ord_atMost_nat
% 0.30/0.65 = ( ^ [U2: nat] :
% 0.30/0.65 ( collect_nat
% 0.30/0.65 @ ^ [X: nat] : ( ord_less_eq_nat @ X @ U2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % atMost_def
% 0.30/0.65 thf(fact_8890_atMost__def,axiom,
% 0.30/0.65 ( set_ord_atMost_int
% 0.30/0.65 = ( ^ [U2: int] :
% 0.30/0.65 ( collect_int
% 0.30/0.65 @ ^ [X: int] : ( ord_less_eq_int @ X @ U2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % atMost_def
% 0.30/0.65 thf(fact_8891_floor__mono,axiom,
% 0.30/0.65 ! [X2: real,Y3: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.30/0.65 => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim6058952711729229775r_real @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_mono
% 0.30/0.65 thf(fact_8892_floor__mono,axiom,
% 0.30/0.65 ! [X2: rat,Y3: rat] :
% 0.30/0.65 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.30/0.65 => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_mono
% 0.30/0.65 thf(fact_8893_of__int__floor__le,axiom,
% 0.30/0.65 ! [X2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X2 ) ) @ X2 ) ).
% 0.30/0.65
% 0.30/0.65 % of_int_floor_le
% 0.30/0.65 thf(fact_8894_of__int__floor__le,axiom,
% 0.30/0.65 ! [X2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X2 ) ) @ X2 ) ).
% 0.30/0.65
% 0.30/0.65 % of_int_floor_le
% 0.30/0.65 thf(fact_8895_floor__less__cancel,axiom,
% 0.30/0.65 ! [X2: real,Y3: real] :
% 0.30/0.65 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim6058952711729229775r_real @ Y3 ) )
% 0.30/0.65 => ( ord_less_real @ X2 @ Y3 ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_less_cancel
% 0.30/0.65 thf(fact_8896_floor__less__cancel,axiom,
% 0.30/0.65 ! [X2: rat,Y3: rat] :
% 0.30/0.65 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( archim3151403230148437115or_rat @ Y3 ) )
% 0.30/0.65 => ( ord_less_rat @ X2 @ Y3 ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_less_cancel
% 0.30/0.65 thf(fact_8897_floor__le__ceiling,axiom,
% 0.30/0.65 ! [X2: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim7802044766580827645g_real @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_le_ceiling
% 0.30/0.65 thf(fact_8898_floor__le__ceiling,axiom,
% 0.30/0.65 ! [X2: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( archim2889992004027027881ng_rat @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_le_ceiling
% 0.30/0.65 thf(fact_8899_lessThan__Suc__atMost,axiom,
% 0.30/0.65 ! [K: nat] :
% 0.30/0.65 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 0.30/0.65 = ( set_ord_atMost_nat @ K ) ) ).
% 0.30/0.65
% 0.30/0.65 % lessThan_Suc_atMost
% 0.30/0.65 thf(fact_8900_pochhammer__pos,axiom,
% 0.30/0.65 ! [X2: real,N: nat] :
% 0.30/0.65 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X2 @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_pos
% 0.30/0.65 thf(fact_8901_pochhammer__pos,axiom,
% 0.30/0.65 ! [X2: rat,N: nat] :
% 0.30/0.65 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 0.30/0.65 => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X2 @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_pos
% 0.30/0.65 thf(fact_8902_pochhammer__pos,axiom,
% 0.30/0.65 ! [X2: nat,N: nat] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 0.30/0.65 => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X2 @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_pos
% 0.30/0.65 thf(fact_8903_pochhammer__pos,axiom,
% 0.30/0.65 ! [X2: int,N: nat] :
% 0.30/0.65 ( ( ord_less_int @ zero_zero_int @ X2 )
% 0.30/0.65 => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X2 @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_pos
% 0.30/0.65 thf(fact_8904_pochhammer__neq__0__mono,axiom,
% 0.30/0.65 ! [A: complex,M: nat,N: nat] :
% 0.30/0.65 ( ( ( comm_s2602460028002588243omplex @ A @ M )
% 0.30/0.65 != zero_zero_complex )
% 0.30/0.65 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.65 => ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.30/0.65 != zero_zero_complex ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_neq_0_mono
% 0.30/0.65 thf(fact_8905_pochhammer__neq__0__mono,axiom,
% 0.30/0.65 ! [A: real,M: nat,N: nat] :
% 0.30/0.65 ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.30/0.65 != zero_zero_real )
% 0.30/0.65 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.65 => ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.30/0.65 != zero_zero_real ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_neq_0_mono
% 0.30/0.65 thf(fact_8906_pochhammer__neq__0__mono,axiom,
% 0.30/0.65 ! [A: rat,M: nat,N: nat] :
% 0.30/0.65 ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.30/0.65 != zero_zero_rat )
% 0.30/0.65 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.65 => ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.30/0.65 != zero_zero_rat ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_neq_0_mono
% 0.30/0.65 thf(fact_8907_pochhammer__eq__0__mono,axiom,
% 0.30/0.65 ! [A: complex,N: nat,M: nat] :
% 0.30/0.65 ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.30/0.65 = zero_zero_complex )
% 0.30/0.65 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.65 => ( ( comm_s2602460028002588243omplex @ A @ M )
% 0.30/0.65 = zero_zero_complex ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_eq_0_mono
% 0.30/0.65 thf(fact_8908_pochhammer__eq__0__mono,axiom,
% 0.30/0.65 ! [A: real,N: nat,M: nat] :
% 0.30/0.65 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.30/0.65 = zero_zero_real )
% 0.30/0.65 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.65 => ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.30/0.65 = zero_zero_real ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_eq_0_mono
% 0.30/0.65 thf(fact_8909_pochhammer__eq__0__mono,axiom,
% 0.30/0.65 ! [A: rat,N: nat,M: nat] :
% 0.30/0.65 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.30/0.65 = zero_zero_rat )
% 0.30/0.65 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.65 => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.30/0.65 = zero_zero_rat ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_eq_0_mono
% 0.30/0.65 thf(fact_8910_floor__le__round,axiom,
% 0.30/0.65 ! [X2: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim8280529875227126926d_real @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_le_round
% 0.30/0.65 thf(fact_8911_floor__le__round,axiom,
% 0.30/0.65 ! [X2: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( archim7778729529865785530nd_rat @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_le_round
% 0.30/0.65 thf(fact_8912_pochhammer__fact,axiom,
% 0.30/0.65 ( semiri5044797733671781792omplex
% 0.30/0.65 = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_fact
% 0.30/0.65 thf(fact_8913_pochhammer__fact,axiom,
% 0.30/0.65 ( semiri773545260158071498ct_rat
% 0.30/0.65 = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_fact
% 0.30/0.65 thf(fact_8914_pochhammer__fact,axiom,
% 0.30/0.65 ( semiri1406184849735516958ct_int
% 0.30/0.65 = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_fact
% 0.30/0.65 thf(fact_8915_pochhammer__fact,axiom,
% 0.30/0.65 ( semiri2265585572941072030t_real
% 0.30/0.65 = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_fact
% 0.30/0.65 thf(fact_8916_pochhammer__fact,axiom,
% 0.30/0.65 ( semiri1408675320244567234ct_nat
% 0.30/0.65 = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_fact
% 0.30/0.65 thf(fact_8917_not__Iic__le__Icc,axiom,
% 0.30/0.65 ! [H2: int,L3: int,H3: int] :
% 0.30/0.65 ~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H2 ) @ ( set_or1266510415728281911st_int @ L3 @ H3 ) ) ).
% 0.30/0.65
% 0.30/0.65 % not_Iic_le_Icc
% 0.30/0.65 thf(fact_8918_not__Iic__le__Icc,axiom,
% 0.30/0.65 ! [H2: real,L3: real,H3: real] :
% 0.30/0.65 ~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H2 ) @ ( set_or1222579329274155063t_real @ L3 @ H3 ) ) ).
% 0.30/0.65
% 0.30/0.65 % not_Iic_le_Icc
% 0.30/0.65 thf(fact_8919_finite__nat__iff__bounded__le,axiom,
% 0.30/0.65 ( finite_finite_nat
% 0.30/0.65 = ( ^ [S5: set_nat] :
% 0.30/0.65 ? [K2: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_atMost_nat @ K2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % finite_nat_iff_bounded_le
% 0.30/0.65 thf(fact_8920_le__floor__iff,axiom,
% 0.30/0.65 ! [Z: int,X2: real] :
% 0.30/0.65 ( ( ord_less_eq_int @ Z @ ( archim6058952711729229775r_real @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % le_floor_iff
% 0.30/0.65 thf(fact_8921_le__floor__iff,axiom,
% 0.30/0.65 ! [Z: int,X2: rat] :
% 0.30/0.65 ( ( ord_less_eq_int @ Z @ ( archim3151403230148437115or_rat @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % le_floor_iff
% 0.30/0.65 thf(fact_8922_floor__less__iff,axiom,
% 0.30/0.65 ! [X2: real,Z: int] :
% 0.30/0.65 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ Z )
% 0.30/0.65 = ( ord_less_real @ X2 @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_less_iff
% 0.30/0.65 thf(fact_8923_floor__less__iff,axiom,
% 0.30/0.65 ! [X2: rat,Z: int] :
% 0.30/0.65 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X2 ) @ Z )
% 0.30/0.65 = ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_less_iff
% 0.30/0.65 thf(fact_8924_le__floor__add,axiom,
% 0.30/0.65 ! [X2: real,Y3: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim6058952711729229775r_real @ Y3 ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X2 @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % le_floor_add
% 0.30/0.65 thf(fact_8925_le__floor__add,axiom,
% 0.30/0.65 ! [X2: rat,Y3: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X2 @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % le_floor_add
% 0.30/0.65 thf(fact_8926_floor__add__int,axiom,
% 0.30/0.65 ! [X2: real,Z: int] :
% 0.30/0.65 ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X2 ) @ Z )
% 0.30/0.65 = ( archim6058952711729229775r_real @ ( plus_plus_real @ X2 @ ( ring_1_of_int_real @ Z ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_add_int
% 0.30/0.65 thf(fact_8927_floor__add__int,axiom,
% 0.30/0.65 ! [X2: rat,Z: int] :
% 0.30/0.65 ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X2 ) @ Z )
% 0.30/0.65 = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_add_int
% 0.30/0.65 thf(fact_8928_int__add__floor,axiom,
% 0.30/0.65 ! [Z: int,X2: real] :
% 0.30/0.65 ( ( plus_plus_int @ Z @ ( archim6058952711729229775r_real @ X2 ) )
% 0.30/0.65 = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % int_add_floor
% 0.30/0.65 thf(fact_8929_int__add__floor,axiom,
% 0.30/0.65 ! [Z: int,X2: rat] :
% 0.30/0.65 ( ( plus_plus_int @ Z @ ( archim3151403230148437115or_rat @ X2 ) )
% 0.30/0.65 = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % int_add_floor
% 0.30/0.65 thf(fact_8930_floor__divide__of__int__eq,axiom,
% 0.30/0.65 ! [K: int,L: int] :
% 0.30/0.65 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( ring_1_of_int_real @ K ) @ ( ring_1_of_int_real @ L ) ) )
% 0.30/0.65 = ( divide_divide_int @ K @ L ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_divide_of_int_eq
% 0.30/0.65 thf(fact_8931_floor__divide__of__int__eq,axiom,
% 0.30/0.65 ! [K: int,L: int] :
% 0.30/0.65 ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ K ) @ ( ring_1_of_int_rat @ L ) ) )
% 0.30/0.65 = ( divide_divide_int @ K @ L ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_divide_of_int_eq
% 0.30/0.65 thf(fact_8932_floor__power,axiom,
% 0.30/0.65 ! [X2: real,N: nat] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X2 ) ) )
% 0.30/0.65 => ( ( archim6058952711729229775r_real @ ( power_power_real @ X2 @ N ) )
% 0.30/0.65 = ( power_power_int @ ( archim6058952711729229775r_real @ X2 ) @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_power
% 0.30/0.65 thf(fact_8933_floor__power,axiom,
% 0.30/0.65 ! [X2: rat,N: nat] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X2 ) ) )
% 0.30/0.65 => ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X2 @ N ) )
% 0.30/0.65 = ( power_power_int @ ( archim3151403230148437115or_rat @ X2 ) @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_power
% 0.30/0.65 thf(fact_8934_pochhammer__nonneg,axiom,
% 0.30/0.65 ! [X2: real,N: nat] :
% 0.30/0.65 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X2 @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_nonneg
% 0.30/0.65 thf(fact_8935_pochhammer__nonneg,axiom,
% 0.30/0.65 ! [X2: rat,N: nat] :
% 0.30/0.65 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 0.30/0.65 => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X2 @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_nonneg
% 0.30/0.65 thf(fact_8936_pochhammer__nonneg,axiom,
% 0.30/0.65 ! [X2: nat,N: nat] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 0.30/0.65 => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X2 @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_nonneg
% 0.30/0.65 thf(fact_8937_pochhammer__nonneg,axiom,
% 0.30/0.65 ! [X2: int,N: nat] :
% 0.30/0.65 ( ( ord_less_int @ zero_zero_int @ X2 )
% 0.30/0.65 => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X2 @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_nonneg
% 0.30/0.65 thf(fact_8938_pochhammer__binomial__sum,axiom,
% 0.30/0.65 ! [A: int,B: int,N: nat] :
% 0.30/0.65 ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B ) @ N )
% 0.30/0.65 = ( groups3539618377306564664at_int
% 0.30/0.65 @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K2 ) ) @ ( comm_s4660882817536571857er_int @ A @ K2 ) ) @ ( comm_s4660882817536571857er_int @ B @ ( minus_minus_nat @ N @ K2 ) ) )
% 0.30/0.65 @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_binomial_sum
% 0.30/0.65 thf(fact_8939_pochhammer__binomial__sum,axiom,
% 0.30/0.65 ! [A: rat,B: rat,N: nat] :
% 0.30/0.65 ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ B ) @ N )
% 0.30/0.65 = ( groups2906978787729119204at_rat
% 0.30/0.65 @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ A @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ B @ ( minus_minus_nat @ N @ K2 ) ) )
% 0.30/0.65 @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_binomial_sum
% 0.30/0.65 thf(fact_8940_pochhammer__binomial__sum,axiom,
% 0.30/0.65 ! [A: real,B: real,N: nat] :
% 0.30/0.65 ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ B ) @ N )
% 0.30/0.65 = ( groups6591440286371151544t_real
% 0.30/0.65 @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K2 ) ) @ ( comm_s7457072308508201937r_real @ A @ K2 ) ) @ ( comm_s7457072308508201937r_real @ B @ ( minus_minus_nat @ N @ K2 ) ) )
% 0.30/0.65 @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_binomial_sum
% 0.30/0.65 thf(fact_8941_pochhammer__0__left,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( ( N = zero_zero_nat )
% 0.30/0.65 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 0.30/0.65 = one_one_complex ) )
% 0.30/0.65 & ( ( N != zero_zero_nat )
% 0.30/0.65 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 0.30/0.65 = zero_zero_complex ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_0_left
% 0.30/0.65 thf(fact_8942_pochhammer__0__left,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( ( N = zero_zero_nat )
% 0.30/0.65 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 0.30/0.65 = one_one_real ) )
% 0.30/0.65 & ( ( N != zero_zero_nat )
% 0.30/0.65 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 0.30/0.65 = zero_zero_real ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_0_left
% 0.30/0.65 thf(fact_8943_pochhammer__0__left,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( ( N = zero_zero_nat )
% 0.30/0.65 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 0.30/0.65 = one_one_rat ) )
% 0.30/0.65 & ( ( N != zero_zero_nat )
% 0.30/0.65 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 0.30/0.65 = zero_zero_rat ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_0_left
% 0.30/0.65 thf(fact_8944_pochhammer__0__left,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( ( N = zero_zero_nat )
% 0.30/0.65 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 0.30/0.65 = one_one_nat ) )
% 0.30/0.65 & ( ( N != zero_zero_nat )
% 0.30/0.65 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 0.30/0.65 = zero_zero_nat ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_0_left
% 0.30/0.65 thf(fact_8945_pochhammer__0__left,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( ( N = zero_zero_nat )
% 0.30/0.65 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 0.30/0.65 = one_one_int ) )
% 0.30/0.65 & ( ( N != zero_zero_nat )
% 0.30/0.65 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 0.30/0.65 = zero_zero_int ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_0_left
% 0.30/0.65 thf(fact_8946_Iic__subset__Iio__iff,axiom,
% 0.30/0.65 ! [A: rat,B: rat] :
% 0.30/0.65 ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A ) @ ( set_ord_lessThan_rat @ B ) )
% 0.30/0.65 = ( ord_less_rat @ A @ B ) ) ).
% 0.30/0.65
% 0.30/0.65 % Iic_subset_Iio_iff
% 0.30/0.65 thf(fact_8947_Iic__subset__Iio__iff,axiom,
% 0.30/0.65 ! [A: num,B: num] :
% 0.30/0.65 ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ A ) @ ( set_ord_lessThan_num @ B ) )
% 0.30/0.65 = ( ord_less_num @ A @ B ) ) ).
% 0.30/0.65
% 0.30/0.65 % Iic_subset_Iio_iff
% 0.30/0.65 thf(fact_8948_Iic__subset__Iio__iff,axiom,
% 0.30/0.65 ! [A: nat,B: nat] :
% 0.30/0.65 ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
% 0.30/0.65 = ( ord_less_nat @ A @ B ) ) ).
% 0.30/0.65
% 0.30/0.65 % Iic_subset_Iio_iff
% 0.30/0.65 thf(fact_8949_Iic__subset__Iio__iff,axiom,
% 0.30/0.65 ! [A: int,B: int] :
% 0.30/0.65 ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B ) )
% 0.30/0.65 = ( ord_less_int @ A @ B ) ) ).
% 0.30/0.65
% 0.30/0.65 % Iic_subset_Iio_iff
% 0.30/0.65 thf(fact_8950_Iic__subset__Iio__iff,axiom,
% 0.30/0.65 ! [A: real,B: real] :
% 0.30/0.65 ( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ A ) @ ( set_or5984915006950818249n_real @ B ) )
% 0.30/0.65 = ( ord_less_real @ A @ B ) ) ).
% 0.30/0.65
% 0.30/0.65 % Iic_subset_Iio_iff
% 0.30/0.65 thf(fact_8951_sum__choose__upper,axiom,
% 0.30/0.65 ! [M: nat,N: nat] :
% 0.30/0.65 ( ( groups3542108847815614940at_nat
% 0.30/0.65 @ ^ [K2: nat] : ( binomial @ K2 @ M )
% 0.30/0.65 @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65 = ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % sum_choose_upper
% 0.30/0.65 thf(fact_8952_one__add__floor,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X2 ) @ one_one_int )
% 0.30/0.65 = ( archim6058952711729229775r_real @ ( plus_plus_real @ X2 @ one_one_real ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % one_add_floor
% 0.30/0.65 thf(fact_8953_one__add__floor,axiom,
% 0.30/0.65 ! [X2: rat] :
% 0.30/0.65 ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X2 ) @ one_one_int )
% 0.30/0.65 = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X2 @ one_one_rat ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % one_add_floor
% 0.30/0.65 thf(fact_8954_floor__divide__of__nat__eq,axiom,
% 0.30/0.65 ! [M: nat,N: nat] :
% 0.30/0.65 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.30/0.65 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_divide_of_nat_eq
% 0.30/0.65 thf(fact_8955_floor__divide__of__nat__eq,axiom,
% 0.30/0.65 ! [M: nat,N: nat] :
% 0.30/0.65 ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) )
% 0.30/0.65 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_divide_of_nat_eq
% 0.30/0.65 thf(fact_8956_ceiling__altdef,axiom,
% 0.30/0.65 ( archim7802044766580827645g_real
% 0.30/0.65 = ( ^ [X: real] :
% 0.30/0.65 ( if_int
% 0.30/0.65 @ ( X
% 0.30/0.65 = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) )
% 0.30/0.65 @ ( archim6058952711729229775r_real @ X )
% 0.30/0.65 @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % ceiling_altdef
% 0.30/0.65 thf(fact_8957_ceiling__altdef,axiom,
% 0.30/0.65 ( archim2889992004027027881ng_rat
% 0.30/0.65 = ( ^ [X: rat] :
% 0.30/0.65 ( if_int
% 0.30/0.65 @ ( X
% 0.30/0.65 = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) )
% 0.30/0.65 @ ( archim3151403230148437115or_rat @ X )
% 0.30/0.65 @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % ceiling_altdef
% 0.30/0.65 thf(fact_8958_ceiling__diff__floor__le__1,axiom,
% 0.30/0.65 ! [X2: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim7802044766580827645g_real @ X2 ) @ ( archim6058952711729229775r_real @ X2 ) ) @ one_one_int ) ).
% 0.30/0.65
% 0.30/0.65 % ceiling_diff_floor_le_1
% 0.30/0.65 thf(fact_8959_ceiling__diff__floor__le__1,axiom,
% 0.30/0.65 ! [X2: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( archim3151403230148437115or_rat @ X2 ) ) @ one_one_int ) ).
% 0.30/0.65
% 0.30/0.65 % ceiling_diff_floor_le_1
% 0.30/0.65 thf(fact_8960_floor__eq,axiom,
% 0.30/0.65 ! [N: int,X2: real] :
% 0.30/0.65 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X2 )
% 0.30/0.65 => ( ( ord_less_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.30/0.65 => ( ( archim6058952711729229775r_real @ X2 )
% 0.30/0.65 = N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_eq
% 0.30/0.65 thf(fact_8961_real__of__int__floor__add__one__gt,axiom,
% 0.30/0.65 ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_of_int_floor_add_one_gt
% 0.30/0.65 thf(fact_8962_real__of__int__floor__add__one__ge,axiom,
% 0.30/0.65 ! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_of_int_floor_add_one_ge
% 0.30/0.65 thf(fact_8963_real__of__int__floor__gt__diff__one,axiom,
% 0.30/0.65 ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_of_int_floor_gt_diff_one
% 0.30/0.65 thf(fact_8964_real__of__int__floor__ge__diff__one,axiom,
% 0.30/0.65 ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_of_int_floor_ge_diff_one
% 0.30/0.65 thf(fact_8965_pochhammer__rec,axiom,
% 0.30/0.65 ! [A: complex,N: nat] :
% 0.30/0.65 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 0.30/0.65 = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_rec
% 0.30/0.65 thf(fact_8966_pochhammer__rec,axiom,
% 0.30/0.65 ! [A: real,N: nat] :
% 0.30/0.65 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 0.30/0.65 = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_rec
% 0.30/0.65 thf(fact_8967_pochhammer__rec,axiom,
% 0.30/0.65 ! [A: rat,N: nat] :
% 0.30/0.65 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 0.30/0.65 = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_rec
% 0.30/0.65 thf(fact_8968_pochhammer__rec,axiom,
% 0.30/0.65 ! [A: nat,N: nat] :
% 0.30/0.65 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 0.30/0.65 = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_rec
% 0.30/0.65 thf(fact_8969_pochhammer__rec,axiom,
% 0.30/0.65 ! [A: int,N: nat] :
% 0.30/0.65 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 0.30/0.65 = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_rec
% 0.30/0.65 thf(fact_8970_pochhammer__Suc,axiom,
% 0.30/0.65 ! [A: int,N: nat] :
% 0.30/0.65 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 0.30/0.65 = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_Suc
% 0.30/0.65 thf(fact_8971_pochhammer__Suc,axiom,
% 0.30/0.65 ! [A: real,N: nat] :
% 0.30/0.65 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 0.30/0.65 = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_Suc
% 0.30/0.65 thf(fact_8972_pochhammer__Suc,axiom,
% 0.30/0.65 ! [A: nat,N: nat] :
% 0.30/0.65 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 0.30/0.65 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_Suc
% 0.30/0.65 thf(fact_8973_pochhammer__Suc,axiom,
% 0.30/0.65 ! [A: rat,N: nat] :
% 0.30/0.65 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 0.30/0.65 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_Suc
% 0.30/0.65 thf(fact_8974_pochhammer__rec_H,axiom,
% 0.30/0.65 ! [Z: int,N: nat] :
% 0.30/0.65 ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
% 0.30/0.65 = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_rec'
% 0.30/0.65 thf(fact_8975_pochhammer__rec_H,axiom,
% 0.30/0.65 ! [Z: real,N: nat] :
% 0.30/0.65 ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) )
% 0.30/0.65 = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_rec'
% 0.30/0.65 thf(fact_8976_pochhammer__rec_H,axiom,
% 0.30/0.65 ! [Z: nat,N: nat] :
% 0.30/0.65 ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
% 0.30/0.65 = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_rec'
% 0.30/0.65 thf(fact_8977_pochhammer__rec_H,axiom,
% 0.30/0.65 ! [Z: rat,N: nat] :
% 0.30/0.65 ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
% 0.30/0.65 = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_rec'
% 0.30/0.65 thf(fact_8978_pochhammer__eq__0__iff,axiom,
% 0.30/0.65 ! [A: complex,N: nat] :
% 0.30/0.65 ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.30/0.65 = zero_zero_complex )
% 0.30/0.65 = ( ? [K2: nat] :
% 0.30/0.65 ( ( ord_less_nat @ K2 @ N )
% 0.30/0.65 & ( A
% 0.30/0.65 = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K2 ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_eq_0_iff
% 0.30/0.65 thf(fact_8979_pochhammer__eq__0__iff,axiom,
% 0.30/0.65 ! [A: real,N: nat] :
% 0.30/0.65 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.30/0.65 = zero_zero_real )
% 0.30/0.65 = ( ? [K2: nat] :
% 0.30/0.65 ( ( ord_less_nat @ K2 @ N )
% 0.30/0.65 & ( A
% 0.30/0.65 = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K2 ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_eq_0_iff
% 0.30/0.65 thf(fact_8980_pochhammer__eq__0__iff,axiom,
% 0.30/0.65 ! [A: rat,N: nat] :
% 0.30/0.65 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.30/0.65 = zero_zero_rat )
% 0.30/0.65 = ( ? [K2: nat] :
% 0.30/0.65 ( ( ord_less_nat @ K2 @ N )
% 0.30/0.65 & ( A
% 0.30/0.65 = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K2 ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_eq_0_iff
% 0.30/0.65 thf(fact_8981_pochhammer__of__nat__eq__0__iff,axiom,
% 0.30/0.65 ! [N: nat,K: nat] :
% 0.30/0.65 ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.30/0.65 = zero_zero_complex )
% 0.30/0.65 = ( ord_less_nat @ N @ K ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_of_nat_eq_0_iff
% 0.30/0.65 thf(fact_8982_pochhammer__of__nat__eq__0__iff,axiom,
% 0.30/0.65 ! [N: nat,K: nat] :
% 0.30/0.65 ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.30/0.65 = zero_z3403309356797280102nteger )
% 0.30/0.65 = ( ord_less_nat @ N @ K ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_of_nat_eq_0_iff
% 0.30/0.65 thf(fact_8983_pochhammer__of__nat__eq__0__iff,axiom,
% 0.30/0.65 ! [N: nat,K: nat] :
% 0.30/0.65 ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.30/0.65 = zero_zero_int )
% 0.30/0.65 = ( ord_less_nat @ N @ K ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_of_nat_eq_0_iff
% 0.30/0.65 thf(fact_8984_pochhammer__of__nat__eq__0__iff,axiom,
% 0.30/0.65 ! [N: nat,K: nat] :
% 0.30/0.65 ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.30/0.65 = zero_zero_real )
% 0.30/0.65 = ( ord_less_nat @ N @ K ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_of_nat_eq_0_iff
% 0.30/0.65 thf(fact_8985_pochhammer__of__nat__eq__0__iff,axiom,
% 0.30/0.65 ! [N: nat,K: nat] :
% 0.30/0.65 ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.30/0.65 = zero_zero_rat )
% 0.30/0.65 = ( ord_less_nat @ N @ K ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_of_nat_eq_0_iff
% 0.30/0.65 thf(fact_8986_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.30/0.65 ! [N: nat,K: nat] :
% 0.30/0.65 ( ( ord_less_nat @ N @ K )
% 0.30/0.65 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.30/0.65 = zero_zero_complex ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_of_nat_eq_0_lemma
% 0.30/0.65 thf(fact_8987_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.30/0.65 ! [N: nat,K: nat] :
% 0.30/0.65 ( ( ord_less_nat @ N @ K )
% 0.30/0.65 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.30/0.65 = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_of_nat_eq_0_lemma
% 0.30/0.65 thf(fact_8988_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.30/0.65 ! [N: nat,K: nat] :
% 0.30/0.65 ( ( ord_less_nat @ N @ K )
% 0.30/0.65 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.30/0.65 = zero_zero_int ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_of_nat_eq_0_lemma
% 0.30/0.65 thf(fact_8989_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.30/0.65 ! [N: nat,K: nat] :
% 0.30/0.65 ( ( ord_less_nat @ N @ K )
% 0.30/0.65 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.30/0.65 = zero_zero_real ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_of_nat_eq_0_lemma
% 0.30/0.65 thf(fact_8990_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.30/0.65 ! [N: nat,K: nat] :
% 0.30/0.65 ( ( ord_less_nat @ N @ K )
% 0.30/0.65 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.30/0.65 = zero_zero_rat ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_of_nat_eq_0_lemma
% 0.30/0.65 thf(fact_8991_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.30/0.65 ! [K: nat,N: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.30/0.65 != zero_zero_complex ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_of_nat_eq_0_lemma'
% 0.30/0.65 thf(fact_8992_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.30/0.65 ! [K: nat,N: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.30/0.65 != zero_z3403309356797280102nteger ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_of_nat_eq_0_lemma'
% 0.30/0.65 thf(fact_8993_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.30/0.65 ! [K: nat,N: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.30/0.65 != zero_zero_int ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_of_nat_eq_0_lemma'
% 0.30/0.65 thf(fact_8994_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.30/0.65 ! [K: nat,N: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.30/0.65 != zero_zero_real ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_of_nat_eq_0_lemma'
% 0.30/0.65 thf(fact_8995_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.30/0.65 ! [K: nat,N: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.30/0.65 != zero_zero_rat ) ) ).
% 0.30/0.65
% 0.30/0.65 % pochhammer_of_nat_eq_0_lemma'
% 0.30/0.65 thf(fact_8996_sum_OatMost__Suc__shift,axiom,
% 0.30/0.65 ! [G: nat > int,N: nat] :
% 0.30/0.65 ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65 = ( plus_plus_int @ ( G @ zero_zero_nat )
% 0.30/0.65 @ ( groups3539618377306564664at_int
% 0.30/0.65 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65 @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % sum.atMost_Suc_shift
% 0.30/0.65 thf(fact_8997_sum_OatMost__Suc__shift,axiom,
% 0.30/0.65 ! [G: nat > nat,N: nat] :
% 0.30/0.65 ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65 = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 0.30/0.65 @ ( groups3542108847815614940at_nat
% 0.30/0.65 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65 @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % sum.atMost_Suc_shift
% 0.30/0.65 thf(fact_8998_sum_OatMost__Suc__shift,axiom,
% 0.30/0.65 ! [G: nat > real,N: nat] :
% 0.30/0.65 ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65 = ( plus_plus_real @ ( G @ zero_zero_nat )
% 0.30/0.65 @ ( groups6591440286371151544t_real
% 0.30/0.65 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65 @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % sum.atMost_Suc_shift
% 0.30/0.65 thf(fact_8999_sum__choose__lower,axiom,
% 0.30/0.65 ! [R2: nat,N: nat] :
% 0.30/0.65 ( ( groups3542108847815614940at_nat
% 0.30/0.65 @ ^ [K2: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K2 ) @ K2 )
% 0.30/0.65 @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65 = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N ) ) @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % sum_choose_lower
% 0.30/0.65 thf(fact_9000_choose__rising__sum_I2_J,axiom,
% 0.30/0.65 ! [N: nat,M: nat] :
% 0.30/0.65 ( ( groups3542108847815614940at_nat
% 0.30/0.65 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 0.30/0.65 @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.65 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).
% 0.30/0.65
% 0.30/0.65 % choose_rising_sum(2)
% 0.30/0.65 thf(fact_9001_choose__rising__sum_I1_J,axiom,
% 0.30/0.65 ! [N: nat,M: nat] :
% 0.30/0.65 ( ( groups3542108847815614940at_nat
% 0.30/0.65 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 0.30/0.65 @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.65 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % choose_rising_sum(1)
% 0.30/0.65 thf(fact_9002_floor__eq2,axiom,
% 0.30/0.65 ! [N: int,X2: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X2 )
% 0.30/0.65 => ( ( ord_less_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.30/0.65 => ( ( archim6058952711729229775r_real @ X2 )
% 0.30/0.65 = N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_eq2
% 0.30/0.65 thf(fact_9003_floor__divide__real__eq__div,axiom,
% 0.30/0.65 ! [B: int,A: real] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.65 => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
% 0.30/0.65 = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_divide_real_eq_div
% 0.30/0.65 thf(fact_9004_sum__choose__diagonal,axiom,
% 0.30/0.65 ! [M: nat,N: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65 => ( ( groups3542108847815614940at_nat
% 0.30/0.65 @ ^ [K2: nat] : ( binomial @ ( minus_minus_nat @ N @ K2 ) @ ( minus_minus_nat @ M @ K2 ) )
% 0.30/0.65 @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.65 = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % sum_choose_diagonal
% 0.30/0.65 thf(fact_9005_vandermonde,axiom,
% 0.30/0.65 ! [M: nat,N: nat,R2: nat] :
% 0.30/0.65 ( ( groups3542108847815614940at_nat
% 0.30/0.65 @ ^ [K2: nat] : ( times_times_nat @ ( binomial @ M @ K2 ) @ ( binomial @ N @ ( minus_minus_nat @ R2 @ K2 ) ) )
% 0.30/0.65 @ ( set_ord_atMost_nat @ R2 ) )
% 0.30/0.65 = ( binomial @ ( plus_plus_nat @ M @ N ) @ R2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % vandermonde
% 0.30/0.65 thf(fact_9006_choose__row__sum,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % choose_row_sum
% 0.30/0.65 thf(fact_9007_binomial,axiom,
% 0.30/0.65 ! [A: nat,B: nat,N: nat] :
% 0.30/0.65 ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
% 0.30/0.65 = ( groups3542108847815614940at_nat
% 0.30/0.65 @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K2 ) ) )
% 0.30/0.65 @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % binomial
% 0.30/0.65 thf(fact_9008_polynomial__product__nat,axiom,
% 0.30/0.65 ! [M: nat,A: nat > nat,N: nat,B: nat > nat,X2: nat] :
% 0.30/0.65 ( ! [I3: nat] :
% 0.30/0.65 ( ( ord_less_nat @ M @ I3 )
% 0.30/0.65 => ( ( A @ I3 )
% 0.30/0.65 = zero_zero_nat ) )
% 0.30/0.65 => ( ! [J2: nat] :
% 0.30/0.65 ( ( ord_less_nat @ N @ J2 )
% 0.30/0.65 => ( ( B @ J2 )
% 0.30/0.65 = zero_zero_nat ) )
% 0.30/0.65 => ( ( times_times_nat
% 0.30/0.65 @ ( groups3542108847815614940at_nat
% 0.30/0.65 @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( power_power_nat @ X2 @ I4 ) )
% 0.30/0.65 @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.65 @ ( groups3542108847815614940at_nat
% 0.30/0.65 @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X2 @ J3 ) )
% 0.30/0.65 @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.65 = ( groups3542108847815614940at_nat
% 0.30/0.65 @ ^ [R5: nat] :
% 0.30/0.65 ( times_times_nat
% 0.30/0.65 @ ( groups3542108847815614940at_nat
% 0.30/0.65 @ ^ [K2: nat] : ( times_times_nat @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
% 0.30/0.65 @ ( set_ord_atMost_nat @ R5 ) )
% 0.30/0.65 @ ( power_power_nat @ X2 @ R5 ) )
% 0.30/0.65 @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % polynomial_product_nat
% 0.30/0.65 thf(fact_9009_choose__square__sum,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( groups3542108847815614940at_nat
% 0.30/0.65 @ ^ [K2: nat] : ( power_power_nat @ ( binomial @ N @ K2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.65 @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65 = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % choose_square_sum
% 0.30/0.65 thf(fact_9010_floor__log__eq__powr__iff,axiom,
% 0.30/0.65 ! [X2: real,B: real,K: int] :
% 0.30/0.65 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.65 => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X2 ) )
% 0.30/0.65 = K )
% 0.30/0.65 = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X2 )
% 0.30/0.65 & ( ord_less_real @ X2 @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_log_eq_powr_iff
% 0.30/0.65 thf(fact_9011_binomial__r__part__sum,axiom,
% 0.30/0.65 ! [M: nat] :
% 0.30/0.65 ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.65 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % binomial_r_part_sum
% 0.30/0.65 thf(fact_9012_choose__linear__sum,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( groups3542108847815614940at_nat
% 0.30/0.65 @ ^ [I4: nat] : ( times_times_nat @ I4 @ ( binomial @ N @ I4 ) )
% 0.30/0.65 @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65 = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % choose_linear_sum
% 0.30/0.65 thf(fact_9013_floor__log2__div2,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.65 => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.30/0.65 = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_log2_div2
% 0.30/0.65 thf(fact_9014_floor__log__nat__eq__if,axiom,
% 0.30/0.65 ! [B: nat,N: nat,K: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.30/0.65 => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 0.30/0.65 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.30/0.65 => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.30/0.65 = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_log_nat_eq_if
% 0.30/0.65 thf(fact_9015_of__nat__id,axiom,
% 0.30/0.65 ( semiri1316708129612266289at_nat
% 0.30/0.65 = ( ^ [N3: nat] : N3 ) ) ).
% 0.30/0.65
% 0.30/0.65 % of_nat_id
% 0.30/0.65 thf(fact_9016_real__scaleR__def,axiom,
% 0.30/0.65 real_V1485227260804924795R_real = times_times_real ).
% 0.30/0.65
% 0.30/0.65 % real_scaleR_def
% 0.30/0.65 thf(fact_9017_complex__scaleR,axiom,
% 0.30/0.65 ! [R2: real,A: real,B: real] :
% 0.30/0.65 ( ( real_V2046097035970521341omplex @ R2 @ ( complex2 @ A @ B ) )
% 0.30/0.65 = ( complex2 @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ B ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % complex_scaleR
% 0.30/0.65 thf(fact_9018_fact__eq__fact__times,axiom,
% 0.30/0.65 ! [N: nat,M: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.65 => ( ( semiri1408675320244567234ct_nat @ M )
% 0.30/0.65 = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
% 0.30/0.65 @ ( groups708209901874060359at_nat
% 0.30/0.65 @ ^ [X: nat] : X
% 0.30/0.65 @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % fact_eq_fact_times
% 0.30/0.65 thf(fact_9019_fact__div__fact,axiom,
% 0.30/0.65 ! [N: nat,M: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.65 => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.30/0.65 = ( groups708209901874060359at_nat
% 0.30/0.65 @ ^ [X: nat] : X
% 0.30/0.65 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % fact_div_fact
% 0.30/0.65 thf(fact_9020_exp__two__pi__i_H,axiom,
% 0.30/0.65 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65 = one_one_complex ) ).
% 0.30/0.65
% 0.30/0.65 % exp_two_pi_i'
% 0.30/0.65 thf(fact_9021_exp__two__pi__i,axiom,
% 0.30/0.65 ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
% 0.30/0.65 = one_one_complex ) ).
% 0.30/0.65
% 0.30/0.65 % exp_two_pi_i
% 0.30/0.65 thf(fact_9022_norm__ii,axiom,
% 0.30/0.65 ( ( real_V1022390504157884413omplex @ imaginary_unit )
% 0.30/0.65 = one_one_real ) ).
% 0.30/0.65
% 0.30/0.65 % norm_ii
% 0.30/0.65 thf(fact_9023_divide__i,axiom,
% 0.30/0.65 ! [X2: complex] :
% 0.30/0.65 ( ( divide1717551699836669952omplex @ X2 @ imaginary_unit )
% 0.30/0.65 = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % divide_i
% 0.30/0.65 thf(fact_9024_complex__i__mult__minus,axiom,
% 0.30/0.65 ! [X2: complex] :
% 0.30/0.65 ( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X2 ) )
% 0.30/0.65 = ( uminus1482373934393186551omplex @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % complex_i_mult_minus
% 0.30/0.65 thf(fact_9025_i__squared,axiom,
% 0.30/0.65 ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
% 0.30/0.65 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.65
% 0.30/0.65 % i_squared
% 0.30/0.65 thf(fact_9026_divide__numeral__i,axiom,
% 0.30/0.65 ! [Z: complex,N: num] :
% 0.30/0.65 ( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
% 0.30/0.65 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % divide_numeral_i
% 0.30/0.65 thf(fact_9027_power2__i,axiom,
% 0.30/0.65 ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.65 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.65
% 0.30/0.65 % power2_i
% 0.30/0.65 thf(fact_9028_exp__pi__i,axiom,
% 0.30/0.65 ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
% 0.30/0.65 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.65
% 0.30/0.65 % exp_pi_i
% 0.30/0.65 thf(fact_9029_exp__pi__i_H,axiom,
% 0.30/0.65 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
% 0.30/0.65 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.65
% 0.30/0.65 % exp_pi_i'
% 0.30/0.65 thf(fact_9030_i__even__power,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.65 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % i_even_power
% 0.30/0.65 thf(fact_9031_complex__i__not__one,axiom,
% 0.30/0.65 imaginary_unit != one_one_complex ).
% 0.30/0.65
% 0.30/0.65 % complex_i_not_one
% 0.30/0.65 thf(fact_9032_complex__i__not__numeral,axiom,
% 0.30/0.65 ! [W: num] :
% 0.30/0.65 ( imaginary_unit
% 0.30/0.65 != ( numera6690914467698888265omplex @ W ) ) ).
% 0.30/0.65
% 0.30/0.65 % complex_i_not_numeral
% 0.30/0.65 thf(fact_9033_i__times__eq__iff,axiom,
% 0.30/0.65 ! [W: complex,Z: complex] :
% 0.30/0.65 ( ( ( times_times_complex @ imaginary_unit @ W )
% 0.30/0.65 = Z )
% 0.30/0.65 = ( W
% 0.30/0.65 = ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % i_times_eq_iff
% 0.30/0.65 thf(fact_9034_complex__i__not__neg__numeral,axiom,
% 0.30/0.65 ! [W: num] :
% 0.30/0.65 ( imaginary_unit
% 0.30/0.65 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % complex_i_not_neg_numeral
% 0.30/0.65 thf(fact_9035_imaginary__unit_Ocode,axiom,
% 0.30/0.65 ( imaginary_unit
% 0.30/0.65 = ( complex2 @ zero_zero_real @ one_one_real ) ) ).
% 0.30/0.65
% 0.30/0.65 % imaginary_unit.code
% 0.30/0.65 thf(fact_9036_Complex__eq__i,axiom,
% 0.30/0.65 ! [X2: real,Y3: real] :
% 0.30/0.65 ( ( ( complex2 @ X2 @ Y3 )
% 0.30/0.65 = imaginary_unit )
% 0.30/0.65 = ( ( X2 = zero_zero_real )
% 0.30/0.65 & ( Y3 = one_one_real ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Complex_eq_i
% 0.30/0.65 thf(fact_9037_Complex__mult__i,axiom,
% 0.30/0.65 ! [A: real,B: real] :
% 0.30/0.65 ( ( times_times_complex @ ( complex2 @ A @ B ) @ imaginary_unit )
% 0.30/0.65 = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.30/0.65
% 0.30/0.65 % Complex_mult_i
% 0.30/0.65 thf(fact_9038_i__mult__Complex,axiom,
% 0.30/0.65 ! [A: real,B: real] :
% 0.30/0.65 ( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B ) )
% 0.30/0.65 = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.30/0.65
% 0.30/0.65 % i_mult_Complex
% 0.30/0.65 thf(fact_9039_prod__int__plus__eq,axiom,
% 0.30/0.65 ! [I2: nat,J: nat] :
% 0.30/0.65 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ ( plus_plus_nat @ I2 @ J ) ) )
% 0.30/0.65 = ( groups1705073143266064639nt_int
% 0.30/0.65 @ ^ [X: int] : X
% 0.30/0.65 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I2 @ J ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % prod_int_plus_eq
% 0.30/0.65 thf(fact_9040_complex__of__real__i,axiom,
% 0.30/0.65 ! [R2: real] :
% 0.30/0.65 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ imaginary_unit )
% 0.30/0.65 = ( complex2 @ zero_zero_real @ R2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % complex_of_real_i
% 0.30/0.65 thf(fact_9041_i__complex__of__real,axiom,
% 0.30/0.65 ! [R2: real] :
% 0.30/0.65 ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R2 ) )
% 0.30/0.65 = ( complex2 @ zero_zero_real @ R2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % i_complex_of_real
% 0.30/0.65 thf(fact_9042_Complex__eq,axiom,
% 0.30/0.65 ( complex2
% 0.30/0.65 = ( ^ [A3: real,B2: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A3 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B2 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Complex_eq
% 0.30/0.65 thf(fact_9043_complex__split__polar,axiom,
% 0.30/0.65 ! [Z: complex] :
% 0.30/0.65 ? [R3: real,A4: real] :
% 0.30/0.65 ( Z
% 0.30/0.65 = ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A4 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A4 ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % complex_split_polar
% 0.30/0.65 thf(fact_9044_cmod__unit__one,axiom,
% 0.30/0.65 ! [A: real] :
% 0.30/0.65 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
% 0.30/0.65 = one_one_real ) ).
% 0.30/0.65
% 0.30/0.65 % cmod_unit_one
% 0.30/0.65 thf(fact_9045_cmod__complex__polar,axiom,
% 0.30/0.65 ! [R2: real,A: real] :
% 0.30/0.65 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
% 0.30/0.65 = ( abs_abs_real @ R2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % cmod_complex_polar
% 0.30/0.65 thf(fact_9046_Arg__minus__ii,axiom,
% 0.30/0.65 ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
% 0.30/0.65 = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Arg_minus_ii
% 0.30/0.65 thf(fact_9047_csqrt__ii,axiom,
% 0.30/0.65 ( ( csqrt @ imaginary_unit )
% 0.30/0.65 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % csqrt_ii
% 0.30/0.65 thf(fact_9048_Arg__ii,axiom,
% 0.30/0.65 ( ( arg @ imaginary_unit )
% 0.30/0.65 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Arg_ii
% 0.30/0.65 thf(fact_9049_cis__minus__pi__half,axiom,
% 0.30/0.65 ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65 = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% 0.30/0.65
% 0.30/0.65 % cis_minus_pi_half
% 0.30/0.65 thf(fact_9050_csqrt__1,axiom,
% 0.30/0.65 ( ( csqrt @ one_one_complex )
% 0.30/0.65 = one_one_complex ) ).
% 0.30/0.65
% 0.30/0.65 % csqrt_1
% 0.30/0.65 thf(fact_9051_csqrt__eq__1,axiom,
% 0.30/0.65 ! [Z: complex] :
% 0.30/0.65 ( ( ( csqrt @ Z )
% 0.30/0.65 = one_one_complex )
% 0.30/0.65 = ( Z = one_one_complex ) ) ).
% 0.30/0.65
% 0.30/0.65 % csqrt_eq_1
% 0.30/0.65 thf(fact_9052_norm__cis,axiom,
% 0.30/0.65 ! [A: real] :
% 0.30/0.65 ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
% 0.30/0.65 = one_one_real ) ).
% 0.30/0.65
% 0.30/0.65 % norm_cis
% 0.30/0.65 thf(fact_9053_cis__zero,axiom,
% 0.30/0.65 ( ( cis @ zero_zero_real )
% 0.30/0.65 = one_one_complex ) ).
% 0.30/0.65
% 0.30/0.65 % cis_zero
% 0.30/0.65 thf(fact_9054_cis__pi,axiom,
% 0.30/0.65 ( ( cis @ pi )
% 0.30/0.65 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.65
% 0.30/0.65 % cis_pi
% 0.30/0.65 thf(fact_9055_power2__csqrt,axiom,
% 0.30/0.65 ! [Z: complex] :
% 0.30/0.65 ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.65 = Z ) ).
% 0.30/0.65
% 0.30/0.65 % power2_csqrt
% 0.30/0.65 thf(fact_9056_cis__pi__half,axiom,
% 0.30/0.65 ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.65 = imaginary_unit ) ).
% 0.30/0.65
% 0.30/0.65 % cis_pi_half
% 0.30/0.65 thf(fact_9057_cis__2pi,axiom,
% 0.30/0.65 ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.30/0.65 = one_one_complex ) ).
% 0.30/0.65
% 0.30/0.65 % cis_2pi
% 0.30/0.65 thf(fact_9058_cis__divide,axiom,
% 0.30/0.65 ! [A: real,B: real] :
% 0.30/0.65 ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B ) )
% 0.30/0.65 = ( cis @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % cis_divide
% 0.30/0.65 thf(fact_9059_cis__mult,axiom,
% 0.30/0.65 ! [A: real,B: real] :
% 0.30/0.65 ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
% 0.30/0.65 = ( cis @ ( plus_plus_real @ A @ B ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % cis_mult
% 0.30/0.65 thf(fact_9060_DeMoivre,axiom,
% 0.30/0.65 ! [A: real,N: nat] :
% 0.30/0.65 ( ( power_power_complex @ ( cis @ A ) @ N )
% 0.30/0.65 = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % DeMoivre
% 0.30/0.65 thf(fact_9061_cis__conv__exp,axiom,
% 0.30/0.65 ( cis
% 0.30/0.65 = ( ^ [B2: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B2 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % cis_conv_exp
% 0.30/0.65 thf(fact_9062_of__real__sqrt,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( real_V4546457046886955230omplex @ ( sqrt @ X2 ) )
% 0.30/0.65 = ( csqrt @ ( real_V4546457046886955230omplex @ X2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % of_real_sqrt
% 0.30/0.65 thf(fact_9063_Arg__bounded,axiom,
% 0.30/0.65 ! [Z: complex] :
% 0.30/0.65 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 0.30/0.65 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).
% 0.30/0.65
% 0.30/0.65 % Arg_bounded
% 0.30/0.65 thf(fact_9064_bij__betw__roots__unity,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( bij_betw_nat_complex
% 0.30/0.65 @ ^ [K2: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.30/0.65 @ ( set_ord_lessThan_nat @ N )
% 0.30/0.65 @ ( collect_complex
% 0.30/0.65 @ ^ [Z3: complex] :
% 0.30/0.65 ( ( power_power_complex @ Z3 @ N )
% 0.30/0.65 = one_one_complex ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % bij_betw_roots_unity
% 0.30/0.65 thf(fact_9065_Maclaurin__sin__bound,axiom,
% 0.30/0.65 ! [X2: real,N: nat] :
% 0.30/0.65 ( ord_less_eq_real
% 0.30/0.65 @ ( abs_abs_real
% 0.30/0.65 @ ( minus_minus_real @ ( sin_real @ X2 )
% 0.30/0.65 @ ( groups6591440286371151544t_real
% 0.30/0.65 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
% 0.30/0.65 @ ( set_ord_lessThan_nat @ N ) ) ) )
% 0.30/0.65 @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X2 ) @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Maclaurin_sin_bound
% 0.30/0.65 thf(fact_9066_real__sqrt__inverse,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( sqrt @ ( inverse_inverse_real @ X2 ) )
% 0.30/0.65 = ( inverse_inverse_real @ ( sqrt @ X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_sqrt_inverse
% 0.30/0.65 thf(fact_9067_divide__real__def,axiom,
% 0.30/0.65 ( divide_divide_real
% 0.30/0.65 = ( ^ [X: real,Y: real] : ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % divide_real_def
% 0.30/0.65 thf(fact_9068_inverse__powr,axiom,
% 0.30/0.65 ! [Y3: real,A: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.65 => ( ( powr_real @ ( inverse_inverse_real @ Y3 ) @ A )
% 0.30/0.65 = ( inverse_inverse_real @ ( powr_real @ Y3 @ A ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % inverse_powr
% 0.30/0.65 thf(fact_9069_forall__pos__mono__1,axiom,
% 0.30/0.65 ! [P: real > $o,E: real] :
% 0.30/0.65 ( ! [D3: real,E2: real] :
% 0.30/0.65 ( ( ord_less_real @ D3 @ E2 )
% 0.30/0.65 => ( ( P @ D3 )
% 0.30/0.65 => ( P @ E2 ) ) )
% 0.30/0.65 => ( ! [N2: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
% 0.30/0.65 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.30/0.65 => ( P @ E ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % forall_pos_mono_1
% 0.30/0.65 thf(fact_9070_forall__pos__mono,axiom,
% 0.30/0.65 ! [P: real > $o,E: real] :
% 0.30/0.65 ( ! [D3: real,E2: real] :
% 0.30/0.65 ( ( ord_less_real @ D3 @ E2 )
% 0.30/0.65 => ( ( P @ D3 )
% 0.30/0.65 => ( P @ E2 ) ) )
% 0.30/0.65 => ( ! [N2: nat] :
% 0.30/0.65 ( ( N2 != zero_zero_nat )
% 0.30/0.65 => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
% 0.30/0.65 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.30/0.65 => ( P @ E ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % forall_pos_mono
% 0.30/0.65 thf(fact_9071_real__arch__inverse,axiom,
% 0.30/0.65 ! [E: real] :
% 0.30/0.65 ( ( ord_less_real @ zero_zero_real @ E )
% 0.30/0.65 = ( ? [N3: nat] :
% 0.30/0.65 ( ( N3 != zero_zero_nat )
% 0.30/0.65 & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) )
% 0.30/0.65 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ E ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_arch_inverse
% 0.30/0.65 thf(fact_9072_sqrt__divide__self__eq,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( divide_divide_real @ ( sqrt @ X2 ) @ X2 )
% 0.30/0.65 = ( inverse_inverse_real @ ( sqrt @ X2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % sqrt_divide_self_eq
% 0.30/0.65 thf(fact_9073_log__inverse,axiom,
% 0.30/0.65 ! [A: real,X2: real] :
% 0.30/0.65 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.65 => ( ( A != one_one_real )
% 0.30/0.65 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( log @ A @ ( inverse_inverse_real @ X2 ) )
% 0.30/0.65 = ( uminus_uminus_real @ ( log @ A @ X2 ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % log_inverse
% 0.30/0.65 thf(fact_9074_exp__plus__inverse__exp,axiom,
% 0.30/0.65 ! [X2: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % exp_plus_inverse_exp
% 0.30/0.65 thf(fact_9075_plus__inverse__ge__2,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % plus_inverse_ge_2
% 0.30/0.65 thf(fact_9076_real__inv__sqrt__pow2,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.65 = ( inverse_inverse_real @ X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_inv_sqrt_pow2
% 0.30/0.65 thf(fact_9077_tan__cot,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) )
% 0.30/0.65 = ( inverse_inverse_real @ ( tan_real @ X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % tan_cot
% 0.30/0.65 thf(fact_9078_real__le__x__sinh,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ord_less_eq_real @ X2 @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_le_x_sinh
% 0.30/0.65 thf(fact_9079_real__le__abs__sinh,axiom,
% 0.30/0.65 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_le_abs_sinh
% 0.30/0.65 thf(fact_9080_VEBT__internal_OminNull_Opelims_I1_J,axiom,
% 0.30/0.65 ! [X2: vEBT_VEBT,Y3: $o] :
% 0.30/0.65 ( ( ( vEBT_VEBT_minNull @ X2 )
% 0.30/0.65 = Y3 )
% 0.30/0.65 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
% 0.30/0.65 => ( ( ( X2
% 0.30/0.65 = ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.65 => ( Y3
% 0.30/0.65 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
% 0.30/0.65 => ( ! [Uv2: $o] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.30/0.65 => ( ~ Y3
% 0.30/0.65 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
% 0.30/0.65 => ( ! [Uu2: $o] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.30/0.65 => ( ~ Y3
% 0.30/0.65 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
% 0.30/0.65 => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.30/0.65 => ( Y3
% 0.30/0.65 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
% 0.30/0.65 => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
% 0.30/0.65 => ( ~ Y3
% 0.30/0.65 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % VEBT_internal.minNull.pelims(1)
% 0.30/0.65 thf(fact_9081_cot__less__zero,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
% 0.30/0.65 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.30/0.65 => ( ord_less_real @ ( cot_real @ X2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % cot_less_zero
% 0.30/0.65 thf(fact_9082_sinh__real__le__iff,axiom,
% 0.30/0.65 ! [X2: real,Y3: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ ( sinh_real @ X2 ) @ ( sinh_real @ Y3 ) )
% 0.30/0.65 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 0.30/0.65
% 0.30/0.65 % sinh_real_le_iff
% 0.30/0.65 thf(fact_9083_sinh__real__nonpos__iff,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ ( sinh_real @ X2 ) @ zero_zero_real )
% 0.30/0.65 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 0.30/0.65
% 0.30/0.65 % sinh_real_nonpos_iff
% 0.30/0.65 thf(fact_9084_sinh__real__nonneg__iff,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % sinh_real_nonneg_iff
% 0.30/0.65 thf(fact_9085_cot__npi,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.30/0.65 = zero_zero_real ) ).
% 0.30/0.65
% 0.30/0.65 % cot_npi
% 0.30/0.65 thf(fact_9086_cot__periodic,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( cot_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.30/0.65 = ( cot_real @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % cot_periodic
% 0.30/0.65 thf(fact_9087_sinh__le__cosh__real,axiom,
% 0.30/0.65 ! [X2: real] : ( ord_less_eq_real @ ( sinh_real @ X2 ) @ ( cosh_real @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % sinh_le_cosh_real
% 0.30/0.65 thf(fact_9088_cosh__real__nonneg,axiom,
% 0.30/0.65 ! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % cosh_real_nonneg
% 0.30/0.65 thf(fact_9089_cosh__real__nonneg__le__iff,axiom,
% 0.30/0.65 ! [X2: real,Y3: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.65 => ( ( ord_less_eq_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
% 0.30/0.65 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % cosh_real_nonneg_le_iff
% 0.30/0.65 thf(fact_9090_cosh__real__nonpos__le__iff,axiom,
% 0.30/0.65 ! [X2: real,Y3: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.30/0.65 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.30/0.65 => ( ( ord_less_eq_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
% 0.30/0.65 = ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % cosh_real_nonpos_le_iff
% 0.30/0.65 thf(fact_9091_cosh__real__ge__1,axiom,
% 0.30/0.65 ! [X2: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % cosh_real_ge_1
% 0.30/0.65 thf(fact_9092_divide__complex__def,axiom,
% 0.30/0.65 ( divide1717551699836669952omplex
% 0.30/0.65 = ( ^ [X: complex,Y: complex] : ( times_times_complex @ X @ ( invers8013647133539491842omplex @ Y ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % divide_complex_def
% 0.30/0.65 thf(fact_9093_cosh__real__strict__mono,axiom,
% 0.30/0.65 ! [X2: real,Y3: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.65 => ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % cosh_real_strict_mono
% 0.30/0.65 thf(fact_9094_cosh__real__nonneg__less__iff,axiom,
% 0.30/0.65 ! [X2: real,Y3: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.65 => ( ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
% 0.30/0.65 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % cosh_real_nonneg_less_iff
% 0.30/0.65 thf(fact_9095_cosh__real__nonpos__less__iff,axiom,
% 0.30/0.65 ! [X2: real,Y3: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.30/0.65 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.30/0.65 => ( ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
% 0.30/0.65 = ( ord_less_real @ Y3 @ X2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % cosh_real_nonpos_less_iff
% 0.30/0.65 thf(fact_9096_arcosh__cosh__real,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( arcosh_real @ ( cosh_real @ X2 ) )
% 0.30/0.65 = X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % arcosh_cosh_real
% 0.30/0.65 thf(fact_9097_complex__inverse,axiom,
% 0.30/0.65 ! [A: real,B: real] :
% 0.30/0.65 ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
% 0.30/0.65 = ( 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 ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % complex_inverse
% 0.30/0.65 thf(fact_9098_VEBT__internal_OminNull_Opelims_I3_J,axiom,
% 0.30/0.65 ! [X2: vEBT_VEBT] :
% 0.30/0.65 ( ~ ( vEBT_VEBT_minNull @ X2 )
% 0.30/0.65 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
% 0.30/0.65 => ( ! [Uv2: $o] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.30/0.65 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
% 0.30/0.65 => ( ! [Uu2: $o] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.30/0.65 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
% 0.30/0.65 => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
% 0.30/0.65 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % VEBT_internal.minNull.pelims(3)
% 0.30/0.65 thf(fact_9099_VEBT__internal_OminNull_Opelims_I2_J,axiom,
% 0.30/0.65 ! [X2: vEBT_VEBT] :
% 0.30/0.65 ( ( vEBT_VEBT_minNull @ X2 )
% 0.30/0.65 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
% 0.30/0.65 => ( ( ( X2
% 0.30/0.65 = ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.65 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
% 0.30/0.65 => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.30/0.65 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % VEBT_internal.minNull.pelims(2)
% 0.30/0.65 thf(fact_9100_cosh__ln__real,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( cosh_real @ ( ln_ln_real @ X2 ) )
% 0.30/0.65 = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % cosh_ln_real
% 0.30/0.65 thf(fact_9101_cot__gt__zero,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.65 => ( ord_less_real @ zero_zero_real @ ( cot_real @ X2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % cot_gt_zero
% 0.30/0.65 thf(fact_9102_sinh__ln__real,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( sinh_real @ ( ln_ln_real @ X2 ) )
% 0.30/0.65 = ( divide_divide_real @ ( minus_minus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % sinh_ln_real
% 0.30/0.65 thf(fact_9103_tan__cot_H,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) )
% 0.30/0.65 = ( cot_real @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % tan_cot'
% 0.30/0.65 thf(fact_9104_bij__betw__nth__root__unity,axiom,
% 0.30/0.65 ! [C: complex,N: nat] :
% 0.30/0.65 ( ( C != zero_zero_complex )
% 0.30/0.65 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 0.30/0.65 @ ( collect_complex
% 0.30/0.65 @ ^ [Z3: complex] :
% 0.30/0.65 ( ( power_power_complex @ Z3 @ N )
% 0.30/0.65 = one_one_complex ) )
% 0.30/0.65 @ ( collect_complex
% 0.30/0.65 @ ^ [Z3: complex] :
% 0.30/0.65 ( ( power_power_complex @ Z3 @ N )
% 0.30/0.65 = C ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % bij_betw_nth_root_unity
% 0.30/0.65 thf(fact_9105_upto_Opinduct,axiom,
% 0.30/0.65 ! [A0: int,A12: int,P: int > int > $o] :
% 0.30/0.65 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 0.30/0.65 => ( ! [I3: int,J2: int] :
% 0.30/0.65 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
% 0.30/0.65 => ( ( ( ord_less_eq_int @ I3 @ J2 )
% 0.30/0.65 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
% 0.30/0.65 => ( P @ I3 @ J2 ) ) )
% 0.30/0.65 => ( P @ A0 @ A12 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % upto.pinduct
% 0.30/0.65 thf(fact_9106_arctan__def,axiom,
% 0.30/0.65 ( arctan
% 0.30/0.65 = ( ^ [Y: real] :
% 0.30/0.65 ( the_real
% 0.30/0.65 @ ^ [X: real] :
% 0.30/0.65 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.30/0.65 & ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.65 & ( ( tan_real @ X )
% 0.30/0.65 = Y ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % arctan_def
% 0.30/0.65 thf(fact_9107_arcsin__def,axiom,
% 0.30/0.65 ( arcsin
% 0.30/0.65 = ( ^ [Y: real] :
% 0.30/0.65 ( the_real
% 0.30/0.65 @ ^ [X: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.30/0.65 & ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.65 & ( ( sin_real @ X )
% 0.30/0.65 = Y ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % arcsin_def
% 0.30/0.65 thf(fact_9108_real__root__zero,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( root @ N @ zero_zero_real )
% 0.30/0.65 = zero_zero_real ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_zero
% 0.30/0.65 thf(fact_9109_real__root__Suc__0,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( root @ ( suc @ zero_zero_nat ) @ X2 )
% 0.30/0.65 = X2 ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_Suc_0
% 0.30/0.65 thf(fact_9110_real__root__eq__iff,axiom,
% 0.30/0.65 ! [N: nat,X2: real,Y3: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ( root @ N @ X2 )
% 0.30/0.65 = ( root @ N @ Y3 ) )
% 0.30/0.65 = ( X2 = Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_eq_iff
% 0.30/0.65 thf(fact_9111_root__0,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( root @ zero_zero_nat @ X2 )
% 0.30/0.65 = zero_zero_real ) ).
% 0.30/0.65
% 0.30/0.65 % root_0
% 0.30/0.65 thf(fact_9112_real__root__eq__0__iff,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ( root @ N @ X2 )
% 0.30/0.65 = zero_zero_real )
% 0.30/0.65 = ( X2 = zero_zero_real ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_eq_0_iff
% 0.30/0.65 thf(fact_9113_real__root__less__iff,axiom,
% 0.30/0.65 ! [N: nat,X2: real,Y3: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) )
% 0.30/0.65 = ( ord_less_real @ X2 @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_less_iff
% 0.30/0.65 thf(fact_9114_real__root__le__iff,axiom,
% 0.30/0.65 ! [N: nat,X2: real,Y3: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) )
% 0.30/0.65 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_le_iff
% 0.30/0.65 thf(fact_9115_real__root__one,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( root @ N @ one_one_real )
% 0.30/0.65 = one_one_real ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_one
% 0.30/0.65 thf(fact_9116_real__root__eq__1__iff,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ( root @ N @ X2 )
% 0.30/0.65 = one_one_real )
% 0.30/0.65 = ( X2 = one_one_real ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_eq_1_iff
% 0.30/0.65 thf(fact_9117_real__root__lt__0__iff,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_real @ ( root @ N @ X2 ) @ zero_zero_real )
% 0.30/0.65 = ( ord_less_real @ X2 @ zero_zero_real ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_lt_0_iff
% 0.30/0.65 thf(fact_9118_real__root__gt__0__iff,axiom,
% 0.30/0.65 ! [N: nat,Y3: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 0.30/0.65 = ( ord_less_real @ zero_zero_real @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_gt_0_iff
% 0.30/0.65 thf(fact_9119_real__root__ge__0__iff,axiom,
% 0.30/0.65 ! [N: nat,Y3: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 0.30/0.65 = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_ge_0_iff
% 0.30/0.65 thf(fact_9120_real__root__le__0__iff,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ zero_zero_real )
% 0.30/0.65 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_le_0_iff
% 0.30/0.65 thf(fact_9121_real__root__gt__1__iff,axiom,
% 0.30/0.65 ! [N: nat,Y3: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y3 ) )
% 0.30/0.65 = ( ord_less_real @ one_one_real @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_gt_1_iff
% 0.30/0.65 thf(fact_9122_real__root__lt__1__iff,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_real @ ( root @ N @ X2 ) @ one_one_real )
% 0.30/0.65 = ( ord_less_real @ X2 @ one_one_real ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_lt_1_iff
% 0.30/0.65 thf(fact_9123_real__root__ge__1__iff,axiom,
% 0.30/0.65 ! [N: nat,Y3: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y3 ) )
% 0.30/0.65 = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_ge_1_iff
% 0.30/0.65 thf(fact_9124_real__root__le__1__iff,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ one_one_real )
% 0.30/0.65 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_le_1_iff
% 0.30/0.65 thf(fact_9125_real__root__pow__pos2,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( power_power_real @ ( root @ N @ X2 ) @ N )
% 0.30/0.65 = X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_pow_pos2
% 0.30/0.65 thf(fact_9126_real__root__inverse,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ( root @ N @ ( inverse_inverse_real @ X2 ) )
% 0.30/0.65 = ( inverse_inverse_real @ ( root @ N @ X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_inverse
% 0.30/0.65 thf(fact_9127_real__root__divide,axiom,
% 0.30/0.65 ! [N: nat,X2: real,Y3: real] :
% 0.30/0.65 ( ( root @ N @ ( divide_divide_real @ X2 @ Y3 ) )
% 0.30/0.65 = ( divide_divide_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_divide
% 0.30/0.65 thf(fact_9128_real__root__mult,axiom,
% 0.30/0.65 ! [N: nat,X2: real,Y3: real] :
% 0.30/0.65 ( ( root @ N @ ( times_times_real @ X2 @ Y3 ) )
% 0.30/0.65 = ( times_times_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_mult
% 0.30/0.65 thf(fact_9129_real__root__mult__exp,axiom,
% 0.30/0.65 ! [M: nat,N: nat,X2: real] :
% 0.30/0.65 ( ( root @ ( times_times_nat @ M @ N ) @ X2 )
% 0.30/0.65 = ( root @ M @ ( root @ N @ X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_mult_exp
% 0.30/0.65 thf(fact_9130_real__root__minus,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ( root @ N @ ( uminus_uminus_real @ X2 ) )
% 0.30/0.65 = ( uminus_uminus_real @ ( root @ N @ X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_minus
% 0.30/0.65 thf(fact_9131_real__root__commute,axiom,
% 0.30/0.65 ! [M: nat,N: nat,X2: real] :
% 0.30/0.65 ( ( root @ M @ ( root @ N @ X2 ) )
% 0.30/0.65 = ( root @ N @ ( root @ M @ X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_commute
% 0.30/0.65 thf(fact_9132_real__root__pos__pos__le,axiom,
% 0.30/0.65 ! [X2: real,N: nat] :
% 0.30/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_pos_pos_le
% 0.30/0.65 thf(fact_9133_real__root__less__mono,axiom,
% 0.30/0.65 ! [N: nat,X2: real,Y3: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.65 => ( ord_less_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_less_mono
% 0.30/0.65 thf(fact_9134_real__root__le__mono,axiom,
% 0.30/0.65 ! [N: nat,X2: real,Y3: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.30/0.65 => ( ord_less_eq_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_le_mono
% 0.30/0.65 thf(fact_9135_real__root__power,axiom,
% 0.30/0.65 ! [N: nat,X2: real,K: nat] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( root @ N @ ( power_power_real @ X2 @ K ) )
% 0.30/0.65 = ( power_power_real @ ( root @ N @ X2 ) @ K ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_power
% 0.30/0.65 thf(fact_9136_real__root__abs,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( root @ N @ ( abs_abs_real @ X2 ) )
% 0.30/0.65 = ( abs_abs_real @ ( root @ N @ X2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_abs
% 0.30/0.65 thf(fact_9137_ln__neg__is__const,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.30/0.65 => ( ( ln_ln_real @ X2 )
% 0.30/0.65 = ( the_real
% 0.30/0.65 @ ^ [X: real] : $false ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % ln_neg_is_const
% 0.30/0.65 thf(fact_9138_real__root__gt__zero,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ord_less_real @ zero_zero_real @ ( root @ N @ X2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_gt_zero
% 0.30/0.65 thf(fact_9139_real__root__strict__decreasing,axiom,
% 0.30/0.65 ! [N: nat,N5: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_nat @ N @ N5 )
% 0.30/0.65 => ( ( ord_less_real @ one_one_real @ X2 )
% 0.30/0.65 => ( ord_less_real @ ( root @ N5 @ X2 ) @ ( root @ N @ X2 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_strict_decreasing
% 0.30/0.65 thf(fact_9140_sqrt__def,axiom,
% 0.30/0.65 ( sqrt
% 0.30/0.65 = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % sqrt_def
% 0.30/0.65 thf(fact_9141_root__abs__power,axiom,
% 0.30/0.65 ! [N: nat,Y3: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y3 @ N ) ) )
% 0.30/0.65 = ( abs_abs_real @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % root_abs_power
% 0.30/0.65 thf(fact_9142_real__root__pos__pos,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_pos_pos
% 0.30/0.65 thf(fact_9143_real__root__strict__increasing,axiom,
% 0.30/0.65 ! [N: nat,N5: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_nat @ N @ N5 )
% 0.30/0.65 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.30/0.65 => ( ord_less_real @ ( root @ N @ X2 ) @ ( root @ N5 @ X2 ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_strict_increasing
% 0.30/0.65 thf(fact_9144_real__root__decreasing,axiom,
% 0.30/0.65 ! [N: nat,N5: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.65 => ( ( ord_less_eq_real @ one_one_real @ X2 )
% 0.30/0.65 => ( ord_less_eq_real @ ( root @ N5 @ X2 ) @ ( root @ N @ X2 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_decreasing
% 0.30/0.65 thf(fact_9145_real__root__pow__pos,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( power_power_real @ ( root @ N @ X2 ) @ N )
% 0.30/0.65 = X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_pow_pos
% 0.30/0.65 thf(fact_9146_real__root__power__cancel,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( root @ N @ ( power_power_real @ X2 @ N ) )
% 0.30/0.65 = X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_power_cancel
% 0.30/0.65 thf(fact_9147_real__root__pos__unique,axiom,
% 0.30/0.65 ! [N: nat,Y3: real,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.65 => ( ( ( power_power_real @ Y3 @ N )
% 0.30/0.65 = X2 )
% 0.30/0.65 => ( ( root @ N @ X2 )
% 0.30/0.65 = Y3 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_pos_unique
% 0.30/0.65 thf(fact_9148_odd__real__root__pow,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.65 => ( ( power_power_real @ ( root @ N @ X2 ) @ N )
% 0.30/0.65 = X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % odd_real_root_pow
% 0.30/0.65 thf(fact_9149_odd__real__root__unique,axiom,
% 0.30/0.65 ! [N: nat,Y3: real,X2: real] :
% 0.30/0.65 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.65 => ( ( ( power_power_real @ Y3 @ N )
% 0.30/0.65 = X2 )
% 0.30/0.65 => ( ( root @ N @ X2 )
% 0.30/0.65 = Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % odd_real_root_unique
% 0.30/0.65 thf(fact_9150_odd__real__root__power__cancel,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.65 => ( ( root @ N @ ( power_power_real @ X2 @ N ) )
% 0.30/0.65 = X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % odd_real_root_power_cancel
% 0.30/0.65 thf(fact_9151_real__root__increasing,axiom,
% 0.30/0.65 ! [N: nat,N5: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.65 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.30/0.65 => ( ord_less_eq_real @ ( root @ N @ X2 ) @ ( root @ N5 @ X2 ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_root_increasing
% 0.30/0.65 thf(fact_9152_arccos__def,axiom,
% 0.30/0.65 ( arccos
% 0.30/0.65 = ( ^ [Y: real] :
% 0.30/0.65 ( the_real
% 0.30/0.65 @ ^ [X: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.65 & ( ord_less_eq_real @ X @ pi )
% 0.30/0.65 & ( ( cos_real @ X )
% 0.30/0.65 = Y ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % arccos_def
% 0.30/0.65 thf(fact_9153_log__root,axiom,
% 0.30/0.65 ! [N: nat,A: real,B: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.65 => ( ( log @ B @ ( root @ N @ A ) )
% 0.30/0.65 = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % log_root
% 0.30/0.65 thf(fact_9154_log__base__root,axiom,
% 0.30/0.65 ! [N: nat,B: real,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.65 => ( ( log @ ( root @ N @ B ) @ X2 )
% 0.30/0.65 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X2 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % log_base_root
% 0.30/0.65 thf(fact_9155_ln__root,axiom,
% 0.30/0.65 ! [N: nat,B: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.65 => ( ( ln_ln_real @ ( root @ N @ B ) )
% 0.30/0.65 = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % ln_root
% 0.30/0.65 thf(fact_9156_root__powr__inverse,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( root @ N @ X2 )
% 0.30/0.65 = ( powr_real @ X2 @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % root_powr_inverse
% 0.30/0.65 thf(fact_9157_pi__half,axiom,
% 0.30/0.65 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.65 = ( the_real
% 0.30/0.65 @ ^ [X: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.65 & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.65 & ( ( cos_real @ X )
% 0.30/0.65 = zero_zero_real ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pi_half
% 0.30/0.65 thf(fact_9158_pi__def,axiom,
% 0.30/0.65 ( pi
% 0.30/0.65 = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.30/0.65 @ ( the_real
% 0.30/0.65 @ ^ [X: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.65 & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.65 & ( ( cos_real @ X )
% 0.30/0.65 = zero_zero_real ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % pi_def
% 0.30/0.65 thf(fact_9159_modulo__int__unfold,axiom,
% 0.30/0.65 ! [L: int,K: int,N: nat,M: nat] :
% 0.30/0.65 ( ( ( ( ( sgn_sgn_int @ L )
% 0.30/0.65 = zero_zero_int )
% 0.30/0.65 | ( ( sgn_sgn_int @ K )
% 0.30/0.65 = zero_zero_int )
% 0.30/0.65 | ( N = zero_zero_nat ) )
% 0.30/0.65 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.30/0.65 = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 0.30/0.65 & ( ~ ( ( ( sgn_sgn_int @ L )
% 0.30/0.65 = zero_zero_int )
% 0.30/0.65 | ( ( sgn_sgn_int @ K )
% 0.30/0.65 = zero_zero_int )
% 0.30/0.65 | ( N = zero_zero_nat ) )
% 0.30/0.65 => ( ( ( ( sgn_sgn_int @ K )
% 0.30/0.65 = ( sgn_sgn_int @ L ) )
% 0.30/0.65 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.30/0.65 = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
% 0.30/0.65 & ( ( ( sgn_sgn_int @ K )
% 0.30/0.65 != ( sgn_sgn_int @ L ) )
% 0.30/0.65 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.30/0.65 = ( times_times_int @ ( sgn_sgn_int @ L )
% 0.30/0.65 @ ( minus_minus_int
% 0.30/0.65 @ ( semiri1314217659103216013at_int
% 0.30/0.65 @ ( times_times_nat @ N
% 0.30/0.65 @ ( zero_n2687167440665602831ol_nat
% 0.30/0.65 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
% 0.30/0.65 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % modulo_int_unfold
% 0.30/0.65 thf(fact_9160_even__set__encode__iff,axiom,
% 0.30/0.65 ! [A2: set_nat] :
% 0.30/0.65 ( ( finite_finite_nat @ A2 )
% 0.30/0.65 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
% 0.30/0.65 = ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % even_set_encode_iff
% 0.30/0.65 thf(fact_9161_powr__int,axiom,
% 0.30/0.65 ! [X2: real,I2: int] :
% 0.30/0.65 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( ( ord_less_eq_int @ zero_zero_int @ I2 )
% 0.30/0.65 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ I2 ) )
% 0.30/0.65 = ( power_power_real @ X2 @ ( nat2 @ I2 ) ) ) )
% 0.30/0.65 & ( ~ ( ord_less_eq_int @ zero_zero_int @ I2 )
% 0.30/0.65 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ I2 ) )
% 0.30/0.65 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X2 @ ( nat2 @ ( uminus_uminus_int @ I2 ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % powr_int
% 0.30/0.65 thf(fact_9162_mask__nat__positive__iff,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.30/0.65 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % mask_nat_positive_iff
% 0.30/0.65 thf(fact_9163_nat__numeral,axiom,
% 0.30/0.65 ! [K: num] :
% 0.30/0.65 ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 0.30/0.65 = ( numeral_numeral_nat @ K ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_numeral
% 0.30/0.65 thf(fact_9164_nat__1,axiom,
% 0.30/0.65 ( ( nat2 @ one_one_int )
% 0.30/0.65 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_1
% 0.30/0.65 thf(fact_9165_nat__0__iff,axiom,
% 0.30/0.65 ! [I2: int] :
% 0.30/0.65 ( ( ( nat2 @ I2 )
% 0.30/0.65 = zero_zero_nat )
% 0.30/0.65 = ( ord_less_eq_int @ I2 @ zero_zero_int ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_0_iff
% 0.30/0.65 thf(fact_9166_nat__le__0,axiom,
% 0.30/0.65 ! [Z: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 0.30/0.65 => ( ( nat2 @ Z )
% 0.30/0.65 = zero_zero_nat ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_le_0
% 0.30/0.65 thf(fact_9167_zless__nat__conj,axiom,
% 0.30/0.65 ! [W: int,Z: int] :
% 0.30/0.65 ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.30/0.65 = ( ( ord_less_int @ zero_zero_int @ Z )
% 0.30/0.65 & ( ord_less_int @ W @ Z ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % zless_nat_conj
% 0.30/0.65 thf(fact_9168_nat__neg__numeral,axiom,
% 0.30/0.65 ! [K: num] :
% 0.30/0.65 ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.65 = zero_zero_nat ) ).
% 0.30/0.65
% 0.30/0.65 % nat_neg_numeral
% 0.30/0.65 thf(fact_9169_int__nat__eq,axiom,
% 0.30/0.65 ! [Z: int] :
% 0.30/0.65 ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.65 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.30/0.65 = Z ) )
% 0.30/0.65 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.65 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.30/0.65 = zero_zero_int ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % int_nat_eq
% 0.30/0.65 thf(fact_9170_dvd__mult__sgn__iff,axiom,
% 0.30/0.65 ! [L: int,K: int,R2: int] :
% 0.30/0.65 ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
% 0.30/0.65 = ( ( dvd_dvd_int @ L @ K )
% 0.30/0.65 | ( R2 = zero_zero_int ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % dvd_mult_sgn_iff
% 0.30/0.65 thf(fact_9171_dvd__sgn__mult__iff,axiom,
% 0.30/0.65 ! [L: int,R2: int,K: int] :
% 0.30/0.65 ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
% 0.30/0.65 = ( ( dvd_dvd_int @ L @ K )
% 0.30/0.65 | ( R2 = zero_zero_int ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % dvd_sgn_mult_iff
% 0.30/0.65 thf(fact_9172_mult__sgn__dvd__iff,axiom,
% 0.30/0.65 ! [L: int,R2: int,K: int] :
% 0.30/0.65 ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K )
% 0.30/0.65 = ( ( dvd_dvd_int @ L @ K )
% 0.30/0.65 & ( ( R2 = zero_zero_int )
% 0.30/0.65 => ( K = zero_zero_int ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % mult_sgn_dvd_iff
% 0.30/0.65 thf(fact_9173_sgn__mult__dvd__iff,axiom,
% 0.30/0.65 ! [R2: int,L: int,K: int] :
% 0.30/0.65 ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K )
% 0.30/0.65 = ( ( dvd_dvd_int @ L @ K )
% 0.30/0.65 & ( ( R2 = zero_zero_int )
% 0.30/0.65 => ( K = zero_zero_int ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % sgn_mult_dvd_iff
% 0.30/0.65 thf(fact_9174_zero__less__nat__eq,axiom,
% 0.30/0.65 ! [Z: int] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
% 0.30/0.65 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.30/0.65
% 0.30/0.65 % zero_less_nat_eq
% 0.30/0.65 thf(fact_9175_diff__nat__numeral,axiom,
% 0.30/0.65 ! [V: num,V3: num] :
% 0.30/0.65 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
% 0.30/0.65 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % diff_nat_numeral
% 0.30/0.65 thf(fact_9176_nat__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.65 ! [Y3: int,X2: num,N: nat] :
% 0.30/0.65 ( ( ( nat2 @ Y3 )
% 0.30/0.65 = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) )
% 0.30/0.65 = ( Y3
% 0.30/0.65 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_eq_numeral_power_cancel_iff
% 0.30/0.65 thf(fact_9177_numeral__power__eq__nat__cancel__iff,axiom,
% 0.30/0.65 ! [X2: num,N: nat,Y3: int] :
% 0.30/0.65 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
% 0.30/0.65 = ( nat2 @ Y3 ) )
% 0.30/0.65 = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N )
% 0.30/0.65 = Y3 ) ) ).
% 0.30/0.65
% 0.30/0.65 % numeral_power_eq_nat_cancel_iff
% 0.30/0.65 thf(fact_9178_nat__ceiling__le__eq,axiom,
% 0.30/0.65 ! [X2: real,A: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X2 ) ) @ A )
% 0.30/0.65 = ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_ceiling_le_eq
% 0.30/0.65 thf(fact_9179_one__less__nat__eq,axiom,
% 0.30/0.65 ! [Z: int] :
% 0.30/0.65 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
% 0.30/0.65 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.30/0.65
% 0.30/0.65 % one_less_nat_eq
% 0.30/0.65 thf(fact_9180_nat__numeral__diff__1,axiom,
% 0.30/0.65 ! [V: num] :
% 0.30/0.65 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
% 0.30/0.65 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_numeral_diff_1
% 0.30/0.65 thf(fact_9181_nat__less__numeral__power__cancel__iff,axiom,
% 0.30/0.65 ! [A: int,X2: num,N: nat] :
% 0.30/0.65 ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) )
% 0.30/0.65 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_less_numeral_power_cancel_iff
% 0.30/0.65 thf(fact_9182_numeral__power__less__nat__cancel__iff,axiom,
% 0.30/0.65 ! [X2: num,N: nat,A: int] :
% 0.30/0.65 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) @ ( nat2 @ A ) )
% 0.30/0.65 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A ) ) ).
% 0.30/0.65
% 0.30/0.65 % numeral_power_less_nat_cancel_iff
% 0.30/0.65 thf(fact_9183_nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.65 ! [A: int,X2: num,N: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) )
% 0.30/0.65 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_le_numeral_power_cancel_iff
% 0.30/0.65 thf(fact_9184_numeral__power__le__nat__cancel__iff,axiom,
% 0.30/0.65 ! [X2: num,N: nat,A: int] :
% 0.30/0.65 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) @ ( nat2 @ A ) )
% 0.30/0.65 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A ) ) ).
% 0.30/0.65
% 0.30/0.65 % numeral_power_le_nat_cancel_iff
% 0.30/0.65 thf(fact_9185_less__eq__mask,axiom,
% 0.30/0.65 ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % less_eq_mask
% 0.30/0.65 thf(fact_9186_nat__mask__eq,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
% 0.30/0.65 = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_mask_eq
% 0.30/0.65 thf(fact_9187_nat__numeral__as__int,axiom,
% 0.30/0.65 ( numeral_numeral_nat
% 0.30/0.65 = ( ^ [I4: num] : ( nat2 @ ( numeral_numeral_int @ I4 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_numeral_as_int
% 0.30/0.65 thf(fact_9188_nat__mono,axiom,
% 0.30/0.65 ! [X2: int,Y3: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.30/0.65 => ( ord_less_eq_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_mono
% 0.30/0.65 thf(fact_9189_int__sgnE,axiom,
% 0.30/0.65 ! [K: int] :
% 0.30/0.65 ~ ! [N2: nat,L4: int] :
% 0.30/0.65 ( K
% 0.30/0.65 != ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % int_sgnE
% 0.30/0.65 thf(fact_9190_ex__nat,axiom,
% 0.30/0.65 ( ( ^ [P2: nat > $o] :
% 0.30/0.65 ? [X6: nat] : ( P2 @ X6 ) )
% 0.30/0.65 = ( ^ [P3: nat > $o] :
% 0.30/0.65 ? [X: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.30/0.65 & ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % ex_nat
% 0.30/0.65 thf(fact_9191_all__nat,axiom,
% 0.30/0.65 ( ( ^ [P2: nat > $o] :
% 0.30/0.65 ! [X6: nat] : ( P2 @ X6 ) )
% 0.30/0.65 = ( ^ [P3: nat > $o] :
% 0.30/0.65 ! [X: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.30/0.65 => ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % all_nat
% 0.30/0.65 thf(fact_9192_eq__nat__nat__iff,axiom,
% 0.30/0.65 ! [Z: int,Z7: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 0.30/0.65 => ( ( ( nat2 @ Z )
% 0.30/0.65 = ( nat2 @ Z7 ) )
% 0.30/0.65 = ( Z = Z7 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % eq_nat_nat_iff
% 0.30/0.65 thf(fact_9193_nat__one__as__int,axiom,
% 0.30/0.65 ( one_one_nat
% 0.30/0.65 = ( nat2 @ one_one_int ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_one_as_int
% 0.30/0.65 thf(fact_9194_unset__bit__nat__def,axiom,
% 0.30/0.65 ( bit_se4205575877204974255it_nat
% 0.30/0.65 = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M3 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % unset_bit_nat_def
% 0.30/0.65 thf(fact_9195_mask__nonnegative__int,axiom,
% 0.30/0.65 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % mask_nonnegative_int
% 0.30/0.65 thf(fact_9196_not__mask__negative__int,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% 0.30/0.65
% 0.30/0.65 % not_mask_negative_int
% 0.30/0.65 thf(fact_9197_nat__mono__iff,axiom,
% 0.30/0.65 ! [Z: int,W: int] :
% 0.30/0.65 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.30/0.65 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.30/0.65 = ( ord_less_int @ W @ Z ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_mono_iff
% 0.30/0.65 thf(fact_9198_zless__nat__eq__int__zless,axiom,
% 0.30/0.65 ! [M: nat,Z: int] :
% 0.30/0.65 ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
% 0.30/0.65 = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% 0.30/0.65
% 0.30/0.65 % zless_nat_eq_int_zless
% 0.30/0.65 thf(fact_9199_nat__le__iff,axiom,
% 0.30/0.65 ! [X2: int,N: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ ( nat2 @ X2 ) @ N )
% 0.30/0.65 = ( ord_less_eq_int @ X2 @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_le_iff
% 0.30/0.65 thf(fact_9200_int__eq__iff,axiom,
% 0.30/0.65 ! [M: nat,Z: int] :
% 0.30/0.65 ( ( ( semiri1314217659103216013at_int @ M )
% 0.30/0.65 = Z )
% 0.30/0.65 = ( ( M
% 0.30/0.65 = ( nat2 @ Z ) )
% 0.30/0.65 & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % int_eq_iff
% 0.30/0.65 thf(fact_9201_nat__0__le,axiom,
% 0.30/0.65 ! [Z: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.65 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.30/0.65 = Z ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_0_le
% 0.30/0.65 thf(fact_9202_nat__int__add,axiom,
% 0.30/0.65 ! [A: nat,B: nat] :
% 0.30/0.65 ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
% 0.30/0.65 = ( plus_plus_nat @ A @ B ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_int_add
% 0.30/0.65 thf(fact_9203_sgn__mod,axiom,
% 0.30/0.65 ! [L: int,K: int] :
% 0.30/0.65 ( ( L != zero_zero_int )
% 0.30/0.65 => ( ~ ( dvd_dvd_int @ L @ K )
% 0.30/0.65 => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
% 0.30/0.65 = ( sgn_sgn_int @ L ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % sgn_mod
% 0.30/0.65 thf(fact_9204_int__minus,axiom,
% 0.30/0.65 ! [N: nat,M: nat] :
% 0.30/0.65 ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
% 0.30/0.65 = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % int_minus
% 0.30/0.65 thf(fact_9205_nat__abs__mult__distrib,axiom,
% 0.30/0.65 ! [W: int,Z: int] :
% 0.30/0.65 ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
% 0.30/0.65 = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_abs_mult_distrib
% 0.30/0.65 thf(fact_9206_real__nat__ceiling__ge,axiom,
% 0.30/0.65 ! [X2: real] : ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % real_nat_ceiling_ge
% 0.30/0.65 thf(fact_9207_nat__plus__as__int,axiom,
% 0.30/0.65 ( plus_plus_nat
% 0.30/0.65 = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_plus_as_int
% 0.30/0.65 thf(fact_9208_nat__times__as__int,axiom,
% 0.30/0.65 ( times_times_nat
% 0.30/0.65 = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_times_as_int
% 0.30/0.65 thf(fact_9209_less__mask,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.65 => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % less_mask
% 0.30/0.65 thf(fact_9210_nat__minus__as__int,axiom,
% 0.30/0.65 ( minus_minus_nat
% 0.30/0.65 = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_minus_as_int
% 0.30/0.65 thf(fact_9211_nat__div__as__int,axiom,
% 0.30/0.65 ( divide_divide_nat
% 0.30/0.65 = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_div_as_int
% 0.30/0.65 thf(fact_9212_zsgn__def,axiom,
% 0.30/0.65 ( sgn_sgn_int
% 0.30/0.65 = ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % zsgn_def
% 0.30/0.65 thf(fact_9213_modulo__int__def,axiom,
% 0.30/0.65 ( modulo_modulo_int
% 0.30/0.65 = ( ^ [K2: int,L2: int] :
% 0.30/0.65 ( if_int @ ( L2 = zero_zero_int ) @ K2
% 0.30/0.65 @ ( if_int
% 0.30/0.65 @ ( ( sgn_sgn_int @ K2 )
% 0.30/0.65 = ( sgn_sgn_int @ L2 ) )
% 0.30/0.65 @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) )
% 0.30/0.65 @ ( times_times_int @ ( sgn_sgn_int @ L2 )
% 0.30/0.65 @ ( minus_minus_int
% 0.30/0.65 @ ( times_times_int @ ( abs_abs_int @ L2 )
% 0.30/0.65 @ ( zero_n2684676970156552555ol_int
% 0.30/0.65 @ ~ ( dvd_dvd_int @ L2 @ K2 ) ) )
% 0.30/0.65 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % modulo_int_def
% 0.30/0.65 thf(fact_9214_nat__less__eq__zless,axiom,
% 0.30/0.65 ! [W: int,Z: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.30/0.65 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.30/0.65 = ( ord_less_int @ W @ Z ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_less_eq_zless
% 0.30/0.65 thf(fact_9215_nat__le__eq__zle,axiom,
% 0.30/0.65 ! [W: int,Z: int] :
% 0.30/0.65 ( ( ( ord_less_int @ zero_zero_int @ W )
% 0.30/0.65 | ( ord_less_eq_int @ zero_zero_int @ Z ) )
% 0.30/0.65 => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.30/0.65 = ( ord_less_eq_int @ W @ Z ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_le_eq_zle
% 0.30/0.65 thf(fact_9216_nat__eq__iff,axiom,
% 0.30/0.65 ! [W: int,M: nat] :
% 0.30/0.65 ( ( ( nat2 @ W )
% 0.30/0.65 = M )
% 0.30/0.65 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.30/0.65 => ( W
% 0.30/0.65 = ( semiri1314217659103216013at_int @ M ) ) )
% 0.30/0.65 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 0.30/0.65 => ( M = zero_zero_nat ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_eq_iff
% 0.30/0.65 thf(fact_9217_nat__eq__iff2,axiom,
% 0.30/0.65 ! [M: nat,W: int] :
% 0.30/0.65 ( ( M
% 0.30/0.65 = ( nat2 @ W ) )
% 0.30/0.65 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.30/0.65 => ( W
% 0.30/0.65 = ( semiri1314217659103216013at_int @ M ) ) )
% 0.30/0.65 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 0.30/0.65 => ( M = zero_zero_nat ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_eq_iff2
% 0.30/0.65 thf(fact_9218_le__nat__iff,axiom,
% 0.30/0.65 ! [K: int,N: nat] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.65 => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 0.30/0.65 = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % le_nat_iff
% 0.30/0.65 thf(fact_9219_nat__add__distrib,axiom,
% 0.30/0.65 ! [Z: int,Z7: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 0.30/0.65 => ( ( nat2 @ ( plus_plus_int @ Z @ Z7 ) )
% 0.30/0.65 = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_add_distrib
% 0.30/0.65 thf(fact_9220_div__sgn__abs__cancel,axiom,
% 0.30/0.65 ! [V: int,K: int,L: int] :
% 0.30/0.65 ( ( V != zero_zero_int )
% 0.30/0.65 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
% 0.30/0.65 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % div_sgn_abs_cancel
% 0.30/0.65 thf(fact_9221_Suc__as__int,axiom,
% 0.30/0.65 ( suc
% 0.30/0.65 = ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Suc_as_int
% 0.30/0.65 thf(fact_9222_nat__mult__distrib,axiom,
% 0.30/0.65 ! [Z: int,Z7: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.65 => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 0.30/0.65 = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_mult_distrib
% 0.30/0.65 thf(fact_9223_nat__diff__distrib_H,axiom,
% 0.30/0.65 ! [X2: int,Y3: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.65 => ( ( nat2 @ ( minus_minus_int @ X2 @ Y3 ) )
% 0.30/0.65 = ( minus_minus_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_diff_distrib'
% 0.30/0.65 thf(fact_9224_nat__diff__distrib,axiom,
% 0.30/0.65 ! [Z7: int,Z: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 0.30/0.65 => ( ( ord_less_eq_int @ Z7 @ Z )
% 0.30/0.65 => ( ( nat2 @ ( minus_minus_int @ Z @ Z7 ) )
% 0.30/0.65 = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_diff_distrib
% 0.30/0.65 thf(fact_9225_nat__abs__triangle__ineq,axiom,
% 0.30/0.65 ! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_abs_triangle_ineq
% 0.30/0.65 thf(fact_9226_nat__div__distrib_H,axiom,
% 0.30/0.65 ! [Y3: int,X2: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.65 => ( ( nat2 @ ( divide_divide_int @ X2 @ Y3 ) )
% 0.30/0.65 = ( divide_divide_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_div_distrib'
% 0.30/0.65 thf(fact_9227_nat__div__distrib,axiom,
% 0.30/0.65 ! [X2: int,Y3: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.65 => ( ( nat2 @ ( divide_divide_int @ X2 @ Y3 ) )
% 0.30/0.65 = ( divide_divide_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_div_distrib
% 0.30/0.65 thf(fact_9228_div__dvd__sgn__abs,axiom,
% 0.30/0.65 ! [L: int,K: int] :
% 0.30/0.65 ( ( dvd_dvd_int @ L @ K )
% 0.30/0.65 => ( ( divide_divide_int @ K @ L )
% 0.30/0.65 = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % div_dvd_sgn_abs
% 0.30/0.65 thf(fact_9229_nat__power__eq,axiom,
% 0.30/0.65 ! [Z: int,N: nat] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.65 => ( ( nat2 @ ( power_power_int @ Z @ N ) )
% 0.30/0.65 = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_power_eq
% 0.30/0.65 thf(fact_9230_nat__floor__neg,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.30/0.65 => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
% 0.30/0.65 = zero_zero_nat ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_floor_neg
% 0.30/0.65 thf(fact_9231_nat__mod__distrib,axiom,
% 0.30/0.65 ! [X2: int,Y3: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.65 => ( ( nat2 @ ( modulo_modulo_int @ X2 @ Y3 ) )
% 0.30/0.65 = ( modulo_modulo_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_mod_distrib
% 0.30/0.65 thf(fact_9232_div__abs__eq__div__nat,axiom,
% 0.30/0.65 ! [K: int,L: int] :
% 0.30/0.65 ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
% 0.30/0.65 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % div_abs_eq_div_nat
% 0.30/0.65 thf(fact_9233_floor__eq3,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X2 )
% 0.30/0.65 => ( ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.30/0.65 => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
% 0.30/0.65 = N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_eq3
% 0.30/0.65 thf(fact_9234_le__nat__floor,axiom,
% 0.30/0.65 ! [X2: nat,A: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ A )
% 0.30/0.65 => ( ord_less_eq_nat @ X2 @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % le_nat_floor
% 0.30/0.65 thf(fact_9235_divide__int__def,axiom,
% 0.30/0.65 ( divide_divide_int
% 0.30/0.65 = ( ^ [K2: int,L2: int] :
% 0.30/0.65 ( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
% 0.30/0.65 @ ( if_int
% 0.30/0.65 @ ( ( sgn_sgn_int @ K2 )
% 0.30/0.65 = ( sgn_sgn_int @ L2 ) )
% 0.30/0.65 @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
% 0.30/0.65 @ ( uminus_uminus_int
% 0.30/0.65 @ ( semiri1314217659103216013at_int
% 0.30/0.65 @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
% 0.30/0.65 @ ( zero_n2687167440665602831ol_nat
% 0.30/0.65 @ ~ ( dvd_dvd_int @ L2 @ K2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % divide_int_def
% 0.30/0.65 thf(fact_9236_nat__2,axiom,
% 0.30/0.65 ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.65 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_2
% 0.30/0.65 thf(fact_9237_Suc__nat__eq__nat__zadd1,axiom,
% 0.30/0.65 ! [Z: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.65 => ( ( suc @ ( nat2 @ Z ) )
% 0.30/0.65 = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Suc_nat_eq_nat_zadd1
% 0.30/0.65 thf(fact_9238_nat__less__iff,axiom,
% 0.30/0.65 ! [W: int,M: nat] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.30/0.65 => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
% 0.30/0.65 = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_less_iff
% 0.30/0.65 thf(fact_9239_nat__mult__distrib__neg,axiom,
% 0.30/0.65 ! [Z: int,Z7: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 0.30/0.65 => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 0.30/0.65 = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_mult_distrib_neg
% 0.30/0.65 thf(fact_9240_nat__abs__int__diff,axiom,
% 0.30/0.65 ! [A: nat,B: nat] :
% 0.30/0.65 ( ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.65 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 0.30/0.65 = ( minus_minus_nat @ B @ A ) ) )
% 0.30/0.65 & ( ~ ( ord_less_eq_nat @ A @ B )
% 0.30/0.65 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 0.30/0.65 = ( minus_minus_nat @ A @ B ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_abs_int_diff
% 0.30/0.65 thf(fact_9241_floor__eq4,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X2 )
% 0.30/0.65 => ( ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.30/0.65 => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
% 0.30/0.65 = N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % floor_eq4
% 0.30/0.65 thf(fact_9242_diff__nat__eq__if,axiom,
% 0.30/0.65 ! [Z7: int,Z: int] :
% 0.30/0.65 ( ( ( ord_less_int @ Z7 @ zero_zero_int )
% 0.30/0.65 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
% 0.30/0.65 = ( nat2 @ Z ) ) )
% 0.30/0.65 & ( ~ ( ord_less_int @ Z7 @ zero_zero_int )
% 0.30/0.65 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
% 0.30/0.65 = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z7 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z7 ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % diff_nat_eq_if
% 0.30/0.65 thf(fact_9243_Suc__mask__eq__exp,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.30/0.65 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % Suc_mask_eq_exp
% 0.30/0.65 thf(fact_9244_mask__nat__less__exp,axiom,
% 0.30/0.65 ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % mask_nat_less_exp
% 0.30/0.65 thf(fact_9245_div__noneq__sgn__abs,axiom,
% 0.30/0.65 ! [L: int,K: int] :
% 0.30/0.65 ( ( L != zero_zero_int )
% 0.30/0.65 => ( ( ( sgn_sgn_int @ K )
% 0.30/0.65 != ( sgn_sgn_int @ L ) )
% 0.30/0.65 => ( ( divide_divide_int @ K @ L )
% 0.30/0.65 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) )
% 0.30/0.65 @ ( zero_n2684676970156552555ol_int
% 0.30/0.65 @ ~ ( dvd_dvd_int @ L @ K ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % div_noneq_sgn_abs
% 0.30/0.65 thf(fact_9246_nat__dvd__iff,axiom,
% 0.30/0.65 ! [Z: int,M: nat] :
% 0.30/0.65 ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
% 0.30/0.65 = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.65 => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
% 0.30/0.65 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.65 => ( M = zero_zero_nat ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_dvd_iff
% 0.30/0.65 thf(fact_9247_eucl__rel__int__remainderI,axiom,
% 0.30/0.65 ! [R2: int,L: int,K: int,Q3: int] :
% 0.30/0.65 ( ( ( sgn_sgn_int @ R2 )
% 0.30/0.65 = ( sgn_sgn_int @ L ) )
% 0.30/0.65 => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
% 0.30/0.65 => ( ( K
% 0.30/0.65 = ( plus_plus_int @ ( times_times_int @ Q3 @ L ) @ R2 ) )
% 0.30/0.65 => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % eucl_rel_int_remainderI
% 0.30/0.65 thf(fact_9248_mask__nat__def,axiom,
% 0.30/0.65 ( bit_se2002935070580805687sk_nat
% 0.30/0.65 = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % mask_nat_def
% 0.30/0.65 thf(fact_9249_mask__half__int,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.65 = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % mask_half_int
% 0.30/0.65 thf(fact_9250_mask__int__def,axiom,
% 0.30/0.65 ( bit_se2000444600071755411sk_int
% 0.30/0.65 = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % mask_int_def
% 0.30/0.65 thf(fact_9251_eucl__rel__int_Ocases,axiom,
% 0.30/0.65 ! [A12: int,A23: int,A32: product_prod_int_int] :
% 0.30/0.65 ( ( eucl_rel_int @ A12 @ A23 @ A32 )
% 0.30/0.65 => ( ( ( A23 = zero_zero_int )
% 0.30/0.65 => ( A32
% 0.30/0.65 != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
% 0.30/0.65 => ( ! [Q2: int] :
% 0.30/0.65 ( ( A32
% 0.30/0.65 = ( product_Pair_int_int @ Q2 @ zero_zero_int ) )
% 0.30/0.65 => ( ( A23 != zero_zero_int )
% 0.30/0.65 => ( A12
% 0.30/0.65 != ( times_times_int @ Q2 @ A23 ) ) ) )
% 0.30/0.65 => ~ ! [R3: int,Q2: int] :
% 0.30/0.65 ( ( A32
% 0.30/0.65 = ( product_Pair_int_int @ Q2 @ R3 ) )
% 0.30/0.65 => ( ( ( sgn_sgn_int @ R3 )
% 0.30/0.65 = ( sgn_sgn_int @ A23 ) )
% 0.30/0.65 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A23 ) )
% 0.30/0.65 => ( A12
% 0.30/0.65 != ( plus_plus_int @ ( times_times_int @ Q2 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % eucl_rel_int.cases
% 0.30/0.65 thf(fact_9252_eucl__rel__int_Osimps,axiom,
% 0.30/0.65 ( eucl_rel_int
% 0.30/0.65 = ( ^ [A1: int,A22: int,A33: product_prod_int_int] :
% 0.30/0.65 ( ? [K2: int] :
% 0.30/0.65 ( ( A1 = K2 )
% 0.30/0.65 & ( A22 = zero_zero_int )
% 0.30/0.65 & ( A33
% 0.30/0.65 = ( product_Pair_int_int @ zero_zero_int @ K2 ) ) )
% 0.30/0.65 | ? [L2: int,K2: int,Q4: int] :
% 0.30/0.65 ( ( A1 = K2 )
% 0.30/0.65 & ( A22 = L2 )
% 0.30/0.65 & ( A33
% 0.30/0.65 = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
% 0.30/0.65 & ( L2 != zero_zero_int )
% 0.30/0.65 & ( K2
% 0.30/0.65 = ( times_times_int @ Q4 @ L2 ) ) )
% 0.30/0.65 | ? [R5: int,L2: int,K2: int,Q4: int] :
% 0.30/0.65 ( ( A1 = K2 )
% 0.30/0.65 & ( A22 = L2 )
% 0.30/0.65 & ( A33
% 0.30/0.65 = ( product_Pair_int_int @ Q4 @ R5 ) )
% 0.30/0.65 & ( ( sgn_sgn_int @ R5 )
% 0.30/0.65 = ( sgn_sgn_int @ L2 ) )
% 0.30/0.65 & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
% 0.30/0.65 & ( K2
% 0.30/0.65 = ( plus_plus_int @ ( times_times_int @ Q4 @ L2 ) @ R5 ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % eucl_rel_int.simps
% 0.30/0.65 thf(fact_9253_even__nat__iff,axiom,
% 0.30/0.65 ! [K: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.65 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
% 0.30/0.65 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % even_nat_iff
% 0.30/0.65 thf(fact_9254_set__encode__def,axiom,
% 0.30/0.65 ( nat_set_encode
% 0.30/0.65 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % set_encode_def
% 0.30/0.65 thf(fact_9255_powr__real__of__int,axiom,
% 0.30/0.65 ! [X2: real,N: int] :
% 0.30/0.65 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.65 => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
% 0.30/0.65 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N ) )
% 0.30/0.65 = ( power_power_real @ X2 @ ( nat2 @ N ) ) ) )
% 0.30/0.65 & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
% 0.30/0.65 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N ) )
% 0.30/0.65 = ( inverse_inverse_real @ ( power_power_real @ X2 @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % powr_real_of_int
% 0.30/0.65 thf(fact_9256_divide__int__unfold,axiom,
% 0.30/0.65 ! [L: int,K: int,N: nat,M: nat] :
% 0.30/0.65 ( ( ( ( ( sgn_sgn_int @ L )
% 0.30/0.65 = zero_zero_int )
% 0.30/0.65 | ( ( sgn_sgn_int @ K )
% 0.30/0.65 = zero_zero_int )
% 0.30/0.65 | ( N = zero_zero_nat ) )
% 0.30/0.65 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.30/0.65 = zero_zero_int ) )
% 0.30/0.65 & ( ~ ( ( ( sgn_sgn_int @ L )
% 0.30/0.65 = zero_zero_int )
% 0.30/0.65 | ( ( sgn_sgn_int @ K )
% 0.30/0.65 = zero_zero_int )
% 0.30/0.65 | ( N = zero_zero_nat ) )
% 0.30/0.65 => ( ( ( ( sgn_sgn_int @ K )
% 0.30/0.65 = ( sgn_sgn_int @ L ) )
% 0.30/0.65 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.30/0.65 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
% 0.30/0.65 & ( ( ( sgn_sgn_int @ K )
% 0.30/0.65 != ( sgn_sgn_int @ L ) )
% 0.30/0.65 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.30/0.65 = ( uminus_uminus_int
% 0.30/0.65 @ ( semiri1314217659103216013at_int
% 0.30/0.65 @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
% 0.30/0.65 @ ( zero_n2687167440665602831ol_nat
% 0.30/0.65 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % divide_int_unfold
% 0.30/0.65 thf(fact_9257_num_Osize__gen_I3_J,axiom,
% 0.30/0.65 ! [X32: num] :
% 0.30/0.65 ( ( size_num @ ( bit1 @ X32 ) )
% 0.30/0.65 = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % num.size_gen(3)
% 0.30/0.65 thf(fact_9258_num_Osize__gen_I2_J,axiom,
% 0.30/0.65 ! [X22: num] :
% 0.30/0.65 ( ( size_num @ ( bit0 @ X22 ) )
% 0.30/0.65 = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % num.size_gen(2)
% 0.30/0.65 thf(fact_9259_zero__le__sgn__iff,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X2 ) )
% 0.30/0.65 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % zero_le_sgn_iff
% 0.30/0.65 thf(fact_9260_sgn__le__0__iff,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ord_less_eq_real @ ( sgn_sgn_real @ X2 ) @ zero_zero_real )
% 0.30/0.65 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 0.30/0.65
% 0.30/0.65 % sgn_le_0_iff
% 0.30/0.65 thf(fact_9261_concat__bit__of__zero__2,axiom,
% 0.30/0.65 ! [N: nat,K: int] :
% 0.30/0.65 ( ( bit_concat_bit @ N @ K @ zero_zero_int )
% 0.30/0.65 = ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 0.30/0.65
% 0.30/0.65 % concat_bit_of_zero_2
% 0.30/0.65 thf(fact_9262_take__bit__of__Suc__0,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.30/0.65 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_of_Suc_0
% 0.30/0.65 thf(fact_9263_nat__take__bit__eq,axiom,
% 0.30/0.65 ! [K: int,N: nat] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.65 => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.30/0.65 = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % nat_take_bit_eq
% 0.30/0.65 thf(fact_9264_take__bit__nat__eq,axiom,
% 0.30/0.65 ! [K: int,N: nat] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.65 => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
% 0.30/0.65 = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_nat_eq
% 0.30/0.65 thf(fact_9265_take__bit__mult,axiom,
% 0.30/0.65 ! [N: nat,K: int,L: int] :
% 0.30/0.65 ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
% 0.30/0.65 = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_mult
% 0.30/0.65 thf(fact_9266_take__bit__diff,axiom,
% 0.30/0.65 ! [N: nat,K: int,L: int] :
% 0.30/0.65 ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
% 0.30/0.65 = ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_diff
% 0.30/0.65 thf(fact_9267_take__bit__minus,axiom,
% 0.30/0.65 ! [N: nat,K: int] :
% 0.30/0.65 ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.30/0.65 = ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_minus
% 0.30/0.65 thf(fact_9268_take__bit__nat__less__eq__self,axiom,
% 0.30/0.65 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_nat_less_eq_self
% 0.30/0.65 thf(fact_9269_take__bit__tightened__less__eq__nat,axiom,
% 0.30/0.65 ! [M: nat,N: nat,Q3: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65 => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q3 ) @ ( bit_se2925701944663578781it_nat @ N @ Q3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_tightened_less_eq_nat
% 0.30/0.65 thf(fact_9270_concat__bit__eq__iff,axiom,
% 0.30/0.65 ! [N: nat,K: int,L: int,R2: int,S: int] :
% 0.30/0.65 ( ( ( bit_concat_bit @ N @ K @ L )
% 0.30/0.65 = ( bit_concat_bit @ N @ R2 @ S ) )
% 0.30/0.65 = ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.30/0.65 = ( bit_se2923211474154528505it_int @ N @ R2 ) )
% 0.30/0.65 & ( L = S ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % concat_bit_eq_iff
% 0.30/0.65 thf(fact_9271_concat__bit__take__bit__eq,axiom,
% 0.30/0.65 ! [N: nat,B: int] :
% 0.30/0.65 ( ( bit_concat_bit @ N @ ( bit_se2923211474154528505it_int @ N @ B ) )
% 0.30/0.65 = ( bit_concat_bit @ N @ B ) ) ).
% 0.30/0.65
% 0.30/0.65 % concat_bit_take_bit_eq
% 0.30/0.65 thf(fact_9272_take__bit__tightened__less__eq__int,axiom,
% 0.30/0.65 ! [M: nat,N: nat,K: int] :
% 0.30/0.65 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_tightened_less_eq_int
% 0.30/0.65 thf(fact_9273_take__bit__nonnegative,axiom,
% 0.30/0.65 ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_nonnegative
% 0.30/0.65 thf(fact_9274_take__bit__int__less__eq__self__iff,axiom,
% 0.30/0.65 ! [N: nat,K: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 0.30/0.65 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_int_less_eq_self_iff
% 0.30/0.65 thf(fact_9275_not__take__bit__negative,axiom,
% 0.30/0.65 ! [N: nat,K: int] :
% 0.30/0.65 ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% 0.30/0.65
% 0.30/0.65 % not_take_bit_negative
% 0.30/0.65 thf(fact_9276_take__bit__int__greater__self__iff,axiom,
% 0.30/0.65 ! [K: int,N: nat] :
% 0.30/0.65 ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.30/0.65 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_int_greater_self_iff
% 0.30/0.65 thf(fact_9277_sgn__root,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( sgn_sgn_real @ ( root @ N @ X2 ) )
% 0.30/0.65 = ( sgn_sgn_real @ X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % sgn_root
% 0.30/0.65 thf(fact_9278_take__bit__eq__mask__iff,axiom,
% 0.30/0.65 ! [N: nat,K: int] :
% 0.30/0.65 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.30/0.65 = ( bit_se2000444600071755411sk_int @ N ) )
% 0.30/0.65 = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 0.30/0.65 = zero_zero_int ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_eq_mask_iff
% 0.30/0.65 thf(fact_9279_take__bit__decr__eq,axiom,
% 0.30/0.65 ! [N: nat,K: int] :
% 0.30/0.65 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.30/0.65 != zero_zero_int )
% 0.30/0.65 => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
% 0.30/0.65 = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_decr_eq
% 0.30/0.65 thf(fact_9280_sgn__real__def,axiom,
% 0.30/0.65 ( sgn_sgn_real
% 0.30/0.65 = ( ^ [A3: real] : ( if_real @ ( A3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % sgn_real_def
% 0.30/0.65 thf(fact_9281_take__bit__nat__eq__self,axiom,
% 0.30/0.65 ! [M: nat,N: nat] :
% 0.30/0.65 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.65 => ( ( bit_se2925701944663578781it_nat @ N @ M )
% 0.30/0.65 = M ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_nat_eq_self
% 0.30/0.65 thf(fact_9282_take__bit__nat__less__exp,axiom,
% 0.30/0.65 ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_nat_less_exp
% 0.30/0.65 thf(fact_9283_take__bit__nat__eq__self__iff,axiom,
% 0.30/0.65 ! [N: nat,M: nat] :
% 0.30/0.65 ( ( ( bit_se2925701944663578781it_nat @ N @ M )
% 0.30/0.65 = M )
% 0.30/0.65 = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_nat_eq_self_iff
% 0.30/0.65 thf(fact_9284_take__bit__nat__def,axiom,
% 0.30/0.65 ( bit_se2925701944663578781it_nat
% 0.30/0.65 = ( ^ [N3: nat,M3: nat] : ( modulo_modulo_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_nat_def
% 0.30/0.65 thf(fact_9285_sgn__power__injE,axiom,
% 0.30/0.65 ! [A: real,N: nat,X2: real,B: real] :
% 0.30/0.65 ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 0.30/0.65 = X2 )
% 0.30/0.65 => ( ( X2
% 0.30/0.65 = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
% 0.30/0.65 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( A = B ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % sgn_power_injE
% 0.30/0.65 thf(fact_9286_take__bit__int__less__exp,axiom,
% 0.30/0.65 ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_int_less_exp
% 0.30/0.65 thf(fact_9287_take__bit__int__def,axiom,
% 0.30/0.65 ( bit_se2923211474154528505it_int
% 0.30/0.65 = ( ^ [N3: nat,K2: int] : ( modulo_modulo_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_int_def
% 0.30/0.65 thf(fact_9288_num_Osize__gen_I1_J,axiom,
% 0.30/0.65 ( ( size_num @ one )
% 0.30/0.65 = zero_zero_nat ) ).
% 0.30/0.65
% 0.30/0.65 % num.size_gen(1)
% 0.30/0.65 thf(fact_9289_take__bit__nat__less__self__iff,axiom,
% 0.30/0.65 ! [N: nat,M: nat] :
% 0.30/0.65 ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
% 0.30/0.65 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_nat_less_self_iff
% 0.30/0.65 thf(fact_9290_take__bit__Suc__minus__bit0,axiom,
% 0.30/0.65 ! [N: nat,K: num] :
% 0.30/0.65 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.30/0.65 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_Suc_minus_bit0
% 0.30/0.65 thf(fact_9291_sgn__power__root,axiom,
% 0.30/0.65 ! [N: nat,X2: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X2 ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X2 ) ) @ N ) )
% 0.30/0.65 = X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % sgn_power_root
% 0.30/0.65 thf(fact_9292_root__sgn__power,axiom,
% 0.30/0.65 ! [N: nat,Y3: real] :
% 0.30/0.65 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) ) )
% 0.30/0.65 = Y3 ) ) ).
% 0.30/0.65
% 0.30/0.65 % root_sgn_power
% 0.30/0.65 thf(fact_9293_take__bit__int__greater__eq__self__iff,axiom,
% 0.30/0.65 ! [K: int,N: nat] :
% 0.30/0.65 ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.30/0.65 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_int_greater_eq_self_iff
% 0.30/0.65 thf(fact_9294_take__bit__int__less__self__iff,axiom,
% 0.30/0.65 ! [N: nat,K: int] :
% 0.30/0.65 ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 0.30/0.65 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_int_less_self_iff
% 0.30/0.65 thf(fact_9295_cis__Arg__unique,axiom,
% 0.30/0.65 ! [Z: complex,X2: real] :
% 0.30/0.65 ( ( ( sgn_sgn_complex @ Z )
% 0.30/0.65 = ( cis @ X2 ) )
% 0.30/0.65 => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X2 )
% 0.30/0.65 => ( ( ord_less_eq_real @ X2 @ pi )
% 0.30/0.65 => ( ( arg @ Z )
% 0.30/0.65 = X2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % cis_Arg_unique
% 0.30/0.65 thf(fact_9296_split__root,axiom,
% 0.30/0.65 ! [P: real > $o,N: nat,X2: real] :
% 0.30/0.65 ( ( P @ ( root @ N @ X2 ) )
% 0.30/0.65 = ( ( ( N = zero_zero_nat )
% 0.30/0.65 => ( P @ zero_zero_real ) )
% 0.30/0.65 & ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ! [Y: real] :
% 0.30/0.65 ( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 0.30/0.65 = X2 )
% 0.30/0.65 => ( P @ Y ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % split_root
% 0.30/0.65 thf(fact_9297_take__bit__int__eq__self,axiom,
% 0.30/0.65 ! [K: int,N: nat] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.65 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.65 => ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.30/0.65 = K ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_int_eq_self
% 0.30/0.65 thf(fact_9298_take__bit__int__eq__self__iff,axiom,
% 0.30/0.65 ! [N: nat,K: int] :
% 0.30/0.65 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.30/0.65 = K )
% 0.30/0.65 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.65 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_int_eq_self_iff
% 0.30/0.65 thf(fact_9299_take__bit__numeral__minus__bit0,axiom,
% 0.30/0.65 ! [L: num,K: num] :
% 0.30/0.65 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.30/0.65 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_numeral_minus_bit0
% 0.30/0.65 thf(fact_9300_take__bit__incr__eq,axiom,
% 0.30/0.65 ! [N: nat,K: int] :
% 0.30/0.65 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.30/0.65 != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 0.30/0.65 => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 0.30/0.65 = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_incr_eq
% 0.30/0.65 thf(fact_9301_take__bit__eq__mask__iff__exp__dvd,axiom,
% 0.30/0.65 ! [N: nat,K: int] :
% 0.30/0.65 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.30/0.65 = ( bit_se2000444600071755411sk_int @ N ) )
% 0.30/0.65 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_eq_mask_iff_exp_dvd
% 0.30/0.65 thf(fact_9302_Arg__correct,axiom,
% 0.30/0.65 ! [Z: complex] :
% 0.30/0.65 ( ( Z != zero_zero_complex )
% 0.30/0.65 => ( ( ( sgn_sgn_complex @ Z )
% 0.30/0.65 = ( cis @ ( arg @ Z ) ) )
% 0.30/0.65 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 0.30/0.65 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Arg_correct
% 0.30/0.65 thf(fact_9303_take__bit__int__less__eq,axiom,
% 0.30/0.65 ! [N: nat,K: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 0.30/0.65 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_int_less_eq
% 0.30/0.65 thf(fact_9304_take__bit__int__greater__eq,axiom,
% 0.30/0.65 ! [K: int,N: nat] :
% 0.30/0.65 ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.65 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_int_greater_eq
% 0.30/0.65 thf(fact_9305_signed__take__bit__eq__take__bit__shift,axiom,
% 0.30/0.65 ( bit_ri631733984087533419it_int
% 0.30/0.65 = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % signed_take_bit_eq_take_bit_shift
% 0.30/0.65 thf(fact_9306_take__bit__minus__small__eq,axiom,
% 0.30/0.65 ! [K: int,N: nat] :
% 0.30/0.65 ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.65 => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.65 => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
% 0.30/0.65 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_minus_small_eq
% 0.30/0.65 thf(fact_9307_arctan__inverse,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( X2 != zero_zero_real )
% 0.30/0.65 => ( ( arctan @ ( divide_divide_real @ one_one_real @ X2 ) )
% 0.30/0.65 = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X2 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % arctan_inverse
% 0.30/0.65 thf(fact_9308_take__bit__numeral__minus__bit1,axiom,
% 0.30/0.65 ! [L: num,K: num] :
% 0.30/0.65 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.30/0.65 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_numeral_minus_bit1
% 0.30/0.65 thf(fact_9309_take__bit__Suc__minus__bit1,axiom,
% 0.30/0.65 ! [N: nat,K: num] :
% 0.30/0.65 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.30/0.65 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_Suc_minus_bit1
% 0.30/0.65 thf(fact_9310_and__int__unfold,axiom,
% 0.30/0.65 ( bit_se725231765392027082nd_int
% 0.30/0.65 = ( ^ [K2: int,L2: int] :
% 0.30/0.65 ( if_int
% 0.30/0.65 @ ( ( K2 = zero_zero_int )
% 0.30/0.65 | ( L2 = zero_zero_int ) )
% 0.30/0.65 @ zero_zero_int
% 0.30/0.65 @ ( if_int
% 0.30/0.65 @ ( K2
% 0.30/0.65 = ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.65 @ L2
% 0.30/0.65 @ ( if_int
% 0.30/0.65 @ ( L2
% 0.30/0.65 = ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.65 @ K2
% 0.30/0.65 @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_int_unfold
% 0.30/0.65 thf(fact_9311_and__nonnegative__int__iff,axiom,
% 0.30/0.65 ! [K: int,L: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 0.30/0.65 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.65 | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_nonnegative_int_iff
% 0.30/0.65 thf(fact_9312_and__negative__int__iff,axiom,
% 0.30/0.65 ! [K: int,L: int] :
% 0.30/0.65 ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
% 0.30/0.65 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.65 & ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_negative_int_iff
% 0.30/0.65 thf(fact_9313_pred__numeral__inc,axiom,
% 0.30/0.65 ! [K: num] :
% 0.30/0.65 ( ( pred_numeral @ ( inc @ K ) )
% 0.30/0.65 = ( numeral_numeral_nat @ K ) ) ).
% 0.30/0.65
% 0.30/0.65 % pred_numeral_inc
% 0.30/0.65 thf(fact_9314_and__minus__numerals_I6_J,axiom,
% 0.30/0.65 ! [N: num] :
% 0.30/0.65 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 0.30/0.65 = one_one_int ) ).
% 0.30/0.65
% 0.30/0.65 % and_minus_numerals(6)
% 0.30/0.65 thf(fact_9315_and__minus__numerals_I2_J,axiom,
% 0.30/0.65 ! [N: num] :
% 0.30/0.65 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.65 = one_one_int ) ).
% 0.30/0.65
% 0.30/0.65 % and_minus_numerals(2)
% 0.30/0.65 thf(fact_9316_and__minus__numerals_I5_J,axiom,
% 0.30/0.65 ! [N: num] :
% 0.30/0.65 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 0.30/0.65 = zero_zero_int ) ).
% 0.30/0.65
% 0.30/0.65 % and_minus_numerals(5)
% 0.30/0.65 thf(fact_9317_and__minus__numerals_I1_J,axiom,
% 0.30/0.65 ! [N: num] :
% 0.30/0.65 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.65 = zero_zero_int ) ).
% 0.30/0.65
% 0.30/0.65 % and_minus_numerals(1)
% 0.30/0.65 thf(fact_9318_num__induct,axiom,
% 0.30/0.65 ! [P: num > $o,X2: num] :
% 0.30/0.65 ( ( P @ one )
% 0.30/0.65 => ( ! [X5: num] :
% 0.30/0.65 ( ( P @ X5 )
% 0.30/0.65 => ( P @ ( inc @ X5 ) ) )
% 0.30/0.65 => ( P @ X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % num_induct
% 0.30/0.65 thf(fact_9319_add__inc,axiom,
% 0.30/0.65 ! [X2: num,Y3: num] :
% 0.30/0.65 ( ( plus_plus_num @ X2 @ ( inc @ Y3 ) )
% 0.30/0.65 = ( inc @ ( plus_plus_num @ X2 @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % add_inc
% 0.30/0.65 thf(fact_9320_AND__upper2_H,axiom,
% 0.30/0.65 ! [Y3: int,Z: int,X2: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.65 => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.30/0.65 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ Z ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % AND_upper2'
% 0.30/0.65 thf(fact_9321_AND__upper1_H,axiom,
% 0.30/0.65 ! [Y3: int,Z: int,Ya: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.65 => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.30/0.65 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % AND_upper1'
% 0.30/0.65 thf(fact_9322_AND__upper2,axiom,
% 0.30/0.65 ! [Y3: int,X2: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.65 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ Y3 ) ) ).
% 0.30/0.65
% 0.30/0.65 % AND_upper2
% 0.30/0.65 thf(fact_9323_AND__upper1,axiom,
% 0.30/0.65 ! [X2: int,Y3: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.65 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % AND_upper1
% 0.30/0.65 thf(fact_9324_AND__lower,axiom,
% 0.30/0.65 ! [X2: int,Y3: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.65 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % AND_lower
% 0.30/0.65 thf(fact_9325_pow_Osimps_I1_J,axiom,
% 0.30/0.65 ! [X2: num] :
% 0.30/0.65 ( ( pow @ X2 @ one )
% 0.30/0.65 = X2 ) ).
% 0.30/0.65
% 0.30/0.65 % pow.simps(1)
% 0.30/0.65 thf(fact_9326_inc_Osimps_I1_J,axiom,
% 0.30/0.65 ( ( inc @ one )
% 0.30/0.65 = ( bit0 @ one ) ) ).
% 0.30/0.65
% 0.30/0.65 % inc.simps(1)
% 0.30/0.65 thf(fact_9327_inc_Osimps_I2_J,axiom,
% 0.30/0.65 ! [X2: num] :
% 0.30/0.65 ( ( inc @ ( bit0 @ X2 ) )
% 0.30/0.65 = ( bit1 @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % inc.simps(2)
% 0.30/0.65 thf(fact_9328_inc_Osimps_I3_J,axiom,
% 0.30/0.65 ! [X2: num] :
% 0.30/0.65 ( ( inc @ ( bit1 @ X2 ) )
% 0.30/0.65 = ( bit0 @ ( inc @ X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % inc.simps(3)
% 0.30/0.65 thf(fact_9329_add__One,axiom,
% 0.30/0.65 ! [X2: num] :
% 0.30/0.65 ( ( plus_plus_num @ X2 @ one )
% 0.30/0.65 = ( inc @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % add_One
% 0.30/0.65 thf(fact_9330_and__less__eq,axiom,
% 0.30/0.65 ! [L: int,K: int] :
% 0.30/0.65 ( ( ord_less_int @ L @ zero_zero_int )
% 0.30/0.65 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_less_eq
% 0.30/0.65 thf(fact_9331_AND__upper1_H_H,axiom,
% 0.30/0.65 ! [Y3: int,Z: int,Ya: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.65 => ( ( ord_less_int @ Y3 @ Z )
% 0.30/0.65 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % AND_upper1''
% 0.30/0.65 thf(fact_9332_AND__upper2_H_H,axiom,
% 0.30/0.65 ! [Y3: int,Z: int,X2: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.65 => ( ( ord_less_int @ Y3 @ Z )
% 0.30/0.65 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ Z ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % AND_upper2''
% 0.30/0.65 thf(fact_9333_inc__BitM__eq,axiom,
% 0.30/0.65 ! [N: num] :
% 0.30/0.65 ( ( inc @ ( bitM @ N ) )
% 0.30/0.65 = ( bit0 @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % inc_BitM_eq
% 0.30/0.65 thf(fact_9334_BitM__inc__eq,axiom,
% 0.30/0.65 ! [N: num] :
% 0.30/0.65 ( ( bitM @ ( inc @ N ) )
% 0.30/0.65 = ( bit1 @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % BitM_inc_eq
% 0.30/0.65 thf(fact_9335_mult__inc,axiom,
% 0.30/0.65 ! [X2: num,Y3: num] :
% 0.30/0.65 ( ( times_times_num @ X2 @ ( inc @ Y3 ) )
% 0.30/0.65 = ( plus_plus_num @ ( times_times_num @ X2 @ Y3 ) @ X2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % mult_inc
% 0.30/0.65 thf(fact_9336_even__and__iff__int,axiom,
% 0.30/0.65 ! [K: int,L: int] :
% 0.30/0.65 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 0.30/0.65 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.30/0.65 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % even_and_iff_int
% 0.30/0.65 thf(fact_9337_and__int__rec,axiom,
% 0.30/0.65 ( bit_se725231765392027082nd_int
% 0.30/0.65 = ( ^ [K2: int,L2: int] :
% 0.30/0.65 ( plus_plus_int
% 0.30/0.65 @ ( zero_n2684676970156552555ol_int
% 0.30/0.65 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.30/0.65 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 0.30/0.65 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_int_rec
% 0.30/0.65 thf(fact_9338_and__int_Oelims,axiom,
% 0.30/0.65 ! [X2: int,Xa2: int,Y3: int] :
% 0.30/0.65 ( ( ( bit_se725231765392027082nd_int @ X2 @ Xa2 )
% 0.30/0.65 = Y3 )
% 0.30/0.65 => ( ( ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.65 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.65 => ( Y3
% 0.30/0.65 = ( uminus_uminus_int
% 0.30/0.65 @ ( zero_n2684676970156552555ol_int
% 0.30/0.65 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
% 0.30/0.65 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 0.30/0.65 & ( ~ ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.65 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.65 => ( Y3
% 0.30/0.65 = ( plus_plus_int
% 0.30/0.65 @ ( zero_n2684676970156552555ol_int
% 0.30/0.65 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
% 0.30/0.65 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 0.30/0.65 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_int.elims
% 0.30/0.65 thf(fact_9339_and__int_Osimps,axiom,
% 0.30/0.65 ( bit_se725231765392027082nd_int
% 0.30/0.65 = ( ^ [K2: int,L2: int] :
% 0.30/0.65 ( if_int
% 0.30/0.65 @ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.65 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.65 @ ( uminus_uminus_int
% 0.30/0.65 @ ( zero_n2684676970156552555ol_int
% 0.30/0.65 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.30/0.65 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 0.30/0.65 @ ( plus_plus_int
% 0.30/0.65 @ ( zero_n2684676970156552555ol_int
% 0.30/0.65 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.30/0.65 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 0.30/0.65 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_int.simps
% 0.30/0.65 thf(fact_9340_Arg__def,axiom,
% 0.30/0.65 ( arg
% 0.30/0.65 = ( ^ [Z3: complex] :
% 0.30/0.65 ( if_real @ ( Z3 = zero_zero_complex ) @ zero_zero_real
% 0.30/0.65 @ ( fChoice_real
% 0.30/0.65 @ ^ [A3: real] :
% 0.30/0.65 ( ( ( sgn_sgn_complex @ Z3 )
% 0.30/0.65 = ( cis @ A3 ) )
% 0.30/0.65 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A3 )
% 0.30/0.65 & ( ord_less_eq_real @ A3 @ pi ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Arg_def
% 0.30/0.65 thf(fact_9341_and__int_Opelims,axiom,
% 0.30/0.65 ! [X2: int,Xa2: int,Y3: int] :
% 0.30/0.65 ( ( ( bit_se725231765392027082nd_int @ X2 @ Xa2 )
% 0.30/0.65 = Y3 )
% 0.30/0.65 => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) )
% 0.30/0.65 => ~ ( ( ( ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.65 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.65 => ( Y3
% 0.30/0.65 = ( uminus_uminus_int
% 0.30/0.65 @ ( zero_n2684676970156552555ol_int
% 0.30/0.65 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
% 0.30/0.65 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 0.30/0.65 & ( ~ ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.65 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.65 => ( Y3
% 0.30/0.65 = ( plus_plus_int
% 0.30/0.65 @ ( zero_n2684676970156552555ol_int
% 0.30/0.65 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
% 0.30/0.65 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 0.30/0.65 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.30/0.65 => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_int.pelims
% 0.30/0.65 thf(fact_9342_and__nat__numerals_I3_J,axiom,
% 0.30/0.65 ! [X2: num] :
% 0.30/0.65 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.65 = zero_zero_nat ) ).
% 0.30/0.65
% 0.30/0.65 % and_nat_numerals(3)
% 0.30/0.65 thf(fact_9343_and__nat__numerals_I1_J,axiom,
% 0.30/0.65 ! [Y3: num] :
% 0.30/0.65 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.30/0.65 = zero_zero_nat ) ).
% 0.30/0.65
% 0.30/0.65 % and_nat_numerals(1)
% 0.30/0.65 thf(fact_9344_and__nat__numerals_I2_J,axiom,
% 0.30/0.65 ! [Y3: num] :
% 0.30/0.65 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.30/0.65 = one_one_nat ) ).
% 0.30/0.65
% 0.30/0.65 % and_nat_numerals(2)
% 0.30/0.65 thf(fact_9345_and__nat__numerals_I4_J,axiom,
% 0.30/0.65 ! [X2: num] :
% 0.30/0.65 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.65 = one_one_nat ) ).
% 0.30/0.65
% 0.30/0.65 % and_nat_numerals(4)
% 0.30/0.65 thf(fact_9346_Suc__0__and__eq,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.65 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Suc_0_and_eq
% 0.30/0.65 thf(fact_9347_and__Suc__0__eq,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.30/0.65 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_Suc_0_eq
% 0.30/0.65 thf(fact_9348_and__nat__def,axiom,
% 0.30/0.65 ( bit_se727722235901077358nd_nat
% 0.30/0.65 = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_nat_def
% 0.30/0.65 thf(fact_9349_atLeastAtMostPlus1__int__conv,axiom,
% 0.30/0.65 ! [M: int,N: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 0.30/0.65 => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 0.30/0.65 = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % atLeastAtMostPlus1_int_conv
% 0.30/0.65 thf(fact_9350_simp__from__to,axiom,
% 0.30/0.65 ( set_or1266510415728281911st_int
% 0.30/0.65 = ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % simp_from_to
% 0.30/0.65 thf(fact_9351_and__int_Opinduct,axiom,
% 0.30/0.65 ! [A0: int,A12: int,P: int > int > $o] :
% 0.30/0.65 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 0.30/0.65 => ( ! [K3: int,L4: int] :
% 0.30/0.65 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L4 ) )
% 0.30/0.65 => ( ( ~ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.65 & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.65 => ( P @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65 => ( P @ K3 @ L4 ) ) )
% 0.30/0.65 => ( P @ A0 @ A12 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_int.pinduct
% 0.30/0.65 thf(fact_9352_and__nat__unfold,axiom,
% 0.30/0.65 ( bit_se727722235901077358nd_nat
% 0.30/0.65 = ( ^ [M3: nat,N3: nat] :
% 0.30/0.65 ( if_nat
% 0.30/0.65 @ ( ( M3 = zero_zero_nat )
% 0.30/0.65 | ( N3 = zero_zero_nat ) )
% 0.30/0.65 @ zero_zero_nat
% 0.30/0.65 @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_nat_unfold
% 0.30/0.65 thf(fact_9353_and__nat__rec,axiom,
% 0.30/0.65 ( bit_se727722235901077358nd_nat
% 0.30/0.65 = ( ^ [M3: nat,N3: nat] :
% 0.30/0.65 ( plus_plus_nat
% 0.30/0.65 @ ( zero_n2687167440665602831ol_nat
% 0.30/0.65 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 0.30/0.65 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 0.30/0.65 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_nat_rec
% 0.30/0.65 thf(fact_9354_and__int_Opsimps,axiom,
% 0.30/0.65 ! [K: int,L: int] :
% 0.30/0.65 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
% 0.30/0.65 => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.65 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.65 => ( ( bit_se725231765392027082nd_int @ K @ L )
% 0.30/0.65 = ( uminus_uminus_int
% 0.30/0.65 @ ( zero_n2684676970156552555ol_int
% 0.30/0.65 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.30/0.65 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
% 0.30/0.65 & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.65 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.65 => ( ( bit_se725231765392027082nd_int @ K @ L )
% 0.30/0.65 = ( plus_plus_int
% 0.30/0.65 @ ( zero_n2684676970156552555ol_int
% 0.30/0.65 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.30/0.65 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 0.30/0.65 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_int.psimps
% 0.30/0.65 thf(fact_9355_signed__take__bit__eq__take__bit__minus,axiom,
% 0.30/0.65 ( bit_ri631733984087533419it_int
% 0.30/0.65 = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ K2 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % signed_take_bit_eq_take_bit_minus
% 0.30/0.65 thf(fact_9356_or__int__unfold,axiom,
% 0.30/0.65 ( bit_se1409905431419307370or_int
% 0.30/0.65 = ( ^ [K2: int,L2: int] :
% 0.30/0.65 ( if_int
% 0.30/0.65 @ ( ( K2
% 0.30/0.65 = ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.65 | ( L2
% 0.30/0.65 = ( uminus_uminus_int @ one_one_int ) ) )
% 0.30/0.65 @ ( uminus_uminus_int @ one_one_int )
% 0.30/0.65 @ ( if_int @ ( K2 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_int_unfold
% 0.30/0.65 thf(fact_9357_cis__multiple__2pi,axiom,
% 0.30/0.65 ! [N: real] :
% 0.30/0.65 ( ( member_real @ N @ ring_1_Ints_real )
% 0.30/0.65 => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.30/0.65 = one_one_complex ) ) ).
% 0.30/0.65
% 0.30/0.65 % cis_multiple_2pi
% 0.30/0.65 thf(fact_9358_or__nonnegative__int__iff,axiom,
% 0.30/0.65 ! [K: int,L: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
% 0.30/0.65 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.65 & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_nonnegative_int_iff
% 0.30/0.65 thf(fact_9359_or__negative__int__iff,axiom,
% 0.30/0.65 ! [K: int,L: int] :
% 0.30/0.65 ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
% 0.30/0.65 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.65 | ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_negative_int_iff
% 0.30/0.65 thf(fact_9360_signed__take__bit__nonnegative__iff,axiom,
% 0.30/0.65 ! [N: nat,K: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.30/0.65 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % signed_take_bit_nonnegative_iff
% 0.30/0.65 thf(fact_9361_signed__take__bit__negative__iff,axiom,
% 0.30/0.65 ! [N: nat,K: int] :
% 0.30/0.65 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
% 0.30/0.65 = ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % signed_take_bit_negative_iff
% 0.30/0.65 thf(fact_9362_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 0.30/0.65 ! [W: num,N: nat] :
% 0.30/0.65 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
% 0.30/0.65 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % bit_minus_numeral_Bit0_Suc_iff
% 0.30/0.65 thf(fact_9363_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 0.30/0.65 ! [W: num,N: nat] :
% 0.30/0.65 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
% 0.30/0.65 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % bit_minus_numeral_Bit1_Suc_iff
% 0.30/0.65 thf(fact_9364_or__minus__numerals_I6_J,axiom,
% 0.30/0.65 ! [N: num] :
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 0.30/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_minus_numerals(6)
% 0.30/0.65 thf(fact_9365_or__minus__numerals_I2_J,axiom,
% 0.30/0.65 ! [N: num] :
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_minus_numerals(2)
% 0.30/0.65 thf(fact_9366_bit__minus__numeral__int_I1_J,axiom,
% 0.30/0.65 ! [W: num,N: num] :
% 0.30/0.65 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.65 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % bit_minus_numeral_int(1)
% 0.30/0.65 thf(fact_9367_bit__minus__numeral__int_I2_J,axiom,
% 0.30/0.65 ! [W: num,N: num] :
% 0.30/0.65 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.65 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % bit_minus_numeral_int(2)
% 0.30/0.65 thf(fact_9368_set__encode__insert,axiom,
% 0.30/0.65 ! [A2: set_nat,N: nat] :
% 0.30/0.65 ( ( finite_finite_nat @ A2 )
% 0.30/0.65 => ( ~ ( member_nat @ N @ A2 )
% 0.30/0.65 => ( ( nat_set_encode @ ( insert_nat @ N @ A2 ) )
% 0.30/0.65 = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % set_encode_insert
% 0.30/0.65 thf(fact_9369_bit__and__int__iff,axiom,
% 0.30/0.65 ! [K: int,L: int,N: nat] :
% 0.30/0.65 ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ N )
% 0.30/0.65 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.30/0.65 & ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % bit_and_int_iff
% 0.30/0.65 thf(fact_9370_bit__or__int__iff,axiom,
% 0.30/0.65 ! [K: int,L: int,N: nat] :
% 0.30/0.65 ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ N )
% 0.30/0.65 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.30/0.65 | ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % bit_or_int_iff
% 0.30/0.65 thf(fact_9371_or__greater__eq,axiom,
% 0.30/0.65 ! [L: int,K: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ L )
% 0.30/0.65 => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_greater_eq
% 0.30/0.65 thf(fact_9372_OR__lower,axiom,
% 0.30/0.65 ! [X2: int,Y3: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.65 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % OR_lower
% 0.30/0.65 thf(fact_9373_plus__and__or,axiom,
% 0.30/0.65 ! [X2: int,Y3: int] :
% 0.30/0.65 ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ ( bit_se1409905431419307370or_int @ X2 @ Y3 ) )
% 0.30/0.65 = ( plus_plus_int @ X2 @ Y3 ) ) ).
% 0.30/0.65
% 0.30/0.65 % plus_and_or
% 0.30/0.65 thf(fact_9374_lessThan__Suc,axiom,
% 0.30/0.65 ! [K: nat] :
% 0.30/0.65 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 0.30/0.65 = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % lessThan_Suc
% 0.30/0.65 thf(fact_9375_atMost__Suc,axiom,
% 0.30/0.65 ! [K: nat] :
% 0.30/0.65 ( ( set_ord_atMost_nat @ ( suc @ K ) )
% 0.30/0.65 = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % atMost_Suc
% 0.30/0.65 thf(fact_9376_bit__not__int__iff_H,axiom,
% 0.30/0.65 ! [K: int,N: nat] :
% 0.30/0.65 ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
% 0.30/0.65 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % bit_not_int_iff'
% 0.30/0.65 thf(fact_9377_atLeast0__atMost__Suc,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.30/0.65 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % atLeast0_atMost_Suc
% 0.30/0.65 thf(fact_9378_Icc__eq__insert__lb__nat,axiom,
% 0.30/0.65 ! [M: nat,N: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65 => ( ( set_or1269000886237332187st_nat @ M @ N )
% 0.30/0.65 = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Icc_eq_insert_lb_nat
% 0.30/0.65 thf(fact_9379_atLeastAtMostSuc__conv,axiom,
% 0.30/0.65 ! [M: nat,N: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.65 => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
% 0.30/0.65 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % atLeastAtMostSuc_conv
% 0.30/0.65 thf(fact_9380_atLeastAtMost__insertL,axiom,
% 0.30/0.65 ! [M: nat,N: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65 => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.30/0.65 = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % atLeastAtMost_insertL
% 0.30/0.65 thf(fact_9381_lessThan__nat__numeral,axiom,
% 0.30/0.65 ! [K: num] :
% 0.30/0.65 ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
% 0.30/0.65 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % lessThan_nat_numeral
% 0.30/0.65 thf(fact_9382_atMost__nat__numeral,axiom,
% 0.30/0.65 ! [K: num] :
% 0.30/0.65 ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
% 0.30/0.65 = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % atMost_nat_numeral
% 0.30/0.65 thf(fact_9383_bit__imp__take__bit__positive,axiom,
% 0.30/0.65 ! [N: nat,M: nat,K: int] :
% 0.30/0.65 ( ( ord_less_nat @ N @ M )
% 0.30/0.65 => ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.30/0.65 => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % bit_imp_take_bit_positive
% 0.30/0.65 thf(fact_9384_bit__concat__bit__iff,axiom,
% 0.30/0.65 ! [M: nat,K: int,L: int,N: nat] :
% 0.30/0.65 ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
% 0.30/0.65 = ( ( ( ord_less_nat @ N @ M )
% 0.30/0.65 & ( bit_se1146084159140164899it_int @ K @ N ) )
% 0.30/0.65 | ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65 & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % bit_concat_bit_iff
% 0.30/0.65 thf(fact_9385_sin__times__pi__eq__0,axiom,
% 0.30/0.65 ! [X2: real] :
% 0.30/0.65 ( ( ( sin_real @ ( times_times_real @ X2 @ pi ) )
% 0.30/0.65 = zero_zero_real )
% 0.30/0.65 = ( member_real @ X2 @ ring_1_Ints_real ) ) ).
% 0.30/0.65
% 0.30/0.65 % sin_times_pi_eq_0
% 0.30/0.65 thf(fact_9386_signed__take__bit__eq__concat__bit,axiom,
% 0.30/0.65 ( bit_ri631733984087533419it_int
% 0.30/0.65 = ( ^ [N3: nat,K2: int] : ( bit_concat_bit @ N3 @ K2 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % signed_take_bit_eq_concat_bit
% 0.30/0.65 thf(fact_9387_int__bit__bound,axiom,
% 0.30/0.65 ! [K: int] :
% 0.30/0.65 ~ ! [N2: nat] :
% 0.30/0.65 ( ! [M2: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ N2 @ M2 )
% 0.30/0.65 => ( ( bit_se1146084159140164899it_int @ K @ M2 )
% 0.30/0.65 = ( bit_se1146084159140164899it_int @ K @ N2 ) ) )
% 0.30/0.65 => ~ ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65 => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 0.30/0.65 = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % int_bit_bound
% 0.30/0.65 thf(fact_9388_atLeast1__atMost__eq__remove0,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.65 = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % atLeast1_atMost_eq_remove0
% 0.30/0.65 thf(fact_9389_bit__int__def,axiom,
% 0.30/0.65 ( bit_se1146084159140164899it_int
% 0.30/0.65 = ( ^ [K2: int,N3: nat] :
% 0.30/0.65 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % bit_int_def
% 0.30/0.65 thf(fact_9390_set__decode__plus__power__2,axiom,
% 0.30/0.65 ! [N: nat,Z: nat] :
% 0.30/0.65 ( ~ ( member_nat @ N @ ( nat_set_decode @ Z ) )
% 0.30/0.65 => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z ) )
% 0.30/0.65 = ( insert_nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % set_decode_plus_power_2
% 0.30/0.65 thf(fact_9391_OR__upper,axiom,
% 0.30/0.65 ! [X2: int,N: nat,Y3: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.65 => ( ( ord_less_int @ X2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.65 => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.65 => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X2 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % OR_upper
% 0.30/0.65 thf(fact_9392_sin__integer__2pi,axiom,
% 0.30/0.65 ! [N: real] :
% 0.30/0.65 ( ( member_real @ N @ ring_1_Ints_real )
% 0.30/0.65 => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.30/0.65 = zero_zero_real ) ) ).
% 0.30/0.65
% 0.30/0.65 % sin_integer_2pi
% 0.30/0.65 thf(fact_9393_cos__integer__2pi,axiom,
% 0.30/0.65 ! [N: real] :
% 0.30/0.65 ( ( member_real @ N @ ring_1_Ints_real )
% 0.30/0.65 => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.30/0.65 = one_one_real ) ) ).
% 0.30/0.65
% 0.30/0.65 % cos_integer_2pi
% 0.30/0.65 thf(fact_9394_or__int__rec,axiom,
% 0.30/0.65 ( bit_se1409905431419307370or_int
% 0.30/0.65 = ( ^ [K2: int,L2: int] :
% 0.30/0.65 ( plus_plus_int
% 0.30/0.65 @ ( zero_n2684676970156552555ol_int
% 0.30/0.65 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.30/0.65 | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 0.30/0.65 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_int_rec
% 0.30/0.65 thf(fact_9395_set__bit__eq,axiom,
% 0.30/0.65 ( bit_se7879613467334960850it_int
% 0.30/0.65 = ( ^ [N3: nat,K2: int] :
% 0.30/0.65 ( plus_plus_int @ K2
% 0.30/0.65 @ ( times_times_int
% 0.30/0.65 @ ( zero_n2684676970156552555ol_int
% 0.30/0.65 @ ~ ( bit_se1146084159140164899it_int @ K2 @ N3 ) )
% 0.30/0.65 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % set_bit_eq
% 0.30/0.65 thf(fact_9396_unset__bit__eq,axiom,
% 0.30/0.65 ( bit_se4203085406695923979it_int
% 0.30/0.65 = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ K2 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % unset_bit_eq
% 0.30/0.65 thf(fact_9397_take__bit__Suc__from__most,axiom,
% 0.30/0.65 ! [N: nat,K: int] :
% 0.30/0.65 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
% 0.30/0.65 = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N ) ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % take_bit_Suc_from_most
% 0.30/0.65 thf(fact_9398_or__minus__numerals_I5_J,axiom,
% 0.30/0.65 ! [N: num] :
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 0.30/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_minus_numerals(5)
% 0.30/0.65 thf(fact_9399_or__minus__numerals_I1_J,axiom,
% 0.30/0.65 ! [N: num] :
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_minus_numerals(1)
% 0.30/0.65 thf(fact_9400_xor__Suc__0__eq,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.30/0.65 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.65 @ ( zero_n2687167440665602831ol_nat
% 0.30/0.65 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % xor_Suc_0_eq
% 0.30/0.65 thf(fact_9401_or__nat__numerals_I2_J,axiom,
% 0.30/0.65 ! [Y3: num] :
% 0.30/0.65 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.30/0.65 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_nat_numerals(2)
% 0.30/0.65 thf(fact_9402_or__nat__numerals_I4_J,axiom,
% 0.30/0.65 ! [X2: num] :
% 0.30/0.65 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.65 = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_nat_numerals(4)
% 0.30/0.65 thf(fact_9403_or__nat__numerals_I1_J,axiom,
% 0.30/0.65 ! [Y3: num] :
% 0.30/0.65 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.30/0.65 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_nat_numerals(1)
% 0.30/0.65 thf(fact_9404_or__nat__numerals_I3_J,axiom,
% 0.30/0.65 ! [X2: num] :
% 0.30/0.65 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.65 = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_nat_numerals(3)
% 0.30/0.65 thf(fact_9405_xor__nat__numerals_I1_J,axiom,
% 0.30/0.65 ! [Y3: num] :
% 0.30/0.65 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.30/0.65 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % xor_nat_numerals(1)
% 0.30/0.65 thf(fact_9406_xor__nat__numerals_I2_J,axiom,
% 0.30/0.65 ! [Y3: num] :
% 0.30/0.65 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.30/0.65 = ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % xor_nat_numerals(2)
% 0.30/0.65 thf(fact_9407_xor__nat__numerals_I3_J,axiom,
% 0.30/0.65 ! [X2: num] :
% 0.30/0.65 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.65 = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % xor_nat_numerals(3)
% 0.30/0.65 thf(fact_9408_xor__nat__numerals_I4_J,axiom,
% 0.30/0.65 ! [X2: num] :
% 0.30/0.65 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.65 = ( numeral_numeral_nat @ ( bit0 @ X2 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % xor_nat_numerals(4)
% 0.30/0.65 thf(fact_9409_or__minus__numerals_I4_J,axiom,
% 0.30/0.65 ! [M: num,N: num] :
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_minus_numerals(4)
% 0.30/0.65 thf(fact_9410_or__minus__numerals_I8_J,axiom,
% 0.30/0.65 ! [N: num,M: num] :
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.30/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_minus_numerals(8)
% 0.30/0.65 thf(fact_9411_or__minus__numerals_I3_J,axiom,
% 0.30/0.65 ! [M: num,N: num] :
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_minus_numerals(3)
% 0.30/0.65 thf(fact_9412_or__minus__numerals_I7_J,axiom,
% 0.30/0.65 ! [N: num,M: num] :
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.30/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_minus_numerals(7)
% 0.30/0.65 thf(fact_9413_or__not__num__neg_Osimps_I1_J,axiom,
% 0.30/0.65 ( ( bit_or_not_num_neg @ one @ one )
% 0.30/0.65 = one ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_num_neg.simps(1)
% 0.30/0.65 thf(fact_9414_bit__Suc__0__iff,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.65 = ( N = zero_zero_nat ) ) ).
% 0.30/0.65
% 0.30/0.65 % bit_Suc_0_iff
% 0.30/0.65 thf(fact_9415_not__bit__Suc__0__Suc,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % not_bit_Suc_0_Suc
% 0.30/0.65 thf(fact_9416_or__not__num__neg_Osimps_I4_J,axiom,
% 0.30/0.65 ! [N: num] :
% 0.30/0.65 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
% 0.30/0.65 = ( bit0 @ one ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_num_neg.simps(4)
% 0.30/0.65 thf(fact_9417_or__not__num__neg_Osimps_I6_J,axiom,
% 0.30/0.65 ! [N: num,M: num] :
% 0.30/0.65 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
% 0.30/0.65 = ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_num_neg.simps(6)
% 0.30/0.65 thf(fact_9418_or__not__num__neg_Osimps_I7_J,axiom,
% 0.30/0.65 ! [N: num] :
% 0.30/0.65 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
% 0.30/0.65 = one ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_num_neg.simps(7)
% 0.30/0.65 thf(fact_9419_or__not__num__neg_Osimps_I3_J,axiom,
% 0.30/0.65 ! [M: num] :
% 0.30/0.65 ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
% 0.30/0.65 = ( bit1 @ M ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_num_neg.simps(3)
% 0.30/0.65 thf(fact_9420_or__not__num__neg_Osimps_I5_J,axiom,
% 0.30/0.65 ! [N: num,M: num] :
% 0.30/0.65 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
% 0.30/0.65 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_num_neg.simps(5)
% 0.30/0.65 thf(fact_9421_not__bit__Suc__0__numeral,axiom,
% 0.30/0.65 ! [N: num] :
% 0.30/0.65 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % not_bit_Suc_0_numeral
% 0.30/0.65 thf(fact_9422_or__not__num__neg_Osimps_I9_J,axiom,
% 0.30/0.65 ! [N: num,M: num] :
% 0.30/0.65 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
% 0.30/0.65 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_num_neg.simps(9)
% 0.30/0.65 thf(fact_9423_or__nat__def,axiom,
% 0.30/0.65 ( bit_se1412395901928357646or_nat
% 0.30/0.65 = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_nat_def
% 0.30/0.65 thf(fact_9424_or__not__num__neg_Osimps_I2_J,axiom,
% 0.30/0.65 ! [M: num] :
% 0.30/0.65 ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
% 0.30/0.65 = ( bit1 @ M ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_num_neg.simps(2)
% 0.30/0.65 thf(fact_9425_or__not__num__neg_Osimps_I8_J,axiom,
% 0.30/0.65 ! [N: num,M: num] :
% 0.30/0.65 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
% 0.30/0.65 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_num_neg.simps(8)
% 0.30/0.65 thf(fact_9426_bit__nat__iff,axiom,
% 0.30/0.65 ! [K: int,N: nat] :
% 0.30/0.65 ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
% 0.30/0.65 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.65 & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % bit_nat_iff
% 0.30/0.65 thf(fact_9427_or__not__num__neg_Oelims,axiom,
% 0.30/0.65 ! [X2: num,Xa2: num,Y3: num] :
% 0.30/0.65 ( ( ( bit_or_not_num_neg @ X2 @ Xa2 )
% 0.30/0.65 = Y3 )
% 0.30/0.65 => ( ( ( X2 = one )
% 0.30/0.65 => ( ( Xa2 = one )
% 0.30/0.65 => ( Y3 != one ) ) )
% 0.30/0.65 => ( ( ( X2 = one )
% 0.30/0.65 => ! [M4: num] :
% 0.30/0.65 ( ( Xa2
% 0.30/0.65 = ( bit0 @ M4 ) )
% 0.30/0.65 => ( Y3
% 0.30/0.65 != ( bit1 @ M4 ) ) ) )
% 0.30/0.65 => ( ( ( X2 = one )
% 0.30/0.65 => ! [M4: num] :
% 0.30/0.65 ( ( Xa2
% 0.30/0.65 = ( bit1 @ M4 ) )
% 0.30/0.65 => ( Y3
% 0.30/0.65 != ( bit1 @ M4 ) ) ) )
% 0.30/0.65 => ( ( ? [N2: num] :
% 0.30/0.65 ( X2
% 0.30/0.65 = ( bit0 @ N2 ) )
% 0.30/0.65 => ( ( Xa2 = one )
% 0.30/0.65 => ( Y3
% 0.30/0.65 != ( bit0 @ one ) ) ) )
% 0.30/0.65 => ( ! [N2: num] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( bit0 @ N2 ) )
% 0.30/0.65 => ! [M4: num] :
% 0.30/0.65 ( ( Xa2
% 0.30/0.65 = ( bit0 @ M4 ) )
% 0.30/0.65 => ( Y3
% 0.30/0.65 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
% 0.30/0.65 => ( ! [N2: num] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( bit0 @ N2 ) )
% 0.30/0.65 => ! [M4: num] :
% 0.30/0.65 ( ( Xa2
% 0.30/0.65 = ( bit1 @ M4 ) )
% 0.30/0.65 => ( Y3
% 0.30/0.65 != ( bit0 @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
% 0.30/0.65 => ( ( ? [N2: num] :
% 0.30/0.65 ( X2
% 0.30/0.65 = ( bit1 @ N2 ) )
% 0.30/0.65 => ( ( Xa2 = one )
% 0.30/0.65 => ( Y3 != one ) ) )
% 0.30/0.65 => ( ! [N2: num] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( bit1 @ N2 ) )
% 0.30/0.65 => ! [M4: num] :
% 0.30/0.65 ( ( Xa2
% 0.30/0.65 = ( bit0 @ M4 ) )
% 0.30/0.65 => ( Y3
% 0.30/0.65 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
% 0.30/0.65 => ~ ! [N2: num] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( bit1 @ N2 ) )
% 0.30/0.65 => ! [M4: num] :
% 0.30/0.65 ( ( Xa2
% 0.30/0.65 = ( bit1 @ M4 ) )
% 0.30/0.65 => ( Y3
% 0.30/0.65 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_num_neg.elims
% 0.30/0.65 thf(fact_9428_bit__nat__def,axiom,
% 0.30/0.65 ( bit_se1148574629649215175it_nat
% 0.30/0.65 = ( ^ [M3: nat,N3: nat] :
% 0.30/0.65 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % bit_nat_def
% 0.30/0.65 thf(fact_9429_Suc__0__or__eq,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.65 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Suc_0_or_eq
% 0.30/0.65 thf(fact_9430_or__Suc__0__eq,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.30/0.65 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_Suc_0_eq
% 0.30/0.65 thf(fact_9431_or__nat__rec,axiom,
% 0.30/0.65 ( bit_se1412395901928357646or_nat
% 0.30/0.65 = ( ^ [M3: nat,N3: nat] :
% 0.30/0.65 ( plus_plus_nat
% 0.30/0.65 @ ( zero_n2687167440665602831ol_nat
% 0.30/0.65 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 0.30/0.65 | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 0.30/0.65 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_nat_rec
% 0.30/0.65 thf(fact_9432_xor__nat__unfold,axiom,
% 0.30/0.65 ( bit_se6528837805403552850or_nat
% 0.30/0.65 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % xor_nat_unfold
% 0.30/0.65 thf(fact_9433_xor__nat__rec,axiom,
% 0.30/0.65 ( bit_se6528837805403552850or_nat
% 0.30/0.65 = ( ^ [M3: nat,N3: nat] :
% 0.30/0.65 ( plus_plus_nat
% 0.30/0.65 @ ( zero_n2687167440665602831ol_nat
% 0.30/0.65 @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 0.30/0.65 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 0.30/0.65 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % xor_nat_rec
% 0.30/0.65 thf(fact_9434_or__nat__unfold,axiom,
% 0.30/0.65 ( bit_se1412395901928357646or_nat
% 0.30/0.65 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_nat_unfold
% 0.30/0.65 thf(fact_9435_Suc__0__xor__eq,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.65 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.65 @ ( zero_n2687167440665602831ol_nat
% 0.30/0.65 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Suc_0_xor_eq
% 0.30/0.65 thf(fact_9436_or__not__num__neg_Opelims,axiom,
% 0.30/0.65 ! [X2: num,Xa2: num,Y3: num] :
% 0.30/0.65 ( ( ( bit_or_not_num_neg @ X2 @ Xa2 )
% 0.30/0.65 = Y3 )
% 0.30/0.65 => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X2 @ Xa2 ) )
% 0.30/0.65 => ( ( ( X2 = one )
% 0.30/0.65 => ( ( Xa2 = one )
% 0.30/0.65 => ( ( Y3 = one )
% 0.30/0.65 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.30/0.65 => ( ( ( X2 = one )
% 0.30/0.65 => ! [M4: num] :
% 0.30/0.65 ( ( Xa2
% 0.30/0.65 = ( bit0 @ M4 ) )
% 0.30/0.65 => ( ( Y3
% 0.30/0.65 = ( bit1 @ M4 ) )
% 0.30/0.65 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M4 ) ) ) ) ) )
% 0.30/0.65 => ( ( ( X2 = one )
% 0.30/0.65 => ! [M4: num] :
% 0.30/0.65 ( ( Xa2
% 0.30/0.65 = ( bit1 @ M4 ) )
% 0.30/0.65 => ( ( Y3
% 0.30/0.65 = ( bit1 @ M4 ) )
% 0.30/0.65 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M4 ) ) ) ) ) )
% 0.30/0.65 => ( ! [N2: num] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( bit0 @ N2 ) )
% 0.30/0.65 => ( ( Xa2 = one )
% 0.30/0.65 => ( ( Y3
% 0.30/0.65 = ( bit0 @ one ) )
% 0.30/0.65 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ one ) ) ) ) )
% 0.30/0.65 => ( ! [N2: num] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( bit0 @ N2 ) )
% 0.30/0.65 => ! [M4: num] :
% 0.30/0.65 ( ( Xa2
% 0.30/0.65 = ( bit0 @ M4 ) )
% 0.30/0.65 => ( ( Y3
% 0.30/0.65 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
% 0.30/0.65 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 0.30/0.65 => ( ! [N2: num] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( bit0 @ N2 ) )
% 0.30/0.65 => ! [M4: num] :
% 0.30/0.65 ( ( Xa2
% 0.30/0.65 = ( bit1 @ M4 ) )
% 0.30/0.65 => ( ( Y3
% 0.30/0.65 = ( bit0 @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
% 0.30/0.65 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ ( bit1 @ M4 ) ) ) ) ) )
% 0.30/0.65 => ( ! [N2: num] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( bit1 @ N2 ) )
% 0.30/0.65 => ( ( Xa2 = one )
% 0.30/0.65 => ( ( Y3 = one )
% 0.30/0.65 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ one ) ) ) ) )
% 0.30/0.65 => ( ! [N2: num] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( bit1 @ N2 ) )
% 0.30/0.65 => ! [M4: num] :
% 0.30/0.65 ( ( Xa2
% 0.30/0.65 = ( bit0 @ M4 ) )
% 0.30/0.65 => ( ( Y3
% 0.30/0.65 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
% 0.30/0.65 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 0.30/0.65 => ~ ! [N2: num] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( bit1 @ N2 ) )
% 0.30/0.65 => ! [M4: num] :
% 0.30/0.65 ( ( Xa2
% 0.30/0.65 = ( bit1 @ M4 ) )
% 0.30/0.65 => ( ( Y3
% 0.30/0.65 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
% 0.30/0.65 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ ( bit1 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_num_neg.pelims
% 0.30/0.65 thf(fact_9437_setceilmax,axiom,
% 0.30/0.65 ! [S: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,N: nat] :
% 0.30/0.65 ( ( vEBT_invar_vebt @ S @ M )
% 0.30/0.65 => ( ! [X5: vEBT_VEBT] :
% 0.30/0.65 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Listy ) )
% 0.30/0.65 => ( vEBT_invar_vebt @ X5 @ N ) )
% 0.30/0.65 => ( ( M
% 0.30/0.65 = ( suc @ N ) )
% 0.30/0.65 => ( ! [X5: vEBT_VEBT] :
% 0.30/0.65 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Listy ) )
% 0.30/0.65 => ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ X5 ) )
% 0.30/0.65 = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 0.30/0.65 => ( ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ S ) )
% 0.30/0.65 = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) )
% 0.30/0.65 => ( ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ S @ ( set_VEBT_VEBT2 @ Listy ) ) ) ) )
% 0.30/0.65 = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % setceilmax
% 0.30/0.65 thf(fact_9438_horner__sum__of__bool__2__less,axiom,
% 0.30/0.65 ! [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 ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % horner_sum_of_bool_2_less
% 0.30/0.65 thf(fact_9439_height__compose__list,axiom,
% 0.30/0.65 ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.65 ( ( member_VEBT_VEBT @ T @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.65 => ( ord_less_eq_nat @ ( vEBT_VEBT_height @ T ) @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary @ ( set_VEBT_VEBT2 @ TreeList ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % height_compose_list
% 0.30/0.65 thf(fact_9440_max__ins__scaled,axiom,
% 0.30/0.65 ! [N: nat,X14: vEBT_VEBT,M: nat,X13: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( vEBT_VEBT_height @ X14 ) ) @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ ( lattic8265883725875713057ax_nat @ ( insert_nat @ ( vEBT_VEBT_height @ X14 ) @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % max_ins_scaled
% 0.30/0.65 thf(fact_9441_height__i__max,axiom,
% 0.30/0.65 ! [I2: nat,X13: list_VEBT_VEBT,Foo: nat] :
% 0.30/0.65 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ X13 ) )
% 0.30/0.65 => ( ord_less_eq_nat @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ X13 @ I2 ) ) @ ( ord_max_nat @ Foo @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % height_i_max
% 0.30/0.65 thf(fact_9442_max__idx__list,axiom,
% 0.30/0.65 ! [I2: nat,X13: list_VEBT_VEBT,N: nat,X14: vEBT_VEBT] :
% 0.30/0.65 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ X13 ) )
% 0.30/0.65 => ( ord_less_eq_nat @ ( times_times_nat @ N @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ X13 @ I2 ) ) ) @ ( suc @ ( suc @ ( times_times_nat @ N @ ( ord_max_nat @ ( vEBT_VEBT_height @ X14 ) @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % max_idx_list
% 0.30/0.65 thf(fact_9443_xor__nonnegative__int__iff,axiom,
% 0.30/0.65 ! [K: int,L: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 0.30/0.65 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.65 = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % xor_nonnegative_int_iff
% 0.30/0.65 thf(fact_9444_xor__negative__int__iff,axiom,
% 0.30/0.65 ! [K: int,L: int] :
% 0.30/0.65 ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
% 0.30/0.65 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.65 != ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % xor_negative_int_iff
% 0.30/0.65 thf(fact_9445_bit__xor__int__iff,axiom,
% 0.30/0.65 ! [K: int,L: int,N: nat] :
% 0.30/0.65 ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ N )
% 0.30/0.65 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.30/0.65 != ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % bit_xor_int_iff
% 0.30/0.65 thf(fact_9446_XOR__lower,axiom,
% 0.30/0.65 ! [X2: int,Y3: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.65 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % XOR_lower
% 0.30/0.65 thf(fact_9447_xor__nat__def,axiom,
% 0.30/0.65 ( bit_se6528837805403552850or_nat
% 0.30/0.65 = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % xor_nat_def
% 0.30/0.65 thf(fact_9448_VEBT__internal_Oheight_Osimps_I2_J,axiom,
% 0.30/0.65 ! [Uu: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.65 ( ( vEBT_VEBT_height @ ( vEBT_Node @ Uu @ Deg @ TreeList @ Summary ) )
% 0.30/0.65 = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary @ ( set_VEBT_VEBT2 @ TreeList ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % VEBT_internal.height.simps(2)
% 0.30/0.65 thf(fact_9449_VEBT__internal_Oheight_Oelims,axiom,
% 0.30/0.65 ! [X2: vEBT_VEBT,Y3: nat] :
% 0.30/0.65 ( ( ( vEBT_VEBT_height @ X2 )
% 0.30/0.65 = Y3 )
% 0.30/0.65 => ( ( ? [A4: $o,B4: $o] :
% 0.30/0.65 ( X2
% 0.30/0.65 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.65 => ( Y3 != zero_zero_nat ) )
% 0.30/0.65 => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.30/0.65 => ( Y3
% 0.30/0.65 != ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary2 @ ( set_VEBT_VEBT2 @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % VEBT_internal.height.elims
% 0.30/0.65 thf(fact_9450_divide__nat__def,axiom,
% 0.30/0.65 ( divide_divide_nat
% 0.30/0.65 = ( ^ [M3: nat,N3: nat] :
% 0.30/0.65 ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat
% 0.30/0.65 @ ( lattic8265883725875713057ax_nat
% 0.30/0.65 @ ( collect_nat
% 0.30/0.65 @ ^ [K2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K2 @ N3 ) @ M3 ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % divide_nat_def
% 0.30/0.65 thf(fact_9451_XOR__upper,axiom,
% 0.30/0.65 ! [X2: int,N: nat,Y3: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.65 => ( ( ord_less_int @ X2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.65 => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.65 => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X2 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % XOR_upper
% 0.30/0.65 thf(fact_9452_xor__int__rec,axiom,
% 0.30/0.65 ( bit_se6526347334894502574or_int
% 0.30/0.65 = ( ^ [K2: int,L2: int] :
% 0.30/0.65 ( plus_plus_int
% 0.30/0.65 @ ( zero_n2684676970156552555ol_int
% 0.30/0.65 @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) )
% 0.30/0.65 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 0.30/0.65 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % xor_int_rec
% 0.30/0.65 thf(fact_9453_VEBT__internal_Oheight_Opelims,axiom,
% 0.30/0.65 ! [X2: vEBT_VEBT,Y3: nat] :
% 0.30/0.65 ( ( ( vEBT_VEBT_height @ X2 )
% 0.30/0.65 = Y3 )
% 0.30/0.65 => ( ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ X2 )
% 0.30/0.65 => ( ! [A4: $o,B4: $o] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.65 => ( ( Y3 = zero_zero_nat )
% 0.30/0.65 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
% 0.30/0.65 => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.65 ( ( X2
% 0.30/0.65 = ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.30/0.65 => ( ( Y3
% 0.30/0.65 = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary2 @ ( set_VEBT_VEBT2 @ TreeList2 ) ) ) ) ) )
% 0.30/0.65 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % VEBT_internal.height.pelims
% 0.30/0.65 thf(fact_9454_xor__int__unfold,axiom,
% 0.30/0.65 ( bit_se6526347334894502574or_int
% 0.30/0.65 = ( ^ [K2: int,L2: int] :
% 0.30/0.65 ( if_int
% 0.30/0.65 @ ( K2
% 0.30/0.65 = ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.65 @ ( bit_ri7919022796975470100ot_int @ L2 )
% 0.30/0.65 @ ( if_int
% 0.30/0.65 @ ( L2
% 0.30/0.65 = ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.65 @ ( bit_ri7919022796975470100ot_int @ K2 )
% 0.30/0.65 @ ( if_int @ ( K2 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % xor_int_unfold
% 0.30/0.65 thf(fact_9455_bij__betw__Suc,axiom,
% 0.30/0.65 ! [M7: set_nat,N5: set_nat] :
% 0.30/0.65 ( ( bij_betw_nat_nat @ suc @ M7 @ N5 )
% 0.30/0.65 = ( ( image_nat_nat @ suc @ M7 )
% 0.30/0.65 = N5 ) ) ).
% 0.30/0.65
% 0.30/0.65 % bij_betw_Suc
% 0.30/0.65 thf(fact_9456_image__Suc__atLeastAtMost,axiom,
% 0.30/0.65 ! [I2: nat,J: nat] :
% 0.30/0.65 ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I2 @ J ) )
% 0.30/0.65 = ( set_or1269000886237332187st_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % image_Suc_atLeastAtMost
% 0.30/0.65 thf(fact_9457_not__nonnegative__int__iff,axiom,
% 0.30/0.65 ! [K: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.30/0.65 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.65
% 0.30/0.65 % not_nonnegative_int_iff
% 0.30/0.65 thf(fact_9458_not__negative__int__iff,axiom,
% 0.30/0.65 ! [K: int] :
% 0.30/0.65 ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 0.30/0.65 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.30/0.65
% 0.30/0.65 % not_negative_int_iff
% 0.30/0.65 thf(fact_9459_or__minus__minus__numerals,axiom,
% 0.30/0.65 ! [M: num,N: num] :
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.65 = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_minus_minus_numerals
% 0.30/0.65 thf(fact_9460_and__minus__minus__numerals,axiom,
% 0.30/0.65 ! [M: num,N: num] :
% 0.30/0.65 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.65 = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_minus_minus_numerals
% 0.30/0.65 thf(fact_9461_bit__not__int__iff,axiom,
% 0.30/0.65 ! [K: int,N: nat] :
% 0.30/0.65 ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N )
% 0.30/0.65 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % bit_not_int_iff
% 0.30/0.65 thf(fact_9462_zero__notin__Suc__image,axiom,
% 0.30/0.65 ! [A2: set_nat] :
% 0.30/0.65 ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% 0.30/0.65
% 0.30/0.65 % zero_notin_Suc_image
% 0.30/0.65 thf(fact_9463_or__int__def,axiom,
% 0.30/0.65 ( bit_se1409905431419307370or_int
% 0.30/0.65 = ( ^ [K2: int,L2: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ ( bit_ri7919022796975470100ot_int @ L2 ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_int_def
% 0.30/0.65 thf(fact_9464_finite__int__iff__bounded,axiom,
% 0.30/0.65 ( finite_finite_int
% 0.30/0.65 = ( ^ [S5: set_int] :
% 0.30/0.65 ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S5 ) @ ( set_ord_lessThan_int @ K2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % finite_int_iff_bounded
% 0.30/0.65 thf(fact_9465_finite__int__iff__bounded__le,axiom,
% 0.30/0.65 ( finite_finite_int
% 0.30/0.65 = ( ^ [S5: set_int] :
% 0.30/0.65 ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S5 ) @ ( set_ord_atMost_int @ K2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % finite_int_iff_bounded_le
% 0.30/0.65 thf(fact_9466_not__int__def,axiom,
% 0.30/0.65 ( bit_ri7919022796975470100ot_int
% 0.30/0.65 = ( ^ [K2: int] : ( minus_minus_int @ ( uminus_uminus_int @ K2 ) @ one_one_int ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % not_int_def
% 0.30/0.65 thf(fact_9467_and__not__numerals_I1_J,axiom,
% 0.30/0.65 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.30/0.65 = zero_zero_int ) ).
% 0.30/0.65
% 0.30/0.65 % and_not_numerals(1)
% 0.30/0.65 thf(fact_9468_or__not__numerals_I1_J,axiom,
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.30/0.65 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_numerals(1)
% 0.30/0.65 thf(fact_9469_xor__int__def,axiom,
% 0.30/0.65 ( bit_se6526347334894502574or_int
% 0.30/0.65 = ( ^ [K2: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ L2 ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ L2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % xor_int_def
% 0.30/0.65 thf(fact_9470_not__int__div__2,axiom,
% 0.30/0.65 ! [K: int] :
% 0.30/0.65 ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.65 = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % not_int_div_2
% 0.30/0.65 thf(fact_9471_even__not__iff__int,axiom,
% 0.30/0.65 ! [K: int] :
% 0.30/0.65 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.30/0.65 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % even_not_iff_int
% 0.30/0.65 thf(fact_9472_and__not__numerals_I2_J,axiom,
% 0.30/0.65 ! [N: num] :
% 0.30/0.65 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.65 = one_one_int ) ).
% 0.30/0.65
% 0.30/0.65 % and_not_numerals(2)
% 0.30/0.65 thf(fact_9473_and__not__numerals_I4_J,axiom,
% 0.30/0.65 ! [M: num] :
% 0.30/0.65 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.30/0.65 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_not_numerals(4)
% 0.30/0.65 thf(fact_9474_or__not__numerals_I2_J,axiom,
% 0.30/0.65 ! [N: num] :
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.65 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_numerals(2)
% 0.30/0.65 thf(fact_9475_or__not__numerals_I4_J,axiom,
% 0.30/0.65 ! [M: num] :
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.30/0.65 = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_numerals(4)
% 0.30/0.65 thf(fact_9476_bit__minus__int__iff,axiom,
% 0.30/0.65 ! [K: int,N: nat] :
% 0.30/0.65 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
% 0.30/0.65 = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % bit_minus_int_iff
% 0.30/0.65 thf(fact_9477_image__Suc__lessThan,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.65 = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % image_Suc_lessThan
% 0.30/0.65 thf(fact_9478_image__Suc__atMost,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65 = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % image_Suc_atMost
% 0.30/0.65 thf(fact_9479_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.30/0.65 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % atLeast0_atMost_Suc_eq_insert_0
% 0.30/0.65 thf(fact_9480_lessThan__Suc__eq__insert__0,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( set_ord_lessThan_nat @ ( suc @ N ) )
% 0.30/0.65 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % lessThan_Suc_eq_insert_0
% 0.30/0.65 thf(fact_9481_numeral__or__not__num__eq,axiom,
% 0.30/0.65 ! [M: num,N: num] :
% 0.30/0.65 ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
% 0.30/0.65 = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % numeral_or_not_num_eq
% 0.30/0.65 thf(fact_9482_int__numeral__not__or__num__neg,axiom,
% 0.30/0.65 ! [M: num,N: num] :
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % int_numeral_not_or_num_neg
% 0.30/0.65 thf(fact_9483_int__numeral__or__not__num__neg,axiom,
% 0.30/0.65 ! [M: num,N: num] :
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % int_numeral_or_not_num_neg
% 0.30/0.65 thf(fact_9484_atMost__Suc__eq__insert__0,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( set_ord_atMost_nat @ ( suc @ N ) )
% 0.30/0.65 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % atMost_Suc_eq_insert_0
% 0.30/0.65 thf(fact_9485_and__not__numerals_I5_J,axiom,
% 0.30/0.65 ! [M: num,N: num] :
% 0.30/0.65 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.65 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_not_numerals(5)
% 0.30/0.65 thf(fact_9486_and__not__numerals_I7_J,axiom,
% 0.30/0.65 ! [M: num] :
% 0.30/0.65 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.30/0.65 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_not_numerals(7)
% 0.30/0.65 thf(fact_9487_or__not__numerals_I3_J,axiom,
% 0.30/0.65 ! [N: num] :
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.65 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_numerals(3)
% 0.30/0.65 thf(fact_9488_and__not__numerals_I3_J,axiom,
% 0.30/0.65 ! [N: num] :
% 0.30/0.65 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.65 = zero_zero_int ) ).
% 0.30/0.65
% 0.30/0.65 % and_not_numerals(3)
% 0.30/0.65 thf(fact_9489_or__not__numerals_I7_J,axiom,
% 0.30/0.65 ! [M: num] :
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.30/0.65 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_numerals(7)
% 0.30/0.65 thf(fact_9490_and__not__numerals_I6_J,axiom,
% 0.30/0.65 ! [M: num,N: num] :
% 0.30/0.65 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.65 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_not_numerals(6)
% 0.30/0.65 thf(fact_9491_and__not__numerals_I9_J,axiom,
% 0.30/0.65 ! [M: num,N: num] :
% 0.30/0.65 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.65 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_not_numerals(9)
% 0.30/0.65 thf(fact_9492_or__not__numerals_I6_J,axiom,
% 0.30/0.65 ! [M: num,N: num] :
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.65 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_numerals(6)
% 0.30/0.65 thf(fact_9493_or__not__numerals_I5_J,axiom,
% 0.30/0.65 ! [M: num,N: num] :
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.65 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_numerals(5)
% 0.30/0.65 thf(fact_9494_and__not__numerals_I8_J,axiom,
% 0.30/0.65 ! [M: num,N: num] :
% 0.30/0.65 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.65 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % and_not_numerals(8)
% 0.30/0.65 thf(fact_9495_or__not__numerals_I8_J,axiom,
% 0.30/0.65 ! [M: num,N: num] :
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.65 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_numerals(8)
% 0.30/0.65 thf(fact_9496_or__not__numerals_I9_J,axiom,
% 0.30/0.65 ! [M: num,N: num] :
% 0.30/0.65 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.65 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % or_not_numerals(9)
% 0.30/0.65 thf(fact_9497_not__int__rec,axiom,
% 0.30/0.65 ( bit_ri7919022796975470100ot_int
% 0.30/0.65 = ( ^ [K2: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % not_int_rec
% 0.30/0.65 thf(fact_9498_push__bit__nonnegative__int__iff,axiom,
% 0.30/0.65 ! [N: nat,K: int] :
% 0.30/0.65 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
% 0.30/0.65 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.30/0.65
% 0.30/0.65 % push_bit_nonnegative_int_iff
% 0.30/0.65 thf(fact_9499_push__bit__negative__int__iff,axiom,
% 0.30/0.65 ! [N: nat,K: int] :
% 0.30/0.65 ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
% 0.30/0.65 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.65
% 0.30/0.65 % push_bit_negative_int_iff
% 0.30/0.65 thf(fact_9500_concat__bit__of__zero__1,axiom,
% 0.30/0.65 ! [N: nat,L: int] :
% 0.30/0.65 ( ( bit_concat_bit @ N @ zero_zero_int @ L )
% 0.30/0.65 = ( bit_se545348938243370406it_int @ N @ L ) ) ).
% 0.30/0.65
% 0.30/0.65 % concat_bit_of_zero_1
% 0.30/0.65 thf(fact_9501_push__bit__of__Suc__0,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.30/0.65 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.65
% 0.30/0.65 % push_bit_of_Suc_0
% 0.30/0.65 thf(fact_9502_push__bit__nat__eq,axiom,
% 0.30/0.65 ! [N: nat,K: int] :
% 0.30/0.65 ( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K ) )
% 0.30/0.65 = ( nat2 @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % push_bit_nat_eq
% 0.30/0.65 thf(fact_9503_flip__bit__nat__def,axiom,
% 0.30/0.65 ( bit_se2161824704523386999it_nat
% 0.30/0.65 = ( ^ [M3: nat,N3: nat] : ( bit_se6528837805403552850or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % flip_bit_nat_def
% 0.30/0.65 thf(fact_9504_set__bit__nat__def,axiom,
% 0.30/0.65 ( bit_se7882103937844011126it_nat
% 0.30/0.65 = ( ^ [M3: nat,N3: nat] : ( bit_se1412395901928357646or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % set_bit_nat_def
% 0.30/0.65 thf(fact_9505_bit__push__bit__iff__int,axiom,
% 0.30/0.65 ! [M: nat,K: int,N: nat] :
% 0.30/0.65 ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
% 0.30/0.65 = ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65 & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % bit_push_bit_iff_int
% 0.30/0.65 thf(fact_9506_bit__push__bit__iff__nat,axiom,
% 0.30/0.65 ! [M: nat,Q3: nat,N: nat] :
% 0.30/0.65 ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q3 ) @ N )
% 0.30/0.65 = ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65 & ( bit_se1148574629649215175it_nat @ Q3 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % bit_push_bit_iff_nat
% 0.30/0.65 thf(fact_9507_concat__bit__eq,axiom,
% 0.30/0.65 ( bit_concat_bit
% 0.30/0.65 = ( ^ [N3: nat,K2: int,L2: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N3 @ K2 ) @ ( bit_se545348938243370406it_int @ N3 @ L2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % concat_bit_eq
% 0.30/0.65 thf(fact_9508_concat__bit__def,axiom,
% 0.30/0.65 ( bit_concat_bit
% 0.30/0.65 = ( ^ [N3: nat,K2: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N3 @ K2 ) @ ( bit_se545348938243370406it_int @ N3 @ L2 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % concat_bit_def
% 0.30/0.65 thf(fact_9509_set__bit__int__def,axiom,
% 0.30/0.65 ( bit_se7879613467334960850it_int
% 0.30/0.65 = ( ^ [N3: nat,K2: int] : ( bit_se1409905431419307370or_int @ K2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % set_bit_int_def
% 0.30/0.65 thf(fact_9510_flip__bit__int__def,axiom,
% 0.30/0.65 ( bit_se2159334234014336723it_int
% 0.30/0.65 = ( ^ [N3: nat,K2: int] : ( bit_se6526347334894502574or_int @ K2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % flip_bit_int_def
% 0.30/0.65 thf(fact_9511_unset__bit__int__def,axiom,
% 0.30/0.65 ( bit_se4203085406695923979it_int
% 0.30/0.65 = ( ^ [N3: nat,K2: int] : ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % unset_bit_int_def
% 0.30/0.65 thf(fact_9512_push__bit__int__def,axiom,
% 0.30/0.65 ( bit_se545348938243370406it_int
% 0.30/0.65 = ( ^ [N3: nat,K2: int] : ( times_times_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % push_bit_int_def
% 0.30/0.65 thf(fact_9513_push__bit__nat__def,axiom,
% 0.30/0.65 ( bit_se547839408752420682it_nat
% 0.30/0.65 = ( ^ [N3: nat,M3: nat] : ( times_times_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % push_bit_nat_def
% 0.30/0.65 thf(fact_9514_push__bit__minus__one,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.65 = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % push_bit_minus_one
% 0.30/0.65 thf(fact_9515_Cauchy__iff2,axiom,
% 0.30/0.65 ( topolo4055970368930404560y_real
% 0.30/0.65 = ( ^ [X4: nat > real] :
% 0.30/0.65 ! [J3: nat] :
% 0.30/0.65 ? [M8: nat] :
% 0.30/0.65 ! [M3: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ M8 @ M3 )
% 0.30/0.65 => ! [N3: nat] :
% 0.30/0.65 ( ( ord_less_eq_nat @ M8 @ N3 )
% 0.30/0.65 => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X4 @ M3 ) @ ( X4 @ N3 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Cauchy_iff2
% 0.30/0.65 thf(fact_9516_Sum__Ico__nat,axiom,
% 0.30/0.65 ! [M: nat,N: nat] :
% 0.30/0.65 ( ( groups3542108847815614940at_nat
% 0.30/0.65 @ ^ [X: nat] : X
% 0.30/0.65 @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 0.30/0.65 = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % Sum_Ico_nat
% 0.30/0.65 thf(fact_9517_image__Suc__atLeastLessThan,axiom,
% 0.30/0.65 ! [I2: nat,J: nat] :
% 0.30/0.65 ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I2 @ J ) )
% 0.30/0.65 = ( set_or4665077453230672383an_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % image_Suc_atLeastLessThan
% 0.30/0.65 thf(fact_9518_atLeastLessThan__singleton,axiom,
% 0.30/0.65 ! [M: nat] :
% 0.30/0.65 ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
% 0.30/0.65 = ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% 0.30/0.65
% 0.30/0.65 % atLeastLessThan_singleton
% 0.30/0.65 thf(fact_9519_all__nat__less__eq,axiom,
% 0.30/0.65 ! [N: nat,P: nat > $o] :
% 0.30/0.65 ( ( ! [M3: nat] :
% 0.30/0.65 ( ( ord_less_nat @ M3 @ N )
% 0.30/0.65 => ( P @ M3 ) ) )
% 0.30/0.65 = ( ! [X: nat] :
% 0.30/0.65 ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.30/0.65 => ( P @ X ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % all_nat_less_eq
% 0.30/0.65 thf(fact_9520_ex__nat__less__eq,axiom,
% 0.30/0.65 ! [N: nat,P: nat > $o] :
% 0.30/0.65 ( ( ? [M3: nat] :
% 0.30/0.65 ( ( ord_less_nat @ M3 @ N )
% 0.30/0.65 & ( P @ M3 ) ) )
% 0.30/0.65 = ( ? [X: nat] :
% 0.30/0.65 ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.30/0.65 & ( P @ X ) ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % ex_nat_less_eq
% 0.30/0.65 thf(fact_9521_atLeastLessThanSuc__atLeastAtMost,axiom,
% 0.30/0.65 ! [L: nat,U: nat] :
% 0.30/0.65 ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
% 0.30/0.65 = ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% 0.30/0.65
% 0.30/0.65 % atLeastLessThanSuc_atLeastAtMost
% 0.30/0.65 thf(fact_9522_atLeast0__lessThan__Suc,axiom,
% 0.30/0.65 ! [N: nat] :
% 0.30/0.65 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.30/0.65 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.65
% 0.30/0.65 % atLeast0_lessThan_Suc
% 0.30/0.65 thf(fact_9523_subset__eq__atLeast0__lessThan__finite,axiom,
% 0.30/0.65 ! [N5: set_nat,N: nat] :
% 0.30/0.65 ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.30/0.65 => ( finite_finite_nat @ N5 ) ) ).
% 0.30/0.65
% 0.30/0.65 % subset_eq_atLeast0_lessThan_finite
% 0.30/0.65 thf(fact_9524_atLeastLessThanSuc,axiom,
% 0.30/0.66 ! [M: nat,N: nat] :
% 0.30/0.66 ( ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.66 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 0.30/0.66 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
% 0.30/0.66 & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.30/0.66 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 0.30/0.66 = bot_bot_set_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % atLeastLessThanSuc
% 0.30/0.66 thf(fact_9525_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 0.30/0.66 ! [N: nat] :
% 0.30/0.66 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.30/0.66 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % atLeast0_lessThan_Suc_eq_insert_0
% 0.30/0.66 thf(fact_9526_prod__Suc__Suc__fact,axiom,
% 0.30/0.66 ! [N: nat] :
% 0.30/0.66 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.30/0.66 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.30/0.66
% 0.30/0.66 % prod_Suc_Suc_fact
% 0.30/0.66 thf(fact_9527_prod__Suc__fact,axiom,
% 0.30/0.66 ! [N: nat] :
% 0.30/0.66 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.30/0.66 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.30/0.66
% 0.30/0.66 % prod_Suc_fact
% 0.30/0.66 thf(fact_9528_atLeastLessThan__nat__numeral,axiom,
% 0.30/0.66 ! [M: nat,K: num] :
% 0.30/0.66 ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.30/0.66 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.30/0.66 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 0.30/0.66 & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.30/0.66 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.30/0.66 = bot_bot_set_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % atLeastLessThan_nat_numeral
% 0.30/0.66 thf(fact_9529_image__minus__const__atLeastLessThan__nat,axiom,
% 0.30/0.66 ! [C: nat,Y3: nat,X2: nat] :
% 0.30/0.66 ( ( ( ord_less_nat @ C @ Y3 )
% 0.30/0.66 => ( ( image_nat_nat
% 0.30/0.66 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.30/0.66 @ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
% 0.30/0.66 = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X2 @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
% 0.30/0.66 & ( ~ ( ord_less_nat @ C @ Y3 )
% 0.30/0.66 => ( ( ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.66 => ( ( image_nat_nat
% 0.30/0.66 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.30/0.66 @ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
% 0.30/0.66 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 0.30/0.66 & ( ~ ( ord_less_nat @ X2 @ Y3 )
% 0.30/0.66 => ( ( image_nat_nat
% 0.30/0.66 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.30/0.66 @ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
% 0.30/0.66 = bot_bot_set_nat ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % image_minus_const_atLeastLessThan_nat
% 0.30/0.66 thf(fact_9530_atLeast1__lessThan__eq__remove0,axiom,
% 0.30/0.66 ! [N: nat] :
% 0.30/0.66 ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.66 = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % atLeast1_lessThan_eq_remove0
% 0.30/0.66 thf(fact_9531_sum__power2,axiom,
% 0.30/0.66 ! [K: nat] :
% 0.30/0.66 ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 0.30/0.66 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % sum_power2
% 0.30/0.66 thf(fact_9532_Chebyshev__sum__upper__nat,axiom,
% 0.30/0.66 ! [N: nat,A: nat > nat,B: nat > nat] :
% 0.30/0.66 ( ! [I3: nat,J2: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ I3 @ J2 )
% 0.30/0.66 => ( ( ord_less_nat @ J2 @ N )
% 0.30/0.66 => ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
% 0.30/0.66 => ( ! [I3: nat,J2: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ I3 @ J2 )
% 0.30/0.66 => ( ( ord_less_nat @ J2 @ N )
% 0.30/0.66 => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I3 ) ) ) )
% 0.30/0.66 => ( ord_less_eq_nat
% 0.30/0.66 @ ( times_times_nat @ N
% 0.30/0.66 @ ( groups3542108847815614940at_nat
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( B @ I4 ) )
% 0.30/0.66 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.66 @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Chebyshev_sum_upper_nat
% 0.30/0.66 thf(fact_9533_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
% 0.30/0.66 ! [L: int,U: int] :
% 0.30/0.66 ( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
% 0.30/0.66 = ( set_or1266510415728281911st_int @ L @ U ) ) ).
% 0.30/0.66
% 0.30/0.66 % atLeastLessThanPlusOne_atLeastAtMost_int
% 0.30/0.66 thf(fact_9534_image__add__int__atLeastLessThan,axiom,
% 0.30/0.66 ! [L: int,U: int] :
% 0.30/0.66 ( ( image_int_int
% 0.30/0.66 @ ^ [X: int] : ( plus_plus_int @ X @ L )
% 0.30/0.66 @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
% 0.30/0.66 = ( set_or4662586982721622107an_int @ L @ U ) ) ).
% 0.30/0.66
% 0.30/0.66 % image_add_int_atLeastLessThan
% 0.30/0.66 thf(fact_9535_image__atLeastZeroLessThan__int,axiom,
% 0.30/0.66 ! [U: int] :
% 0.30/0.66 ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.30/0.66 => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
% 0.30/0.66 = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % image_atLeastZeroLessThan_int
% 0.30/0.66 thf(fact_9536_VEBT_Osize_I3_J,axiom,
% 0.30/0.66 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.30/0.66 ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.30/0.66 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % VEBT.size(3)
% 0.30/0.66 thf(fact_9537_valid__eq2,axiom,
% 0.30/0.66 ! [T: vEBT_VEBT,D: nat] :
% 0.30/0.66 ( ( vEBT_VEBT_valid @ T @ D )
% 0.30/0.66 => ( vEBT_invar_vebt @ T @ D ) ) ).
% 0.30/0.66
% 0.30/0.66 % valid_eq2
% 0.30/0.66 thf(fact_9538_valid__eq1,axiom,
% 0.30/0.66 ! [T: vEBT_VEBT,D: nat] :
% 0.30/0.66 ( ( vEBT_invar_vebt @ T @ D )
% 0.30/0.66 => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 0.30/0.66
% 0.30/0.66 % valid_eq1
% 0.30/0.66 thf(fact_9539_valid__eq,axiom,
% 0.30/0.66 vEBT_VEBT_valid = vEBT_invar_vebt ).
% 0.30/0.66
% 0.30/0.66 % valid_eq
% 0.30/0.66 thf(fact_9540_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 0.30/0.66 ! [Uu: $o,Uv: $o,D: nat] :
% 0.30/0.66 ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
% 0.30/0.66 = ( D = one_one_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % VEBT_internal.valid'.simps(1)
% 0.30/0.66 thf(fact_9541_VEBT_Osize__gen_I1_J,axiom,
% 0.30/0.66 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.30/0.66 ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.30/0.66 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % VEBT.size_gen(1)
% 0.30/0.66 thf(fact_9542_VEBT_Osize__gen_I2_J,axiom,
% 0.30/0.66 ! [X21: $o,X222: $o] :
% 0.30/0.66 ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 0.30/0.66 = zero_zero_nat ) ).
% 0.30/0.66
% 0.30/0.66 % VEBT.size_gen(2)
% 0.30/0.66 thf(fact_9543_Code__Target__Int_Opositive__def,axiom,
% 0.30/0.66 code_Target_positive = numeral_numeral_int ).
% 0.30/0.66
% 0.30/0.66 % Code_Target_Int.positive_def
% 0.30/0.66 thf(fact_9544_csqrt_Osimps_I1_J,axiom,
% 0.30/0.66 ! [Z: complex] :
% 0.30/0.66 ( ( re @ ( csqrt @ Z ) )
% 0.30/0.66 = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % csqrt.simps(1)
% 0.30/0.66 thf(fact_9545_complex__Re__numeral,axiom,
% 0.30/0.66 ! [V: num] :
% 0.30/0.66 ( ( re @ ( numera6690914467698888265omplex @ V ) )
% 0.30/0.66 = ( numeral_numeral_real @ V ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_Re_numeral
% 0.30/0.66 thf(fact_9546_Re__divide__numeral,axiom,
% 0.30/0.66 ! [Z: complex,W: num] :
% 0.30/0.66 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.66 = ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Re_divide_numeral
% 0.30/0.66 thf(fact_9547_complex__Re__le__cmod,axiom,
% 0.30/0.66 ! [X2: complex] : ( ord_less_eq_real @ ( re @ X2 ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_Re_le_cmod
% 0.30/0.66 thf(fact_9548_one__complex_Osimps_I1_J,axiom,
% 0.30/0.66 ( ( re @ one_one_complex )
% 0.30/0.66 = one_one_real ) ).
% 0.30/0.66
% 0.30/0.66 % one_complex.simps(1)
% 0.30/0.66 thf(fact_9549_scaleR__complex_Osimps_I1_J,axiom,
% 0.30/0.66 ! [R2: real,X2: complex] :
% 0.30/0.66 ( ( re @ ( real_V2046097035970521341omplex @ R2 @ X2 ) )
% 0.30/0.66 = ( times_times_real @ R2 @ ( re @ X2 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % scaleR_complex.simps(1)
% 0.30/0.66 thf(fact_9550_minus__complex_Osimps_I1_J,axiom,
% 0.30/0.66 ! [X2: complex,Y3: complex] :
% 0.30/0.66 ( ( re @ ( minus_minus_complex @ X2 @ Y3 ) )
% 0.30/0.66 = ( minus_minus_real @ ( re @ X2 ) @ ( re @ Y3 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % minus_complex.simps(1)
% 0.30/0.66 thf(fact_9551_abs__Re__le__cmod,axiom,
% 0.30/0.66 ! [X2: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X2 ) ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 0.30/0.66
% 0.30/0.66 % abs_Re_le_cmod
% 0.30/0.66 thf(fact_9552_Re__csqrt,axiom,
% 0.30/0.66 ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Re_csqrt
% 0.30/0.66 thf(fact_9553_cmod__plus__Re__le__0__iff,axiom,
% 0.30/0.66 ! [Z: complex] :
% 0.30/0.66 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
% 0.30/0.66 = ( ( re @ Z )
% 0.30/0.66 = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % cmod_plus_Re_le_0_iff
% 0.30/0.66 thf(fact_9554_cos__n__Re__cis__pow__n,axiom,
% 0.30/0.66 ! [N: nat,A: real] :
% 0.30/0.66 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 0.30/0.66 = ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % cos_n_Re_cis_pow_n
% 0.30/0.66 thf(fact_9555_csqrt_Ocode,axiom,
% 0.30/0.66 ( csqrt
% 0.30/0.66 = ( ^ [Z3: complex] :
% 0.30/0.66 ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.66 @ ( times_times_real
% 0.30/0.66 @ ( if_real
% 0.30/0.66 @ ( ( im @ Z3 )
% 0.30/0.66 = zero_zero_real )
% 0.30/0.66 @ one_one_real
% 0.30/0.66 @ ( sgn_sgn_real @ ( im @ Z3 ) ) )
% 0.30/0.66 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % csqrt.code
% 0.30/0.66 thf(fact_9556_csqrt_Osimps_I2_J,axiom,
% 0.30/0.66 ! [Z: complex] :
% 0.30/0.66 ( ( im @ ( csqrt @ Z ) )
% 0.30/0.66 = ( times_times_real
% 0.30/0.66 @ ( if_real
% 0.30/0.66 @ ( ( im @ Z )
% 0.30/0.66 = zero_zero_real )
% 0.30/0.66 @ one_one_real
% 0.30/0.66 @ ( sgn_sgn_real @ ( im @ Z ) ) )
% 0.30/0.66 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % csqrt.simps(2)
% 0.30/0.66 thf(fact_9557_csqrt__of__real__nonpos,axiom,
% 0.30/0.66 ! [X2: complex] :
% 0.30/0.66 ( ( ( im @ X2 )
% 0.30/0.66 = zero_zero_real )
% 0.30/0.66 => ( ( ord_less_eq_real @ ( re @ X2 ) @ zero_zero_real )
% 0.30/0.66 => ( ( csqrt @ X2 )
% 0.30/0.66 = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X2 ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % csqrt_of_real_nonpos
% 0.30/0.66 thf(fact_9558_Im__power__real,axiom,
% 0.30/0.66 ! [X2: complex,N: nat] :
% 0.30/0.66 ( ( ( im @ X2 )
% 0.30/0.66 = zero_zero_real )
% 0.30/0.66 => ( ( im @ ( power_power_complex @ X2 @ N ) )
% 0.30/0.66 = zero_zero_real ) ) ).
% 0.30/0.66
% 0.30/0.66 % Im_power_real
% 0.30/0.66 thf(fact_9559_complex__Im__numeral,axiom,
% 0.30/0.66 ! [V: num] :
% 0.30/0.66 ( ( im @ ( numera6690914467698888265omplex @ V ) )
% 0.30/0.66 = zero_zero_real ) ).
% 0.30/0.66
% 0.30/0.66 % complex_Im_numeral
% 0.30/0.66 thf(fact_9560_Im__i__times,axiom,
% 0.30/0.66 ! [Z: complex] :
% 0.30/0.66 ( ( im @ ( times_times_complex @ imaginary_unit @ Z ) )
% 0.30/0.66 = ( re @ Z ) ) ).
% 0.30/0.66
% 0.30/0.66 % Im_i_times
% 0.30/0.66 thf(fact_9561_Re__power__real,axiom,
% 0.30/0.66 ! [X2: complex,N: nat] :
% 0.30/0.66 ( ( ( im @ X2 )
% 0.30/0.66 = zero_zero_real )
% 0.30/0.66 => ( ( re @ ( power_power_complex @ X2 @ N ) )
% 0.30/0.66 = ( power_power_real @ ( re @ X2 ) @ N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Re_power_real
% 0.30/0.66 thf(fact_9562_Re__i__times,axiom,
% 0.30/0.66 ! [Z: complex] :
% 0.30/0.66 ( ( re @ ( times_times_complex @ imaginary_unit @ Z ) )
% 0.30/0.66 = ( uminus_uminus_real @ ( im @ Z ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Re_i_times
% 0.30/0.66 thf(fact_9563_Im__divide__numeral,axiom,
% 0.30/0.66 ! [Z: complex,W: num] :
% 0.30/0.66 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.66 = ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Im_divide_numeral
% 0.30/0.66 thf(fact_9564_csqrt__of__real__nonneg,axiom,
% 0.30/0.66 ! [X2: complex] :
% 0.30/0.66 ( ( ( im @ X2 )
% 0.30/0.66 = zero_zero_real )
% 0.30/0.66 => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X2 ) )
% 0.30/0.66 => ( ( csqrt @ X2 )
% 0.30/0.66 = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X2 ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % csqrt_of_real_nonneg
% 0.30/0.66 thf(fact_9565_csqrt__minus,axiom,
% 0.30/0.66 ! [X2: complex] :
% 0.30/0.66 ( ( ( ord_less_real @ ( im @ X2 ) @ zero_zero_real )
% 0.30/0.66 | ( ( ( im @ X2 )
% 0.30/0.66 = zero_zero_real )
% 0.30/0.66 & ( ord_less_eq_real @ zero_zero_real @ ( re @ X2 ) ) ) )
% 0.30/0.66 => ( ( csqrt @ ( uminus1482373934393186551omplex @ X2 ) )
% 0.30/0.66 = ( times_times_complex @ imaginary_unit @ ( csqrt @ X2 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % csqrt_minus
% 0.30/0.66 thf(fact_9566_imaginary__unit_Osimps_I2_J,axiom,
% 0.30/0.66 ( ( im @ imaginary_unit )
% 0.30/0.66 = one_one_real ) ).
% 0.30/0.66
% 0.30/0.66 % imaginary_unit.simps(2)
% 0.30/0.66 thf(fact_9567_one__complex_Osimps_I2_J,axiom,
% 0.30/0.66 ( ( im @ one_one_complex )
% 0.30/0.66 = zero_zero_real ) ).
% 0.30/0.66
% 0.30/0.66 % one_complex.simps(2)
% 0.30/0.66 thf(fact_9568_scaleR__complex_Osimps_I2_J,axiom,
% 0.30/0.66 ! [R2: real,X2: complex] :
% 0.30/0.66 ( ( im @ ( real_V2046097035970521341omplex @ R2 @ X2 ) )
% 0.30/0.66 = ( times_times_real @ R2 @ ( im @ X2 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % scaleR_complex.simps(2)
% 0.30/0.66 thf(fact_9569_minus__complex_Osimps_I2_J,axiom,
% 0.30/0.66 ! [X2: complex,Y3: complex] :
% 0.30/0.66 ( ( im @ ( minus_minus_complex @ X2 @ Y3 ) )
% 0.30/0.66 = ( minus_minus_real @ ( im @ X2 ) @ ( im @ Y3 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % minus_complex.simps(2)
% 0.30/0.66 thf(fact_9570_abs__Im__le__cmod,axiom,
% 0.30/0.66 ! [X2: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X2 ) ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 0.30/0.66
% 0.30/0.66 % abs_Im_le_cmod
% 0.30/0.66 thf(fact_9571_times__complex_Osimps_I2_J,axiom,
% 0.30/0.66 ! [X2: complex,Y3: complex] :
% 0.30/0.66 ( ( im @ ( times_times_complex @ X2 @ Y3 ) )
% 0.30/0.66 = ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y3 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y3 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % times_complex.simps(2)
% 0.30/0.66 thf(fact_9572_cmod__Im__le__iff,axiom,
% 0.30/0.66 ! [X2: complex,Y3: complex] :
% 0.30/0.66 ( ( ( re @ X2 )
% 0.30/0.66 = ( re @ Y3 ) )
% 0.30/0.66 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 0.30/0.66 = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X2 ) ) @ ( abs_abs_real @ ( im @ Y3 ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % cmod_Im_le_iff
% 0.30/0.66 thf(fact_9573_cmod__Re__le__iff,axiom,
% 0.30/0.66 ! [X2: complex,Y3: complex] :
% 0.30/0.66 ( ( ( im @ X2 )
% 0.30/0.66 = ( im @ Y3 ) )
% 0.30/0.66 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 0.30/0.66 = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X2 ) ) @ ( abs_abs_real @ ( re @ Y3 ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % cmod_Re_le_iff
% 0.30/0.66 thf(fact_9574_times__complex_Osimps_I1_J,axiom,
% 0.30/0.66 ! [X2: complex,Y3: complex] :
% 0.30/0.66 ( ( re @ ( times_times_complex @ X2 @ Y3 ) )
% 0.30/0.66 = ( minus_minus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y3 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % times_complex.simps(1)
% 0.30/0.66 thf(fact_9575_scaleR__complex_Ocode,axiom,
% 0.30/0.66 ( real_V2046097035970521341omplex
% 0.30/0.66 = ( ^ [R5: real,X: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X ) ) @ ( times_times_real @ R5 @ ( im @ X ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % scaleR_complex.code
% 0.30/0.66 thf(fact_9576_minus__complex_Ocode,axiom,
% 0.30/0.66 ( minus_minus_complex
% 0.30/0.66 = ( ^ [X: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( re @ X ) @ ( re @ Y ) ) @ ( minus_minus_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % minus_complex.code
% 0.30/0.66 thf(fact_9577_csqrt__principal,axiom,
% 0.30/0.66 ! [Z: complex] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
% 0.30/0.66 | ( ( ( re @ ( csqrt @ Z ) )
% 0.30/0.66 = zero_zero_real )
% 0.30/0.66 & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % csqrt_principal
% 0.30/0.66 thf(fact_9578_cmod__le,axiom,
% 0.30/0.66 ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % cmod_le
% 0.30/0.66 thf(fact_9579_sin__n__Im__cis__pow__n,axiom,
% 0.30/0.66 ! [N: nat,A: real] :
% 0.30/0.66 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 0.30/0.66 = ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % sin_n_Im_cis_pow_n
% 0.30/0.66 thf(fact_9580_Re__exp,axiom,
% 0.30/0.66 ! [Z: complex] :
% 0.30/0.66 ( ( re @ ( exp_complex @ Z ) )
% 0.30/0.66 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Re_exp
% 0.30/0.66 thf(fact_9581_Im__exp,axiom,
% 0.30/0.66 ! [Z: complex] :
% 0.30/0.66 ( ( im @ ( exp_complex @ Z ) )
% 0.30/0.66 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Im_exp
% 0.30/0.66 thf(fact_9582_complex__eq,axiom,
% 0.30/0.66 ! [A: complex] :
% 0.30/0.66 ( A
% 0.30/0.66 = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_eq
% 0.30/0.66 thf(fact_9583_times__complex_Ocode,axiom,
% 0.30/0.66 ( times_times_complex
% 0.30/0.66 = ( ^ [X: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % times_complex.code
% 0.30/0.66 thf(fact_9584_exp__eq__polar,axiom,
% 0.30/0.66 ( exp_complex
% 0.30/0.66 = ( ^ [Z3: complex] : ( times_times_complex @ ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z3 ) ) ) @ ( cis @ ( im @ Z3 ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % exp_eq_polar
% 0.30/0.66 thf(fact_9585_cmod__power2,axiom,
% 0.30/0.66 ! [Z: complex] :
% 0.30/0.66 ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.66 = ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % cmod_power2
% 0.30/0.66 thf(fact_9586_Im__power2,axiom,
% 0.30/0.66 ! [X2: complex] :
% 0.30/0.66 ( ( im @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.66 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X2 ) ) @ ( im @ X2 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Im_power2
% 0.30/0.66 thf(fact_9587_Re__power2,axiom,
% 0.30/0.66 ! [X2: complex] :
% 0.30/0.66 ( ( re @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.66 = ( minus_minus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Re_power2
% 0.30/0.66 thf(fact_9588_complex__eq__0,axiom,
% 0.30/0.66 ! [Z: complex] :
% 0.30/0.66 ( ( Z = zero_zero_complex )
% 0.30/0.66 = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.66 = zero_zero_real ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_eq_0
% 0.30/0.66 thf(fact_9589_norm__complex__def,axiom,
% 0.30/0.66 ( real_V1022390504157884413omplex
% 0.30/0.66 = ( ^ [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 ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % norm_complex_def
% 0.30/0.66 thf(fact_9590_inverse__complex_Osimps_I1_J,axiom,
% 0.30/0.66 ! [X2: complex] :
% 0.30/0.66 ( ( re @ ( invers8013647133539491842omplex @ X2 ) )
% 0.30/0.66 = ( 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 ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % inverse_complex.simps(1)
% 0.30/0.66 thf(fact_9591_complex__neq__0,axiom,
% 0.30/0.66 ! [Z: complex] :
% 0.30/0.66 ( ( Z != zero_zero_complex )
% 0.30/0.66 = ( 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 ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_neq_0
% 0.30/0.66 thf(fact_9592_Re__divide,axiom,
% 0.30/0.66 ! [X2: complex,Y3: complex] :
% 0.30/0.66 ( ( re @ ( divide1717551699836669952omplex @ X2 @ Y3 ) )
% 0.30/0.66 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Re_divide
% 0.30/0.66 thf(fact_9593_csqrt__unique,axiom,
% 0.30/0.66 ! [W: complex,Z: complex] :
% 0.30/0.66 ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.66 = Z )
% 0.30/0.66 => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
% 0.30/0.66 | ( ( ( re @ W )
% 0.30/0.66 = zero_zero_real )
% 0.30/0.66 & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
% 0.30/0.66 => ( ( csqrt @ Z )
% 0.30/0.66 = W ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % csqrt_unique
% 0.30/0.66 thf(fact_9594_csqrt__square,axiom,
% 0.30/0.66 ! [B: complex] :
% 0.30/0.66 ( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
% 0.30/0.66 | ( ( ( re @ B )
% 0.30/0.66 = zero_zero_real )
% 0.30/0.66 & ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
% 0.30/0.66 => ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.66 = B ) ) ).
% 0.30/0.66
% 0.30/0.66 % csqrt_square
% 0.30/0.66 thf(fact_9595_inverse__complex_Osimps_I2_J,axiom,
% 0.30/0.66 ! [X2: complex] :
% 0.30/0.66 ( ( im @ ( invers8013647133539491842omplex @ X2 ) )
% 0.30/0.66 = ( 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 ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % inverse_complex.simps(2)
% 0.30/0.66 thf(fact_9596_Im__divide,axiom,
% 0.30/0.66 ! [X2: complex,Y3: complex] :
% 0.30/0.66 ( ( im @ ( divide1717551699836669952omplex @ X2 @ Y3 ) )
% 0.30/0.66 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Im_divide
% 0.30/0.66 thf(fact_9597_complex__abs__le__norm,axiom,
% 0.30/0.66 ! [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 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_abs_le_norm
% 0.30/0.66 thf(fact_9598_complex__unit__circle,axiom,
% 0.30/0.66 ! [Z: complex] :
% 0.30/0.66 ( ( Z != zero_zero_complex )
% 0.30/0.66 => ( ( 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 ) ) ) )
% 0.30/0.66 = one_one_real ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_unit_circle
% 0.30/0.66 thf(fact_9599_inverse__complex_Ocode,axiom,
% 0.30/0.66 ( invers8013647133539491842omplex
% 0.30/0.66 = ( ^ [X: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % inverse_complex.code
% 0.30/0.66 thf(fact_9600_Complex__divide,axiom,
% 0.30/0.66 ( divide1717551699836669952omplex
% 0.30/0.66 = ( ^ [X: complex,Y: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Complex_divide
% 0.30/0.66 thf(fact_9601_Im__Reals__divide,axiom,
% 0.30/0.66 ! [R2: complex,Z: complex] :
% 0.30/0.66 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 0.30/0.66 => ( ( im @ ( divide1717551699836669952omplex @ R2 @ Z ) )
% 0.30/0.66 = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R2 ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Im_Reals_divide
% 0.30/0.66 thf(fact_9602_Re__Reals__divide,axiom,
% 0.30/0.66 ! [R2: complex,Z: complex] :
% 0.30/0.66 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 0.30/0.66 => ( ( re @ ( divide1717551699836669952omplex @ R2 @ Z ) )
% 0.30/0.66 = ( divide_divide_real @ ( times_times_real @ ( re @ R2 ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Re_Reals_divide
% 0.30/0.66 thf(fact_9603_complex__diff__cnj,axiom,
% 0.30/0.66 ! [Z: complex] :
% 0.30/0.66 ( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
% 0.30/0.66 = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_diff_cnj
% 0.30/0.66 thf(fact_9604_complex__cnj__mult,axiom,
% 0.30/0.66 ! [X2: complex,Y3: complex] :
% 0.30/0.66 ( ( cnj @ ( times_times_complex @ X2 @ Y3 ) )
% 0.30/0.66 = ( times_times_complex @ ( cnj @ X2 ) @ ( cnj @ Y3 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_cnj_mult
% 0.30/0.66 thf(fact_9605_complex__cnj__one,axiom,
% 0.30/0.66 ( ( cnj @ one_one_complex )
% 0.30/0.66 = one_one_complex ) ).
% 0.30/0.66
% 0.30/0.66 % complex_cnj_one
% 0.30/0.66 thf(fact_9606_complex__cnj__one__iff,axiom,
% 0.30/0.66 ! [Z: complex] :
% 0.30/0.66 ( ( ( cnj @ Z )
% 0.30/0.66 = one_one_complex )
% 0.30/0.66 = ( Z = one_one_complex ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_cnj_one_iff
% 0.30/0.66 thf(fact_9607_complex__cnj__power,axiom,
% 0.30/0.66 ! [X2: complex,N: nat] :
% 0.30/0.66 ( ( cnj @ ( power_power_complex @ X2 @ N ) )
% 0.30/0.66 = ( power_power_complex @ ( cnj @ X2 ) @ N ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_cnj_power
% 0.30/0.66 thf(fact_9608_complex__cnj__numeral,axiom,
% 0.30/0.66 ! [W: num] :
% 0.30/0.66 ( ( cnj @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.66 = ( numera6690914467698888265omplex @ W ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_cnj_numeral
% 0.30/0.66 thf(fact_9609_complex__cnj__diff,axiom,
% 0.30/0.66 ! [X2: complex,Y3: complex] :
% 0.30/0.66 ( ( cnj @ ( minus_minus_complex @ X2 @ Y3 ) )
% 0.30/0.66 = ( minus_minus_complex @ ( cnj @ X2 ) @ ( cnj @ Y3 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_cnj_diff
% 0.30/0.66 thf(fact_9610_complex__cnj__neg__numeral,axiom,
% 0.30/0.66 ! [W: num] :
% 0.30/0.66 ( ( cnj @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.66 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_cnj_neg_numeral
% 0.30/0.66 thf(fact_9611_complex__In__mult__cnj__zero,axiom,
% 0.30/0.66 ! [Z: complex] :
% 0.30/0.66 ( ( im @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 0.30/0.66 = zero_zero_real ) ).
% 0.30/0.66
% 0.30/0.66 % complex_In_mult_cnj_zero
% 0.30/0.66 thf(fact_9612_real__eq__imaginary__iff,axiom,
% 0.30/0.66 ! [Y3: complex,X2: complex] :
% 0.30/0.66 ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
% 0.30/0.66 => ( ( member_complex @ X2 @ real_V2521375963428798218omplex )
% 0.30/0.66 => ( ( X2
% 0.30/0.66 = ( times_times_complex @ imaginary_unit @ Y3 ) )
% 0.30/0.66 = ( ( X2 = zero_zero_complex )
% 0.30/0.66 & ( Y3 = zero_zero_complex ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % real_eq_imaginary_iff
% 0.30/0.66 thf(fact_9613_imaginary__eq__real__iff,axiom,
% 0.30/0.66 ! [Y3: complex,X2: complex] :
% 0.30/0.66 ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
% 0.30/0.66 => ( ( member_complex @ X2 @ real_V2521375963428798218omplex )
% 0.30/0.66 => ( ( ( times_times_complex @ imaginary_unit @ Y3 )
% 0.30/0.66 = X2 )
% 0.30/0.66 = ( ( X2 = zero_zero_complex )
% 0.30/0.66 & ( Y3 = zero_zero_complex ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % imaginary_eq_real_iff
% 0.30/0.66 thf(fact_9614_Re__complex__div__eq__0,axiom,
% 0.30/0.66 ! [A: complex,B: complex] :
% 0.30/0.66 ( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.30/0.66 = zero_zero_real )
% 0.30/0.66 = ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 0.30/0.66 = zero_zero_real ) ) ).
% 0.30/0.66
% 0.30/0.66 % Re_complex_div_eq_0
% 0.30/0.66 thf(fact_9615_Im__complex__div__eq__0,axiom,
% 0.30/0.66 ! [A: complex,B: complex] :
% 0.30/0.66 ( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.30/0.66 = zero_zero_real )
% 0.30/0.66 = ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 0.30/0.66 = zero_zero_real ) ) ).
% 0.30/0.66
% 0.30/0.66 % Im_complex_div_eq_0
% 0.30/0.66 thf(fact_9616_complex__mod__sqrt__Re__mult__cnj,axiom,
% 0.30/0.66 ( real_V1022390504157884413omplex
% 0.30/0.66 = ( ^ [Z3: complex] : ( sqrt @ ( re @ ( times_times_complex @ Z3 @ ( cnj @ Z3 ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_mod_sqrt_Re_mult_cnj
% 0.30/0.66 thf(fact_9617_Re__complex__div__lt__0,axiom,
% 0.30/0.66 ! [A: complex,B: complex] :
% 0.30/0.66 ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.30/0.66 = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.30/0.66
% 0.30/0.66 % Re_complex_div_lt_0
% 0.30/0.66 thf(fact_9618_Re__complex__div__gt__0,axiom,
% 0.30/0.66 ! [A: complex,B: complex] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.30/0.66 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Re_complex_div_gt_0
% 0.30/0.66 thf(fact_9619_Re__complex__div__ge__0,axiom,
% 0.30/0.66 ! [A: complex,B: complex] :
% 0.30/0.66 ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.30/0.66 = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Re_complex_div_ge_0
% 0.30/0.66 thf(fact_9620_Re__complex__div__le__0,axiom,
% 0.30/0.66 ! [A: complex,B: complex] :
% 0.30/0.66 ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.30/0.66 = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.30/0.66
% 0.30/0.66 % Re_complex_div_le_0
% 0.30/0.66 thf(fact_9621_Im__complex__div__lt__0,axiom,
% 0.30/0.66 ! [A: complex,B: complex] :
% 0.30/0.66 ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.30/0.66 = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.30/0.66
% 0.30/0.66 % Im_complex_div_lt_0
% 0.30/0.66 thf(fact_9622_Im__complex__div__gt__0,axiom,
% 0.30/0.66 ! [A: complex,B: complex] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.30/0.66 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Im_complex_div_gt_0
% 0.30/0.66 thf(fact_9623_Im__complex__div__ge__0,axiom,
% 0.30/0.66 ! [A: complex,B: complex] :
% 0.30/0.66 ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.30/0.66 = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Im_complex_div_ge_0
% 0.30/0.66 thf(fact_9624_Im__complex__div__le__0,axiom,
% 0.30/0.66 ! [A: complex,B: complex] :
% 0.30/0.66 ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.30/0.66 = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.30/0.66
% 0.30/0.66 % Im_complex_div_le_0
% 0.30/0.66 thf(fact_9625_complex__mod__mult__cnj,axiom,
% 0.30/0.66 ! [Z: complex] :
% 0.30/0.66 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 0.30/0.66 = ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_mod_mult_cnj
% 0.30/0.66 thf(fact_9626_complex__div__gt__0,axiom,
% 0.30/0.66 ! [A: complex,B: complex] :
% 0.30/0.66 ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.30/0.66 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
% 0.30/0.66 & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.30/0.66 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_div_gt_0
% 0.30/0.66 thf(fact_9627_complex__norm__square,axiom,
% 0.30/0.66 ! [Z: complex] :
% 0.30/0.66 ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.66 = ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_norm_square
% 0.30/0.66 thf(fact_9628_complex__add__cnj,axiom,
% 0.30/0.66 ! [Z: complex] :
% 0.30/0.66 ( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
% 0.30/0.66 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_add_cnj
% 0.30/0.66 thf(fact_9629_complex__div__cnj,axiom,
% 0.30/0.66 ( divide1717551699836669952omplex
% 0.30/0.66 = ( ^ [A3: complex,B2: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ ( cnj @ B2 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_div_cnj
% 0.30/0.66 thf(fact_9630_cnj__add__mult__eq__Re,axiom,
% 0.30/0.66 ! [Z: complex,W: complex] :
% 0.30/0.66 ( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
% 0.30/0.66 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % cnj_add_mult_eq_Re
% 0.30/0.66 thf(fact_9631_complex__mult__cnj,axiom,
% 0.30/0.66 ! [Z: complex] :
% 0.30/0.66 ( ( times_times_complex @ Z @ ( cnj @ Z ) )
% 0.30/0.66 = ( 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 ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % complex_mult_cnj
% 0.30/0.66 thf(fact_9632_divmod__step__integer__def,axiom,
% 0.30/0.66 ( unique4921790084139445826nteger
% 0.30/0.66 = ( ^ [L2: num] :
% 0.30/0.66 ( produc6916734918728496179nteger
% 0.30/0.66 @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % divmod_step_integer_def
% 0.30/0.66 thf(fact_9633_card__Collect__less__nat,axiom,
% 0.30/0.66 ! [N: nat] :
% 0.30/0.66 ( ( finite_card_nat
% 0.30/0.66 @ ( collect_nat
% 0.30/0.66 @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
% 0.30/0.66 = N ) ).
% 0.30/0.66
% 0.30/0.66 % card_Collect_less_nat
% 0.30/0.66 thf(fact_9634_card__atMost,axiom,
% 0.30/0.66 ! [U: nat] :
% 0.30/0.66 ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
% 0.30/0.66 = ( suc @ U ) ) ).
% 0.30/0.66
% 0.30/0.66 % card_atMost
% 0.30/0.66 thf(fact_9635_card__atLeastLessThan,axiom,
% 0.30/0.66 ! [L: nat,U: nat] :
% 0.30/0.66 ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
% 0.30/0.66 = ( minus_minus_nat @ U @ L ) ) ).
% 0.30/0.66
% 0.30/0.66 % card_atLeastLessThan
% 0.30/0.66 thf(fact_9636_card__Collect__le__nat,axiom,
% 0.30/0.66 ! [N: nat] :
% 0.30/0.66 ( ( finite_card_nat
% 0.30/0.66 @ ( collect_nat
% 0.30/0.66 @ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
% 0.30/0.66 = ( suc @ N ) ) ).
% 0.30/0.66
% 0.30/0.66 % card_Collect_le_nat
% 0.30/0.66 thf(fact_9637_card__atLeastAtMost,axiom,
% 0.30/0.66 ! [L: nat,U: nat] :
% 0.30/0.66 ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 0.30/0.66 = ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% 0.30/0.66
% 0.30/0.66 % card_atLeastAtMost
% 0.30/0.66 thf(fact_9638_card__atLeastLessThan__int,axiom,
% 0.30/0.66 ! [L: int,U: int] :
% 0.30/0.66 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L @ U ) )
% 0.30/0.66 = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % card_atLeastLessThan_int
% 0.30/0.66 thf(fact_9639_card__atLeastAtMost__int,axiom,
% 0.30/0.66 ! [L: int,U: int] :
% 0.30/0.66 ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
% 0.30/0.66 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % card_atLeastAtMost_int
% 0.30/0.66 thf(fact_9640_minus__integer__code_I1_J,axiom,
% 0.30/0.66 ! [K: code_integer] :
% 0.30/0.66 ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
% 0.30/0.66 = K ) ).
% 0.30/0.66
% 0.30/0.66 % minus_integer_code(1)
% 0.30/0.66 thf(fact_9641_minus__integer__code_I2_J,axiom,
% 0.30/0.66 ! [L: code_integer] :
% 0.30/0.66 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
% 0.30/0.66 = ( uminus1351360451143612070nteger @ L ) ) ).
% 0.30/0.66
% 0.30/0.66 % minus_integer_code(2)
% 0.30/0.66 thf(fact_9642_times__integer__code_I1_J,axiom,
% 0.30/0.66 ! [K: code_integer] :
% 0.30/0.66 ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
% 0.30/0.66 = zero_z3403309356797280102nteger ) ).
% 0.30/0.66
% 0.30/0.66 % times_integer_code(1)
% 0.30/0.66 thf(fact_9643_times__integer__code_I2_J,axiom,
% 0.30/0.66 ! [L: code_integer] :
% 0.30/0.66 ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
% 0.30/0.66 = zero_z3403309356797280102nteger ) ).
% 0.30/0.66
% 0.30/0.66 % times_integer_code(2)
% 0.30/0.66 thf(fact_9644_divmod__integer_H__def,axiom,
% 0.30/0.66 ( unique3479559517661332726nteger
% 0.30/0.66 = ( ^ [M3: num,N3: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N3 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N3 ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % divmod_integer'_def
% 0.30/0.66 thf(fact_9645_sgn__integer__code,axiom,
% 0.30/0.66 ( sgn_sgn_Code_integer
% 0.30/0.66 = ( ^ [K2: code_integer] : ( if_Code_integer @ ( K2 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % sgn_integer_code
% 0.30/0.66 thf(fact_9646_less__eq__integer__code_I1_J,axiom,
% 0.30/0.66 ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).
% 0.30/0.66
% 0.30/0.66 % less_eq_integer_code(1)
% 0.30/0.66 thf(fact_9647_card__less__Suc2,axiom,
% 0.30/0.66 ! [M7: set_nat,I2: nat] :
% 0.30/0.66 ( ~ ( member_nat @ zero_zero_nat @ M7 )
% 0.30/0.66 => ( ( finite_card_nat
% 0.30/0.66 @ ( collect_nat
% 0.30/0.66 @ ^ [K2: nat] :
% 0.30/0.66 ( ( member_nat @ ( suc @ K2 ) @ M7 )
% 0.30/0.66 & ( ord_less_nat @ K2 @ I2 ) ) ) )
% 0.30/0.66 = ( finite_card_nat
% 0.30/0.66 @ ( collect_nat
% 0.30/0.66 @ ^ [K2: nat] :
% 0.30/0.66 ( ( member_nat @ K2 @ M7 )
% 0.30/0.66 & ( ord_less_nat @ K2 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % card_less_Suc2
% 0.30/0.66 thf(fact_9648_card__less__Suc,axiom,
% 0.30/0.66 ! [M7: set_nat,I2: nat] :
% 0.30/0.66 ( ( member_nat @ zero_zero_nat @ M7 )
% 0.30/0.66 => ( ( suc
% 0.30/0.66 @ ( finite_card_nat
% 0.30/0.66 @ ( collect_nat
% 0.30/0.66 @ ^ [K2: nat] :
% 0.30/0.66 ( ( member_nat @ ( suc @ K2 ) @ M7 )
% 0.30/0.66 & ( ord_less_nat @ K2 @ I2 ) ) ) ) )
% 0.30/0.66 = ( finite_card_nat
% 0.30/0.66 @ ( collect_nat
% 0.30/0.66 @ ^ [K2: nat] :
% 0.30/0.66 ( ( member_nat @ K2 @ M7 )
% 0.30/0.66 & ( ord_less_nat @ K2 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % card_less_Suc
% 0.30/0.66 thf(fact_9649_card__less,axiom,
% 0.30/0.66 ! [M7: set_nat,I2: nat] :
% 0.30/0.66 ( ( member_nat @ zero_zero_nat @ M7 )
% 0.30/0.66 => ( ( finite_card_nat
% 0.30/0.66 @ ( collect_nat
% 0.30/0.66 @ ^ [K2: nat] :
% 0.30/0.66 ( ( member_nat @ K2 @ M7 )
% 0.30/0.66 & ( ord_less_nat @ K2 @ ( suc @ I2 ) ) ) ) )
% 0.30/0.66 != zero_zero_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % card_less
% 0.30/0.66 thf(fact_9650_one__natural_Orsp,axiom,
% 0.30/0.66 one_one_nat = one_one_nat ).
% 0.30/0.66
% 0.30/0.66 % one_natural.rsp
% 0.30/0.66 thf(fact_9651_one__integer_Orsp,axiom,
% 0.30/0.66 one_one_int = one_one_int ).
% 0.30/0.66
% 0.30/0.66 % one_integer.rsp
% 0.30/0.66 thf(fact_9652_subset__card__intvl__is__intvl,axiom,
% 0.30/0.66 ! [A2: set_nat,K: nat] :
% 0.30/0.66 ( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
% 0.30/0.66 => ( A2
% 0.30/0.66 = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % subset_card_intvl_is_intvl
% 0.30/0.66 thf(fact_9653_card__le__Suc__Max,axiom,
% 0.30/0.66 ! [S3: set_nat] :
% 0.30/0.66 ( ( finite_finite_nat @ S3 )
% 0.30/0.66 => ( ord_less_eq_nat @ ( finite_card_nat @ S3 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S3 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % card_le_Suc_Max
% 0.30/0.66 thf(fact_9654_subset__eq__atLeast0__lessThan__card,axiom,
% 0.30/0.66 ! [N5: set_nat,N: nat] :
% 0.30/0.66 ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.30/0.66 => ( ord_less_eq_nat @ ( finite_card_nat @ N5 ) @ N ) ) ).
% 0.30/0.66
% 0.30/0.66 % subset_eq_atLeast0_lessThan_card
% 0.30/0.66 thf(fact_9655_card__sum__le__nat__sum,axiom,
% 0.30/0.66 ! [S3: set_nat] :
% 0.30/0.66 ( ord_less_eq_nat
% 0.30/0.66 @ ( groups3542108847815614940at_nat
% 0.30/0.66 @ ^ [X: nat] : X
% 0.30/0.66 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S3 ) ) )
% 0.30/0.66 @ ( groups3542108847815614940at_nat
% 0.30/0.66 @ ^ [X: nat] : X
% 0.30/0.66 @ S3 ) ) ).
% 0.30/0.66
% 0.30/0.66 % card_sum_le_nat_sum
% 0.30/0.66 thf(fact_9656_card__nth__roots,axiom,
% 0.30/0.66 ! [C: complex,N: nat] :
% 0.30/0.66 ( ( C != zero_zero_complex )
% 0.30/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( finite_card_complex
% 0.30/0.66 @ ( collect_complex
% 0.30/0.66 @ ^ [Z3: complex] :
% 0.30/0.66 ( ( power_power_complex @ Z3 @ N )
% 0.30/0.66 = C ) ) )
% 0.30/0.66 = N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % card_nth_roots
% 0.30/0.66 thf(fact_9657_card__roots__unity__eq,axiom,
% 0.30/0.66 ! [N: nat] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( finite_card_complex
% 0.30/0.66 @ ( collect_complex
% 0.30/0.66 @ ^ [Z3: complex] :
% 0.30/0.66 ( ( power_power_complex @ Z3 @ N )
% 0.30/0.66 = one_one_complex ) ) )
% 0.30/0.66 = N ) ) ).
% 0.30/0.66
% 0.30/0.66 % card_roots_unity_eq
% 0.30/0.66 thf(fact_9658_integer__of__int__code,axiom,
% 0.30/0.66 ( code_integer_of_int
% 0.30/0.66 = ( ^ [K2: int] :
% 0.30/0.66 ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K2 ) ) )
% 0.30/0.66 @ ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 0.30/0.66 @ ( if_Code_integer
% 0.30/0.66 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.66 = zero_zero_int )
% 0.30/0.66 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % integer_of_int_code
% 0.30/0.66 thf(fact_9659_times__integer_Oabs__eq,axiom,
% 0.30/0.66 ! [Xa2: int,X2: int] :
% 0.30/0.66 ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X2 ) )
% 0.30/0.66 = ( code_integer_of_int @ ( times_times_int @ Xa2 @ X2 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % times_integer.abs_eq
% 0.30/0.66 thf(fact_9660_one__integer__def,axiom,
% 0.30/0.66 ( one_one_Code_integer
% 0.30/0.66 = ( code_integer_of_int @ one_one_int ) ) ).
% 0.30/0.66
% 0.30/0.66 % one_integer_def
% 0.30/0.66 thf(fact_9661_less__eq__integer_Oabs__eq,axiom,
% 0.30/0.66 ! [Xa2: int,X2: int] :
% 0.30/0.66 ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X2 ) )
% 0.30/0.66 = ( ord_less_eq_int @ Xa2 @ X2 ) ) ).
% 0.30/0.66
% 0.30/0.66 % less_eq_integer.abs_eq
% 0.30/0.66 thf(fact_9662_minus__integer_Oabs__eq,axiom,
% 0.30/0.66 ! [Xa2: int,X2: int] :
% 0.30/0.66 ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X2 ) )
% 0.30/0.66 = ( code_integer_of_int @ ( minus_minus_int @ Xa2 @ X2 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % minus_integer.abs_eq
% 0.30/0.66 thf(fact_9663_Code__Numeral_Opositive__def,axiom,
% 0.30/0.66 code_positive = numera6620942414471956472nteger ).
% 0.30/0.66
% 0.30/0.66 % Code_Numeral.positive_def
% 0.30/0.66 thf(fact_9664_integer__of__num_I3_J,axiom,
% 0.30/0.66 ! [N: num] :
% 0.30/0.66 ( ( code_integer_of_num @ ( bit1 @ N ) )
% 0.30/0.66 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).
% 0.30/0.66
% 0.30/0.66 % integer_of_num(3)
% 0.30/0.66 thf(fact_9665_int__of__integer__code,axiom,
% 0.30/0.66 ( code_int_of_integer
% 0.30/0.66 = ( ^ [K2: code_integer] :
% 0.30/0.66 ( if_int @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K2 ) ) )
% 0.30/0.66 @ ( if_int @ ( K2 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 0.30/0.66 @ ( produc1553301316500091796er_int
% 0.30/0.66 @ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
% 0.30/0.66 @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % int_of_integer_code
% 0.30/0.66 thf(fact_9666_int__of__integer__numeral,axiom,
% 0.30/0.66 ! [K: num] :
% 0.30/0.66 ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 0.30/0.66 = ( numeral_numeral_int @ K ) ) ).
% 0.30/0.66
% 0.30/0.66 % int_of_integer_numeral
% 0.30/0.66 thf(fact_9667_times__integer_Orep__eq,axiom,
% 0.30/0.66 ! [X2: code_integer,Xa2: code_integer] :
% 0.30/0.66 ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X2 @ Xa2 ) )
% 0.30/0.66 = ( times_times_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % times_integer.rep_eq
% 0.30/0.66 thf(fact_9668_one__integer_Orep__eq,axiom,
% 0.30/0.66 ( ( code_int_of_integer @ one_one_Code_integer )
% 0.30/0.66 = one_one_int ) ).
% 0.30/0.66
% 0.30/0.66 % one_integer.rep_eq
% 0.30/0.66 thf(fact_9669_minus__integer_Orep__eq,axiom,
% 0.30/0.66 ! [X2: code_integer,Xa2: code_integer] :
% 0.30/0.66 ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X2 @ Xa2 ) )
% 0.30/0.66 = ( minus_minus_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % minus_integer.rep_eq
% 0.30/0.66 thf(fact_9670_less__eq__integer_Orep__eq,axiom,
% 0.30/0.66 ( ord_le3102999989581377725nteger
% 0.30/0.66 = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % less_eq_integer.rep_eq
% 0.30/0.66 thf(fact_9671_integer__less__eq__iff,axiom,
% 0.30/0.66 ( ord_le3102999989581377725nteger
% 0.30/0.66 = ( ^ [K2: code_integer,L2: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % integer_less_eq_iff
% 0.30/0.66 thf(fact_9672_integer__of__num__def,axiom,
% 0.30/0.66 code_integer_of_num = numera6620942414471956472nteger ).
% 0.30/0.66
% 0.30/0.66 % integer_of_num_def
% 0.30/0.66 thf(fact_9673_integer__of__num__triv_I1_J,axiom,
% 0.30/0.66 ( ( code_integer_of_num @ one )
% 0.30/0.66 = one_one_Code_integer ) ).
% 0.30/0.66
% 0.30/0.66 % integer_of_num_triv(1)
% 0.30/0.66 thf(fact_9674_integer__of__num_I2_J,axiom,
% 0.30/0.66 ! [N: num] :
% 0.30/0.66 ( ( code_integer_of_num @ ( bit0 @ N ) )
% 0.30/0.66 = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % integer_of_num(2)
% 0.30/0.66 thf(fact_9675_integer__of__num__triv_I2_J,axiom,
% 0.30/0.66 ( ( code_integer_of_num @ ( bit0 @ one ) )
% 0.30/0.66 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % integer_of_num_triv(2)
% 0.30/0.66 thf(fact_9676_num__of__integer__code,axiom,
% 0.30/0.66 ( code_num_of_integer
% 0.30/0.66 = ( ^ [K2: code_integer] :
% 0.30/0.66 ( if_num @ ( ord_le3102999989581377725nteger @ K2 @ one_one_Code_integer ) @ one
% 0.30/0.66 @ ( produc7336495610019696514er_num
% 0.30/0.66 @ ^ [L2: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one ) )
% 0.30/0.66 @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % num_of_integer_code
% 0.30/0.66 thf(fact_9677_nat__of__integer__code,axiom,
% 0.30/0.66 ( code_nat_of_integer
% 0.30/0.66 = ( ^ [K2: code_integer] :
% 0.30/0.66 ( if_nat @ ( ord_le3102999989581377725nteger @ K2 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 0.30/0.66 @ ( produc1555791787009142072er_nat
% 0.30/0.66 @ ^ [L2: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ one_one_nat ) )
% 0.30/0.66 @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % nat_of_integer_code
% 0.30/0.66 thf(fact_9678_bit__cut__integer__def,axiom,
% 0.30/0.66 ( code_bit_cut_integer
% 0.30/0.66 = ( ^ [K2: code_integer] :
% 0.30/0.66 ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.66 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K2 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % bit_cut_integer_def
% 0.30/0.66 thf(fact_9679_nat__of__integer__non__positive,axiom,
% 0.30/0.66 ! [K: code_integer] :
% 0.30/0.66 ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
% 0.30/0.66 => ( ( code_nat_of_integer @ K )
% 0.30/0.66 = zero_zero_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % nat_of_integer_non_positive
% 0.30/0.66 thf(fact_9680_nat__of__integer__code__post_I3_J,axiom,
% 0.30/0.66 ! [K: num] :
% 0.30/0.66 ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 0.30/0.66 = ( numeral_numeral_nat @ K ) ) ).
% 0.30/0.66
% 0.30/0.66 % nat_of_integer_code_post(3)
% 0.30/0.66 thf(fact_9681_nat__of__integer__code__post_I2_J,axiom,
% 0.30/0.66 ( ( code_nat_of_integer @ one_one_Code_integer )
% 0.30/0.66 = one_one_nat ) ).
% 0.30/0.66
% 0.30/0.66 % nat_of_integer_code_post(2)
% 0.30/0.66 thf(fact_9682_bit__cut__integer__code,axiom,
% 0.30/0.66 ( code_bit_cut_integer
% 0.30/0.66 = ( ^ [K2: code_integer] :
% 0.30/0.66 ( if_Pro5737122678794959658eger_o @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 0.30/0.66 @ ( produc9125791028180074456eger_o
% 0.30/0.66 @ ^ [R5: code_integer,S6: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S6 ) ) @ ( S6 = one_one_Code_integer ) )
% 0.30/0.66 @ ( code_divmod_abs @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % bit_cut_integer_code
% 0.30/0.66 thf(fact_9683_divmod__integer__code,axiom,
% 0.30/0.66 ( code_divmod_integer
% 0.30/0.66 = ( ^ [K2: code_integer,L2: code_integer] :
% 0.30/0.66 ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.30/0.66 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
% 0.30/0.66 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ ( code_divmod_abs @ K2 @ L2 )
% 0.30/0.66 @ ( produc6916734918728496179nteger
% 0.30/0.66 @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S6 ) ) )
% 0.30/0.66 @ ( code_divmod_abs @ K2 @ L2 ) ) )
% 0.30/0.66 @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 0.30/0.66 @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 0.30/0.66 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K2 @ L2 )
% 0.30/0.66 @ ( produc6916734918728496179nteger
% 0.30/0.66 @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S6 ) ) )
% 0.30/0.66 @ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % divmod_integer_code
% 0.30/0.66 thf(fact_9684_floor__real__def,axiom,
% 0.30/0.66 ( archim6058952711729229775r_real
% 0.30/0.66 = ( ^ [X: real] :
% 0.30/0.66 ( the_int
% 0.30/0.66 @ ^ [Z3: int] :
% 0.30/0.66 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X )
% 0.30/0.66 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % floor_real_def
% 0.30/0.66 thf(fact_9685_floor__rat__def,axiom,
% 0.30/0.66 ( archim3151403230148437115or_rat
% 0.30/0.66 = ( ^ [X: rat] :
% 0.30/0.66 ( the_int
% 0.30/0.66 @ ^ [Z3: int] :
% 0.30/0.66 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X )
% 0.30/0.66 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % floor_rat_def
% 0.30/0.66 thf(fact_9686_less__eq__rat__def,axiom,
% 0.30/0.66 ( ord_less_eq_rat
% 0.30/0.66 = ( ^ [X: rat,Y: rat] :
% 0.30/0.66 ( ( ord_less_rat @ X @ Y )
% 0.30/0.66 | ( X = Y ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % less_eq_rat_def
% 0.30/0.66 thf(fact_9687_obtain__pos__sum,axiom,
% 0.30/0.66 ! [R2: rat] :
% 0.30/0.66 ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 0.30/0.66 => ~ ! [S2: rat] :
% 0.30/0.66 ( ( ord_less_rat @ zero_zero_rat @ S2 )
% 0.30/0.66 => ! [T5: rat] :
% 0.30/0.66 ( ( ord_less_rat @ zero_zero_rat @ T5 )
% 0.30/0.66 => ( R2
% 0.30/0.66 != ( plus_plus_rat @ S2 @ T5 ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % obtain_pos_sum
% 0.30/0.66 thf(fact_9688_sgn__rat__def,axiom,
% 0.30/0.66 ( sgn_sgn_rat
% 0.30/0.66 = ( ^ [A3: rat] : ( if_rat @ ( A3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % sgn_rat_def
% 0.30/0.66 thf(fact_9689_nat_Odisc__eq__case_I1_J,axiom,
% 0.30/0.66 ! [Nat: nat] :
% 0.30/0.66 ( ( Nat = zero_zero_nat )
% 0.30/0.66 = ( case_nat_o @ $true
% 0.30/0.66 @ ^ [Uu3: nat] : $false
% 0.30/0.66 @ Nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % nat.disc_eq_case(1)
% 0.30/0.66 thf(fact_9690_nat_Odisc__eq__case_I2_J,axiom,
% 0.30/0.66 ! [Nat: nat] :
% 0.30/0.66 ( ( Nat != zero_zero_nat )
% 0.30/0.66 = ( case_nat_o @ $false
% 0.30/0.66 @ ^ [Uu3: nat] : $true
% 0.30/0.66 @ Nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % nat.disc_eq_case(2)
% 0.30/0.66 thf(fact_9691_less__eq__nat_Osimps_I2_J,axiom,
% 0.30/0.66 ! [M: nat,N: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.30/0.66 = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 0.30/0.66
% 0.30/0.66 % less_eq_nat.simps(2)
% 0.30/0.66 thf(fact_9692_max__Suc2,axiom,
% 0.30/0.66 ! [M: nat,N: nat] :
% 0.30/0.66 ( ( ord_max_nat @ M @ ( suc @ N ) )
% 0.30/0.66 = ( case_nat_nat @ ( suc @ N )
% 0.30/0.66 @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ M5 @ N ) )
% 0.30/0.66 @ M ) ) ).
% 0.30/0.66
% 0.30/0.66 % max_Suc2
% 0.30/0.66 thf(fact_9693_max__Suc1,axiom,
% 0.30/0.66 ! [N: nat,M: nat] :
% 0.30/0.66 ( ( ord_max_nat @ ( suc @ N ) @ M )
% 0.30/0.66 = ( case_nat_nat @ ( suc @ N )
% 0.30/0.66 @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ N @ M5 ) )
% 0.30/0.66 @ M ) ) ).
% 0.30/0.66
% 0.30/0.66 % max_Suc1
% 0.30/0.66 thf(fact_9694_diff__Suc,axiom,
% 0.30/0.66 ! [M: nat,N: nat] :
% 0.30/0.66 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.30/0.66 = ( case_nat_nat @ zero_zero_nat
% 0.30/0.66 @ ^ [K2: nat] : K2
% 0.30/0.66 @ ( minus_minus_nat @ M @ N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % diff_Suc
% 0.30/0.66 thf(fact_9695_rat__inverse__code,axiom,
% 0.30/0.66 ! [P6: rat] :
% 0.30/0.66 ( ( quotient_of @ ( inverse_inverse_rat @ P6 ) )
% 0.30/0.66 = ( produc4245557441103728435nt_int
% 0.30/0.66 @ ^ [A3: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( A3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A3 ) @ B2 ) @ ( abs_abs_int @ A3 ) ) )
% 0.30/0.66 @ ( quotient_of @ P6 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % rat_inverse_code
% 0.30/0.66 thf(fact_9696_quotient__of__number_I3_J,axiom,
% 0.30/0.66 ! [K: num] :
% 0.30/0.66 ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
% 0.30/0.66 = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% 0.30/0.66
% 0.30/0.66 % quotient_of_number(3)
% 0.30/0.66 thf(fact_9697_rat__one__code,axiom,
% 0.30/0.66 ( ( quotient_of @ one_one_rat )
% 0.30/0.66 = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% 0.30/0.66
% 0.30/0.66 % rat_one_code
% 0.30/0.66 thf(fact_9698_rat__zero__code,axiom,
% 0.30/0.66 ( ( quotient_of @ zero_zero_rat )
% 0.30/0.66 = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 0.30/0.66
% 0.30/0.66 % rat_zero_code
% 0.30/0.66 thf(fact_9699_quotient__of__number_I5_J,axiom,
% 0.30/0.66 ! [K: num] :
% 0.30/0.66 ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.30/0.66 = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 0.30/0.66
% 0.30/0.66 % quotient_of_number(5)
% 0.30/0.66 thf(fact_9700_quotient__of__number_I4_J,axiom,
% 0.30/0.66 ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.66 = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).
% 0.30/0.66
% 0.30/0.66 % quotient_of_number(4)
% 0.30/0.66 thf(fact_9701_diff__rat__def,axiom,
% 0.30/0.66 ( minus_minus_rat
% 0.30/0.66 = ( ^ [Q4: rat,R5: rat] : ( plus_plus_rat @ Q4 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % diff_rat_def
% 0.30/0.66 thf(fact_9702_divide__rat__def,axiom,
% 0.30/0.66 ( divide_divide_rat
% 0.30/0.66 = ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % divide_rat_def
% 0.30/0.66 thf(fact_9703_pred__def,axiom,
% 0.30/0.66 ( pred
% 0.30/0.66 = ( case_nat_nat @ zero_zero_nat
% 0.30/0.66 @ ^ [X24: nat] : X24 ) ) ).
% 0.30/0.66
% 0.30/0.66 % pred_def
% 0.30/0.66 thf(fact_9704_rat__less__code,axiom,
% 0.30/0.66 ( ord_less_rat
% 0.30/0.66 = ( ^ [P4: rat,Q4: rat] :
% 0.30/0.66 ( produc4947309494688390418_int_o
% 0.30/0.66 @ ^ [A3: int,C2: int] :
% 0.30/0.66 ( produc4947309494688390418_int_o
% 0.30/0.66 @ ^ [B2: int,D2: int] : ( ord_less_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C2 @ B2 ) )
% 0.30/0.66 @ ( quotient_of @ Q4 ) )
% 0.30/0.66 @ ( quotient_of @ P4 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % rat_less_code
% 0.30/0.66 thf(fact_9705_rat__less__eq__code,axiom,
% 0.30/0.66 ( ord_less_eq_rat
% 0.30/0.66 = ( ^ [P4: rat,Q4: rat] :
% 0.30/0.66 ( produc4947309494688390418_int_o
% 0.30/0.66 @ ^ [A3: int,C2: int] :
% 0.30/0.66 ( produc4947309494688390418_int_o
% 0.30/0.66 @ ^ [B2: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C2 @ B2 ) )
% 0.30/0.66 @ ( quotient_of @ Q4 ) )
% 0.30/0.66 @ ( quotient_of @ P4 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % rat_less_eq_code
% 0.30/0.66 thf(fact_9706_quotient__of__int,axiom,
% 0.30/0.66 ! [A: int] :
% 0.30/0.66 ( ( quotient_of @ ( of_int @ A ) )
% 0.30/0.66 = ( product_Pair_int_int @ A @ one_one_int ) ) ).
% 0.30/0.66
% 0.30/0.66 % quotient_of_int
% 0.30/0.66 thf(fact_9707_rat__plus__code,axiom,
% 0.30/0.66 ! [P6: rat,Q3: rat] :
% 0.30/0.66 ( ( quotient_of @ ( plus_plus_rat @ P6 @ Q3 ) )
% 0.30/0.66 = ( produc4245557441103728435nt_int
% 0.30/0.66 @ ^ [A3: int,C2: int] :
% 0.30/0.66 ( produc4245557441103728435nt_int
% 0.30/0.66 @ ^ [B2: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ B2 @ C2 ) ) @ ( times_times_int @ C2 @ D2 ) ) )
% 0.30/0.66 @ ( quotient_of @ Q3 ) )
% 0.30/0.66 @ ( quotient_of @ P6 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % rat_plus_code
% 0.30/0.66 thf(fact_9708_rat__minus__code,axiom,
% 0.30/0.66 ! [P6: rat,Q3: rat] :
% 0.30/0.66 ( ( quotient_of @ ( minus_minus_rat @ P6 @ Q3 ) )
% 0.30/0.66 = ( produc4245557441103728435nt_int
% 0.30/0.66 @ ^ [A3: int,C2: int] :
% 0.30/0.66 ( produc4245557441103728435nt_int
% 0.30/0.66 @ ^ [B2: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ B2 @ C2 ) ) @ ( times_times_int @ C2 @ D2 ) ) )
% 0.30/0.66 @ ( quotient_of @ Q3 ) )
% 0.30/0.66 @ ( quotient_of @ P6 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % rat_minus_code
% 0.30/0.66 thf(fact_9709_normalize__denom__zero,axiom,
% 0.30/0.66 ! [P6: int] :
% 0.30/0.66 ( ( normalize @ ( product_Pair_int_int @ P6 @ zero_zero_int ) )
% 0.30/0.66 = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 0.30/0.66
% 0.30/0.66 % normalize_denom_zero
% 0.30/0.66 thf(fact_9710_normalize__crossproduct,axiom,
% 0.30/0.66 ! [Q3: int,S: int,P6: int,R2: int] :
% 0.30/0.66 ( ( Q3 != zero_zero_int )
% 0.30/0.66 => ( ( S != zero_zero_int )
% 0.30/0.66 => ( ( ( normalize @ ( product_Pair_int_int @ P6 @ Q3 ) )
% 0.30/0.66 = ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
% 0.30/0.66 => ( ( times_times_int @ P6 @ S )
% 0.30/0.66 = ( times_times_int @ R2 @ Q3 ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % normalize_crossproduct
% 0.30/0.66 thf(fact_9711_rat__divide__code,axiom,
% 0.30/0.66 ! [P6: rat,Q3: rat] :
% 0.30/0.66 ( ( quotient_of @ ( divide_divide_rat @ P6 @ Q3 ) )
% 0.30/0.66 = ( produc4245557441103728435nt_int
% 0.30/0.66 @ ^ [A3: int,C2: int] :
% 0.30/0.66 ( produc4245557441103728435nt_int
% 0.30/0.66 @ ^ [B2: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C2 @ B2 ) ) )
% 0.30/0.66 @ ( quotient_of @ Q3 ) )
% 0.30/0.66 @ ( quotient_of @ P6 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % rat_divide_code
% 0.30/0.66 thf(fact_9712_rat__times__code,axiom,
% 0.30/0.66 ! [P6: rat,Q3: rat] :
% 0.30/0.66 ( ( quotient_of @ ( times_times_rat @ P6 @ Q3 ) )
% 0.30/0.66 = ( produc4245557441103728435nt_int
% 0.30/0.66 @ ^ [A3: int,C2: int] :
% 0.30/0.66 ( produc4245557441103728435nt_int
% 0.30/0.66 @ ^ [B2: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ B2 ) @ ( times_times_int @ C2 @ D2 ) ) )
% 0.30/0.66 @ ( quotient_of @ Q3 ) )
% 0.30/0.66 @ ( quotient_of @ P6 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % rat_times_code
% 0.30/0.66 thf(fact_9713_Frct__code__post_I5_J,axiom,
% 0.30/0.66 ! [K: num] :
% 0.30/0.66 ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.66 = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Frct_code_post(5)
% 0.30/0.66 thf(fact_9714_Frct__code__post_I6_J,axiom,
% 0.30/0.66 ! [K: num,L: num] :
% 0.30/0.66 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
% 0.30/0.66 = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Frct_code_post(6)
% 0.30/0.66 thf(fact_9715_bezw__0,axiom,
% 0.30/0.66 ! [X2: nat] :
% 0.30/0.66 ( ( bezw @ X2 @ zero_zero_nat )
% 0.30/0.66 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% 0.30/0.66
% 0.30/0.66 % bezw_0
% 0.30/0.66 thf(fact_9716_Frct__code__post_I3_J,axiom,
% 0.30/0.66 ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
% 0.30/0.66 = one_one_rat ) ).
% 0.30/0.66
% 0.30/0.66 % Frct_code_post(3)
% 0.30/0.66 thf(fact_9717_Frct__code__post_I4_J,axiom,
% 0.30/0.66 ! [K: num] :
% 0.30/0.66 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
% 0.30/0.66 = ( numeral_numeral_rat @ K ) ) ).
% 0.30/0.66
% 0.30/0.66 % Frct_code_post(4)
% 0.30/0.66 thf(fact_9718_drop__bit__numeral__minus__bit1,axiom,
% 0.30/0.66 ! [L: num,K: num] :
% 0.30/0.66 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.30/0.66 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % drop_bit_numeral_minus_bit1
% 0.30/0.66 thf(fact_9719_prod__decode__aux_Osimps,axiom,
% 0.30/0.66 ( nat_prod_decode_aux
% 0.30/0.66 = ( ^ [K2: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K2 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K2 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M3 @ ( suc @ K2 ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % prod_decode_aux.simps
% 0.30/0.66 thf(fact_9720_prod__decode__aux_Oelims,axiom,
% 0.30/0.66 ! [X2: nat,Xa2: nat,Y3: product_prod_nat_nat] :
% 0.30/0.66 ( ( ( nat_prod_decode_aux @ X2 @ Xa2 )
% 0.30/0.66 = Y3 )
% 0.30/0.66 => ( ( ( ord_less_eq_nat @ Xa2 @ X2 )
% 0.30/0.66 => ( Y3
% 0.30/0.66 = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X2 @ Xa2 ) ) ) )
% 0.30/0.66 & ( ~ ( ord_less_eq_nat @ Xa2 @ X2 )
% 0.30/0.66 => ( Y3
% 0.30/0.66 = ( nat_prod_decode_aux @ ( suc @ X2 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % prod_decode_aux.elims
% 0.30/0.66 thf(fact_9721_drop__bit__nonnegative__int__iff,axiom,
% 0.30/0.66 ! [N: nat,K: int] :
% 0.30/0.66 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
% 0.30/0.66 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.30/0.66
% 0.30/0.66 % drop_bit_nonnegative_int_iff
% 0.30/0.66 thf(fact_9722_drop__bit__negative__int__iff,axiom,
% 0.30/0.66 ! [N: nat,K: int] :
% 0.30/0.66 ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
% 0.30/0.66 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.66
% 0.30/0.66 % drop_bit_negative_int_iff
% 0.30/0.66 thf(fact_9723_drop__bit__minus__one,axiom,
% 0.30/0.66 ! [N: nat] :
% 0.30/0.66 ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.66 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.66
% 0.30/0.66 % drop_bit_minus_one
% 0.30/0.66 thf(fact_9724_drop__bit__Suc__minus__bit0,axiom,
% 0.30/0.66 ! [N: nat,K: num] :
% 0.30/0.66 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.30/0.66 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % drop_bit_Suc_minus_bit0
% 0.30/0.66 thf(fact_9725_drop__bit__numeral__minus__bit0,axiom,
% 0.30/0.66 ! [L: num,K: num] :
% 0.30/0.66 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.30/0.66 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % drop_bit_numeral_minus_bit0
% 0.30/0.66 thf(fact_9726_drop__bit__Suc__minus__bit1,axiom,
% 0.30/0.66 ! [N: nat,K: num] :
% 0.30/0.66 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.30/0.66 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % drop_bit_Suc_minus_bit1
% 0.30/0.66 thf(fact_9727_drop__bit__push__bit__int,axiom,
% 0.30/0.66 ! [M: nat,N: nat,K: int] :
% 0.30/0.66 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N @ K ) )
% 0.30/0.66 = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M ) @ K ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % drop_bit_push_bit_int
% 0.30/0.66 thf(fact_9728_drop__bit__int__def,axiom,
% 0.30/0.66 ( bit_se8568078237143864401it_int
% 0.30/0.66 = ( ^ [N3: nat,K2: int] : ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % drop_bit_int_def
% 0.30/0.66 thf(fact_9729_prod__decode__aux_Opelims,axiom,
% 0.30/0.66 ! [X2: nat,Xa2: nat,Y3: product_prod_nat_nat] :
% 0.30/0.66 ( ( ( nat_prod_decode_aux @ X2 @ Xa2 )
% 0.30/0.66 = Y3 )
% 0.30/0.66 => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) )
% 0.30/0.66 => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X2 )
% 0.30/0.66 => ( Y3
% 0.30/0.66 = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X2 @ Xa2 ) ) ) )
% 0.30/0.66 & ( ~ ( ord_less_eq_nat @ Xa2 @ X2 )
% 0.30/0.66 => ( Y3
% 0.30/0.66 = ( nat_prod_decode_aux @ ( suc @ X2 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X2 ) ) ) ) ) )
% 0.30/0.66 => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % prod_decode_aux.pelims
% 0.30/0.66 thf(fact_9730_Suc__0__mod__numeral,axiom,
% 0.30/0.66 ! [K: num] :
% 0.30/0.66 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.66 = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Suc_0_mod_numeral
% 0.30/0.66 thf(fact_9731_Suc__0__div__numeral,axiom,
% 0.30/0.66 ! [K: num] :
% 0.30/0.66 ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.66 = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Suc_0_div_numeral
% 0.30/0.66 thf(fact_9732_drop__bit__of__Suc__0,axiom,
% 0.30/0.66 ! [N: nat] :
% 0.30/0.66 ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.30/0.66 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % drop_bit_of_Suc_0
% 0.30/0.66 thf(fact_9733_fst__divmod__nat,axiom,
% 0.30/0.66 ! [M: nat,N: nat] :
% 0.30/0.66 ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
% 0.30/0.66 = ( divide_divide_nat @ M @ N ) ) ).
% 0.30/0.66
% 0.30/0.66 % fst_divmod_nat
% 0.30/0.66 thf(fact_9734_drop__bit__nat__eq,axiom,
% 0.30/0.66 ! [N: nat,K: int] :
% 0.30/0.66 ( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K ) )
% 0.30/0.66 = ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % drop_bit_nat_eq
% 0.30/0.66 thf(fact_9735_drop__bit__nat__def,axiom,
% 0.30/0.66 ( bit_se8570568707652914677it_nat
% 0.30/0.66 = ( ^ [N3: nat,M3: nat] : ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % drop_bit_nat_def
% 0.30/0.66 thf(fact_9736_bezw__non__0,axiom,
% 0.30/0.66 ! [Y3: nat,X2: nat] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ Y3 )
% 0.30/0.66 => ( ( bezw @ X2 @ Y3 )
% 0.30/0.66 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % bezw_non_0
% 0.30/0.66 thf(fact_9737_bezw_Osimps,axiom,
% 0.30/0.66 ( bezw
% 0.30/0.66 = ( ^ [X: nat,Y: nat] : ( if_Pro3027730157355071871nt_int @ ( Y = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % bezw.simps
% 0.30/0.66 thf(fact_9738_bezw_Oelims,axiom,
% 0.30/0.66 ! [X2: nat,Xa2: nat,Y3: product_prod_int_int] :
% 0.30/0.66 ( ( ( bezw @ X2 @ Xa2 )
% 0.30/0.66 = Y3 )
% 0.30/0.66 => ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.66 => ( Y3
% 0.30/0.66 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.30/0.66 & ( ( Xa2 != zero_zero_nat )
% 0.30/0.66 => ( Y3
% 0.30/0.66 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Xa2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % bezw.elims
% 0.30/0.66 thf(fact_9739_rat__sgn__code,axiom,
% 0.30/0.66 ! [P6: rat] :
% 0.30/0.66 ( ( quotient_of @ ( sgn_sgn_rat @ P6 ) )
% 0.30/0.66 = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P6 ) ) ) @ one_one_int ) ) ).
% 0.30/0.66
% 0.30/0.66 % rat_sgn_code
% 0.30/0.66 thf(fact_9740_bezw_Opelims,axiom,
% 0.30/0.66 ! [X2: nat,Xa2: nat,Y3: product_prod_int_int] :
% 0.30/0.66 ( ( ( bezw @ X2 @ Xa2 )
% 0.30/0.66 = Y3 )
% 0.30/0.66 => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) )
% 0.30/0.66 => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.66 => ( Y3
% 0.30/0.66 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.30/0.66 & ( ( Xa2 != zero_zero_nat )
% 0.30/0.66 => ( Y3
% 0.30/0.66 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Xa2 ) ) ) ) ) ) ) )
% 0.30/0.66 => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % bezw.pelims
% 0.30/0.66 thf(fact_9741_one__mod__minus__numeral,axiom,
% 0.30/0.66 ! [N: num] :
% 0.30/0.66 ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.66 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % one_mod_minus_numeral
% 0.30/0.66 thf(fact_9742_minus__one__mod__numeral,axiom,
% 0.30/0.66 ! [N: num] :
% 0.30/0.66 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.66 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % minus_one_mod_numeral
% 0.30/0.66 thf(fact_9743_minus__numeral__mod__numeral,axiom,
% 0.30/0.66 ! [M: num,N: num] :
% 0.30/0.66 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.66 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % minus_numeral_mod_numeral
% 0.30/0.66 thf(fact_9744_numeral__mod__minus__numeral,axiom,
% 0.30/0.66 ! [M: num,N: num] :
% 0.30/0.66 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.66 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % numeral_mod_minus_numeral
% 0.30/0.66 thf(fact_9745_Divides_Oadjust__mod__def,axiom,
% 0.30/0.66 ( adjust_mod
% 0.30/0.66 = ( ^ [L2: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L2 @ R5 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Divides.adjust_mod_def
% 0.30/0.66 thf(fact_9746_normalize__def,axiom,
% 0.30/0.66 ( normalize
% 0.30/0.66 = ( ^ [P4: product_prod_int_int] :
% 0.30/0.66 ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P4 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P4 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P4 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) )
% 0.30/0.66 @ ( if_Pro3027730157355071871nt_int
% 0.30/0.66 @ ( ( product_snd_int_int @ P4 )
% 0.30/0.66 = zero_zero_int )
% 0.30/0.66 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.30/0.66 @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P4 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P4 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % normalize_def
% 0.30/0.66 thf(fact_9747_finite__enumerate,axiom,
% 0.30/0.66 ! [S3: set_nat] :
% 0.30/0.66 ( ( finite_finite_nat @ S3 )
% 0.30/0.66 => ? [R3: nat > nat] :
% 0.30/0.66 ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S3 ) ) )
% 0.30/0.66 & ! [N9: nat] :
% 0.30/0.66 ( ( ord_less_nat @ N9 @ ( finite_card_nat @ S3 ) )
% 0.30/0.66 => ( member_nat @ ( R3 @ N9 ) @ S3 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % finite_enumerate
% 0.30/0.66 thf(fact_9748_gcd__1__int,axiom,
% 0.30/0.66 ! [M: int] :
% 0.30/0.66 ( ( gcd_gcd_int @ M @ one_one_int )
% 0.30/0.66 = one_one_int ) ).
% 0.30/0.66
% 0.30/0.66 % gcd_1_int
% 0.30/0.66 thf(fact_9749_gcd__neg__numeral__1__int,axiom,
% 0.30/0.66 ! [N: num,X2: int] :
% 0.30/0.66 ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ X2 )
% 0.30/0.66 = ( gcd_gcd_int @ ( numeral_numeral_int @ N ) @ X2 ) ) ).
% 0.30/0.66
% 0.30/0.66 % gcd_neg_numeral_1_int
% 0.30/0.66 thf(fact_9750_gcd__neg__numeral__2__int,axiom,
% 0.30/0.66 ! [X2: int,N: num] :
% 0.30/0.66 ( ( gcd_gcd_int @ X2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.66 = ( gcd_gcd_int @ X2 @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % gcd_neg_numeral_2_int
% 0.30/0.66 thf(fact_9751_gcd__ge__0__int,axiom,
% 0.30/0.66 ! [X2: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X2 @ Y3 ) ) ).
% 0.30/0.66
% 0.30/0.66 % gcd_ge_0_int
% 0.30/0.66 thf(fact_9752_bezout__int,axiom,
% 0.30/0.66 ! [X2: int,Y3: int] :
% 0.30/0.66 ? [U3: int,V2: int] :
% 0.30/0.66 ( ( plus_plus_int @ ( times_times_int @ U3 @ X2 ) @ ( times_times_int @ V2 @ Y3 ) )
% 0.30/0.66 = ( gcd_gcd_int @ X2 @ Y3 ) ) ).
% 0.30/0.66
% 0.30/0.66 % bezout_int
% 0.30/0.66 thf(fact_9753_gcd__mult__distrib__int,axiom,
% 0.30/0.66 ! [K: int,M: int,N: int] :
% 0.30/0.66 ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
% 0.30/0.66 = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % gcd_mult_distrib_int
% 0.30/0.66 thf(fact_9754_gcd__le2__int,axiom,
% 0.30/0.66 ! [B: int,A: int] :
% 0.30/0.66 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.66 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% 0.30/0.66
% 0.30/0.66 % gcd_le2_int
% 0.30/0.66 thf(fact_9755_gcd__le1__int,axiom,
% 0.30/0.66 ! [A: int,B: int] :
% 0.30/0.66 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.66 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% 0.30/0.66
% 0.30/0.66 % gcd_le1_int
% 0.30/0.66 thf(fact_9756_gcd__cases__int,axiom,
% 0.30/0.66 ! [X2: int,Y3: int,P: int > $o] :
% 0.30/0.66 ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.66 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.66 => ( P @ ( gcd_gcd_int @ X2 @ Y3 ) ) ) )
% 0.30/0.66 => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.30/0.66 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 0.30/0.66 => ( P @ ( gcd_gcd_int @ X2 @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 0.30/0.66 => ( ( ( ord_less_eq_int @ X2 @ zero_zero_int )
% 0.30/0.66 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.66 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X2 ) @ Y3 ) ) ) )
% 0.30/0.66 => ( ( ( ord_less_eq_int @ X2 @ zero_zero_int )
% 0.30/0.66 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 0.30/0.66 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X2 ) @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 0.30/0.66 => ( P @ ( gcd_gcd_int @ X2 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % gcd_cases_int
% 0.30/0.66 thf(fact_9757_gcd__unique__int,axiom,
% 0.30/0.66 ! [D: int,A: int,B: int] :
% 0.30/0.66 ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 0.30/0.66 & ( dvd_dvd_int @ D @ A )
% 0.30/0.66 & ( dvd_dvd_int @ D @ B )
% 0.30/0.66 & ! [E3: int] :
% 0.30/0.66 ( ( ( dvd_dvd_int @ E3 @ A )
% 0.30/0.66 & ( dvd_dvd_int @ E3 @ B ) )
% 0.30/0.66 => ( dvd_dvd_int @ E3 @ D ) ) )
% 0.30/0.66 = ( D
% 0.30/0.66 = ( gcd_gcd_int @ A @ B ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % gcd_unique_int
% 0.30/0.66 thf(fact_9758_nat__descend__induct,axiom,
% 0.30/0.66 ! [N: nat,P: nat > $o,M: nat] :
% 0.30/0.66 ( ! [K3: nat] :
% 0.30/0.66 ( ( ord_less_nat @ N @ K3 )
% 0.30/0.66 => ( P @ K3 ) )
% 0.30/0.66 => ( ! [K3: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ K3 @ N )
% 0.30/0.66 => ( ! [I: nat] :
% 0.30/0.66 ( ( ord_less_nat @ K3 @ I )
% 0.30/0.66 => ( P @ I ) )
% 0.30/0.66 => ( P @ K3 ) ) )
% 0.30/0.66 => ( P @ M ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % nat_descend_induct
% 0.30/0.66 thf(fact_9759_gcd__1__nat,axiom,
% 0.30/0.66 ! [M: nat] :
% 0.30/0.66 ( ( gcd_gcd_nat @ M @ one_one_nat )
% 0.30/0.66 = one_one_nat ) ).
% 0.30/0.66
% 0.30/0.66 % gcd_1_nat
% 0.30/0.66 thf(fact_9760_gcd__Suc__0,axiom,
% 0.30/0.66 ! [M: nat] :
% 0.30/0.66 ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.30/0.66 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % gcd_Suc_0
% 0.30/0.66 thf(fact_9761_gcd__pos__nat,axiom,
% 0.30/0.66 ! [M: nat,N: nat] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
% 0.30/0.66 = ( ( M != zero_zero_nat )
% 0.30/0.66 | ( N != zero_zero_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % gcd_pos_nat
% 0.30/0.66 thf(fact_9762_gcd__le2__nat,axiom,
% 0.30/0.66 ! [B: nat,A: nat] :
% 0.30/0.66 ( ( B != zero_zero_nat )
% 0.30/0.66 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% 0.30/0.66
% 0.30/0.66 % gcd_le2_nat
% 0.30/0.66 thf(fact_9763_gcd__le1__nat,axiom,
% 0.30/0.66 ! [A: nat,B: nat] :
% 0.30/0.66 ( ( A != zero_zero_nat )
% 0.30/0.66 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% 0.30/0.66
% 0.30/0.66 % gcd_le1_nat
% 0.30/0.66 thf(fact_9764_gcd__mult__distrib__nat,axiom,
% 0.30/0.66 ! [K: nat,M: nat,N: nat] :
% 0.30/0.66 ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
% 0.30/0.66 = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % gcd_mult_distrib_nat
% 0.30/0.66 thf(fact_9765_gcd__diff1__nat,axiom,
% 0.30/0.66 ! [N: nat,M: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.66 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 0.30/0.66 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % gcd_diff1_nat
% 0.30/0.66 thf(fact_9766_gcd__diff2__nat,axiom,
% 0.30/0.66 ! [M: nat,N: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.66 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
% 0.30/0.66 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % gcd_diff2_nat
% 0.30/0.66 thf(fact_9767_bezout__nat,axiom,
% 0.30/0.66 ! [A: nat,B: nat] :
% 0.30/0.66 ( ( A != zero_zero_nat )
% 0.30/0.66 => ? [X5: nat,Y4: nat] :
% 0.30/0.66 ( ( times_times_nat @ A @ X5 )
% 0.30/0.66 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % bezout_nat
% 0.30/0.66 thf(fact_9768_bezout__gcd__nat_H,axiom,
% 0.30/0.66 ! [B: nat,A: nat] :
% 0.30/0.66 ? [X5: nat,Y4: nat] :
% 0.30/0.66 ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y4 ) @ ( times_times_nat @ A @ X5 ) )
% 0.30/0.66 & ( ( minus_minus_nat @ ( times_times_nat @ A @ X5 ) @ ( times_times_nat @ B @ Y4 ) )
% 0.30/0.66 = ( gcd_gcd_nat @ A @ B ) ) )
% 0.30/0.66 | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B @ X5 ) )
% 0.30/0.66 & ( ( minus_minus_nat @ ( times_times_nat @ B @ X5 ) @ ( times_times_nat @ A @ Y4 ) )
% 0.30/0.66 = ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % bezout_gcd_nat'
% 0.30/0.66 thf(fact_9769_gcd__is__Max__divisors__nat,axiom,
% 0.30/0.66 ! [N: nat,M: nat] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( gcd_gcd_nat @ M @ N )
% 0.30/0.66 = ( lattic8265883725875713057ax_nat
% 0.30/0.66 @ ( collect_nat
% 0.30/0.66 @ ^ [D2: nat] :
% 0.30/0.66 ( ( dvd_dvd_nat @ D2 @ M )
% 0.30/0.66 & ( dvd_dvd_nat @ D2 @ N ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % gcd_is_Max_divisors_nat
% 0.30/0.66 thf(fact_9770_bezw__aux,axiom,
% 0.30/0.66 ! [X2: nat,Y3: nat] :
% 0.30/0.66 ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X2 @ Y3 ) )
% 0.30/0.66 = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X2 @ Y3 ) ) @ ( semiri1314217659103216013at_int @ X2 ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X2 @ Y3 ) ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % bezw_aux
% 0.30/0.66 thf(fact_9771_gcd__nat_Opelims,axiom,
% 0.30/0.66 ! [X2: nat,Xa2: nat,Y3: nat] :
% 0.30/0.66 ( ( ( gcd_gcd_nat @ X2 @ Xa2 )
% 0.30/0.66 = Y3 )
% 0.30/0.66 => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) )
% 0.30/0.66 => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.66 => ( Y3 = X2 ) )
% 0.30/0.66 & ( ( Xa2 != zero_zero_nat )
% 0.30/0.66 => ( Y3
% 0.30/0.66 = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) ) )
% 0.30/0.66 => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % gcd_nat.pelims
% 0.30/0.66 thf(fact_9772_card__greaterThanLessThan__int,axiom,
% 0.30/0.66 ! [L: int,U: int] :
% 0.30/0.66 ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
% 0.30/0.66 = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % card_greaterThanLessThan_int
% 0.30/0.66 thf(fact_9773_xor__minus__numerals_I1_J,axiom,
% 0.30/0.66 ! [N: num,K: int] :
% 0.30/0.66 ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
% 0.30/0.66 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % xor_minus_numerals(1)
% 0.30/0.66 thf(fact_9774_xor__minus__numerals_I2_J,axiom,
% 0.30/0.66 ! [K: int,N: num] :
% 0.30/0.66 ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.66 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % xor_minus_numerals(2)
% 0.30/0.66 thf(fact_9775_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
% 0.30/0.66 ! [L: int,U: int] :
% 0.30/0.66 ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
% 0.30/0.66 = ( set_or5832277885323065728an_int @ L @ U ) ) ).
% 0.30/0.66
% 0.30/0.66 % atLeastPlusOneLessThan_greaterThanLessThan_int
% 0.30/0.66 thf(fact_9776_sub__BitM__One__eq,axiom,
% 0.30/0.66 ! [N: num] :
% 0.30/0.66 ( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
% 0.30/0.66 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % sub_BitM_One_eq
% 0.30/0.66 thf(fact_9777_card__greaterThanLessThan,axiom,
% 0.30/0.66 ! [L: nat,U: nat] :
% 0.30/0.66 ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
% 0.30/0.66 = ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % card_greaterThanLessThan
% 0.30/0.66 thf(fact_9778_atLeastSucLessThan__greaterThanLessThan,axiom,
% 0.30/0.66 ! [L: nat,U: nat] :
% 0.30/0.66 ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
% 0.30/0.66 = ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% 0.30/0.66
% 0.30/0.66 % atLeastSucLessThan_greaterThanLessThan
% 0.30/0.66 thf(fact_9779_tanh__real__bounds,axiom,
% 0.30/0.66 ! [X2: real] : ( member_real @ ( tanh_real @ X2 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).
% 0.30/0.66
% 0.30/0.66 % tanh_real_bounds
% 0.30/0.66 thf(fact_9780_Suc__funpow,axiom,
% 0.30/0.66 ! [N: nat] :
% 0.30/0.66 ( ( compow_nat_nat @ N @ suc )
% 0.30/0.66 = ( plus_plus_nat @ N ) ) ).
% 0.30/0.66
% 0.30/0.66 % Suc_funpow
% 0.30/0.66 thf(fact_9781_divmod__integer__eq__cases,axiom,
% 0.30/0.66 ( code_divmod_integer
% 0.30/0.66 = ( ^ [K2: code_integer,L2: code_integer] :
% 0.30/0.66 ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.30/0.66 @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 0.30/0.66 @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
% 0.30/0.66 @ ( if_Pro6119634080678213985nteger
% 0.30/0.66 @ ( ( sgn_sgn_Code_integer @ K2 )
% 0.30/0.66 = ( sgn_sgn_Code_integer @ L2 ) )
% 0.30/0.66 @ ( code_divmod_abs @ K2 @ L2 )
% 0.30/0.66 @ ( produc6916734918728496179nteger
% 0.30/0.66 @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S6 ) ) )
% 0.30/0.66 @ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % divmod_integer_eq_cases
% 0.30/0.66 thf(fact_9782_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.30/0.66 ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 0.30/0.66 @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
% 0.30/0.66 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ).
% 0.30/0.66
% 0.30/0.66 % max_nat.semilattice_neutr_order_axioms
% 0.30/0.66 thf(fact_9783_card_Ocomp__fun__commute__on,axiom,
% 0.30/0.66 ( ( comp_nat_nat_nat @ suc @ suc )
% 0.30/0.66 = ( comp_nat_nat_nat @ suc @ suc ) ) ).
% 0.30/0.66
% 0.30/0.66 % card.comp_fun_commute_on
% 0.30/0.66 thf(fact_9784_Inf__real__def,axiom,
% 0.30/0.66 ( comple4887499456419720421f_real
% 0.30/0.66 = ( ^ [X4: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X4 ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Inf_real_def
% 0.30/0.66 thf(fact_9785_Code__Numeral_Onegative__def,axiom,
% 0.30/0.66 ( code_negative
% 0.30/0.66 = ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).
% 0.30/0.66
% 0.30/0.66 % Code_Numeral.negative_def
% 0.30/0.66 thf(fact_9786_Code__Target__Int_Onegative__def,axiom,
% 0.30/0.66 ( code_Target_negative
% 0.30/0.66 = ( comp_int_int_num @ uminus_uminus_int @ numeral_numeral_int ) ) ).
% 0.30/0.66
% 0.30/0.66 % Code_Target_Int.negative_def
% 0.30/0.66 thf(fact_9787_suminf__eq__SUP__real,axiom,
% 0.30/0.66 ! [X8: nat > real] :
% 0.30/0.66 ( ( summable_real @ X8 )
% 0.30/0.66 => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X8 @ I3 ) )
% 0.30/0.66 => ( ( suminf_real @ X8 )
% 0.30/0.66 = ( comple1385675409528146559p_real
% 0.30/0.66 @ ( image_nat_real
% 0.30/0.66 @ ^ [I4: nat] : ( groups6591440286371151544t_real @ X8 @ ( set_ord_lessThan_nat @ I4 ) )
% 0.30/0.66 @ top_top_set_nat ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % suminf_eq_SUP_real
% 0.30/0.66 thf(fact_9788_range__mod,axiom,
% 0.30/0.66 ! [N: nat] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( image_nat_nat
% 0.30/0.66 @ ^ [M3: nat] : ( modulo_modulo_nat @ M3 @ N )
% 0.30/0.66 @ top_top_set_nat )
% 0.30/0.66 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % range_mod
% 0.30/0.66 thf(fact_9789_UNIV__nat__eq,axiom,
% 0.30/0.66 ( top_top_set_nat
% 0.30/0.66 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % UNIV_nat_eq
% 0.30/0.66 thf(fact_9790_card__UNIV__unit,axiom,
% 0.30/0.66 ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
% 0.30/0.66 = one_one_nat ) ).
% 0.30/0.66
% 0.30/0.66 % card_UNIV_unit
% 0.30/0.66 thf(fact_9791_card__UNIV__bool,axiom,
% 0.30/0.66 ( ( finite_card_o @ top_top_set_o )
% 0.30/0.66 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % card_UNIV_bool
% 0.30/0.66 thf(fact_9792_range__mult,axiom,
% 0.30/0.66 ! [A: real] :
% 0.30/0.66 ( ( ( A = zero_zero_real )
% 0.30/0.66 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.30/0.66 = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 0.30/0.66 & ( ( A != zero_zero_real )
% 0.30/0.66 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.30/0.66 = top_top_set_real ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % range_mult
% 0.30/0.66 thf(fact_9793_root__def,axiom,
% 0.30/0.66 ( root
% 0.30/0.66 = ( ^ [N3: nat,X: real] :
% 0.30/0.66 ( if_real @ ( N3 = zero_zero_nat ) @ zero_zero_real
% 0.30/0.66 @ ( the_in5290026491893676941l_real @ top_top_set_real
% 0.30/0.66 @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N3 ) )
% 0.30/0.66 @ X ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % root_def
% 0.30/0.66 thf(fact_9794_card__UNIV__char,axiom,
% 0.30/0.66 ( ( finite_card_char @ top_top_set_char )
% 0.30/0.66 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % card_UNIV_char
% 0.30/0.66 thf(fact_9795_UNIV__char__of__nat,axiom,
% 0.30/0.66 ( top_top_set_char
% 0.30/0.66 = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % UNIV_char_of_nat
% 0.30/0.66 thf(fact_9796_nat__of__char__less__256,axiom,
% 0.30/0.66 ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % nat_of_char_less_256
% 0.30/0.66 thf(fact_9797_range__nat__of__char,axiom,
% 0.30/0.66 ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
% 0.30/0.66 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % range_nat_of_char
% 0.30/0.66 thf(fact_9798_integer__of__char__code,axiom,
% 0.30/0.66 ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 0.30/0.66 ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 0.30/0.66 = ( 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 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % integer_of_char_code
% 0.30/0.66 thf(fact_9799_String_Ochar__of__ascii__of,axiom,
% 0.30/0.66 ! [C: char] :
% 0.30/0.66 ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
% 0.30/0.66 = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % String.char_of_ascii_of
% 0.30/0.66 thf(fact_9800_DERIV__real__root__generic,axiom,
% 0.30/0.66 ! [N: nat,X2: real,D4: real] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( X2 != zero_zero_real )
% 0.30/0.66 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.66 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.66 => ( D4
% 0.30/0.66 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 0.30/0.66 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.66 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.30/0.66 => ( D4
% 0.30/0.66 = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 0.30/0.66 => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.66 => ( D4
% 0.30/0.66 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_real_root_generic
% 0.30/0.66 thf(fact_9801_deriv__nonneg__imp__mono,axiom,
% 0.30/0.66 ! [A: real,B: real,G: real > real,G2: real > real] :
% 0.30/0.66 ( ! [X5: real] :
% 0.30/0.66 ( ( member_real @ X5 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 0.30/0.66 => ( ! [X5: real] :
% 0.30/0.66 ( ( member_real @ X5 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 0.30/0.66 => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X5 ) ) )
% 0.30/0.66 => ( ( ord_less_eq_real @ A @ B )
% 0.30/0.66 => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % deriv_nonneg_imp_mono
% 0.30/0.66 thf(fact_9802_DERIV__nonpos__imp__nonincreasing,axiom,
% 0.30/0.66 ! [A: real,B: real,F: real > real] :
% 0.30/0.66 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.66 => ( ! [X5: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.66 => ( ( ord_less_eq_real @ X5 @ B )
% 0.30/0.66 => ? [Y5: real] :
% 0.30/0.66 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.66 & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
% 0.30/0.66 => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_nonpos_imp_nonincreasing
% 0.30/0.66 thf(fact_9803_DERIV__nonneg__imp__nondecreasing,axiom,
% 0.30/0.66 ! [A: real,B: real,F: real > real] :
% 0.30/0.66 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.66 => ( ! [X5: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.66 => ( ( ord_less_eq_real @ X5 @ B )
% 0.30/0.66 => ? [Y5: real] :
% 0.30/0.66 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.66 & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
% 0.30/0.66 => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_nonneg_imp_nondecreasing
% 0.30/0.66 thf(fact_9804_DERIV__const__ratio__const,axiom,
% 0.30/0.66 ! [A: real,B: real,F: real > real,K: real] :
% 0.30/0.66 ( ( A != B )
% 0.30/0.66 => ( ! [X5: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.66 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.30/0.66 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ K ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_const_ratio_const
% 0.30/0.66 thf(fact_9805_DERIV__const__ratio__const2,axiom,
% 0.30/0.66 ! [A: real,B: real,F: real > real,K: real] :
% 0.30/0.66 ( ( A != B )
% 0.30/0.66 => ( ! [X5: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.66 => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
% 0.30/0.66 = K ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_const_ratio_const2
% 0.30/0.66 thf(fact_9806_DERIV__neg__dec__left,axiom,
% 0.30/0.66 ! [F: real > real,L: real,X2: real] :
% 0.30/0.66 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.30/0.66 => ( ( ord_less_real @ L @ zero_zero_real )
% 0.30/0.66 => ? [D3: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.30/0.66 & ! [H4: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.66 => ( ( ord_less_real @ H4 @ D3 )
% 0.30/0.66 => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_neg_dec_left
% 0.30/0.66 thf(fact_9807_DERIV__pos__inc__left,axiom,
% 0.30/0.66 ! [F: real > real,L: real,X2: real] :
% 0.30/0.66 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.30/0.66 => ( ( ord_less_real @ zero_zero_real @ L )
% 0.30/0.66 => ? [D3: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.30/0.66 & ! [H4: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.66 => ( ( ord_less_real @ H4 @ D3 )
% 0.30/0.66 => ( ord_less_real @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_pos_inc_left
% 0.30/0.66 thf(fact_9808_has__real__derivative__neg__dec__left,axiom,
% 0.30/0.66 ! [F: real > real,L: real,X2: real,S3: set_real] :
% 0.30/0.66 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S3 ) )
% 0.30/0.66 => ( ( ord_less_real @ L @ zero_zero_real )
% 0.30/0.66 => ? [D3: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.30/0.66 & ! [H4: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.66 => ( ( member_real @ ( minus_minus_real @ X2 @ H4 ) @ S3 )
% 0.30/0.66 => ( ( ord_less_real @ H4 @ D3 )
% 0.30/0.66 => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % has_real_derivative_neg_dec_left
% 0.30/0.66 thf(fact_9809_has__real__derivative__pos__inc__left,axiom,
% 0.30/0.66 ! [F: real > real,L: real,X2: real,S3: set_real] :
% 0.30/0.66 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S3 ) )
% 0.30/0.66 => ( ( ord_less_real @ zero_zero_real @ L )
% 0.30/0.66 => ? [D3: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.30/0.66 & ! [H4: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.66 => ( ( member_real @ ( minus_minus_real @ X2 @ H4 ) @ S3 )
% 0.30/0.66 => ( ( ord_less_real @ H4 @ D3 )
% 0.30/0.66 => ( ord_less_real @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % has_real_derivative_pos_inc_left
% 0.30/0.66 thf(fact_9810_DERIV__neg__imp__decreasing,axiom,
% 0.30/0.66 ! [A: real,B: real,F: real > real] :
% 0.30/0.66 ( ( ord_less_real @ A @ B )
% 0.30/0.66 => ( ! [X5: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.66 => ( ( ord_less_eq_real @ X5 @ B )
% 0.30/0.66 => ? [Y5: real] :
% 0.30/0.66 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.66 & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 0.30/0.66 => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_neg_imp_decreasing
% 0.30/0.66 thf(fact_9811_DERIV__pos__imp__increasing,axiom,
% 0.30/0.66 ! [A: real,B: real,F: real > real] :
% 0.30/0.66 ( ( ord_less_real @ A @ B )
% 0.30/0.66 => ( ! [X5: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.66 => ( ( ord_less_eq_real @ X5 @ B )
% 0.30/0.66 => ? [Y5: real] :
% 0.30/0.66 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.66 & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 0.30/0.66 => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_pos_imp_increasing
% 0.30/0.66 thf(fact_9812_MVT2,axiom,
% 0.30/0.66 ! [A: real,B: real,F: real > real,F4: real > real] :
% 0.30/0.66 ( ( ord_less_real @ A @ B )
% 0.30/0.66 => ( ! [X5: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.66 => ( ( ord_less_eq_real @ X5 @ B )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 0.30/0.66 => ? [Z2: real] :
% 0.30/0.66 ( ( ord_less_real @ A @ Z2 )
% 0.30/0.66 & ( ord_less_real @ Z2 @ B )
% 0.30/0.66 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.30/0.66 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F4 @ Z2 ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % MVT2
% 0.30/0.66 thf(fact_9813_DERIV__local__const,axiom,
% 0.30/0.66 ! [F: real > real,L: real,X2: real,D: real] :
% 0.30/0.66 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.30/0.66 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.30/0.66 => ( ! [Y4: real] :
% 0.30/0.66 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ D )
% 0.30/0.66 => ( ( F @ X2 )
% 0.30/0.66 = ( F @ Y4 ) ) )
% 0.30/0.66 => ( L = zero_zero_real ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_local_const
% 0.30/0.66 thf(fact_9814_DERIV__const__average,axiom,
% 0.30/0.66 ! [A: real,B: real,V: real > real,K: real] :
% 0.30/0.66 ( ( A != B )
% 0.30/0.66 => ( ! [X5: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.66 => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.66 = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_const_average
% 0.30/0.66 thf(fact_9815_DERIV__local__min,axiom,
% 0.30/0.66 ! [F: real > real,L: real,X2: real,D: real] :
% 0.30/0.66 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.30/0.66 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.30/0.66 => ( ! [Y4: real] :
% 0.30/0.66 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ D )
% 0.30/0.66 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.30/0.66 => ( L = zero_zero_real ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_local_min
% 0.30/0.66 thf(fact_9816_DERIV__local__max,axiom,
% 0.30/0.66 ! [F: real > real,L: real,X2: real,D: real] :
% 0.30/0.66 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.30/0.66 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.30/0.66 => ( ! [Y4: real] :
% 0.30/0.66 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ D )
% 0.30/0.66 => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X2 ) ) )
% 0.30/0.66 => ( L = zero_zero_real ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_local_max
% 0.30/0.66 thf(fact_9817_DERIV__ln__divide,axiom,
% 0.30/0.66 ! [X2: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_ln_divide
% 0.30/0.66 thf(fact_9818_DERIV__pow,axiom,
% 0.30/0.66 ! [N: nat,X2: real,S: set_real] :
% 0.30/0.66 ( has_fi5821293074295781190e_real
% 0.30/0.66 @ ^ [X: real] : ( power_power_real @ X @ N )
% 0.30/0.66 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 0.30/0.66 @ ( topolo2177554685111907308n_real @ X2 @ S ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_pow
% 0.30/0.66 thf(fact_9819_DERIV__fun__pow,axiom,
% 0.30/0.66 ! [G: real > real,M: real,X2: real,N: nat] :
% 0.30/0.66 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.30/0.66 => ( has_fi5821293074295781190e_real
% 0.30/0.66 @ ^ [X: real] : ( power_power_real @ ( G @ X ) @ N )
% 0.30/0.66 @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X2 ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
% 0.30/0.66 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_fun_pow
% 0.30/0.66 thf(fact_9820_has__real__derivative__powr,axiom,
% 0.30/0.66 ! [Z: real,R2: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.30/0.66 => ( has_fi5821293074295781190e_real
% 0.30/0.66 @ ^ [Z3: real] : ( powr_real @ Z3 @ R2 )
% 0.30/0.66 @ ( times_times_real @ R2 @ ( powr_real @ Z @ ( minus_minus_real @ R2 @ one_one_real ) ) )
% 0.30/0.66 @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % has_real_derivative_powr
% 0.30/0.66 thf(fact_9821_DERIV__log,axiom,
% 0.30/0.66 ! [X2: real,B: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X2 ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_log
% 0.30/0.66 thf(fact_9822_DERIV__fun__powr,axiom,
% 0.30/0.66 ! [G: real > real,M: real,X2: real,R2: real] :
% 0.30/0.66 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.30/0.66 => ( ( ord_less_real @ zero_zero_real @ ( G @ X2 ) )
% 0.30/0.66 => ( has_fi5821293074295781190e_real
% 0.30/0.66 @ ^ [X: real] : ( powr_real @ ( G @ X ) @ R2 )
% 0.30/0.66 @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X2 ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 0.30/0.66 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_fun_powr
% 0.30/0.66 thf(fact_9823_DERIV__powr,axiom,
% 0.30/0.66 ! [G: real > real,M: real,X2: real,F: real > real,R2: real] :
% 0.30/0.66 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.30/0.66 => ( ( ord_less_real @ zero_zero_real @ ( G @ X2 ) )
% 0.30/0.66 => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.30/0.66 => ( has_fi5821293074295781190e_real
% 0.30/0.66 @ ^ [X: real] : ( powr_real @ ( G @ X ) @ ( F @ X ) )
% 0.30/0.66 @ ( times_times_real @ ( powr_real @ ( G @ X2 ) @ ( F @ X2 ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X2 ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X2 ) ) @ ( G @ X2 ) ) ) )
% 0.30/0.66 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_powr
% 0.30/0.66 thf(fact_9824_DERIV__real__sqrt,axiom,
% 0.30/0.66 ! [X2: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_real_sqrt
% 0.30/0.66 thf(fact_9825_DERIV__series_H,axiom,
% 0.30/0.66 ! [F: real > nat > real,F4: real > nat > real,X0: real,A: real,B: real,L5: nat > real] :
% 0.30/0.66 ( ! [N2: nat] :
% 0.30/0.66 ( has_fi5821293074295781190e_real
% 0.30/0.66 @ ^ [X: real] : ( F @ X @ N2 )
% 0.30/0.66 @ ( F4 @ X0 @ N2 )
% 0.30/0.66 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
% 0.30/0.66 => ( ! [X5: real] :
% 0.30/0.66 ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.30/0.66 => ( summable_real @ ( F @ X5 ) ) )
% 0.30/0.66 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.30/0.66 => ( ( summable_real @ ( F4 @ X0 ) )
% 0.30/0.66 => ( ( summable_real @ L5 )
% 0.30/0.66 => ( ! [N2: nat,X5: real,Y4: real] :
% 0.30/0.66 ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.30/0.66 => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.30/0.66 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X5 @ N2 ) @ ( F @ Y4 @ N2 ) ) ) @ ( times_times_real @ ( L5 @ N2 ) @ ( abs_abs_real @ ( minus_minus_real @ X5 @ Y4 ) ) ) ) ) )
% 0.30/0.66 => ( has_fi5821293074295781190e_real
% 0.30/0.66 @ ^ [X: real] : ( suminf_real @ ( F @ X ) )
% 0.30/0.66 @ ( suminf_real @ ( F4 @ X0 ) )
% 0.30/0.66 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_series'
% 0.30/0.66 thf(fact_9826_DERIV__arctan,axiom,
% 0.30/0.66 ! [X2: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_arctan
% 0.30/0.66 thf(fact_9827_arsinh__real__has__field__derivative,axiom,
% 0.30/0.66 ! [X2: real,A2: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ A2 ) ) ).
% 0.30/0.66
% 0.30/0.66 % arsinh_real_has_field_derivative
% 0.30/0.66 thf(fact_9828_DERIV__real__sqrt__generic,axiom,
% 0.30/0.66 ! [X2: real,D4: real] :
% 0.30/0.66 ( ( X2 != zero_zero_real )
% 0.30/0.66 => ( ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.66 => ( D4
% 0.30/0.66 = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66 => ( ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.30/0.66 => ( D4
% 0.30/0.66 = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_real_sqrt_generic
% 0.30/0.66 thf(fact_9829_arcosh__real__has__field__derivative,axiom,
% 0.30/0.66 ! [X2: real,A2: set_real] :
% 0.30/0.66 ( ( ord_less_real @ one_one_real @ X2 )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ A2 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % arcosh_real_has_field_derivative
% 0.30/0.66 thf(fact_9830_artanh__real__has__field__derivative,axiom,
% 0.30/0.66 ! [X2: real,A2: set_real] :
% 0.30/0.66 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ A2 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % artanh_real_has_field_derivative
% 0.30/0.66 thf(fact_9831_DERIV__power__series_H,axiom,
% 0.30/0.66 ! [R: real,F: nat > real,X0: real] :
% 0.30/0.66 ( ! [X5: real] :
% 0.30/0.66 ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 0.30/0.66 => ( summable_real
% 0.30/0.66 @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X5 @ N3 ) ) ) )
% 0.30/0.66 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 0.30/0.66 => ( ( ord_less_real @ zero_zero_real @ R )
% 0.30/0.66 => ( has_fi5821293074295781190e_real
% 0.30/0.66 @ ^ [X: real] :
% 0.30/0.66 ( suminf_real
% 0.30/0.66 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X @ ( suc @ N3 ) ) ) )
% 0.30/0.66 @ ( suminf_real
% 0.30/0.66 @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X0 @ N3 ) ) )
% 0.30/0.66 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_power_series'
% 0.30/0.66 thf(fact_9832_DERIV__real__root,axiom,
% 0.30/0.66 ! [N: nat,X2: real] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_real_root
% 0.30/0.66 thf(fact_9833_DERIV__arccos,axiom,
% 0.30/0.66 ! [X2: real] :
% 0.30/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.66 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_arccos
% 0.30/0.66 thf(fact_9834_DERIV__arcsin,axiom,
% 0.30/0.66 ! [X2: real] :
% 0.30/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.66 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_arcsin
% 0.30/0.66 thf(fact_9835_Maclaurin__all__le__objl,axiom,
% 0.30/0.66 ! [Diff: nat > real > real,F: real > real,X2: real,N: nat] :
% 0.30/0.66 ( ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66 = F )
% 0.30/0.66 & ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 0.30/0.66 => ? [T5: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X2 ) )
% 0.30/0.66 & ( ( F @ X2 )
% 0.30/0.66 = ( plus_plus_real
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Maclaurin_all_le_objl
% 0.30/0.66 thf(fact_9836_Maclaurin__all__le,axiom,
% 0.30/0.66 ! [Diff: nat > real > real,F: real > real,X2: real,N: nat] :
% 0.30/0.66 ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66 = F )
% 0.30/0.66 => ( ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.66 => ? [T5: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X2 ) )
% 0.30/0.66 & ( ( F @ X2 )
% 0.30/0.66 = ( plus_plus_real
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Maclaurin_all_le
% 0.30/0.66 thf(fact_9837_DERIV__odd__real__root,axiom,
% 0.30/0.66 ! [N: nat,X2: real] :
% 0.30/0.66 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.66 => ( ( X2 != zero_zero_real )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_odd_real_root
% 0.30/0.66 thf(fact_9838_Maclaurin__minus,axiom,
% 0.30/0.66 ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 0.30/0.66 ( ( ord_less_real @ H2 @ zero_zero_real )
% 0.30/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66 = F )
% 0.30/0.66 => ( ! [M4: nat,T5: real] :
% 0.30/0.66 ( ( ( ord_less_nat @ M4 @ N )
% 0.30/0.66 & ( ord_less_eq_real @ H2 @ T5 )
% 0.30/0.66 & ( ord_less_eq_real @ T5 @ zero_zero_real ) )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.30/0.66 => ? [T5: real] :
% 0.30/0.66 ( ( ord_less_real @ H2 @ T5 )
% 0.30/0.66 & ( ord_less_real @ T5 @ zero_zero_real )
% 0.30/0.66 & ( ( F @ H2 )
% 0.30/0.66 = ( plus_plus_real
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Maclaurin_minus
% 0.30/0.66 thf(fact_9839_Maclaurin2,axiom,
% 0.30/0.66 ! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.30/0.66 => ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66 = F )
% 0.30/0.66 => ( ! [M4: nat,T5: real] :
% 0.30/0.66 ( ( ( ord_less_nat @ M4 @ N )
% 0.30/0.66 & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.30/0.66 & ( ord_less_eq_real @ T5 @ H2 ) )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.30/0.66 => ? [T5: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ T5 )
% 0.30/0.66 & ( ord_less_eq_real @ T5 @ H2 )
% 0.30/0.66 & ( ( F @ H2 )
% 0.30/0.66 = ( plus_plus_real
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Maclaurin2
% 0.30/0.66 thf(fact_9840_Maclaurin,axiom,
% 0.30/0.66 ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.30/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66 = F )
% 0.30/0.66 => ( ! [M4: nat,T5: real] :
% 0.30/0.66 ( ( ( ord_less_nat @ M4 @ N )
% 0.30/0.66 & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.30/0.66 & ( ord_less_eq_real @ T5 @ H2 ) )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.30/0.66 => ? [T5: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ T5 )
% 0.30/0.66 & ( ord_less_real @ T5 @ H2 )
% 0.30/0.66 & ( ( F @ H2 )
% 0.30/0.66 = ( plus_plus_real
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Maclaurin
% 0.30/0.66 thf(fact_9841_Maclaurin__all__lt,axiom,
% 0.30/0.66 ! [Diff: nat > real > real,F: real > real,N: nat,X2: real] :
% 0.30/0.66 ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66 = F )
% 0.30/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( X2 != zero_zero_real )
% 0.30/0.66 => ( ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.66 => ? [T5: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
% 0.30/0.66 & ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X2 ) )
% 0.30/0.66 & ( ( F @ X2 )
% 0.30/0.66 = ( plus_plus_real
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Maclaurin_all_lt
% 0.30/0.66 thf(fact_9842_Maclaurin__bi__le,axiom,
% 0.30/0.66 ! [Diff: nat > real > real,F: real > real,N: nat,X2: real] :
% 0.30/0.66 ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66 = F )
% 0.30/0.66 => ( ! [M4: nat,T5: real] :
% 0.30/0.66 ( ( ( ord_less_nat @ M4 @ N )
% 0.30/0.66 & ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X2 ) ) )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.30/0.66 => ? [T5: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X2 ) )
% 0.30/0.66 & ( ( F @ X2 )
% 0.30/0.66 = ( plus_plus_real
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Maclaurin_bi_le
% 0.30/0.66 thf(fact_9843_Taylor__down,axiom,
% 0.30/0.66 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66 = F )
% 0.30/0.66 => ( ! [M4: nat,T5: real] :
% 0.30/0.66 ( ( ( ord_less_nat @ M4 @ N )
% 0.30/0.66 & ( ord_less_eq_real @ A @ T5 )
% 0.30/0.66 & ( ord_less_eq_real @ T5 @ B ) )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.30/0.66 => ( ( ord_less_real @ A @ C )
% 0.30/0.66 => ( ( ord_less_eq_real @ C @ B )
% 0.30/0.66 => ? [T5: real] :
% 0.30/0.66 ( ( ord_less_real @ A @ T5 )
% 0.30/0.66 & ( ord_less_real @ T5 @ C )
% 0.30/0.66 & ( ( F @ A )
% 0.30/0.66 = ( plus_plus_real
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M3 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Taylor_down
% 0.30/0.66 thf(fact_9844_Taylor__up,axiom,
% 0.30/0.66 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66 = F )
% 0.30/0.66 => ( ! [M4: nat,T5: real] :
% 0.30/0.66 ( ( ( ord_less_nat @ M4 @ N )
% 0.30/0.66 & ( ord_less_eq_real @ A @ T5 )
% 0.30/0.66 & ( ord_less_eq_real @ T5 @ B ) )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.30/0.66 => ( ( ord_less_eq_real @ A @ C )
% 0.30/0.66 => ( ( ord_less_real @ C @ B )
% 0.30/0.66 => ? [T5: real] :
% 0.30/0.66 ( ( ord_less_real @ C @ T5 )
% 0.30/0.66 & ( ord_less_real @ T5 @ B )
% 0.30/0.66 & ( ( F @ B )
% 0.30/0.66 = ( plus_plus_real
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M3 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Taylor_up
% 0.30/0.66 thf(fact_9845_Taylor,axiom,
% 0.30/0.66 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X2: real] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66 = F )
% 0.30/0.66 => ( ! [M4: nat,T5: real] :
% 0.30/0.66 ( ( ( ord_less_nat @ M4 @ N )
% 0.30/0.66 & ( ord_less_eq_real @ A @ T5 )
% 0.30/0.66 & ( ord_less_eq_real @ T5 @ B ) )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.30/0.66 => ( ( ord_less_eq_real @ A @ C )
% 0.30/0.66 => ( ( ord_less_eq_real @ C @ B )
% 0.30/0.66 => ( ( ord_less_eq_real @ A @ X2 )
% 0.30/0.66 => ( ( ord_less_eq_real @ X2 @ B )
% 0.30/0.66 => ( ( X2 != C )
% 0.30/0.66 => ? [T5: real] :
% 0.30/0.66 ( ( ( ord_less_real @ X2 @ C )
% 0.30/0.66 => ( ( ord_less_real @ X2 @ T5 )
% 0.30/0.66 & ( ord_less_real @ T5 @ C ) ) )
% 0.30/0.66 & ( ~ ( ord_less_real @ X2 @ C )
% 0.30/0.66 => ( ( ord_less_real @ C @ T5 )
% 0.30/0.66 & ( ord_less_real @ T5 @ X2 ) ) )
% 0.30/0.66 & ( ( F @ X2 )
% 0.30/0.66 = ( plus_plus_real
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ X2 @ C ) @ M3 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.66 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X2 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Taylor
% 0.30/0.66 thf(fact_9846_Maclaurin__lemma2,axiom,
% 0.30/0.66 ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B3: real] :
% 0.30/0.66 ( ! [M4: nat,T5: real] :
% 0.30/0.66 ( ( ( ord_less_nat @ M4 @ N )
% 0.30/0.66 & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.30/0.66 & ( ord_less_eq_real @ T5 @ H2 ) )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.30/0.66 => ( ( N
% 0.30/0.66 = ( suc @ K ) )
% 0.30/0.66 => ! [M2: nat,T6: real] :
% 0.30/0.66 ( ( ( ord_less_nat @ M2 @ N )
% 0.30/0.66 & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.30/0.66 & ( ord_less_eq_real @ T6 @ H2 ) )
% 0.30/0.66 => ( has_fi5821293074295781190e_real
% 0.30/0.66 @ ^ [U2: real] :
% 0.30/0.66 ( minus_minus_real @ ( Diff @ M2 @ U2 )
% 0.30/0.66 @ ( plus_plus_real
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [P4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P4 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ U2 @ P4 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M2 ) ) )
% 0.30/0.66 @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) )
% 0.30/0.66 @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T6 )
% 0.30/0.66 @ ( plus_plus_real
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [P4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P4 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ T6 @ P4 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) )
% 0.30/0.66 @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ T6 @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) ) ) ) )
% 0.30/0.66 @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Maclaurin_lemma2
% 0.30/0.66 thf(fact_9847_DERIV__arctan__series,axiom,
% 0.30/0.66 ! [X2: real] :
% 0.30/0.66 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.30/0.66 => ( has_fi5821293074295781190e_real
% 0.30/0.66 @ ^ [X9: real] :
% 0.30/0.66 ( suminf_real
% 0.30/0.66 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 0.30/0.66 @ ( suminf_real
% 0.30/0.66 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( power_power_real @ X2 @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.66 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_arctan_series
% 0.30/0.66 thf(fact_9848_DERIV__even__real__root,axiom,
% 0.30/0.66 ! [N: nat,X2: real] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.66 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_even_real_root
% 0.30/0.66 thf(fact_9849_isCont__Lb__Ub,axiom,
% 0.30/0.66 ! [A: real,B: real,F: real > real] :
% 0.30/0.66 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.66 => ( ! [X5: real] :
% 0.30/0.66 ( ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.66 & ( ord_less_eq_real @ X5 @ B ) )
% 0.30/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ F ) )
% 0.30/0.66 => ? [L6: real,M9: real] :
% 0.30/0.66 ( ! [X3: real] :
% 0.30/0.66 ( ( ( ord_less_eq_real @ A @ X3 )
% 0.30/0.66 & ( ord_less_eq_real @ X3 @ B ) )
% 0.30/0.66 => ( ( ord_less_eq_real @ L6 @ ( F @ X3 ) )
% 0.30/0.66 & ( ord_less_eq_real @ ( F @ X3 ) @ M9 ) ) )
% 0.30/0.66 & ! [Y5: real] :
% 0.30/0.66 ( ( ( ord_less_eq_real @ L6 @ Y5 )
% 0.30/0.66 & ( ord_less_eq_real @ Y5 @ M9 ) )
% 0.30/0.66 => ? [X5: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.66 & ( ord_less_eq_real @ X5 @ B )
% 0.30/0.66 & ( ( F @ X5 )
% 0.30/0.66 = Y5 ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % isCont_Lb_Ub
% 0.30/0.66 thf(fact_9850_isCont__real__sqrt,axiom,
% 0.30/0.66 ! [X2: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ sqrt ) ).
% 0.30/0.66
% 0.30/0.66 % isCont_real_sqrt
% 0.30/0.66 thf(fact_9851_isCont__real__root,axiom,
% 0.30/0.66 ! [X2: real,N: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ ( root @ N ) ) ).
% 0.30/0.66
% 0.30/0.66 % isCont_real_root
% 0.30/0.66 thf(fact_9852_isCont__inverse__function2,axiom,
% 0.30/0.66 ! [A: real,X2: real,B: real,G: real > real,F: real > real] :
% 0.30/0.66 ( ( ord_less_real @ A @ X2 )
% 0.30/0.66 => ( ( ord_less_real @ X2 @ B )
% 0.30/0.66 => ( ! [Z2: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ A @ Z2 )
% 0.30/0.66 => ( ( ord_less_eq_real @ Z2 @ B )
% 0.30/0.66 => ( ( G @ ( F @ Z2 ) )
% 0.30/0.66 = Z2 ) ) )
% 0.30/0.66 => ( ! [Z2: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ A @ Z2 )
% 0.30/0.66 => ( ( ord_less_eq_real @ Z2 @ B )
% 0.30/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
% 0.30/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X2 ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % isCont_inverse_function2
% 0.30/0.66 thf(fact_9853_isCont__arcosh,axiom,
% 0.30/0.66 ! [X2: real] :
% 0.30/0.66 ( ( ord_less_real @ one_one_real @ X2 )
% 0.30/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arcosh_real ) ) ).
% 0.30/0.66
% 0.30/0.66 % isCont_arcosh
% 0.30/0.66 thf(fact_9854_isCont__arccos,axiom,
% 0.30/0.66 ! [X2: real] :
% 0.30/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.66 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.30/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arccos ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % isCont_arccos
% 0.30/0.66 thf(fact_9855_isCont__arcsin,axiom,
% 0.30/0.66 ! [X2: real] :
% 0.30/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.66 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.30/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arcsin ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % isCont_arcsin
% 0.30/0.66 thf(fact_9856_LIM__less__bound,axiom,
% 0.30/0.66 ! [B: real,X2: real,F: real > real] :
% 0.30/0.66 ( ( ord_less_real @ B @ X2 )
% 0.30/0.66 => ( ! [X5: real] :
% 0.30/0.66 ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ B @ X2 ) )
% 0.30/0.66 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.30/0.66 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ F )
% 0.30/0.66 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % LIM_less_bound
% 0.30/0.66 thf(fact_9857_isCont__artanh,axiom,
% 0.30/0.66 ! [X2: real] :
% 0.30/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.30/0.66 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.30/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ artanh_real ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % isCont_artanh
% 0.30/0.66 thf(fact_9858_isCont__inverse__function,axiom,
% 0.30/0.66 ! [D: real,X2: real,G: real > real,F: real > real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ D )
% 0.30/0.66 => ( ! [Z2: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X2 ) ) @ D )
% 0.30/0.66 => ( ( G @ ( F @ Z2 ) )
% 0.30/0.66 = Z2 ) )
% 0.30/0.66 => ( ! [Z2: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X2 ) ) @ D )
% 0.30/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) )
% 0.30/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X2 ) @ top_top_set_real ) @ G ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % isCont_inverse_function
% 0.30/0.66 thf(fact_9859_GMVT_H,axiom,
% 0.30/0.66 ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F4: real > real] :
% 0.30/0.66 ( ( ord_less_real @ A @ B )
% 0.30/0.66 => ( ! [Z2: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ A @ Z2 )
% 0.30/0.66 => ( ( ord_less_eq_real @ Z2 @ B )
% 0.30/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
% 0.30/0.66 => ( ! [Z2: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ A @ Z2 )
% 0.30/0.66 => ( ( ord_less_eq_real @ Z2 @ B )
% 0.30/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ G ) ) )
% 0.30/0.66 => ( ! [Z2: real] :
% 0.30/0.66 ( ( ord_less_real @ A @ Z2 )
% 0.30/0.66 => ( ( ord_less_real @ Z2 @ B )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
% 0.30/0.66 => ( ! [Z2: real] :
% 0.30/0.66 ( ( ord_less_real @ A @ Z2 )
% 0.30/0.66 => ( ( ord_less_real @ Z2 @ B )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ F @ ( F4 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
% 0.30/0.66 => ? [C3: real] :
% 0.30/0.66 ( ( ord_less_real @ A @ C3 )
% 0.30/0.66 & ( ord_less_real @ C3 @ B )
% 0.30/0.66 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C3 ) )
% 0.30/0.66 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F4 @ C3 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % GMVT'
% 0.30/0.66 thf(fact_9860_upto__aux__rec,axiom,
% 0.30/0.66 ( upto_aux
% 0.30/0.66 = ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % upto_aux_rec
% 0.30/0.66 thf(fact_9861_LIM__fun__gt__zero,axiom,
% 0.30/0.66 ! [F: real > real,L: real,C: real] :
% 0.30/0.66 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.30/0.66 => ( ( ord_less_real @ zero_zero_real @ L )
% 0.30/0.66 => ? [R3: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.30/0.66 & ! [X3: real] :
% 0.30/0.66 ( ( ( X3 != C )
% 0.30/0.66 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X3 ) ) @ R3 ) )
% 0.30/0.66 => ( ord_less_real @ zero_zero_real @ ( F @ X3 ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % LIM_fun_gt_zero
% 0.30/0.66 thf(fact_9862_LIM__fun__not__zero,axiom,
% 0.30/0.66 ! [F: real > real,L: real,C: real] :
% 0.30/0.66 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.30/0.66 => ( ( L != zero_zero_real )
% 0.30/0.66 => ? [R3: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.30/0.66 & ! [X3: real] :
% 0.30/0.66 ( ( ( X3 != C )
% 0.30/0.66 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X3 ) ) @ R3 ) )
% 0.30/0.66 => ( ( F @ X3 )
% 0.30/0.66 != zero_zero_real ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % LIM_fun_not_zero
% 0.30/0.66 thf(fact_9863_LIM__fun__less__zero,axiom,
% 0.30/0.66 ! [F: real > real,L: real,C: real] :
% 0.30/0.66 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.30/0.66 => ( ( ord_less_real @ L @ zero_zero_real )
% 0.30/0.66 => ? [R3: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.30/0.66 & ! [X3: real] :
% 0.30/0.66 ( ( ( X3 != C )
% 0.30/0.66 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X3 ) ) @ R3 ) )
% 0.30/0.66 => ( ord_less_real @ ( F @ X3 ) @ zero_zero_real ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % LIM_fun_less_zero
% 0.30/0.66 thf(fact_9864_LIM__cos__div__sin,axiom,
% 0.30/0.66 ( filterlim_real_real
% 0.30/0.66 @ ^ [X: real] : ( divide_divide_real @ ( cos_real @ X ) @ ( sin_real @ X ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.66 @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).
% 0.30/0.66
% 0.30/0.66 % LIM_cos_div_sin
% 0.30/0.66 thf(fact_9865_summable__Leibniz_I3_J,axiom,
% 0.30/0.66 ! [A: nat > real] :
% 0.30/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66 => ( ( topolo6980174941875973593q_real @ A )
% 0.30/0.66 => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 0.30/0.66 => ! [N9: nat] :
% 0.30/0.66 ( member_real
% 0.30/0.66 @ ( suminf_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 0.30/0.66 @ ( set_or1222579329274155063t_real
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) )
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % summable_Leibniz(3)
% 0.30/0.66 thf(fact_9866_summable__Leibniz_I2_J,axiom,
% 0.30/0.66 ! [A: nat > real] :
% 0.30/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66 => ( ( topolo6980174941875973593q_real @ A )
% 0.30/0.66 => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 0.30/0.66 => ! [N9: nat] :
% 0.30/0.66 ( member_real
% 0.30/0.66 @ ( suminf_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 0.30/0.66 @ ( set_or1222579329274155063t_real
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % summable_Leibniz(2)
% 0.30/0.66 thf(fact_9867_filterlim__Suc,axiom,
% 0.30/0.66 filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% 0.30/0.66
% 0.30/0.66 % filterlim_Suc
% 0.30/0.66 thf(fact_9868_mult__nat__left__at__top,axiom,
% 0.30/0.66 ! [C: nat] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.66 => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % mult_nat_left_at_top
% 0.30/0.66 thf(fact_9869_mult__nat__right__at__top,axiom,
% 0.30/0.66 ! [C: nat] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.66 => ( filterlim_nat_nat
% 0.30/0.66 @ ^ [X: nat] : ( times_times_nat @ X @ C )
% 0.30/0.66 @ at_top_nat
% 0.30/0.66 @ at_top_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % mult_nat_right_at_top
% 0.30/0.66 thf(fact_9870_monoseq__convergent,axiom,
% 0.30/0.66 ! [X8: nat > real,B3: real] :
% 0.30/0.66 ( ( topolo6980174941875973593q_real @ X8 )
% 0.30/0.66 => ( ! [I3: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X8 @ I3 ) ) @ B3 )
% 0.30/0.66 => ~ ! [L6: real] :
% 0.30/0.66 ~ ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % monoseq_convergent
% 0.30/0.66 thf(fact_9871_LIMSEQ__root,axiom,
% 0.30/0.66 ( filterlim_nat_real
% 0.30/0.66 @ ^ [N3: nat] : ( root @ N3 @ ( semiri5074537144036343181t_real @ N3 ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.30/0.66 @ at_top_nat ) ).
% 0.30/0.66
% 0.30/0.66 % LIMSEQ_root
% 0.30/0.66 thf(fact_9872_nested__sequence__unique,axiom,
% 0.30/0.66 ! [F: nat > real,G: nat > real] :
% 0.30/0.66 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N2 ) ) @ ( G @ N2 ) )
% 0.30/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.30/0.66 => ( ( filterlim_nat_real
% 0.30/0.66 @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.66 @ at_top_nat )
% 0.30/0.66 => ? [L4: real] :
% 0.30/0.66 ( ! [N9: nat] : ( ord_less_eq_real @ ( F @ N9 ) @ L4 )
% 0.30/0.66 & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
% 0.30/0.66 & ! [N9: nat] : ( ord_less_eq_real @ L4 @ ( G @ N9 ) )
% 0.30/0.66 & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % nested_sequence_unique
% 0.30/0.66 thf(fact_9873_LIMSEQ__inverse__zero,axiom,
% 0.30/0.66 ! [X8: nat > real] :
% 0.30/0.66 ( ! [R3: real] :
% 0.30/0.66 ? [N7: nat] :
% 0.30/0.66 ! [N2: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ N7 @ N2 )
% 0.30/0.66 => ( ord_less_real @ R3 @ ( X8 @ N2 ) ) )
% 0.30/0.66 => ( filterlim_nat_real
% 0.30/0.66 @ ^ [N3: nat] : ( inverse_inverse_real @ ( X8 @ N3 ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.66 @ at_top_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % LIMSEQ_inverse_zero
% 0.30/0.66 thf(fact_9874_lim__inverse__n_H,axiom,
% 0.30/0.66 ( filterlim_nat_real
% 0.30/0.66 @ ^ [N3: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N3 ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.66 @ at_top_nat ) ).
% 0.30/0.66
% 0.30/0.66 % lim_inverse_n'
% 0.30/0.66 thf(fact_9875_LIMSEQ__root__const,axiom,
% 0.30/0.66 ! [C: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.66 => ( filterlim_nat_real
% 0.30/0.66 @ ^ [N3: nat] : ( root @ N3 @ C )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.30/0.66 @ at_top_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % LIMSEQ_root_const
% 0.30/0.66 thf(fact_9876_LIMSEQ__inverse__real__of__nat,axiom,
% 0.30/0.66 ( filterlim_nat_real
% 0.30/0.66 @ ^ [N3: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.66 @ at_top_nat ) ).
% 0.30/0.66
% 0.30/0.66 % LIMSEQ_inverse_real_of_nat
% 0.30/0.66 thf(fact_9877_LIMSEQ__inverse__real__of__nat__add,axiom,
% 0.30/0.66 ! [R2: real] :
% 0.30/0.66 ( filterlim_nat_real
% 0.30/0.66 @ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ R2 )
% 0.30/0.66 @ at_top_nat ) ).
% 0.30/0.66
% 0.30/0.66 % LIMSEQ_inverse_real_of_nat_add
% 0.30/0.66 thf(fact_9878_increasing__LIMSEQ,axiom,
% 0.30/0.66 ! [F: nat > real,L: real] :
% 0.30/0.66 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ L )
% 0.30/0.66 => ( ! [E2: real] :
% 0.30/0.66 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.30/0.66 => ? [N9: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N9 ) @ E2 ) ) )
% 0.30/0.66 => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % increasing_LIMSEQ
% 0.30/0.66 thf(fact_9879_LIMSEQ__realpow__zero,axiom,
% 0.30/0.66 ! [X2: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.66 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.30/0.66 => ( filterlim_nat_real @ ( power_power_real @ X2 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % LIMSEQ_realpow_zero
% 0.30/0.66 thf(fact_9880_LIMSEQ__divide__realpow__zero,axiom,
% 0.30/0.66 ! [X2: real,A: real] :
% 0.30/0.66 ( ( ord_less_real @ one_one_real @ X2 )
% 0.30/0.66 => ( filterlim_nat_real
% 0.30/0.66 @ ^ [N3: nat] : ( divide_divide_real @ A @ ( power_power_real @ X2 @ N3 ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.66 @ at_top_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % LIMSEQ_divide_realpow_zero
% 0.30/0.66 thf(fact_9881_LIMSEQ__abs__realpow__zero,axiom,
% 0.30/0.66 ! [C: real] :
% 0.30/0.66 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.30/0.66 => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % LIMSEQ_abs_realpow_zero
% 0.30/0.66 thf(fact_9882_LIMSEQ__abs__realpow__zero2,axiom,
% 0.30/0.66 ! [C: real] :
% 0.30/0.66 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.30/0.66 => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % LIMSEQ_abs_realpow_zero2
% 0.30/0.66 thf(fact_9883_LIMSEQ__inverse__realpow__zero,axiom,
% 0.30/0.66 ! [X2: real] :
% 0.30/0.66 ( ( ord_less_real @ one_one_real @ X2 )
% 0.30/0.66 => ( filterlim_nat_real
% 0.30/0.66 @ ^ [N3: nat] : ( inverse_inverse_real @ ( power_power_real @ X2 @ N3 ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.66 @ at_top_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % LIMSEQ_inverse_realpow_zero
% 0.30/0.66 thf(fact_9884_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
% 0.30/0.66 ! [R2: real] :
% 0.30/0.66 ( filterlim_nat_real
% 0.30/0.66 @ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ R2 )
% 0.30/0.66 @ at_top_nat ) ).
% 0.30/0.66
% 0.30/0.66 % LIMSEQ_inverse_real_of_nat_add_minus
% 0.30/0.66 thf(fact_9885_tendsto__exp__limit__sequentially,axiom,
% 0.30/0.66 ! [X2: real] :
% 0.30/0.66 ( filterlim_nat_real
% 0.30/0.66 @ ^ [N3: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N3 ) ) ) @ N3 )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ ( exp_real @ X2 ) )
% 0.30/0.66 @ at_top_nat ) ).
% 0.30/0.66
% 0.30/0.66 % tendsto_exp_limit_sequentially
% 0.30/0.66 thf(fact_9886_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
% 0.30/0.66 ! [R2: real] :
% 0.30/0.66 ( filterlim_nat_real
% 0.30/0.66 @ ^ [N3: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ R2 )
% 0.30/0.66 @ at_top_nat ) ).
% 0.30/0.66
% 0.30/0.66 % LIMSEQ_inverse_real_of_nat_add_minus_mult
% 0.30/0.66 thf(fact_9887_summable__Leibniz_I1_J,axiom,
% 0.30/0.66 ! [A: nat > real] :
% 0.30/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66 => ( ( topolo6980174941875973593q_real @ A )
% 0.30/0.66 => ( summable_real
% 0.30/0.66 @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % summable_Leibniz(1)
% 0.30/0.66 thf(fact_9888_summable,axiom,
% 0.30/0.66 ! [A: nat > real] :
% 0.30/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.30/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.30/0.66 => ( summable_real
% 0.30/0.66 @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % summable
% 0.30/0.66 thf(fact_9889_cos__diff__limit__1,axiom,
% 0.30/0.66 ! [Theta: nat > real,Theta2: real] :
% 0.30/0.66 ( ( filterlim_nat_real
% 0.30/0.66 @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.30/0.66 @ at_top_nat )
% 0.30/0.66 => ~ ! [K3: nat > int] :
% 0.30/0.66 ~ ( filterlim_nat_real
% 0.30/0.66 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ Theta2 )
% 0.30/0.66 @ at_top_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % cos_diff_limit_1
% 0.30/0.66 thf(fact_9890_cos__limit__1,axiom,
% 0.30/0.66 ! [Theta: nat > real] :
% 0.30/0.66 ( ( filterlim_nat_real
% 0.30/0.66 @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.30/0.66 @ at_top_nat )
% 0.30/0.66 => ? [K3: nat > int] :
% 0.30/0.66 ( filterlim_nat_real
% 0.30/0.66 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.66 @ at_top_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % cos_limit_1
% 0.30/0.66 thf(fact_9891_summable__Leibniz_I4_J,axiom,
% 0.30/0.66 ! [A: nat > real] :
% 0.30/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66 => ( ( topolo6980174941875973593q_real @ A )
% 0.30/0.66 => ( filterlim_nat_real
% 0.30/0.66 @ ^ [N3: nat] :
% 0.30/0.66 ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real
% 0.30/0.66 @ ( suminf_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 0.30/0.66 @ at_top_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % summable_Leibniz(4)
% 0.30/0.66 thf(fact_9892_zeroseq__arctan__series,axiom,
% 0.30/0.66 ! [X2: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.30/0.66 => ( filterlim_nat_real
% 0.30/0.66 @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.66 @ at_top_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % zeroseq_arctan_series
% 0.30/0.66 thf(fact_9893_summable__Leibniz_H_I3_J,axiom,
% 0.30/0.66 ! [A: nat > real] :
% 0.30/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.30/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.30/0.66 => ( filterlim_nat_real
% 0.30/0.66 @ ^ [N3: nat] :
% 0.30/0.66 ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real
% 0.30/0.66 @ ( suminf_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 0.30/0.66 @ at_top_nat ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % summable_Leibniz'(3)
% 0.30/0.66 thf(fact_9894_summable__Leibniz_H_I2_J,axiom,
% 0.30/0.66 ! [A: nat > real,N: nat] :
% 0.30/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.30/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.30/0.66 => ( ord_less_eq_real
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.66 @ ( suminf_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % summable_Leibniz'(2)
% 0.30/0.66 thf(fact_9895_sums__alternating__upper__lower,axiom,
% 0.30/0.66 ! [A: nat > real] :
% 0.30/0.66 ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.30/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.30/0.66 => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66 => ? [L4: real] :
% 0.30/0.66 ( ! [N9: nat] :
% 0.30/0.66 ( ord_less_eq_real
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
% 0.30/0.66 @ L4 )
% 0.30/0.66 & ( filterlim_nat_real
% 0.30/0.66 @ ^ [N3: nat] :
% 0.30/0.66 ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ L4 )
% 0.30/0.66 @ at_top_nat )
% 0.30/0.66 & ! [N9: nat] :
% 0.30/0.66 ( ord_less_eq_real @ L4
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) )
% 0.30/0.66 & ( filterlim_nat_real
% 0.30/0.66 @ ^ [N3: nat] :
% 0.30/0.66 ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ L4 )
% 0.30/0.66 @ at_top_nat ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % sums_alternating_upper_lower
% 0.30/0.66 thf(fact_9896_summable__Leibniz_I5_J,axiom,
% 0.30/0.66 ! [A: nat > real] :
% 0.30/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66 => ( ( topolo6980174941875973593q_real @ A )
% 0.30/0.66 => ( filterlim_nat_real
% 0.30/0.66 @ ^ [N3: nat] :
% 0.30/0.66 ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real
% 0.30/0.66 @ ( suminf_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 0.30/0.66 @ at_top_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % summable_Leibniz(5)
% 0.30/0.66 thf(fact_9897_summable__Leibniz_H_I5_J,axiom,
% 0.30/0.66 ! [A: nat > real] :
% 0.30/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.30/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.30/0.66 => ( filterlim_nat_real
% 0.30/0.66 @ ^ [N3: nat] :
% 0.30/0.66 ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real
% 0.30/0.66 @ ( suminf_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 0.30/0.66 @ at_top_nat ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % summable_Leibniz'(5)
% 0.30/0.66 thf(fact_9898_summable__Leibniz_H_I4_J,axiom,
% 0.30/0.66 ! [A: nat > real,N: nat] :
% 0.30/0.66 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 0.30/0.66 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 0.30/0.66 => ( ord_less_eq_real
% 0.30/0.66 @ ( suminf_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 0.30/0.66 @ ( groups6591440286371151544t_real
% 0.30/0.66 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.66 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % summable_Leibniz'(4)
% 0.30/0.66 thf(fact_9899_real__bounded__linear,axiom,
% 0.30/0.66 ( real_V5970128139526366754l_real
% 0.30/0.66 = ( ^ [F3: real > real] :
% 0.30/0.66 ? [C2: real] :
% 0.30/0.66 ( F3
% 0.30/0.66 = ( ^ [X: real] : ( times_times_real @ X @ C2 ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % real_bounded_linear
% 0.30/0.66 thf(fact_9900_tendsto__exp__limit__at__right,axiom,
% 0.30/0.66 ! [X2: real] :
% 0.30/0.66 ( filterlim_real_real
% 0.30/0.66 @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X2 @ Y ) ) @ ( divide_divide_real @ one_one_real @ Y ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ ( exp_real @ X2 ) )
% 0.30/0.66 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % tendsto_exp_limit_at_right
% 0.30/0.66 thf(fact_9901_dist__real__def,axiom,
% 0.30/0.66 ( real_V975177566351809787t_real
% 0.30/0.66 = ( ^ [X: real,Y: real] : ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % dist_real_def
% 0.30/0.66 thf(fact_9902_dist__complex__def,axiom,
% 0.30/0.66 ( real_V3694042436643373181omplex
% 0.30/0.66 = ( ^ [X: complex,Y: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % dist_complex_def
% 0.30/0.66 thf(fact_9903_tendsto__arcosh__at__left__1,axiom,
% 0.30/0.66 filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).
% 0.30/0.66
% 0.30/0.66 % tendsto_arcosh_at_left_1
% 0.30/0.66 thf(fact_9904_greaterThan__0,axiom,
% 0.30/0.66 ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
% 0.30/0.66 = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % greaterThan_0
% 0.30/0.66 thf(fact_9905_greaterThan__Suc,axiom,
% 0.30/0.66 ! [K: nat] :
% 0.30/0.66 ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 0.30/0.66 = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % greaterThan_Suc
% 0.30/0.66 thf(fact_9906_filterlim__tan__at__right,axiom,
% 0.30/0.66 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 ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % filterlim_tan_at_right
% 0.30/0.66 thf(fact_9907_tendsto__arctan__at__bot,axiom,
% 0.30/0.66 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).
% 0.30/0.66
% 0.30/0.66 % tendsto_arctan_at_bot
% 0.30/0.66 thf(fact_9908_tanh__real__at__bot,axiom,
% 0.30/0.66 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ one_one_real ) ) @ at_bot_real ).
% 0.30/0.66
% 0.30/0.66 % tanh_real_at_bot
% 0.30/0.66 thf(fact_9909_artanh__real__at__right__1,axiom,
% 0.30/0.66 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 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % artanh_real_at_right_1
% 0.30/0.66 thf(fact_9910_DERIV__pos__imp__increasing__at__bot,axiom,
% 0.30/0.66 ! [B: real,F: real > real,Flim: real] :
% 0.30/0.66 ( ! [X5: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ X5 @ B )
% 0.30/0.66 => ? [Y5: real] :
% 0.30/0.66 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.66 & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
% 0.30/0.66 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 0.30/0.66 => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_pos_imp_increasing_at_bot
% 0.30/0.66 thf(fact_9911_filterlim__pow__at__bot__odd,axiom,
% 0.30/0.66 ! [N: nat,F: real > real,F5: filter_real] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
% 0.30/0.66 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.66 => ( filterlim_real_real
% 0.30/0.66 @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
% 0.30/0.66 @ at_bot_real
% 0.30/0.66 @ F5 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % filterlim_pow_at_bot_odd
% 0.30/0.66 thf(fact_9912_filterlim__pow__at__bot__even,axiom,
% 0.30/0.66 ! [N: nat,F: real > real,F5: filter_real] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
% 0.30/0.66 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.66 => ( filterlim_real_real
% 0.30/0.66 @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
% 0.30/0.66 @ at_top_real
% 0.30/0.66 @ F5 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % filterlim_pow_at_bot_even
% 0.30/0.66 thf(fact_9913_at__bot__le__at__infinity,axiom,
% 0.30/0.66 ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).
% 0.30/0.66
% 0.30/0.66 % at_bot_le_at_infinity
% 0.30/0.66 thf(fact_9914_at__top__le__at__infinity,axiom,
% 0.30/0.66 ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).
% 0.30/0.66
% 0.30/0.66 % at_top_le_at_infinity
% 0.30/0.66 thf(fact_9915_sqrt__at__top,axiom,
% 0.30/0.66 filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).
% 0.30/0.66
% 0.30/0.66 % sqrt_at_top
% 0.30/0.66 thf(fact_9916_tanh__real__at__top,axiom,
% 0.30/0.66 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).
% 0.30/0.66
% 0.30/0.66 % tanh_real_at_top
% 0.30/0.66 thf(fact_9917_artanh__real__at__left__1,axiom,
% 0.30/0.66 filterlim_real_real @ artanh_real @ at_top_real @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5984915006950818249n_real @ one_one_real ) ) ).
% 0.30/0.66
% 0.30/0.66 % artanh_real_at_left_1
% 0.30/0.66 thf(fact_9918_tendsto__power__div__exp__0,axiom,
% 0.30/0.66 ! [K: nat] :
% 0.30/0.66 ( filterlim_real_real
% 0.30/0.66 @ ^ [X: real] : ( divide_divide_real @ ( power_power_real @ X @ K ) @ ( exp_real @ X ) )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.66 @ at_top_real ) ).
% 0.30/0.66
% 0.30/0.66 % tendsto_power_div_exp_0
% 0.30/0.66 thf(fact_9919_tendsto__exp__limit__at__top,axiom,
% 0.30/0.66 ! [X2: real] :
% 0.30/0.66 ( filterlim_real_real
% 0.30/0.66 @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X2 @ Y ) ) @ Y )
% 0.30/0.66 @ ( topolo2815343760600316023s_real @ ( exp_real @ X2 ) )
% 0.30/0.66 @ at_top_real ) ).
% 0.30/0.66
% 0.30/0.66 % tendsto_exp_limit_at_top
% 0.30/0.66 thf(fact_9920_filterlim__tan__at__left,axiom,
% 0.30/0.66 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 ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % filterlim_tan_at_left
% 0.30/0.66 thf(fact_9921_DERIV__neg__imp__decreasing__at__top,axiom,
% 0.30/0.66 ! [B: real,F: real > real,Flim: real] :
% 0.30/0.66 ( ! [X5: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ B @ X5 )
% 0.30/0.66 => ? [Y5: real] :
% 0.30/0.66 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.66 & ( ord_less_real @ Y5 @ zero_zero_real ) ) )
% 0.30/0.66 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 0.30/0.66 => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_neg_imp_decreasing_at_top
% 0.30/0.66 thf(fact_9922_tendsto__arctan__at__top,axiom,
% 0.30/0.66 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).
% 0.30/0.66
% 0.30/0.66 % tendsto_arctan_at_top
% 0.30/0.66 thf(fact_9923_upto_Opelims,axiom,
% 0.30/0.66 ! [X2: int,Xa2: int,Y3: list_int] :
% 0.30/0.66 ( ( ( upto @ X2 @ Xa2 )
% 0.30/0.66 = Y3 )
% 0.30/0.66 => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) )
% 0.30/0.66 => ~ ( ( ( ( ord_less_eq_int @ X2 @ Xa2 )
% 0.30/0.66 => ( Y3
% 0.30/0.66 = ( cons_int @ X2 @ ( upto @ ( plus_plus_int @ X2 @ one_one_int ) @ Xa2 ) ) ) )
% 0.30/0.66 & ( ~ ( ord_less_eq_int @ X2 @ Xa2 )
% 0.30/0.66 => ( Y3 = nil_int ) ) )
% 0.30/0.66 => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % upto.pelims
% 0.30/0.66 thf(fact_9924_eventually__sequentially__Suc,axiom,
% 0.30/0.66 ! [P: nat > $o] :
% 0.30/0.66 ( ( eventually_nat
% 0.30/0.66 @ ^ [I4: nat] : ( P @ ( suc @ I4 ) )
% 0.30/0.66 @ at_top_nat )
% 0.30/0.66 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % eventually_sequentially_Suc
% 0.30/0.66 thf(fact_9925_eventually__sequentially__seg,axiom,
% 0.30/0.66 ! [P: nat > $o,K: nat] :
% 0.30/0.66 ( ( eventually_nat
% 0.30/0.66 @ ^ [N3: nat] : ( P @ ( plus_plus_nat @ N3 @ K ) )
% 0.30/0.66 @ at_top_nat )
% 0.30/0.66 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % eventually_sequentially_seg
% 0.30/0.66 thf(fact_9926_nth__upto,axiom,
% 0.30/0.66 ! [I2: int,K: nat,J: int] :
% 0.30/0.66 ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 0.30/0.66 => ( ( nth_int @ ( upto @ I2 @ J ) @ K )
% 0.30/0.66 = ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % nth_upto
% 0.30/0.66 thf(fact_9927_length__upto,axiom,
% 0.30/0.66 ! [I2: int,J: int] :
% 0.30/0.66 ( ( size_size_list_int @ ( upto @ I2 @ J ) )
% 0.30/0.66 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I2 ) @ one_one_int ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % length_upto
% 0.30/0.66 thf(fact_9928_upto__rec__numeral_I1_J,axiom,
% 0.30/0.66 ! [M: num,N: num] :
% 0.30/0.66 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.66 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.66 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 0.30/0.66 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.66 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.66 = nil_int ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % upto_rec_numeral(1)
% 0.30/0.66 thf(fact_9929_upto__rec__numeral_I4_J,axiom,
% 0.30/0.66 ! [M: num,N: num] :
% 0.30/0.66 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.66 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.66 = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 0.30/0.66 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.66 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.66 = nil_int ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % upto_rec_numeral(4)
% 0.30/0.66 thf(fact_9930_upto__rec__numeral_I3_J,axiom,
% 0.30/0.66 ! [M: num,N: num] :
% 0.30/0.66 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.66 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.66 = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 0.30/0.66 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.66 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.66 = nil_int ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % upto_rec_numeral(3)
% 0.30/0.66 thf(fact_9931_upto__rec__numeral_I2_J,axiom,
% 0.30/0.66 ! [M: num,N: num] :
% 0.30/0.66 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.66 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.66 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 0.30/0.66 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.66 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.66 = nil_int ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % upto_rec_numeral(2)
% 0.30/0.66 thf(fact_9932_sequentially__offset,axiom,
% 0.30/0.66 ! [P: nat > $o,K: nat] :
% 0.30/0.66 ( ( eventually_nat @ P @ at_top_nat )
% 0.30/0.66 => ( eventually_nat
% 0.30/0.66 @ ^ [I4: nat] : ( P @ ( plus_plus_nat @ I4 @ K ) )
% 0.30/0.66 @ at_top_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % sequentially_offset
% 0.30/0.66 thf(fact_9933_eventually__sequentially,axiom,
% 0.30/0.66 ! [P: nat > $o] :
% 0.30/0.66 ( ( eventually_nat @ P @ at_top_nat )
% 0.30/0.66 = ( ? [N6: nat] :
% 0.30/0.66 ! [N3: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ N6 @ N3 )
% 0.30/0.66 => ( P @ N3 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % eventually_sequentially
% 0.30/0.66 thf(fact_9934_eventually__sequentiallyI,axiom,
% 0.30/0.66 ! [C: nat,P: nat > $o] :
% 0.30/0.66 ( ! [X5: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ C @ X5 )
% 0.30/0.66 => ( P @ X5 ) )
% 0.30/0.66 => ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % eventually_sequentiallyI
% 0.30/0.66 thf(fact_9935_le__sequentially,axiom,
% 0.30/0.66 ! [F5: filter_nat] :
% 0.30/0.66 ( ( ord_le2510731241096832064er_nat @ F5 @ at_top_nat )
% 0.30/0.66 = ( ! [N6: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N6 ) @ F5 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % le_sequentially
% 0.30/0.66 thf(fact_9936_upto__split2,axiom,
% 0.30/0.66 ! [I2: int,J: int,K: int] :
% 0.30/0.66 ( ( ord_less_eq_int @ I2 @ J )
% 0.30/0.66 => ( ( ord_less_eq_int @ J @ K )
% 0.30/0.66 => ( ( upto @ I2 @ K )
% 0.30/0.66 = ( append_int @ ( upto @ I2 @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % upto_split2
% 0.30/0.66 thf(fact_9937_upto__split1,axiom,
% 0.30/0.66 ! [I2: int,J: int,K: int] :
% 0.30/0.66 ( ( ord_less_eq_int @ I2 @ J )
% 0.30/0.66 => ( ( ord_less_eq_int @ J @ K )
% 0.30/0.66 => ( ( upto @ I2 @ K )
% 0.30/0.66 = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % upto_split1
% 0.30/0.66 thf(fact_9938_atLeastLessThan__upto,axiom,
% 0.30/0.66 ( set_or4662586982721622107an_int
% 0.30/0.66 = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % atLeastLessThan_upto
% 0.30/0.66 thf(fact_9939_upto_Osimps,axiom,
% 0.30/0.66 ( upto
% 0.30/0.66 = ( ^ [I4: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J3 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % upto.simps
% 0.30/0.66 thf(fact_9940_upto_Oelims,axiom,
% 0.30/0.66 ! [X2: int,Xa2: int,Y3: list_int] :
% 0.30/0.66 ( ( ( upto @ X2 @ Xa2 )
% 0.30/0.66 = Y3 )
% 0.30/0.66 => ( ( ( ord_less_eq_int @ X2 @ Xa2 )
% 0.30/0.66 => ( Y3
% 0.30/0.66 = ( cons_int @ X2 @ ( upto @ ( plus_plus_int @ X2 @ one_one_int ) @ Xa2 ) ) ) )
% 0.30/0.66 & ( ~ ( ord_less_eq_int @ X2 @ Xa2 )
% 0.30/0.66 => ( Y3 = nil_int ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % upto.elims
% 0.30/0.66 thf(fact_9941_upto__rec1,axiom,
% 0.30/0.66 ! [I2: int,J: int] :
% 0.30/0.66 ( ( ord_less_eq_int @ I2 @ J )
% 0.30/0.66 => ( ( upto @ I2 @ J )
% 0.30/0.66 = ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % upto_rec1
% 0.30/0.66 thf(fact_9942_upto__rec2,axiom,
% 0.30/0.66 ! [I2: int,J: int] :
% 0.30/0.66 ( ( ord_less_eq_int @ I2 @ J )
% 0.30/0.66 => ( ( upto @ I2 @ J )
% 0.30/0.66 = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % upto_rec2
% 0.30/0.66 thf(fact_9943_greaterThanLessThan__upto,axiom,
% 0.30/0.66 ( set_or5832277885323065728an_int
% 0.30/0.66 = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % greaterThanLessThan_upto
% 0.30/0.66 thf(fact_9944_upto__split3,axiom,
% 0.30/0.66 ! [I2: int,J: int,K: int] :
% 0.30/0.66 ( ( ord_less_eq_int @ I2 @ J )
% 0.30/0.66 => ( ( ord_less_eq_int @ J @ K )
% 0.30/0.66 => ( ( upto @ I2 @ K )
% 0.30/0.66 = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % upto_split3
% 0.30/0.66 thf(fact_9945_upto_Opsimps,axiom,
% 0.30/0.66 ! [I2: int,J: int] :
% 0.30/0.66 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J ) )
% 0.30/0.66 => ( ( ( ord_less_eq_int @ I2 @ J )
% 0.30/0.66 => ( ( upto @ I2 @ J )
% 0.30/0.66 = ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) )
% 0.30/0.66 & ( ~ ( ord_less_eq_int @ I2 @ J )
% 0.30/0.66 => ( ( upto @ I2 @ J )
% 0.30/0.66 = nil_int ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % upto.psimps
% 0.30/0.66 thf(fact_9946_Bseq__eq__bounded,axiom,
% 0.30/0.66 ! [F: nat > real,A: real,B: real] :
% 0.30/0.66 ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A @ B ) )
% 0.30/0.66 => ( bfun_nat_real @ F @ at_top_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % Bseq_eq_bounded
% 0.30/0.66 thf(fact_9947_Bseq__realpow,axiom,
% 0.30/0.66 ! [X2: real] :
% 0.30/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.66 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.30/0.66 => ( bfun_nat_real @ ( power_power_real @ X2 ) @ at_top_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Bseq_realpow
% 0.30/0.66 thf(fact_9948_card__greaterThanAtMost,axiom,
% 0.30/0.66 ! [L: nat,U: nat] :
% 0.30/0.66 ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
% 0.30/0.66 = ( minus_minus_nat @ U @ L ) ) ).
% 0.30/0.66
% 0.30/0.66 % card_greaterThanAtMost
% 0.30/0.66 thf(fact_9949_GreatestI__ex__nat,axiom,
% 0.30/0.66 ! [P: nat > $o,B: nat] :
% 0.30/0.66 ( ? [X_1: nat] : ( P @ X_1 )
% 0.30/0.66 => ( ! [Y4: nat] :
% 0.30/0.66 ( ( P @ Y4 )
% 0.30/0.66 => ( ord_less_eq_nat @ Y4 @ B ) )
% 0.30/0.66 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % GreatestI_ex_nat
% 0.30/0.66 thf(fact_9950_Greatest__le__nat,axiom,
% 0.30/0.66 ! [P: nat > $o,K: nat,B: nat] :
% 0.30/0.66 ( ( P @ K )
% 0.30/0.66 => ( ! [Y4: nat] :
% 0.30/0.66 ( ( P @ Y4 )
% 0.30/0.66 => ( ord_less_eq_nat @ Y4 @ B ) )
% 0.30/0.66 => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Greatest_le_nat
% 0.30/0.66 thf(fact_9951_GreatestI__nat,axiom,
% 0.30/0.66 ! [P: nat > $o,K: nat,B: nat] :
% 0.30/0.66 ( ( P @ K )
% 0.30/0.66 => ( ! [Y4: nat] :
% 0.30/0.66 ( ( P @ Y4 )
% 0.30/0.66 => ( ord_less_eq_nat @ Y4 @ B ) )
% 0.30/0.66 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % GreatestI_nat
% 0.30/0.66 thf(fact_9952_atLeastSucAtMost__greaterThanAtMost,axiom,
% 0.30/0.66 ! [L: nat,U: nat] :
% 0.30/0.66 ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
% 0.30/0.66 = ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% 0.30/0.66
% 0.30/0.66 % atLeastSucAtMost_greaterThanAtMost
% 0.30/0.66 thf(fact_9953_card__greaterThanAtMost__int,axiom,
% 0.30/0.66 ! [L: int,U: int] :
% 0.30/0.66 ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L @ U ) )
% 0.30/0.66 = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % card_greaterThanAtMost_int
% 0.30/0.66 thf(fact_9954_atLeast__Suc__greaterThan,axiom,
% 0.30/0.66 ! [K: nat] :
% 0.30/0.66 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 0.30/0.66 = ( set_or1210151606488870762an_nat @ K ) ) ).
% 0.30/0.66
% 0.30/0.66 % atLeast_Suc_greaterThan
% 0.30/0.66 thf(fact_9955_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
% 0.30/0.66 ! [L: int,U: int] :
% 0.30/0.66 ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
% 0.30/0.66 = ( set_or6656581121297822940st_int @ L @ U ) ) ).
% 0.30/0.66
% 0.30/0.66 % atLeastPlusOneAtMost_greaterThanAtMost_int
% 0.30/0.66 thf(fact_9956_decseq__bounded,axiom,
% 0.30/0.66 ! [X8: nat > real,B3: real] :
% 0.30/0.66 ( ( order_9091379641038594480t_real @ X8 )
% 0.30/0.66 => ( ! [I3: nat] : ( ord_less_eq_real @ B3 @ ( X8 @ I3 ) )
% 0.30/0.66 => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % decseq_bounded
% 0.30/0.66 thf(fact_9957_greaterThanAtMost__upto,axiom,
% 0.30/0.66 ( set_or6656581121297822940st_int
% 0.30/0.66 = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % greaterThanAtMost_upto
% 0.30/0.66 thf(fact_9958_decseq__convergent,axiom,
% 0.30/0.66 ! [X8: nat > real,B3: real] :
% 0.30/0.66 ( ( order_9091379641038594480t_real @ X8 )
% 0.30/0.66 => ( ! [I3: nat] : ( ord_less_eq_real @ B3 @ ( X8 @ I3 ) )
% 0.30/0.66 => ~ ! [L6: real] :
% 0.30/0.66 ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 0.30/0.66 => ~ ! [I: nat] : ( ord_less_eq_real @ L6 @ ( X8 @ I ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % decseq_convergent
% 0.30/0.66 thf(fact_9959_atLeast__Suc,axiom,
% 0.30/0.66 ! [K: nat] :
% 0.30/0.66 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 0.30/0.66 = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % atLeast_Suc
% 0.30/0.66 thf(fact_9960_GMVT,axiom,
% 0.30/0.66 ! [A: real,B: real,F: real > real,G: real > real] :
% 0.30/0.66 ( ( ord_less_real @ A @ B )
% 0.30/0.66 => ( ! [X5: real] :
% 0.30/0.66 ( ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.66 & ( ord_less_eq_real @ X5 @ B ) )
% 0.30/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ F ) )
% 0.30/0.66 => ( ! [X5: real] :
% 0.30/0.66 ( ( ( ord_less_real @ A @ X5 )
% 0.30/0.66 & ( ord_less_real @ X5 @ B ) )
% 0.30/0.66 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 0.30/0.66 => ( ! [X5: real] :
% 0.30/0.66 ( ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.66 & ( ord_less_eq_real @ X5 @ B ) )
% 0.30/0.66 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ G ) )
% 0.30/0.66 => ( ! [X5: real] :
% 0.30/0.66 ( ( ( ord_less_real @ A @ X5 )
% 0.30/0.66 & ( ord_less_real @ X5 @ B ) )
% 0.30/0.66 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 0.30/0.66 => ? [G_c: real,F_c: real,C3: real] :
% 0.30/0.66 ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 0.30/0.66 & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 0.30/0.66 & ( ord_less_real @ A @ C3 )
% 0.30/0.66 & ( ord_less_real @ C3 @ B )
% 0.30/0.66 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
% 0.30/0.66 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % GMVT
% 0.30/0.66 thf(fact_9961_MVT,axiom,
% 0.30/0.66 ! [A: real,B: real,F: real > real] :
% 0.30/0.66 ( ( ord_less_real @ A @ B )
% 0.30/0.66 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.30/0.66 => ( ! [X5: real] :
% 0.30/0.66 ( ( ord_less_real @ A @ X5 )
% 0.30/0.66 => ( ( ord_less_real @ X5 @ B )
% 0.30/0.66 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 0.30/0.66 => ? [L4: real,Z2: real] :
% 0.30/0.66 ( ( ord_less_real @ A @ Z2 )
% 0.30/0.66 & ( ord_less_real @ Z2 @ B )
% 0.30/0.66 & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) )
% 0.30/0.66 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.30/0.66 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % MVT
% 0.30/0.66 thf(fact_9962_continuous__on__arcosh_H,axiom,
% 0.30/0.66 ! [A2: set_real,F: real > real] :
% 0.30/0.66 ( ( topolo5044208981011980120l_real @ A2 @ F )
% 0.30/0.66 => ( ! [X5: real] :
% 0.30/0.66 ( ( member_real @ X5 @ A2 )
% 0.30/0.66 => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
% 0.30/0.66 => ( topolo5044208981011980120l_real @ A2
% 0.30/0.66 @ ^ [X: real] : ( arcosh_real @ ( F @ X ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % continuous_on_arcosh'
% 0.30/0.66 thf(fact_9963_continuous__image__closed__interval,axiom,
% 0.30/0.66 ! [A: real,B: real,F: real > real] :
% 0.30/0.66 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.66 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.30/0.66 => ? [C3: real,D3: real] :
% 0.30/0.66 ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
% 0.30/0.66 = ( set_or1222579329274155063t_real @ C3 @ D3 ) )
% 0.30/0.66 & ( ord_less_eq_real @ C3 @ D3 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % continuous_image_closed_interval
% 0.30/0.66 thf(fact_9964_continuous__on__arcosh,axiom,
% 0.30/0.66 ! [A2: set_real] :
% 0.30/0.66 ( ( ord_less_eq_set_real @ A2 @ ( set_ord_atLeast_real @ one_one_real ) )
% 0.30/0.66 => ( topolo5044208981011980120l_real @ A2 @ arcosh_real ) ) ).
% 0.30/0.66
% 0.30/0.66 % continuous_on_arcosh
% 0.30/0.66 thf(fact_9965_continuous__on__arccos_H,axiom,
% 0.30/0.66 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arccos ).
% 0.30/0.66
% 0.30/0.66 % continuous_on_arccos'
% 0.30/0.66 thf(fact_9966_continuous__on__arcsin_H,axiom,
% 0.30/0.66 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arcsin ).
% 0.30/0.66
% 0.30/0.66 % continuous_on_arcsin'
% 0.30/0.66 thf(fact_9967_continuous__on__artanh_H,axiom,
% 0.30/0.66 ! [A2: set_real,F: real > real] :
% 0.30/0.66 ( ( topolo5044208981011980120l_real @ A2 @ F )
% 0.30/0.66 => ( ! [X5: real] :
% 0.30/0.66 ( ( member_real @ X5 @ A2 )
% 0.30/0.66 => ( member_real @ ( F @ X5 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
% 0.30/0.66 => ( topolo5044208981011980120l_real @ A2
% 0.30/0.66 @ ^ [X: real] : ( artanh_real @ ( F @ X ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % continuous_on_artanh'
% 0.30/0.66 thf(fact_9968_mvt,axiom,
% 0.30/0.66 ! [A: real,B: real,F: real > real,F4: real > real > real] :
% 0.30/0.66 ( ( ord_less_real @ A @ B )
% 0.30/0.66 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.30/0.66 => ( ! [X5: real] :
% 0.30/0.66 ( ( ord_less_real @ A @ X5 )
% 0.30/0.66 => ( ( ord_less_real @ X5 @ B )
% 0.30/0.66 => ( has_de1759254742604945161l_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 0.30/0.66 => ~ ! [Xi: real] :
% 0.30/0.66 ( ( ord_less_real @ A @ Xi )
% 0.30/0.66 => ( ( ord_less_real @ Xi @ B )
% 0.30/0.66 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.30/0.66 != ( F4 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % mvt
% 0.30/0.66 thf(fact_9969_continuous__on__artanh,axiom,
% 0.30/0.66 ! [A2: set_real] :
% 0.30/0.66 ( ( ord_less_eq_set_real @ A2 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
% 0.30/0.66 => ( topolo5044208981011980120l_real @ A2 @ artanh_real ) ) ).
% 0.30/0.66
% 0.30/0.66 % continuous_on_artanh
% 0.30/0.66 thf(fact_9970_DERIV__isconst2,axiom,
% 0.30/0.66 ! [A: real,B: real,F: real > real,X2: real] :
% 0.30/0.66 ( ( ord_less_real @ A @ B )
% 0.30/0.66 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.30/0.66 => ( ! [X5: real] :
% 0.30/0.66 ( ( ord_less_real @ A @ X5 )
% 0.30/0.66 => ( ( ord_less_real @ X5 @ B )
% 0.30/0.66 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 0.30/0.66 => ( ( ord_less_eq_real @ A @ X2 )
% 0.30/0.66 => ( ( ord_less_eq_real @ X2 @ B )
% 0.30/0.66 => ( ( F @ X2 )
% 0.30/0.66 = ( F @ A ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % DERIV_isconst2
% 0.30/0.66 thf(fact_9971_mono__times__nat,axiom,
% 0.30/0.66 ! [N: nat] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % mono_times_nat
% 0.30/0.66 thf(fact_9972_mono__Suc,axiom,
% 0.30/0.66 order_mono_nat_nat @ suc ).
% 0.30/0.66
% 0.30/0.66 % mono_Suc
% 0.30/0.66 thf(fact_9973_incseq__bounded,axiom,
% 0.30/0.66 ! [X8: nat > real,B3: real] :
% 0.30/0.66 ( ( order_mono_nat_real @ X8 )
% 0.30/0.66 => ( ! [I3: nat] : ( ord_less_eq_real @ ( X8 @ I3 ) @ B3 )
% 0.30/0.66 => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % incseq_bounded
% 0.30/0.66 thf(fact_9974_incseq__convergent,axiom,
% 0.30/0.66 ! [X8: nat > real,B3: real] :
% 0.30/0.66 ( ( order_mono_nat_real @ X8 )
% 0.30/0.66 => ( ! [I3: nat] : ( ord_less_eq_real @ ( X8 @ I3 ) @ B3 )
% 0.30/0.66 => ~ ! [L6: real] :
% 0.30/0.66 ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 0.30/0.66 => ~ ! [I: nat] : ( ord_less_eq_real @ ( X8 @ I ) @ L6 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % incseq_convergent
% 0.30/0.66 thf(fact_9975_mono__ge2__power__minus__self,axiom,
% 0.30/0.66 ! [K: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.30/0.66 => ( order_mono_nat_nat
% 0.30/0.66 @ ^ [M3: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M3 ) @ M3 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % mono_ge2_power_minus_self
% 0.30/0.66 thf(fact_9976_nonneg__incseq__Bseq__subseq__iff,axiom,
% 0.30/0.66 ! [F: nat > real,G: nat > nat] :
% 0.30/0.66 ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 0.30/0.66 => ( ( order_mono_nat_real @ F )
% 0.30/0.66 => ( ( order_5726023648592871131at_nat @ G )
% 0.30/0.66 => ( ( bfun_nat_real
% 0.30/0.66 @ ^ [X: nat] : ( F @ ( G @ X ) )
% 0.30/0.66 @ at_top_nat )
% 0.30/0.66 = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % nonneg_incseq_Bseq_subseq_iff
% 0.30/0.66 thf(fact_9977_inj__sgn__power,axiom,
% 0.30/0.66 ! [N: nat] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( inj_on_real_real
% 0.30/0.66 @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 0.30/0.66 @ top_top_set_real ) ) ).
% 0.30/0.66
% 0.30/0.66 % inj_sgn_power
% 0.30/0.66 thf(fact_9978_strict__mono__imp__increasing,axiom,
% 0.30/0.66 ! [F: nat > nat,N: nat] :
% 0.30/0.66 ( ( order_5726023648592871131at_nat @ F )
% 0.30/0.66 => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % strict_mono_imp_increasing
% 0.30/0.66 thf(fact_9979_log__inj,axiom,
% 0.30/0.66 ! [B: real] :
% 0.30/0.66 ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.66 => ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % log_inj
% 0.30/0.66 thf(fact_9980_inj__on__diff__nat,axiom,
% 0.30/0.66 ! [N5: set_nat,K: nat] :
% 0.30/0.66 ( ! [N2: nat] :
% 0.30/0.66 ( ( member_nat @ N2 @ N5 )
% 0.30/0.66 => ( ord_less_eq_nat @ K @ N2 ) )
% 0.30/0.66 => ( inj_on_nat_nat
% 0.30/0.66 @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K )
% 0.30/0.66 @ N5 ) ) ).
% 0.30/0.66
% 0.30/0.66 % inj_on_diff_nat
% 0.30/0.66 thf(fact_9981_inj__Suc,axiom,
% 0.30/0.66 ! [N5: set_nat] : ( inj_on_nat_nat @ suc @ N5 ) ).
% 0.30/0.66
% 0.30/0.66 % inj_Suc
% 0.30/0.66 thf(fact_9982_summable__reindex,axiom,
% 0.30/0.66 ! [F: nat > real,G: nat > nat] :
% 0.30/0.66 ( ( summable_real @ F )
% 0.30/0.66 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.30/0.66 => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 0.30/0.66 => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % summable_reindex
% 0.30/0.66 thf(fact_9983_suminf__reindex__mono,axiom,
% 0.30/0.66 ! [F: nat > real,G: nat > nat] :
% 0.30/0.66 ( ( summable_real @ F )
% 0.30/0.66 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.30/0.66 => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 0.30/0.66 => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % suminf_reindex_mono
% 0.30/0.66 thf(fact_9984_inj__on__char__of__nat,axiom,
% 0.30/0.66 inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % inj_on_char_of_nat
% 0.30/0.66 thf(fact_9985_suminf__reindex,axiom,
% 0.30/0.66 ! [F: nat > real,G: nat > nat] :
% 0.30/0.66 ( ( summable_real @ F )
% 0.30/0.66 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.30/0.66 => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 0.30/0.66 => ( ! [X5: nat] :
% 0.30/0.66 ( ~ ( member_nat @ X5 @ ( image_nat_nat @ G @ top_top_set_nat ) )
% 0.30/0.66 => ( ( F @ X5 )
% 0.30/0.66 = zero_zero_real ) )
% 0.30/0.66 => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
% 0.30/0.66 = ( suminf_real @ F ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % suminf_reindex
% 0.30/0.66 thf(fact_9986_times__int_Oabs__eq,axiom,
% 0.30/0.66 ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
% 0.30/0.66 ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
% 0.30/0.66 = ( abs_Integ
% 0.30/0.66 @ ( produc27273713700761075at_nat
% 0.30/0.66 @ ^ [X: nat,Y: nat] :
% 0.30/0.66 ( produc2626176000494625587at_nat
% 0.30/0.66 @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) )
% 0.30/0.66 @ Xa2
% 0.30/0.66 @ X2 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % times_int.abs_eq
% 0.30/0.66 thf(fact_9987_Gcd__remove0__nat,axiom,
% 0.30/0.66 ! [M7: set_nat] :
% 0.30/0.66 ( ( finite_finite_nat @ M7 )
% 0.30/0.66 => ( ( gcd_Gcd_nat @ M7 )
% 0.30/0.66 = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M7 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Gcd_remove0_nat
% 0.30/0.66 thf(fact_9988_nat_Oabs__eq,axiom,
% 0.30/0.66 ! [X2: product_prod_nat_nat] :
% 0.30/0.66 ( ( nat2 @ ( abs_Integ @ X2 ) )
% 0.30/0.66 = ( produc6842872674320459806at_nat @ minus_minus_nat @ X2 ) ) ).
% 0.30/0.66
% 0.30/0.66 % nat.abs_eq
% 0.30/0.66 thf(fact_9989_eq__Abs__Integ,axiom,
% 0.30/0.66 ! [Z: int] :
% 0.30/0.66 ~ ! [X5: nat,Y4: nat] :
% 0.30/0.66 ( Z
% 0.30/0.66 != ( abs_Integ @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % eq_Abs_Integ
% 0.30/0.66 thf(fact_9990_Gcd__nat__eq__one,axiom,
% 0.30/0.66 ! [N5: set_nat] :
% 0.30/0.66 ( ( member_nat @ one_one_nat @ N5 )
% 0.30/0.66 => ( ( gcd_Gcd_nat @ N5 )
% 0.30/0.66 = one_one_nat ) ) ).
% 0.30/0.66
% 0.30/0.66 % Gcd_nat_eq_one
% 0.30/0.66 thf(fact_9991_zero__int__def,axiom,
% 0.30/0.66 ( zero_zero_int
% 0.30/0.66 = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % zero_int_def
% 0.30/0.66 thf(fact_9992_int__def,axiom,
% 0.30/0.66 ( semiri1314217659103216013at_int
% 0.30/0.66 = ( ^ [N3: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N3 @ zero_zero_nat ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % int_def
% 0.30/0.66 thf(fact_9993_uminus__int_Oabs__eq,axiom,
% 0.30/0.66 ! [X2: product_prod_nat_nat] :
% 0.30/0.66 ( ( uminus_uminus_int @ ( abs_Integ @ X2 ) )
% 0.30/0.66 = ( abs_Integ
% 0.30/0.66 @ ( produc2626176000494625587at_nat
% 0.30/0.66 @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X )
% 0.30/0.66 @ X2 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % uminus_int.abs_eq
% 0.30/0.66 thf(fact_9994_one__int__def,axiom,
% 0.30/0.66 ( one_one_int
% 0.30/0.66 = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % one_int_def
% 0.30/0.66 thf(fact_9995_less__int_Oabs__eq,axiom,
% 0.30/0.66 ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
% 0.30/0.66 ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
% 0.30/0.66 = ( produc8739625826339149834_nat_o
% 0.30/0.66 @ ^ [X: nat,Y: nat] :
% 0.30/0.66 ( produc6081775807080527818_nat_o
% 0.30/0.66 @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
% 0.30/0.66 @ Xa2
% 0.30/0.66 @ X2 ) ) ).
% 0.30/0.66
% 0.30/0.66 % less_int.abs_eq
% 0.30/0.66 thf(fact_9996_less__eq__int_Oabs__eq,axiom,
% 0.30/0.66 ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
% 0.30/0.66 ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
% 0.30/0.66 = ( produc8739625826339149834_nat_o
% 0.30/0.66 @ ^ [X: nat,Y: nat] :
% 0.30/0.66 ( produc6081775807080527818_nat_o
% 0.30/0.66 @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
% 0.30/0.66 @ Xa2
% 0.30/0.66 @ X2 ) ) ).
% 0.30/0.66
% 0.30/0.66 % less_eq_int.abs_eq
% 0.30/0.66 thf(fact_9997_plus__int_Oabs__eq,axiom,
% 0.30/0.66 ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
% 0.30/0.66 ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
% 0.30/0.66 = ( abs_Integ
% 0.30/0.66 @ ( produc27273713700761075at_nat
% 0.30/0.66 @ ^ [X: nat,Y: nat] :
% 0.30/0.66 ( produc2626176000494625587at_nat
% 0.30/0.66 @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) )
% 0.30/0.66 @ Xa2
% 0.30/0.66 @ X2 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % plus_int.abs_eq
% 0.30/0.66 thf(fact_9998_minus__int_Oabs__eq,axiom,
% 0.30/0.66 ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
% 0.30/0.66 ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
% 0.30/0.66 = ( abs_Integ
% 0.30/0.66 @ ( produc27273713700761075at_nat
% 0.30/0.66 @ ^ [X: nat,Y: nat] :
% 0.30/0.66 ( produc2626176000494625587at_nat
% 0.30/0.66 @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) )
% 0.30/0.66 @ Xa2
% 0.30/0.66 @ X2 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % minus_int.abs_eq
% 0.30/0.66 thf(fact_9999_Gcd__int__greater__eq__0,axiom,
% 0.30/0.66 ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).
% 0.30/0.66
% 0.30/0.66 % Gcd_int_greater_eq_0
% 0.30/0.66 thf(fact_10000_powr__real__of__int_H,axiom,
% 0.30/0.66 ! [X2: real,N: int] :
% 0.30/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.30/0.66 => ( ( ( X2 != zero_zero_real )
% 0.30/0.66 | ( ord_less_int @ zero_zero_int @ N ) )
% 0.30/0.66 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N ) )
% 0.30/0.66 = ( power_int_real @ X2 @ N ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % powr_real_of_int'
% 0.30/0.66 thf(fact_10001_less__eq__int_Orep__eq,axiom,
% 0.30/0.66 ( ord_less_eq_int
% 0.30/0.66 = ( ^ [X: int,Xa4: int] :
% 0.30/0.66 ( produc8739625826339149834_nat_o
% 0.30/0.66 @ ^ [Y: nat,Z3: nat] :
% 0.30/0.66 ( produc6081775807080527818_nat_o
% 0.30/0.66 @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
% 0.30/0.66 @ ( rep_Integ @ X )
% 0.30/0.66 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % less_eq_int.rep_eq
% 0.30/0.66 thf(fact_10002_less__int_Orep__eq,axiom,
% 0.30/0.66 ( ord_less_int
% 0.30/0.66 = ( ^ [X: int,Xa4: int] :
% 0.30/0.66 ( produc8739625826339149834_nat_o
% 0.30/0.66 @ ^ [Y: nat,Z3: nat] :
% 0.30/0.66 ( produc6081775807080527818_nat_o
% 0.30/0.66 @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
% 0.30/0.66 @ ( rep_Integ @ X )
% 0.30/0.66 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % less_int.rep_eq
% 0.30/0.66 thf(fact_10003_nat_Orep__eq,axiom,
% 0.30/0.66 ( nat2
% 0.30/0.66 = ( ^ [X: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % nat.rep_eq
% 0.30/0.66 thf(fact_10004_prod__encode__def,axiom,
% 0.30/0.66 ( nat_prod_encode
% 0.30/0.66 = ( produc6842872674320459806at_nat
% 0.30/0.66 @ ^ [M3: nat,N3: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M3 @ N3 ) ) @ M3 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % prod_encode_def
% 0.30/0.66 thf(fact_10005_le__prod__encode__1,axiom,
% 0.30/0.66 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % le_prod_encode_1
% 0.30/0.66 thf(fact_10006_le__prod__encode__2,axiom,
% 0.30/0.66 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % le_prod_encode_2
% 0.30/0.66 thf(fact_10007_prod__encode__prod__decode__aux,axiom,
% 0.30/0.66 ! [K: nat,M: nat] :
% 0.30/0.66 ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
% 0.30/0.66 = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).
% 0.30/0.66
% 0.30/0.66 % prod_encode_prod_decode_aux
% 0.30/0.66 thf(fact_10008_uminus__int__def,axiom,
% 0.30/0.66 ( uminus_uminus_int
% 0.30/0.66 = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
% 0.30/0.66 @ ( produc2626176000494625587at_nat
% 0.30/0.66 @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % uminus_int_def
% 0.30/0.66 thf(fact_10009_times__int__def,axiom,
% 0.30/0.66 ( times_times_int
% 0.30/0.66 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.30/0.66 @ ( produc27273713700761075at_nat
% 0.30/0.66 @ ^ [X: nat,Y: nat] :
% 0.30/0.66 ( produc2626176000494625587at_nat
% 0.30/0.66 @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % times_int_def
% 0.30/0.66 thf(fact_10010_minus__int__def,axiom,
% 0.30/0.66 ( minus_minus_int
% 0.30/0.66 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.30/0.66 @ ( produc27273713700761075at_nat
% 0.30/0.66 @ ^ [X: nat,Y: nat] :
% 0.30/0.66 ( produc2626176000494625587at_nat
% 0.30/0.66 @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % minus_int_def
% 0.30/0.66 thf(fact_10011_plus__int__def,axiom,
% 0.30/0.66 ( plus_plus_int
% 0.30/0.66 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.30/0.66 @ ( produc27273713700761075at_nat
% 0.30/0.66 @ ^ [X: nat,Y: nat] :
% 0.30/0.66 ( produc2626176000494625587at_nat
% 0.30/0.66 @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % plus_int_def
% 0.30/0.66 thf(fact_10012_list__encode_Oelims,axiom,
% 0.30/0.66 ! [X2: list_nat,Y3: nat] :
% 0.30/0.66 ( ( ( nat_list_encode @ X2 )
% 0.30/0.66 = Y3 )
% 0.30/0.66 => ( ( ( X2 = nil_nat )
% 0.30/0.66 => ( Y3 != zero_zero_nat ) )
% 0.30/0.66 => ~ ! [X5: nat,Xs3: list_nat] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( cons_nat @ X5 @ Xs3 ) )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % list_encode.elims
% 0.30/0.66 thf(fact_10013_list__encode_Osimps_I2_J,axiom,
% 0.30/0.66 ! [X2: nat,Xs2: list_nat] :
% 0.30/0.66 ( ( nat_list_encode @ ( cons_nat @ X2 @ Xs2 ) )
% 0.30/0.66 = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X2 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % list_encode.simps(2)
% 0.30/0.66 thf(fact_10014_list__encode_Opelims,axiom,
% 0.30/0.66 ! [X2: list_nat,Y3: nat] :
% 0.30/0.66 ( ( ( nat_list_encode @ X2 )
% 0.30/0.66 = Y3 )
% 0.30/0.66 => ( ( accp_list_nat @ nat_list_encode_rel @ X2 )
% 0.30/0.66 => ( ( ( X2 = nil_nat )
% 0.30/0.66 => ( ( Y3 = zero_zero_nat )
% 0.30/0.66 => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
% 0.30/0.66 => ~ ! [X5: nat,Xs3: list_nat] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( cons_nat @ X5 @ Xs3 ) )
% 0.30/0.66 => ( ( Y3
% 0.30/0.66 = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) )
% 0.30/0.66 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X5 @ Xs3 ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % list_encode.pelims
% 0.30/0.66 thf(fact_10015_pred__nat__def,axiom,
% 0.30/0.66 ( pred_nat
% 0.30/0.66 = ( collec3392354462482085612at_nat
% 0.30/0.66 @ ( produc6081775807080527818_nat_o
% 0.30/0.66 @ ^ [M3: nat,N3: nat] :
% 0.30/0.66 ( N3
% 0.30/0.66 = ( suc @ M3 ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % pred_nat_def
% 0.30/0.66 thf(fact_10016_num__of__nat_Osimps_I2_J,axiom,
% 0.30/0.66 ! [N: nat] :
% 0.30/0.66 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( num_of_nat @ ( suc @ N ) )
% 0.30/0.66 = ( inc @ ( num_of_nat @ N ) ) ) )
% 0.30/0.66 & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( num_of_nat @ ( suc @ N ) )
% 0.30/0.66 = one ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % num_of_nat.simps(2)
% 0.30/0.66 thf(fact_10017_num__of__nat__numeral__eq,axiom,
% 0.30/0.66 ! [Q3: num] :
% 0.30/0.66 ( ( num_of_nat @ ( numeral_numeral_nat @ Q3 ) )
% 0.30/0.66 = Q3 ) ).
% 0.30/0.66
% 0.30/0.66 % num_of_nat_numeral_eq
% 0.30/0.66 thf(fact_10018_num__of__nat_Osimps_I1_J,axiom,
% 0.30/0.66 ( ( num_of_nat @ zero_zero_nat )
% 0.30/0.66 = one ) ).
% 0.30/0.66
% 0.30/0.66 % num_of_nat.simps(1)
% 0.30/0.66 thf(fact_10019_numeral__num__of__nat,axiom,
% 0.30/0.66 ! [N: nat] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
% 0.30/0.66 = N ) ) ).
% 0.30/0.66
% 0.30/0.66 % numeral_num_of_nat
% 0.30/0.66 thf(fact_10020_num__of__nat__One,axiom,
% 0.30/0.66 ! [N: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ N @ one_one_nat )
% 0.30/0.66 => ( ( num_of_nat @ N )
% 0.30/0.66 = one ) ) ).
% 0.30/0.66
% 0.30/0.66 % num_of_nat_One
% 0.30/0.66 thf(fact_10021_num__of__nat__double,axiom,
% 0.30/0.66 ! [N: nat] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
% 0.30/0.66 = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % num_of_nat_double
% 0.30/0.66 thf(fact_10022_num__of__nat__plus__distrib,axiom,
% 0.30/0.66 ! [M: nat,N: nat] :
% 0.30/0.66 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66 => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.66 = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % num_of_nat_plus_distrib
% 0.30/0.66 thf(fact_10023_pow_Osimps_I3_J,axiom,
% 0.30/0.66 ! [X2: num,Y3: num] :
% 0.30/0.66 ( ( pow @ X2 @ ( bit1 @ Y3 ) )
% 0.30/0.66 = ( times_times_num @ ( sqr @ ( pow @ X2 @ Y3 ) ) @ X2 ) ) ).
% 0.30/0.66
% 0.30/0.66 % pow.simps(3)
% 0.30/0.66 thf(fact_10024_sorted__list__of__set__lessThan__Suc,axiom,
% 0.30/0.66 ! [K: nat] :
% 0.30/0.66 ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
% 0.30/0.66 = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % sorted_list_of_set_lessThan_Suc
% 0.30/0.66 thf(fact_10025_sorted__list__of__set__atMost__Suc,axiom,
% 0.30/0.66 ! [K: nat] :
% 0.30/0.66 ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
% 0.30/0.66 = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % sorted_list_of_set_atMost_Suc
% 0.30/0.66 thf(fact_10026_sqr__conv__mult,axiom,
% 0.30/0.66 ( sqr
% 0.30/0.66 = ( ^ [X: num] : ( times_times_num @ X @ X ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % sqr_conv_mult
% 0.30/0.66 thf(fact_10027_sqr_Osimps_I1_J,axiom,
% 0.30/0.66 ( ( sqr @ one )
% 0.30/0.66 = one ) ).
% 0.30/0.66
% 0.30/0.66 % sqr.simps(1)
% 0.30/0.66 thf(fact_10028_sqr_Osimps_I2_J,axiom,
% 0.30/0.66 ! [N: num] :
% 0.30/0.66 ( ( sqr @ ( bit0 @ N ) )
% 0.30/0.66 = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % sqr.simps(2)
% 0.30/0.66 thf(fact_10029_pow_Osimps_I2_J,axiom,
% 0.30/0.66 ! [X2: num,Y3: num] :
% 0.30/0.66 ( ( pow @ X2 @ ( bit0 @ Y3 ) )
% 0.30/0.66 = ( sqr @ ( pow @ X2 @ Y3 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % pow.simps(2)
% 0.30/0.66 thf(fact_10030_sorted__list__of__set__greaterThanAtMost,axiom,
% 0.30/0.66 ! [I2: nat,J: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ ( suc @ I2 ) @ J )
% 0.30/0.66 => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J ) )
% 0.30/0.66 = ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % sorted_list_of_set_greaterThanAtMost
% 0.30/0.66 thf(fact_10031_sorted__list__of__set__greaterThanLessThan,axiom,
% 0.30/0.66 ! [I2: nat,J: nat] :
% 0.30/0.66 ( ( ord_less_nat @ ( suc @ I2 ) @ J )
% 0.30/0.66 => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J ) )
% 0.30/0.66 = ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % sorted_list_of_set_greaterThanLessThan
% 0.30/0.66 thf(fact_10032_sqr_Osimps_I3_J,axiom,
% 0.30/0.66 ! [N: num] :
% 0.30/0.66 ( ( sqr @ ( bit1 @ N ) )
% 0.30/0.66 = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % sqr.simps(3)
% 0.30/0.66 thf(fact_10033_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 0.30/0.66 ! [N: nat,J: nat,I2: nat] :
% 0.30/0.66 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I2 ) )
% 0.30/0.66 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J ) ) @ N )
% 0.30/0.66 = ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % nth_sorted_list_of_set_greaterThanAtMost
% 0.30/0.66 thf(fact_10034_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 0.30/0.66 ! [N: nat,J: nat,I2: nat] :
% 0.30/0.66 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I2 ) ) )
% 0.30/0.66 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J ) ) @ N )
% 0.30/0.66 = ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % nth_sorted_list_of_set_greaterThanLessThan
% 0.30/0.66 thf(fact_10035_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 0.30/0.66 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.30/0.66 ( ~ ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 0.30/0.66 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.30/0.66 ( X2
% 0.30/0.66 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.66 => ( Xa2 = one_one_nat ) )
% 0.30/0.66 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.30/0.66 => ( ( Deg2 = Xa2 )
% 0.30/0.66 & ! [X5: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.66 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.66 & ( case_o184042715313410164at_nat
% 0.30/0.66 @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X4 )
% 0.30/0.66 & ! [X: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.66 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 0.30/0.66 @ ( produc6081775807080527818_nat_o
% 0.30/0.66 @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.66 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.66 & ! [I4: nat] :
% 0.30/0.66 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.66 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X4 ) )
% 0.30/0.66 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.30/0.66 & ( ( Mi3 = Ma3 )
% 0.30/0.66 => ! [X: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.66 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 0.30/0.66 & ( ( Mi3 != Ma3 )
% 0.30/0.66 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 0.30/0.66 & ! [X: nat] :
% 0.30/0.66 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.66 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 0.30/0.66 => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.66 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.66 @ Mima ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % VEBT_internal.valid'.elims(3)
% 0.30/0.66 thf(fact_10036_atLeastLessThan__add__Un,axiom,
% 0.30/0.66 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.66 => ( ( set_or4665077453230672383an_nat @ I2 @ ( plus_plus_nat @ J @ K ) )
% 0.30/0.66 = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % atLeastLessThan_add_Un
% 0.30/0.66 thf(fact_10037_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 0.30/0.66 ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
% 0.30/0.66 ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg4 )
% 0.30/0.66 = ( ( Deg = Deg4 )
% 0.30/0.66 & ! [X: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.66 => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.30/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.66 & ( case_o184042715313410164at_nat
% 0.30/0.66 @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X4 )
% 0.30/0.66 & ! [X: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.66 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 0.30/0.66 @ ( produc6081775807080527818_nat_o
% 0.30/0.66 @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.66 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.30/0.66 & ! [I4: nat] :
% 0.30/0.66 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.66 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X4 ) )
% 0.30/0.66 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 0.30/0.66 & ( ( Mi3 = Ma3 )
% 0.30/0.66 => ! [X: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.66 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 0.30/0.66 & ( ( Mi3 != Ma3 )
% 0.30/0.66 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
% 0.30/0.66 & ! [X: nat] :
% 0.30/0.66 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.30/0.66 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
% 0.30/0.66 => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.66 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.66 @ Mima2 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % VEBT_internal.valid'.simps(2)
% 0.30/0.66 thf(fact_10038_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 0.30/0.66 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.66 ( ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 0.30/0.66 = Y3 )
% 0.30/0.66 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.30/0.66 ( X2
% 0.30/0.66 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.66 => ( Y3
% 0.30/0.66 = ( Xa2 != one_one_nat ) ) )
% 0.30/0.66 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.30/0.66 => ( Y3
% 0.30/0.66 = ( ~ ( ( Deg2 = Xa2 )
% 0.30/0.66 & ! [X: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.66 => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.66 & ( case_o184042715313410164at_nat
% 0.30/0.66 @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X4 )
% 0.30/0.66 & ! [X: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.66 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 0.30/0.66 @ ( produc6081775807080527818_nat_o
% 0.30/0.66 @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.66 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.66 & ! [I4: nat] :
% 0.30/0.66 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.66 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X4 ) )
% 0.30/0.66 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.30/0.66 & ( ( Mi3 = Ma3 )
% 0.30/0.66 => ! [X: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.66 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 0.30/0.66 & ( ( Mi3 != Ma3 )
% 0.30/0.66 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 0.30/0.66 & ! [X: nat] :
% 0.30/0.66 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.66 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 0.30/0.66 => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.66 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.66 @ Mima ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % VEBT_internal.valid'.elims(1)
% 0.30/0.66 thf(fact_10039_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 0.30/0.66 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.30/0.66 ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 0.30/0.66 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.30/0.66 ( X2
% 0.30/0.66 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.66 => ( Xa2 != one_one_nat ) )
% 0.30/0.66 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.30/0.66 => ~ ( ( Deg2 = Xa2 )
% 0.30/0.66 & ! [X3: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.66 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.66 & ( case_o184042715313410164at_nat
% 0.30/0.66 @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X4 )
% 0.30/0.66 & ! [X: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.66 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 0.30/0.66 @ ( produc6081775807080527818_nat_o
% 0.30/0.66 @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.66 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.66 & ! [I4: nat] :
% 0.30/0.66 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.66 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X4 ) )
% 0.30/0.66 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.30/0.66 & ( ( Mi3 = Ma3 )
% 0.30/0.66 => ! [X: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.66 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 0.30/0.66 & ( ( Mi3 != Ma3 )
% 0.30/0.66 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 0.30/0.66 & ! [X: nat] :
% 0.30/0.66 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.66 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 0.30/0.66 => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.66 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.66 @ Mima ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % VEBT_internal.valid'.elims(2)
% 0.30/0.66 thf(fact_10040_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 0.30/0.66 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.66 ( ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 0.30/0.66 = Y3 )
% 0.30/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.30/0.66 => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.66 => ( ( Y3
% 0.30/0.66 = ( Xa2 = one_one_nat ) )
% 0.30/0.66 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 0.30/0.66 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.30/0.66 => ( ( Y3
% 0.30/0.66 = ( ( Deg2 = Xa2 )
% 0.30/0.66 & ! [X: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.66 => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.66 & ( case_o184042715313410164at_nat
% 0.30/0.66 @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X4 )
% 0.30/0.66 & ! [X: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.66 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 0.30/0.66 @ ( produc6081775807080527818_nat_o
% 0.30/0.66 @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.66 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.66 & ! [I4: nat] :
% 0.30/0.66 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.66 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X4 ) )
% 0.30/0.66 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.30/0.66 & ( ( Mi3 = Ma3 )
% 0.30/0.66 => ! [X: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.66 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 0.30/0.66 & ( ( Mi3 != Ma3 )
% 0.30/0.66 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 0.30/0.66 & ! [X: nat] :
% 0.30/0.66 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.66 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 0.30/0.66 => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.66 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.66 @ Mima ) ) )
% 0.30/0.66 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % VEBT_internal.valid'.pelims(1)
% 0.30/0.66 thf(fact_10041_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 0.30/0.66 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.30/0.66 ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 0.30/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.30/0.66 => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 0.30/0.66 => ( Xa2 != one_one_nat ) ) )
% 0.30/0.66 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.30/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) )
% 0.30/0.66 => ~ ( ( Deg2 = Xa2 )
% 0.30/0.66 & ! [X3: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.66 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.66 & ( case_o184042715313410164at_nat
% 0.30/0.66 @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X4 )
% 0.30/0.66 & ! [X: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.66 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 0.30/0.66 @ ( produc6081775807080527818_nat_o
% 0.30/0.66 @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.66 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.66 & ! [I4: nat] :
% 0.30/0.66 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.66 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X4 ) )
% 0.30/0.66 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.30/0.66 & ( ( Mi3 = Ma3 )
% 0.30/0.66 => ! [X: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.66 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 0.30/0.66 & ( ( Mi3 != Ma3 )
% 0.30/0.66 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 0.30/0.66 & ! [X: nat] :
% 0.30/0.66 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.66 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 0.30/0.66 => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.66 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.66 @ Mima ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % VEBT_internal.valid'.pelims(2)
% 0.30/0.66 thf(fact_10042_sup__nat__def,axiom,
% 0.30/0.66 sup_sup_nat = ord_max_nat ).
% 0.30/0.66
% 0.30/0.66 % sup_nat_def
% 0.30/0.66 thf(fact_10043_sup__enat__def,axiom,
% 0.30/0.66 sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).
% 0.30/0.66
% 0.30/0.66 % sup_enat_def
% 0.30/0.66 thf(fact_10044_Sup__int__def,axiom,
% 0.30/0.66 ( complete_Sup_Sup_int
% 0.30/0.66 = ( ^ [X4: set_int] :
% 0.30/0.66 ( the_int
% 0.30/0.66 @ ^ [X: int] :
% 0.30/0.66 ( ( member_int @ X @ X4 )
% 0.30/0.66 & ! [Y: int] :
% 0.30/0.66 ( ( member_int @ Y @ X4 )
% 0.30/0.66 => ( ord_less_eq_int @ Y @ X ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Sup_int_def
% 0.30/0.66 thf(fact_10045_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 0.30/0.66 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.30/0.66 ( ~ ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 0.30/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.30/0.66 => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 0.30/0.66 => ( Xa2 = one_one_nat ) ) )
% 0.30/0.66 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 0.30/0.66 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) )
% 0.30/0.66 => ( ( Deg2 = Xa2 )
% 0.30/0.66 & ! [X5: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.66 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.66 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.66 & ( case_o184042715313410164at_nat
% 0.30/0.66 @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X4 )
% 0.30/0.66 & ! [X: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.66 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 0.30/0.66 @ ( produc6081775807080527818_nat_o
% 0.30/0.66 @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.66 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.66 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.66 & ! [I4: nat] :
% 0.30/0.66 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.66 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X4 ) )
% 0.30/0.66 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.30/0.66 & ( ( Mi3 = Ma3 )
% 0.30/0.66 => ! [X: vEBT_VEBT] :
% 0.30/0.66 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.66 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 0.30/0.66 & ( ( Mi3 != Ma3 )
% 0.30/0.66 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 0.30/0.66 & ! [X: nat] :
% 0.30/0.66 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.66 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 0.30/0.66 => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.66 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.66 @ Mima ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % VEBT_internal.valid'.pelims(3)
% 0.30/0.66 thf(fact_10046_take__bit__numeral__minus__numeral__int,axiom,
% 0.30/0.66 ! [M: num,N: num] :
% 0.30/0.66 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.66 = ( case_option_int_num @ zero_zero_int
% 0.30/0.66 @ ^ [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 ) ) )
% 0.30/0.66 @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % take_bit_numeral_minus_numeral_int
% 0.30/0.66 thf(fact_10047_take__bit__num__simps_I1_J,axiom,
% 0.30/0.66 ! [M: num] :
% 0.30/0.66 ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 0.30/0.66 = none_num ) ).
% 0.30/0.66
% 0.30/0.66 % take_bit_num_simps(1)
% 0.30/0.66 thf(fact_10048_take__bit__num__simps_I2_J,axiom,
% 0.30/0.66 ! [N: nat] :
% 0.30/0.66 ( ( bit_take_bit_num @ ( suc @ N ) @ one )
% 0.30/0.66 = ( some_num @ one ) ) ).
% 0.30/0.66
% 0.30/0.66 % take_bit_num_simps(2)
% 0.30/0.66 thf(fact_10049_take__bit__num__simps_I5_J,axiom,
% 0.30/0.66 ! [R2: num] :
% 0.30/0.66 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ one )
% 0.30/0.66 = ( some_num @ one ) ) ).
% 0.30/0.66
% 0.30/0.66 % take_bit_num_simps(5)
% 0.30/0.66 thf(fact_10050_take__bit__num__simps_I3_J,axiom,
% 0.30/0.66 ! [N: nat,M: num] :
% 0.30/0.66 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
% 0.30/0.66 = ( case_o6005452278849405969um_num @ none_num
% 0.30/0.66 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.30/0.66 @ ( bit_take_bit_num @ N @ M ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % take_bit_num_simps(3)
% 0.30/0.66 thf(fact_10051_take__bit__num__simps_I4_J,axiom,
% 0.30/0.66 ! [N: nat,M: num] :
% 0.30/0.66 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
% 0.30/0.66 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % take_bit_num_simps(4)
% 0.30/0.66 thf(fact_10052_take__bit__num__simps_I6_J,axiom,
% 0.30/0.66 ! [R2: num,M: num] :
% 0.30/0.66 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
% 0.30/0.66 = ( case_o6005452278849405969um_num @ none_num
% 0.30/0.66 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.30/0.66 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % take_bit_num_simps(6)
% 0.30/0.66 thf(fact_10053_take__bit__num__simps_I7_J,axiom,
% 0.30/0.66 ! [R2: num,M: num] :
% 0.30/0.66 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M ) )
% 0.30/0.66 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % take_bit_num_simps(7)
% 0.30/0.66 thf(fact_10054_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 0.30/0.66 ! [N: nat,M: num] :
% 0.30/0.66 ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
% 0.30/0.66 = ( case_nat_option_num @ none_num
% 0.30/0.66 @ ^ [N3: nat] :
% 0.30/0.66 ( case_o6005452278849405969um_num @ none_num
% 0.30/0.66 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.30/0.66 @ ( bit_take_bit_num @ N3 @ M ) )
% 0.30/0.66 @ N ) ) ).
% 0.30/0.66
% 0.30/0.66 % Code_Abstract_Nat.take_bit_num_code(2)
% 0.30/0.66 thf(fact_10055_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 0.30/0.66 ! [N: nat] :
% 0.30/0.66 ( ( bit_take_bit_num @ N @ one )
% 0.30/0.66 = ( case_nat_option_num @ none_num
% 0.30/0.66 @ ^ [N3: nat] : ( some_num @ one )
% 0.30/0.66 @ N ) ) ).
% 0.30/0.66
% 0.30/0.66 % Code_Abstract_Nat.take_bit_num_code(1)
% 0.30/0.66 thf(fact_10056_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 0.30/0.66 ! [N: nat,M: num] :
% 0.30/0.66 ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
% 0.30/0.66 = ( case_nat_option_num @ none_num
% 0.30/0.66 @ ^ [N3: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N3 @ M ) ) )
% 0.30/0.66 @ N ) ) ).
% 0.30/0.66
% 0.30/0.66 % Code_Abstract_Nat.take_bit_num_code(3)
% 0.30/0.66 thf(fact_10057_take__bit__num__def,axiom,
% 0.30/0.66 ( bit_take_bit_num
% 0.30/0.66 = ( ^ [N3: nat,M3: num] :
% 0.30/0.66 ( if_option_num
% 0.30/0.66 @ ( ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M3 ) )
% 0.30/0.66 = zero_zero_nat )
% 0.30/0.66 @ none_num
% 0.30/0.66 @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M3 ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % take_bit_num_def
% 0.30/0.66 thf(fact_10058_and__minus__numerals_I3_J,axiom,
% 0.30/0.66 ! [M: num,N: num] :
% 0.30/0.66 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.66 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % and_minus_numerals(3)
% 0.30/0.66 thf(fact_10059_and__minus__numerals_I8_J,axiom,
% 0.30/0.66 ! [N: num,M: num] :
% 0.30/0.66 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.30/0.66 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % and_minus_numerals(8)
% 0.30/0.66 thf(fact_10060_and__minus__numerals_I4_J,axiom,
% 0.30/0.66 ! [M: num,N: num] :
% 0.30/0.66 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.66 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % and_minus_numerals(4)
% 0.30/0.66 thf(fact_10061_and__minus__numerals_I7_J,axiom,
% 0.30/0.66 ! [N: num,M: num] :
% 0.30/0.66 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.30/0.66 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % and_minus_numerals(7)
% 0.30/0.66 thf(fact_10062_and__not__num_Osimps_I1_J,axiom,
% 0.30/0.66 ( ( bit_and_not_num @ one @ one )
% 0.30/0.66 = none_num ) ).
% 0.30/0.66
% 0.30/0.66 % and_not_num.simps(1)
% 0.30/0.66 thf(fact_10063_and__not__num_Osimps_I3_J,axiom,
% 0.30/0.66 ! [N: num] :
% 0.30/0.66 ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
% 0.30/0.66 = none_num ) ).
% 0.30/0.66
% 0.30/0.66 % and_not_num.simps(3)
% 0.30/0.66 thf(fact_10064_and__not__num_Osimps_I2_J,axiom,
% 0.30/0.66 ! [N: num] :
% 0.30/0.66 ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
% 0.30/0.66 = ( some_num @ one ) ) ).
% 0.30/0.66
% 0.30/0.66 % and_not_num.simps(2)
% 0.30/0.66 thf(fact_10065_and__not__num_Osimps_I4_J,axiom,
% 0.30/0.66 ! [M: num] :
% 0.30/0.66 ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
% 0.30/0.66 = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % and_not_num.simps(4)
% 0.30/0.66 thf(fact_10066_and__not__num_Osimps_I7_J,axiom,
% 0.30/0.66 ! [M: num] :
% 0.30/0.66 ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
% 0.30/0.66 = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % and_not_num.simps(7)
% 0.30/0.66 thf(fact_10067_and__not__num__eq__Some__iff,axiom,
% 0.30/0.66 ! [M: num,N: num,Q3: num] :
% 0.30/0.66 ( ( ( bit_and_not_num @ M @ N )
% 0.30/0.66 = ( some_num @ Q3 ) )
% 0.30/0.66 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.66 = ( numeral_numeral_int @ Q3 ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % and_not_num_eq_Some_iff
% 0.30/0.66 thf(fact_10068_and__not__num_Osimps_I8_J,axiom,
% 0.30/0.66 ! [M: num,N: num] :
% 0.30/0.66 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.66 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.30/0.66 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.30/0.66 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % and_not_num.simps(8)
% 0.30/0.66 thf(fact_10069_and__not__num__eq__None__iff,axiom,
% 0.30/0.66 ! [M: num,N: num] :
% 0.30/0.66 ( ( ( bit_and_not_num @ M @ N )
% 0.30/0.66 = none_num )
% 0.30/0.66 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.66 = zero_zero_int ) ) ).
% 0.30/0.66
% 0.30/0.66 % and_not_num_eq_None_iff
% 0.30/0.66 thf(fact_10070_int__numeral__and__not__num,axiom,
% 0.30/0.66 ! [M: num,N: num] :
% 0.30/0.66 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.66 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % int_numeral_and_not_num
% 0.30/0.66 thf(fact_10071_int__numeral__not__and__num,axiom,
% 0.30/0.66 ! [M: num,N: num] :
% 0.30/0.66 ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.66 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % int_numeral_not_and_num
% 0.30/0.66 thf(fact_10072_Bit__Operations_Otake__bit__num__code,axiom,
% 0.30/0.66 ( bit_take_bit_num
% 0.30/0.66 = ( ^ [N3: nat,M3: num] :
% 0.30/0.66 ( produc478579273971653890on_num
% 0.30/0.66 @ ^ [A3: nat,X: num] :
% 0.30/0.66 ( case_nat_option_num @ none_num
% 0.30/0.66 @ ^ [O: nat] :
% 0.30/0.66 ( case_num_option_num @ ( some_num @ one )
% 0.30/0.66 @ ^ [P4: num] :
% 0.30/0.66 ( case_o6005452278849405969um_num @ none_num
% 0.30/0.66 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.30/0.66 @ ( bit_take_bit_num @ O @ P4 ) )
% 0.30/0.66 @ ^ [P4: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P4 ) ) )
% 0.30/0.66 @ X )
% 0.30/0.66 @ A3 )
% 0.30/0.66 @ ( product_Pair_nat_num @ N3 @ M3 ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % Bit_Operations.take_bit_num_code
% 0.30/0.66 thf(fact_10073_and__not__num_Oelims,axiom,
% 0.30/0.66 ! [X2: num,Xa2: num,Y3: option_num] :
% 0.30/0.66 ( ( ( bit_and_not_num @ X2 @ Xa2 )
% 0.30/0.66 = Y3 )
% 0.30/0.66 => ( ( ( X2 = one )
% 0.30/0.66 => ( ( Xa2 = one )
% 0.30/0.66 => ( Y3 != none_num ) ) )
% 0.30/0.66 => ( ( ( X2 = one )
% 0.30/0.66 => ( ? [N2: num] :
% 0.30/0.66 ( Xa2
% 0.30/0.66 = ( bit0 @ N2 ) )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( some_num @ one ) ) ) )
% 0.30/0.66 => ( ( ( X2 = one )
% 0.30/0.66 => ( ? [N2: num] :
% 0.30/0.66 ( Xa2
% 0.30/0.66 = ( bit1 @ N2 ) )
% 0.30/0.66 => ( Y3 != none_num ) ) )
% 0.30/0.66 => ( ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit0 @ M4 ) )
% 0.30/0.66 => ( ( Xa2 = one )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 0.30/0.66 => ( ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit0 @ M4 ) )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit0 @ N2 ) )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) )
% 0.30/0.66 => ( ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit0 @ M4 ) )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit1 @ N2 ) )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) )
% 0.30/0.66 => ( ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit1 @ M4 ) )
% 0.30/0.66 => ( ( Xa2 = one )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 0.30/0.66 => ( ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit1 @ M4 ) )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit0 @ N2 ) )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.30/0.66 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.30/0.66 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) )
% 0.30/0.66 => ~ ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit1 @ M4 ) )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit1 @ N2 ) )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % and_not_num.elims
% 0.30/0.66 thf(fact_10074_and__not__num_Osimps_I9_J,axiom,
% 0.30/0.66 ! [M: num,N: num] :
% 0.30/0.66 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.66 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % and_not_num.simps(9)
% 0.30/0.66 thf(fact_10075_and__not__num_Osimps_I6_J,axiom,
% 0.30/0.66 ! [M: num,N: num] :
% 0.30/0.66 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.66 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % and_not_num.simps(6)
% 0.30/0.66 thf(fact_10076_and__not__num_Osimps_I5_J,axiom,
% 0.30/0.66 ! [M: num,N: num] :
% 0.30/0.66 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.66 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % and_not_num.simps(5)
% 0.30/0.66 thf(fact_10077_and__not__num_Opelims,axiom,
% 0.30/0.66 ! [X2: num,Xa2: num,Y3: option_num] :
% 0.30/0.66 ( ( ( bit_and_not_num @ X2 @ Xa2 )
% 0.30/0.66 = Y3 )
% 0.30/0.66 => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X2 @ Xa2 ) )
% 0.30/0.66 => ( ( ( X2 = one )
% 0.30/0.66 => ( ( Xa2 = one )
% 0.30/0.66 => ( ( Y3 = none_num )
% 0.30/0.66 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.30/0.66 => ( ( ( X2 = one )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit0 @ N2 ) )
% 0.30/0.66 => ( ( Y3
% 0.30/0.66 = ( some_num @ one ) )
% 0.30/0.66 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 0.30/0.66 => ( ( ( X2 = one )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit1 @ N2 ) )
% 0.30/0.66 => ( ( Y3 = none_num )
% 0.30/0.66 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 0.30/0.66 => ( ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit0 @ M4 ) )
% 0.30/0.66 => ( ( Xa2 = one )
% 0.30/0.66 => ( ( Y3
% 0.30/0.66 = ( some_num @ ( bit0 @ M4 ) ) )
% 0.30/0.66 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 0.30/0.66 => ( ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit0 @ M4 ) )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit0 @ N2 ) )
% 0.30/0.66 => ( ( Y3
% 0.30/0.66 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) )
% 0.30/0.66 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.30/0.66 => ( ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit0 @ M4 ) )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit1 @ N2 ) )
% 0.30/0.66 => ( ( Y3
% 0.30/0.66 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) )
% 0.30/0.66 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 0.30/0.66 => ( ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit1 @ M4 ) )
% 0.30/0.66 => ( ( Xa2 = one )
% 0.30/0.66 => ( ( Y3
% 0.30/0.66 = ( some_num @ ( bit0 @ M4 ) ) )
% 0.30/0.66 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 0.30/0.66 => ( ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit1 @ M4 ) )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit0 @ N2 ) )
% 0.30/0.66 => ( ( Y3
% 0.30/0.66 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.30/0.66 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.30/0.66 @ ( bit_and_not_num @ M4 @ N2 ) ) )
% 0.30/0.66 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.30/0.66 => ~ ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit1 @ M4 ) )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit1 @ N2 ) )
% 0.30/0.66 => ( ( Y3
% 0.30/0.66 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) )
% 0.30/0.66 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % and_not_num.pelims
% 0.30/0.66 thf(fact_10078_and__num_Oelims,axiom,
% 0.30/0.66 ! [X2: num,Xa2: num,Y3: option_num] :
% 0.30/0.66 ( ( ( bit_un7362597486090784418nd_num @ X2 @ Xa2 )
% 0.30/0.66 = Y3 )
% 0.30/0.66 => ( ( ( X2 = one )
% 0.30/0.66 => ( ( Xa2 = one )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( some_num @ one ) ) ) )
% 0.30/0.66 => ( ( ( X2 = one )
% 0.30/0.66 => ( ? [N2: num] :
% 0.30/0.66 ( Xa2
% 0.30/0.66 = ( bit0 @ N2 ) )
% 0.30/0.66 => ( Y3 != none_num ) ) )
% 0.30/0.66 => ( ( ( X2 = one )
% 0.30/0.66 => ( ? [N2: num] :
% 0.30/0.66 ( Xa2
% 0.30/0.66 = ( bit1 @ N2 ) )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( some_num @ one ) ) ) )
% 0.30/0.66 => ( ( ? [M4: num] :
% 0.30/0.66 ( X2
% 0.30/0.66 = ( bit0 @ M4 ) )
% 0.30/0.66 => ( ( Xa2 = one )
% 0.30/0.66 => ( Y3 != none_num ) ) )
% 0.30/0.66 => ( ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit0 @ M4 ) )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit0 @ N2 ) )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) )
% 0.30/0.66 => ( ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit0 @ M4 ) )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit1 @ N2 ) )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) )
% 0.30/0.66 => ( ( ? [M4: num] :
% 0.30/0.66 ( X2
% 0.30/0.66 = ( bit1 @ M4 ) )
% 0.30/0.66 => ( ( Xa2 = one )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( some_num @ one ) ) ) )
% 0.30/0.66 => ( ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit1 @ M4 ) )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit0 @ N2 ) )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) )
% 0.30/0.66 => ~ ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit1 @ M4 ) )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit1 @ N2 ) )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.30/0.66 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.30/0.66 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66
% 0.30/0.66 % and_num.elims
% 0.30/0.66 thf(fact_10079_xor__num_Oelims,axiom,
% 0.30/0.66 ! [X2: num,Xa2: num,Y3: option_num] :
% 0.30/0.66 ( ( ( bit_un2480387367778600638or_num @ X2 @ Xa2 )
% 0.30/0.66 = Y3 )
% 0.30/0.66 => ( ( ( X2 = one )
% 0.30/0.66 => ( ( Xa2 = one )
% 0.30/0.66 => ( Y3 != none_num ) ) )
% 0.30/0.66 => ( ( ( X2 = one )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit0 @ N2 ) )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( some_num @ ( bit1 @ N2 ) ) ) ) )
% 0.30/0.66 => ( ( ( X2 = one )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit1 @ N2 ) )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( some_num @ ( bit0 @ N2 ) ) ) ) )
% 0.30/0.66 => ( ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit0 @ M4 ) )
% 0.30/0.66 => ( ( Xa2 = one )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( some_num @ ( bit1 @ M4 ) ) ) ) )
% 0.30/0.66 => ( ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit0 @ M4 ) )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit0 @ N2 ) )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) )
% 0.30/0.66 => ( ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit0 @ M4 ) )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit1 @ N2 ) )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) ) )
% 0.30/0.66 => ( ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit1 @ M4 ) )
% 0.30/0.66 => ( ( Xa2 = one )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 0.30/0.66 => ( ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit1 @ M4 ) )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit0 @ N2 ) )
% 0.30/0.66 => ( Y3
% 0.30/0.66 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) ) )
% 0.30/0.66 => ~ ! [M4: num] :
% 0.30/0.66 ( ( X2
% 0.30/0.66 = ( bit1 @ M4 ) )
% 0.30/0.66 => ! [N2: num] :
% 0.30/0.66 ( ( Xa2
% 0.30/0.66 = ( bit1 @ N2 ) )
% 0.30/0.66 => ( Y3
% 0.30/0.67 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % xor_num.elims
% 0.30/0.67 thf(fact_10080_and__num_Osimps_I1_J,axiom,
% 0.30/0.67 ( ( bit_un7362597486090784418nd_num @ one @ one )
% 0.30/0.67 = ( some_num @ one ) ) ).
% 0.30/0.67
% 0.30/0.67 % and_num.simps(1)
% 0.30/0.67 thf(fact_10081_xor__num_Osimps_I1_J,axiom,
% 0.30/0.67 ( ( bit_un2480387367778600638or_num @ one @ one )
% 0.30/0.67 = none_num ) ).
% 0.30/0.67
% 0.30/0.67 % xor_num.simps(1)
% 0.30/0.67 thf(fact_10082_and__num_Osimps_I5_J,axiom,
% 0.30/0.67 ! [M: num,N: num] :
% 0.30/0.67 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.67 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % and_num.simps(5)
% 0.30/0.67 thf(fact_10083_xor__num_Osimps_I5_J,axiom,
% 0.30/0.67 ! [M: num,N: num] :
% 0.30/0.67 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.67 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % xor_num.simps(5)
% 0.30/0.67 thf(fact_10084_and__num_Osimps_I7_J,axiom,
% 0.30/0.67 ! [M: num] :
% 0.30/0.67 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
% 0.30/0.67 = ( some_num @ one ) ) ).
% 0.30/0.67
% 0.30/0.67 % and_num.simps(7)
% 0.30/0.67 thf(fact_10085_and__num_Osimps_I3_J,axiom,
% 0.30/0.67 ! [N: num] :
% 0.30/0.67 ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
% 0.30/0.67 = ( some_num @ one ) ) ).
% 0.30/0.67
% 0.30/0.67 % and_num.simps(3)
% 0.30/0.67 thf(fact_10086_and__num_Osimps_I2_J,axiom,
% 0.30/0.67 ! [N: num] :
% 0.30/0.67 ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
% 0.30/0.67 = none_num ) ).
% 0.30/0.67
% 0.30/0.67 % and_num.simps(2)
% 0.30/0.67 thf(fact_10087_and__num_Osimps_I4_J,axiom,
% 0.30/0.67 ! [M: num] :
% 0.30/0.67 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 0.30/0.67 = none_num ) ).
% 0.30/0.67
% 0.30/0.67 % and_num.simps(4)
% 0.30/0.67 thf(fact_10088_xor__num_Osimps_I9_J,axiom,
% 0.30/0.67 ! [M: num,N: num] :
% 0.30/0.67 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.67 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % xor_num.simps(9)
% 0.30/0.67 thf(fact_10089_and__num_Osimps_I6_J,axiom,
% 0.30/0.67 ! [M: num,N: num] :
% 0.30/0.67 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.67 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % and_num.simps(6)
% 0.30/0.67 thf(fact_10090_and__num_Osimps_I8_J,axiom,
% 0.30/0.67 ! [M: num,N: num] :
% 0.30/0.67 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.67 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % and_num.simps(8)
% 0.30/0.67 thf(fact_10091_xor__num_Osimps_I7_J,axiom,
% 0.30/0.67 ! [M: num] :
% 0.30/0.67 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
% 0.30/0.67 = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % xor_num.simps(7)
% 0.30/0.67 thf(fact_10092_xor__num_Osimps_I4_J,axiom,
% 0.30/0.67 ! [M: num] :
% 0.30/0.67 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
% 0.30/0.67 = ( some_num @ ( bit1 @ M ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % xor_num.simps(4)
% 0.30/0.67 thf(fact_10093_xor__num_Osimps_I3_J,axiom,
% 0.30/0.67 ! [N: num] :
% 0.30/0.67 ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
% 0.30/0.67 = ( some_num @ ( bit0 @ N ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % xor_num.simps(3)
% 0.30/0.67 thf(fact_10094_xor__num_Osimps_I2_J,axiom,
% 0.30/0.67 ! [N: num] :
% 0.30/0.67 ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
% 0.30/0.67 = ( some_num @ ( bit1 @ N ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % xor_num.simps(2)
% 0.30/0.67 thf(fact_10095_and__num_Osimps_I9_J,axiom,
% 0.30/0.67 ! [M: num,N: num] :
% 0.30/0.67 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.67 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.30/0.67 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.30/0.67 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % and_num.simps(9)
% 0.30/0.67 thf(fact_10096_xor__num_Osimps_I6_J,axiom,
% 0.30/0.67 ! [M: num,N: num] :
% 0.30/0.67 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.67 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % xor_num.simps(6)
% 0.30/0.67 thf(fact_10097_xor__num_Osimps_I8_J,axiom,
% 0.30/0.67 ! [M: num,N: num] :
% 0.30/0.67 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.67 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % xor_num.simps(8)
% 0.30/0.67 thf(fact_10098_and__num_Opelims,axiom,
% 0.30/0.67 ! [X2: num,Xa2: num,Y3: option_num] :
% 0.30/0.67 ( ( ( bit_un7362597486090784418nd_num @ X2 @ Xa2 )
% 0.30/0.67 = Y3 )
% 0.30/0.67 => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X2 @ Xa2 ) )
% 0.30/0.67 => ( ( ( X2 = one )
% 0.30/0.67 => ( ( Xa2 = one )
% 0.30/0.67 => ( ( Y3
% 0.30/0.67 = ( some_num @ one ) )
% 0.30/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.30/0.67 => ( ( ( X2 = one )
% 0.30/0.67 => ! [N2: num] :
% 0.30/0.67 ( ( Xa2
% 0.30/0.67 = ( bit0 @ N2 ) )
% 0.30/0.67 => ( ( Y3 = none_num )
% 0.30/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 0.30/0.67 => ( ( ( X2 = one )
% 0.30/0.67 => ! [N2: num] :
% 0.30/0.67 ( ( Xa2
% 0.30/0.67 = ( bit1 @ N2 ) )
% 0.30/0.67 => ( ( Y3
% 0.30/0.67 = ( some_num @ one ) )
% 0.30/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 0.30/0.67 => ( ! [M4: num] :
% 0.30/0.67 ( ( X2
% 0.30/0.67 = ( bit0 @ M4 ) )
% 0.30/0.67 => ( ( Xa2 = one )
% 0.30/0.67 => ( ( Y3 = none_num )
% 0.30/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 0.30/0.67 => ( ! [M4: num] :
% 0.30/0.67 ( ( X2
% 0.30/0.67 = ( bit0 @ M4 ) )
% 0.30/0.67 => ! [N2: num] :
% 0.30/0.67 ( ( Xa2
% 0.30/0.67 = ( bit0 @ N2 ) )
% 0.30/0.67 => ( ( Y3
% 0.30/0.67 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
% 0.30/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.30/0.67 => ( ! [M4: num] :
% 0.30/0.67 ( ( X2
% 0.30/0.67 = ( bit0 @ M4 ) )
% 0.30/0.67 => ! [N2: num] :
% 0.30/0.67 ( ( Xa2
% 0.30/0.67 = ( bit1 @ N2 ) )
% 0.30/0.67 => ( ( Y3
% 0.30/0.67 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
% 0.30/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 0.30/0.67 => ( ! [M4: num] :
% 0.30/0.67 ( ( X2
% 0.30/0.67 = ( bit1 @ M4 ) )
% 0.30/0.67 => ( ( Xa2 = one )
% 0.30/0.67 => ( ( Y3
% 0.30/0.67 = ( some_num @ one ) )
% 0.30/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 0.30/0.67 => ( ! [M4: num] :
% 0.30/0.67 ( ( X2
% 0.30/0.67 = ( bit1 @ M4 ) )
% 0.30/0.67 => ! [N2: num] :
% 0.30/0.67 ( ( Xa2
% 0.30/0.67 = ( bit0 @ N2 ) )
% 0.30/0.67 => ( ( Y3
% 0.30/0.67 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
% 0.30/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.30/0.67 => ~ ! [M4: num] :
% 0.30/0.67 ( ( X2
% 0.30/0.67 = ( bit1 @ M4 ) )
% 0.30/0.67 => ! [N2: num] :
% 0.30/0.67 ( ( Xa2
% 0.30/0.67 = ( bit1 @ N2 ) )
% 0.30/0.67 => ( ( Y3
% 0.30/0.67 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.30/0.67 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.30/0.67 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
% 0.30/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % and_num.pelims
% 0.30/0.67 thf(fact_10099_xor__num_Opelims,axiom,
% 0.30/0.67 ! [X2: num,Xa2: num,Y3: option_num] :
% 0.30/0.67 ( ( ( bit_un2480387367778600638or_num @ X2 @ Xa2 )
% 0.30/0.67 = Y3 )
% 0.30/0.67 => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X2 @ Xa2 ) )
% 0.30/0.67 => ( ( ( X2 = one )
% 0.30/0.67 => ( ( Xa2 = one )
% 0.30/0.67 => ( ( Y3 = none_num )
% 0.30/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.30/0.67 => ( ( ( X2 = one )
% 0.30/0.67 => ! [N2: num] :
% 0.30/0.67 ( ( Xa2
% 0.30/0.67 = ( bit0 @ N2 ) )
% 0.30/0.67 => ( ( Y3
% 0.30/0.67 = ( some_num @ ( bit1 @ N2 ) ) )
% 0.30/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 0.30/0.67 => ( ( ( X2 = one )
% 0.30/0.67 => ! [N2: num] :
% 0.30/0.67 ( ( Xa2
% 0.30/0.67 = ( bit1 @ N2 ) )
% 0.30/0.67 => ( ( Y3
% 0.30/0.67 = ( some_num @ ( bit0 @ N2 ) ) )
% 0.30/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 0.30/0.67 => ( ! [M4: num] :
% 0.30/0.67 ( ( X2
% 0.30/0.67 = ( bit0 @ M4 ) )
% 0.30/0.67 => ( ( Xa2 = one )
% 0.30/0.67 => ( ( Y3
% 0.30/0.67 = ( some_num @ ( bit1 @ M4 ) ) )
% 0.30/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 0.30/0.67 => ( ! [M4: num] :
% 0.30/0.67 ( ( X2
% 0.30/0.67 = ( bit0 @ M4 ) )
% 0.30/0.67 => ! [N2: num] :
% 0.30/0.67 ( ( Xa2
% 0.30/0.67 = ( bit0 @ N2 ) )
% 0.30/0.67 => ( ( Y3
% 0.30/0.67 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) )
% 0.30/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.30/0.67 => ( ! [M4: num] :
% 0.30/0.67 ( ( X2
% 0.30/0.67 = ( bit0 @ M4 ) )
% 0.30/0.67 => ! [N2: num] :
% 0.30/0.67 ( ( Xa2
% 0.30/0.67 = ( bit1 @ N2 ) )
% 0.30/0.67 => ( ( Y3
% 0.30/0.67 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) )
% 0.30/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 0.30/0.67 => ( ! [M4: num] :
% 0.30/0.67 ( ( X2
% 0.30/0.67 = ( bit1 @ M4 ) )
% 0.30/0.67 => ( ( Xa2 = one )
% 0.30/0.67 => ( ( Y3
% 0.30/0.67 = ( some_num @ ( bit0 @ M4 ) ) )
% 0.30/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 0.30/0.67 => ( ! [M4: num] :
% 0.30/0.67 ( ( X2
% 0.30/0.67 = ( bit1 @ M4 ) )
% 0.30/0.67 => ! [N2: num] :
% 0.30/0.67 ( ( Xa2
% 0.30/0.67 = ( bit0 @ N2 ) )
% 0.30/0.67 => ( ( Y3
% 0.30/0.67 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) )
% 0.30/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 0.30/0.67 => ~ ! [M4: num] :
% 0.30/0.67 ( ( X2
% 0.30/0.67 = ( bit1 @ M4 ) )
% 0.30/0.67 => ! [N2: num] :
% 0.30/0.67 ( ( Xa2
% 0.30/0.67 = ( bit1 @ N2 ) )
% 0.30/0.67 => ( ( Y3
% 0.30/0.67 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) )
% 0.30/0.67 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % xor_num.pelims
% 0.30/0.67 thf(fact_10100_and__num__rel__dict,axiom,
% 0.30/0.67 bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).
% 0.30/0.67
% 0.30/0.67 % and_num_rel_dict
% 0.30/0.67 thf(fact_10101_xor__num__rel__dict,axiom,
% 0.30/0.67 bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).
% 0.30/0.67
% 0.30/0.67 % xor_num_rel_dict
% 0.30/0.67 thf(fact_10102_xor__num__dict,axiom,
% 0.30/0.67 bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).
% 0.30/0.67
% 0.30/0.67 % xor_num_dict
% 0.30/0.67 thf(fact_10103_and__num__dict,axiom,
% 0.30/0.67 bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).
% 0.30/0.67
% 0.30/0.67 % and_num_dict
% 0.30/0.67 thf(fact_10104_less__eq,axiom,
% 0.30/0.67 ! [M: nat,N: nat] :
% 0.30/0.67 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
% 0.30/0.67 = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.67
% 0.30/0.67 % less_eq
% 0.30/0.67 thf(fact_10105_pred__nat__trancl__eq__le,axiom,
% 0.30/0.67 ! [M: nat,N: nat] :
% 0.30/0.67 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
% 0.30/0.67 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.67
% 0.30/0.67 % pred_nat_trancl_eq_le
% 0.30/0.67 thf(fact_10106_Rats__eq__int__div__nat,axiom,
% 0.30/0.67 ( field_5140801741446780682s_real
% 0.30/0.67 = ( collect_real
% 0.30/0.67 @ ^ [Uu3: real] :
% 0.30/0.67 ? [I4: int,N3: nat] :
% 0.30/0.67 ( ( Uu3
% 0.30/0.67 = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
% 0.30/0.67 & ( N3 != zero_zero_nat ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % Rats_eq_int_div_nat
% 0.30/0.67 thf(fact_10107_Rats__abs__iff,axiom,
% 0.30/0.67 ! [X2: real] :
% 0.30/0.67 ( ( member_real @ ( abs_abs_real @ X2 ) @ field_5140801741446780682s_real )
% 0.30/0.67 = ( member_real @ X2 @ field_5140801741446780682s_real ) ) ).
% 0.30/0.67
% 0.30/0.67 % Rats_abs_iff
% 0.30/0.67 thf(fact_10108_Rats__no__bot__less,axiom,
% 0.30/0.67 ! [X2: real] :
% 0.30/0.67 ? [X5: real] :
% 0.30/0.67 ( ( member_real @ X5 @ field_5140801741446780682s_real )
% 0.30/0.67 & ( ord_less_real @ X5 @ X2 ) ) ).
% 0.30/0.67
% 0.30/0.67 % Rats_no_bot_less
% 0.30/0.67 thf(fact_10109_Rats__dense__in__real,axiom,
% 0.30/0.67 ! [X2: real,Y3: real] :
% 0.30/0.67 ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.67 => ? [X5: real] :
% 0.30/0.67 ( ( member_real @ X5 @ field_5140801741446780682s_real )
% 0.30/0.67 & ( ord_less_real @ X2 @ X5 )
% 0.30/0.67 & ( ord_less_real @ X5 @ Y3 ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % Rats_dense_in_real
% 0.30/0.67 thf(fact_10110_Rats__no__top__le,axiom,
% 0.30/0.67 ! [X2: real] :
% 0.30/0.67 ? [X5: real] :
% 0.30/0.67 ( ( member_real @ X5 @ field_5140801741446780682s_real )
% 0.30/0.67 & ( ord_less_eq_real @ X2 @ X5 ) ) ).
% 0.30/0.67
% 0.30/0.67 % Rats_no_top_le
% 0.30/0.67 thf(fact_10111_Rats__eq__int__div__int,axiom,
% 0.30/0.67 ( field_5140801741446780682s_real
% 0.30/0.67 = ( collect_real
% 0.30/0.67 @ ^ [Uu3: real] :
% 0.30/0.67 ? [I4: int,J3: int] :
% 0.30/0.67 ( ( Uu3
% 0.30/0.67 = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( ring_1_of_int_real @ J3 ) ) )
% 0.30/0.67 & ( J3 != zero_zero_int ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % Rats_eq_int_div_int
% 0.30/0.67 thf(fact_10112_rat__floor__lemma,axiom,
% 0.30/0.67 ! [A: int,B: int] :
% 0.30/0.67 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
% 0.30/0.67 & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % rat_floor_lemma
% 0.30/0.67 thf(fact_10113_mult__rat,axiom,
% 0.30/0.67 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.67 ( ( times_times_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.30/0.67 = ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % mult_rat
% 0.30/0.67 thf(fact_10114_divide__rat,axiom,
% 0.30/0.67 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.67 ( ( divide_divide_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.30/0.67 = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % divide_rat
% 0.30/0.67 thf(fact_10115_less__rat,axiom,
% 0.30/0.67 ! [B: int,D: int,A: int,C: int] :
% 0.30/0.67 ( ( B != zero_zero_int )
% 0.30/0.67 => ( ( D != zero_zero_int )
% 0.30/0.67 => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.30/0.67 = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % less_rat
% 0.30/0.67 thf(fact_10116_add__rat,axiom,
% 0.30/0.67 ! [B: int,D: int,A: int,C: int] :
% 0.30/0.67 ( ( B != zero_zero_int )
% 0.30/0.67 => ( ( D != zero_zero_int )
% 0.30/0.67 => ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.30/0.67 = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % add_rat
% 0.30/0.67 thf(fact_10117_le__rat,axiom,
% 0.30/0.67 ! [B: int,D: int,A: int,C: int] :
% 0.30/0.67 ( ( B != zero_zero_int )
% 0.30/0.67 => ( ( D != zero_zero_int )
% 0.30/0.67 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.30/0.67 = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % le_rat
% 0.30/0.67 thf(fact_10118_diff__rat,axiom,
% 0.30/0.67 ! [B: int,D: int,A: int,C: int] :
% 0.30/0.67 ( ( B != zero_zero_int )
% 0.30/0.67 => ( ( D != zero_zero_int )
% 0.30/0.67 => ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.30/0.67 = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % diff_rat
% 0.30/0.67 thf(fact_10119_sgn__rat,axiom,
% 0.30/0.67 ! [A: int,B: int] :
% 0.30/0.67 ( ( sgn_sgn_rat @ ( fract @ A @ B ) )
% 0.30/0.67 = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % sgn_rat
% 0.30/0.67 thf(fact_10120_Zero__rat__def,axiom,
% 0.30/0.67 ( zero_zero_rat
% 0.30/0.67 = ( fract @ zero_zero_int @ one_one_int ) ) ).
% 0.30/0.67
% 0.30/0.67 % Zero_rat_def
% 0.30/0.67 thf(fact_10121_Fract__of__int__eq,axiom,
% 0.30/0.67 ! [K: int] :
% 0.30/0.67 ( ( fract @ K @ one_one_int )
% 0.30/0.67 = ( ring_1_of_int_rat @ K ) ) ).
% 0.30/0.67
% 0.30/0.67 % Fract_of_int_eq
% 0.30/0.67 thf(fact_10122_Fract__of__nat__eq,axiom,
% 0.30/0.67 ! [K: nat] :
% 0.30/0.67 ( ( fract @ ( semiri1314217659103216013at_int @ K ) @ one_one_int )
% 0.30/0.67 = ( semiri681578069525770553at_rat @ K ) ) ).
% 0.30/0.67
% 0.30/0.67 % Fract_of_nat_eq
% 0.30/0.67 thf(fact_10123_mult__rat__cancel,axiom,
% 0.30/0.67 ! [C: int,A: int,B: int] :
% 0.30/0.67 ( ( C != zero_zero_int )
% 0.30/0.67 => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.67 = ( fract @ A @ B ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % mult_rat_cancel
% 0.30/0.67 thf(fact_10124_eq__rat_I1_J,axiom,
% 0.30/0.67 ! [B: int,D: int,A: int,C: int] :
% 0.30/0.67 ( ( B != zero_zero_int )
% 0.30/0.67 => ( ( D != zero_zero_int )
% 0.30/0.67 => ( ( ( fract @ A @ B )
% 0.30/0.67 = ( fract @ C @ D ) )
% 0.30/0.67 = ( ( times_times_int @ A @ D )
% 0.30/0.67 = ( times_times_int @ C @ B ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % eq_rat(1)
% 0.30/0.67 thf(fact_10125_eq__rat_I2_J,axiom,
% 0.30/0.67 ! [A: int] :
% 0.30/0.67 ( ( fract @ A @ zero_zero_int )
% 0.30/0.67 = ( fract @ zero_zero_int @ one_one_int ) ) ).
% 0.30/0.67
% 0.30/0.67 % eq_rat(2)
% 0.30/0.67 thf(fact_10126_One__rat__def,axiom,
% 0.30/0.67 ( one_one_rat
% 0.30/0.67 = ( fract @ one_one_int @ one_one_int ) ) ).
% 0.30/0.67
% 0.30/0.67 % One_rat_def
% 0.30/0.67 thf(fact_10127_rat__number__expand_I3_J,axiom,
% 0.30/0.67 ( numeral_numeral_rat
% 0.30/0.67 = ( ^ [K2: num] : ( fract @ ( numeral_numeral_int @ K2 ) @ one_one_int ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % rat_number_expand(3)
% 0.30/0.67 thf(fact_10128_rat__number__collapse_I3_J,axiom,
% 0.30/0.67 ! [W: num] :
% 0.30/0.67 ( ( fract @ ( numeral_numeral_int @ W ) @ one_one_int )
% 0.30/0.67 = ( numeral_numeral_rat @ W ) ) ).
% 0.30/0.67
% 0.30/0.67 % rat_number_collapse(3)
% 0.30/0.67 thf(fact_10129_one__less__Fract__iff,axiom,
% 0.30/0.67 ! [B: int,A: int] :
% 0.30/0.67 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.67 => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
% 0.30/0.67 = ( ord_less_int @ B @ A ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % one_less_Fract_iff
% 0.30/0.67 thf(fact_10130_Fract__less__one__iff,axiom,
% 0.30/0.67 ! [B: int,A: int] :
% 0.30/0.67 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.67 => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
% 0.30/0.67 = ( ord_less_int @ A @ B ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % Fract_less_one_iff
% 0.30/0.67 thf(fact_10131_rat__number__collapse_I5_J,axiom,
% 0.30/0.67 ( ( fract @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.30/0.67 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.67
% 0.30/0.67 % rat_number_collapse(5)
% 0.30/0.67 thf(fact_10132_Fract__add__one,axiom,
% 0.30/0.67 ! [N: int,M: int] :
% 0.30/0.67 ( ( N != zero_zero_int )
% 0.30/0.67 => ( ( fract @ ( plus_plus_int @ M @ N ) @ N )
% 0.30/0.67 = ( plus_plus_rat @ ( fract @ M @ N ) @ one_one_rat ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % Fract_add_one
% 0.30/0.67 thf(fact_10133_Fract__le__zero__iff,axiom,
% 0.30/0.67 ! [B: int,A: int] :
% 0.30/0.67 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.67 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 0.30/0.67 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % Fract_le_zero_iff
% 0.30/0.67 thf(fact_10134_zero__le__Fract__iff,axiom,
% 0.30/0.67 ! [B: int,A: int] :
% 0.30/0.67 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.67 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 0.30/0.67 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % zero_le_Fract_iff
% 0.30/0.67 thf(fact_10135_Fract__le__one__iff,axiom,
% 0.30/0.67 ! [B: int,A: int] :
% 0.30/0.67 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.67 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
% 0.30/0.67 = ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % Fract_le_one_iff
% 0.30/0.67 thf(fact_10136_one__le__Fract__iff,axiom,
% 0.30/0.67 ! [B: int,A: int] :
% 0.30/0.67 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.67 => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
% 0.30/0.67 = ( ord_less_eq_int @ B @ A ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % one_le_Fract_iff
% 0.30/0.67 thf(fact_10137_rat__number__expand_I5_J,axiom,
% 0.30/0.67 ! [K: num] :
% 0.30/0.67 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) )
% 0.30/0.67 = ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 0.30/0.67
% 0.30/0.67 % rat_number_expand(5)
% 0.30/0.67 thf(fact_10138_rat__number__collapse_I4_J,axiom,
% 0.30/0.67 ! [W: num] :
% 0.30/0.67 ( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ one_one_int )
% 0.30/0.67 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % rat_number_collapse(4)
% 0.30/0.67 thf(fact_10139_of__nat__eq__id,axiom,
% 0.30/0.67 semiri1316708129612266289at_nat = id_nat ).
% 0.30/0.67
% 0.30/0.67 % of_nat_eq_id
% 0.30/0.67 thf(fact_10140_less__eq__int__def,axiom,
% 0.30/0.67 ( ord_less_eq_int
% 0.30/0.67 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 0.30/0.67 @ ( produc8739625826339149834_nat_o
% 0.30/0.67 @ ^ [X: nat,Y: nat] :
% 0.30/0.67 ( produc6081775807080527818_nat_o
% 0.30/0.67 @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % less_eq_int_def
% 0.30/0.67 thf(fact_10141_less__int__def,axiom,
% 0.30/0.67 ( ord_less_int
% 0.30/0.67 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 0.30/0.67 @ ( produc8739625826339149834_nat_o
% 0.30/0.67 @ ^ [X: nat,Y: nat] :
% 0.30/0.67 ( produc6081775807080527818_nat_o
% 0.30/0.67 @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % less_int_def
% 0.30/0.67 thf(fact_10142_nat__def,axiom,
% 0.30/0.67 ( nat2
% 0.30/0.67 = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % nat_def
% 0.30/0.67 thf(fact_10143_positive__rat,axiom,
% 0.30/0.67 ! [A: int,B: int] :
% 0.30/0.67 ( ( positive @ ( fract @ A @ B ) )
% 0.30/0.67 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % positive_rat
% 0.30/0.67 thf(fact_10144_Rat_Opositive__mult,axiom,
% 0.30/0.67 ! [X2: rat,Y3: rat] :
% 0.30/0.67 ( ( positive @ X2 )
% 0.30/0.67 => ( ( positive @ Y3 )
% 0.30/0.67 => ( positive @ ( times_times_rat @ X2 @ Y3 ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % Rat.positive_mult
% 0.30/0.67 thf(fact_10145_less__rat__def,axiom,
% 0.30/0.67 ( ord_less_rat
% 0.30/0.67 = ( ^ [X: rat,Y: rat] : ( positive @ ( minus_minus_rat @ Y @ X ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % less_rat_def
% 0.30/0.67 thf(fact_10146_Rat_Opositive_Orep__eq,axiom,
% 0.30/0.67 ( positive
% 0.30/0.67 = ( ^ [X: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X ) ) @ ( product_snd_int_int @ ( rep_Rat @ X ) ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % Rat.positive.rep_eq
% 0.30/0.67 thf(fact_10147_Rat_Opositive__def,axiom,
% 0.30/0.67 ( positive
% 0.30/0.67 = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
% 0.30/0.67 @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % Rat.positive_def
% 0.30/0.67 thf(fact_10148_of__rat__dense,axiom,
% 0.30/0.67 ! [X2: real,Y3: real] :
% 0.30/0.67 ( ( ord_less_real @ X2 @ Y3 )
% 0.30/0.67 => ? [Q2: rat] :
% 0.30/0.67 ( ( ord_less_real @ X2 @ ( field_7254667332652039916t_real @ Q2 ) )
% 0.30/0.67 & ( ord_less_real @ ( field_7254667332652039916t_real @ Q2 ) @ Y3 ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % of_rat_dense
% 0.30/0.67 thf(fact_10149_plus__rat__def,axiom,
% 0.30/0.67 ( plus_plus_rat
% 0.30/0.67 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 0.30/0.67 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % plus_rat_def
% 0.30/0.67 thf(fact_10150_inverse__rat__def,axiom,
% 0.30/0.67 ( inverse_inverse_rat
% 0.30/0.67 = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
% 0.30/0.67 @ ^ [X: product_prod_int_int] :
% 0.30/0.67 ( if_Pro3027730157355071871nt_int
% 0.30/0.67 @ ( ( product_fst_int_int @ X )
% 0.30/0.67 = zero_zero_int )
% 0.30/0.67 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.30/0.67 @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % inverse_rat_def
% 0.30/0.67 thf(fact_10151_one__rat__def,axiom,
% 0.30/0.67 ( one_one_rat
% 0.30/0.67 = ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % one_rat_def
% 0.30/0.67 thf(fact_10152_Fract_Oabs__eq,axiom,
% 0.30/0.67 ( fract
% 0.30/0.67 = ( ^ [Xa4: int,X: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % Fract.abs_eq
% 0.30/0.67 thf(fact_10153_zero__rat__def,axiom,
% 0.30/0.67 ( zero_zero_rat
% 0.30/0.67 = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % zero_rat_def
% 0.30/0.67 thf(fact_10154_times__rat__def,axiom,
% 0.30/0.67 ( times_times_rat
% 0.30/0.67 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 0.30/0.67 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % times_rat_def
% 0.30/0.67 thf(fact_10155_plus__rat_Oabs__eq,axiom,
% 0.30/0.67 ! [Xa2: product_prod_int_int,X2: product_prod_int_int] :
% 0.30/0.67 ( ( ratrel @ Xa2 @ Xa2 )
% 0.30/0.67 => ( ( ratrel @ X2 @ X2 )
% 0.30/0.67 => ( ( plus_plus_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X2 ) )
% 0.30/0.67 = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_snd_int_int @ X2 ) ) @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % plus_rat.abs_eq
% 0.30/0.67 thf(fact_10156_ratrel__iff,axiom,
% 0.30/0.67 ( ratrel
% 0.30/0.67 = ( ^ [X: product_prod_int_int,Y: product_prod_int_int] :
% 0.30/0.67 ( ( ( product_snd_int_int @ X )
% 0.30/0.67 != zero_zero_int )
% 0.30/0.67 & ( ( product_snd_int_int @ Y )
% 0.30/0.67 != zero_zero_int )
% 0.30/0.67 & ( ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) )
% 0.30/0.67 = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % ratrel_iff
% 0.30/0.67 thf(fact_10157_one__rat_Orsp,axiom,
% 0.30/0.67 ratrel @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ).
% 0.30/0.67
% 0.30/0.67 % one_rat.rsp
% 0.30/0.67 thf(fact_10158_zero__rat_Orsp,axiom,
% 0.30/0.67 ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).
% 0.30/0.67
% 0.30/0.67 % zero_rat.rsp
% 0.30/0.67 thf(fact_10159_ratrel__def,axiom,
% 0.30/0.67 ( ratrel
% 0.30/0.67 = ( ^ [X: product_prod_int_int,Y: product_prod_int_int] :
% 0.30/0.67 ( ( ( product_snd_int_int @ X )
% 0.30/0.67 != zero_zero_int )
% 0.30/0.67 & ( ( product_snd_int_int @ Y )
% 0.30/0.67 != zero_zero_int )
% 0.30/0.67 & ( ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) )
% 0.30/0.67 = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % ratrel_def
% 0.30/0.67 thf(fact_10160_times__rat_Oabs__eq,axiom,
% 0.30/0.67 ! [Xa2: product_prod_int_int,X2: product_prod_int_int] :
% 0.30/0.67 ( ( ratrel @ Xa2 @ Xa2 )
% 0.30/0.67 => ( ( ratrel @ X2 @ X2 )
% 0.30/0.67 => ( ( times_times_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X2 ) )
% 0.30/0.67 = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_fst_int_int @ X2 ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % times_rat.abs_eq
% 0.30/0.67 thf(fact_10161_Rat_Opositive_Oabs__eq,axiom,
% 0.30/0.67 ! [X2: product_prod_int_int] :
% 0.30/0.67 ( ( ratrel @ X2 @ X2 )
% 0.30/0.67 => ( ( positive @ ( abs_Rat @ X2 ) )
% 0.30/0.67 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % Rat.positive.abs_eq
% 0.30/0.67 thf(fact_10162_inverse__rat_Oabs__eq,axiom,
% 0.30/0.67 ! [X2: product_prod_int_int] :
% 0.30/0.67 ( ( ratrel @ X2 @ X2 )
% 0.30/0.67 => ( ( inverse_inverse_rat @ ( abs_Rat @ X2 ) )
% 0.30/0.67 = ( abs_Rat
% 0.30/0.67 @ ( if_Pro3027730157355071871nt_int
% 0.30/0.67 @ ( ( product_fst_int_int @ X2 )
% 0.30/0.67 = zero_zero_int )
% 0.30/0.67 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.30/0.67 @ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % inverse_rat.abs_eq
% 0.30/0.67 thf(fact_10163_min__Suc__Suc,axiom,
% 0.30/0.67 ! [M: nat,N: nat] :
% 0.30/0.67 ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.67 = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % min_Suc_Suc
% 0.30/0.67 thf(fact_10164_min__0R,axiom,
% 0.30/0.67 ! [N: nat] :
% 0.30/0.67 ( ( ord_min_nat @ N @ zero_zero_nat )
% 0.30/0.67 = zero_zero_nat ) ).
% 0.30/0.67
% 0.30/0.67 % min_0R
% 0.30/0.67 thf(fact_10165_min__0L,axiom,
% 0.30/0.67 ! [N: nat] :
% 0.30/0.67 ( ( ord_min_nat @ zero_zero_nat @ N )
% 0.30/0.67 = zero_zero_nat ) ).
% 0.30/0.67
% 0.30/0.67 % min_0L
% 0.30/0.67 thf(fact_10166_min__Suc__numeral,axiom,
% 0.30/0.67 ! [N: nat,K: num] :
% 0.30/0.67 ( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.67 = ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % min_Suc_numeral
% 0.30/0.67 thf(fact_10167_min__numeral__Suc,axiom,
% 0.30/0.67 ! [K: num,N: nat] :
% 0.30/0.67 ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.30/0.67 = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % min_numeral_Suc
% 0.30/0.67 thf(fact_10168_nat__mult__min__left,axiom,
% 0.30/0.67 ! [M: nat,N: nat,Q3: nat] :
% 0.30/0.67 ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q3 )
% 0.30/0.67 = ( ord_min_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % nat_mult_min_left
% 0.30/0.67 thf(fact_10169_nat__mult__min__right,axiom,
% 0.30/0.67 ! [M: nat,N: nat,Q3: nat] :
% 0.30/0.67 ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q3 ) )
% 0.30/0.67 = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q3 ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % nat_mult_min_right
% 0.30/0.67 thf(fact_10170_inf__nat__def,axiom,
% 0.30/0.67 inf_inf_nat = ord_min_nat ).
% 0.30/0.67
% 0.30/0.67 % inf_nat_def
% 0.30/0.67 thf(fact_10171_min__diff,axiom,
% 0.30/0.67 ! [M: nat,I2: nat,N: nat] :
% 0.30/0.67 ( ( ord_min_nat @ ( minus_minus_nat @ M @ I2 ) @ ( minus_minus_nat @ N @ I2 ) )
% 0.30/0.67 = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I2 ) ) ).
% 0.30/0.67
% 0.30/0.67 % min_diff
% 0.30/0.67 thf(fact_10172_concat__bit__assoc__sym,axiom,
% 0.30/0.67 ! [M: nat,N: nat,K: int,L: int,R2: int] :
% 0.30/0.67 ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L ) @ R2 )
% 0.30/0.67 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N ) @ L @ R2 ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % concat_bit_assoc_sym
% 0.30/0.67 thf(fact_10173_take__bit__concat__bit__eq,axiom,
% 0.30/0.67 ! [M: nat,N: nat,K: int,L: int] :
% 0.30/0.67 ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N @ K @ L ) )
% 0.30/0.67 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ L ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % take_bit_concat_bit_eq
% 0.30/0.67 thf(fact_10174_min__Suc2,axiom,
% 0.30/0.67 ! [M: nat,N: nat] :
% 0.30/0.67 ( ( ord_min_nat @ M @ ( suc @ N ) )
% 0.30/0.67 = ( case_nat_nat @ zero_zero_nat
% 0.30/0.67 @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ M5 @ N ) )
% 0.30/0.67 @ M ) ) ).
% 0.30/0.67
% 0.30/0.67 % min_Suc2
% 0.30/0.67 thf(fact_10175_min__Suc1,axiom,
% 0.30/0.67 ! [N: nat,M: nat] :
% 0.30/0.67 ( ( ord_min_nat @ ( suc @ N ) @ M )
% 0.30/0.67 = ( case_nat_nat @ zero_zero_nat
% 0.30/0.67 @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ N @ M5 ) )
% 0.30/0.67 @ M ) ) ).
% 0.30/0.67
% 0.30/0.67 % min_Suc1
% 0.30/0.67 thf(fact_10176_min__enat__simps_I2_J,axiom,
% 0.30/0.67 ! [Q3: extended_enat] :
% 0.30/0.67 ( ( ord_mi8085742599997312461d_enat @ Q3 @ zero_z5237406670263579293d_enat )
% 0.30/0.67 = zero_z5237406670263579293d_enat ) ).
% 0.30/0.67
% 0.30/0.67 % min_enat_simps(2)
% 0.30/0.67 thf(fact_10177_min__enat__simps_I3_J,axiom,
% 0.30/0.67 ! [Q3: extended_enat] :
% 0.30/0.67 ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q3 )
% 0.30/0.67 = zero_z5237406670263579293d_enat ) ).
% 0.30/0.67
% 0.30/0.67 % min_enat_simps(3)
% 0.30/0.67 thf(fact_10178_inf__enat__def,axiom,
% 0.30/0.67 inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).
% 0.30/0.67
% 0.30/0.67 % inf_enat_def
% 0.30/0.67 thf(fact_10179_tl__upt,axiom,
% 0.30/0.67 ! [M: nat,N: nat] :
% 0.30/0.67 ( ( tl_nat @ ( upt @ M @ N ) )
% 0.30/0.67 = ( upt @ ( suc @ M ) @ N ) ) ).
% 0.30/0.67
% 0.30/0.67 % tl_upt
% 0.30/0.67 thf(fact_10180_hd__upt,axiom,
% 0.30/0.67 ! [I2: nat,J: nat] :
% 0.30/0.67 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.67 => ( ( hd_nat @ ( upt @ I2 @ J ) )
% 0.30/0.67 = I2 ) ) ).
% 0.30/0.67
% 0.30/0.67 % hd_upt
% 0.30/0.67 thf(fact_10181_drop__upt,axiom,
% 0.30/0.67 ! [M: nat,I2: nat,J: nat] :
% 0.30/0.67 ( ( drop_nat @ M @ ( upt @ I2 @ J ) )
% 0.30/0.67 = ( upt @ ( plus_plus_nat @ I2 @ M ) @ J ) ) ).
% 0.30/0.67
% 0.30/0.67 % drop_upt
% 0.30/0.67 thf(fact_10182_length__upt,axiom,
% 0.30/0.67 ! [I2: nat,J: nat] :
% 0.30/0.67 ( ( size_size_list_nat @ ( upt @ I2 @ J ) )
% 0.30/0.67 = ( minus_minus_nat @ J @ I2 ) ) ).
% 0.30/0.67
% 0.30/0.67 % length_upt
% 0.30/0.67 thf(fact_10183_take__upt,axiom,
% 0.30/0.67 ! [I2: nat,M: nat,N: nat] :
% 0.30/0.67 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ M ) @ N )
% 0.30/0.67 => ( ( take_nat @ M @ ( upt @ I2 @ N ) )
% 0.30/0.67 = ( upt @ I2 @ ( plus_plus_nat @ I2 @ M ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % take_upt
% 0.30/0.67 thf(fact_10184_upt__conv__Nil,axiom,
% 0.30/0.67 ! [J: nat,I2: nat] :
% 0.30/0.67 ( ( ord_less_eq_nat @ J @ I2 )
% 0.30/0.67 => ( ( upt @ I2 @ J )
% 0.30/0.67 = nil_nat ) ) ).
% 0.30/0.67
% 0.30/0.67 % upt_conv_Nil
% 0.30/0.67 thf(fact_10185_upt__eq__Nil__conv,axiom,
% 0.30/0.67 ! [I2: nat,J: nat] :
% 0.30/0.67 ( ( ( upt @ I2 @ J )
% 0.30/0.67 = nil_nat )
% 0.30/0.67 = ( ( J = zero_zero_nat )
% 0.30/0.67 | ( ord_less_eq_nat @ J @ I2 ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % upt_eq_Nil_conv
% 0.30/0.67 thf(fact_10186_nth__upt,axiom,
% 0.30/0.67 ! [I2: nat,K: nat,J: nat] :
% 0.30/0.67 ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J )
% 0.30/0.67 => ( ( nth_nat @ ( upt @ I2 @ J ) @ K )
% 0.30/0.67 = ( plus_plus_nat @ I2 @ K ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % nth_upt
% 0.30/0.67 thf(fact_10187_upt__rec__numeral,axiom,
% 0.30/0.67 ! [M: num,N: num] :
% 0.30/0.67 ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.67 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.67 = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
% 0.30/0.67 & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.67 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.67 = nil_nat ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % upt_rec_numeral
% 0.30/0.67 thf(fact_10188_map__add__upt,axiom,
% 0.30/0.67 ! [N: nat,M: nat] :
% 0.30/0.67 ( ( map_nat_nat
% 0.30/0.67 @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
% 0.30/0.67 @ ( upt @ zero_zero_nat @ M ) )
% 0.30/0.67 = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % map_add_upt
% 0.30/0.67 thf(fact_10189_map__Suc__upt,axiom,
% 0.30/0.67 ! [M: nat,N: nat] :
% 0.30/0.67 ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
% 0.30/0.67 = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % map_Suc_upt
% 0.30/0.67 thf(fact_10190_map__decr__upt,axiom,
% 0.30/0.67 ! [M: nat,N: nat] :
% 0.30/0.67 ( ( map_nat_nat
% 0.30/0.67 @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) )
% 0.30/0.67 @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.67 = ( upt @ M @ N ) ) ).
% 0.30/0.67
% 0.30/0.67 % map_decr_upt
% 0.30/0.67 thf(fact_10191_upt__conv__Cons__Cons,axiom,
% 0.30/0.67 ! [M: nat,N: nat,Ns: list_nat,Q3: nat] :
% 0.30/0.67 ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
% 0.30/0.67 = ( upt @ M @ Q3 ) )
% 0.30/0.67 = ( ( cons_nat @ N @ Ns )
% 0.30/0.67 = ( upt @ ( suc @ M ) @ Q3 ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % upt_conv_Cons_Cons
% 0.30/0.67 thf(fact_10192_atMost__upto,axiom,
% 0.30/0.67 ( set_ord_atMost_nat
% 0.30/0.67 = ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N3 ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % atMost_upto
% 0.30/0.67 thf(fact_10193_atLeastAtMost__upt,axiom,
% 0.30/0.67 ( set_or1269000886237332187st_nat
% 0.30/0.67 = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ N3 @ ( suc @ M3 ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % atLeastAtMost_upt
% 0.30/0.67 thf(fact_10194_greaterThanLessThan__upt,axiom,
% 0.30/0.67 ( set_or5834768355832116004an_nat
% 0.30/0.67 = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ M3 ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % greaterThanLessThan_upt
% 0.30/0.67 thf(fact_10195_greaterThanAtMost__upt,axiom,
% 0.30/0.67 ( set_or6659071591806873216st_nat
% 0.30/0.67 = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ ( suc @ M3 ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % greaterThanAtMost_upt
% 0.30/0.67 thf(fact_10196_upt__conv__Cons,axiom,
% 0.30/0.67 ! [I2: nat,J: nat] :
% 0.30/0.67 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.67 => ( ( upt @ I2 @ J )
% 0.30/0.67 = ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % upt_conv_Cons
% 0.30/0.67 thf(fact_10197_upt__add__eq__append,axiom,
% 0.30/0.67 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.67 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.67 => ( ( upt @ I2 @ ( plus_plus_nat @ J @ K ) )
% 0.30/0.67 = ( append_nat @ ( upt @ I2 @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % upt_add_eq_append
% 0.30/0.67 thf(fact_10198_upt__eq__Cons__conv,axiom,
% 0.30/0.67 ! [I2: nat,J: nat,X2: nat,Xs2: list_nat] :
% 0.30/0.67 ( ( ( upt @ I2 @ J )
% 0.30/0.67 = ( cons_nat @ X2 @ Xs2 ) )
% 0.30/0.67 = ( ( ord_less_nat @ I2 @ J )
% 0.30/0.67 & ( I2 = X2 )
% 0.30/0.67 & ( ( upt @ ( plus_plus_nat @ I2 @ one_one_nat ) @ J )
% 0.30/0.67 = Xs2 ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % upt_eq_Cons_conv
% 0.30/0.67 thf(fact_10199_upt__rec,axiom,
% 0.30/0.67 ( upt
% 0.30/0.67 = ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % upt_rec
% 0.30/0.67 thf(fact_10200_upt__Suc,axiom,
% 0.30/0.67 ! [I2: nat,J: nat] :
% 0.30/0.67 ( ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.67 => ( ( upt @ I2 @ ( suc @ J ) )
% 0.30/0.67 = ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
% 0.30/0.67 & ( ~ ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.67 => ( ( upt @ I2 @ ( suc @ J ) )
% 0.30/0.67 = nil_nat ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % upt_Suc
% 0.30/0.67 thf(fact_10201_upt__Suc__append,axiom,
% 0.30/0.67 ! [I2: nat,J: nat] :
% 0.30/0.67 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.67 => ( ( upt @ I2 @ ( suc @ J ) )
% 0.30/0.67 = ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % upt_Suc_append
% 0.30/0.67 thf(fact_10202_sum__list__upt,axiom,
% 0.30/0.67 ! [M: nat,N: nat] :
% 0.30/0.67 ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.67 => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
% 0.30/0.67 = ( groups3542108847815614940at_nat
% 0.30/0.67 @ ^ [X: nat] : X
% 0.30/0.67 @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % sum_list_upt
% 0.30/0.67 thf(fact_10203_card__length__sum__list__rec,axiom,
% 0.30/0.67 ! [M: nat,N5: nat] :
% 0.30/0.67 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.30/0.67 => ( ( finite_card_list_nat
% 0.30/0.67 @ ( collect_list_nat
% 0.30/0.67 @ ^ [L2: list_nat] :
% 0.30/0.67 ( ( ( size_size_list_nat @ L2 )
% 0.30/0.67 = M )
% 0.30/0.67 & ( ( groups4561878855575611511st_nat @ L2 )
% 0.30/0.67 = N5 ) ) ) )
% 0.30/0.67 = ( plus_plus_nat
% 0.30/0.67 @ ( finite_card_list_nat
% 0.30/0.67 @ ( collect_list_nat
% 0.30/0.67 @ ^ [L2: list_nat] :
% 0.30/0.67 ( ( ( size_size_list_nat @ L2 )
% 0.30/0.67 = ( minus_minus_nat @ M @ one_one_nat ) )
% 0.30/0.67 & ( ( groups4561878855575611511st_nat @ L2 )
% 0.30/0.67 = N5 ) ) ) )
% 0.30/0.67 @ ( finite_card_list_nat
% 0.30/0.67 @ ( collect_list_nat
% 0.30/0.67 @ ^ [L2: list_nat] :
% 0.30/0.67 ( ( ( size_size_list_nat @ L2 )
% 0.30/0.67 = M )
% 0.30/0.67 & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
% 0.30/0.67 = N5 ) ) ) ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % card_length_sum_list_rec
% 0.30/0.67 thf(fact_10204_card__length__sum__list,axiom,
% 0.30/0.67 ! [M: nat,N5: nat] :
% 0.30/0.67 ( ( finite_card_list_nat
% 0.30/0.67 @ ( collect_list_nat
% 0.30/0.67 @ ^ [L2: list_nat] :
% 0.30/0.67 ( ( ( size_size_list_nat @ L2 )
% 0.30/0.67 = M )
% 0.30/0.67 & ( ( groups4561878855575611511st_nat @ L2 )
% 0.30/0.67 = N5 ) ) ) )
% 0.30/0.67 = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N5 @ M ) @ one_one_nat ) @ N5 ) ) ).
% 0.30/0.67
% 0.30/0.67 % card_length_sum_list
% 0.30/0.67 thf(fact_10205_sorted__upto,axiom,
% 0.30/0.67 ! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).
% 0.30/0.67
% 0.30/0.67 % sorted_upto
% 0.30/0.67 thf(fact_10206_sorted__upt,axiom,
% 0.30/0.67 ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% 0.30/0.67
% 0.30/0.67 % sorted_upt
% 0.30/0.67 thf(fact_10207_sorted__wrt__upt,axiom,
% 0.30/0.67 ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% 0.30/0.67
% 0.30/0.67 % sorted_wrt_upt
% 0.30/0.67 thf(fact_10208_sorted__wrt__less__idx,axiom,
% 0.30/0.67 ! [Ns: list_nat,I2: nat] :
% 0.30/0.67 ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
% 0.30/0.67 => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ns ) )
% 0.30/0.67 => ( ord_less_eq_nat @ I2 @ ( nth_nat @ Ns @ I2 ) ) ) ) ).
% 0.30/0.67
% 0.30/0.67 % sorted_wrt_less_idx
% 0.30/0.67
% 0.30/0.67 % Helper facts (38)
% 0.30/0.67 thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 0.30/0.67 ! [X2: int,Y3: int] :
% 0.30/0.67 ( ( if_int @ $false @ X2 @ Y3 )
% 0.30/0.67 = Y3 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 0.30/0.67 ! [X2: int,Y3: int] :
% 0.30/0.67 ( ( if_int @ $true @ X2 @ Y3 )
% 0.30/0.67 = X2 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.30/0.67 ! [X2: nat,Y3: nat] :
% 0.30/0.67 ( ( if_nat @ $false @ X2 @ Y3 )
% 0.30/0.67 = Y3 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.30/0.67 ! [X2: nat,Y3: nat] :
% 0.30/0.67 ( ( if_nat @ $true @ X2 @ Y3 )
% 0.30/0.67 = X2 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
% 0.30/0.67 ! [X2: num,Y3: num] :
% 0.30/0.67 ( ( if_num @ $false @ X2 @ Y3 )
% 0.30/0.67 = Y3 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
% 0.30/0.67 ! [X2: num,Y3: num] :
% 0.30/0.67 ( ( if_num @ $true @ X2 @ Y3 )
% 0.30/0.67 = X2 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 0.30/0.67 ! [X2: rat,Y3: rat] :
% 0.30/0.67 ( ( if_rat @ $false @ X2 @ Y3 )
% 0.30/0.67 = Y3 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 0.30/0.67 ! [X2: rat,Y3: rat] :
% 0.30/0.67 ( ( if_rat @ $true @ X2 @ Y3 )
% 0.30/0.67 = X2 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 0.30/0.67 ! [X2: real,Y3: real] :
% 0.30/0.67 ( ( if_real @ $false @ X2 @ Y3 )
% 0.30/0.67 = Y3 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 0.30/0.67 ! [X2: real,Y3: real] :
% 0.30/0.67 ( ( if_real @ $true @ X2 @ Y3 )
% 0.30/0.67 = X2 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 0.30/0.67 ! [P: real > $o] :
% 0.30/0.67 ( ( P @ ( fChoice_real @ P ) )
% 0.30/0.67 = ( ? [X4: real] : ( P @ X4 ) ) ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.30/0.67 ! [X2: complex,Y3: complex] :
% 0.30/0.67 ( ( if_complex @ $false @ X2 @ Y3 )
% 0.30/0.67 = Y3 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.30/0.67 ! [X2: complex,Y3: complex] :
% 0.30/0.67 ( ( if_complex @ $true @ X2 @ Y3 )
% 0.30/0.67 = X2 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 0.30/0.67 ! [X2: extended_enat,Y3: extended_enat] :
% 0.30/0.67 ( ( if_Extended_enat @ $false @ X2 @ Y3 )
% 0.30/0.67 = Y3 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 0.30/0.67 ! [X2: extended_enat,Y3: extended_enat] :
% 0.30/0.67 ( ( if_Extended_enat @ $true @ X2 @ Y3 )
% 0.30/0.67 = X2 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.30/0.67 ! [X2: code_integer,Y3: code_integer] :
% 0.30/0.67 ( ( if_Code_integer @ $false @ X2 @ Y3 )
% 0.30/0.67 = Y3 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.30/0.67 ! [X2: code_integer,Y3: code_integer] :
% 0.30/0.67 ( ( if_Code_integer @ $true @ X2 @ Y3 )
% 0.30/0.67 = X2 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.30/0.67 ! [X2: set_int,Y3: set_int] :
% 0.30/0.67 ( ( if_set_int @ $false @ X2 @ Y3 )
% 0.30/0.67 = Y3 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.30/0.67 ! [X2: set_int,Y3: set_int] :
% 0.30/0.67 ( ( if_set_int @ $true @ X2 @ Y3 )
% 0.30/0.67 = X2 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 0.30/0.67 ! [X2: vEBT_VEBT,Y3: vEBT_VEBT] :
% 0.30/0.67 ( ( if_VEBT_VEBT @ $false @ X2 @ Y3 )
% 0.30/0.67 = Y3 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 0.30/0.67 ! [X2: vEBT_VEBT,Y3: vEBT_VEBT] :
% 0.30/0.67 ( ( if_VEBT_VEBT @ $true @ X2 @ Y3 )
% 0.30/0.67 = X2 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.30/0.67 ! [X2: list_int,Y3: list_int] :
% 0.30/0.67 ( ( if_list_int @ $false @ X2 @ Y3 )
% 0.30/0.67 = Y3 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.30/0.67 ! [X2: list_int,Y3: list_int] :
% 0.30/0.67 ( ( if_list_int @ $true @ X2 @ Y3 )
% 0.30/0.67 = X2 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 0.30/0.67 ! [X2: list_nat,Y3: list_nat] :
% 0.30/0.67 ( ( if_list_nat @ $false @ X2 @ Y3 )
% 0.30/0.67 = Y3 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 0.30/0.67 ! [X2: list_nat,Y3: list_nat] :
% 0.30/0.67 ( ( if_list_nat @ $true @ X2 @ Y3 )
% 0.30/0.67 = X2 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 0.30/0.67 ! [X2: option_nat,Y3: option_nat] :
% 0.30/0.67 ( ( if_option_nat @ $false @ X2 @ Y3 )
% 0.30/0.67 = Y3 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 0.30/0.67 ! [X2: option_nat,Y3: option_nat] :
% 0.30/0.67 ( ( if_option_nat @ $true @ X2 @ Y3 )
% 0.30/0.67 = X2 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.30/0.67 ! [X2: option_num,Y3: option_num] :
% 0.30/0.67 ( ( if_option_num @ $false @ X2 @ Y3 )
% 0.30/0.67 = Y3 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.30/0.67 ! [X2: option_num,Y3: option_num] :
% 0.30/0.67 ( ( if_option_num @ $true @ X2 @ Y3 )
% 0.30/0.67 = X2 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.30/0.67 ! [X2: product_prod_int_int,Y3: product_prod_int_int] :
% 0.30/0.67 ( ( if_Pro3027730157355071871nt_int @ $false @ X2 @ Y3 )
% 0.30/0.67 = Y3 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.30/0.67 ! [X2: product_prod_int_int,Y3: product_prod_int_int] :
% 0.30/0.67 ( ( if_Pro3027730157355071871nt_int @ $true @ X2 @ Y3 )
% 0.30/0.67 = X2 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.30/0.67 ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 0.30/0.67 ( ( if_Pro6206227464963214023at_nat @ $false @ X2 @ Y3 )
% 0.30/0.67 = Y3 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.30/0.67 ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 0.30/0.67 ( ( if_Pro6206227464963214023at_nat @ $true @ X2 @ Y3 )
% 0.30/0.67 = X2 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.30/0.67 ! [X2: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 0.30/0.67 ( ( if_Pro5737122678794959658eger_o @ $false @ X2 @ Y3 )
% 0.30/0.67 = Y3 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.30/0.67 ! [X2: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 0.30/0.67 ( ( if_Pro5737122678794959658eger_o @ $true @ X2 @ Y3 )
% 0.30/0.67 = X2 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.30/0.67 ! [P: $o] :
% 0.30/0.67 ( ( P = $true )
% 0.30/0.67 | ( P = $false ) ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.30/0.67 ! [X2: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 0.30/0.67 ( ( if_Pro6119634080678213985nteger @ $false @ X2 @ Y3 )
% 0.30/0.67 = Y3 ) ).
% 0.30/0.67
% 0.30/0.67 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.30/0.67 ! [X2: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 0.30/0.67 ( ( if_Pro6119634080678213985nteger @ $true @ X2 @ Y3 )
% 0.30/0.67 = X2 ) ).
% 0.30/0.67
% 0.30/0.67 % Conjectures (1)
% 0.30/0.67 thf(conj_0,conjecture,
% 0.30/0.67 ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) ) ) ).
% 0.30/0.67
% 0.30/0.67 %------------------------------------------------------------------------------
% 0.30/0.67 ------------------------
% 6.24/2.59 ---- Embedded in THF ---
% 6.24/2.59 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 6.24/2.59 %%% Generated on Fri Feb 27 00:31:58 EST 2026
% 6.24/2.59 %%% using '$$dhol' embedding, version 1.3.0.
% 6.24/2.59 %%% Logic specification used:
% 6.24/2.59 %%% thf(spec, logic, $$dhol).
% 6.24/2.59
% 6.24/2.59 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.jDIxpPE1PX/DTF2DTF16103.p
% See solution above
% 6.54/2.64 ------------------------
% 8.83/4.96 ---- Cleaned THF ---
% 8.83/4.96 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,
% 8.83/4.96 produc5542196010084753463at_nat: $tType ).
% 8.83/4.96
% 8.83/4.96 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,
% 8.83/4.96 produc5491161045314408544at_nat: $tType ).
% 8.83/4.96
% 8.83/4.96 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,
% 8.83/4.96 produc1193250871479095198on_num: $tType ).
% 8.83/4.96
% 8.83/4.96 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,
% 8.83/4.96 produc8306885398267862888on_nat: $tType ).
% 8.83/4.96
% 8.83/4.96 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,
% 8.83/4.96 produc6121120109295599847at_nat: $tType ).
% 8.83/4.96
% 8.83/4.96 thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J_J,type,
% 8.83/4.96 produc3368934014287244435at_num: $tType ).
% 8.83/4.96
% 8.83/4.96 thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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% 8.83/4.96 cons_int: int > list_int > list_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
% 8.83/4.96 cons_nat: nat > list_nat > list_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
% 8.83/4.96 nil_int: list_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
% 8.83/4.96 nil_nat: list_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
% 8.83/4.96 hd_nat: list_nat > nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
% 8.83/4.96 map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist_Oset_001_Eo,type,
% 8.83/4.96 set_o2: list_o > set_o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
% 8.83/4.96 set_complex2: list_complex > set_complex ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
% 8.83/4.96 set_int2: list_int > set_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
% 8.83/4.96 set_nat2: list_nat > set_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.83/4.96 set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
% 8.83/4.96 set_real2: list_real > set_real ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
% 8.83/4.96 set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
% 8.83/4.96 size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
% 8.83/4.96 tl_nat: list_nat > list_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist__update_001_Eo,type,
% 8.83/4.96 list_update_o: list_o > nat > $o > list_o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist__update_001t__Complex__Ocomplex,type,
% 8.83/4.96 list_update_complex: list_complex > nat > complex > list_complex ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist__update_001t__Int__Oint,type,
% 8.83/4.96 list_update_int: list_int > nat > int > list_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
% 8.83/4.96 list_update_nat: list_nat > nat > nat > list_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.83/4.96 list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
% 8.83/4.96 list_update_real: list_real > nat > real > list_real ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
% 8.83/4.96 list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Onth_001_Eo,type,
% 8.83/4.96 nth_o: list_o > nat > $o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Onth_001t__Code____Numeral__Ointeger,type,
% 8.83/4.96 nth_Code_integer: list_Code_integer > nat > code_integer ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
% 8.83/4.96 nth_complex: list_complex > nat > complex ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Onth_001t__Int__Oint,type,
% 8.83/4.96 nth_int: list_int > nat > int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 8.83/4.96 nth_nat: list_nat > nat > nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Onth_001t__Num__Onum,type,
% 8.83/4.96 nth_num: list_num > nat > num ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 8.83/4.96 nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 8.83/4.96 nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
% 8.83/4.96 nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 8.83/4.96 nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 8.83/4.96 nth_Pr8522763379788166057eger_o: list_P8526636022914148096eger_o > nat > produc6271795597528267376eger_o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.83/4.96 nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 8.83/4.96 nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 8.83/4.96 nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 8.83/4.96 nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 8.83/4.96 nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 8.83/4.96 nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Onth_001t__Real__Oreal,type,
% 8.83/4.96 nth_real: list_real > nat > real ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
% 8.83/4.96 nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
% 8.83/4.96 product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
% 8.83/4.96 product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
% 8.83/4.96 product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 8.83/4.96 product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001_Eo,type,
% 8.83/4.96 produc3607205314601156340eger_o: list_Code_integer > list_o > list_P8526636022914148096eger_o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
% 8.83/4.96 product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
% 8.83/4.96 produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oproduct_001t__Num__Onum_001t__Num__Onum,type,
% 8.83/4.96 product_num_num: list_num > list_num > list_P3744719386663036955um_num ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 8.83/4.96 product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 8.83/4.96 produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 8.83/4.96 produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 8.83/4.96 produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oreplicate_001_Eo,type,
% 8.83/4.96 replicate_o: nat > $o > list_o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oreplicate_001t__Complex__Ocomplex,type,
% 8.83/4.96 replicate_complex: nat > complex > list_complex ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
% 8.83/4.96 replicate_int: nat > int > list_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
% 8.83/4.96 replicate_nat: nat > nat > list_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.83/4.96 replic4235873036481779905at_nat: nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
% 8.83/4.96 replicate_real: nat > real > list_real ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 8.83/4.96 replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
% 8.83/4.96 sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
% 8.83/4.96 sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Otake_001t__Nat__Onat,type,
% 8.83/4.96 take_nat: nat > list_nat > list_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oupt,type,
% 8.83/4.96 upt: nat > nat > list_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oupto,type,
% 8.83/4.96 upto: int > int > list_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oupto__aux,type,
% 8.83/4.96 upto_aux: int > int > list_int > list_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_List_Oupto__rel,type,
% 8.83/4.96 upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_OSuc,type,
% 8.83/4.96 suc: nat > nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.83/4.96 compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 8.83/4.96 case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 8.83/4.96 case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
% 8.83/4.96 case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Onat_Opred,type,
% 8.83/4.96 pred: nat > nat ).
% 8.83/4.96
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% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
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% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
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% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
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% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
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% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 8.83/4.96 semiri5074537144036343181t_real: nat > real ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
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% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
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% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
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% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
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% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
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% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
% 8.83/4.96 size_size_list_o: list_o > nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
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% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
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% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
% 8.83/4.96 size_size_list_int: list_int > nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
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% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
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% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
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% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
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% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
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% 8.83/4.96
% 8.83/4.96 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
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% 8.83/4.96 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,
% 8.83/4.96 produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).
% 8.83/4.96
% 8.83/4.96 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,
% 8.83/4.96 produc851828971589881931at_num: ( nat > num > num ) > produc2963631642982155120at_num > produc3368934014287244435at_num ).
% 8.83/4.96
% 8.83/4.96 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,
% 8.83/4.96 produc3576312749637752826on_num: ( num > num > $o ) > produc3447558737645232053on_num > produc7036089656553540234on_num ).
% 8.83/4.96
% 8.83/4.96 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,
% 8.83/4.96 produc5778274026573060048on_num: ( num > num > num ) > produc3447558737645232053on_num > produc1193250871479095198on_num ).
% 8.83/4.96
% 8.83/4.96 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,
% 8.83/4.96 produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).
% 8.83/4.96
% 8.83/4.96 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,
% 8.83/4.96 produc2899441246263362727at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > produc6121120109295599847at_nat > produc5542196010084753463at_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
% 8.83/4.96 product_Pair_o_o: $o > $o > product_prod_o_o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
% 8.83/4.96 product_Pair_o_int: $o > int > product_prod_o_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
% 8.83/4.96 product_Pair_o_nat: $o > nat > product_prod_o_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 8.83/4.96 produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
% 8.83/4.96 produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 8.83/4.96 produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
% 8.83/4.96 product_Pair_int_int: int > int > product_prod_int_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
% 8.83/4.96 product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
% 8.83/4.96 product_Pair_nat_num: nat > num > product_prod_nat_num ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.83/4.96 produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 8.83/4.96 produc1195630363706982562at_num: nat > product_prod_nat_num > produc2963631642982155120at_num ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
% 8.83/4.96 product_Pair_num_num: num > num > product_prod_num_num ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
% 8.83/4.96 produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
% 8.83/4.96 produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num ).
% 8.83/4.96
% 8.83/4.96 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,
% 8.83/4.96 produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 8.83/4.96 produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 8.83/4.96 produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 8.83/4.96 produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 8.83/4.96 produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 8.83/4.96 produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001_Eo,type,
% 8.83/4.96 produc7828578312038201481er_o_o: ( code_integer > $o > $o ) > produc6271795597528267376eger_o > $o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 8.83/4.96 produc1043322548047392435omplex: ( code_integer > $o > set_complex ) > produc6271795597528267376eger_o > set_complex ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__Set__Oset_It__Int__Oint_J,type,
% 8.83/4.96 produc1253318751659547953et_int: ( code_integer > $o > set_int ) > produc6271795597528267376eger_o > set_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
% 8.83/4.96 produc5431169771168744661et_nat: ( code_integer > $o > set_nat ) > produc6271795597528267376eger_o > set_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__Set__Oset_It__Real__Oreal_J,type,
% 8.83/4.96 produc242741666403216561t_real: ( code_integer > $o > set_real ) > produc6271795597528267376eger_o > set_real ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
% 8.83/4.96 produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
% 8.83/4.96 produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
% 8.83/4.96 produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 8.83/4.96 produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 8.83/4.96 produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
% 8.83/4.96 produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 8.83/4.96 produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 8.83/4.96 produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
% 8.83/4.96
% 8.83/4.96 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,
% 8.83/4.96 produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
% 8.83/4.96 produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 8.83/4.96 produc1917071388513777916omplex: ( nat > nat > complex ) > product_prod_nat_nat > complex ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
% 8.83/4.96 produc6840382203811409530at_int: ( nat > nat > int ) > product_prod_nat_nat > int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 8.83/4.96 produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.83/4.96 produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
% 8.83/4.96 produc6207742614233964070at_rat: ( nat > nat > rat ) > product_prod_nat_nat > rat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
% 8.83/4.96 produc1703576794950452218t_real: ( nat > nat > real ) > product_prod_nat_nat > real ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001_Eo,type,
% 8.83/4.96 produc4927758841916487424_num_o: ( nat > num > $o ) > product_prod_nat_num > $o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
% 8.83/4.96 produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 8.83/4.96 produc6231982587499038204omplex: ( nat > num > set_complex ) > product_prod_nat_num > set_complex ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Set__Oset_It__Real__Oreal_J,type,
% 8.83/4.96 produc1435849484188172666t_real: ( nat > num > set_real ) > product_prod_nat_num > set_real ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001_Eo,type,
% 8.83/4.96 produc5703948589228662326_num_o: ( num > num > $o ) > product_prod_num_num > $o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 8.83/4.96 produc2866383454006189126omplex: ( num > num > set_complex ) > product_prod_num_num > set_complex ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Int__Oint_J,type,
% 8.83/4.96 produc6406642877701697732et_int: ( num > num > set_int ) > product_prod_num_num > set_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Nat__Onat_J,type,
% 8.83/4.96 produc1361121860356118632et_nat: ( num > num > set_nat ) > product_prod_num_num > set_nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Real__Oreal_J,type,
% 8.83/4.96 produc8296048397933160132t_real: ( num > num > set_real ) > product_prod_num_num > set_real ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
% 8.83/4.96 product_fst_int_int: product_prod_int_int > int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
% 8.83/4.96 product_fst_nat_nat: product_prod_nat_nat > nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
% 8.83/4.96 product_snd_int_int: product_prod_int_int > int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
% 8.83/4.96 product_snd_nat_nat: product_prod_nat_nat > nat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Rat_OAbs__Rat,type,
% 8.83/4.96 abs_Rat: product_prod_int_int > rat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Rat_OFract,type,
% 8.83/4.96 fract: int > int > rat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Rat_OFrct,type,
% 8.83/4.96 frct: product_prod_int_int > rat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Rat_ORep__Rat,type,
% 8.83/4.96 rep_Rat: rat > product_prod_int_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
% 8.83/4.96 field_5140801741446780682s_real: set_real ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
% 8.83/4.96 field_7254667332652039916t_real: rat > real ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Rat_Onormalize,type,
% 8.83/4.96 normalize: product_prod_int_int > product_prod_int_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Rat_Oof__int,type,
% 8.83/4.96 of_int: int > rat ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Rat_Opositive,type,
% 8.83/4.96 positive: rat > $o ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Rat_Oquotient__of,type,
% 8.83/4.96 quotient_of: rat > product_prod_int_int ).
% 8.83/4.96
% 8.83/4.96 thf(sy_c_Rat_Oratrel,type,
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% 8.83/4.96
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% 8.83/4.97 set_or1633881224788618240n_real: real > real > set_real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
% 8.83/4.97 set_or1210151606488870762an_nat: nat > set_nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
% 8.83/4.97 set_or5849166863359141190n_real: real > set_real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
% 8.83/4.97 set_ord_lessThan_int: int > set_int ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
% 8.83/4.97 set_ord_lessThan_nat: nat > set_nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
% 8.83/4.97 set_ord_lessThan_num: num > set_num ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
% 8.83/4.97 set_ord_lessThan_rat: rat > set_rat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
% 8.83/4.97 set_or5984915006950818249n_real: real > set_real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_String_Oascii__of,type,
% 8.83/4.97 ascii_of: char > char ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_String_Ochar_OChar,type,
% 8.83/4.97 char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
% 8.83/4.97 comm_s629917340098488124ar_nat: char > nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_String_Ointeger__of__char,type,
% 8.83/4.97 integer_of_char: char > code_integer ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
% 8.83/4.97 unique3096191561947761185of_nat: nat > char ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
% 8.83/4.97 topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 8.83/4.97 topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
% 8.83/4.97 topolo4899668324122417113eq_int: ( nat > int ) > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
% 8.83/4.97 topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
% 8.83/4.97 topolo1459490580787246023eq_num: ( nat > num ) > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
% 8.83/4.97 topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
% 8.83/4.97 topolo6980174941875973593q_real: ( nat > real ) > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Int__Oint_J,type,
% 8.83/4.97 topolo3100542954746470799et_int: ( nat > set_int ) > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
% 8.83/4.97 topolo2177554685111907308n_real: real > set_real > filter_real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
% 8.83/4.97 topolo2815343760600316023s_real: real > filter_real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
% 8.83/4.97 topolo4055970368930404560y_real: ( nat > real ) > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Oarccos,type,
% 8.83/4.97 arccos: real > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
% 8.83/4.97 arcosh_real: real > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Oarcsin,type,
% 8.83/4.97 arcsin: real > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Oarctan,type,
% 8.83/4.97 arctan: real > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
% 8.83/4.97 arsinh_real: real > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
% 8.83/4.97 artanh_real: real > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
% 8.83/4.97 cos_complex: complex > complex ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
% 8.83/4.97 cos_real: real > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Ocos__coeff,type,
% 8.83/4.97 cos_coeff: nat > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
% 8.83/4.97 cosh_real: real > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
% 8.83/4.97 cot_real: real > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
% 8.83/4.97 diffs_complex: ( nat > complex ) > nat > complex ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
% 8.83/4.97 diffs_int: ( nat > int ) > nat > int ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
% 8.83/4.97 diffs_rat: ( nat > rat ) > nat > rat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
% 8.83/4.97 diffs_real: ( nat > real ) > nat > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
% 8.83/4.97 exp_complex: complex > complex ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
% 8.83/4.97 exp_real: real > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
% 8.83/4.97 ln_ln_real: real > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Olog,type,
% 8.83/4.97 log: real > real > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Opi,type,
% 8.83/4.97 pi: real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 8.83/4.97 powr_real: real > real > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
% 8.83/4.97 sin_complex: complex > complex ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
% 8.83/4.97 sin_real: real > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Osin__coeff,type,
% 8.83/4.97 sin_coeff: nat > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
% 8.83/4.97 sinh_real: real > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
% 8.83/4.97 tan_complex: complex > complex ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
% 8.83/4.97 tan_real: real > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
% 8.83/4.97 tanh_complex: complex > complex ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
% 8.83/4.97 tanh_real: real > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
% 8.83/4.97 transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
% 8.83/4.97 transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
% 8.83/4.97 vEBT_T_i_n_s_e_r_t: vEBT_VEBT > nat > nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
% 8.83/4.97 vEBT_T_i_n_s_e_r_t2: vEBT_VEBT > nat > nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
% 8.83/4.97 vEBT_T5076183648494686801_t_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
% 8.83/4.97 vEBT_T9217963907923527482_t_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
% 8.83/4.97 vEBT_T_m_a_x_t: vEBT_VEBT > nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
% 8.83/4.97 vEBT_T_m_a_x_t_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
% 8.83/4.97 vEBT_T_m_e_m_b_e_r: vEBT_VEBT > nat > nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
% 8.83/4.97 vEBT_T_m_e_m_b_e_r2: vEBT_VEBT > nat > nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
% 8.83/4.97 vEBT_T8099345112685741742_r_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
% 8.83/4.97 vEBT_T5837161174952499735_r_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
% 8.83/4.97 vEBT_T_m_i_n_N_u_l_l: vEBT_VEBT > nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l__rel,type,
% 8.83/4.97 vEBT_T5462971552011256508_l_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
% 8.83/4.97 vEBT_T_m_i_n_t: vEBT_VEBT > nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
% 8.83/4.97 vEBT_T_m_i_n_t_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
% 8.83/4.97 vEBT_Leaf: $o > $o > vEBT_VEBT ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
% 8.83/4.97 vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
% 8.83/4.97 vEBT_size_VEBT: vEBT_VEBT > nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
% 8.83/4.97 vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
% 8.83/4.97 vEBT_VEBT_high: nat > nat > nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
% 8.83/4.97 vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
% 8.83/4.97 vEBT_VEBT_low: nat > nat > nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
% 8.83/4.97 vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
% 8.83/4.97 vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
% 8.83/4.97 vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
% 8.83/4.97 vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
% 8.83/4.97 vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
% 8.83/4.97 vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
% 8.83/4.97 vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Definitions_Oset__vebt,type,
% 8.83/4.97 vEBT_set_vebt: vEBT_VEBT > set_nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
% 8.83/4.97 vEBT_vebt_buildup: nat > vEBT_VEBT ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
% 8.83/4.97 vEBT_v4011308405150292612up_rel: nat > nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
% 8.83/4.97 vEBT_VEBT_height: vEBT_VEBT > nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
% 8.83/4.97 vEBT_VEBT_height_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Insert_Ovebt__insert,type,
% 8.83/4.97 vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
% 8.83/4.97 vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
% 8.83/4.97 vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
% 8.83/4.97 vEBT_VEBT_minNull: vEBT_VEBT > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
% 8.83/4.97 vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
% 8.83/4.97 vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Member_Ovebt__member,type,
% 8.83/4.97 vEBT_vebt_member: vEBT_VEBT > nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
% 8.83/4.97 vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
% 8.83/4.97 vEBT_VEBT_add: option_nat > option_nat > option_nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
% 8.83/4.97 vEBT_VEBT_greater: option_nat > option_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
% 8.83/4.97 vEBT_VEBT_less: option_nat > option_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
% 8.83/4.97 vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
% 8.83/4.97 vEBT_VEBT_max_in_set: set_nat > nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
% 8.83/4.97 vEBT_VEBT_min_in_set: set_nat > nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
% 8.83/4.97 vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
% 8.83/4.97 vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
% 8.83/4.97 vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.83/4.97 vEBT_V1502963449132264192at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Nat__Onat,type,
% 8.83/4.97 vEBT_V3895251965096974666el_nat: produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Num__Onum,type,
% 8.83/4.97 vEBT_V452583751252753300el_num: produc1193250871479095198on_num > produc1193250871479095198on_num > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.83/4.97 vEBT_V7235779383477046023at_nat: produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
% 8.83/4.97 vEBT_VEBT_power: option_nat > option_nat > option_nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
% 8.83/4.97 vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
% 8.83/4.97 vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
% 8.83/4.97 vEBT_vebt_mint: vEBT_VEBT > option_nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
% 8.83/4.97 vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Pred_Ois__pred__in__set,type,
% 8.83/4.97 vEBT_is_pred_in_set: set_nat > nat > nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Pred_Ovebt__pred,type,
% 8.83/4.97 vEBT_vebt_pred: vEBT_VEBT > nat > option_nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
% 8.83/4.97 vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Succ_Ois__succ__in__set,type,
% 8.83/4.97 vEBT_is_succ_in_set: set_nat > nat > nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Succ_Ovebt__succ,type,
% 8.83/4.97 vEBT_vebt_succ: vEBT_VEBT > nat > option_nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
% 8.83/4.97 vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
% 8.83/4.97 accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
% 8.83/4.97 accp_nat: ( nat > nat > $o ) > nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 8.83/4.97 accp_P6019419558468335806at_nat: ( produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ) > produc4471711990508489141at_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Wellfounded_Oaccp_001t__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,
% 8.83/4.97 accp_P5496254298877145759on_nat: ( produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ) > produc8306885398267862888on_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J_J,type,
% 8.83/4.97 accp_P4916641582247091100at_num: ( produc3368934014287244435at_num > produc3368934014287244435at_num > $o ) > produc3368934014287244435at_num > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Wellfounded_Oaccp_001t__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,
% 8.83/4.97 accp_P7605991808943153877on_num: ( produc1193250871479095198on_num > produc1193250871479095198on_num > $o ) > produc1193250871479095198on_num > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Wellfounded_Oaccp_001t__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,
% 8.83/4.97 accp_P3267385326087170368at_nat: ( produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ) > produc5542196010084753463at_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 8.83/4.97 accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.83/4.97 accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 8.83/4.97 accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 8.83/4.97 accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
% 8.83/4.97 accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Wellfounded_Omeasure_001t__Int__Oint,type,
% 8.83/4.97 measure_int: ( int > nat ) > set_Pr958786334691620121nt_int ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Wellfounded_Omeasure_001t__Nat__Onat,type,
% 8.83/4.97 measure_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Wellfounded_Omeasure_001t__Num__Onum,type,
% 8.83/4.97 measure_num: ( num > nat ) > set_Pr8218934625190621173um_num ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_Wellfounded_Opred__nat,type,
% 8.83/4.97 pred_nat: set_Pr1261947904930325089at_nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_fChoice_001t__Real__Oreal,type,
% 8.83/4.97 fChoice_real: ( real > $o ) > real ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_member_001_Eo,type,
% 8.83/4.97 member_o: $o > set_o > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_member_001t__Complex__Ocomplex,type,
% 8.83/4.97 member_complex: complex > set_complex > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_member_001t__Int__Oint,type,
% 8.83/4.97 member_int: int > set_int > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
% 8.83/4.97 member_list_o: list_o > set_list_o > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
% 8.83/4.97 member_list_int: list_int > set_list_int > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
% 8.83/4.97 member_list_nat: list_nat > set_list_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 8.83/4.97 member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_member_001t__Nat__Onat,type,
% 8.83/4.97 member_nat: nat > set_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_member_001t__Num__Onum,type,
% 8.83/4.97 member_num: num > set_num > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 8.83/4.97 member1379723562493234055eger_o: produc6271795597528267376eger_o > set_Pr448751882837621926eger_o > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 8.83/4.97 member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.83/4.97 member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 8.83/4.97 member9148766508732265716at_num: product_prod_nat_num > set_Pr6200539531224447659at_num > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_member_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 8.83/4.97 member7279096912039735102um_num: product_prod_num_num > set_Pr8218934625190621173um_num > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_member_001t__Rat__Orat,type,
% 8.83/4.97 member_rat: rat > set_rat > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_member_001t__Real__Oreal,type,
% 8.83/4.97 member_real: real > set_real > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
% 8.83/4.97 member_set_int: set_int > set_set_int > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
% 8.83/4.97 member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
% 8.83/4.97
% 8.83/4.97 thf(sy_v_deg____,type,
% 8.83/4.97 deg: nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_v_m____,type,
% 8.83/4.97 m: nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_v_ma____,type,
% 8.83/4.97 ma: nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_v_mi____,type,
% 8.83/4.97 mi: nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_v_na____,type,
% 8.83/4.97 na: nat ).
% 8.83/4.97
% 8.83/4.97 thf(sy_v_summary____,type,
% 8.83/4.97 summary: vEBT_VEBT ).
% 8.83/4.97
% 8.83/4.97 thf(sy_v_treeList____,type,
% 8.83/4.97 treeList: list_VEBT_VEBT ).
% 8.83/4.97
% 8.83/4.97 thf(sy_v_xa____,type,
% 8.83/4.97 xa: nat ).
% 8.83/4.97
% 8.83/4.97 thf(fact_0__C5_Ohyps_C_I7_J,axiom,
% 8.83/4.97 ord_less_eq_nat @ mi @ ma ).
% 8.83/4.97
% 8.83/4.97 thf(fact_1_max__in__set__def,axiom,
% 8.83/4.97 ( vEBT_VEBT_max_in_set
% 8.83/4.97 = ( ^ [Xs: set_nat,X: nat] :
% 8.83/4.97 ( ( member_nat @ X @ Xs )
% 8.83/4.97 & ! [Y: nat] :
% 8.83/4.97 ( ( member_nat @ Y @ Xs )
% 8.83/4.97 => ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_2_min__in__set__def,axiom,
% 8.83/4.97 ( vEBT_VEBT_min_in_set
% 8.83/4.97 = ( ^ [Xs: set_nat,X: nat] :
% 8.83/4.97 ( ( member_nat @ X @ Xs )
% 8.83/4.97 & ! [Y: nat] :
% 8.83/4.97 ( ( member_nat @ Y @ Xs )
% 8.83/4.97 => ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_3_height__compose__summary,axiom,
% 8.83/4.97 ! [Summary: vEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ Summary ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_4__C5_OIH_C_I2_J,axiom,
% 8.83/4.97 ! [X2: nat] : ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ summary @ X2 ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ summary ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_5_height__compose__child,axiom,
% 8.83/4.97 ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,Summary: vEBT_VEBT] :
% 8.83/4.97 ( ( member_VEBT_VEBT @ T @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.83/4.97 => ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_6__092_060open_062T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_092_060le_062_A1_A_L_A_I1_A_L_Aheight_A_ItreeList_A_B_Ahigh_A_Iif_Ax_A_060_Ami_Athen_Ami_Aelse_Ax_J_A_Ideg_Adiv_A2_J_J_J_092_060close_062,axiom,
% 8.83/4.97 ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_7__C5_Ohyps_C_I6_J,axiom,
% 8.83/4.97 ( ( mi = ma )
% 8.83/4.97 => ! [X3: vEBT_VEBT] :
% 8.83/4.97 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ treeList ) )
% 8.83/4.97 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_8__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,axiom,
% 8.83/4.97 ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).
% 8.83/4.97
% 8.83/4.97 thf(fact_9__092_060open_062treeList_A_B_Ahigh_A_Iif_Ax_A_060_Ami_Athen_Ami_Aelse_Ax_J_A_Ideg_Adiv_A2_J_A_092_060in_062_Aset_AtreeList_092_060close_062,axiom,
% 8.83/4.97 member_VEBT_VEBT @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( set_VEBT_VEBT2 @ treeList ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_10_nat__add__left__cancel__le,axiom,
% 8.83/4.97 ! [K: nat,M: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 8.83/4.97 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_11_add__le__cancel__left,axiom,
% 8.83/4.97 ! [C: real,A: real,B: real] :
% 8.83/4.97 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 8.83/4.97 = ( ord_less_eq_real @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_12_add__le__cancel__left,axiom,
% 8.83/4.97 ! [C: rat,A: rat,B: rat] :
% 8.83/4.97 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 8.83/4.97 = ( ord_less_eq_rat @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_13_add__le__cancel__left,axiom,
% 8.83/4.97 ! [C: nat,A: nat,B: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 8.83/4.97 = ( ord_less_eq_nat @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_14_add__le__cancel__left,axiom,
% 8.83/4.97 ! [C: int,A: int,B: int] :
% 8.83/4.97 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 8.83/4.97 = ( ord_less_eq_int @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_15_add__le__cancel__right,axiom,
% 8.83/4.97 ! [A: real,C: real,B: real] :
% 8.83/4.97 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 8.83/4.97 = ( ord_less_eq_real @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_16_add__le__cancel__right,axiom,
% 8.83/4.97 ! [A: rat,C: rat,B: rat] :
% 8.83/4.97 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 8.83/4.97 = ( ord_less_eq_rat @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_17_add__le__cancel__right,axiom,
% 8.83/4.97 ! [A: nat,C: nat,B: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 8.83/4.97 = ( ord_less_eq_nat @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_18_add__le__cancel__right,axiom,
% 8.83/4.97 ! [A: int,C: int,B: int] :
% 8.83/4.97 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 8.83/4.97 = ( ord_less_eq_int @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_19__C5_Ohyps_C_I4_J,axiom,
% 8.83/4.97 ( deg
% 8.83/4.97 = ( plus_plus_nat @ na @ m ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_20_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
% 8.83/4.97 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 8.83/4.97 ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 8.83/4.97 = one_one_nat ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_21_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
% 8.83/4.97 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 8.83/4.97 ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 8.83/4.97 = one_one_nat ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_22_VEBT_Oinject_I1_J,axiom,
% 8.83/4.97 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
% 8.83/4.97 ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 8.83/4.97 = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
% 8.83/4.97 = ( ( X11 = Y11 )
% 8.83/4.97 & ( X12 = Y12 )
% 8.83/4.97 & ( X13 = Y13 )
% 8.83/4.97 & ( X14 = Y14 ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_23_option_Oinject,axiom,
% 8.83/4.97 ! [X22: product_prod_nat_nat,Y2: product_prod_nat_nat] :
% 8.83/4.97 ( ( ( some_P7363390416028606310at_nat @ X22 )
% 8.83/4.97 = ( some_P7363390416028606310at_nat @ Y2 ) )
% 8.83/4.97 = ( X22 = Y2 ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_24_option_Oinject,axiom,
% 8.83/4.97 ! [X22: nat,Y2: nat] :
% 8.83/4.97 ( ( ( some_nat @ X22 )
% 8.83/4.97 = ( some_nat @ Y2 ) )
% 8.83/4.97 = ( X22 = Y2 ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_25_option_Oinject,axiom,
% 8.83/4.97 ! [X22: num,Y2: num] :
% 8.83/4.97 ( ( ( some_num @ X22 )
% 8.83/4.97 = ( some_num @ Y2 ) )
% 8.83/4.97 = ( X22 = Y2 ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_26_add__right__cancel,axiom,
% 8.83/4.97 ! [B: real,A: real,C: real] :
% 8.83/4.97 ( ( ( plus_plus_real @ B @ A )
% 8.83/4.97 = ( plus_plus_real @ C @ A ) )
% 8.83/4.97 = ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_27_add__right__cancel,axiom,
% 8.83/4.97 ! [B: rat,A: rat,C: rat] :
% 8.83/4.97 ( ( ( plus_plus_rat @ B @ A )
% 8.83/4.97 = ( plus_plus_rat @ C @ A ) )
% 8.83/4.97 = ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_28_add__right__cancel,axiom,
% 8.83/4.97 ! [B: nat,A: nat,C: nat] :
% 8.83/4.97 ( ( ( plus_plus_nat @ B @ A )
% 8.83/4.97 = ( plus_plus_nat @ C @ A ) )
% 8.83/4.97 = ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_29_add__right__cancel,axiom,
% 8.83/4.97 ! [B: int,A: int,C: int] :
% 8.83/4.97 ( ( ( plus_plus_int @ B @ A )
% 8.83/4.97 = ( plus_plus_int @ C @ A ) )
% 8.83/4.97 = ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_30_add__left__cancel,axiom,
% 8.83/4.97 ! [A: real,B: real,C: real] :
% 8.83/4.97 ( ( ( plus_plus_real @ A @ B )
% 8.83/4.97 = ( plus_plus_real @ A @ C ) )
% 8.83/4.97 = ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_31_add__left__cancel,axiom,
% 8.83/4.97 ! [A: rat,B: rat,C: rat] :
% 8.83/4.97 ( ( ( plus_plus_rat @ A @ B )
% 8.83/4.97 = ( plus_plus_rat @ A @ C ) )
% 8.83/4.97 = ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_32_add__left__cancel,axiom,
% 8.83/4.97 ! [A: nat,B: nat,C: nat] :
% 8.83/4.97 ( ( ( plus_plus_nat @ A @ B )
% 8.83/4.97 = ( plus_plus_nat @ A @ C ) )
% 8.83/4.97 = ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_33_add__left__cancel,axiom,
% 8.83/4.97 ! [A: int,B: int,C: int] :
% 8.83/4.97 ( ( ( plus_plus_int @ A @ B )
% 8.83/4.97 = ( plus_plus_int @ A @ C ) )
% 8.83/4.97 = ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_34_add__less__cancel__right,axiom,
% 8.83/4.97 ! [A: real,C: real,B: real] :
% 8.83/4.97 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 8.83/4.97 = ( ord_less_real @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_35_add__less__cancel__right,axiom,
% 8.83/4.97 ! [A: rat,C: rat,B: rat] :
% 8.83/4.97 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 8.83/4.97 = ( ord_less_rat @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_36_add__less__cancel__right,axiom,
% 8.83/4.97 ! [A: nat,C: nat,B: nat] :
% 8.83/4.97 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 8.83/4.97 = ( ord_less_nat @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_37_add__less__cancel__right,axiom,
% 8.83/4.97 ! [A: int,C: int,B: int] :
% 8.83/4.97 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 8.83/4.97 = ( ord_less_int @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_38_add__less__cancel__left,axiom,
% 8.83/4.97 ! [C: real,A: real,B: real] :
% 8.83/4.97 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 8.83/4.97 = ( ord_less_real @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_39_add__less__cancel__left,axiom,
% 8.83/4.97 ! [C: rat,A: rat,B: rat] :
% 8.83/4.97 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 8.83/4.97 = ( ord_less_rat @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_40_add__less__cancel__left,axiom,
% 8.83/4.97 ! [C: nat,A: nat,B: nat] :
% 8.83/4.97 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 8.83/4.97 = ( ord_less_nat @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_41_add__less__cancel__left,axiom,
% 8.83/4.97 ! [C: int,A: int,B: int] :
% 8.83/4.97 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 8.83/4.97 = ( ord_less_int @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_42_nat__add__left__cancel__less,axiom,
% 8.83/4.97 ! [K: nat,M: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 8.83/4.97 = ( ord_less_nat @ M @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_43_lesseq__shift,axiom,
% 8.83/4.97 ( ord_less_eq_nat
% 8.83/4.97 = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_44__C5_Ohyps_C_I3_J,axiom,
% 8.83/4.97 ( m
% 8.83/4.97 = ( suc @ na ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_45_False,axiom,
% 8.83/4.97 ~ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_46__C5_Ohyps_C_I8_J,axiom,
% 8.83/4.97 ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_47__C5_Ohyps_C_I5_J,axiom,
% 8.83/4.97 ! [I: nat] :
% 8.83/4.97 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 8.83/4.97 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ X4 ) )
% 8.83/4.97 = ( vEBT_V8194947554948674370ptions @ summary @ I ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_48__C5_Ohyps_C_I1_J,axiom,
% 8.83/4.97 vEBT_invar_vebt @ summary @ m ).
% 8.83/4.97
% 8.83/4.97 thf(fact_49_True,axiom,
% 8.83/4.97 ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) )
% 8.83/4.97 & ~ ( ( xa = mi )
% 8.83/4.97 | ( xa = ma ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_50__C5_OIH_C_I1_J,axiom,
% 8.83/4.97 ! [X3: vEBT_VEBT] :
% 8.83/4.97 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ treeList ) )
% 8.83/4.97 => ( ( vEBT_invar_vebt @ X3 @ na )
% 8.83/4.97 & ! [Xa: nat] : ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ X3 @ Xa ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ X3 ) ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_51_nat__neq__iff,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( M != N )
% 8.83/4.97 = ( ( ord_less_nat @ M @ N )
% 8.83/4.97 | ( ord_less_nat @ N @ M ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_52_less__not__refl,axiom,
% 8.83/4.97 ! [N: nat] :
% 8.83/4.97 ~ ( ord_less_nat @ N @ N ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_53_less__not__refl2,axiom,
% 8.83/4.97 ! [N: nat,M: nat] :
% 8.83/4.97 ( ( ord_less_nat @ N @ M )
% 8.83/4.97 => ( M != N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_54_less__not__refl3,axiom,
% 8.83/4.97 ! [S: nat,T: nat] :
% 8.83/4.97 ( ( ord_less_nat @ S @ T )
% 8.83/4.97 => ( S != T ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_55_less__irrefl__nat,axiom,
% 8.83/4.97 ! [N: nat] :
% 8.83/4.97 ~ ( ord_less_nat @ N @ N ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_56_nat__less__induct,axiom,
% 8.83/4.97 ! [P: nat > $o,N: nat] :
% 8.83/4.97 ( ! [N2: nat] :
% 8.83/4.97 ( ! [M2: nat] :
% 8.83/4.97 ( ( ord_less_nat @ M2 @ N2 )
% 8.83/4.97 => ( P @ M2 ) )
% 8.83/4.97 => ( P @ N2 ) )
% 8.83/4.97 => ( P @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_57_infinite__descent,axiom,
% 8.83/4.97 ! [P: nat > $o,N: nat] :
% 8.83/4.97 ( ! [N2: nat] :
% 8.83/4.97 ( ~ ( P @ N2 )
% 8.83/4.97 => ? [M2: nat] :
% 8.83/4.97 ( ( ord_less_nat @ M2 @ N2 )
% 8.83/4.97 & ~ ( P @ M2 ) ) )
% 8.83/4.97 => ( P @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_58_linorder__neqE__nat,axiom,
% 8.83/4.97 ! [X2: nat,Y3: nat] :
% 8.83/4.97 ( ( X2 != Y3 )
% 8.83/4.97 => ( ~ ( ord_less_nat @ X2 @ Y3 )
% 8.83/4.97 => ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_59_add__less__imp__less__right,axiom,
% 8.83/4.97 ! [A: real,C: real,B: real] :
% 8.83/4.97 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 8.83/4.97 => ( ord_less_real @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_60_add__less__imp__less__right,axiom,
% 8.83/4.97 ! [A: rat,C: rat,B: rat] :
% 8.83/4.97 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 8.83/4.97 => ( ord_less_rat @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_61_add__less__imp__less__right,axiom,
% 8.83/4.97 ! [A: nat,C: nat,B: nat] :
% 8.83/4.97 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 8.83/4.97 => ( ord_less_nat @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_62_add__less__imp__less__right,axiom,
% 8.83/4.97 ! [A: int,C: int,B: int] :
% 8.83/4.97 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 8.83/4.97 => ( ord_less_int @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_63_add__less__imp__less__left,axiom,
% 8.83/4.97 ! [C: real,A: real,B: real] :
% 8.83/4.97 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 8.83/4.97 => ( ord_less_real @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_64_add__less__imp__less__left,axiom,
% 8.83/4.97 ! [C: rat,A: rat,B: rat] :
% 8.83/4.97 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 8.83/4.97 => ( ord_less_rat @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_65_add__less__imp__less__left,axiom,
% 8.83/4.97 ! [C: nat,A: nat,B: nat] :
% 8.83/4.97 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 8.83/4.97 => ( ord_less_nat @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_66_add__less__imp__less__left,axiom,
% 8.83/4.97 ! [C: int,A: int,B: int] :
% 8.83/4.97 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 8.83/4.97 => ( ord_less_int @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_67_mem__Collect__eq,axiom,
% 8.83/4.97 ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
% 8.83/4.97 ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
% 8.83/4.97 = ( P @ A ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_68_mem__Collect__eq,axiom,
% 8.83/4.97 ! [A: complex,P: complex > $o] :
% 8.83/4.97 ( ( member_complex @ A @ ( collect_complex @ P ) )
% 8.83/4.97 = ( P @ A ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_69_mem__Collect__eq,axiom,
% 8.83/4.97 ! [A: real,P: real > $o] :
% 8.83/4.97 ( ( member_real @ A @ ( collect_real @ P ) )
% 8.83/4.97 = ( P @ A ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_70_mem__Collect__eq,axiom,
% 8.83/4.97 ! [A: list_nat,P: list_nat > $o] :
% 8.83/4.97 ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
% 8.83/4.97 = ( P @ A ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_71_mem__Collect__eq,axiom,
% 8.83/4.97 ! [A: nat,P: nat > $o] :
% 8.83/4.97 ( ( member_nat @ A @ ( collect_nat @ P ) )
% 8.83/4.97 = ( P @ A ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_72_mem__Collect__eq,axiom,
% 8.83/4.97 ! [A: int,P: int > $o] :
% 8.83/4.97 ( ( member_int @ A @ ( collect_int @ P ) )
% 8.83/4.97 = ( P @ A ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_73_Collect__mem__eq,axiom,
% 8.83/4.97 ! [A2: set_Pr1261947904930325089at_nat] :
% 8.83/4.97 ( ( collec3392354462482085612at_nat
% 8.83/4.97 @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A2 ) )
% 8.83/4.97 = A2 ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_74_Collect__mem__eq,axiom,
% 8.83/4.97 ! [A2: set_complex] :
% 8.83/4.97 ( ( collect_complex
% 8.83/4.97 @ ^ [X: complex] : ( member_complex @ X @ A2 ) )
% 8.83/4.97 = A2 ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_75_Collect__mem__eq,axiom,
% 8.83/4.97 ! [A2: set_real] :
% 8.83/4.97 ( ( collect_real
% 8.83/4.97 @ ^ [X: real] : ( member_real @ X @ A2 ) )
% 8.83/4.97 = A2 ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_76_Collect__mem__eq,axiom,
% 8.83/4.97 ! [A2: set_list_nat] :
% 8.83/4.97 ( ( collect_list_nat
% 8.83/4.97 @ ^ [X: list_nat] : ( member_list_nat @ X @ A2 ) )
% 8.83/4.97 = A2 ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_77_Collect__mem__eq,axiom,
% 8.83/4.97 ! [A2: set_nat] :
% 8.83/4.97 ( ( collect_nat
% 8.83/4.97 @ ^ [X: nat] : ( member_nat @ X @ A2 ) )
% 8.83/4.97 = A2 ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_78_Collect__mem__eq,axiom,
% 8.83/4.97 ! [A2: set_int] :
% 8.83/4.97 ( ( collect_int
% 8.83/4.97 @ ^ [X: int] : ( member_int @ X @ A2 ) )
% 8.83/4.97 = A2 ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_79_Collect__cong,axiom,
% 8.83/4.97 ! [P: complex > $o,Q: complex > $o] :
% 8.83/4.97 ( ! [X5: complex] :
% 8.83/4.97 ( ( P @ X5 )
% 8.83/4.97 = ( Q @ X5 ) )
% 8.83/4.97 => ( ( collect_complex @ P )
% 8.83/4.97 = ( collect_complex @ Q ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_80_Collect__cong,axiom,
% 8.83/4.97 ! [P: real > $o,Q: real > $o] :
% 8.83/4.97 ( ! [X5: real] :
% 8.83/4.97 ( ( P @ X5 )
% 8.83/4.97 = ( Q @ X5 ) )
% 8.83/4.97 => ( ( collect_real @ P )
% 8.83/4.97 = ( collect_real @ Q ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_81_Collect__cong,axiom,
% 8.83/4.97 ! [P: list_nat > $o,Q: list_nat > $o] :
% 8.83/4.97 ( ! [X5: list_nat] :
% 8.83/4.97 ( ( P @ X5 )
% 8.83/4.97 = ( Q @ X5 ) )
% 8.83/4.97 => ( ( collect_list_nat @ P )
% 8.83/4.97 = ( collect_list_nat @ Q ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_82_Collect__cong,axiom,
% 8.83/4.97 ! [P: nat > $o,Q: nat > $o] :
% 8.83/4.97 ( ! [X5: nat] :
% 8.83/4.97 ( ( P @ X5 )
% 8.83/4.97 = ( Q @ X5 ) )
% 8.83/4.97 => ( ( collect_nat @ P )
% 8.83/4.97 = ( collect_nat @ Q ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_83_Collect__cong,axiom,
% 8.83/4.97 ! [P: int > $o,Q: int > $o] :
% 8.83/4.97 ( ! [X5: int] :
% 8.83/4.97 ( ( P @ X5 )
% 8.83/4.97 = ( Q @ X5 ) )
% 8.83/4.97 => ( ( collect_int @ P )
% 8.83/4.97 = ( collect_int @ Q ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_84_add__strict__right__mono,axiom,
% 8.83/4.97 ! [A: real,B: real,C: real] :
% 8.83/4.97 ( ( ord_less_real @ A @ B )
% 8.83/4.97 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_85_add__strict__right__mono,axiom,
% 8.83/4.97 ! [A: rat,B: rat,C: rat] :
% 8.83/4.97 ( ( ord_less_rat @ A @ B )
% 8.83/4.97 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_86_add__strict__right__mono,axiom,
% 8.83/4.97 ! [A: nat,B: nat,C: nat] :
% 8.83/4.97 ( ( ord_less_nat @ A @ B )
% 8.83/4.97 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_87_add__strict__right__mono,axiom,
% 8.83/4.97 ! [A: int,B: int,C: int] :
% 8.83/4.97 ( ( ord_less_int @ A @ B )
% 8.83/4.97 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_88_add__strict__left__mono,axiom,
% 8.83/4.97 ! [A: real,B: real,C: real] :
% 8.83/4.97 ( ( ord_less_real @ A @ B )
% 8.83/4.97 => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_89_add__strict__left__mono,axiom,
% 8.83/4.97 ! [A: rat,B: rat,C: rat] :
% 8.83/4.97 ( ( ord_less_rat @ A @ B )
% 8.83/4.97 => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_90_add__strict__left__mono,axiom,
% 8.83/4.97 ! [A: nat,B: nat,C: nat] :
% 8.83/4.97 ( ( ord_less_nat @ A @ B )
% 8.83/4.97 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_91_add__strict__left__mono,axiom,
% 8.83/4.97 ! [A: int,B: int,C: int] :
% 8.83/4.97 ( ( ord_less_int @ A @ B )
% 8.83/4.97 => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_92_add__strict__mono,axiom,
% 8.83/4.97 ! [A: real,B: real,C: real,D: real] :
% 8.83/4.97 ( ( ord_less_real @ A @ B )
% 8.83/4.97 => ( ( ord_less_real @ C @ D )
% 8.83/4.97 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_93_add__strict__mono,axiom,
% 8.83/4.97 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.83/4.97 ( ( ord_less_rat @ A @ B )
% 8.83/4.97 => ( ( ord_less_rat @ C @ D )
% 8.83/4.97 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_94_add__strict__mono,axiom,
% 8.83/4.97 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.83/4.97 ( ( ord_less_nat @ A @ B )
% 8.83/4.97 => ( ( ord_less_nat @ C @ D )
% 8.83/4.97 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_95_add__strict__mono,axiom,
% 8.83/4.97 ! [A: int,B: int,C: int,D: int] :
% 8.83/4.97 ( ( ord_less_int @ A @ B )
% 8.83/4.97 => ( ( ord_less_int @ C @ D )
% 8.83/4.97 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_96_add__mono__thms__linordered__field_I1_J,axiom,
% 8.83/4.97 ! [I2: real,J: real,K: real,L: real] :
% 8.83/4.97 ( ( ( ord_less_real @ I2 @ J )
% 8.83/4.97 & ( K = L ) )
% 8.83/4.97 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_97_add__mono__thms__linordered__field_I1_J,axiom,
% 8.83/4.97 ! [I2: rat,J: rat,K: rat,L: rat] :
% 8.83/4.97 ( ( ( ord_less_rat @ I2 @ J )
% 8.83/4.97 & ( K = L ) )
% 8.83/4.97 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_98_add__mono__thms__linordered__field_I1_J,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,K: nat,L: nat] :
% 8.83/4.97 ( ( ( ord_less_nat @ I2 @ J )
% 8.83/4.97 & ( K = L ) )
% 8.83/4.97 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_99_add__mono__thms__linordered__field_I1_J,axiom,
% 8.83/4.97 ! [I2: int,J: int,K: int,L: int] :
% 8.83/4.97 ( ( ( ord_less_int @ I2 @ J )
% 8.83/4.97 & ( K = L ) )
% 8.83/4.97 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_100_add__mono__thms__linordered__field_I2_J,axiom,
% 8.83/4.97 ! [I2: real,J: real,K: real,L: real] :
% 8.83/4.97 ( ( ( I2 = J )
% 8.83/4.97 & ( ord_less_real @ K @ L ) )
% 8.83/4.97 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_101_add__mono__thms__linordered__field_I2_J,axiom,
% 8.83/4.97 ! [I2: rat,J: rat,K: rat,L: rat] :
% 8.83/4.97 ( ( ( I2 = J )
% 8.83/4.97 & ( ord_less_rat @ K @ L ) )
% 8.83/4.97 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_102_add__mono__thms__linordered__field_I2_J,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,K: nat,L: nat] :
% 8.83/4.97 ( ( ( I2 = J )
% 8.83/4.97 & ( ord_less_nat @ K @ L ) )
% 8.83/4.97 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_103_add__mono__thms__linordered__field_I2_J,axiom,
% 8.83/4.97 ! [I2: int,J: int,K: int,L: int] :
% 8.83/4.97 ( ( ( I2 = J )
% 8.83/4.97 & ( ord_less_int @ K @ L ) )
% 8.83/4.97 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_104_add__mono__thms__linordered__field_I5_J,axiom,
% 8.83/4.97 ! [I2: real,J: real,K: real,L: real] :
% 8.83/4.97 ( ( ( ord_less_real @ I2 @ J )
% 8.83/4.97 & ( ord_less_real @ K @ L ) )
% 8.83/4.97 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_105_add__mono__thms__linordered__field_I5_J,axiom,
% 8.83/4.97 ! [I2: rat,J: rat,K: rat,L: rat] :
% 8.83/4.97 ( ( ( ord_less_rat @ I2 @ J )
% 8.83/4.97 & ( ord_less_rat @ K @ L ) )
% 8.83/4.97 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_106_add__mono__thms__linordered__field_I5_J,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,K: nat,L: nat] :
% 8.83/4.97 ( ( ( ord_less_nat @ I2 @ J )
% 8.83/4.97 & ( ord_less_nat @ K @ L ) )
% 8.83/4.97 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_107_add__mono__thms__linordered__field_I5_J,axiom,
% 8.83/4.97 ! [I2: int,J: int,K: int,L: int] :
% 8.83/4.97 ( ( ( ord_less_int @ I2 @ J )
% 8.83/4.97 & ( ord_less_int @ K @ L ) )
% 8.83/4.97 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_108_less__mono__imp__le__mono,axiom,
% 8.83/4.97 ! [F: nat > nat,I2: nat,J: nat] :
% 8.83/4.97 ( ! [I3: nat,J2: nat] :
% 8.83/4.97 ( ( ord_less_nat @ I3 @ J2 )
% 8.83/4.97 => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
% 8.83/4.97 => ( ( ord_less_eq_nat @ I2 @ J )
% 8.83/4.97 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_109_le__neq__implies__less,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ M @ N )
% 8.83/4.97 => ( ( M != N )
% 8.83/4.97 => ( ord_less_nat @ M @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_110_less__or__eq__imp__le,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( ( ord_less_nat @ M @ N )
% 8.83/4.97 | ( M = N ) )
% 8.83/4.97 => ( ord_less_eq_nat @ M @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_111_le__eq__less__or__eq,axiom,
% 8.83/4.97 ( ord_less_eq_nat
% 8.83/4.97 = ( ^ [M3: nat,N3: nat] :
% 8.83/4.97 ( ( ord_less_nat @ M3 @ N3 )
% 8.83/4.97 | ( M3 = N3 ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_112_less__imp__le__nat,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_nat @ M @ N )
% 8.83/4.97 => ( ord_less_eq_nat @ M @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_113_nat__less__le,axiom,
% 8.83/4.97 ( ord_less_nat
% 8.83/4.97 = ( ^ [M3: nat,N3: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ M3 @ N3 )
% 8.83/4.97 & ( M3 != N3 ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_114_less__add__eq__less,axiom,
% 8.83/4.97 ! [K: nat,L: nat,M: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_nat @ K @ L )
% 8.83/4.97 => ( ( ( plus_plus_nat @ M @ L )
% 8.83/4.97 = ( plus_plus_nat @ K @ N ) )
% 8.83/4.97 => ( ord_less_nat @ M @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_115_trans__less__add2,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,M: nat] :
% 8.83/4.97 ( ( ord_less_nat @ I2 @ J )
% 8.83/4.97 => ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_116_trans__less__add1,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,M: nat] :
% 8.83/4.97 ( ( ord_less_nat @ I2 @ J )
% 8.83/4.97 => ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_117_add__less__mono1,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,K: nat] :
% 8.83/4.97 ( ( ord_less_nat @ I2 @ J )
% 8.83/4.97 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_118_not__add__less2,axiom,
% 8.83/4.97 ! [J: nat,I2: nat] :
% 8.83/4.97 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_119_not__add__less1,axiom,
% 8.83/4.97 ! [I2: nat,J: nat] :
% 8.83/4.97 ~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_120_add__less__mono,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,K: nat,L: nat] :
% 8.83/4.97 ( ( ord_less_nat @ I2 @ J )
% 8.83/4.97 => ( ( ord_less_nat @ K @ L )
% 8.83/4.97 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_121_add__lessD1,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,K: nat] :
% 8.83/4.97 ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
% 8.83/4.97 => ( ord_less_nat @ I2 @ K ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_122_one__reorient,axiom,
% 8.83/4.97 ! [X2: complex] :
% 8.83/4.97 ( ( one_one_complex = X2 )
% 8.83/4.97 = ( X2 = one_one_complex ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_123_one__reorient,axiom,
% 8.83/4.97 ! [X2: real] :
% 8.83/4.97 ( ( one_one_real = X2 )
% 8.83/4.97 = ( X2 = one_one_real ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_124_one__reorient,axiom,
% 8.83/4.97 ! [X2: rat] :
% 8.83/4.97 ( ( one_one_rat = X2 )
% 8.83/4.97 = ( X2 = one_one_rat ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_125_one__reorient,axiom,
% 8.83/4.97 ! [X2: nat] :
% 8.83/4.97 ( ( one_one_nat = X2 )
% 8.83/4.97 = ( X2 = one_one_nat ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_126_one__reorient,axiom,
% 8.83/4.97 ! [X2: int] :
% 8.83/4.97 ( ( one_one_int = X2 )
% 8.83/4.97 = ( X2 = one_one_int ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_127_add__right__imp__eq,axiom,
% 8.83/4.97 ! [B: real,A: real,C: real] :
% 8.83/4.97 ( ( ( plus_plus_real @ B @ A )
% 8.83/4.97 = ( plus_plus_real @ C @ A ) )
% 8.83/4.97 => ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_128_add__right__imp__eq,axiom,
% 8.83/4.97 ! [B: rat,A: rat,C: rat] :
% 8.83/4.97 ( ( ( plus_plus_rat @ B @ A )
% 8.83/4.97 = ( plus_plus_rat @ C @ A ) )
% 8.83/4.97 => ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_129_add__right__imp__eq,axiom,
% 8.83/4.97 ! [B: nat,A: nat,C: nat] :
% 8.83/4.97 ( ( ( plus_plus_nat @ B @ A )
% 8.83/4.97 = ( plus_plus_nat @ C @ A ) )
% 8.83/4.97 => ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_130_add__right__imp__eq,axiom,
% 8.83/4.97 ! [B: int,A: int,C: int] :
% 8.83/4.97 ( ( ( plus_plus_int @ B @ A )
% 8.83/4.97 = ( plus_plus_int @ C @ A ) )
% 8.83/4.97 => ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_131_add__left__imp__eq,axiom,
% 8.83/4.97 ! [A: real,B: real,C: real] :
% 8.83/4.97 ( ( ( plus_plus_real @ A @ B )
% 8.83/4.97 = ( plus_plus_real @ A @ C ) )
% 8.83/4.97 => ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_132_add__left__imp__eq,axiom,
% 8.83/4.97 ! [A: rat,B: rat,C: rat] :
% 8.83/4.97 ( ( ( plus_plus_rat @ A @ B )
% 8.83/4.97 = ( plus_plus_rat @ A @ C ) )
% 8.83/4.97 => ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_133_add__left__imp__eq,axiom,
% 8.83/4.97 ! [A: nat,B: nat,C: nat] :
% 8.83/4.97 ( ( ( plus_plus_nat @ A @ B )
% 8.83/4.97 = ( plus_plus_nat @ A @ C ) )
% 8.83/4.97 => ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_134_add__left__imp__eq,axiom,
% 8.83/4.97 ! [A: int,B: int,C: int] :
% 8.83/4.97 ( ( ( plus_plus_int @ A @ B )
% 8.83/4.97 = ( plus_plus_int @ A @ C ) )
% 8.83/4.97 => ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_135_add_Oleft__commute,axiom,
% 8.83/4.97 ! [B: real,A: real,C: real] :
% 8.83/4.97 ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 8.83/4.97 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_136_add_Oleft__commute,axiom,
% 8.83/4.97 ! [B: rat,A: rat,C: rat] :
% 8.83/4.97 ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
% 8.83/4.97 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_137_add_Oleft__commute,axiom,
% 8.83/4.97 ! [B: nat,A: nat,C: nat] :
% 8.83/4.97 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 8.83/4.97 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_138_add_Oleft__commute,axiom,
% 8.83/4.97 ! [B: int,A: int,C: int] :
% 8.83/4.97 ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 8.83/4.97 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_139_add_Ocommute,axiom,
% 8.83/4.97 ( plus_plus_real
% 8.83/4.97 = ( ^ [A3: real,B2: real] : ( plus_plus_real @ B2 @ A3 ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_140_add_Ocommute,axiom,
% 8.83/4.97 ( plus_plus_rat
% 8.83/4.97 = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ B2 @ A3 ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_141_add_Ocommute,axiom,
% 8.83/4.97 ( plus_plus_nat
% 8.83/4.97 = ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_142_add_Ocommute,axiom,
% 8.83/4.97 ( plus_plus_int
% 8.83/4.97 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_143_add_Oright__cancel,axiom,
% 8.83/4.97 ! [B: real,A: real,C: real] :
% 8.83/4.97 ( ( ( plus_plus_real @ B @ A )
% 8.83/4.97 = ( plus_plus_real @ C @ A ) )
% 8.83/4.97 = ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_144_add_Oright__cancel,axiom,
% 8.83/4.97 ! [B: rat,A: rat,C: rat] :
% 8.83/4.97 ( ( ( plus_plus_rat @ B @ A )
% 8.83/4.97 = ( plus_plus_rat @ C @ A ) )
% 8.83/4.97 = ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_145_add_Oright__cancel,axiom,
% 8.83/4.97 ! [B: int,A: int,C: int] :
% 8.83/4.97 ( ( ( plus_plus_int @ B @ A )
% 8.83/4.97 = ( plus_plus_int @ C @ A ) )
% 8.83/4.97 = ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_146_add_Oleft__cancel,axiom,
% 8.83/4.97 ! [A: real,B: real,C: real] :
% 8.83/4.97 ( ( ( plus_plus_real @ A @ B )
% 8.83/4.97 = ( plus_plus_real @ A @ C ) )
% 8.83/4.97 = ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_147_add_Oleft__cancel,axiom,
% 8.83/4.97 ! [A: rat,B: rat,C: rat] :
% 8.83/4.97 ( ( ( plus_plus_rat @ A @ B )
% 8.83/4.97 = ( plus_plus_rat @ A @ C ) )
% 8.83/4.97 = ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_148_add_Oleft__cancel,axiom,
% 8.83/4.97 ! [A: int,B: int,C: int] :
% 8.83/4.97 ( ( ( plus_plus_int @ A @ B )
% 8.83/4.97 = ( plus_plus_int @ A @ C ) )
% 8.83/4.97 = ( B = C ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_149_add_Oassoc,axiom,
% 8.83/4.97 ! [A: real,B: real,C: real] :
% 8.83/4.97 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 8.83/4.97 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_150_add_Oassoc,axiom,
% 8.83/4.97 ! [A: rat,B: rat,C: rat] :
% 8.83/4.97 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 8.83/4.97 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_151_add_Oassoc,axiom,
% 8.83/4.97 ! [A: nat,B: nat,C: nat] :
% 8.83/4.97 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 8.83/4.97 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_152_add_Oassoc,axiom,
% 8.83/4.97 ! [A: int,B: int,C: int] :
% 8.83/4.97 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.83/4.97 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_153_group__cancel_Oadd2,axiom,
% 8.83/4.97 ! [B3: real,K: real,B: real,A: real] :
% 8.83/4.97 ( ( B3
% 8.83/4.97 = ( plus_plus_real @ K @ B ) )
% 8.83/4.97 => ( ( plus_plus_real @ A @ B3 )
% 8.83/4.97 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_154_group__cancel_Oadd2,axiom,
% 8.83/4.97 ! [B3: rat,K: rat,B: rat,A: rat] :
% 8.83/4.97 ( ( B3
% 8.83/4.97 = ( plus_plus_rat @ K @ B ) )
% 8.83/4.97 => ( ( plus_plus_rat @ A @ B3 )
% 8.83/4.97 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_155_group__cancel_Oadd2,axiom,
% 8.83/4.97 ! [B3: nat,K: nat,B: nat,A: nat] :
% 8.83/4.97 ( ( B3
% 8.83/4.97 = ( plus_plus_nat @ K @ B ) )
% 8.83/4.97 => ( ( plus_plus_nat @ A @ B3 )
% 8.83/4.97 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_156_group__cancel_Oadd2,axiom,
% 8.83/4.97 ! [B3: int,K: int,B: int,A: int] :
% 8.83/4.97 ( ( B3
% 8.83/4.97 = ( plus_plus_int @ K @ B ) )
% 8.83/4.97 => ( ( plus_plus_int @ A @ B3 )
% 8.83/4.97 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_157_group__cancel_Oadd1,axiom,
% 8.83/4.97 ! [A2: real,K: real,A: real,B: real] :
% 8.83/4.97 ( ( A2
% 8.83/4.97 = ( plus_plus_real @ K @ A ) )
% 8.83/4.97 => ( ( plus_plus_real @ A2 @ B )
% 8.83/4.97 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_158_group__cancel_Oadd1,axiom,
% 8.83/4.97 ! [A2: rat,K: rat,A: rat,B: rat] :
% 8.83/4.97 ( ( A2
% 8.83/4.97 = ( plus_plus_rat @ K @ A ) )
% 8.83/4.97 => ( ( plus_plus_rat @ A2 @ B )
% 8.83/4.97 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_159_group__cancel_Oadd1,axiom,
% 8.83/4.97 ! [A2: nat,K: nat,A: nat,B: nat] :
% 8.83/4.97 ( ( A2
% 8.83/4.97 = ( plus_plus_nat @ K @ A ) )
% 8.83/4.97 => ( ( plus_plus_nat @ A2 @ B )
% 8.83/4.97 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_160_group__cancel_Oadd1,axiom,
% 8.83/4.97 ! [A2: int,K: int,A: int,B: int] :
% 8.83/4.97 ( ( A2
% 8.83/4.97 = ( plus_plus_int @ K @ A ) )
% 8.83/4.97 => ( ( plus_plus_int @ A2 @ B )
% 8.83/4.97 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_161_add__mono__thms__linordered__semiring_I4_J,axiom,
% 8.83/4.97 ! [I2: real,J: real,K: real,L: real] :
% 8.83/4.97 ( ( ( I2 = J )
% 8.83/4.97 & ( K = L ) )
% 8.83/4.97 => ( ( plus_plus_real @ I2 @ K )
% 8.83/4.97 = ( plus_plus_real @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_162_add__mono__thms__linordered__semiring_I4_J,axiom,
% 8.83/4.97 ! [I2: rat,J: rat,K: rat,L: rat] :
% 8.83/4.97 ( ( ( I2 = J )
% 8.83/4.97 & ( K = L ) )
% 8.83/4.97 => ( ( plus_plus_rat @ I2 @ K )
% 8.83/4.97 = ( plus_plus_rat @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_163_add__mono__thms__linordered__semiring_I4_J,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,K: nat,L: nat] :
% 8.83/4.97 ( ( ( I2 = J )
% 8.83/4.97 & ( K = L ) )
% 8.83/4.97 => ( ( plus_plus_nat @ I2 @ K )
% 8.83/4.97 = ( plus_plus_nat @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_164_add__mono__thms__linordered__semiring_I4_J,axiom,
% 8.83/4.97 ! [I2: int,J: int,K: int,L: int] :
% 8.83/4.97 ( ( ( I2 = J )
% 8.83/4.97 & ( K = L ) )
% 8.83/4.97 => ( ( plus_plus_int @ I2 @ K )
% 8.83/4.97 = ( plus_plus_int @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_165_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 8.83/4.97 ! [A: real,B: real,C: real] :
% 8.83/4.97 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 8.83/4.97 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_166_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 8.83/4.97 ! [A: rat,B: rat,C: rat] :
% 8.83/4.97 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 8.83/4.97 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_167_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 8.83/4.97 ! [A: nat,B: nat,C: nat] :
% 8.83/4.97 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 8.83/4.97 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_168_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 8.83/4.97 ! [A: int,B: int,C: int] :
% 8.83/4.97 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.83/4.97 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_169_Nat_Oex__has__greatest__nat,axiom,
% 8.83/4.97 ! [P: nat > $o,K: nat,B: nat] :
% 8.83/4.97 ( ( P @ K )
% 8.83/4.97 => ( ! [Y4: nat] :
% 8.83/4.97 ( ( P @ Y4 )
% 8.83/4.97 => ( ord_less_eq_nat @ Y4 @ B ) )
% 8.83/4.97 => ? [X5: nat] :
% 8.83/4.97 ( ( P @ X5 )
% 8.83/4.97 & ! [Y5: nat] :
% 8.83/4.97 ( ( P @ Y5 )
% 8.83/4.97 => ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_170_nat__le__linear,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ M @ N )
% 8.83/4.97 | ( ord_less_eq_nat @ N @ M ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_171_le__antisym,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ M @ N )
% 8.83/4.97 => ( ( ord_less_eq_nat @ N @ M )
% 8.83/4.97 => ( M = N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_172_eq__imp__le,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( M = N )
% 8.83/4.97 => ( ord_less_eq_nat @ M @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_173_le__trans,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,K: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ I2 @ J )
% 8.83/4.97 => ( ( ord_less_eq_nat @ J @ K )
% 8.83/4.97 => ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_174_le__refl,axiom,
% 8.83/4.97 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_175_add__less__le__mono,axiom,
% 8.83/4.97 ! [A: real,B: real,C: real,D: real] :
% 8.83/4.97 ( ( ord_less_real @ A @ B )
% 8.83/4.97 => ( ( ord_less_eq_real @ C @ D )
% 8.83/4.97 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_176_add__less__le__mono,axiom,
% 8.83/4.97 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.83/4.97 ( ( ord_less_rat @ A @ B )
% 8.83/4.97 => ( ( ord_less_eq_rat @ C @ D )
% 8.83/4.97 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_177_add__less__le__mono,axiom,
% 8.83/4.97 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.83/4.97 ( ( ord_less_nat @ A @ B )
% 8.83/4.97 => ( ( ord_less_eq_nat @ C @ D )
% 8.83/4.97 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_178_add__less__le__mono,axiom,
% 8.83/4.97 ! [A: int,B: int,C: int,D: int] :
% 8.83/4.97 ( ( ord_less_int @ A @ B )
% 8.83/4.97 => ( ( ord_less_eq_int @ C @ D )
% 8.83/4.97 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_179_add__le__less__mono,axiom,
% 8.83/4.97 ! [A: real,B: real,C: real,D: real] :
% 8.83/4.97 ( ( ord_less_eq_real @ A @ B )
% 8.83/4.97 => ( ( ord_less_real @ C @ D )
% 8.83/4.97 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_180_add__le__less__mono,axiom,
% 8.83/4.97 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.83/4.97 ( ( ord_less_eq_rat @ A @ B )
% 8.83/4.97 => ( ( ord_less_rat @ C @ D )
% 8.83/4.97 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_181_add__le__less__mono,axiom,
% 8.83/4.97 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ A @ B )
% 8.83/4.97 => ( ( ord_less_nat @ C @ D )
% 8.83/4.97 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_182_add__le__less__mono,axiom,
% 8.83/4.97 ! [A: int,B: int,C: int,D: int] :
% 8.83/4.97 ( ( ord_less_eq_int @ A @ B )
% 8.83/4.97 => ( ( ord_less_int @ C @ D )
% 8.83/4.97 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_183_add__mono__thms__linordered__field_I3_J,axiom,
% 8.83/4.97 ! [I2: real,J: real,K: real,L: real] :
% 8.83/4.97 ( ( ( ord_less_real @ I2 @ J )
% 8.83/4.97 & ( ord_less_eq_real @ K @ L ) )
% 8.83/4.97 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_184_add__mono__thms__linordered__field_I3_J,axiom,
% 8.83/4.97 ! [I2: rat,J: rat,K: rat,L: rat] :
% 8.83/4.97 ( ( ( ord_less_rat @ I2 @ J )
% 8.83/4.97 & ( ord_less_eq_rat @ K @ L ) )
% 8.83/4.97 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_185_add__mono__thms__linordered__field_I3_J,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,K: nat,L: nat] :
% 8.83/4.97 ( ( ( ord_less_nat @ I2 @ J )
% 8.83/4.97 & ( ord_less_eq_nat @ K @ L ) )
% 8.83/4.97 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_186_add__mono__thms__linordered__field_I3_J,axiom,
% 8.83/4.97 ! [I2: int,J: int,K: int,L: int] :
% 8.83/4.97 ( ( ( ord_less_int @ I2 @ J )
% 8.83/4.97 & ( ord_less_eq_int @ K @ L ) )
% 8.83/4.97 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_187_add__mono__thms__linordered__field_I4_J,axiom,
% 8.83/4.97 ! [I2: real,J: real,K: real,L: real] :
% 8.83/4.97 ( ( ( ord_less_eq_real @ I2 @ J )
% 8.83/4.97 & ( ord_less_real @ K @ L ) )
% 8.83/4.97 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_188_add__mono__thms__linordered__field_I4_J,axiom,
% 8.83/4.97 ! [I2: rat,J: rat,K: rat,L: rat] :
% 8.83/4.97 ( ( ( ord_less_eq_rat @ I2 @ J )
% 8.83/4.97 & ( ord_less_rat @ K @ L ) )
% 8.83/4.97 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_189_add__mono__thms__linordered__field_I4_J,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,K: nat,L: nat] :
% 8.83/4.97 ( ( ( ord_less_eq_nat @ I2 @ J )
% 8.83/4.97 & ( ord_less_nat @ K @ L ) )
% 8.83/4.97 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_190_add__mono__thms__linordered__field_I4_J,axiom,
% 8.83/4.97 ! [I2: int,J: int,K: int,L: int] :
% 8.83/4.97 ( ( ( ord_less_eq_int @ I2 @ J )
% 8.83/4.97 & ( ord_less_int @ K @ L ) )
% 8.83/4.97 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_191_mono__nat__linear__lb,axiom,
% 8.83/4.97 ! [F: nat > nat,M: nat,K: nat] :
% 8.83/4.97 ( ! [M4: nat,N2: nat] :
% 8.83/4.97 ( ( ord_less_nat @ M4 @ N2 )
% 8.83/4.97 => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
% 8.83/4.97 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_192_add__le__imp__le__right,axiom,
% 8.83/4.97 ! [A: real,C: real,B: real] :
% 8.83/4.97 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 8.83/4.97 => ( ord_less_eq_real @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_193_add__le__imp__le__right,axiom,
% 8.83/4.97 ! [A: rat,C: rat,B: rat] :
% 8.83/4.97 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 8.83/4.97 => ( ord_less_eq_rat @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_194_add__le__imp__le__right,axiom,
% 8.83/4.97 ! [A: nat,C: nat,B: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 8.83/4.97 => ( ord_less_eq_nat @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_195_add__le__imp__le__right,axiom,
% 8.83/4.97 ! [A: int,C: int,B: int] :
% 8.83/4.97 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 8.83/4.97 => ( ord_less_eq_int @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_196_add__le__imp__le__left,axiom,
% 8.83/4.97 ! [C: real,A: real,B: real] :
% 8.83/4.97 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 8.83/4.97 => ( ord_less_eq_real @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_197_add__le__imp__le__left,axiom,
% 8.83/4.97 ! [C: rat,A: rat,B: rat] :
% 8.83/4.97 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 8.83/4.97 => ( ord_less_eq_rat @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_198_add__le__imp__le__left,axiom,
% 8.83/4.97 ! [C: nat,A: nat,B: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 8.83/4.97 => ( ord_less_eq_nat @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_199_add__le__imp__le__left,axiom,
% 8.83/4.97 ! [C: int,A: int,B: int] :
% 8.83/4.97 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 8.83/4.97 => ( ord_less_eq_int @ A @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_200_le__iff__add,axiom,
% 8.83/4.97 ( ord_less_eq_nat
% 8.83/4.97 = ( ^ [A3: nat,B2: nat] :
% 8.83/4.97 ? [C2: nat] :
% 8.83/4.97 ( B2
% 8.83/4.97 = ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_201_add__right__mono,axiom,
% 8.83/4.97 ! [A: real,B: real,C: real] :
% 8.83/4.97 ( ( ord_less_eq_real @ A @ B )
% 8.83/4.97 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_202_add__right__mono,axiom,
% 8.83/4.97 ! [A: rat,B: rat,C: rat] :
% 8.83/4.97 ( ( ord_less_eq_rat @ A @ B )
% 8.83/4.97 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_203_add__right__mono,axiom,
% 8.83/4.97 ! [A: nat,B: nat,C: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ A @ B )
% 8.83/4.97 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_204_add__right__mono,axiom,
% 8.83/4.97 ! [A: int,B: int,C: int] :
% 8.83/4.97 ( ( ord_less_eq_int @ A @ B )
% 8.83/4.97 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_205_less__eqE,axiom,
% 8.83/4.97 ! [A: nat,B: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ A @ B )
% 8.83/4.97 => ~ ! [C3: nat] :
% 8.83/4.97 ( B
% 8.83/4.97 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_206_add__left__mono,axiom,
% 8.83/4.97 ! [A: real,B: real,C: real] :
% 8.83/4.97 ( ( ord_less_eq_real @ A @ B )
% 8.83/4.97 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_207_add__left__mono,axiom,
% 8.83/4.97 ! [A: rat,B: rat,C: rat] :
% 8.83/4.97 ( ( ord_less_eq_rat @ A @ B )
% 8.83/4.97 => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_208_add__left__mono,axiom,
% 8.83/4.97 ! [A: nat,B: nat,C: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ A @ B )
% 8.83/4.97 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_209_add__left__mono,axiom,
% 8.83/4.97 ! [A: int,B: int,C: int] :
% 8.83/4.97 ( ( ord_less_eq_int @ A @ B )
% 8.83/4.97 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_210_add__mono,axiom,
% 8.83/4.97 ! [A: real,B: real,C: real,D: real] :
% 8.83/4.97 ( ( ord_less_eq_real @ A @ B )
% 8.83/4.97 => ( ( ord_less_eq_real @ C @ D )
% 8.83/4.97 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_211_add__mono,axiom,
% 8.83/4.97 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.83/4.97 ( ( ord_less_eq_rat @ A @ B )
% 8.83/4.97 => ( ( ord_less_eq_rat @ C @ D )
% 8.83/4.97 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_212_add__mono,axiom,
% 8.83/4.97 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ A @ B )
% 8.83/4.97 => ( ( ord_less_eq_nat @ C @ D )
% 8.83/4.97 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_213_add__mono,axiom,
% 8.83/4.97 ! [A: int,B: int,C: int,D: int] :
% 8.83/4.97 ( ( ord_less_eq_int @ A @ B )
% 8.83/4.97 => ( ( ord_less_eq_int @ C @ D )
% 8.83/4.97 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_214_add__mono__thms__linordered__semiring_I1_J,axiom,
% 8.83/4.97 ! [I2: real,J: real,K: real,L: real] :
% 8.83/4.97 ( ( ( ord_less_eq_real @ I2 @ J )
% 8.83/4.97 & ( ord_less_eq_real @ K @ L ) )
% 8.83/4.97 => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_215_add__mono__thms__linordered__semiring_I1_J,axiom,
% 8.83/4.97 ! [I2: rat,J: rat,K: rat,L: rat] :
% 8.83/4.97 ( ( ( ord_less_eq_rat @ I2 @ J )
% 8.83/4.97 & ( ord_less_eq_rat @ K @ L ) )
% 8.83/4.97 => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_216_add__mono__thms__linordered__semiring_I1_J,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,K: nat,L: nat] :
% 8.83/4.97 ( ( ( ord_less_eq_nat @ I2 @ J )
% 8.83/4.97 & ( ord_less_eq_nat @ K @ L ) )
% 8.83/4.97 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_217_add__mono__thms__linordered__semiring_I1_J,axiom,
% 8.83/4.97 ! [I2: int,J: int,K: int,L: int] :
% 8.83/4.97 ( ( ( ord_less_eq_int @ I2 @ J )
% 8.83/4.97 & ( ord_less_eq_int @ K @ L ) )
% 8.83/4.97 => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_218_add__mono__thms__linordered__semiring_I2_J,axiom,
% 8.83/4.97 ! [I2: real,J: real,K: real,L: real] :
% 8.83/4.97 ( ( ( I2 = J )
% 8.83/4.97 & ( ord_less_eq_real @ K @ L ) )
% 8.83/4.97 => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_219_add__mono__thms__linordered__semiring_I2_J,axiom,
% 8.83/4.97 ! [I2: rat,J: rat,K: rat,L: rat] :
% 8.83/4.97 ( ( ( I2 = J )
% 8.83/4.97 & ( ord_less_eq_rat @ K @ L ) )
% 8.83/4.97 => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_220_add__mono__thms__linordered__semiring_I2_J,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,K: nat,L: nat] :
% 8.83/4.97 ( ( ( I2 = J )
% 8.83/4.97 & ( ord_less_eq_nat @ K @ L ) )
% 8.83/4.97 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_221_add__mono__thms__linordered__semiring_I2_J,axiom,
% 8.83/4.97 ! [I2: int,J: int,K: int,L: int] :
% 8.83/4.97 ( ( ( I2 = J )
% 8.83/4.97 & ( ord_less_eq_int @ K @ L ) )
% 8.83/4.97 => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_222_add__mono__thms__linordered__semiring_I3_J,axiom,
% 8.83/4.97 ! [I2: real,J: real,K: real,L: real] :
% 8.83/4.97 ( ( ( ord_less_eq_real @ I2 @ J )
% 8.83/4.97 & ( K = L ) )
% 8.83/4.97 => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_223_add__mono__thms__linordered__semiring_I3_J,axiom,
% 8.83/4.97 ! [I2: rat,J: rat,K: rat,L: rat] :
% 8.83/4.97 ( ( ( ord_less_eq_rat @ I2 @ J )
% 8.83/4.97 & ( K = L ) )
% 8.83/4.97 => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_224_add__mono__thms__linordered__semiring_I3_J,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,K: nat,L: nat] :
% 8.83/4.97 ( ( ( ord_less_eq_nat @ I2 @ J )
% 8.83/4.97 & ( K = L ) )
% 8.83/4.97 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_225_add__mono__thms__linordered__semiring_I3_J,axiom,
% 8.83/4.97 ! [I2: int,J: int,K: int,L: int] :
% 8.83/4.97 ( ( ( ord_less_eq_int @ I2 @ J )
% 8.83/4.97 & ( K = L ) )
% 8.83/4.97 => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_226_nat__le__iff__add,axiom,
% 8.83/4.97 ( ord_less_eq_nat
% 8.83/4.97 = ( ^ [M3: nat,N3: nat] :
% 8.83/4.97 ? [K2: nat] :
% 8.83/4.97 ( N3
% 8.83/4.97 = ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_227_trans__le__add2,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,M: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ I2 @ J )
% 8.83/4.97 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_228_trans__le__add1,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,M: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ I2 @ J )
% 8.83/4.97 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_229_add__le__mono1,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,K: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ I2 @ J )
% 8.83/4.97 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_230_add__le__mono,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,K: nat,L: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ I2 @ J )
% 8.83/4.97 => ( ( ord_less_eq_nat @ K @ L )
% 8.83/4.97 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_231_le__Suc__ex,axiom,
% 8.83/4.97 ! [K: nat,L: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ K @ L )
% 8.83/4.97 => ? [N2: nat] :
% 8.83/4.97 ( L
% 8.83/4.97 = ( plus_plus_nat @ K @ N2 ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_232_add__leD2,axiom,
% 8.83/4.97 ! [M: nat,K: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 8.83/4.97 => ( ord_less_eq_nat @ K @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_233_add__leD1,axiom,
% 8.83/4.97 ! [M: nat,K: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 8.83/4.97 => ( ord_less_eq_nat @ M @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_234_le__add2,axiom,
% 8.83/4.97 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_235_le__add1,axiom,
% 8.83/4.97 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_236_add__leE,axiom,
% 8.83/4.97 ! [M: nat,K: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 8.83/4.97 => ~ ( ( ord_less_eq_nat @ M @ N )
% 8.83/4.97 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_237__C2_C,axiom,
% 8.83/4.97 ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 8.83/4.97 = ( plus_plus_nat @ one_one_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_238_insert__simp__mima,axiom,
% 8.83/4.97 ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 8.83/4.97 ( ( ( X2 = Mi )
% 8.83/4.97 | ( X2 = Ma ) )
% 8.83/4.97 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 8.83/4.97 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 8.83/4.97 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_239_add__self__div__2,axiom,
% 8.83/4.97 ! [M: nat] :
% 8.83/4.97 ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.97 = M ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_240_succ__min,axiom,
% 8.83/4.97 ! [Deg: nat,X2: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 8.83/4.97 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 8.83/4.97 => ( ( ord_less_nat @ X2 @ Mi )
% 8.83/4.97 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 8.83/4.97 = ( some_nat @ Mi ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_241_pred__max,axiom,
% 8.83/4.97 ! [Deg: nat,Ma: nat,X2: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 8.83/4.97 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 8.83/4.97 => ( ( ord_less_nat @ Ma @ X2 )
% 8.83/4.97 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 8.83/4.97 = ( some_nat @ Ma ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_242_one__add__one,axiom,
% 8.83/4.97 ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
% 8.83/4.97 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_243_one__add__one,axiom,
% 8.83/4.97 ( ( plus_plus_real @ one_one_real @ one_one_real )
% 8.83/4.97 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_244_one__add__one,axiom,
% 8.83/4.97 ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
% 8.83/4.97 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_245_one__add__one,axiom,
% 8.83/4.97 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 8.83/4.97 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_246_one__add__one,axiom,
% 8.83/4.97 ( ( plus_plus_int @ one_one_int @ one_one_int )
% 8.83/4.97 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_247_one__plus__numeral,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
% 8.83/4.97 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_248_one__plus__numeral,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 8.83/4.97 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_249_one__plus__numeral,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 8.83/4.97 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_250_one__plus__numeral,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 8.83/4.97 = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_251_one__plus__numeral,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 8.83/4.97 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_252_numeral__plus__one,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
% 8.83/4.97 = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_253_numeral__plus__one,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 8.83/4.97 = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_254_numeral__plus__one,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 8.83/4.97 = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_255_numeral__plus__one,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 8.83/4.97 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_256_numeral__plus__one,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 8.83/4.97 = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_257_one__less__numeral__iff,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 8.83/4.97 = ( ord_less_num @ one @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_258_one__less__numeral__iff,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 8.83/4.97 = ( ord_less_num @ one @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_259_one__less__numeral__iff,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 8.83/4.97 = ( ord_less_num @ one @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_260_one__less__numeral__iff,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 8.83/4.97 = ( ord_less_num @ one @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_261_numeral__le__one__iff,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 8.83/4.97 = ( ord_less_eq_num @ N @ one ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_262_numeral__le__one__iff,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 8.83/4.97 = ( ord_less_eq_num @ N @ one ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_263_numeral__le__one__iff,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 8.83/4.97 = ( ord_less_eq_num @ N @ one ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_264_numeral__le__one__iff,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 8.83/4.97 = ( ord_less_eq_num @ N @ one ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_265_one__eq__numeral__iff,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( one_one_complex
% 8.83/4.97 = ( numera6690914467698888265omplex @ N ) )
% 8.83/4.97 = ( one = N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_266_one__eq__numeral__iff,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( one_one_real
% 8.83/4.97 = ( numeral_numeral_real @ N ) )
% 8.83/4.97 = ( one = N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_267_one__eq__numeral__iff,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( one_one_rat
% 8.83/4.97 = ( numeral_numeral_rat @ N ) )
% 8.83/4.97 = ( one = N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_268_one__eq__numeral__iff,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( one_one_nat
% 8.83/4.97 = ( numeral_numeral_nat @ N ) )
% 8.83/4.97 = ( one = N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_269_one__eq__numeral__iff,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( one_one_int
% 8.83/4.97 = ( numeral_numeral_int @ N ) )
% 8.83/4.97 = ( one = N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_270_numeral__eq__one__iff,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( ( numera6690914467698888265omplex @ N )
% 8.83/4.97 = one_one_complex )
% 8.83/4.97 = ( N = one ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_271_numeral__eq__one__iff,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( ( numeral_numeral_real @ N )
% 8.83/4.97 = one_one_real )
% 8.83/4.97 = ( N = one ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_272_numeral__eq__one__iff,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( ( numeral_numeral_rat @ N )
% 8.83/4.97 = one_one_rat )
% 8.83/4.97 = ( N = one ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_273_numeral__eq__one__iff,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( ( numeral_numeral_nat @ N )
% 8.83/4.97 = one_one_nat )
% 8.83/4.97 = ( N = one ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_274_numeral__eq__one__iff,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( ( numeral_numeral_int @ N )
% 8.83/4.97 = one_one_int )
% 8.83/4.97 = ( N = one ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_275_even__odd__cases,axiom,
% 8.83/4.97 ! [X2: nat] :
% 8.83/4.97 ( ! [N2: nat] :
% 8.83/4.97 ( X2
% 8.83/4.97 != ( plus_plus_nat @ N2 @ N2 ) )
% 8.83/4.97 => ~ ! [N2: nat] :
% 8.83/4.97 ( X2
% 8.83/4.97 != ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_276_deg__deg__n,axiom,
% 8.83/4.97 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 8.83/4.97 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
% 8.83/4.97 => ( Deg = N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_277_not__min__Null__member,axiom,
% 8.83/4.97 ! [T: vEBT_VEBT] :
% 8.83/4.97 ( ~ ( vEBT_VEBT_minNull @ T )
% 8.83/4.97 => ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_278_deg__SUcn__Node,axiom,
% 8.83/4.97 ! [Tree: vEBT_VEBT,N: nat] :
% 8.83/4.97 ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
% 8.83/4.97 => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 8.83/4.97 ( Tree
% 8.83/4.97 = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList2 @ S2 ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_279_inthall,axiom,
% 8.83/4.97 ! [Xs2: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,N: nat] :
% 8.83/4.97 ( ! [X5: product_prod_nat_nat] :
% 8.83/4.97 ( ( member8440522571783428010at_nat @ X5 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
% 8.83/4.97 => ( P @ X5 ) )
% 8.83/4.97 => ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
% 8.83/4.97 => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ N ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_280_inthall,axiom,
% 8.83/4.97 ! [Xs2: list_complex,P: complex > $o,N: nat] :
% 8.83/4.97 ( ! [X5: complex] :
% 8.83/4.97 ( ( member_complex @ X5 @ ( set_complex2 @ Xs2 ) )
% 8.83/4.97 => ( P @ X5 ) )
% 8.83/4.97 => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 8.83/4.97 => ( P @ ( nth_complex @ Xs2 @ N ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_281_inthall,axiom,
% 8.83/4.97 ! [Xs2: list_real,P: real > $o,N: nat] :
% 8.83/4.97 ( ! [X5: real] :
% 8.83/4.97 ( ( member_real @ X5 @ ( set_real2 @ Xs2 ) )
% 8.83/4.97 => ( P @ X5 ) )
% 8.83/4.97 => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 8.83/4.97 => ( P @ ( nth_real @ Xs2 @ N ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_282_inthall,axiom,
% 8.83/4.97 ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
% 8.83/4.97 ( ! [X5: vEBT_VEBT] :
% 8.83/4.97 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 8.83/4.97 => ( P @ X5 ) )
% 8.83/4.97 => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 8.83/4.97 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_283_inthall,axiom,
% 8.83/4.97 ! [Xs2: list_o,P: $o > $o,N: nat] :
% 8.83/4.97 ( ! [X5: $o] :
% 8.83/4.97 ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
% 8.83/4.97 => ( P @ X5 ) )
% 8.83/4.97 => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 8.83/4.97 => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_284_inthall,axiom,
% 8.83/4.97 ! [Xs2: list_nat,P: nat > $o,N: nat] :
% 8.83/4.97 ( ! [X5: nat] :
% 8.83/4.97 ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
% 8.83/4.97 => ( P @ X5 ) )
% 8.83/4.97 => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 8.83/4.97 => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_285_inthall,axiom,
% 8.83/4.97 ! [Xs2: list_int,P: int > $o,N: nat] :
% 8.83/4.97 ( ! [X5: int] :
% 8.83/4.97 ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
% 8.83/4.97 => ( P @ X5 ) )
% 8.83/4.97 => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 8.83/4.97 => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_286_power__shift,axiom,
% 8.83/4.97 ! [X2: nat,Y3: nat,Z: nat] :
% 8.83/4.97 ( ( ( power_power_nat @ X2 @ Y3 )
% 8.83/4.97 = Z )
% 8.83/4.97 = ( ( vEBT_VEBT_power @ ( some_nat @ X2 ) @ ( some_nat @ Y3 ) )
% 8.83/4.97 = ( some_nat @ Z ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_287_bit__split__inv,axiom,
% 8.83/4.97 ! [X2: nat,D: nat] :
% 8.83/4.97 ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X2 @ D ) @ ( vEBT_VEBT_low @ X2 @ D ) @ D )
% 8.83/4.97 = X2 ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_288_numeral__eq__iff,axiom,
% 8.83/4.97 ! [M: num,N: num] :
% 8.83/4.97 ( ( ( numera6690914467698888265omplex @ M )
% 8.83/4.97 = ( numera6690914467698888265omplex @ N ) )
% 8.83/4.97 = ( M = N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_289_numeral__eq__iff,axiom,
% 8.83/4.97 ! [M: num,N: num] :
% 8.83/4.97 ( ( ( numeral_numeral_real @ M )
% 8.83/4.97 = ( numeral_numeral_real @ N ) )
% 8.83/4.97 = ( M = N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_290_numeral__eq__iff,axiom,
% 8.83/4.97 ! [M: num,N: num] :
% 8.83/4.97 ( ( ( numeral_numeral_rat @ M )
% 8.83/4.97 = ( numeral_numeral_rat @ N ) )
% 8.83/4.97 = ( M = N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_291_numeral__eq__iff,axiom,
% 8.83/4.97 ! [M: num,N: num] :
% 8.83/4.97 ( ( ( numeral_numeral_nat @ M )
% 8.83/4.97 = ( numeral_numeral_nat @ N ) )
% 8.83/4.97 = ( M = N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_292_numeral__eq__iff,axiom,
% 8.83/4.97 ! [M: num,N: num] :
% 8.83/4.97 ( ( ( numeral_numeral_int @ M )
% 8.83/4.97 = ( numeral_numeral_int @ N ) )
% 8.83/4.97 = ( M = N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_293_nat_Oinject,axiom,
% 8.83/4.97 ! [X22: nat,Y2: nat] :
% 8.83/4.97 ( ( ( suc @ X22 )
% 8.83/4.97 = ( suc @ Y2 ) )
% 8.83/4.97 = ( X22 = Y2 ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_294_old_Onat_Oinject,axiom,
% 8.83/4.97 ! [Nat: nat,Nat2: nat] :
% 8.83/4.97 ( ( ( suc @ Nat )
% 8.83/4.97 = ( suc @ Nat2 ) )
% 8.83/4.97 = ( Nat = Nat2 ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_295__C5_Ohyps_C_I2_J,axiom,
% 8.83/4.97 ( ( size_s6755466524823107622T_VEBT @ treeList )
% 8.83/4.97 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_296_pow__sum,axiom,
% 8.83/4.97 ! [A: nat,B: nat] :
% 8.83/4.97 ( ( 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 ) )
% 8.83/4.97 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_297_high__def,axiom,
% 8.83/4.97 ( vEBT_VEBT_high
% 8.83/4.97 = ( ^ [X: nat,N3: nat] : ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_298_mi__eq__ma__no__ch,axiom,
% 8.83/4.97 ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 8.83/4.97 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
% 8.83/4.97 => ( ( Mi = Ma )
% 8.83/4.97 => ( ! [X3: vEBT_VEBT] :
% 8.83/4.97 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.83/4.97 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
% 8.83/4.97 & ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_299_high__bound__aux,axiom,
% 8.83/4.97 ! [Ma: nat,N: nat,M: nat] :
% 8.83/4.97 ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 8.83/4.97 => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_300_Suc__less__eq,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 8.83/4.97 = ( ord_less_nat @ M @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_301_Suc__mono,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_nat @ M @ N )
% 8.83/4.97 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_302_lessI,axiom,
% 8.83/4.97 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_303_Suc__le__mono,axiom,
% 8.83/4.97 ! [N: nat,M: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 8.83/4.97 = ( ord_less_eq_nat @ N @ M ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_304_add__Suc__right,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 8.83/4.97 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_305_helpypredd,axiom,
% 8.83/4.97 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 8.83/4.97 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.97 => ( ( ( vEBT_vebt_pred @ T @ X2 )
% 8.83/4.97 = ( some_nat @ Y3 ) )
% 8.83/4.97 => ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_306_helpyd,axiom,
% 8.83/4.97 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 8.83/4.97 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.97 => ( ( ( vEBT_vebt_succ @ T @ X2 )
% 8.83/4.97 = ( some_nat @ Y3 ) )
% 8.83/4.97 => ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_307_valid__insert__both__member__options__pres,axiom,
% 8.83/4.97 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 8.83/4.97 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.97 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.83/4.97 => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.83/4.97 => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
% 8.83/4.97 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y3 ) @ X2 ) ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_308_valid__insert__both__member__options__add,axiom,
% 8.83/4.97 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 8.83/4.97 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.97 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.83/4.97 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X2 ) @ X2 ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_309_set__n__deg__not__0,axiom,
% 8.83/4.97 ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
% 8.83/4.97 ( ! [X5: vEBT_VEBT] :
% 8.83/4.97 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.83/4.97 => ( vEBT_invar_vebt @ X5 @ N ) )
% 8.83/4.97 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 8.83/4.97 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.83/4.97 => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_310_mi__ma__2__deg,axiom,
% 8.83/4.97 ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 8.83/4.97 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 8.83/4.97 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 8.83/4.97 & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_311__C5_Ohyps_C_I9_J,axiom,
% 8.83/4.97 ( ( mi != ma )
% 8.83/4.97 => ! [I: nat] :
% 8.83/4.97 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 8.83/4.97 => ( ( ( ( vEBT_VEBT_high @ ma @ na )
% 8.83/4.97 = I )
% 8.83/4.97 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
% 8.83/4.97 & ! [X3: nat] :
% 8.83/4.97 ( ( ( ( vEBT_VEBT_high @ X3 @ na )
% 8.83/4.97 = I )
% 8.83/4.97 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ ( vEBT_VEBT_low @ X3 @ na ) ) )
% 8.83/4.97 => ( ( ord_less_nat @ mi @ X3 )
% 8.83/4.97 & ( ord_less_eq_nat @ X3 @ ma ) ) ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_312_numeral__plus__numeral,axiom,
% 8.83/4.97 ! [M: num,N: num] :
% 8.83/4.97 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 8.83/4.97 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_313_numeral__plus__numeral,axiom,
% 8.83/4.97 ! [M: num,N: num] :
% 8.83/4.97 ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 8.83/4.97 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_314_numeral__plus__numeral,axiom,
% 8.83/4.97 ! [M: num,N: num] :
% 8.83/4.97 ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 8.83/4.97 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_315_numeral__plus__numeral,axiom,
% 8.83/4.97 ! [M: num,N: num] :
% 8.83/4.97 ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 8.83/4.97 = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_316_numeral__plus__numeral,axiom,
% 8.83/4.97 ! [M: num,N: num] :
% 8.83/4.97 ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 8.83/4.97 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_317_add__numeral__left,axiom,
% 8.83/4.97 ! [V: num,W: num,Z: complex] :
% 8.83/4.97 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 8.83/4.97 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_318_add__numeral__left,axiom,
% 8.83/4.97 ! [V: num,W: num,Z: real] :
% 8.83/4.97 ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 8.83/4.97 = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_319_add__numeral__left,axiom,
% 8.83/4.97 ! [V: num,W: num,Z: rat] :
% 8.83/4.97 ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 8.83/4.97 = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_320_add__numeral__left,axiom,
% 8.83/4.97 ! [V: num,W: num,Z: nat] :
% 8.83/4.97 ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 8.83/4.97 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_321_add__numeral__left,axiom,
% 8.83/4.97 ! [V: num,W: num,Z: int] :
% 8.83/4.97 ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 8.83/4.97 = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_322_both__member__options__ding,axiom,
% 8.83/4.97 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X2: nat] :
% 8.83/4.97 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
% 8.83/4.97 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 8.83/4.97 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.83/4.97 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X2 ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_323_numeral__le__iff,axiom,
% 8.83/4.97 ! [M: num,N: num] :
% 8.83/4.97 ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 8.83/4.97 = ( ord_less_eq_num @ M @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_324_numeral__le__iff,axiom,
% 8.83/4.97 ! [M: num,N: num] :
% 8.83/4.97 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 8.83/4.97 = ( ord_less_eq_num @ M @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_325_numeral__le__iff,axiom,
% 8.83/4.97 ! [M: num,N: num] :
% 8.83/4.97 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 8.83/4.97 = ( ord_less_eq_num @ M @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_326_numeral__le__iff,axiom,
% 8.83/4.97 ! [M: num,N: num] :
% 8.83/4.97 ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 8.83/4.97 = ( ord_less_eq_num @ M @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_327_numeral__less__iff,axiom,
% 8.83/4.97 ! [M: num,N: num] :
% 8.83/4.97 ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 8.83/4.97 = ( ord_less_num @ M @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_328_numeral__less__iff,axiom,
% 8.83/4.97 ! [M: num,N: num] :
% 8.83/4.97 ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 8.83/4.97 = ( ord_less_num @ M @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_329_numeral__less__iff,axiom,
% 8.83/4.97 ! [M: num,N: num] :
% 8.83/4.97 ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 8.83/4.97 = ( ord_less_num @ M @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_330_numeral__less__iff,axiom,
% 8.83/4.97 ! [M: num,N: num] :
% 8.83/4.97 ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 8.83/4.97 = ( ord_less_num @ M @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_331_both__member__options__from__complete__tree__to__child,axiom,
% 8.83/4.97 ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 8.83/4.97 => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 8.83/4.97 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.83/4.97 | ( X2 = Mi )
% 8.83/4.97 | ( X2 = Ma ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_332_both__member__options__from__chilf__to__complete__tree,axiom,
% 8.83/4.97 ! [X2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 8.83/4.97 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 8.83/4.97 => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 8.83/4.97 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.83/4.97 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_333_Suc__numeral,axiom,
% 8.83/4.97 ! [N: num] :
% 8.83/4.97 ( ( suc @ ( numeral_numeral_nat @ N ) )
% 8.83/4.97 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_334_add__2__eq__Suc_H,axiom,
% 8.83/4.97 ! [N: nat] :
% 8.83/4.97 ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.97 = ( suc @ ( suc @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_335_add__2__eq__Suc,axiom,
% 8.83/4.97 ! [N: nat] :
% 8.83/4.97 ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.83/4.97 = ( suc @ ( suc @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_336_Suc__1,axiom,
% 8.83/4.97 ( ( suc @ one_one_nat )
% 8.83/4.97 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_337_div2__Suc__Suc,axiom,
% 8.83/4.97 ! [M: nat] :
% 8.83/4.97 ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.97 = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_338__C1_C,axiom,
% 8.83/4.97 ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 8.83/4.97 = ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_339__C0_C,axiom,
% 8.83/4.97 ( ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) )
% 8.83/4.97 & ~ ( ( xa = mi )
% 8.83/4.97 | ( xa = ma ) ) )
% 8.83/4.97 => ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 8.83/4.97 = ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) ) ) )
% 8.83/4.97 & ( ~ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) )
% 8.83/4.97 & ~ ( ( xa = mi )
% 8.83/4.97 | ( xa = ma ) ) )
% 8.83/4.97 => ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 8.83/4.97 = one_one_nat ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_340_Suc__inject,axiom,
% 8.83/4.97 ! [X2: nat,Y3: nat] :
% 8.83/4.97 ( ( ( suc @ X2 )
% 8.83/4.97 = ( suc @ Y3 ) )
% 8.83/4.97 => ( X2 = Y3 ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_341_n__not__Suc__n,axiom,
% 8.83/4.97 ! [N: nat] :
% 8.83/4.97 ( N
% 8.83/4.97 != ( suc @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_342_size__neq__size__imp__neq,axiom,
% 8.83/4.97 ! [X2: list_VEBT_VEBT,Y3: list_VEBT_VEBT] :
% 8.83/4.97 ( ( ( size_s6755466524823107622T_VEBT @ X2 )
% 8.83/4.97 != ( size_s6755466524823107622T_VEBT @ Y3 ) )
% 8.83/4.97 => ( X2 != Y3 ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_343_size__neq__size__imp__neq,axiom,
% 8.83/4.97 ! [X2: list_o,Y3: list_o] :
% 8.83/4.97 ( ( ( size_size_list_o @ X2 )
% 8.83/4.97 != ( size_size_list_o @ Y3 ) )
% 8.83/4.97 => ( X2 != Y3 ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_344_size__neq__size__imp__neq,axiom,
% 8.83/4.97 ! [X2: list_nat,Y3: list_nat] :
% 8.83/4.97 ( ( ( size_size_list_nat @ X2 )
% 8.83/4.97 != ( size_size_list_nat @ Y3 ) )
% 8.83/4.97 => ( X2 != Y3 ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_345_size__neq__size__imp__neq,axiom,
% 8.83/4.97 ! [X2: list_int,Y3: list_int] :
% 8.83/4.97 ( ( ( size_size_list_int @ X2 )
% 8.83/4.97 != ( size_size_list_int @ Y3 ) )
% 8.83/4.97 => ( X2 != Y3 ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_346_size__neq__size__imp__neq,axiom,
% 8.83/4.97 ! [X2: num,Y3: num] :
% 8.83/4.97 ( ( ( size_size_num @ X2 )
% 8.83/4.97 != ( size_size_num @ Y3 ) )
% 8.83/4.97 => ( X2 != Y3 ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_347_not__less__less__Suc__eq,axiom,
% 8.83/4.97 ! [N: nat,M: nat] :
% 8.83/4.97 ( ~ ( ord_less_nat @ N @ M )
% 8.83/4.97 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 8.83/4.97 = ( N = M ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_348_strict__inc__induct,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,P: nat > $o] :
% 8.83/4.97 ( ( ord_less_nat @ I2 @ J )
% 8.83/4.97 => ( ! [I3: nat] :
% 8.83/4.97 ( ( J
% 8.83/4.97 = ( suc @ I3 ) )
% 8.83/4.97 => ( P @ I3 ) )
% 8.83/4.97 => ( ! [I3: nat] :
% 8.83/4.97 ( ( ord_less_nat @ I3 @ J )
% 8.83/4.97 => ( ( P @ ( suc @ I3 ) )
% 8.83/4.97 => ( P @ I3 ) ) )
% 8.83/4.97 => ( P @ I2 ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_349_less__Suc__induct,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,P: nat > nat > $o] :
% 8.83/4.97 ( ( ord_less_nat @ I2 @ J )
% 8.83/4.97 => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
% 8.83/4.97 => ( ! [I3: nat,J2: nat,K3: nat] :
% 8.83/4.97 ( ( ord_less_nat @ I3 @ J2 )
% 8.83/4.97 => ( ( ord_less_nat @ J2 @ K3 )
% 8.83/4.97 => ( ( P @ I3 @ J2 )
% 8.83/4.97 => ( ( P @ J2 @ K3 )
% 8.83/4.97 => ( P @ I3 @ K3 ) ) ) ) )
% 8.83/4.97 => ( P @ I2 @ J ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_350_less__trans__Suc,axiom,
% 8.83/4.97 ! [I2: nat,J: nat,K: nat] :
% 8.83/4.97 ( ( ord_less_nat @ I2 @ J )
% 8.83/4.97 => ( ( ord_less_nat @ J @ K )
% 8.83/4.97 => ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_351_Suc__less__SucD,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 8.83/4.97 => ( ord_less_nat @ M @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_352_less__antisym,axiom,
% 8.83/4.97 ! [N: nat,M: nat] :
% 8.83/4.97 ( ~ ( ord_less_nat @ N @ M )
% 8.83/4.97 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 8.83/4.97 => ( M = N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_353_Suc__less__eq2,axiom,
% 8.83/4.97 ! [N: nat,M: nat] :
% 8.83/4.97 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 8.83/4.97 = ( ? [M5: nat] :
% 8.83/4.97 ( ( M
% 8.83/4.97 = ( suc @ M5 ) )
% 8.83/4.97 & ( ord_less_nat @ N @ M5 ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_354_All__less__Suc,axiom,
% 8.83/4.97 ! [N: nat,P: nat > $o] :
% 8.83/4.97 ( ( ! [I4: nat] :
% 8.83/4.97 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 8.83/4.97 => ( P @ I4 ) ) )
% 8.83/4.97 = ( ( P @ N )
% 8.83/4.97 & ! [I4: nat] :
% 8.83/4.97 ( ( ord_less_nat @ I4 @ N )
% 8.83/4.97 => ( P @ I4 ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_355_not__less__eq,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( ~ ( ord_less_nat @ M @ N ) )
% 8.83/4.97 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_356_less__Suc__eq,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 8.83/4.97 = ( ( ord_less_nat @ M @ N )
% 8.83/4.97 | ( M = N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_357_Ex__less__Suc,axiom,
% 8.83/4.97 ! [N: nat,P: nat > $o] :
% 8.83/4.97 ( ( ? [I4: nat] :
% 8.83/4.97 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 8.83/4.97 & ( P @ I4 ) ) )
% 8.83/4.97 = ( ( P @ N )
% 8.83/4.97 | ? [I4: nat] :
% 8.83/4.97 ( ( ord_less_nat @ I4 @ N )
% 8.83/4.97 & ( P @ I4 ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_358_less__SucI,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_nat @ M @ N )
% 8.83/4.97 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_359_less__SucE,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 8.83/4.97 => ( ~ ( ord_less_nat @ M @ N )
% 8.83/4.97 => ( M = N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_360_Suc__lessI,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_nat @ M @ N )
% 8.83/4.97 => ( ( ( suc @ M )
% 8.83/4.97 != N )
% 8.83/4.97 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_361_Suc__lessE,axiom,
% 8.83/4.97 ! [I2: nat,K: nat] :
% 8.83/4.97 ( ( ord_less_nat @ ( suc @ I2 ) @ K )
% 8.83/4.97 => ~ ! [J2: nat] :
% 8.83/4.97 ( ( ord_less_nat @ I2 @ J2 )
% 8.83/4.97 => ( K
% 8.83/4.97 != ( suc @ J2 ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_362_Suc__lessD,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 8.83/4.97 => ( ord_less_nat @ M @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_363_Nat_OlessE,axiom,
% 8.83/4.97 ! [I2: nat,K: nat] :
% 8.83/4.97 ( ( ord_less_nat @ I2 @ K )
% 8.83/4.97 => ( ( K
% 8.83/4.97 != ( suc @ I2 ) )
% 8.83/4.97 => ~ ! [J2: nat] :
% 8.83/4.97 ( ( ord_less_nat @ I2 @ J2 )
% 8.83/4.97 => ( K
% 8.83/4.97 != ( suc @ J2 ) ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_364_Suc__leD,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 8.83/4.97 => ( ord_less_eq_nat @ M @ N ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_365_le__SucE,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 8.83/4.97 => ( ~ ( ord_less_eq_nat @ M @ N )
% 8.83/4.97 => ( M
% 8.83/4.97 = ( suc @ N ) ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_366_le__SucI,axiom,
% 8.83/4.97 ! [M: nat,N: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ M @ N )
% 8.83/4.97 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 8.83/4.97
% 8.83/4.97 thf(fact_367_Suc__le__D,axiom,
% 8.83/4.97 ! [N: nat,M6: nat] :
% 8.83/4.97 ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
% 8.83/4.98 => ? [M4: nat] :
% 8.83/4.98 ( M6
% 8.83/4.98 = ( suc @ M4 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_368_le__Suc__eq,axiom,
% 8.83/4.98 ! [M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 8.83/4.98 = ( ( ord_less_eq_nat @ M @ N )
% 8.83/4.98 | ( M
% 8.83/4.98 = ( suc @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_369_Suc__n__not__le__n,axiom,
% 8.83/4.98 ! [N: nat] :
% 8.83/4.98 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_370_not__less__eq__eq,axiom,
% 8.83/4.98 ! [M: nat,N: nat] :
% 8.83/4.98 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 8.83/4.98 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_371_full__nat__induct,axiom,
% 8.83/4.98 ! [P: nat > $o,N: nat] :
% 8.83/4.98 ( ! [N2: nat] :
% 8.83/4.98 ( ! [M2: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
% 8.83/4.98 => ( P @ M2 ) )
% 8.83/4.98 => ( P @ N2 ) )
% 8.83/4.98 => ( P @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_372_nat__induct__at__least,axiom,
% 8.83/4.98 ! [M: nat,N: nat,P: nat > $o] :
% 8.83/4.98 ( ( ord_less_eq_nat @ M @ N )
% 8.83/4.98 => ( ( P @ M )
% 8.83/4.98 => ( ! [N2: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ M @ N2 )
% 8.83/4.98 => ( ( P @ N2 )
% 8.83/4.98 => ( P @ ( suc @ N2 ) ) ) )
% 8.83/4.98 => ( P @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_373_transitive__stepwise__le,axiom,
% 8.83/4.98 ! [M: nat,N: nat,R: nat > nat > $o] :
% 8.83/4.98 ( ( ord_less_eq_nat @ M @ N )
% 8.83/4.98 => ( ! [X5: nat] : ( R @ X5 @ X5 )
% 8.83/4.98 => ( ! [X5: nat,Y4: nat,Z2: nat] :
% 8.83/4.98 ( ( R @ X5 @ Y4 )
% 8.83/4.98 => ( ( R @ Y4 @ Z2 )
% 8.83/4.98 => ( R @ X5 @ Z2 ) ) )
% 8.83/4.98 => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
% 8.83/4.98 => ( R @ M @ N ) ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_374_nat__arith_Osuc1,axiom,
% 8.83/4.98 ! [A2: nat,K: nat,A: nat] :
% 8.83/4.98 ( ( A2
% 8.83/4.98 = ( plus_plus_nat @ K @ A ) )
% 8.83/4.98 => ( ( suc @ A2 )
% 8.83/4.98 = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_375_add__Suc,axiom,
% 8.83/4.98 ! [M: nat,N: nat] :
% 8.83/4.98 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 8.83/4.98 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_376_add__Suc__shift,axiom,
% 8.83/4.98 ! [M: nat,N: nat] :
% 8.83/4.98 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 8.83/4.98 = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_377_Suc__div__le__mono,axiom,
% 8.83/4.98 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_378_insertsimp_H,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat,L: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ( vEBT_VEBT_minNull @ T )
% 8.83/4.98 => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ L ) @ one_one_nat ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_379_Suc__nat__number__of__add,axiom,
% 8.83/4.98 ! [V: num,N: nat] :
% 8.83/4.98 ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
% 8.83/4.98 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_380_lift__Suc__mono__less__iff,axiom,
% 8.83/4.98 ! [F: nat > real,N: nat,M: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.83/4.98 => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 8.83/4.98 = ( ord_less_nat @ N @ M ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_381_lift__Suc__mono__less__iff,axiom,
% 8.83/4.98 ! [F: nat > rat,N: nat,M: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.83/4.98 => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
% 8.83/4.98 = ( ord_less_nat @ N @ M ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_382_lift__Suc__mono__less__iff,axiom,
% 8.83/4.98 ! [F: nat > num,N: nat,M: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.83/4.98 => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
% 8.83/4.98 = ( ord_less_nat @ N @ M ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_383_lift__Suc__mono__less__iff,axiom,
% 8.83/4.98 ! [F: nat > nat,N: nat,M: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.83/4.98 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 8.83/4.98 = ( ord_less_nat @ N @ M ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_384_lift__Suc__mono__less__iff,axiom,
% 8.83/4.98 ! [F: nat > int,N: nat,M: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.83/4.98 => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 8.83/4.98 = ( ord_less_nat @ N @ M ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_385_lift__Suc__mono__less,axiom,
% 8.83/4.98 ! [F: nat > real,N: nat,N4: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.83/4.98 => ( ( ord_less_nat @ N @ N4 )
% 8.83/4.98 => ( ord_less_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_386_lift__Suc__mono__less,axiom,
% 8.83/4.98 ! [F: nat > rat,N: nat,N4: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.83/4.98 => ( ( ord_less_nat @ N @ N4 )
% 8.83/4.98 => ( ord_less_rat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_387_lift__Suc__mono__less,axiom,
% 8.83/4.98 ! [F: nat > num,N: nat,N4: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.83/4.98 => ( ( ord_less_nat @ N @ N4 )
% 8.83/4.98 => ( ord_less_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_388_lift__Suc__mono__less,axiom,
% 8.83/4.98 ! [F: nat > nat,N: nat,N4: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.83/4.98 => ( ( ord_less_nat @ N @ N4 )
% 8.83/4.98 => ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_389_lift__Suc__mono__less,axiom,
% 8.83/4.98 ! [F: nat > int,N: nat,N4: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.83/4.98 => ( ( ord_less_nat @ N @ N4 )
% 8.83/4.98 => ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_390_lift__Suc__mono__le,axiom,
% 8.83/4.98 ! [F: nat > set_int,N: nat,N4: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.83/4.98 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.83/4.98 => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_391_lift__Suc__mono__le,axiom,
% 8.83/4.98 ! [F: nat > rat,N: nat,N4: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.83/4.98 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.83/4.98 => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_392_lift__Suc__mono__le,axiom,
% 8.83/4.98 ! [F: nat > num,N: nat,N4: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.83/4.98 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.83/4.98 => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_393_lift__Suc__mono__le,axiom,
% 8.83/4.98 ! [F: nat > nat,N: nat,N4: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.83/4.98 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.83/4.98 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_394_lift__Suc__mono__le,axiom,
% 8.83/4.98 ! [F: nat > int,N: nat,N4: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.83/4.98 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.83/4.98 => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_395_lift__Suc__antimono__le,axiom,
% 8.83/4.98 ! [F: nat > set_int,N: nat,N4: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 8.83/4.98 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.83/4.98 => ( ord_less_eq_set_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_396_lift__Suc__antimono__le,axiom,
% 8.83/4.98 ! [F: nat > rat,N: nat,N4: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 8.83/4.98 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.83/4.98 => ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_397_lift__Suc__antimono__le,axiom,
% 8.83/4.98 ! [F: nat > num,N: nat,N4: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 8.83/4.98 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.83/4.98 => ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_398_lift__Suc__antimono__le,axiom,
% 8.83/4.98 ! [F: nat > nat,N: nat,N4: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 8.83/4.98 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.83/4.98 => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_399_lift__Suc__antimono__le,axiom,
% 8.83/4.98 ! [F: nat > int,N: nat,N4: nat] :
% 8.83/4.98 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 8.83/4.98 => ( ( ord_less_eq_nat @ N @ N4 )
% 8.83/4.98 => ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_400_Suc__leI,axiom,
% 8.83/4.98 ! [M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_nat @ M @ N )
% 8.83/4.98 => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_401_Suc__le__eq,axiom,
% 8.83/4.98 ! [M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 8.83/4.98 = ( ord_less_nat @ M @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_402_dec__induct,axiom,
% 8.83/4.98 ! [I2: nat,J: nat,P: nat > $o] :
% 8.83/4.98 ( ( ord_less_eq_nat @ I2 @ J )
% 8.83/4.98 => ( ( P @ I2 )
% 8.83/4.98 => ( ! [N2: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ I2 @ N2 )
% 8.83/4.98 => ( ( ord_less_nat @ N2 @ J )
% 8.83/4.98 => ( ( P @ N2 )
% 8.83/4.98 => ( P @ ( suc @ N2 ) ) ) ) )
% 8.83/4.98 => ( P @ J ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_403_inc__induct,axiom,
% 8.83/4.98 ! [I2: nat,J: nat,P: nat > $o] :
% 8.83/4.98 ( ( ord_less_eq_nat @ I2 @ J )
% 8.83/4.98 => ( ( P @ J )
% 8.83/4.98 => ( ! [N2: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ I2 @ N2 )
% 8.83/4.98 => ( ( ord_less_nat @ N2 @ J )
% 8.83/4.98 => ( ( P @ ( suc @ N2 ) )
% 8.83/4.98 => ( P @ N2 ) ) ) )
% 8.83/4.98 => ( P @ I2 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_404_Suc__le__lessD,axiom,
% 8.83/4.98 ! [M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 8.83/4.98 => ( ord_less_nat @ M @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_405_le__less__Suc__eq,axiom,
% 8.83/4.98 ! [M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ M @ N )
% 8.83/4.98 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 8.83/4.98 = ( N = M ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_406_less__Suc__eq__le,axiom,
% 8.83/4.98 ! [M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 8.83/4.98 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_407_less__eq__Suc__le,axiom,
% 8.83/4.98 ( ord_less_nat
% 8.83/4.98 = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_408_le__imp__less__Suc,axiom,
% 8.83/4.98 ! [M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ M @ N )
% 8.83/4.98 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_409_less__natE,axiom,
% 8.83/4.98 ! [M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_nat @ M @ N )
% 8.83/4.98 => ~ ! [Q2: nat] :
% 8.83/4.98 ( N
% 8.83/4.98 != ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_410_less__add__Suc1,axiom,
% 8.83/4.98 ! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_411_less__add__Suc2,axiom,
% 8.83/4.98 ! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_412_less__iff__Suc__add,axiom,
% 8.83/4.98 ( ord_less_nat
% 8.83/4.98 = ( ^ [M3: nat,N3: nat] :
% 8.83/4.98 ? [K2: nat] :
% 8.83/4.98 ( N3
% 8.83/4.98 = ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_413_less__imp__Suc__add,axiom,
% 8.83/4.98 ! [M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_nat @ M @ N )
% 8.83/4.98 => ? [K3: nat] :
% 8.83/4.98 ( N
% 8.83/4.98 = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_414_Suc__eq__plus1,axiom,
% 8.83/4.98 ( suc
% 8.83/4.98 = ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_415_plus__1__eq__Suc,axiom,
% 8.83/4.98 ( ( plus_plus_nat @ one_one_nat )
% 8.83/4.98 = suc ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_416_Suc__eq__plus1__left,axiom,
% 8.83/4.98 ( suc
% 8.83/4.98 = ( plus_plus_nat @ one_one_nat ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_417_invar__vebt_Ointros_I5_J,axiom,
% 8.83/4.98 ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 8.83/4.98 ( ! [X5: vEBT_VEBT] :
% 8.83/4.98 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.83/4.98 => ( vEBT_invar_vebt @ X5 @ N ) )
% 8.83/4.98 => ( ( vEBT_invar_vebt @ Summary @ M )
% 8.83/4.98 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 8.83/4.98 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.83/4.98 => ( ( M
% 8.83/4.98 = ( suc @ N ) )
% 8.83/4.98 => ( ( Deg
% 8.83/4.98 = ( plus_plus_nat @ N @ M ) )
% 8.83/4.98 => ( ! [I3: nat] :
% 8.83/4.98 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.83/4.98 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X4 ) )
% 8.83/4.98 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 8.83/4.98 => ( ( ( Mi = Ma )
% 8.83/4.98 => ! [X5: vEBT_VEBT] :
% 8.83/4.98 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.83/4.98 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 8.83/4.98 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 8.83/4.98 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 8.83/4.98 => ( ( ( Mi != Ma )
% 8.83/4.98 => ! [I3: nat] :
% 8.83/4.98 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.83/4.98 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 8.83/4.98 = I3 )
% 8.83/4.98 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 8.83/4.98 & ! [X5: nat] :
% 8.83/4.98 ( ( ( ( vEBT_VEBT_high @ X5 @ N )
% 8.83/4.98 = I3 )
% 8.83/4.98 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
% 8.83/4.98 => ( ( ord_less_nat @ Mi @ X5 )
% 8.83/4.98 & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
% 8.83/4.98 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_418_invar__vebt_Ointros_I4_J,axiom,
% 8.83/4.98 ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 8.83/4.98 ( ! [X5: vEBT_VEBT] :
% 8.83/4.98 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.83/4.98 => ( vEBT_invar_vebt @ X5 @ N ) )
% 8.83/4.98 => ( ( vEBT_invar_vebt @ Summary @ M )
% 8.83/4.98 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 8.83/4.98 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.83/4.98 => ( ( M = N )
% 8.83/4.98 => ( ( Deg
% 8.83/4.98 = ( plus_plus_nat @ N @ M ) )
% 8.83/4.98 => ( ! [I3: nat] :
% 8.83/4.98 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.83/4.98 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X4 ) )
% 8.83/4.98 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 8.83/4.98 => ( ( ( Mi = Ma )
% 8.83/4.98 => ! [X5: vEBT_VEBT] :
% 8.83/4.98 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.83/4.98 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 8.83/4.98 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 8.83/4.98 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 8.83/4.98 => ( ( ( Mi != Ma )
% 8.83/4.98 => ! [I3: nat] :
% 8.83/4.98 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.83/4.98 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 8.83/4.98 = I3 )
% 8.83/4.98 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 8.83/4.98 & ! [X5: nat] :
% 8.83/4.98 ( ( ( ( vEBT_VEBT_high @ X5 @ N )
% 8.83/4.98 = I3 )
% 8.83/4.98 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
% 8.83/4.98 => ( ( ord_less_nat @ Mi @ X5 )
% 8.83/4.98 & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
% 8.83/4.98 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_419_is__num__normalize_I1_J,axiom,
% 8.83/4.98 ! [A: real,B: real,C: real] :
% 8.83/4.98 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 8.83/4.98 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_420_is__num__normalize_I1_J,axiom,
% 8.83/4.98 ! [A: rat,B: rat,C: rat] :
% 8.83/4.98 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 8.83/4.98 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_421_is__num__normalize_I1_J,axiom,
% 8.83/4.98 ! [A: int,B: int,C: int] :
% 8.83/4.98 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.83/4.98 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_422_add__One__commute,axiom,
% 8.83/4.98 ! [N: num] :
% 8.83/4.98 ( ( plus_plus_num @ one @ N )
% 8.83/4.98 = ( plus_plus_num @ N @ one ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_423_le__num__One__iff,axiom,
% 8.83/4.98 ! [X2: num] :
% 8.83/4.98 ( ( ord_less_eq_num @ X2 @ one )
% 8.83/4.98 = ( X2 = one ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_424_insersimp_H,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat,Y3: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 )
% 8.83/4.98 => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ Y3 ) @ one_one_nat ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_425_height__node,axiom,
% 8.83/4.98 ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 8.83/4.98 => ( ord_less_eq_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_426_le__numeral__extra_I4_J,axiom,
% 8.83/4.98 ord_less_eq_real @ one_one_real @ one_one_real ).
% 8.83/4.98
% 8.83/4.98 thf(fact_427_le__numeral__extra_I4_J,axiom,
% 8.83/4.98 ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 8.83/4.98
% 8.83/4.98 thf(fact_428_le__numeral__extra_I4_J,axiom,
% 8.83/4.98 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 8.83/4.98
% 8.83/4.98 thf(fact_429_le__numeral__extra_I4_J,axiom,
% 8.83/4.98 ord_less_eq_int @ one_one_int @ one_one_int ).
% 8.83/4.98
% 8.83/4.98 thf(fact_430_less__numeral__extra_I4_J,axiom,
% 8.83/4.98 ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_431_less__numeral__extra_I4_J,axiom,
% 8.83/4.98 ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_432_less__numeral__extra_I4_J,axiom,
% 8.83/4.98 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_433_less__numeral__extra_I4_J,axiom,
% 8.83/4.98 ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_434_div__le__mono,axiom,
% 8.83/4.98 ! [M: nat,N: nat,K: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ M @ N )
% 8.83/4.98 => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_435_div__le__dividend,axiom,
% 8.83/4.98 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_436_one__le__numeral,axiom,
% 8.83/4.98 ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_437_one__le__numeral,axiom,
% 8.83/4.98 ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_438_one__le__numeral,axiom,
% 8.83/4.98 ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_439_one__le__numeral,axiom,
% 8.83/4.98 ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_440_not__numeral__less__one,axiom,
% 8.83/4.98 ! [N: num] :
% 8.83/4.98 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_441_not__numeral__less__one,axiom,
% 8.83/4.98 ! [N: num] :
% 8.83/4.98 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_442_not__numeral__less__one,axiom,
% 8.83/4.98 ! [N: num] :
% 8.83/4.98 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_443_not__numeral__less__one,axiom,
% 8.83/4.98 ! [N: num] :
% 8.83/4.98 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_444_numeral__Bit0,axiom,
% 8.83/4.98 ! [N: num] :
% 8.83/4.98 ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 8.83/4.98 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_445_numeral__Bit0,axiom,
% 8.83/4.98 ! [N: num] :
% 8.83/4.98 ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 8.83/4.98 = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_446_numeral__Bit0,axiom,
% 8.83/4.98 ! [N: num] :
% 8.83/4.98 ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 8.83/4.98 = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_447_numeral__Bit0,axiom,
% 8.83/4.98 ! [N: num] :
% 8.83/4.98 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 8.83/4.98 = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_448_numeral__Bit0,axiom,
% 8.83/4.98 ! [N: num] :
% 8.83/4.98 ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 8.83/4.98 = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_449_one__plus__numeral__commute,axiom,
% 8.83/4.98 ! [X2: num] :
% 8.83/4.98 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X2 ) )
% 8.83/4.98 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X2 ) @ one_one_complex ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_450_one__plus__numeral__commute,axiom,
% 8.83/4.98 ! [X2: num] :
% 8.83/4.98 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X2 ) )
% 8.83/4.98 = ( plus_plus_real @ ( numeral_numeral_real @ X2 ) @ one_one_real ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_451_one__plus__numeral__commute,axiom,
% 8.83/4.98 ! [X2: num] :
% 8.83/4.98 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X2 ) )
% 8.83/4.98 = ( plus_plus_rat @ ( numeral_numeral_rat @ X2 ) @ one_one_rat ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_452_one__plus__numeral__commute,axiom,
% 8.83/4.98 ! [X2: num] :
% 8.83/4.98 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X2 ) )
% 8.83/4.98 = ( plus_plus_nat @ ( numeral_numeral_nat @ X2 ) @ one_one_nat ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_453_one__plus__numeral__commute,axiom,
% 8.83/4.98 ! [X2: num] :
% 8.83/4.98 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X2 ) )
% 8.83/4.98 = ( plus_plus_int @ ( numeral_numeral_int @ X2 ) @ one_one_int ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_454_numeral__One,axiom,
% 8.83/4.98 ( ( numera6690914467698888265omplex @ one )
% 8.83/4.98 = one_one_complex ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_455_numeral__One,axiom,
% 8.83/4.98 ( ( numeral_numeral_real @ one )
% 8.83/4.98 = one_one_real ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_456_numeral__One,axiom,
% 8.83/4.98 ( ( numeral_numeral_rat @ one )
% 8.83/4.98 = one_one_rat ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_457_numeral__One,axiom,
% 8.83/4.98 ( ( numeral_numeral_nat @ one )
% 8.83/4.98 = one_one_nat ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_458_numeral__One,axiom,
% 8.83/4.98 ( ( numeral_numeral_int @ one )
% 8.83/4.98 = one_one_int ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_459_divide__numeral__1,axiom,
% 8.83/4.98 ! [A: complex] :
% 8.83/4.98 ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_460_divide__numeral__1,axiom,
% 8.83/4.98 ! [A: real] :
% 8.83/4.98 ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_461_divide__numeral__1,axiom,
% 8.83/4.98 ! [A: rat] :
% 8.83/4.98 ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_462_numerals_I1_J,axiom,
% 8.83/4.98 ( ( numeral_numeral_nat @ one )
% 8.83/4.98 = one_one_nat ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_463_numeral__Bit0__div__2,axiom,
% 8.83/4.98 ! [N: num] :
% 8.83/4.98 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.98 = ( numeral_numeral_nat @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_464_numeral__Bit0__div__2,axiom,
% 8.83/4.98 ! [N: num] :
% 8.83/4.98 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.83/4.98 = ( numeral_numeral_int @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_465_nat__1__add__1,axiom,
% 8.83/4.98 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 8.83/4.98 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_466_in__children__def,axiom,
% 8.83/4.98 ( vEBT_V5917875025757280293ildren
% 8.83/4.98 = ( ^ [N3: nat,TreeList3: list_VEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X @ N3 ) ) @ ( vEBT_VEBT_low @ X @ N3 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_467_member__inv,axiom,
% 8.83/4.98 ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 8.83/4.98 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 8.83/4.98 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 8.83/4.98 & ( ( X2 = Mi )
% 8.83/4.98 | ( X2 = Ma )
% 8.83/4.98 | ( ( ord_less_nat @ X2 @ Ma )
% 8.83/4.98 & ( ord_less_nat @ Mi @ X2 )
% 8.83/4.98 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 8.83/4.98 & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_468_power__increasing__iff,axiom,
% 8.83/4.98 ! [B: real,X2: nat,Y3: nat] :
% 8.83/4.98 ( ( ord_less_real @ one_one_real @ B )
% 8.83/4.98 => ( ( ord_less_eq_real @ ( power_power_real @ B @ X2 ) @ ( power_power_real @ B @ Y3 ) )
% 8.83/4.98 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_469_power__increasing__iff,axiom,
% 8.83/4.98 ! [B: rat,X2: nat,Y3: nat] :
% 8.83/4.98 ( ( ord_less_rat @ one_one_rat @ B )
% 8.83/4.98 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X2 ) @ ( power_power_rat @ B @ Y3 ) )
% 8.83/4.98 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_470_power__increasing__iff,axiom,
% 8.83/4.98 ! [B: nat,X2: nat,Y3: nat] :
% 8.83/4.98 ( ( ord_less_nat @ one_one_nat @ B )
% 8.83/4.98 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X2 ) @ ( power_power_nat @ B @ Y3 ) )
% 8.83/4.98 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_471_power__increasing__iff,axiom,
% 8.83/4.98 ! [B: int,X2: nat,Y3: nat] :
% 8.83/4.98 ( ( ord_less_int @ one_one_int @ B )
% 8.83/4.98 => ( ( ord_less_eq_int @ ( power_power_int @ B @ X2 ) @ ( power_power_int @ B @ Y3 ) )
% 8.83/4.98 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_472_less__shift,axiom,
% 8.83/4.98 ( ord_less_nat
% 8.83/4.98 = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_473_greater__shift,axiom,
% 8.83/4.98 ( ord_less_nat
% 8.83/4.98 = ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_474_misiz,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat,M: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ( ( some_nat @ M )
% 8.83/4.98 = ( vEBT_vebt_mint @ T ) )
% 8.83/4.98 => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_475_power__strict__increasing__iff,axiom,
% 8.83/4.98 ! [B: real,X2: nat,Y3: nat] :
% 8.83/4.98 ( ( ord_less_real @ one_one_real @ B )
% 8.83/4.98 => ( ( ord_less_real @ ( power_power_real @ B @ X2 ) @ ( power_power_real @ B @ Y3 ) )
% 8.83/4.98 = ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_476_power__strict__increasing__iff,axiom,
% 8.83/4.98 ! [B: rat,X2: nat,Y3: nat] :
% 8.83/4.98 ( ( ord_less_rat @ one_one_rat @ B )
% 8.83/4.98 => ( ( ord_less_rat @ ( power_power_rat @ B @ X2 ) @ ( power_power_rat @ B @ Y3 ) )
% 8.83/4.98 = ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_477_power__strict__increasing__iff,axiom,
% 8.83/4.98 ! [B: nat,X2: nat,Y3: nat] :
% 8.83/4.98 ( ( ord_less_nat @ one_one_nat @ B )
% 8.83/4.98 => ( ( ord_less_nat @ ( power_power_nat @ B @ X2 ) @ ( power_power_nat @ B @ Y3 ) )
% 8.83/4.98 = ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_478_power__strict__increasing__iff,axiom,
% 8.83/4.98 ! [B: int,X2: nat,Y3: nat] :
% 8.83/4.98 ( ( ord_less_int @ one_one_int @ B )
% 8.83/4.98 => ( ( ord_less_int @ ( power_power_int @ B @ X2 ) @ ( power_power_int @ B @ Y3 ) )
% 8.83/4.98 = ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_479_pred__list__to__short,axiom,
% 8.83/4.98 ! [Deg: nat,X2: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
% 8.83/4.98 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 8.83/4.98 => ( ( ord_less_eq_nat @ X2 @ Ma )
% 8.83/4.98 => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.83/4.98 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 8.83/4.98 = none_nat ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_480_succ__list__to__short,axiom,
% 8.83/4.98 ! [Deg: nat,Mi: nat,X2: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 8.83/4.98 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 8.83/4.98 => ( ( ord_less_eq_nat @ Mi @ X2 )
% 8.83/4.98 => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.83/4.98 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 8.83/4.98 = none_nat ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_481_ex__power__ivl1,axiom,
% 8.83/4.98 ! [B: nat,K: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 8.83/4.98 => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 8.83/4.98 => ? [N2: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 8.83/4.98 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_482_ex__power__ivl2,axiom,
% 8.83/4.98 ! [B: nat,K: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 8.83/4.98 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 8.83/4.98 => ? [N2: nat] :
% 8.83/4.98 ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 8.83/4.98 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_483_semiring__norm_I76_J,axiom,
% 8.83/4.98 ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_484_min__Null__member,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,X2: nat] :
% 8.83/4.98 ( ( vEBT_VEBT_minNull @ T )
% 8.83/4.98 => ~ ( vEBT_vebt_member @ T @ X2 ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_485_both__member__options__equiv__member,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
% 8.83/4.98 = ( vEBT_vebt_member @ T @ X2 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_486_valid__member__both__member__options,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
% 8.83/4.98 => ( vEBT_vebt_member @ T @ X2 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_487_minNullmin,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT] :
% 8.83/4.98 ( ( vEBT_VEBT_minNull @ T )
% 8.83/4.98 => ( ( vEBT_vebt_mint @ T )
% 8.83/4.98 = none_nat ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_488_minminNull,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT] :
% 8.83/4.98 ( ( ( vEBT_vebt_mint @ T )
% 8.83/4.98 = none_nat )
% 8.83/4.98 => ( vEBT_VEBT_minNull @ T ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_489_semiring__norm_I87_J,axiom,
% 8.83/4.98 ! [M: num,N: num] :
% 8.83/4.98 ( ( ( bit0 @ M )
% 8.83/4.98 = ( bit0 @ N ) )
% 8.83/4.98 = ( M = N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_490_mint__member,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ( ( vEBT_vebt_mint @ T )
% 8.83/4.98 = ( some_nat @ Maxi ) )
% 8.83/4.98 => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_491_mint__corr__help,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat,Mini: nat,X2: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ( ( vEBT_vebt_mint @ T )
% 8.83/4.98 = ( some_nat @ Mini ) )
% 8.83/4.98 => ( ( vEBT_vebt_member @ T @ X2 )
% 8.83/4.98 => ( ord_less_eq_nat @ Mini @ X2 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_492_member__bound,axiom,
% 8.83/4.98 ! [Tree: vEBT_VEBT,X2: nat,N: nat] :
% 8.83/4.98 ( ( vEBT_vebt_member @ Tree @ X2 )
% 8.83/4.98 => ( ( vEBT_invar_vebt @ Tree @ N )
% 8.83/4.98 => ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_493_geqmaxNone,axiom,
% 8.83/4.98 ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X2: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 8.83/4.98 => ( ( ord_less_eq_nat @ Ma @ X2 )
% 8.83/4.98 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 8.83/4.98 = none_nat ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_494_semiring__norm_I85_J,axiom,
% 8.83/4.98 ! [M: num] :
% 8.83/4.98 ( ( bit0 @ M )
% 8.83/4.98 != one ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_495_semiring__norm_I83_J,axiom,
% 8.83/4.98 ! [N: num] :
% 8.83/4.98 ( one
% 8.83/4.98 != ( bit0 @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_496_power__one,axiom,
% 8.83/4.98 ! [N: nat] :
% 8.83/4.98 ( ( power_power_rat @ one_one_rat @ N )
% 8.83/4.98 = one_one_rat ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_497_power__one,axiom,
% 8.83/4.98 ! [N: nat] :
% 8.83/4.98 ( ( power_power_nat @ one_one_nat @ N )
% 8.83/4.98 = one_one_nat ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_498_power__one,axiom,
% 8.83/4.98 ! [N: nat] :
% 8.83/4.98 ( ( power_power_real @ one_one_real @ N )
% 8.83/4.98 = one_one_real ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_499_power__one,axiom,
% 8.83/4.98 ! [N: nat] :
% 8.83/4.98 ( ( power_power_int @ one_one_int @ N )
% 8.83/4.98 = one_one_int ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_500_power__one,axiom,
% 8.83/4.98 ! [N: nat] :
% 8.83/4.98 ( ( power_power_complex @ one_one_complex @ N )
% 8.83/4.98 = one_one_complex ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_501_power__one__right,axiom,
% 8.83/4.98 ! [A: nat] :
% 8.83/4.98 ( ( power_power_nat @ A @ one_one_nat )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_502_power__one__right,axiom,
% 8.83/4.98 ! [A: real] :
% 8.83/4.98 ( ( power_power_real @ A @ one_one_nat )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_503_power__one__right,axiom,
% 8.83/4.98 ! [A: int] :
% 8.83/4.98 ( ( power_power_int @ A @ one_one_nat )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_504_power__one__right,axiom,
% 8.83/4.98 ! [A: complex] :
% 8.83/4.98 ( ( power_power_complex @ A @ one_one_nat )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_505_not__Some__eq,axiom,
% 8.83/4.98 ! [X2: option4927543243414619207at_nat] :
% 8.83/4.98 ( ( ! [Y: product_prod_nat_nat] :
% 8.83/4.98 ( X2
% 8.83/4.98 != ( some_P7363390416028606310at_nat @ Y ) ) )
% 8.83/4.98 = ( X2 = none_P5556105721700978146at_nat ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_506_not__Some__eq,axiom,
% 8.83/4.98 ! [X2: option_nat] :
% 8.83/4.98 ( ( ! [Y: nat] :
% 8.83/4.98 ( X2
% 8.83/4.98 != ( some_nat @ Y ) ) )
% 8.83/4.98 = ( X2 = none_nat ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_507_not__Some__eq,axiom,
% 8.83/4.98 ! [X2: option_num] :
% 8.83/4.98 ( ( ! [Y: num] :
% 8.83/4.98 ( X2
% 8.83/4.98 != ( some_num @ Y ) ) )
% 8.83/4.98 = ( X2 = none_num ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_508_not__None__eq,axiom,
% 8.83/4.98 ! [X2: option4927543243414619207at_nat] :
% 8.83/4.98 ( ( X2 != none_P5556105721700978146at_nat )
% 8.83/4.98 = ( ? [Y: product_prod_nat_nat] :
% 8.83/4.98 ( X2
% 8.83/4.98 = ( some_P7363390416028606310at_nat @ Y ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_509_not__None__eq,axiom,
% 8.83/4.98 ! [X2: option_nat] :
% 8.83/4.98 ( ( X2 != none_nat )
% 8.83/4.98 = ( ? [Y: nat] :
% 8.83/4.98 ( X2
% 8.83/4.98 = ( some_nat @ Y ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_510_not__None__eq,axiom,
% 8.83/4.98 ! [X2: option_num] :
% 8.83/4.98 ( ( X2 != none_num )
% 8.83/4.98 = ( ? [Y: num] :
% 8.83/4.98 ( X2
% 8.83/4.98 = ( some_num @ Y ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_511_post__member__pre__member,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.83/4.98 => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.83/4.98 => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X2 ) @ Y3 )
% 8.83/4.98 => ( ( vEBT_vebt_member @ T @ Y3 )
% 8.83/4.98 | ( X2 = Y3 ) ) ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_512_member__correct,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ( vEBT_vebt_member @ T @ X2 )
% 8.83/4.98 = ( member_nat @ X2 @ ( vEBT_set_vebt @ T ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_513_semiring__norm_I6_J,axiom,
% 8.83/4.98 ! [M: num,N: num] :
% 8.83/4.98 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.83/4.98 = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_514_semiring__norm_I71_J,axiom,
% 8.83/4.98 ! [M: num,N: num] :
% 8.83/4.98 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.83/4.98 = ( ord_less_eq_num @ M @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_515_semiring__norm_I78_J,axiom,
% 8.83/4.98 ! [M: num,N: num] :
% 8.83/4.98 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.83/4.98 = ( ord_less_num @ M @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_516_semiring__norm_I68_J,axiom,
% 8.83/4.98 ! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_517_semiring__norm_I75_J,axiom,
% 8.83/4.98 ! [M: num] :
% 8.83/4.98 ~ ( ord_less_num @ M @ one ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_518_power__inject__exp,axiom,
% 8.83/4.98 ! [A: real,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_real @ one_one_real @ A )
% 8.83/4.98 => ( ( ( power_power_real @ A @ M )
% 8.83/4.98 = ( power_power_real @ A @ N ) )
% 8.83/4.98 = ( M = N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_519_power__inject__exp,axiom,
% 8.83/4.98 ! [A: rat,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_rat @ one_one_rat @ A )
% 8.83/4.98 => ( ( ( power_power_rat @ A @ M )
% 8.83/4.98 = ( power_power_rat @ A @ N ) )
% 8.83/4.98 = ( M = N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_520_power__inject__exp,axiom,
% 8.83/4.98 ! [A: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_nat @ one_one_nat @ A )
% 8.83/4.98 => ( ( ( power_power_nat @ A @ M )
% 8.83/4.98 = ( power_power_nat @ A @ N ) )
% 8.83/4.98 = ( M = N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_521_power__inject__exp,axiom,
% 8.83/4.98 ! [A: int,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_int @ one_one_int @ A )
% 8.83/4.98 => ( ( ( power_power_int @ A @ M )
% 8.83/4.98 = ( power_power_int @ A @ N ) )
% 8.83/4.98 = ( M = N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_522_semiring__norm_I2_J,axiom,
% 8.83/4.98 ( ( plus_plus_num @ one @ one )
% 8.83/4.98 = ( bit0 @ one ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_523_semiring__norm_I69_J,axiom,
% 8.83/4.98 ! [M: num] :
% 8.83/4.98 ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_524_local_Opower__def,axiom,
% 8.83/4.98 ( vEBT_VEBT_power
% 8.83/4.98 = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_525_combine__options__cases,axiom,
% 8.83/4.98 ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 8.83/4.98 ( ( ( X2 = none_P5556105721700978146at_nat )
% 8.83/4.98 => ( P @ X2 @ Y3 ) )
% 8.83/4.98 => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 8.83/4.98 => ( P @ X2 @ Y3 ) )
% 8.83/4.98 => ( ! [A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
% 8.83/4.98 ( ( X2
% 8.83/4.98 = ( some_P7363390416028606310at_nat @ A4 ) )
% 8.83/4.98 => ( ( Y3
% 8.83/4.98 = ( some_P7363390416028606310at_nat @ B4 ) )
% 8.83/4.98 => ( P @ X2 @ Y3 ) ) )
% 8.83/4.98 => ( P @ X2 @ Y3 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_526_combine__options__cases,axiom,
% 8.83/4.98 ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y3: option_nat] :
% 8.83/4.98 ( ( ( X2 = none_P5556105721700978146at_nat )
% 8.83/4.98 => ( P @ X2 @ Y3 ) )
% 8.83/4.98 => ( ( ( Y3 = none_nat )
% 8.83/4.98 => ( P @ X2 @ Y3 ) )
% 8.83/4.98 => ( ! [A4: product_prod_nat_nat,B4: nat] :
% 8.83/4.98 ( ( X2
% 8.83/4.98 = ( some_P7363390416028606310at_nat @ A4 ) )
% 8.83/4.98 => ( ( Y3
% 8.83/4.98 = ( some_nat @ B4 ) )
% 8.83/4.98 => ( P @ X2 @ Y3 ) ) )
% 8.83/4.98 => ( P @ X2 @ Y3 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_527_combine__options__cases,axiom,
% 8.83/4.98 ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y3: option_num] :
% 8.83/4.98 ( ( ( X2 = none_P5556105721700978146at_nat )
% 8.83/4.98 => ( P @ X2 @ Y3 ) )
% 8.83/4.98 => ( ( ( Y3 = none_num )
% 8.83/4.98 => ( P @ X2 @ Y3 ) )
% 8.83/4.98 => ( ! [A4: product_prod_nat_nat,B4: num] :
% 8.83/4.98 ( ( X2
% 8.83/4.98 = ( some_P7363390416028606310at_nat @ A4 ) )
% 8.83/4.98 => ( ( Y3
% 8.83/4.98 = ( some_num @ B4 ) )
% 8.83/4.98 => ( P @ X2 @ Y3 ) ) )
% 8.83/4.98 => ( P @ X2 @ Y3 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_528_combine__options__cases,axiom,
% 8.83/4.98 ! [X2: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 8.83/4.98 ( ( ( X2 = none_nat )
% 8.83/4.98 => ( P @ X2 @ Y3 ) )
% 8.83/4.98 => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 8.83/4.98 => ( P @ X2 @ Y3 ) )
% 8.83/4.98 => ( ! [A4: nat,B4: product_prod_nat_nat] :
% 8.83/4.98 ( ( X2
% 8.83/4.98 = ( some_nat @ A4 ) )
% 8.83/4.98 => ( ( Y3
% 8.83/4.98 = ( some_P7363390416028606310at_nat @ B4 ) )
% 8.83/4.98 => ( P @ X2 @ Y3 ) ) )
% 8.83/4.98 => ( P @ X2 @ Y3 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_529_combine__options__cases,axiom,
% 8.83/4.98 ! [X2: option_nat,P: option_nat > option_nat > $o,Y3: option_nat] :
% 8.83/4.98 ( ( ( X2 = none_nat )
% 8.83/4.98 => ( P @ X2 @ Y3 ) )
% 8.83/4.98 => ( ( ( Y3 = none_nat )
% 8.83/4.98 => ( P @ X2 @ Y3 ) )
% 8.83/4.98 => ( ! [A4: nat,B4: nat] :
% 8.83/4.98 ( ( X2
% 8.83/4.98 = ( some_nat @ A4 ) )
% 8.83/4.98 => ( ( Y3
% 8.83/4.98 = ( some_nat @ B4 ) )
% 8.83/4.98 => ( P @ X2 @ Y3 ) ) )
% 8.83/4.98 => ( P @ X2 @ Y3 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_530_combine__options__cases,axiom,
% 8.83/4.98 ! [X2: option_nat,P: option_nat > option_num > $o,Y3: option_num] :
% 8.83/4.98 ( ( ( X2 = none_nat )
% 8.83/4.98 => ( P @ X2 @ Y3 ) )
% 8.83/4.98 => ( ( ( Y3 = none_num )
% 8.83/4.98 => ( P @ X2 @ Y3 ) )
% 8.83/4.98 => ( ! [A4: nat,B4: num] :
% 8.83/4.98 ( ( X2
% 8.83/4.98 = ( some_nat @ A4 ) )
% 8.83/4.98 => ( ( Y3
% 8.83/4.98 = ( some_num @ B4 ) )
% 8.83/4.98 => ( P @ X2 @ Y3 ) ) )
% 8.83/4.98 => ( P @ X2 @ Y3 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_531_combine__options__cases,axiom,
% 8.83/4.98 ! [X2: option_num,P: option_num > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 8.83/4.98 ( ( ( X2 = none_num )
% 8.83/4.98 => ( P @ X2 @ Y3 ) )
% 8.83/4.98 => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 8.83/4.98 => ( P @ X2 @ Y3 ) )
% 8.83/4.98 => ( ! [A4: num,B4: product_prod_nat_nat] :
% 8.83/4.98 ( ( X2
% 8.83/4.98 = ( some_num @ A4 ) )
% 8.83/4.98 => ( ( Y3
% 8.83/4.98 = ( some_P7363390416028606310at_nat @ B4 ) )
% 8.83/4.98 => ( P @ X2 @ Y3 ) ) )
% 8.83/4.98 => ( P @ X2 @ Y3 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_532_combine__options__cases,axiom,
% 8.83/4.98 ! [X2: option_num,P: option_num > option_nat > $o,Y3: option_nat] :
% 8.83/4.98 ( ( ( X2 = none_num )
% 8.83/4.98 => ( P @ X2 @ Y3 ) )
% 8.83/4.98 => ( ( ( Y3 = none_nat )
% 8.83/4.98 => ( P @ X2 @ Y3 ) )
% 8.83/4.98 => ( ! [A4: num,B4: nat] :
% 8.83/4.98 ( ( X2
% 8.83/4.98 = ( some_num @ A4 ) )
% 8.83/4.98 => ( ( Y3
% 8.83/4.98 = ( some_nat @ B4 ) )
% 8.83/4.98 => ( P @ X2 @ Y3 ) ) )
% 8.83/4.98 => ( P @ X2 @ Y3 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_533_combine__options__cases,axiom,
% 8.83/4.98 ! [X2: option_num,P: option_num > option_num > $o,Y3: option_num] :
% 8.83/4.98 ( ( ( X2 = none_num )
% 8.83/4.98 => ( P @ X2 @ Y3 ) )
% 8.83/4.98 => ( ( ( Y3 = none_num )
% 8.83/4.98 => ( P @ X2 @ Y3 ) )
% 8.83/4.98 => ( ! [A4: num,B4: num] :
% 8.83/4.98 ( ( X2
% 8.83/4.98 = ( some_num @ A4 ) )
% 8.83/4.98 => ( ( Y3
% 8.83/4.98 = ( some_num @ B4 ) )
% 8.83/4.98 => ( P @ X2 @ Y3 ) ) )
% 8.83/4.98 => ( P @ X2 @ Y3 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_534_split__option__all,axiom,
% 8.83/4.98 ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 8.83/4.98 ! [X6: option4927543243414619207at_nat] : ( P2 @ X6 ) )
% 8.83/4.98 = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 8.83/4.98 ( ( P3 @ none_P5556105721700978146at_nat )
% 8.83/4.98 & ! [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_535_split__option__all,axiom,
% 8.83/4.98 ( ( ^ [P2: option_nat > $o] :
% 8.83/4.98 ! [X6: option_nat] : ( P2 @ X6 ) )
% 8.83/4.98 = ( ^ [P3: option_nat > $o] :
% 8.83/4.98 ( ( P3 @ none_nat )
% 8.83/4.98 & ! [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_536_split__option__all,axiom,
% 8.83/4.98 ( ( ^ [P2: option_num > $o] :
% 8.83/4.98 ! [X6: option_num] : ( P2 @ X6 ) )
% 8.83/4.98 = ( ^ [P3: option_num > $o] :
% 8.83/4.98 ( ( P3 @ none_num )
% 8.83/4.98 & ! [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_537_split__option__ex,axiom,
% 8.83/4.98 ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 8.83/4.98 ? [X6: option4927543243414619207at_nat] : ( P2 @ X6 ) )
% 8.83/4.98 = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 8.83/4.98 ( ( P3 @ none_P5556105721700978146at_nat )
% 8.83/4.98 | ? [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_538_split__option__ex,axiom,
% 8.83/4.98 ( ( ^ [P2: option_nat > $o] :
% 8.83/4.98 ? [X6: option_nat] : ( P2 @ X6 ) )
% 8.83/4.98 = ( ^ [P3: option_nat > $o] :
% 8.83/4.98 ( ( P3 @ none_nat )
% 8.83/4.98 | ? [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_539_split__option__ex,axiom,
% 8.83/4.98 ( ( ^ [P2: option_num > $o] :
% 8.83/4.98 ? [X6: option_num] : ( P2 @ X6 ) )
% 8.83/4.98 = ( ^ [P3: option_num > $o] :
% 8.83/4.98 ( ( P3 @ none_num )
% 8.83/4.98 | ? [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_540_option_Oexhaust,axiom,
% 8.83/4.98 ! [Y3: option4927543243414619207at_nat] :
% 8.83/4.98 ( ( Y3 != none_P5556105721700978146at_nat )
% 8.83/4.98 => ~ ! [X23: product_prod_nat_nat] :
% 8.83/4.98 ( Y3
% 8.83/4.98 != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_541_option_Oexhaust,axiom,
% 8.83/4.98 ! [Y3: option_nat] :
% 8.83/4.98 ( ( Y3 != none_nat )
% 8.83/4.98 => ~ ! [X23: nat] :
% 8.83/4.98 ( Y3
% 8.83/4.98 != ( some_nat @ X23 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_542_option_Oexhaust,axiom,
% 8.83/4.98 ! [Y3: option_num] :
% 8.83/4.98 ( ( Y3 != none_num )
% 8.83/4.98 => ~ ! [X23: num] :
% 8.83/4.98 ( Y3
% 8.83/4.98 != ( some_num @ X23 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_543_option_OdiscI,axiom,
% 8.83/4.98 ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
% 8.83/4.98 ( ( Option
% 8.83/4.98 = ( some_P7363390416028606310at_nat @ X22 ) )
% 8.83/4.98 => ( Option != none_P5556105721700978146at_nat ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_544_option_OdiscI,axiom,
% 8.83/4.98 ! [Option: option_nat,X22: nat] :
% 8.83/4.98 ( ( Option
% 8.83/4.98 = ( some_nat @ X22 ) )
% 8.83/4.98 => ( Option != none_nat ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_545_option_OdiscI,axiom,
% 8.83/4.98 ! [Option: option_num,X22: num] :
% 8.83/4.98 ( ( Option
% 8.83/4.98 = ( some_num @ X22 ) )
% 8.83/4.98 => ( Option != none_num ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_546_option_Odistinct_I1_J,axiom,
% 8.83/4.98 ! [X22: product_prod_nat_nat] :
% 8.83/4.98 ( none_P5556105721700978146at_nat
% 8.83/4.98 != ( some_P7363390416028606310at_nat @ X22 ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_547_option_Odistinct_I1_J,axiom,
% 8.83/4.98 ! [X22: nat] :
% 8.83/4.98 ( none_nat
% 8.83/4.98 != ( some_nat @ X22 ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_548_option_Odistinct_I1_J,axiom,
% 8.83/4.98 ! [X22: num] :
% 8.83/4.98 ( none_num
% 8.83/4.98 != ( some_num @ X22 ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_549_power__divide,axiom,
% 8.83/4.98 ! [A: complex,B: complex,N: nat] :
% 8.83/4.98 ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N )
% 8.83/4.98 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_550_power__divide,axiom,
% 8.83/4.98 ! [A: real,B: real,N: nat] :
% 8.83/4.98 ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
% 8.83/4.98 = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_551_power__divide,axiom,
% 8.83/4.98 ! [A: rat,B: rat,N: nat] :
% 8.83/4.98 ( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N )
% 8.83/4.98 = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_552_one__le__power,axiom,
% 8.83/4.98 ! [A: real,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_real @ one_one_real @ A )
% 8.83/4.98 => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_553_one__le__power,axiom,
% 8.83/4.98 ! [A: rat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 8.83/4.98 => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_554_one__le__power,axiom,
% 8.83/4.98 ! [A: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 8.83/4.98 => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_555_one__le__power,axiom,
% 8.83/4.98 ! [A: int,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_int @ one_one_int @ A )
% 8.83/4.98 => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_556_power__one__over,axiom,
% 8.83/4.98 ! [A: complex,N: nat] :
% 8.83/4.98 ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
% 8.83/4.98 = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_557_power__one__over,axiom,
% 8.83/4.98 ! [A: real,N: nat] :
% 8.83/4.98 ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
% 8.83/4.98 = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_558_power__one__over,axiom,
% 8.83/4.98 ! [A: rat,N: nat] :
% 8.83/4.98 ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
% 8.83/4.98 = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_559_power__gt1,axiom,
% 8.83/4.98 ! [A: real,N: nat] :
% 8.83/4.98 ( ( ord_less_real @ one_one_real @ A )
% 8.83/4.98 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_560_power__gt1,axiom,
% 8.83/4.98 ! [A: rat,N: nat] :
% 8.83/4.98 ( ( ord_less_rat @ one_one_rat @ A )
% 8.83/4.98 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_561_power__gt1,axiom,
% 8.83/4.98 ! [A: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_nat @ one_one_nat @ A )
% 8.83/4.98 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_562_power__gt1,axiom,
% 8.83/4.98 ! [A: int,N: nat] :
% 8.83/4.98 ( ( ord_less_int @ one_one_int @ A )
% 8.83/4.98 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_563_power__less__imp__less__exp,axiom,
% 8.83/4.98 ! [A: real,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_real @ one_one_real @ A )
% 8.83/4.98 => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 8.83/4.98 => ( ord_less_nat @ M @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_564_power__less__imp__less__exp,axiom,
% 8.83/4.98 ! [A: rat,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_rat @ one_one_rat @ A )
% 8.83/4.98 => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 8.83/4.98 => ( ord_less_nat @ M @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_565_power__less__imp__less__exp,axiom,
% 8.83/4.98 ! [A: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_nat @ one_one_nat @ A )
% 8.83/4.98 => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 8.83/4.98 => ( ord_less_nat @ M @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_566_power__less__imp__less__exp,axiom,
% 8.83/4.98 ! [A: int,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_int @ one_one_int @ A )
% 8.83/4.98 => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 8.83/4.98 => ( ord_less_nat @ M @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_567_power__strict__increasing,axiom,
% 8.83/4.98 ! [N: nat,N5: nat,A: real] :
% 8.83/4.98 ( ( ord_less_nat @ N @ N5 )
% 8.83/4.98 => ( ( ord_less_real @ one_one_real @ A )
% 8.83/4.98 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_568_power__strict__increasing,axiom,
% 8.83/4.98 ! [N: nat,N5: nat,A: rat] :
% 8.83/4.98 ( ( ord_less_nat @ N @ N5 )
% 8.83/4.98 => ( ( ord_less_rat @ one_one_rat @ A )
% 8.83/4.98 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_569_power__strict__increasing,axiom,
% 8.83/4.98 ! [N: nat,N5: nat,A: nat] :
% 8.83/4.98 ( ( ord_less_nat @ N @ N5 )
% 8.83/4.98 => ( ( ord_less_nat @ one_one_nat @ A )
% 8.83/4.98 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_570_power__strict__increasing,axiom,
% 8.83/4.98 ! [N: nat,N5: nat,A: int] :
% 8.83/4.98 ( ( ord_less_nat @ N @ N5 )
% 8.83/4.98 => ( ( ord_less_int @ one_one_int @ A )
% 8.83/4.98 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_571_power__increasing,axiom,
% 8.83/4.98 ! [N: nat,N5: nat,A: real] :
% 8.83/4.98 ( ( ord_less_eq_nat @ N @ N5 )
% 8.83/4.98 => ( ( ord_less_eq_real @ one_one_real @ A )
% 8.83/4.98 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_572_power__increasing,axiom,
% 8.83/4.98 ! [N: nat,N5: nat,A: rat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ N @ N5 )
% 8.83/4.98 => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 8.83/4.98 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_573_power__increasing,axiom,
% 8.83/4.98 ! [N: nat,N5: nat,A: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ N @ N5 )
% 8.83/4.98 => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 8.83/4.98 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_574_power__increasing,axiom,
% 8.83/4.98 ! [N: nat,N5: nat,A: int] :
% 8.83/4.98 ( ( ord_less_eq_nat @ N @ N5 )
% 8.83/4.98 => ( ( ord_less_eq_int @ one_one_int @ A )
% 8.83/4.98 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_575_power__le__imp__le__exp,axiom,
% 8.83/4.98 ! [A: real,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_real @ one_one_real @ A )
% 8.83/4.98 => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 8.83/4.98 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_576_power__le__imp__le__exp,axiom,
% 8.83/4.98 ! [A: rat,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_rat @ one_one_rat @ A )
% 8.83/4.98 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 8.83/4.98 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_577_power__le__imp__le__exp,axiom,
% 8.83/4.98 ! [A: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_nat @ one_one_nat @ A )
% 8.83/4.98 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 8.83/4.98 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_578_power__le__imp__le__exp,axiom,
% 8.83/4.98 ! [A: int,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_int @ one_one_int @ A )
% 8.83/4.98 => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 8.83/4.98 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_579_one__power2,axiom,
% 8.83/4.98 ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.98 = one_one_rat ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_580_one__power2,axiom,
% 8.83/4.98 ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.98 = one_one_nat ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_581_one__power2,axiom,
% 8.83/4.98 ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.98 = one_one_real ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_582_one__power2,axiom,
% 8.83/4.98 ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.98 = one_one_int ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_583_one__power2,axiom,
% 8.83/4.98 ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.98 = one_one_complex ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_584_less__exp,axiom,
% 8.83/4.98 ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_585_self__le__ge2__pow,axiom,
% 8.83/4.98 ! [K: nat,M: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 8.83/4.98 => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_586_power2__nat__le__eq__le,axiom,
% 8.83/4.98 ! [M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/4.98 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_587_power2__nat__le__imp__le,axiom,
% 8.83/4.98 ! [M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 8.83/4.98 => ( ord_less_eq_nat @ M @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_588_mint__corr,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ( ( vEBT_vebt_mint @ T )
% 8.83/4.98 = ( some_nat @ X2 ) )
% 8.83/4.98 => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_589_mint__sound,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 )
% 8.83/4.98 => ( ( vEBT_vebt_mint @ T )
% 8.83/4.98 = ( some_nat @ X2 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_590_mintlistlength,axiom,
% 8.83/4.98 ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 8.83/4.98 => ( ( Mi != Ma )
% 8.83/4.98 => ( ( ord_less_nat @ Mi @ Ma )
% 8.83/4.98 & ? [M4: nat] :
% 8.83/4.98 ( ( ( some_nat @ M4 )
% 8.83/4.98 = ( vEBT_vebt_mint @ Summary ) )
% 8.83/4.98 & ( ord_less_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_591_maxt__corr__help,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat,Maxi: nat,X2: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ( ( vEBT_vebt_maxt @ T )
% 8.83/4.98 = ( some_nat @ Maxi ) )
% 8.83/4.98 => ( ( vEBT_vebt_member @ T @ X2 )
% 8.83/4.98 => ( ord_less_eq_nat @ X2 @ Maxi ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_592_maxt__member,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ( ( vEBT_vebt_maxt @ T )
% 8.83/4.98 = ( some_nat @ Maxi ) )
% 8.83/4.98 => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_593_div__exp__eq,axiom,
% 8.83/4.98 ! [A: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.83/4.98 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_594_div__exp__eq,axiom,
% 8.83/4.98 ! [A: int,M: nat,N: nat] :
% 8.83/4.98 ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.83/4.98 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_595_field__less__half__sum,axiom,
% 8.83/4.98 ! [X2: real,Y3: real] :
% 8.83/4.98 ( ( ord_less_real @ X2 @ Y3 )
% 8.83/4.98 => ( ord_less_real @ X2 @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_596_field__less__half__sum,axiom,
% 8.83/4.98 ! [X2: rat,Y3: rat] :
% 8.83/4.98 ( ( ord_less_rat @ X2 @ Y3 )
% 8.83/4.98 => ( ord_less_rat @ X2 @ ( divide_divide_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_597_two__powr__height__bound__deg,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_VEBT_height @ T ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_598_bit__concat__def,axiom,
% 8.83/4.98 ( vEBT_VEBT_bit_concat
% 8.83/4.98 = ( ^ [H: nat,L2: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L2 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_599_low__inv,axiom,
% 8.83/4.98 ! [X2: nat,N: nat,Y3: nat] :
% 8.83/4.98 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.83/4.98 => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X2 ) @ N )
% 8.83/4.98 = X2 ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_600_high__inv,axiom,
% 8.83/4.98 ! [X2: nat,N: nat,Y3: nat] :
% 8.83/4.98 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.83/4.98 => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X2 ) @ N )
% 8.83/4.98 = Y3 ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_601_maxbmo,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,X2: nat] :
% 8.83/4.98 ( ( ( vEBT_vebt_maxt @ T )
% 8.83/4.98 = ( some_nat @ X2 ) )
% 8.83/4.98 => ( vEBT_V8194947554948674370ptions @ T @ X2 ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_602_set__vebt__set__vebt_H__valid,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ( vEBT_set_vebt @ T )
% 8.83/4.98 = ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_603_maxt__sound,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 )
% 8.83/4.98 => ( ( vEBT_vebt_maxt @ T )
% 8.83/4.98 = ( some_nat @ X2 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_604_maxt__corr,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ( ( vEBT_vebt_maxt @ T )
% 8.83/4.98 = ( some_nat @ X2 ) )
% 8.83/4.98 => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_605_power__minus__is__div,axiom,
% 8.83/4.98 ! [B: nat,A: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ B @ A )
% 8.83/4.98 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B ) )
% 8.83/4.98 = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_606_numeral__times__numeral,axiom,
% 8.83/4.98 ! [M: num,N: num] :
% 8.83/4.98 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 8.83/4.98 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_607_numeral__times__numeral,axiom,
% 8.83/4.98 ! [M: num,N: num] :
% 8.83/4.98 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 8.83/4.98 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_608_numeral__times__numeral,axiom,
% 8.83/4.98 ! [M: num,N: num] :
% 8.83/4.98 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 8.83/4.98 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_609_numeral__times__numeral,axiom,
% 8.83/4.98 ! [M: num,N: num] :
% 8.83/4.98 ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 8.83/4.98 = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_610_numeral__times__numeral,axiom,
% 8.83/4.98 ! [M: num,N: num] :
% 8.83/4.98 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 8.83/4.98 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_611_mult__numeral__left__semiring__numeral,axiom,
% 8.83/4.98 ! [V: num,W: num,Z: complex] :
% 8.83/4.98 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 8.83/4.98 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_612_mult__numeral__left__semiring__numeral,axiom,
% 8.83/4.98 ! [V: num,W: num,Z: real] :
% 8.83/4.98 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 8.83/4.98 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_613_mult__numeral__left__semiring__numeral,axiom,
% 8.83/4.98 ! [V: num,W: num,Z: rat] :
% 8.83/4.98 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 8.83/4.98 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_614_mult__numeral__left__semiring__numeral,axiom,
% 8.83/4.98 ! [V: num,W: num,Z: nat] :
% 8.83/4.98 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 8.83/4.98 = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_615_mult__numeral__left__semiring__numeral,axiom,
% 8.83/4.98 ! [V: num,W: num,Z: int] :
% 8.83/4.98 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 8.83/4.98 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_616_mult_Oright__neutral,axiom,
% 8.83/4.98 ! [A: complex] :
% 8.83/4.98 ( ( times_times_complex @ A @ one_one_complex )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_617_mult_Oright__neutral,axiom,
% 8.83/4.98 ! [A: real] :
% 8.83/4.98 ( ( times_times_real @ A @ one_one_real )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_618_mult_Oright__neutral,axiom,
% 8.83/4.98 ! [A: rat] :
% 8.83/4.98 ( ( times_times_rat @ A @ one_one_rat )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_619_mult_Oright__neutral,axiom,
% 8.83/4.98 ! [A: nat] :
% 8.83/4.98 ( ( times_times_nat @ A @ one_one_nat )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_620_mult_Oright__neutral,axiom,
% 8.83/4.98 ! [A: int] :
% 8.83/4.98 ( ( times_times_int @ A @ one_one_int )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_621_mult__1,axiom,
% 8.83/4.98 ! [A: complex] :
% 8.83/4.98 ( ( times_times_complex @ one_one_complex @ A )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_622_mult__1,axiom,
% 8.83/4.98 ! [A: real] :
% 8.83/4.98 ( ( times_times_real @ one_one_real @ A )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_623_mult__1,axiom,
% 8.83/4.98 ! [A: rat] :
% 8.83/4.98 ( ( times_times_rat @ one_one_rat @ A )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_624_mult__1,axiom,
% 8.83/4.98 ! [A: nat] :
% 8.83/4.98 ( ( times_times_nat @ one_one_nat @ A )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_625_mult__1,axiom,
% 8.83/4.98 ! [A: int] :
% 8.83/4.98 ( ( times_times_int @ one_one_int @ A )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_626_add__diff__cancel,axiom,
% 8.83/4.98 ! [A: real,B: real] :
% 8.83/4.98 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_627_add__diff__cancel,axiom,
% 8.83/4.98 ! [A: rat,B: rat] :
% 8.83/4.98 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_628_add__diff__cancel,axiom,
% 8.83/4.98 ! [A: int,B: int] :
% 8.83/4.98 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_629_diff__add__cancel,axiom,
% 8.83/4.98 ! [A: real,B: real] :
% 8.83/4.98 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_630_diff__add__cancel,axiom,
% 8.83/4.98 ! [A: rat,B: rat] :
% 8.83/4.98 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_631_diff__add__cancel,axiom,
% 8.83/4.98 ! [A: int,B: int] :
% 8.83/4.98 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_632_add__diff__cancel__left,axiom,
% 8.83/4.98 ! [C: real,A: real,B: real] :
% 8.83/4.98 ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 8.83/4.98 = ( minus_minus_real @ A @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_633_add__diff__cancel__left,axiom,
% 8.83/4.98 ! [C: rat,A: rat,B: rat] :
% 8.83/4.98 ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 8.83/4.98 = ( minus_minus_rat @ A @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_634_add__diff__cancel__left,axiom,
% 8.83/4.98 ! [C: nat,A: nat,B: nat] :
% 8.83/4.98 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 8.83/4.98 = ( minus_minus_nat @ A @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_635_add__diff__cancel__left,axiom,
% 8.83/4.98 ! [C: int,A: int,B: int] :
% 8.83/4.98 ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 8.83/4.98 = ( minus_minus_int @ A @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_636_add__diff__cancel__left_H,axiom,
% 8.83/4.98 ! [A: real,B: real] :
% 8.83/4.98 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 8.83/4.98 = B ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_637_add__diff__cancel__left_H,axiom,
% 8.83/4.98 ! [A: rat,B: rat] :
% 8.83/4.98 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
% 8.83/4.98 = B ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_638_add__diff__cancel__left_H,axiom,
% 8.83/4.98 ! [A: nat,B: nat] :
% 8.83/4.98 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 8.83/4.98 = B ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_639_add__diff__cancel__left_H,axiom,
% 8.83/4.98 ! [A: int,B: int] :
% 8.83/4.98 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 8.83/4.98 = B ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_640_add__diff__cancel__right,axiom,
% 8.83/4.98 ! [A: real,C: real,B: real] :
% 8.83/4.98 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 8.83/4.98 = ( minus_minus_real @ A @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_641_add__diff__cancel__right,axiom,
% 8.83/4.98 ! [A: rat,C: rat,B: rat] :
% 8.83/4.98 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 8.83/4.98 = ( minus_minus_rat @ A @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_642_add__diff__cancel__right,axiom,
% 8.83/4.98 ! [A: nat,C: nat,B: nat] :
% 8.83/4.98 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 8.83/4.98 = ( minus_minus_nat @ A @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_643_add__diff__cancel__right,axiom,
% 8.83/4.98 ! [A: int,C: int,B: int] :
% 8.83/4.98 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 8.83/4.98 = ( minus_minus_int @ A @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_644_add__diff__cancel__right_H,axiom,
% 8.83/4.98 ! [A: real,B: real] :
% 8.83/4.98 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_645_add__diff__cancel__right_H,axiom,
% 8.83/4.98 ! [A: rat,B: rat] :
% 8.83/4.98 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_646_add__diff__cancel__right_H,axiom,
% 8.83/4.98 ! [A: nat,B: nat] :
% 8.83/4.98 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_647_add__diff__cancel__right_H,axiom,
% 8.83/4.98 ! [A: int,B: int] :
% 8.83/4.98 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_648_bits__div__by__1,axiom,
% 8.83/4.98 ! [A: nat] :
% 8.83/4.98 ( ( divide_divide_nat @ A @ one_one_nat )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_649_bits__div__by__1,axiom,
% 8.83/4.98 ! [A: int] :
% 8.83/4.98 ( ( divide_divide_int @ A @ one_one_int )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_650_diff__Suc__Suc,axiom,
% 8.83/4.98 ! [M: nat,N: nat] :
% 8.83/4.98 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 8.83/4.98 = ( minus_minus_nat @ M @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_651_Suc__diff__diff,axiom,
% 8.83/4.98 ! [M: nat,N: nat,K: nat] :
% 8.83/4.98 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 8.83/4.98 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_652_diff__diff__cancel,axiom,
% 8.83/4.98 ! [I2: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ I2 @ N )
% 8.83/4.98 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
% 8.83/4.98 = I2 ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_653_diff__diff__left,axiom,
% 8.83/4.98 ! [I2: nat,J: nat,K: nat] :
% 8.83/4.98 ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
% 8.83/4.98 = ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_654_nat__1__eq__mult__iff,axiom,
% 8.83/4.98 ! [M: nat,N: nat] :
% 8.83/4.98 ( ( one_one_nat
% 8.83/4.98 = ( times_times_nat @ M @ N ) )
% 8.83/4.98 = ( ( M = one_one_nat )
% 8.83/4.98 & ( N = one_one_nat ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_655_nat__mult__eq__1__iff,axiom,
% 8.83/4.98 ! [M: nat,N: nat] :
% 8.83/4.98 ( ( ( times_times_nat @ M @ N )
% 8.83/4.98 = one_one_nat )
% 8.83/4.98 = ( ( M = one_one_nat )
% 8.83/4.98 & ( N = one_one_nat ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_656_distrib__left__numeral,axiom,
% 8.83/4.98 ! [V: num,B: complex,C: complex] :
% 8.83/4.98 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
% 8.83/4.98 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_657_distrib__left__numeral,axiom,
% 8.83/4.98 ! [V: num,B: real,C: real] :
% 8.83/4.98 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
% 8.83/4.98 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_658_distrib__left__numeral,axiom,
% 8.83/4.98 ! [V: num,B: rat,C: rat] :
% 8.83/4.98 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
% 8.83/4.98 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_659_distrib__left__numeral,axiom,
% 8.83/4.98 ! [V: num,B: nat,C: nat] :
% 8.83/4.98 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
% 8.83/4.98 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_660_distrib__left__numeral,axiom,
% 8.83/4.98 ! [V: num,B: int,C: int] :
% 8.83/4.98 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
% 8.83/4.98 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_661_distrib__right__numeral,axiom,
% 8.83/4.98 ! [A: complex,B: complex,V: num] :
% 8.83/4.98 ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 8.83/4.98 = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_662_distrib__right__numeral,axiom,
% 8.83/4.98 ! [A: real,B: real,V: num] :
% 8.83/4.98 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 8.83/4.98 = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_663_distrib__right__numeral,axiom,
% 8.83/4.98 ! [A: rat,B: rat,V: num] :
% 8.83/4.98 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 8.83/4.98 = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_664_distrib__right__numeral,axiom,
% 8.83/4.98 ! [A: nat,B: nat,V: num] :
% 8.83/4.98 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
% 8.83/4.98 = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_665_distrib__right__numeral,axiom,
% 8.83/4.98 ! [A: int,B: int,V: num] :
% 8.83/4.98 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 8.83/4.98 = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_666_left__diff__distrib__numeral,axiom,
% 8.83/4.98 ! [A: complex,B: complex,V: num] :
% 8.83/4.98 ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 8.83/4.98 = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_667_left__diff__distrib__numeral,axiom,
% 8.83/4.98 ! [A: real,B: real,V: num] :
% 8.83/4.98 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 8.83/4.98 = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_668_left__diff__distrib__numeral,axiom,
% 8.83/4.98 ! [A: rat,B: rat,V: num] :
% 8.83/4.98 ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 8.83/4.98 = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_669_left__diff__distrib__numeral,axiom,
% 8.83/4.98 ! [A: int,B: int,V: num] :
% 8.83/4.98 ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 8.83/4.98 = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_670_right__diff__distrib__numeral,axiom,
% 8.83/4.98 ! [V: num,B: complex,C: complex] :
% 8.83/4.98 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B @ C ) )
% 8.83/4.98 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_671_right__diff__distrib__numeral,axiom,
% 8.83/4.98 ! [V: num,B: real,C: real] :
% 8.83/4.98 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
% 8.83/4.98 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_672_right__diff__distrib__numeral,axiom,
% 8.83/4.98 ! [V: num,B: rat,C: rat] :
% 8.83/4.98 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
% 8.83/4.98 = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_673_right__diff__distrib__numeral,axiom,
% 8.83/4.98 ! [V: num,B: int,C: int] :
% 8.83/4.98 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
% 8.83/4.98 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_674_mult__Suc__right,axiom,
% 8.83/4.98 ! [M: nat,N: nat] :
% 8.83/4.98 ( ( times_times_nat @ M @ ( suc @ N ) )
% 8.83/4.98 = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_675_diff__Suc__1,axiom,
% 8.83/4.98 ! [N: nat] :
% 8.83/4.98 ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 8.83/4.98 = N ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_676_Nat_Oadd__diff__assoc,axiom,
% 8.83/4.98 ! [K: nat,J: nat,I2: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ K @ J )
% 8.83/4.98 => ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 8.83/4.98 = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_677_Nat_Oadd__diff__assoc2,axiom,
% 8.83/4.98 ! [K: nat,J: nat,I2: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ K @ J )
% 8.83/4.98 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
% 8.83/4.98 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_678_Nat_Odiff__diff__right,axiom,
% 8.83/4.98 ! [K: nat,J: nat,I2: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ K @ J )
% 8.83/4.98 => ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 8.83/4.98 = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_679_divide__le__eq__numeral1_I1_J,axiom,
% 8.83/4.98 ! [B: real,W: num,A: real] :
% 8.83/4.98 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 8.83/4.98 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_680_divide__le__eq__numeral1_I1_J,axiom,
% 8.83/4.98 ! [B: rat,W: num,A: rat] :
% 8.83/4.98 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 8.83/4.98 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_681_le__divide__eq__numeral1_I1_J,axiom,
% 8.83/4.98 ! [A: real,B: real,W: num] :
% 8.83/4.98 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 8.83/4.98 = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_682_le__divide__eq__numeral1_I1_J,axiom,
% 8.83/4.98 ! [A: rat,B: rat,W: num] :
% 8.83/4.98 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 8.83/4.98 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_683_less__divide__eq__numeral1_I1_J,axiom,
% 8.83/4.98 ! [A: real,B: real,W: num] :
% 8.83/4.98 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 8.83/4.98 = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_684_less__divide__eq__numeral1_I1_J,axiom,
% 8.83/4.98 ! [A: rat,B: rat,W: num] :
% 8.83/4.98 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 8.83/4.98 = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_685_divide__less__eq__numeral1_I1_J,axiom,
% 8.83/4.98 ! [B: real,W: num,A: real] :
% 8.83/4.98 ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 8.83/4.98 = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_686_divide__less__eq__numeral1_I1_J,axiom,
% 8.83/4.98 ! [B: rat,W: num,A: rat] :
% 8.83/4.98 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 8.83/4.98 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_687_diff__Suc__diff__eq1,axiom,
% 8.83/4.98 ! [K: nat,J: nat,I2: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ K @ J )
% 8.83/4.98 => ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 8.83/4.98 = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_688_diff__Suc__diff__eq2,axiom,
% 8.83/4.98 ! [K: nat,J: nat,I2: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ K @ J )
% 8.83/4.98 => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
% 8.83/4.98 = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_689_power__add__numeral,axiom,
% 8.83/4.98 ! [A: complex,M: num,N: num] :
% 8.83/4.98 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
% 8.83/4.98 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_690_power__add__numeral,axiom,
% 8.83/4.98 ! [A: real,M: num,N: num] :
% 8.83/4.98 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
% 8.83/4.98 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_691_power__add__numeral,axiom,
% 8.83/4.98 ! [A: rat,M: num,N: num] :
% 8.83/4.98 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
% 8.83/4.98 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_692_power__add__numeral,axiom,
% 8.83/4.98 ! [A: nat,M: num,N: num] :
% 8.83/4.98 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
% 8.83/4.98 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_693_power__add__numeral,axiom,
% 8.83/4.98 ! [A: int,M: num,N: num] :
% 8.83/4.98 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
% 8.83/4.98 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_694_power__add__numeral2,axiom,
% 8.83/4.98 ! [A: complex,M: num,N: num,B: complex] :
% 8.83/4.98 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 8.83/4.98 = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_695_power__add__numeral2,axiom,
% 8.83/4.98 ! [A: real,M: num,N: num,B: real] :
% 8.83/4.98 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 8.83/4.98 = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_696_power__add__numeral2,axiom,
% 8.83/4.98 ! [A: rat,M: num,N: num,B: rat] :
% 8.83/4.98 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 8.83/4.98 = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_697_power__add__numeral2,axiom,
% 8.83/4.98 ! [A: nat,M: num,N: num,B: nat] :
% 8.83/4.98 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 8.83/4.98 = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_698_power__add__numeral2,axiom,
% 8.83/4.98 ! [A: int,M: num,N: num,B: int] :
% 8.83/4.98 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 8.83/4.98 = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_699_pred__member,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,X2: nat,Y3: nat] :
% 8.83/4.98 ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Y3 )
% 8.83/4.98 = ( ( vEBT_vebt_member @ T @ Y3 )
% 8.83/4.98 & ( ord_less_nat @ Y3 @ X2 )
% 8.83/4.98 & ! [Z3: nat] :
% 8.83/4.98 ( ( ( vEBT_vebt_member @ T @ Z3 )
% 8.83/4.98 & ( ord_less_nat @ Z3 @ X2 ) )
% 8.83/4.98 => ( ord_less_eq_nat @ Z3 @ Y3 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_700_succ__member,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,X2: nat,Y3: nat] :
% 8.83/4.98 ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Y3 )
% 8.83/4.98 = ( ( vEBT_vebt_member @ T @ Y3 )
% 8.83/4.98 & ( ord_less_nat @ X2 @ Y3 )
% 8.83/4.98 & ! [Z3: nat] :
% 8.83/4.98 ( ( ( vEBT_vebt_member @ T @ Z3 )
% 8.83/4.98 & ( ord_less_nat @ X2 @ Z3 ) )
% 8.83/4.98 => ( ord_less_eq_nat @ Y3 @ Z3 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_701_pred__corr,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat,X2: nat,Px: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ( ( vEBT_vebt_pred @ T @ X2 )
% 8.83/4.98 = ( some_nat @ Px ) )
% 8.83/4.98 = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Px ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_702_succ__corr,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ( ( vEBT_vebt_succ @ T @ X2 )
% 8.83/4.98 = ( some_nat @ Sx ) )
% 8.83/4.98 = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_703_pred__correct,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ( ( vEBT_vebt_pred @ T @ X2 )
% 8.83/4.98 = ( some_nat @ Sx ) )
% 8.83/4.98 = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_704_succ__correct,axiom,
% 8.83/4.98 ! [T: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
% 8.83/4.98 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.98 => ( ( ( vEBT_vebt_succ @ T @ X2 )
% 8.83/4.98 = ( some_nat @ Sx ) )
% 8.83/4.98 = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_705_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 8.83/4.98 ! [A: real,B: real,C: real] :
% 8.83/4.98 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 8.83/4.98 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_706_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 8.83/4.98 ! [A: rat,B: rat,C: rat] :
% 8.83/4.98 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 8.83/4.98 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_707_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 8.83/4.98 ! [A: nat,B: nat,C: nat] :
% 8.83/4.98 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 8.83/4.98 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_708_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 8.83/4.98 ! [A: int,B: int,C: int] :
% 8.83/4.98 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 8.83/4.98 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_709_diff__eq__diff__eq,axiom,
% 8.83/4.98 ! [A: real,B: real,C: real,D: real] :
% 8.83/4.98 ( ( ( minus_minus_real @ A @ B )
% 8.83/4.98 = ( minus_minus_real @ C @ D ) )
% 8.83/4.98 => ( ( A = B )
% 8.83/4.98 = ( C = D ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_710_diff__eq__diff__eq,axiom,
% 8.83/4.98 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.83/4.98 ( ( ( minus_minus_rat @ A @ B )
% 8.83/4.98 = ( minus_minus_rat @ C @ D ) )
% 8.83/4.98 => ( ( A = B )
% 8.83/4.98 = ( C = D ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_711_diff__eq__diff__eq,axiom,
% 8.83/4.98 ! [A: int,B: int,C: int,D: int] :
% 8.83/4.98 ( ( ( minus_minus_int @ A @ B )
% 8.83/4.98 = ( minus_minus_int @ C @ D ) )
% 8.83/4.98 => ( ( A = B )
% 8.83/4.98 = ( C = D ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_712_mult_Oassoc,axiom,
% 8.83/4.98 ! [A: real,B: real,C: real] :
% 8.83/4.98 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 8.83/4.98 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_713_mult_Oassoc,axiom,
% 8.83/4.98 ! [A: rat,B: rat,C: rat] :
% 8.83/4.98 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 8.83/4.98 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_714_mult_Oassoc,axiom,
% 8.83/4.98 ! [A: nat,B: nat,C: nat] :
% 8.83/4.98 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 8.83/4.98 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_715_mult_Oassoc,axiom,
% 8.83/4.98 ! [A: int,B: int,C: int] :
% 8.83/4.98 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 8.83/4.98 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_716_mult_Ocommute,axiom,
% 8.83/4.98 ( times_times_real
% 8.83/4.98 = ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_717_mult_Ocommute,axiom,
% 8.83/4.98 ( times_times_rat
% 8.83/4.98 = ( ^ [A3: rat,B2: rat] : ( times_times_rat @ B2 @ A3 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_718_mult_Ocommute,axiom,
% 8.83/4.98 ( times_times_nat
% 8.83/4.98 = ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_719_mult_Ocommute,axiom,
% 8.83/4.98 ( times_times_int
% 8.83/4.98 = ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_720_mult_Oleft__commute,axiom,
% 8.83/4.98 ! [B: real,A: real,C: real] :
% 8.83/4.98 ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 8.83/4.98 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_721_mult_Oleft__commute,axiom,
% 8.83/4.98 ! [B: rat,A: rat,C: rat] :
% 8.83/4.98 ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
% 8.83/4.98 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_722_mult_Oleft__commute,axiom,
% 8.83/4.98 ! [B: nat,A: nat,C: nat] :
% 8.83/4.98 ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 8.83/4.98 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_723_mult_Oleft__commute,axiom,
% 8.83/4.98 ! [B: int,A: int,C: int] :
% 8.83/4.98 ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
% 8.83/4.98 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_724_diff__right__commute,axiom,
% 8.83/4.98 ! [A: real,C: real,B: real] :
% 8.83/4.98 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 8.83/4.98 = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_725_diff__right__commute,axiom,
% 8.83/4.98 ! [A: rat,C: rat,B: rat] :
% 8.83/4.98 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
% 8.83/4.98 = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_726_diff__right__commute,axiom,
% 8.83/4.98 ! [A: nat,C: nat,B: nat] :
% 8.83/4.98 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 8.83/4.98 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_727_diff__right__commute,axiom,
% 8.83/4.98 ! [A: int,C: int,B: int] :
% 8.83/4.98 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 8.83/4.98 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_728_diff__commute,axiom,
% 8.83/4.98 ! [I2: nat,J: nat,K: nat] :
% 8.83/4.98 ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
% 8.83/4.98 = ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_729_diff__mult__distrib,axiom,
% 8.83/4.98 ! [M: nat,N: nat,K: nat] :
% 8.83/4.98 ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 8.83/4.98 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_730_diff__mult__distrib2,axiom,
% 8.83/4.98 ! [K: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 8.83/4.98 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_731_mult__diff__mult,axiom,
% 8.83/4.98 ! [X2: real,Y3: real,A: real,B: real] :
% 8.83/4.98 ( ( minus_minus_real @ ( times_times_real @ X2 @ Y3 ) @ ( times_times_real @ A @ B ) )
% 8.83/4.98 = ( plus_plus_real @ ( times_times_real @ X2 @ ( minus_minus_real @ Y3 @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X2 @ A ) @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_732_mult__diff__mult,axiom,
% 8.83/4.98 ! [X2: rat,Y3: rat,A: rat,B: rat] :
% 8.83/4.98 ( ( minus_minus_rat @ ( times_times_rat @ X2 @ Y3 ) @ ( times_times_rat @ A @ B ) )
% 8.83/4.98 = ( plus_plus_rat @ ( times_times_rat @ X2 @ ( minus_minus_rat @ Y3 @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X2 @ A ) @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_733_mult__diff__mult,axiom,
% 8.83/4.98 ! [X2: int,Y3: int,A: int,B: int] :
% 8.83/4.98 ( ( minus_minus_int @ ( times_times_int @ X2 @ Y3 ) @ ( times_times_int @ A @ B ) )
% 8.83/4.98 = ( plus_plus_int @ ( times_times_int @ X2 @ ( minus_minus_int @ Y3 @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X2 @ A ) @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_734_add__diff__add,axiom,
% 8.83/4.98 ! [A: real,C: real,B: real,D: real] :
% 8.83/4.98 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
% 8.83/4.98 = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_735_add__diff__add,axiom,
% 8.83/4.98 ! [A: rat,C: rat,B: rat,D: rat] :
% 8.83/4.98 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
% 8.83/4.98 = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_736_add__diff__add,axiom,
% 8.83/4.98 ! [A: int,C: int,B: int,D: int] :
% 8.83/4.98 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
% 8.83/4.98 = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_737_nat__eq__add__iff1,axiom,
% 8.83/4.98 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ J @ I2 )
% 8.83/4.98 => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
% 8.83/4.98 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 8.83/4.98 = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M )
% 8.83/4.98 = N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_738_nat__eq__add__iff2,axiom,
% 8.83/4.98 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ I2 @ J )
% 8.83/4.98 => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
% 8.83/4.98 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 8.83/4.98 = ( M
% 8.83/4.98 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_739_nat__le__add__iff1,axiom,
% 8.83/4.98 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ J @ I2 )
% 8.83/4.98 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 8.83/4.98 = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_740_nat__le__add__iff2,axiom,
% 8.83/4.98 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ I2 @ J )
% 8.83/4.98 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 8.83/4.98 = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_741_nat__diff__add__eq1,axiom,
% 8.83/4.98 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ J @ I2 )
% 8.83/4.98 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 8.83/4.98 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_742_nat__diff__add__eq2,axiom,
% 8.83/4.98 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ I2 @ J )
% 8.83/4.98 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 8.83/4.98 = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_743_diff__mono,axiom,
% 8.83/4.98 ! [A: real,B: real,D: real,C: real] :
% 8.83/4.98 ( ( ord_less_eq_real @ A @ B )
% 8.83/4.98 => ( ( ord_less_eq_real @ D @ C )
% 8.83/4.98 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_744_diff__mono,axiom,
% 8.83/4.98 ! [A: rat,B: rat,D: rat,C: rat] :
% 8.83/4.98 ( ( ord_less_eq_rat @ A @ B )
% 8.83/4.98 => ( ( ord_less_eq_rat @ D @ C )
% 8.83/4.98 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_745_diff__mono,axiom,
% 8.83/4.98 ! [A: int,B: int,D: int,C: int] :
% 8.83/4.98 ( ( ord_less_eq_int @ A @ B )
% 8.83/4.98 => ( ( ord_less_eq_int @ D @ C )
% 8.83/4.98 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_746_diff__left__mono,axiom,
% 8.83/4.98 ! [B: real,A: real,C: real] :
% 8.83/4.98 ( ( ord_less_eq_real @ B @ A )
% 8.83/4.98 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_747_diff__left__mono,axiom,
% 8.83/4.98 ! [B: rat,A: rat,C: rat] :
% 8.83/4.98 ( ( ord_less_eq_rat @ B @ A )
% 8.83/4.98 => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_748_diff__left__mono,axiom,
% 8.83/4.98 ! [B: int,A: int,C: int] :
% 8.83/4.98 ( ( ord_less_eq_int @ B @ A )
% 8.83/4.98 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_749_diff__right__mono,axiom,
% 8.83/4.98 ! [A: real,B: real,C: real] :
% 8.83/4.98 ( ( ord_less_eq_real @ A @ B )
% 8.83/4.98 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_750_diff__right__mono,axiom,
% 8.83/4.98 ! [A: rat,B: rat,C: rat] :
% 8.83/4.98 ( ( ord_less_eq_rat @ A @ B )
% 8.83/4.98 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_751_diff__right__mono,axiom,
% 8.83/4.98 ! [A: int,B: int,C: int] :
% 8.83/4.98 ( ( ord_less_eq_int @ A @ B )
% 8.83/4.98 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_752_diff__eq__diff__less__eq,axiom,
% 8.83/4.98 ! [A: real,B: real,C: real,D: real] :
% 8.83/4.98 ( ( ( minus_minus_real @ A @ B )
% 8.83/4.98 = ( minus_minus_real @ C @ D ) )
% 8.83/4.98 => ( ( ord_less_eq_real @ A @ B )
% 8.83/4.98 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_753_diff__eq__diff__less__eq,axiom,
% 8.83/4.98 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.83/4.98 ( ( ( minus_minus_rat @ A @ B )
% 8.83/4.98 = ( minus_minus_rat @ C @ D ) )
% 8.83/4.98 => ( ( ord_less_eq_rat @ A @ B )
% 8.83/4.98 = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_754_diff__eq__diff__less__eq,axiom,
% 8.83/4.98 ! [A: int,B: int,C: int,D: int] :
% 8.83/4.98 ( ( ( minus_minus_int @ A @ B )
% 8.83/4.98 = ( minus_minus_int @ C @ D ) )
% 8.83/4.98 => ( ( ord_less_eq_int @ A @ B )
% 8.83/4.98 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_755_diff__strict__right__mono,axiom,
% 8.83/4.98 ! [A: real,B: real,C: real] :
% 8.83/4.98 ( ( ord_less_real @ A @ B )
% 8.83/4.98 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_756_diff__strict__right__mono,axiom,
% 8.83/4.98 ! [A: rat,B: rat,C: rat] :
% 8.83/4.98 ( ( ord_less_rat @ A @ B )
% 8.83/4.98 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_757_diff__strict__right__mono,axiom,
% 8.83/4.98 ! [A: int,B: int,C: int] :
% 8.83/4.98 ( ( ord_less_int @ A @ B )
% 8.83/4.98 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_758_diff__strict__left__mono,axiom,
% 8.83/4.98 ! [B: real,A: real,C: real] :
% 8.83/4.98 ( ( ord_less_real @ B @ A )
% 8.83/4.98 => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_759_diff__strict__left__mono,axiom,
% 8.83/4.98 ! [B: rat,A: rat,C: rat] :
% 8.83/4.98 ( ( ord_less_rat @ B @ A )
% 8.83/4.98 => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_760_diff__strict__left__mono,axiom,
% 8.83/4.98 ! [B: int,A: int,C: int] :
% 8.83/4.98 ( ( ord_less_int @ B @ A )
% 8.83/4.98 => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_761_diff__eq__diff__less,axiom,
% 8.83/4.98 ! [A: real,B: real,C: real,D: real] :
% 8.83/4.98 ( ( ( minus_minus_real @ A @ B )
% 8.83/4.98 = ( minus_minus_real @ C @ D ) )
% 8.83/4.98 => ( ( ord_less_real @ A @ B )
% 8.83/4.98 = ( ord_less_real @ C @ D ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_762_diff__eq__diff__less,axiom,
% 8.83/4.98 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.83/4.98 ( ( ( minus_minus_rat @ A @ B )
% 8.83/4.98 = ( minus_minus_rat @ C @ D ) )
% 8.83/4.98 => ( ( ord_less_rat @ A @ B )
% 8.83/4.98 = ( ord_less_rat @ C @ D ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_763_diff__eq__diff__less,axiom,
% 8.83/4.98 ! [A: int,B: int,C: int,D: int] :
% 8.83/4.98 ( ( ( minus_minus_int @ A @ B )
% 8.83/4.98 = ( minus_minus_int @ C @ D ) )
% 8.83/4.98 => ( ( ord_less_int @ A @ B )
% 8.83/4.98 = ( ord_less_int @ C @ D ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_764_diff__strict__mono,axiom,
% 8.83/4.98 ! [A: real,B: real,D: real,C: real] :
% 8.83/4.98 ( ( ord_less_real @ A @ B )
% 8.83/4.98 => ( ( ord_less_real @ D @ C )
% 8.83/4.98 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_765_diff__strict__mono,axiom,
% 8.83/4.98 ! [A: rat,B: rat,D: rat,C: rat] :
% 8.83/4.98 ( ( ord_less_rat @ A @ B )
% 8.83/4.98 => ( ( ord_less_rat @ D @ C )
% 8.83/4.98 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_766_diff__strict__mono,axiom,
% 8.83/4.98 ! [A: int,B: int,D: int,C: int] :
% 8.83/4.98 ( ( ord_less_int @ A @ B )
% 8.83/4.98 => ( ( ord_less_int @ D @ C )
% 8.83/4.98 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_767_group__cancel_Osub1,axiom,
% 8.83/4.98 ! [A2: real,K: real,A: real,B: real] :
% 8.83/4.98 ( ( A2
% 8.83/4.98 = ( plus_plus_real @ K @ A ) )
% 8.83/4.98 => ( ( minus_minus_real @ A2 @ B )
% 8.83/4.98 = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_768_group__cancel_Osub1,axiom,
% 8.83/4.98 ! [A2: rat,K: rat,A: rat,B: rat] :
% 8.83/4.98 ( ( A2
% 8.83/4.98 = ( plus_plus_rat @ K @ A ) )
% 8.83/4.98 => ( ( minus_minus_rat @ A2 @ B )
% 8.83/4.98 = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_769_group__cancel_Osub1,axiom,
% 8.83/4.98 ! [A2: int,K: int,A: int,B: int] :
% 8.83/4.98 ( ( A2
% 8.83/4.98 = ( plus_plus_int @ K @ A ) )
% 8.83/4.98 => ( ( minus_minus_int @ A2 @ B )
% 8.83/4.98 = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_770_diff__eq__eq,axiom,
% 8.83/4.98 ! [A: real,B: real,C: real] :
% 8.83/4.98 ( ( ( minus_minus_real @ A @ B )
% 8.83/4.98 = C )
% 8.83/4.98 = ( A
% 8.83/4.98 = ( plus_plus_real @ C @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_771_diff__eq__eq,axiom,
% 8.83/4.98 ! [A: rat,B: rat,C: rat] :
% 8.83/4.98 ( ( ( minus_minus_rat @ A @ B )
% 8.83/4.98 = C )
% 8.83/4.98 = ( A
% 8.83/4.98 = ( plus_plus_rat @ C @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_772_diff__eq__eq,axiom,
% 8.83/4.98 ! [A: int,B: int,C: int] :
% 8.83/4.98 ( ( ( minus_minus_int @ A @ B )
% 8.83/4.98 = C )
% 8.83/4.98 = ( A
% 8.83/4.98 = ( plus_plus_int @ C @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_773_eq__diff__eq,axiom,
% 8.83/4.98 ! [A: real,C: real,B: real] :
% 8.83/4.98 ( ( A
% 8.83/4.98 = ( minus_minus_real @ C @ B ) )
% 8.83/4.98 = ( ( plus_plus_real @ A @ B )
% 8.83/4.98 = C ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_774_eq__diff__eq,axiom,
% 8.83/4.98 ! [A: rat,C: rat,B: rat] :
% 8.83/4.98 ( ( A
% 8.83/4.98 = ( minus_minus_rat @ C @ B ) )
% 8.83/4.98 = ( ( plus_plus_rat @ A @ B )
% 8.83/4.98 = C ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_775_eq__diff__eq,axiom,
% 8.83/4.98 ! [A: int,C: int,B: int] :
% 8.83/4.98 ( ( A
% 8.83/4.98 = ( minus_minus_int @ C @ B ) )
% 8.83/4.98 = ( ( plus_plus_int @ A @ B )
% 8.83/4.98 = C ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_776_add__diff__eq,axiom,
% 8.83/4.98 ! [A: real,B: real,C: real] :
% 8.83/4.98 ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 8.83/4.98 = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_777_add__diff__eq,axiom,
% 8.83/4.98 ! [A: rat,B: rat,C: rat] :
% 8.83/4.98 ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 8.83/4.98 = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_778_add__diff__eq,axiom,
% 8.83/4.98 ! [A: int,B: int,C: int] :
% 8.83/4.98 ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 8.83/4.98 = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_779_diff__diff__eq2,axiom,
% 8.83/4.98 ! [A: real,B: real,C: real] :
% 8.83/4.98 ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 8.83/4.98 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_780_diff__diff__eq2,axiom,
% 8.83/4.98 ! [A: rat,B: rat,C: rat] :
% 8.83/4.98 ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 8.83/4.98 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_781_diff__diff__eq2,axiom,
% 8.83/4.98 ! [A: int,B: int,C: int] :
% 8.83/4.98 ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 8.83/4.98 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_782_diff__add__eq,axiom,
% 8.83/4.98 ! [A: real,B: real,C: real] :
% 8.83/4.98 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 8.83/4.98 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_783_diff__add__eq,axiom,
% 8.83/4.98 ! [A: rat,B: rat,C: rat] :
% 8.83/4.98 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 8.83/4.98 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_784_diff__add__eq,axiom,
% 8.83/4.98 ! [A: int,B: int,C: int] :
% 8.83/4.98 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 8.83/4.98 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_785_diff__add__eq__diff__diff__swap,axiom,
% 8.83/4.98 ! [A: real,B: real,C: real] :
% 8.83/4.98 ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 8.83/4.98 = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_786_diff__add__eq__diff__diff__swap,axiom,
% 8.83/4.98 ! [A: rat,B: rat,C: rat] :
% 8.83/4.98 ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 8.83/4.98 = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_787_diff__add__eq__diff__diff__swap,axiom,
% 8.83/4.98 ! [A: int,B: int,C: int] :
% 8.83/4.98 ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 8.83/4.98 = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_788_add__implies__diff,axiom,
% 8.83/4.98 ! [C: real,B: real,A: real] :
% 8.83/4.98 ( ( ( plus_plus_real @ C @ B )
% 8.83/4.98 = A )
% 8.83/4.98 => ( C
% 8.83/4.98 = ( minus_minus_real @ A @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_789_add__implies__diff,axiom,
% 8.83/4.98 ! [C: rat,B: rat,A: rat] :
% 8.83/4.98 ( ( ( plus_plus_rat @ C @ B )
% 8.83/4.98 = A )
% 8.83/4.98 => ( C
% 8.83/4.98 = ( minus_minus_rat @ A @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_790_add__implies__diff,axiom,
% 8.83/4.98 ! [C: nat,B: nat,A: nat] :
% 8.83/4.98 ( ( ( plus_plus_nat @ C @ B )
% 8.83/4.98 = A )
% 8.83/4.98 => ( C
% 8.83/4.98 = ( minus_minus_nat @ A @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_791_add__implies__diff,axiom,
% 8.83/4.98 ! [C: int,B: int,A: int] :
% 8.83/4.98 ( ( ( plus_plus_int @ C @ B )
% 8.83/4.98 = A )
% 8.83/4.98 => ( C
% 8.83/4.98 = ( minus_minus_int @ A @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_792_diff__diff__eq,axiom,
% 8.83/4.98 ! [A: real,B: real,C: real] :
% 8.83/4.98 ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 8.83/4.98 = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_793_diff__diff__eq,axiom,
% 8.83/4.98 ! [A: rat,B: rat,C: rat] :
% 8.83/4.98 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 8.83/4.98 = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_794_diff__diff__eq,axiom,
% 8.83/4.98 ! [A: nat,B: nat,C: nat] :
% 8.83/4.98 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 8.83/4.98 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_795_diff__diff__eq,axiom,
% 8.83/4.98 ! [A: int,B: int,C: int] :
% 8.83/4.98 ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 8.83/4.98 = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_796_zero__induct__lemma,axiom,
% 8.83/4.98 ! [P: nat > $o,K: nat,I2: nat] :
% 8.83/4.98 ( ( P @ K )
% 8.83/4.98 => ( ! [N2: nat] :
% 8.83/4.98 ( ( P @ ( suc @ N2 ) )
% 8.83/4.98 => ( P @ N2 ) )
% 8.83/4.98 => ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_797_comm__monoid__mult__class_Omult__1,axiom,
% 8.83/4.98 ! [A: complex] :
% 8.83/4.98 ( ( times_times_complex @ one_one_complex @ A )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_798_comm__monoid__mult__class_Omult__1,axiom,
% 8.83/4.98 ! [A: real] :
% 8.83/4.98 ( ( times_times_real @ one_one_real @ A )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_799_comm__monoid__mult__class_Omult__1,axiom,
% 8.83/4.98 ! [A: rat] :
% 8.83/4.98 ( ( times_times_rat @ one_one_rat @ A )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_800_comm__monoid__mult__class_Omult__1,axiom,
% 8.83/4.98 ! [A: nat] :
% 8.83/4.98 ( ( times_times_nat @ one_one_nat @ A )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_801_comm__monoid__mult__class_Omult__1,axiom,
% 8.83/4.98 ! [A: int] :
% 8.83/4.98 ( ( times_times_int @ one_one_int @ A )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_802_mult_Ocomm__neutral,axiom,
% 8.83/4.98 ! [A: complex] :
% 8.83/4.98 ( ( times_times_complex @ A @ one_one_complex )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_803_mult_Ocomm__neutral,axiom,
% 8.83/4.98 ! [A: real] :
% 8.83/4.98 ( ( times_times_real @ A @ one_one_real )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_804_mult_Ocomm__neutral,axiom,
% 8.83/4.98 ! [A: rat] :
% 8.83/4.98 ( ( times_times_rat @ A @ one_one_rat )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_805_mult_Ocomm__neutral,axiom,
% 8.83/4.98 ! [A: nat] :
% 8.83/4.98 ( ( times_times_nat @ A @ one_one_nat )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_806_mult_Ocomm__neutral,axiom,
% 8.83/4.98 ! [A: int] :
% 8.83/4.98 ( ( times_times_int @ A @ one_one_int )
% 8.83/4.98 = A ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_807_less__imp__diff__less,axiom,
% 8.83/4.98 ! [J: nat,K: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_nat @ J @ K )
% 8.83/4.98 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_808_diff__less__mono2,axiom,
% 8.83/4.98 ! [M: nat,N: nat,L: nat] :
% 8.83/4.98 ( ( ord_less_nat @ M @ N )
% 8.83/4.98 => ( ( ord_less_nat @ M @ L )
% 8.83/4.98 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_809_power__commuting__commutes,axiom,
% 8.83/4.98 ! [X2: complex,Y3: complex,N: nat] :
% 8.83/4.98 ( ( ( times_times_complex @ X2 @ Y3 )
% 8.83/4.98 = ( times_times_complex @ Y3 @ X2 ) )
% 8.83/4.98 => ( ( times_times_complex @ ( power_power_complex @ X2 @ N ) @ Y3 )
% 8.83/4.98 = ( times_times_complex @ Y3 @ ( power_power_complex @ X2 @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_810_power__commuting__commutes,axiom,
% 8.83/4.98 ! [X2: real,Y3: real,N: nat] :
% 8.83/4.98 ( ( ( times_times_real @ X2 @ Y3 )
% 8.83/4.98 = ( times_times_real @ Y3 @ X2 ) )
% 8.83/4.98 => ( ( times_times_real @ ( power_power_real @ X2 @ N ) @ Y3 )
% 8.83/4.98 = ( times_times_real @ Y3 @ ( power_power_real @ X2 @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_811_power__commuting__commutes,axiom,
% 8.83/4.98 ! [X2: rat,Y3: rat,N: nat] :
% 8.83/4.98 ( ( ( times_times_rat @ X2 @ Y3 )
% 8.83/4.98 = ( times_times_rat @ Y3 @ X2 ) )
% 8.83/4.98 => ( ( times_times_rat @ ( power_power_rat @ X2 @ N ) @ Y3 )
% 8.83/4.98 = ( times_times_rat @ Y3 @ ( power_power_rat @ X2 @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_812_power__commuting__commutes,axiom,
% 8.83/4.98 ! [X2: nat,Y3: nat,N: nat] :
% 8.83/4.98 ( ( ( times_times_nat @ X2 @ Y3 )
% 8.83/4.98 = ( times_times_nat @ Y3 @ X2 ) )
% 8.83/4.98 => ( ( times_times_nat @ ( power_power_nat @ X2 @ N ) @ Y3 )
% 8.83/4.98 = ( times_times_nat @ Y3 @ ( power_power_nat @ X2 @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_813_power__commuting__commutes,axiom,
% 8.83/4.98 ! [X2: int,Y3: int,N: nat] :
% 8.83/4.98 ( ( ( times_times_int @ X2 @ Y3 )
% 8.83/4.98 = ( times_times_int @ Y3 @ X2 ) )
% 8.83/4.98 => ( ( times_times_int @ ( power_power_int @ X2 @ N ) @ Y3 )
% 8.83/4.98 = ( times_times_int @ Y3 @ ( power_power_int @ X2 @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_814_power__mult__distrib,axiom,
% 8.83/4.98 ! [A: complex,B: complex,N: nat] :
% 8.83/4.98 ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
% 8.83/4.98 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_815_power__mult__distrib,axiom,
% 8.83/4.98 ! [A: real,B: real,N: nat] :
% 8.83/4.98 ( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
% 8.83/4.98 = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_816_power__mult__distrib,axiom,
% 8.83/4.98 ! [A: rat,B: rat,N: nat] :
% 8.83/4.98 ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
% 8.83/4.98 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_817_power__mult__distrib,axiom,
% 8.83/4.98 ! [A: nat,B: nat,N: nat] :
% 8.83/4.98 ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
% 8.83/4.98 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_818_power__mult__distrib,axiom,
% 8.83/4.98 ! [A: int,B: int,N: nat] :
% 8.83/4.98 ( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
% 8.83/4.98 = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_819_power__commutes,axiom,
% 8.83/4.98 ! [A: complex,N: nat] :
% 8.83/4.98 ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
% 8.83/4.98 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_820_power__commutes,axiom,
% 8.83/4.98 ! [A: real,N: nat] :
% 8.83/4.98 ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
% 8.83/4.98 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_821_power__commutes,axiom,
% 8.83/4.98 ! [A: rat,N: nat] :
% 8.83/4.98 ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
% 8.83/4.98 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_822_power__commutes,axiom,
% 8.83/4.98 ! [A: nat,N: nat] :
% 8.83/4.98 ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
% 8.83/4.98 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_823_power__commutes,axiom,
% 8.83/4.98 ! [A: int,N: nat] :
% 8.83/4.98 ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
% 8.83/4.98 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_824_Suc__mult__cancel1,axiom,
% 8.83/4.98 ! [K: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 8.83/4.98 = ( times_times_nat @ ( suc @ K ) @ N ) )
% 8.83/4.98 = ( M = N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_825_eq__diff__iff,axiom,
% 8.83/4.98 ! [K: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ K @ M )
% 8.83/4.98 => ( ( ord_less_eq_nat @ K @ N )
% 8.83/4.98 => ( ( ( minus_minus_nat @ M @ K )
% 8.83/4.98 = ( minus_minus_nat @ N @ K ) )
% 8.83/4.98 = ( M = N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_826_le__diff__iff,axiom,
% 8.83/4.98 ! [K: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ K @ M )
% 8.83/4.98 => ( ( ord_less_eq_nat @ K @ N )
% 8.83/4.98 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 8.83/4.98 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_827_Nat_Odiff__diff__eq,axiom,
% 8.83/4.98 ! [K: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ K @ M )
% 8.83/4.98 => ( ( ord_less_eq_nat @ K @ N )
% 8.83/4.98 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 8.83/4.98 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_828_diff__le__mono,axiom,
% 8.83/4.98 ! [M: nat,N: nat,L: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ M @ N )
% 8.83/4.98 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_829_diff__le__self,axiom,
% 8.83/4.98 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_830_le__diff__iff_H,axiom,
% 8.83/4.98 ! [A: nat,C: nat,B: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ A @ C )
% 8.83/4.98 => ( ( ord_less_eq_nat @ B @ C )
% 8.83/4.98 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 8.83/4.98 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_831_diff__le__mono2,axiom,
% 8.83/4.98 ! [M: nat,N: nat,L: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ M @ N )
% 8.83/4.98 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_832_power__mult,axiom,
% 8.83/4.98 ! [A: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
% 8.83/4.98 = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_833_power__mult,axiom,
% 8.83/4.98 ! [A: real,M: nat,N: nat] :
% 8.83/4.98 ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
% 8.83/4.98 = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_834_power__mult,axiom,
% 8.83/4.98 ! [A: int,M: nat,N: nat] :
% 8.83/4.98 ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
% 8.83/4.98 = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_835_power__mult,axiom,
% 8.83/4.98 ! [A: complex,M: nat,N: nat] :
% 8.83/4.98 ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
% 8.83/4.98 = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_836_Nat_Odiff__cancel,axiom,
% 8.83/4.98 ! [K: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 8.83/4.98 = ( minus_minus_nat @ M @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_837_diff__cancel2,axiom,
% 8.83/4.98 ! [M: nat,K: nat,N: nat] :
% 8.83/4.98 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 8.83/4.98 = ( minus_minus_nat @ M @ N ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_838_diff__add__inverse,axiom,
% 8.83/4.98 ! [N: nat,M: nat] :
% 8.83/4.98 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 8.83/4.98 = M ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_839_diff__add__inverse2,axiom,
% 8.83/4.98 ! [M: nat,N: nat] :
% 8.83/4.98 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 8.83/4.98 = M ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_840_le__cube,axiom,
% 8.83/4.98 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_841_le__square,axiom,
% 8.83/4.98 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_842_mult__le__mono,axiom,
% 8.83/4.98 ! [I2: nat,J: nat,K: nat,L: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ I2 @ J )
% 8.83/4.98 => ( ( ord_less_eq_nat @ K @ L )
% 8.83/4.98 => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_843_mult__le__mono1,axiom,
% 8.83/4.98 ! [I2: nat,J: nat,K: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ I2 @ J )
% 8.83/4.98 => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_844_mult__le__mono2,axiom,
% 8.83/4.98 ! [I2: nat,J: nat,K: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ I2 @ J )
% 8.83/4.98 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_845_add__mult__distrib,axiom,
% 8.83/4.98 ! [M: nat,N: nat,K: nat] :
% 8.83/4.98 ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 8.83/4.98 = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_846_add__mult__distrib2,axiom,
% 8.83/4.98 ! [K: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 8.83/4.98 = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_847_left__add__mult__distrib,axiom,
% 8.83/4.98 ! [I2: nat,U: nat,J: nat,K: nat] :
% 8.83/4.98 ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 8.83/4.98 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U ) @ K ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_848_nat__mult__1,axiom,
% 8.83/4.98 ! [N: nat] :
% 8.83/4.98 ( ( times_times_nat @ one_one_nat @ N )
% 8.83/4.98 = N ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_849_nat__mult__1__right,axiom,
% 8.83/4.98 ! [N: nat] :
% 8.83/4.98 ( ( times_times_nat @ N @ one_one_nat )
% 8.83/4.98 = N ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_850_div__mult2__eq,axiom,
% 8.83/4.98 ! [M: nat,N: nat,Q3: nat] :
% 8.83/4.98 ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q3 ) )
% 8.83/4.98 = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q3 ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_851_nat__less__add__iff1,axiom,
% 8.83/4.98 ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ J @ I2 )
% 8.83/4.98 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 8.83/4.98 = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_852_nat__less__add__iff2,axiom,
% 8.83/4.98 ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ I2 @ J )
% 8.83/4.98 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 8.83/4.98 = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_853_diff__le__eq,axiom,
% 8.83/4.98 ! [A: real,B: real,C: real] :
% 8.83/4.98 ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 8.83/4.98 = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_854_diff__le__eq,axiom,
% 8.83/4.98 ! [A: rat,B: rat,C: rat] :
% 8.83/4.98 ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 8.83/4.98 = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_855_diff__le__eq,axiom,
% 8.83/4.98 ! [A: int,B: int,C: int] :
% 8.83/4.98 ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
% 8.83/4.98 = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_856_le__diff__eq,axiom,
% 8.83/4.98 ! [A: real,C: real,B: real] :
% 8.83/4.98 ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 8.83/4.98 = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_857_le__diff__eq,axiom,
% 8.83/4.98 ! [A: rat,C: rat,B: rat] :
% 8.83/4.98 ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 8.83/4.98 = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_858_le__diff__eq,axiom,
% 8.83/4.98 ! [A: int,C: int,B: int] :
% 8.83/4.98 ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
% 8.83/4.98 = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_859_diff__add,axiom,
% 8.83/4.98 ! [A: nat,B: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ A @ B )
% 8.83/4.98 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 8.83/4.98 = B ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_860_le__add__diff,axiom,
% 8.83/4.98 ! [A: nat,B: nat,C: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ A @ B )
% 8.83/4.98 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_861_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 8.83/4.98 ! [A: nat,B: nat,C: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ A @ B )
% 8.83/4.98 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 8.83/4.98 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_862_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 8.83/4.98 ! [A: nat,B: nat,C: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ A @ B )
% 8.83/4.98 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 8.83/4.98 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_863_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 8.83/4.98 ! [A: nat,B: nat,C: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ A @ B )
% 8.83/4.98 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 8.83/4.98 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_864_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 8.83/4.98 ! [A: nat,B: nat,C: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ A @ B )
% 8.83/4.98 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 8.83/4.98 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_865_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 8.83/4.98 ! [A: nat,B: nat,C: nat] :
% 8.83/4.98 ( ( ord_less_eq_nat @ A @ B )
% 8.83/4.98 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 8.83/4.98 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 8.83/4.98
% 8.83/4.98 thf(fact_866_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 8.83/4.99 ! [A: nat,B: nat,C: nat] :
% 8.83/4.99 ( ( ord_less_eq_nat @ A @ B )
% 8.83/4.99 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 8.83/4.99 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_867_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 8.83/4.99 ! [A: nat,B: nat] :
% 8.83/4.99 ( ( ord_less_eq_nat @ A @ B )
% 8.83/4.99 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 8.83/4.99 = B ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_868_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 8.83/4.99 ! [A: nat,B: nat,C: nat] :
% 8.83/4.99 ( ( ord_less_eq_nat @ A @ B )
% 8.83/4.99 => ( ( ord_less_eq_nat @ A @ B )
% 8.83/4.99 => ( ( ( minus_minus_nat @ B @ A )
% 8.83/4.99 = C )
% 8.83/4.99 = ( B
% 8.83/4.99 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_869_diff__less__eq,axiom,
% 8.83/4.99 ! [A: real,B: real,C: real] :
% 8.83/4.99 ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
% 8.83/4.99 = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_870_diff__less__eq,axiom,
% 8.83/4.99 ! [A: rat,B: rat,C: rat] :
% 8.83/4.99 ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 8.83/4.99 = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_871_diff__less__eq,axiom,
% 8.83/4.99 ! [A: int,B: int,C: int] :
% 8.83/4.99 ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
% 8.83/4.99 = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_872_less__diff__eq,axiom,
% 8.83/4.99 ! [A: real,C: real,B: real] :
% 8.83/4.99 ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
% 8.83/4.99 = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_873_less__diff__eq,axiom,
% 8.83/4.99 ! [A: rat,C: rat,B: rat] :
% 8.83/4.99 ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 8.83/4.99 = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_874_less__diff__eq,axiom,
% 8.83/4.99 ! [A: int,C: int,B: int] :
% 8.83/4.99 ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
% 8.83/4.99 = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_875_mult__numeral__1__right,axiom,
% 8.83/4.99 ! [A: complex] :
% 8.83/4.99 ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_876_mult__numeral__1__right,axiom,
% 8.83/4.99 ! [A: real] :
% 8.83/4.99 ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_877_mult__numeral__1__right,axiom,
% 8.83/4.99 ! [A: rat] :
% 8.83/4.99 ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_878_mult__numeral__1__right,axiom,
% 8.83/4.99 ! [A: nat] :
% 8.83/4.99 ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_879_mult__numeral__1__right,axiom,
% 8.83/4.99 ! [A: int] :
% 8.83/4.99 ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_880_mult__numeral__1,axiom,
% 8.83/4.99 ! [A: complex] :
% 8.83/4.99 ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_881_mult__numeral__1,axiom,
% 8.83/4.99 ! [A: real] :
% 8.83/4.99 ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_882_mult__numeral__1,axiom,
% 8.83/4.99 ! [A: rat] :
% 8.83/4.99 ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_883_mult__numeral__1,axiom,
% 8.83/4.99 ! [A: nat] :
% 8.83/4.99 ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_884_mult__numeral__1,axiom,
% 8.83/4.99 ! [A: int] :
% 8.83/4.99 ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_885_diff__less__Suc,axiom,
% 8.83/4.99 ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_886_Suc__diff__Suc,axiom,
% 8.83/4.99 ! [N: nat,M: nat] :
% 8.83/4.99 ( ( ord_less_nat @ N @ M )
% 8.83/4.99 => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 8.83/4.99 = ( minus_minus_nat @ M @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_887_left__right__inverse__power,axiom,
% 8.83/4.99 ! [X2: complex,Y3: complex,N: nat] :
% 8.83/4.99 ( ( ( times_times_complex @ X2 @ Y3 )
% 8.83/4.99 = one_one_complex )
% 8.83/4.99 => ( ( times_times_complex @ ( power_power_complex @ X2 @ N ) @ ( power_power_complex @ Y3 @ N ) )
% 8.83/4.99 = one_one_complex ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_888_left__right__inverse__power,axiom,
% 8.83/4.99 ! [X2: real,Y3: real,N: nat] :
% 8.83/4.99 ( ( ( times_times_real @ X2 @ Y3 )
% 8.83/4.99 = one_one_real )
% 8.83/4.99 => ( ( times_times_real @ ( power_power_real @ X2 @ N ) @ ( power_power_real @ Y3 @ N ) )
% 8.83/4.99 = one_one_real ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_889_left__right__inverse__power,axiom,
% 8.83/4.99 ! [X2: rat,Y3: rat,N: nat] :
% 8.83/4.99 ( ( ( times_times_rat @ X2 @ Y3 )
% 8.83/4.99 = one_one_rat )
% 8.83/4.99 => ( ( times_times_rat @ ( power_power_rat @ X2 @ N ) @ ( power_power_rat @ Y3 @ N ) )
% 8.83/4.99 = one_one_rat ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_890_left__right__inverse__power,axiom,
% 8.83/4.99 ! [X2: nat,Y3: nat,N: nat] :
% 8.83/4.99 ( ( ( times_times_nat @ X2 @ Y3 )
% 8.83/4.99 = one_one_nat )
% 8.83/4.99 => ( ( times_times_nat @ ( power_power_nat @ X2 @ N ) @ ( power_power_nat @ Y3 @ N ) )
% 8.83/4.99 = one_one_nat ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_891_left__right__inverse__power,axiom,
% 8.83/4.99 ! [X2: int,Y3: int,N: nat] :
% 8.83/4.99 ( ( ( times_times_int @ X2 @ Y3 )
% 8.83/4.99 = one_one_int )
% 8.83/4.99 => ( ( times_times_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y3 @ N ) )
% 8.83/4.99 = one_one_int ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_892_Suc__diff__le,axiom,
% 8.83/4.99 ! [N: nat,M: nat] :
% 8.83/4.99 ( ( ord_less_eq_nat @ N @ M )
% 8.83/4.99 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 8.83/4.99 = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_893_power__Suc2,axiom,
% 8.83/4.99 ! [A: complex,N: nat] :
% 8.83/4.99 ( ( power_power_complex @ A @ ( suc @ N ) )
% 8.83/4.99 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_894_power__Suc2,axiom,
% 8.83/4.99 ! [A: real,N: nat] :
% 8.83/4.99 ( ( power_power_real @ A @ ( suc @ N ) )
% 8.83/4.99 = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_895_power__Suc2,axiom,
% 8.83/4.99 ! [A: rat,N: nat] :
% 8.83/4.99 ( ( power_power_rat @ A @ ( suc @ N ) )
% 8.83/4.99 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_896_power__Suc2,axiom,
% 8.83/4.99 ! [A: nat,N: nat] :
% 8.83/4.99 ( ( power_power_nat @ A @ ( suc @ N ) )
% 8.83/4.99 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_897_power__Suc2,axiom,
% 8.83/4.99 ! [A: int,N: nat] :
% 8.83/4.99 ( ( power_power_int @ A @ ( suc @ N ) )
% 8.83/4.99 = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_898_power__Suc,axiom,
% 8.83/4.99 ! [A: complex,N: nat] :
% 8.83/4.99 ( ( power_power_complex @ A @ ( suc @ N ) )
% 8.83/4.99 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_899_power__Suc,axiom,
% 8.83/4.99 ! [A: real,N: nat] :
% 8.83/4.99 ( ( power_power_real @ A @ ( suc @ N ) )
% 8.83/4.99 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_900_power__Suc,axiom,
% 8.83/4.99 ! [A: rat,N: nat] :
% 8.83/4.99 ( ( power_power_rat @ A @ ( suc @ N ) )
% 8.83/4.99 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_901_power__Suc,axiom,
% 8.83/4.99 ! [A: nat,N: nat] :
% 8.83/4.99 ( ( power_power_nat @ A @ ( suc @ N ) )
% 8.83/4.99 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_902_power__Suc,axiom,
% 8.83/4.99 ! [A: int,N: nat] :
% 8.83/4.99 ( ( power_power_int @ A @ ( suc @ N ) )
% 8.83/4.99 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_903_less__diff__iff,axiom,
% 8.83/4.99 ! [K: nat,M: nat,N: nat] :
% 8.83/4.99 ( ( ord_less_eq_nat @ K @ M )
% 8.83/4.99 => ( ( ord_less_eq_nat @ K @ N )
% 8.83/4.99 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 8.83/4.99 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_904_diff__less__mono,axiom,
% 8.83/4.99 ! [A: nat,B: nat,C: nat] :
% 8.83/4.99 ( ( ord_less_nat @ A @ B )
% 8.83/4.99 => ( ( ord_less_eq_nat @ C @ A )
% 8.83/4.99 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_905_less__diff__conv,axiom,
% 8.83/4.99 ! [I2: nat,J: nat,K: nat] :
% 8.83/4.99 ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 8.83/4.99 = ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_906_add__diff__inverse__nat,axiom,
% 8.83/4.99 ! [M: nat,N: nat] :
% 8.83/4.99 ( ~ ( ord_less_nat @ M @ N )
% 8.83/4.99 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 8.83/4.99 = M ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_907_diff__Suc__eq__diff__pred,axiom,
% 8.83/4.99 ! [M: nat,N: nat] :
% 8.83/4.99 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 8.83/4.99 = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_908_Suc__mult__less__cancel1,axiom,
% 8.83/4.99 ! [K: nat,M: nat,N: nat] :
% 8.83/4.99 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 8.83/4.99 = ( ord_less_nat @ M @ N ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_909_le__diff__conv,axiom,
% 8.83/4.99 ! [J: nat,K: nat,I2: nat] :
% 8.83/4.99 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
% 8.83/4.99 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_910_Nat_Ole__diff__conv2,axiom,
% 8.83/4.99 ! [K: nat,J: nat,I2: nat] :
% 8.83/4.99 ( ( ord_less_eq_nat @ K @ J )
% 8.83/4.99 => ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 8.83/4.99 = ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_911_Nat_Odiff__add__assoc,axiom,
% 8.83/4.99 ! [K: nat,J: nat,I2: nat] :
% 8.83/4.99 ( ( ord_less_eq_nat @ K @ J )
% 8.83/4.99 => ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
% 8.83/4.99 = ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_912_Nat_Odiff__add__assoc2,axiom,
% 8.83/4.99 ! [K: nat,J: nat,I2: nat] :
% 8.83/4.99 ( ( ord_less_eq_nat @ K @ J )
% 8.83/4.99 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
% 8.83/4.99 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_913_Nat_Ole__imp__diff__is__add,axiom,
% 8.83/4.99 ! [I2: nat,J: nat,K: nat] :
% 8.83/4.99 ( ( ord_less_eq_nat @ I2 @ J )
% 8.83/4.99 => ( ( ( minus_minus_nat @ J @ I2 )
% 8.83/4.99 = K )
% 8.83/4.99 = ( J
% 8.83/4.99 = ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_914_power__add,axiom,
% 8.83/4.99 ! [A: complex,M: nat,N: nat] :
% 8.83/4.99 ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
% 8.83/4.99 = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_915_power__add,axiom,
% 8.83/4.99 ! [A: real,M: nat,N: nat] :
% 8.83/4.99 ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
% 8.83/4.99 = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_916_power__add,axiom,
% 8.83/4.99 ! [A: rat,M: nat,N: nat] :
% 8.83/4.99 ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
% 8.83/4.99 = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_917_power__add,axiom,
% 8.83/4.99 ! [A: nat,M: nat,N: nat] :
% 8.83/4.99 ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
% 8.83/4.99 = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_918_power__add,axiom,
% 8.83/4.99 ! [A: int,M: nat,N: nat] :
% 8.83/4.99 ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
% 8.83/4.99 = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_919_Suc__mult__le__cancel1,axiom,
% 8.83/4.99 ! [K: nat,M: nat,N: nat] :
% 8.83/4.99 ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 8.83/4.99 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_920_mult__Suc,axiom,
% 8.83/4.99 ! [M: nat,N: nat] :
% 8.83/4.99 ( ( times_times_nat @ ( suc @ M ) @ N )
% 8.83/4.99 = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_921_less__mult__imp__div__less,axiom,
% 8.83/4.99 ! [M: nat,I2: nat,N: nat] :
% 8.83/4.99 ( ( ord_less_nat @ M @ ( times_times_nat @ I2 @ N ) )
% 8.83/4.99 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I2 ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_922_div__times__less__eq__dividend,axiom,
% 8.83/4.99 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_923_times__div__less__eq__dividend,axiom,
% 8.83/4.99 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_924_power__odd__eq,axiom,
% 8.83/4.99 ! [A: complex,N: nat] :
% 8.83/4.99 ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.83/4.99 = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_925_power__odd__eq,axiom,
% 8.83/4.99 ! [A: real,N: nat] :
% 8.83/4.99 ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.83/4.99 = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_926_power__odd__eq,axiom,
% 8.83/4.99 ! [A: rat,N: nat] :
% 8.83/4.99 ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.83/4.99 = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_927_power__odd__eq,axiom,
% 8.83/4.99 ! [A: nat,N: nat] :
% 8.83/4.99 ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.83/4.99 = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_928_power__odd__eq,axiom,
% 8.83/4.99 ! [A: int,N: nat] :
% 8.83/4.99 ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.83/4.99 = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_929_power2__diff,axiom,
% 8.83/4.99 ! [X2: complex,Y3: complex] :
% 8.83/4.99 ( ( power_power_complex @ ( minus_minus_complex @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_930_power2__diff,axiom,
% 8.83/4.99 ! [X2: real,Y3: real] :
% 8.83/4.99 ( ( power_power_real @ ( minus_minus_real @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_931_power2__diff,axiom,
% 8.83/4.99 ! [X2: rat,Y3: rat] :
% 8.83/4.99 ( ( power_power_rat @ ( minus_minus_rat @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_932_power2__diff,axiom,
% 8.83/4.99 ! [X2: int,Y3: int] :
% 8.83/4.99 ( ( power_power_int @ ( minus_minus_int @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_933_power__less__power__Suc,axiom,
% 8.83/4.99 ! [A: real,N: nat] :
% 8.83/4.99 ( ( ord_less_real @ one_one_real @ A )
% 8.83/4.99 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_934_power__less__power__Suc,axiom,
% 8.83/4.99 ! [A: rat,N: nat] :
% 8.83/4.99 ( ( ord_less_rat @ one_one_rat @ A )
% 8.83/4.99 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_935_power__less__power__Suc,axiom,
% 8.83/4.99 ! [A: nat,N: nat] :
% 8.83/4.99 ( ( ord_less_nat @ one_one_nat @ A )
% 8.83/4.99 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_936_power__less__power__Suc,axiom,
% 8.83/4.99 ! [A: int,N: nat] :
% 8.83/4.99 ( ( ord_less_int @ one_one_int @ A )
% 8.83/4.99 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_937_power__gt1__lemma,axiom,
% 8.83/4.99 ! [A: real,N: nat] :
% 8.83/4.99 ( ( ord_less_real @ one_one_real @ A )
% 8.83/4.99 => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_938_power__gt1__lemma,axiom,
% 8.83/4.99 ! [A: rat,N: nat] :
% 8.83/4.99 ( ( ord_less_rat @ one_one_rat @ A )
% 8.83/4.99 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_939_power__gt1__lemma,axiom,
% 8.83/4.99 ! [A: nat,N: nat] :
% 8.83/4.99 ( ( ord_less_nat @ one_one_nat @ A )
% 8.83/4.99 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_940_power__gt1__lemma,axiom,
% 8.83/4.99 ! [A: int,N: nat] :
% 8.83/4.99 ( ( ord_less_int @ one_one_int @ A )
% 8.83/4.99 => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_941_less__diff__conv2,axiom,
% 8.83/4.99 ! [K: nat,J: nat,I2: nat] :
% 8.83/4.99 ( ( ord_less_eq_nat @ K @ J )
% 8.83/4.99 => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
% 8.83/4.99 = ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_942_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
% 8.83/4.99 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 8.83/4.99 ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 8.83/4.99 = one_one_nat ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_943_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
% 8.83/4.99 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 8.83/4.99 ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 8.83/4.99 = one_one_nat ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_944_power2__commute,axiom,
% 8.83/4.99 ! [X2: complex,Y3: complex] :
% 8.83/4.99 ( ( power_power_complex @ ( minus_minus_complex @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( power_power_complex @ ( minus_minus_complex @ Y3 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_945_power2__commute,axiom,
% 8.83/4.99 ! [X2: real,Y3: real] :
% 8.83/4.99 ( ( power_power_real @ ( minus_minus_real @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( power_power_real @ ( minus_minus_real @ Y3 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_946_power2__commute,axiom,
% 8.83/4.99 ! [X2: rat,Y3: rat] :
% 8.83/4.99 ( ( power_power_rat @ ( minus_minus_rat @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( power_power_rat @ ( minus_minus_rat @ Y3 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_947_power2__commute,axiom,
% 8.83/4.99 ! [X2: int,Y3: int] :
% 8.83/4.99 ( ( power_power_int @ ( minus_minus_int @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( power_power_int @ ( minus_minus_int @ Y3 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_948_left__add__twice,axiom,
% 8.83/4.99 ! [A: complex,B: complex] :
% 8.83/4.99 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
% 8.83/4.99 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_949_left__add__twice,axiom,
% 8.83/4.99 ! [A: real,B: real] :
% 8.83/4.99 ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
% 8.83/4.99 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_950_left__add__twice,axiom,
% 8.83/4.99 ! [A: rat,B: rat] :
% 8.83/4.99 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 8.83/4.99 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_951_left__add__twice,axiom,
% 8.83/4.99 ! [A: nat,B: nat] :
% 8.83/4.99 ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 8.83/4.99 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_952_left__add__twice,axiom,
% 8.83/4.99 ! [A: int,B: int] :
% 8.83/4.99 ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
% 8.83/4.99 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_953_mult__2__right,axiom,
% 8.83/4.99 ! [Z: complex] :
% 8.83/4.99 ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( plus_plus_complex @ Z @ Z ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_954_mult__2__right,axiom,
% 8.83/4.99 ! [Z: real] :
% 8.83/4.99 ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( plus_plus_real @ Z @ Z ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_955_mult__2__right,axiom,
% 8.83/4.99 ! [Z: rat] :
% 8.83/4.99 ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( plus_plus_rat @ Z @ Z ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_956_mult__2__right,axiom,
% 8.83/4.99 ! [Z: nat] :
% 8.83/4.99 ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( plus_plus_nat @ Z @ Z ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_957_mult__2__right,axiom,
% 8.83/4.99 ! [Z: int] :
% 8.83/4.99 ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( plus_plus_int @ Z @ Z ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_958_mult__2,axiom,
% 8.83/4.99 ! [Z: complex] :
% 8.83/4.99 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
% 8.83/4.99 = ( plus_plus_complex @ Z @ Z ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_959_mult__2,axiom,
% 8.83/4.99 ! [Z: real] :
% 8.83/4.99 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
% 8.83/4.99 = ( plus_plus_real @ Z @ Z ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_960_mult__2,axiom,
% 8.83/4.99 ! [Z: rat] :
% 8.83/4.99 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
% 8.83/4.99 = ( plus_plus_rat @ Z @ Z ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_961_mult__2,axiom,
% 8.83/4.99 ! [Z: nat] :
% 8.83/4.99 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
% 8.83/4.99 = ( plus_plus_nat @ Z @ Z ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_962_mult__2,axiom,
% 8.83/4.99 ! [Z: int] :
% 8.83/4.99 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
% 8.83/4.99 = ( plus_plus_int @ Z @ Z ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_963_power2__eq__square,axiom,
% 8.83/4.99 ! [A: complex] :
% 8.83/4.99 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( times_times_complex @ A @ A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_964_power2__eq__square,axiom,
% 8.83/4.99 ! [A: real] :
% 8.83/4.99 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( times_times_real @ A @ A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_965_power2__eq__square,axiom,
% 8.83/4.99 ! [A: rat] :
% 8.83/4.99 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( times_times_rat @ A @ A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_966_power2__eq__square,axiom,
% 8.83/4.99 ! [A: nat] :
% 8.83/4.99 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( times_times_nat @ A @ A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_967_power2__eq__square,axiom,
% 8.83/4.99 ! [A: int] :
% 8.83/4.99 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( times_times_int @ A @ A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_968_power4__eq__xxxx,axiom,
% 8.83/4.99 ! [X2: complex] :
% 8.83/4.99 ( ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.83/4.99 = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_969_power4__eq__xxxx,axiom,
% 8.83/4.99 ! [X2: real] :
% 8.83/4.99 ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.83/4.99 = ( times_times_real @ ( times_times_real @ ( times_times_real @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_970_power4__eq__xxxx,axiom,
% 8.83/4.99 ! [X2: rat] :
% 8.83/4.99 ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.83/4.99 = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_971_power4__eq__xxxx,axiom,
% 8.83/4.99 ! [X2: nat] :
% 8.83/4.99 ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.83/4.99 = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_972_power4__eq__xxxx,axiom,
% 8.83/4.99 ! [X2: int] :
% 8.83/4.99 ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.83/4.99 = ( times_times_int @ ( times_times_int @ ( times_times_int @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_973_power__even__eq,axiom,
% 8.83/4.99 ! [A: nat,N: nat] :
% 8.83/4.99 ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.83/4.99 = ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_974_power__even__eq,axiom,
% 8.83/4.99 ! [A: real,N: nat] :
% 8.83/4.99 ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.83/4.99 = ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_975_power__even__eq,axiom,
% 8.83/4.99 ! [A: int,N: nat] :
% 8.83/4.99 ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.83/4.99 = ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_976_power__even__eq,axiom,
% 8.83/4.99 ! [A: complex,N: nat] :
% 8.83/4.99 ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.83/4.99 = ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_977_div__nat__eqI,axiom,
% 8.83/4.99 ! [N: nat,Q3: nat,M: nat] :
% 8.83/4.99 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
% 8.83/4.99 => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
% 8.83/4.99 => ( ( divide_divide_nat @ M @ N )
% 8.83/4.99 = Q3 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_978_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
% 8.83/4.99 ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 8.83/4.99 ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X2 )
% 8.83/4.99 = one_one_nat ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_979_diff__le__diff__pow,axiom,
% 8.83/4.99 ! [K: nat,M: nat,N: nat] :
% 8.83/4.99 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 8.83/4.99 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_980_power2__sum,axiom,
% 8.83/4.99 ! [X2: complex,Y3: complex] :
% 8.83/4.99 ( ( power_power_complex @ ( plus_plus_complex @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_981_power2__sum,axiom,
% 8.83/4.99 ! [X2: real,Y3: real] :
% 8.83/4.99 ( ( power_power_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_982_power2__sum,axiom,
% 8.83/4.99 ! [X2: rat,Y3: rat] :
% 8.83/4.99 ( ( power_power_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_983_power2__sum,axiom,
% 8.83/4.99 ! [X2: nat,Y3: nat] :
% 8.83/4.99 ( ( power_power_nat @ ( plus_plus_nat @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_984_power2__sum,axiom,
% 8.83/4.99 ! [X2: int,Y3: int] :
% 8.83/4.99 ( ( power_power_int @ ( plus_plus_int @ X2 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/4.99 = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_985_two__realpow__ge__one,axiom,
% 8.83/4.99 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_986_invar__vebt_Ointros_I2_J,axiom,
% 8.83/4.99 ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 8.83/4.99 ( ! [X5: vEBT_VEBT] :
% 8.83/4.99 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.83/4.99 => ( vEBT_invar_vebt @ X5 @ N ) )
% 8.83/4.99 => ( ( vEBT_invar_vebt @ Summary @ M )
% 8.83/4.99 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 8.83/4.99 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.83/4.99 => ( ( M = N )
% 8.83/4.99 => ( ( Deg
% 8.83/4.99 = ( plus_plus_nat @ N @ M ) )
% 8.83/4.99 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
% 8.83/4.99 => ( ! [X5: vEBT_VEBT] :
% 8.83/4.99 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.83/4.99 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
% 8.83/4.99 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_987_invar__vebt_Ointros_I3_J,axiom,
% 8.83/4.99 ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 8.83/4.99 ( ! [X5: vEBT_VEBT] :
% 8.83/4.99 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.83/4.99 => ( vEBT_invar_vebt @ X5 @ N ) )
% 8.83/4.99 => ( ( vEBT_invar_vebt @ Summary @ M )
% 8.83/4.99 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 8.83/4.99 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.83/4.99 => ( ( M
% 8.83/4.99 = ( suc @ N ) )
% 8.83/4.99 => ( ( Deg
% 8.83/4.99 = ( plus_plus_nat @ N @ M ) )
% 8.83/4.99 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
% 8.83/4.99 => ( ! [X5: vEBT_VEBT] :
% 8.83/4.99 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.83/4.99 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
% 8.83/4.99 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_988_field__sum__of__halves,axiom,
% 8.83/4.99 ! [X2: real] :
% 8.83/4.99 ( ( plus_plus_real @ ( divide_divide_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.83/4.99 = X2 ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_989_field__sum__of__halves,axiom,
% 8.83/4.99 ! [X2: rat] :
% 8.83/4.99 ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 8.83/4.99 = X2 ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_990_nested__mint,axiom,
% 8.83/4.99 ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
% 8.83/4.99 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
% 8.83/4.99 => ( ( N
% 8.83/4.99 = ( suc @ ( suc @ Va ) ) )
% 8.83/4.99 => ( ~ ( ord_less_nat @ Ma @ Mi )
% 8.83/4.99 => ( ( Ma != Mi )
% 8.83/4.99 => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_991_summaxma,axiom,
% 8.83/4.99 ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 8.83/4.99 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
% 8.83/4.99 => ( ( Mi != Ma )
% 8.83/4.99 => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
% 8.83/4.99 = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_992_sum__squares__bound,axiom,
% 8.83/4.99 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_993_sum__squares__bound,axiom,
% 8.83/4.99 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_994_le__add__diff__inverse,axiom,
% 8.83/4.99 ! [B: real,A: real] :
% 8.83/4.99 ( ( ord_less_eq_real @ B @ A )
% 8.83/4.99 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 8.83/4.99 = A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_995_le__add__diff__inverse,axiom,
% 8.83/4.99 ! [B: rat,A: rat] :
% 8.83/4.99 ( ( ord_less_eq_rat @ B @ A )
% 8.83/4.99 => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 8.83/4.99 = A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_996_le__add__diff__inverse,axiom,
% 8.83/4.99 ! [B: nat,A: nat] :
% 8.83/4.99 ( ( ord_less_eq_nat @ B @ A )
% 8.83/4.99 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 8.83/4.99 = A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_997_le__add__diff__inverse,axiom,
% 8.83/4.99 ! [B: int,A: int] :
% 8.83/4.99 ( ( ord_less_eq_int @ B @ A )
% 8.83/4.99 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 8.83/4.99 = A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_998_le__add__diff__inverse2,axiom,
% 8.83/4.99 ! [B: real,A: real] :
% 8.83/4.99 ( ( ord_less_eq_real @ B @ A )
% 8.83/4.99 => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 8.83/4.99 = A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_999_le__add__diff__inverse2,axiom,
% 8.83/4.99 ! [B: rat,A: rat] :
% 8.83/4.99 ( ( ord_less_eq_rat @ B @ A )
% 8.83/4.99 => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 8.83/4.99 = A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1000_le__add__diff__inverse2,axiom,
% 8.83/4.99 ! [B: nat,A: nat] :
% 8.83/4.99 ( ( ord_less_eq_nat @ B @ A )
% 8.83/4.99 => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
% 8.83/4.99 = A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1001_le__add__diff__inverse2,axiom,
% 8.83/4.99 ! [B: int,A: int] :
% 8.83/4.99 ( ( ord_less_eq_int @ B @ A )
% 8.83/4.99 => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 8.83/4.99 = A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1002_mul__def,axiom,
% 8.83/4.99 ( vEBT_VEBT_mul
% 8.83/4.99 = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1003_add__def,axiom,
% 8.83/4.99 ( vEBT_VEBT_add
% 8.83/4.99 = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1004_vebt__maxt_Osimps_I3_J,axiom,
% 8.83/4.99 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 8.83/4.99 ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 8.83/4.99 = ( some_nat @ Ma ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1005_vebt__mint_Osimps_I3_J,axiom,
% 8.83/4.99 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 8.83/4.99 ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 8.83/4.99 = ( some_nat @ Mi ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1006_vebt__insert_Osimps_I4_J,axiom,
% 8.83/4.99 ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 8.83/4.99 ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X2 )
% 8.83/4.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1007_enat__ord__number_I1_J,axiom,
% 8.83/4.99 ! [M: num,N: num] :
% 8.83/4.99 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 8.83/4.99 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1008_add__shift,axiom,
% 8.83/4.99 ! [X2: nat,Y3: nat,Z: nat] :
% 8.83/4.99 ( ( ( plus_plus_nat @ X2 @ Y3 )
% 8.83/4.99 = Z )
% 8.83/4.99 = ( ( vEBT_VEBT_add @ ( some_nat @ X2 ) @ ( some_nat @ Y3 ) )
% 8.83/4.99 = ( some_nat @ Z ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1009_mul__shift,axiom,
% 8.83/4.99 ! [X2: nat,Y3: nat,Z: nat] :
% 8.83/4.99 ( ( ( times_times_nat @ X2 @ Y3 )
% 8.83/4.99 = Z )
% 8.83/4.99 = ( ( vEBT_VEBT_mul @ ( some_nat @ X2 ) @ ( some_nat @ Y3 ) )
% 8.83/4.99 = ( some_nat @ Z ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1010_div__by__1,axiom,
% 8.83/4.99 ! [A: complex] :
% 8.83/4.99 ( ( divide1717551699836669952omplex @ A @ one_one_complex )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1011_div__by__1,axiom,
% 8.83/4.99 ! [A: real] :
% 8.83/4.99 ( ( divide_divide_real @ A @ one_one_real )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1012_div__by__1,axiom,
% 8.83/4.99 ! [A: rat] :
% 8.83/4.99 ( ( divide_divide_rat @ A @ one_one_rat )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1013_div__by__1,axiom,
% 8.83/4.99 ! [A: nat] :
% 8.83/4.99 ( ( divide_divide_nat @ A @ one_one_nat )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1014_div__by__1,axiom,
% 8.83/4.99 ! [A: int] :
% 8.83/4.99 ( ( divide_divide_int @ A @ one_one_int )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1015_semiring__norm_I13_J,axiom,
% 8.83/4.99 ! [M: num,N: num] :
% 8.83/4.99 ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.83/4.99 = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1016_semiring__norm_I12_J,axiom,
% 8.83/4.99 ! [N: num] :
% 8.83/4.99 ( ( times_times_num @ one @ N )
% 8.83/4.99 = N ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1017_semiring__norm_I11_J,axiom,
% 8.83/4.99 ! [M: num] :
% 8.83/4.99 ( ( times_times_num @ M @ one )
% 8.83/4.99 = M ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1018_num__double,axiom,
% 8.83/4.99 ! [N: num] :
% 8.83/4.99 ( ( times_times_num @ ( bit0 @ one ) @ N )
% 8.83/4.99 = ( bit0 @ N ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1019_power__mult__numeral,axiom,
% 8.83/4.99 ! [A: nat,M: num,N: num] :
% 8.83/4.99 ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 8.83/4.99 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1020_power__mult__numeral,axiom,
% 8.83/4.99 ! [A: real,M: num,N: num] :
% 8.83/4.99 ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 8.83/4.99 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1021_power__mult__numeral,axiom,
% 8.83/4.99 ! [A: int,M: num,N: num] :
% 8.83/4.99 ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 8.83/4.99 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1022_power__mult__numeral,axiom,
% 8.83/4.99 ! [A: complex,M: num,N: num] :
% 8.83/4.99 ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 8.83/4.99 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1023_option_Ocollapse,axiom,
% 8.83/4.99 ! [Option: option4927543243414619207at_nat] :
% 8.83/4.99 ( ( Option != none_P5556105721700978146at_nat )
% 8.83/4.99 => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
% 8.83/4.99 = Option ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1024_option_Ocollapse,axiom,
% 8.83/4.99 ! [Option: option_nat] :
% 8.83/4.99 ( ( Option != none_nat )
% 8.83/4.99 => ( ( some_nat @ ( the_nat @ Option ) )
% 8.83/4.99 = Option ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1025_option_Ocollapse,axiom,
% 8.83/4.99 ! [Option: option_num] :
% 8.83/4.99 ( ( Option != none_num )
% 8.83/4.99 => ( ( some_num @ ( the_num @ Option ) )
% 8.83/4.99 = Option ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1026_enat__ord__number_I2_J,axiom,
% 8.83/4.99 ! [M: num,N: num] :
% 8.83/4.99 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 8.83/4.99 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1027_real__arch__pow,axiom,
% 8.83/4.99 ! [X2: real,Y3: real] :
% 8.83/4.99 ( ( ord_less_real @ one_one_real @ X2 )
% 8.83/4.99 => ? [N2: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X2 @ N2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1028_add__diff__assoc__enat,axiom,
% 8.83/4.99 ! [Z: extended_enat,Y3: extended_enat,X2: extended_enat] :
% 8.83/4.99 ( ( ord_le2932123472753598470d_enat @ Z @ Y3 )
% 8.83/4.99 => ( ( plus_p3455044024723400733d_enat @ X2 @ ( minus_3235023915231533773d_enat @ Y3 @ Z ) )
% 8.83/4.99 = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X2 @ Y3 ) @ Z ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1029_less__eq__real__def,axiom,
% 8.83/4.99 ( ord_less_eq_real
% 8.83/4.99 = ( ^ [X: real,Y: real] :
% 8.83/4.99 ( ( ord_less_real @ X @ Y )
% 8.83/4.99 | ( X = Y ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1030_complete__real,axiom,
% 8.83/4.99 ! [S3: set_real] :
% 8.83/4.99 ( ? [X3: real] : ( member_real @ X3 @ S3 )
% 8.83/4.99 => ( ? [Z4: real] :
% 8.83/4.99 ! [X5: real] :
% 8.83/4.99 ( ( member_real @ X5 @ S3 )
% 8.83/4.99 => ( ord_less_eq_real @ X5 @ Z4 ) )
% 8.83/4.99 => ? [Y4: real] :
% 8.83/4.99 ( ! [X3: real] :
% 8.83/4.99 ( ( member_real @ X3 @ S3 )
% 8.83/4.99 => ( ord_less_eq_real @ X3 @ Y4 ) )
% 8.83/4.99 & ! [Z4: real] :
% 8.83/4.99 ( ! [X5: real] :
% 8.83/4.99 ( ( member_real @ X5 @ S3 )
% 8.83/4.99 => ( ord_less_eq_real @ X5 @ Z4 ) )
% 8.83/4.99 => ( ord_less_eq_real @ Y4 @ Z4 ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1031_option_Osel,axiom,
% 8.83/4.99 ! [X22: product_prod_nat_nat] :
% 8.83/4.99 ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 8.83/4.99 = X22 ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1032_option_Osel,axiom,
% 8.83/4.99 ! [X22: nat] :
% 8.83/4.99 ( ( the_nat @ ( some_nat @ X22 ) )
% 8.83/4.99 = X22 ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1033_option_Osel,axiom,
% 8.83/4.99 ! [X22: num] :
% 8.83/4.99 ( ( the_num @ ( some_num @ X22 ) )
% 8.83/4.99 = X22 ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1034_option_Oexpand,axiom,
% 8.83/4.99 ! [Option: option_nat,Option2: option_nat] :
% 8.83/4.99 ( ( ( Option = none_nat )
% 8.83/4.99 = ( Option2 = none_nat ) )
% 8.83/4.99 => ( ( ( Option != none_nat )
% 8.83/4.99 => ( ( Option2 != none_nat )
% 8.83/4.99 => ( ( the_nat @ Option )
% 8.83/4.99 = ( the_nat @ Option2 ) ) ) )
% 8.83/4.99 => ( Option = Option2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1035_option_Oexpand,axiom,
% 8.83/4.99 ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
% 8.83/4.99 ( ( ( Option = none_P5556105721700978146at_nat )
% 8.83/4.99 = ( Option2 = none_P5556105721700978146at_nat ) )
% 8.83/4.99 => ( ( ( Option != none_P5556105721700978146at_nat )
% 8.83/4.99 => ( ( Option2 != none_P5556105721700978146at_nat )
% 8.83/4.99 => ( ( the_Pr8591224930841456533at_nat @ Option )
% 8.83/4.99 = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
% 8.83/4.99 => ( Option = Option2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1036_option_Oexpand,axiom,
% 8.83/4.99 ! [Option: option_num,Option2: option_num] :
% 8.83/4.99 ( ( ( Option = none_num )
% 8.83/4.99 = ( Option2 = none_num ) )
% 8.83/4.99 => ( ( ( Option != none_num )
% 8.83/4.99 => ( ( Option2 != none_num )
% 8.83/4.99 => ( ( the_num @ Option )
% 8.83/4.99 = ( the_num @ Option2 ) ) ) )
% 8.83/4.99 => ( Option = Option2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1037_four__x__squared,axiom,
% 8.83/4.99 ! [X2: real] :
% 8.83/4.99 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/4.99 = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1038_L2__set__mult__ineq__lemma,axiom,
% 8.83/4.99 ! [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 ) ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1039_div__mult2__numeral__eq,axiom,
% 8.83/4.99 ! [A: nat,K: num,L: num] :
% 8.83/4.99 ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
% 8.83/4.99 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1040_div__mult2__numeral__eq,axiom,
% 8.83/4.99 ! [A: int,K: num,L: num] :
% 8.83/4.99 ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
% 8.83/4.99 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1041_option_Oexhaust__sel,axiom,
% 8.83/4.99 ! [Option: option4927543243414619207at_nat] :
% 8.83/4.99 ( ( Option != none_P5556105721700978146at_nat )
% 8.83/4.99 => ( Option
% 8.83/4.99 = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1042_option_Oexhaust__sel,axiom,
% 8.83/4.99 ! [Option: option_nat] :
% 8.83/4.99 ( ( Option != none_nat )
% 8.83/4.99 => ( Option
% 8.83/4.99 = ( some_nat @ ( the_nat @ Option ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1043_option_Oexhaust__sel,axiom,
% 8.83/4.99 ! [Option: option_num] :
% 8.83/4.99 ( ( Option != none_num )
% 8.83/4.99 => ( Option
% 8.83/4.99 = ( some_num @ ( the_num @ Option ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1044_linorder__neqE__linordered__idom,axiom,
% 8.83/4.99 ! [X2: real,Y3: real] :
% 8.83/4.99 ( ( X2 != Y3 )
% 8.83/4.99 => ( ~ ( ord_less_real @ X2 @ Y3 )
% 8.83/4.99 => ( ord_less_real @ Y3 @ X2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1045_linorder__neqE__linordered__idom,axiom,
% 8.83/4.99 ! [X2: rat,Y3: rat] :
% 8.83/4.99 ( ( X2 != Y3 )
% 8.83/4.99 => ( ~ ( ord_less_rat @ X2 @ Y3 )
% 8.83/4.99 => ( ord_less_rat @ Y3 @ X2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1046_linorder__neqE__linordered__idom,axiom,
% 8.83/4.99 ! [X2: int,Y3: int] :
% 8.83/4.99 ( ( X2 != Y3 )
% 8.83/4.99 => ( ~ ( ord_less_int @ X2 @ Y3 )
% 8.83/4.99 => ( ord_less_int @ Y3 @ X2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1047_combine__common__factor,axiom,
% 8.83/4.99 ! [A: real,E: real,B: real,C: real] :
% 8.83/4.99 ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
% 8.83/4.99 = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1048_combine__common__factor,axiom,
% 8.83/4.99 ! [A: rat,E: rat,B: rat,C: rat] :
% 8.83/4.99 ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C ) )
% 8.83/4.99 = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1049_combine__common__factor,axiom,
% 8.83/4.99 ! [A: nat,E: nat,B: nat,C: nat] :
% 8.83/4.99 ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
% 8.83/4.99 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1050_combine__common__factor,axiom,
% 8.83/4.99 ! [A: int,E: int,B: int,C: int] :
% 8.83/4.99 ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
% 8.83/4.99 = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1051_distrib__right,axiom,
% 8.83/4.99 ! [A: real,B: real,C: real] :
% 8.83/4.99 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 8.83/4.99 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1052_distrib__right,axiom,
% 8.83/4.99 ! [A: rat,B: rat,C: rat] :
% 8.83/4.99 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 8.83/4.99 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1053_distrib__right,axiom,
% 8.83/4.99 ! [A: nat,B: nat,C: nat] :
% 8.83/4.99 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 8.83/4.99 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1054_distrib__right,axiom,
% 8.83/4.99 ! [A: int,B: int,C: int] :
% 8.83/4.99 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.83/4.99 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1055_distrib__left,axiom,
% 8.83/4.99 ! [A: real,B: real,C: real] :
% 8.83/4.99 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 8.83/4.99 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1056_distrib__left,axiom,
% 8.83/4.99 ! [A: rat,B: rat,C: rat] :
% 8.83/4.99 ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 8.83/4.99 = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1057_distrib__left,axiom,
% 8.83/4.99 ! [A: nat,B: nat,C: nat] :
% 8.83/4.99 ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 8.83/4.99 = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1058_distrib__left,axiom,
% 8.83/4.99 ! [A: int,B: int,C: int] :
% 8.83/4.99 ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 8.83/4.99 = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1059_comm__semiring__class_Odistrib,axiom,
% 8.83/4.99 ! [A: real,B: real,C: real] :
% 8.83/4.99 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 8.83/4.99 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1060_comm__semiring__class_Odistrib,axiom,
% 8.83/4.99 ! [A: rat,B: rat,C: rat] :
% 8.83/4.99 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 8.83/4.99 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1061_comm__semiring__class_Odistrib,axiom,
% 8.83/4.99 ! [A: nat,B: nat,C: nat] :
% 8.83/4.99 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 8.83/4.99 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1062_comm__semiring__class_Odistrib,axiom,
% 8.83/4.99 ! [A: int,B: int,C: int] :
% 8.83/4.99 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.83/4.99 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1063_ring__class_Oring__distribs_I1_J,axiom,
% 8.83/4.99 ! [A: real,B: real,C: real] :
% 8.83/4.99 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 8.83/4.99 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1064_ring__class_Oring__distribs_I1_J,axiom,
% 8.83/4.99 ! [A: rat,B: rat,C: rat] :
% 8.83/4.99 ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 8.83/4.99 = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1065_ring__class_Oring__distribs_I1_J,axiom,
% 8.83/4.99 ! [A: int,B: int,C: int] :
% 8.83/4.99 ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 8.83/4.99 = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1066_ring__class_Oring__distribs_I2_J,axiom,
% 8.83/4.99 ! [A: real,B: real,C: real] :
% 8.83/4.99 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 8.83/4.99 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1067_ring__class_Oring__distribs_I2_J,axiom,
% 8.83/4.99 ! [A: rat,B: rat,C: rat] :
% 8.83/4.99 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 8.83/4.99 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1068_ring__class_Oring__distribs_I2_J,axiom,
% 8.83/4.99 ! [A: int,B: int,C: int] :
% 8.83/4.99 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.83/4.99 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1069_left__diff__distrib,axiom,
% 8.83/4.99 ! [A: real,B: real,C: real] :
% 8.83/4.99 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
% 8.83/4.99 = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1070_left__diff__distrib,axiom,
% 8.83/4.99 ! [A: rat,B: rat,C: rat] :
% 8.83/4.99 ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 8.83/4.99 = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1071_left__diff__distrib,axiom,
% 8.83/4.99 ! [A: int,B: int,C: int] :
% 8.83/4.99 ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
% 8.83/4.99 = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1072_right__diff__distrib,axiom,
% 8.83/4.99 ! [A: real,B: real,C: real] :
% 8.83/4.99 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 8.83/4.99 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1073_right__diff__distrib,axiom,
% 8.83/4.99 ! [A: rat,B: rat,C: rat] :
% 8.83/4.99 ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 8.83/4.99 = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1074_right__diff__distrib,axiom,
% 8.83/4.99 ! [A: int,B: int,C: int] :
% 8.83/4.99 ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 8.83/4.99 = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1075_left__diff__distrib_H,axiom,
% 8.83/4.99 ! [B: real,C: real,A: real] :
% 8.83/4.99 ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
% 8.83/4.99 = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1076_left__diff__distrib_H,axiom,
% 8.83/4.99 ! [B: rat,C: rat,A: rat] :
% 8.83/4.99 ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
% 8.83/4.99 = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1077_left__diff__distrib_H,axiom,
% 8.83/4.99 ! [B: nat,C: nat,A: nat] :
% 8.83/4.99 ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
% 8.83/4.99 = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1078_left__diff__distrib_H,axiom,
% 8.83/4.99 ! [B: int,C: int,A: int] :
% 8.83/4.99 ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
% 8.83/4.99 = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1079_right__diff__distrib_H,axiom,
% 8.83/4.99 ! [A: real,B: real,C: real] :
% 8.83/4.99 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 8.83/4.99 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1080_right__diff__distrib_H,axiom,
% 8.83/4.99 ! [A: rat,B: rat,C: rat] :
% 8.83/4.99 ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 8.83/4.99 = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1081_right__diff__distrib_H,axiom,
% 8.83/4.99 ! [A: nat,B: nat,C: nat] :
% 8.83/4.99 ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
% 8.83/4.99 = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1082_right__diff__distrib_H,axiom,
% 8.83/4.99 ! [A: int,B: int,C: int] :
% 8.83/4.99 ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 8.83/4.99 = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1083_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 8.83/4.99 ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
% 8.83/4.99 ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
% 8.83/4.99 = ( some_P7363390416028606310at_nat @ ( F @ A @ B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1084_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 8.83/4.99 ! [F: num > num > num,A: num,B: num] :
% 8.83/4.99 ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
% 8.83/4.99 = ( some_num @ ( F @ A @ B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1085_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 8.83/4.99 ! [F: nat > nat > nat,A: nat,B: nat] :
% 8.83/4.99 ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
% 8.83/4.99 = ( some_nat @ ( F @ A @ B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1086_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 8.83/4.99 ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
% 8.83/4.99 ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
% 8.83/4.99 = none_P5556105721700978146at_nat ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1087_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 8.83/4.99 ! [Uu: num > num > num,Uv: option_num] :
% 8.83/4.99 ( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
% 8.83/4.99 = none_num ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1088_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 8.83/4.99 ! [Uu: nat > nat > nat,Uv: option_nat] :
% 8.83/4.99 ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
% 8.83/4.99 = none_nat ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1089_less__1__mult,axiom,
% 8.83/4.99 ! [M: real,N: real] :
% 8.83/4.99 ( ( ord_less_real @ one_one_real @ M )
% 8.83/4.99 => ( ( ord_less_real @ one_one_real @ N )
% 8.83/4.99 => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1090_less__1__mult,axiom,
% 8.83/4.99 ! [M: rat,N: rat] :
% 8.83/4.99 ( ( ord_less_rat @ one_one_rat @ M )
% 8.83/4.99 => ( ( ord_less_rat @ one_one_rat @ N )
% 8.83/4.99 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1091_less__1__mult,axiom,
% 8.83/4.99 ! [M: nat,N: nat] :
% 8.83/4.99 ( ( ord_less_nat @ one_one_nat @ M )
% 8.83/4.99 => ( ( ord_less_nat @ one_one_nat @ N )
% 8.83/4.99 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1092_less__1__mult,axiom,
% 8.83/4.99 ! [M: int,N: int] :
% 8.83/4.99 ( ( ord_less_int @ one_one_int @ M )
% 8.83/4.99 => ( ( ord_less_int @ one_one_int @ N )
% 8.83/4.99 => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1093_add__mono1,axiom,
% 8.83/4.99 ! [A: real,B: real] :
% 8.83/4.99 ( ( ord_less_real @ A @ B )
% 8.83/4.99 => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1094_add__mono1,axiom,
% 8.83/4.99 ! [A: rat,B: rat] :
% 8.83/4.99 ( ( ord_less_rat @ A @ B )
% 8.83/4.99 => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1095_add__mono1,axiom,
% 8.83/4.99 ! [A: nat,B: nat] :
% 8.83/4.99 ( ( ord_less_nat @ A @ B )
% 8.83/4.99 => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1096_add__mono1,axiom,
% 8.83/4.99 ! [A: int,B: int] :
% 8.83/4.99 ( ( ord_less_int @ A @ B )
% 8.83/4.99 => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1097_less__add__one,axiom,
% 8.83/4.99 ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1098_less__add__one,axiom,
% 8.83/4.99 ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1099_less__add__one,axiom,
% 8.83/4.99 ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1100_less__add__one,axiom,
% 8.83/4.99 ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1101_add__le__add__imp__diff__le,axiom,
% 8.83/4.99 ! [I2: real,K: real,N: real,J: real] :
% 8.83/4.99 ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
% 8.83/4.99 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 8.83/4.99 => ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
% 8.83/4.99 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 8.83/4.99 => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1102_add__le__add__imp__diff__le,axiom,
% 8.83/4.99 ! [I2: rat,K: rat,N: rat,J: rat] :
% 8.83/4.99 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N )
% 8.83/4.99 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 8.83/4.99 => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N )
% 8.83/4.99 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 8.83/4.99 => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1103_add__le__add__imp__diff__le,axiom,
% 8.83/4.99 ! [I2: nat,K: nat,N: nat,J: nat] :
% 8.83/4.99 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
% 8.83/4.99 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 8.83/4.99 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
% 8.83/4.99 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 8.83/4.99 => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1104_add__le__add__imp__diff__le,axiom,
% 8.83/4.99 ! [I2: int,K: int,N: int,J: int] :
% 8.83/4.99 ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
% 8.83/4.99 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 8.83/4.99 => ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
% 8.83/4.99 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 8.83/4.99 => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1105_add__le__imp__le__diff,axiom,
% 8.83/4.99 ! [I2: real,K: real,N: real] :
% 8.83/4.99 ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
% 8.83/4.99 => ( ord_less_eq_real @ I2 @ ( minus_minus_real @ N @ K ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1106_add__le__imp__le__diff,axiom,
% 8.83/4.99 ! [I2: rat,K: rat,N: rat] :
% 8.83/4.99 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N )
% 8.83/4.99 => ( ord_less_eq_rat @ I2 @ ( minus_minus_rat @ N @ K ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1107_add__le__imp__le__diff,axiom,
% 8.83/4.99 ! [I2: nat,K: nat,N: nat] :
% 8.83/4.99 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
% 8.83/4.99 => ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1108_add__le__imp__le__diff,axiom,
% 8.83/4.99 ! [I2: int,K: int,N: int] :
% 8.83/4.99 ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
% 8.83/4.99 => ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N @ K ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1109_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 8.83/4.99 ! [A: real,B: real] :
% 8.83/4.99 ( ~ ( ord_less_real @ A @ B )
% 8.83/4.99 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 8.83/4.99 = A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1110_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 8.83/4.99 ! [A: rat,B: rat] :
% 8.83/4.99 ( ~ ( ord_less_rat @ A @ B )
% 8.83/4.99 => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 8.83/4.99 = A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1111_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 8.83/4.99 ! [A: nat,B: nat] :
% 8.83/4.99 ( ~ ( ord_less_nat @ A @ B )
% 8.83/4.99 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 8.83/4.99 = A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1112_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 8.83/4.99 ! [A: int,B: int] :
% 8.83/4.99 ( ~ ( ord_less_int @ A @ B )
% 8.83/4.99 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 8.83/4.99 = A ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1113_eq__add__iff1,axiom,
% 8.83/4.99 ! [A: real,E: real,C: real,B: real,D: real] :
% 8.83/4.99 ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 8.83/4.99 = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 8.83/4.99 = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
% 8.83/4.99 = D ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1114_eq__add__iff1,axiom,
% 8.83/4.99 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 8.83/4.99 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 8.83/4.99 = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 8.83/4.99 = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C )
% 8.83/4.99 = D ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1115_eq__add__iff1,axiom,
% 8.83/4.99 ! [A: int,E: int,C: int,B: int,D: int] :
% 8.83/4.99 ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 8.83/4.99 = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 8.83/4.99 = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
% 8.83/4.99 = D ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1116_eq__add__iff2,axiom,
% 8.83/4.99 ! [A: real,E: real,C: real,B: real,D: real] :
% 8.83/4.99 ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 8.83/4.99 = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 8.83/4.99 = ( C
% 8.83/4.99 = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1117_eq__add__iff2,axiom,
% 8.83/4.99 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 8.83/4.99 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 8.83/4.99 = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 8.83/4.99 = ( C
% 8.83/4.99 = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1118_eq__add__iff2,axiom,
% 8.83/4.99 ! [A: int,E: int,C: int,B: int,D: int] :
% 8.83/4.99 ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 8.83/4.99 = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 8.83/4.99 = ( C
% 8.83/4.99 = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1119_square__diff__square__factored,axiom,
% 8.83/4.99 ! [X2: real,Y3: real] :
% 8.83/4.99 ( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) )
% 8.83/4.99 = ( times_times_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( minus_minus_real @ X2 @ Y3 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1120_square__diff__square__factored,axiom,
% 8.83/4.99 ! [X2: rat,Y3: rat] :
% 8.83/4.99 ( ( minus_minus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 8.83/4.99 = ( times_times_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ ( minus_minus_rat @ X2 @ Y3 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1121_square__diff__square__factored,axiom,
% 8.83/4.99 ! [X2: int,Y3: int] :
% 8.83/4.99 ( ( minus_minus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) )
% 8.83/4.99 = ( times_times_int @ ( plus_plus_int @ X2 @ Y3 ) @ ( minus_minus_int @ X2 @ Y3 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1122_vebt__mint_Osimps_I2_J,axiom,
% 8.83/4.99 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 8.83/4.99 ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 8.83/4.99 = none_nat ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1123_vebt__maxt_Osimps_I2_J,axiom,
% 8.83/4.99 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 8.83/4.99 ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 8.83/4.99 = none_nat ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1124_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 8.83/4.99 ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
% 8.83/4.99 ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
% 8.83/4.99 = none_P5556105721700978146at_nat ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1125_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 8.83/4.99 ! [Uw: num > num > num,V: num] :
% 8.83/4.99 ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
% 8.83/4.99 = none_num ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1126_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 8.83/4.99 ! [Uw: nat > nat > nat,V: nat] :
% 8.83/4.99 ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
% 8.83/4.99 = none_nat ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1127_VEBT__internal_Ooption__shift_Oelims,axiom,
% 8.83/4.99 ! [X2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
% 8.83/4.99 ( ( ( vEBT_V1502963449132264192at_nat @ X2 @ Xa2 @ Xb )
% 8.83/4.99 = Y3 )
% 8.83/4.99 => ( ( ( Xa2 = none_P5556105721700978146at_nat )
% 8.83/4.99 => ( Y3 != none_P5556105721700978146at_nat ) )
% 8.83/4.99 => ( ( ? [V2: product_prod_nat_nat] :
% 8.83/4.99 ( Xa2
% 8.83/4.99 = ( some_P7363390416028606310at_nat @ V2 ) )
% 8.83/4.99 => ( ( Xb = none_P5556105721700978146at_nat )
% 8.83/4.99 => ( Y3 != none_P5556105721700978146at_nat ) ) )
% 8.83/4.99 => ~ ! [A4: product_prod_nat_nat] :
% 8.83/4.99 ( ( Xa2
% 8.83/4.99 = ( some_P7363390416028606310at_nat @ A4 ) )
% 8.83/4.99 => ! [B4: product_prod_nat_nat] :
% 8.83/4.99 ( ( Xb
% 8.83/4.99 = ( some_P7363390416028606310at_nat @ B4 ) )
% 8.83/4.99 => ( Y3
% 8.83/4.99 != ( some_P7363390416028606310at_nat @ ( X2 @ A4 @ B4 ) ) ) ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1128_VEBT__internal_Ooption__shift_Oelims,axiom,
% 8.83/4.99 ! [X2: num > num > num,Xa2: option_num,Xb: option_num,Y3: option_num] :
% 8.83/4.99 ( ( ( vEBT_V819420779217536731ft_num @ X2 @ Xa2 @ Xb )
% 8.83/4.99 = Y3 )
% 8.83/4.99 => ( ( ( Xa2 = none_num )
% 8.83/4.99 => ( Y3 != none_num ) )
% 8.83/4.99 => ( ( ? [V2: num] :
% 8.83/4.99 ( Xa2
% 8.83/4.99 = ( some_num @ V2 ) )
% 8.83/4.99 => ( ( Xb = none_num )
% 8.83/4.99 => ( Y3 != none_num ) ) )
% 8.83/4.99 => ~ ! [A4: num] :
% 8.83/4.99 ( ( Xa2
% 8.83/4.99 = ( some_num @ A4 ) )
% 8.83/4.99 => ! [B4: num] :
% 8.83/4.99 ( ( Xb
% 8.83/4.99 = ( some_num @ B4 ) )
% 8.83/4.99 => ( Y3
% 8.83/4.99 != ( some_num @ ( X2 @ A4 @ B4 ) ) ) ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1129_VEBT__internal_Ooption__shift_Oelims,axiom,
% 8.83/4.99 ! [X2: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y3: option_nat] :
% 8.83/4.99 ( ( ( vEBT_V4262088993061758097ft_nat @ X2 @ Xa2 @ Xb )
% 8.83/4.99 = Y3 )
% 8.83/4.99 => ( ( ( Xa2 = none_nat )
% 8.83/4.99 => ( Y3 != none_nat ) )
% 8.83/4.99 => ( ( ? [V2: nat] :
% 8.83/4.99 ( Xa2
% 8.83/4.99 = ( some_nat @ V2 ) )
% 8.83/4.99 => ( ( Xb = none_nat )
% 8.83/4.99 => ( Y3 != none_nat ) ) )
% 8.83/4.99 => ~ ! [A4: nat] :
% 8.83/4.99 ( ( Xa2
% 8.83/4.99 = ( some_nat @ A4 ) )
% 8.83/4.99 => ! [B4: nat] :
% 8.83/4.99 ( ( Xb
% 8.83/4.99 = ( some_nat @ B4 ) )
% 8.83/4.99 => ( Y3
% 8.83/4.99 != ( some_nat @ ( X2 @ A4 @ B4 ) ) ) ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1130_ordered__ring__class_Ole__add__iff2,axiom,
% 8.83/4.99 ! [A: real,E: real,C: real,B: real,D: real] :
% 8.83/4.99 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 8.83/4.99 = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1131_ordered__ring__class_Ole__add__iff2,axiom,
% 8.83/4.99 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 8.83/4.99 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 8.83/4.99 = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1132_ordered__ring__class_Ole__add__iff2,axiom,
% 8.83/4.99 ! [A: int,E: int,C: int,B: int,D: int] :
% 8.83/4.99 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 8.83/4.99 = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1133_ordered__ring__class_Ole__add__iff1,axiom,
% 8.83/4.99 ! [A: real,E: real,C: real,B: real,D: real] :
% 8.83/4.99 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 8.83/4.99 = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1134_ordered__ring__class_Ole__add__iff1,axiom,
% 8.83/4.99 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 8.83/4.99 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 8.83/4.99 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1135_ordered__ring__class_Ole__add__iff1,axiom,
% 8.83/4.99 ! [A: int,E: int,C: int,B: int,D: int] :
% 8.83/4.99 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 8.83/4.99 = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1136_less__add__iff1,axiom,
% 8.83/4.99 ! [A: real,E: real,C: real,B: real,D: real] :
% 8.83/4.99 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 8.83/4.99 = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1137_less__add__iff1,axiom,
% 8.83/4.99 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 8.83/4.99 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 8.83/4.99 = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1138_less__add__iff1,axiom,
% 8.83/4.99 ! [A: int,E: int,C: int,B: int,D: int] :
% 8.83/4.99 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 8.83/4.99 = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1139_less__add__iff2,axiom,
% 8.83/4.99 ! [A: real,E: real,C: real,B: real,D: real] :
% 8.83/4.99 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 8.83/4.99 = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1140_less__add__iff2,axiom,
% 8.83/4.99 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 8.83/4.99 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 8.83/4.99 = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1141_less__add__iff2,axiom,
% 8.83/4.99 ! [A: int,E: int,C: int,B: int,D: int] :
% 8.83/4.99 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 8.83/4.99 = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1142_square__diff__one__factored,axiom,
% 8.83/4.99 ! [X2: complex] :
% 8.83/4.99 ( ( minus_minus_complex @ ( times_times_complex @ X2 @ X2 ) @ one_one_complex )
% 8.83/4.99 = ( times_times_complex @ ( plus_plus_complex @ X2 @ one_one_complex ) @ ( minus_minus_complex @ X2 @ one_one_complex ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1143_square__diff__one__factored,axiom,
% 8.83/4.99 ! [X2: real] :
% 8.83/4.99 ( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ one_one_real )
% 8.83/4.99 = ( times_times_real @ ( plus_plus_real @ X2 @ one_one_real ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1144_square__diff__one__factored,axiom,
% 8.83/4.99 ! [X2: rat] :
% 8.83/4.99 ( ( minus_minus_rat @ ( times_times_rat @ X2 @ X2 ) @ one_one_rat )
% 8.83/4.99 = ( times_times_rat @ ( plus_plus_rat @ X2 @ one_one_rat ) @ ( minus_minus_rat @ X2 @ one_one_rat ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1145_square__diff__one__factored,axiom,
% 8.83/4.99 ! [X2: int] :
% 8.83/4.99 ( ( minus_minus_int @ ( times_times_int @ X2 @ X2 ) @ one_one_int )
% 8.83/4.99 = ( times_times_int @ ( plus_plus_int @ X2 @ one_one_int ) @ ( minus_minus_int @ X2 @ one_one_int ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1146_real__average__minus__first,axiom,
% 8.83/4.99 ! [A: real,B: real] :
% 8.83/4.99 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 8.83/4.99 = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1147_real__average__minus__second,axiom,
% 8.83/4.99 ! [B: real,A: real] :
% 8.83/4.99 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 8.83/4.99 = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1148_Suc__double__not__eq__double,axiom,
% 8.83/4.99 ! [M: nat,N: nat] :
% 8.83/4.99 ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.83/4.99 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1149_double__not__eq__Suc__double,axiom,
% 8.83/4.99 ! [M: nat,N: nat] :
% 8.83/4.99 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 8.83/4.99 != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1150_times__divide__eq__left,axiom,
% 8.83/4.99 ! [B: complex,C: complex,A: complex] :
% 8.83/4.99 ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
% 8.83/4.99 = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1151_times__divide__eq__left,axiom,
% 8.83/4.99 ! [B: real,C: real,A: real] :
% 8.83/4.99 ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
% 8.83/4.99 = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1152_times__divide__eq__left,axiom,
% 8.83/4.99 ! [B: rat,C: rat,A: rat] :
% 8.83/4.99 ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 8.83/4.99 = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1153_divide__divide__eq__left,axiom,
% 8.83/4.99 ! [A: complex,B: complex,C: complex] :
% 8.83/4.99 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 8.83/4.99 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1154_divide__divide__eq__left,axiom,
% 8.83/4.99 ! [A: real,B: real,C: real] :
% 8.83/4.99 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 8.83/4.99 = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1155_divide__divide__eq__left,axiom,
% 8.83/4.99 ! [A: rat,B: rat,C: rat] :
% 8.83/4.99 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 8.83/4.99 = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1156_divide__divide__eq__right,axiom,
% 8.83/4.99 ! [A: complex,B: complex,C: complex] :
% 8.83/4.99 ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 8.83/4.99 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1157_divide__divide__eq__right,axiom,
% 8.83/4.99 ! [A: real,B: real,C: real] :
% 8.83/4.99 ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
% 8.83/4.99 = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1158_divide__divide__eq__right,axiom,
% 8.83/4.99 ! [A: rat,B: rat,C: rat] :
% 8.83/4.99 ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 8.83/4.99 = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1159_times__divide__eq__right,axiom,
% 8.83/4.99 ! [A: complex,B: complex,C: complex] :
% 8.83/4.99 ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 8.83/4.99 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1160_times__divide__eq__right,axiom,
% 8.83/4.99 ! [A: real,B: real,C: real] :
% 8.83/4.99 ( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
% 8.83/4.99 = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1161_times__divide__eq__right,axiom,
% 8.83/4.99 ! [A: rat,B: rat,C: rat] :
% 8.83/4.99 ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 8.83/4.99 = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1162_pred__less__length__list,axiom,
% 8.83/4.99 ! [Deg: nat,X2: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
% 8.83/4.99 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 8.83/4.99 => ( ( ord_less_eq_nat @ X2 @ Ma )
% 8.83/4.99 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 8.83/4.99 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 8.83/4.99 = ( if_option_nat
% 8.83/4.99 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 != none_nat )
% 8.83/4.99 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 8.83/4.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 @ ( if_option_nat
% 8.83/4.99 @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.83/4.99 = none_nat )
% 8.83/4.99 @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( some_nat @ Mi ) @ none_nat )
% 8.83/4.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1163_pred__lesseq__max,axiom,
% 8.83/4.99 ! [Deg: nat,X2: nat,Ma: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 8.83/4.99 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 8.83/4.99 => ( ( ord_less_eq_nat @ X2 @ Ma )
% 8.83/4.99 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 8.83/4.99 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 8.83/4.99 @ ( if_option_nat
% 8.83/4.99 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 != none_nat )
% 8.83/4.99 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 8.83/4.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 @ ( if_option_nat
% 8.83/4.99 @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.83/4.99 = none_nat )
% 8.83/4.99 @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( some_nat @ Mi ) @ none_nat )
% 8.83/4.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 8.83/4.99 @ none_nat ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1164_set__vebt_H__def,axiom,
% 8.83/4.99 ( vEBT_VEBT_set_vebt
% 8.83/4.99 = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1165_real__divide__square__eq,axiom,
% 8.83/4.99 ! [R2: real,A: real] :
% 8.83/4.99 ( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
% 8.83/4.99 = ( divide_divide_real @ A @ R2 ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1166_succ__less__length__list,axiom,
% 8.83/4.99 ! [Deg: nat,Mi: nat,X2: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 8.83/4.99 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 8.83/4.99 => ( ( ord_less_eq_nat @ Mi @ X2 )
% 8.83/4.99 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 8.83/4.99 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 8.83/4.99 = ( if_option_nat
% 8.83/4.99 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 != none_nat )
% 8.83/4.99 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 8.83/4.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 @ ( if_option_nat
% 8.83/4.99 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.83/4.99 = none_nat )
% 8.83/4.99 @ none_nat
% 8.83/4.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1167_succ__greatereq__min,axiom,
% 8.83/4.99 ! [Deg: nat,Mi: nat,X2: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 8.83/4.99 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 8.83/4.99 => ( ( ord_less_eq_nat @ Mi @ X2 )
% 8.83/4.99 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 8.83/4.99 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 8.83/4.99 @ ( if_option_nat
% 8.83/4.99 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 != none_nat )
% 8.83/4.99 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 8.83/4.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 @ ( if_option_nat
% 8.83/4.99 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.83/4.99 = none_nat )
% 8.83/4.99 @ none_nat
% 8.83/4.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 8.83/4.99 @ none_nat ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1168_mult__commute__abs,axiom,
% 8.83/4.99 ! [C: real] :
% 8.83/4.99 ( ( ^ [X: real] : ( times_times_real @ X @ C ) )
% 8.83/4.99 = ( times_times_real @ C ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1169_mult__commute__abs,axiom,
% 8.83/4.99 ! [C: rat] :
% 8.83/4.99 ( ( ^ [X: rat] : ( times_times_rat @ X @ C ) )
% 8.83/4.99 = ( times_times_rat @ C ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1170_mult__commute__abs,axiom,
% 8.83/4.99 ! [C: nat] :
% 8.83/4.99 ( ( ^ [X: nat] : ( times_times_nat @ X @ C ) )
% 8.83/4.99 = ( times_times_nat @ C ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1171_mult__commute__abs,axiom,
% 8.83/4.99 ! [C: int] :
% 8.83/4.99 ( ( ^ [X: int] : ( times_times_int @ X @ C ) )
% 8.83/4.99 = ( times_times_int @ C ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1172_enat__less__induct,axiom,
% 8.83/4.99 ! [P: extended_enat > $o,N: extended_enat] :
% 8.83/4.99 ( ! [N2: extended_enat] :
% 8.83/4.99 ( ! [M2: extended_enat] :
% 8.83/4.99 ( ( ord_le72135733267957522d_enat @ M2 @ N2 )
% 8.83/4.99 => ( P @ M2 ) )
% 8.83/4.99 => ( P @ N2 ) )
% 8.83/4.99 => ( P @ N ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1173_lambda__one,axiom,
% 8.83/4.99 ( ( ^ [X: complex] : X )
% 8.83/4.99 = ( times_times_complex @ one_one_complex ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1174_lambda__one,axiom,
% 8.83/4.99 ( ( ^ [X: real] : X )
% 8.83/4.99 = ( times_times_real @ one_one_real ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1175_lambda__one,axiom,
% 8.83/4.99 ( ( ^ [X: rat] : X )
% 8.83/4.99 = ( times_times_rat @ one_one_rat ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1176_lambda__one,axiom,
% 8.83/4.99 ( ( ^ [X: nat] : X )
% 8.83/4.99 = ( times_times_nat @ one_one_nat ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1177_lambda__one,axiom,
% 8.83/4.99 ( ( ^ [X: int] : X )
% 8.83/4.99 = ( times_times_int @ one_one_int ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1178_set__vebt__def,axiom,
% 8.83/4.99 ( vEBT_set_vebt
% 8.83/4.99 = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1179_numeral__code_I2_J,axiom,
% 8.83/4.99 ! [N: num] :
% 8.83/4.99 ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 8.83/4.99 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1180_numeral__code_I2_J,axiom,
% 8.83/4.99 ! [N: num] :
% 8.83/4.99 ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 8.83/4.99 = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1181_numeral__code_I2_J,axiom,
% 8.83/4.99 ! [N: num] :
% 8.83/4.99 ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 8.83/4.99 = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1182_numeral__code_I2_J,axiom,
% 8.83/4.99 ! [N: num] :
% 8.83/4.99 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 8.83/4.99 = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1183_numeral__code_I2_J,axiom,
% 8.83/4.99 ! [N: num] :
% 8.83/4.99 ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 8.83/4.99 = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1184_power__numeral__even,axiom,
% 8.83/4.99 ! [Z: complex,W: num] :
% 8.83/4.99 ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 8.83/4.99 = ( times_times_complex @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1185_power__numeral__even,axiom,
% 8.83/4.99 ! [Z: real,W: num] :
% 8.83/4.99 ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 8.83/4.99 = ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1186_power__numeral__even,axiom,
% 8.83/4.99 ! [Z: rat,W: num] :
% 8.83/4.99 ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 8.83/4.99 = ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1187_power__numeral__even,axiom,
% 8.83/4.99 ! [Z: nat,W: num] :
% 8.83/4.99 ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 8.83/4.99 = ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1188_power__numeral__even,axiom,
% 8.83/4.99 ! [Z: int,W: num] :
% 8.83/4.99 ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 8.83/4.99 = ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1189_linordered__field__no__ub,axiom,
% 8.83/4.99 ! [X3: real] :
% 8.83/4.99 ? [X_12: real] : ( ord_less_real @ X3 @ X_12 ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1190_linordered__field__no__ub,axiom,
% 8.83/4.99 ! [X3: rat] :
% 8.83/4.99 ? [X_12: rat] : ( ord_less_rat @ X3 @ X_12 ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1191_linordered__field__no__lb,axiom,
% 8.83/4.99 ! [X3: real] :
% 8.83/4.99 ? [Y4: real] : ( ord_less_real @ Y4 @ X3 ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1192_linordered__field__no__lb,axiom,
% 8.83/4.99 ! [X3: rat] :
% 8.83/4.99 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X3 ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1193_divide__divide__eq__left_H,axiom,
% 8.83/4.99 ! [A: complex,B: complex,C: complex] :
% 8.83/4.99 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 8.83/4.99 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1194_divide__divide__eq__left_H,axiom,
% 8.83/4.99 ! [A: real,B: real,C: real] :
% 8.83/4.99 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 8.83/4.99 = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1195_divide__divide__eq__left_H,axiom,
% 8.83/4.99 ! [A: rat,B: rat,C: rat] :
% 8.83/4.99 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 8.83/4.99 = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1196_divide__divide__times__eq,axiom,
% 8.83/4.99 ! [X2: complex,Y3: complex,Z: complex,W: complex] :
% 8.83/4.99 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X2 @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 8.83/4.99 = ( divide1717551699836669952omplex @ ( times_times_complex @ X2 @ W ) @ ( times_times_complex @ Y3 @ Z ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1197_divide__divide__times__eq,axiom,
% 8.83/4.99 ! [X2: real,Y3: real,Z: real,W: real] :
% 8.83/4.99 ( ( divide_divide_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
% 8.83/4.99 = ( divide_divide_real @ ( times_times_real @ X2 @ W ) @ ( times_times_real @ Y3 @ Z ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1198_divide__divide__times__eq,axiom,
% 8.83/4.99 ! [X2: rat,Y3: rat,Z: rat,W: rat] :
% 8.83/4.99 ( ( divide_divide_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
% 8.83/4.99 = ( divide_divide_rat @ ( times_times_rat @ X2 @ W ) @ ( times_times_rat @ Y3 @ Z ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1199_times__divide__times__eq,axiom,
% 8.83/4.99 ! [X2: complex,Y3: complex,Z: complex,W: complex] :
% 8.83/4.99 ( ( times_times_complex @ ( divide1717551699836669952omplex @ X2 @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 8.83/4.99 = ( divide1717551699836669952omplex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ Y3 @ W ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1200_times__divide__times__eq,axiom,
% 8.83/4.99 ! [X2: real,Y3: real,Z: real,W: real] :
% 8.83/4.99 ( ( times_times_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
% 8.83/4.99 = ( divide_divide_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y3 @ W ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1201_times__divide__times__eq,axiom,
% 8.83/4.99 ! [X2: rat,Y3: rat,Z: rat,W: rat] :
% 8.83/4.99 ( ( times_times_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
% 8.83/4.99 = ( divide_divide_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y3 @ W ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1202_add__divide__distrib,axiom,
% 8.83/4.99 ! [A: complex,B: complex,C: complex] :
% 8.83/4.99 ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
% 8.83/4.99 = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1203_add__divide__distrib,axiom,
% 8.83/4.99 ! [A: real,B: real,C: real] :
% 8.83/4.99 ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
% 8.83/4.99 = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1204_add__divide__distrib,axiom,
% 8.83/4.99 ! [A: rat,B: rat,C: rat] :
% 8.83/4.99 ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 8.83/4.99 = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1205_diff__divide__distrib,axiom,
% 8.83/4.99 ! [A: complex,B: complex,C: complex] :
% 8.83/4.99 ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
% 8.83/4.99 = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1206_diff__divide__distrib,axiom,
% 8.83/4.99 ! [A: real,B: real,C: real] :
% 8.83/4.99 ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
% 8.83/4.99 = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1207_diff__divide__distrib,axiom,
% 8.83/4.99 ! [A: rat,B: rat,C: rat] :
% 8.83/4.99 ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 8.83/4.99 = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1208_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
% 8.83/4.99 ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 8.83/4.99 ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 8.83/4.99 = ( if_nat
% 8.83/4.99 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 8.83/4.99 & ~ ( ( X2 = Mi )
% 8.83/4.99 | ( X2 = Ma ) ) )
% 8.83/4.99 @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
% 8.83/4.99 @ one_one_nat ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1209_gt__half__sum,axiom,
% 8.83/4.99 ! [A: real,B: real] :
% 8.83/4.99 ( ( ord_less_real @ A @ B )
% 8.83/4.99 => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1210_gt__half__sum,axiom,
% 8.83/4.99 ! [A: rat,B: rat] :
% 8.83/4.99 ( ( ord_less_rat @ A @ B )
% 8.83/4.99 => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1211_less__half__sum,axiom,
% 8.83/4.99 ! [A: real,B: real] :
% 8.83/4.99 ( ( ord_less_real @ A @ B )
% 8.83/4.99 => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1212_less__half__sum,axiom,
% 8.83/4.99 ! [A: rat,B: rat] :
% 8.83/4.99 ( ( ord_less_rat @ A @ B )
% 8.83/4.99 => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1213_vebt__succ_Osimps_I6_J,axiom,
% 8.83/4.99 ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 8.83/4.99 ( ( ( ord_less_nat @ X2 @ Mi )
% 8.83/4.99 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 8.83/4.99 = ( some_nat @ Mi ) ) )
% 8.83/4.99 & ( ~ ( ord_less_nat @ X2 @ Mi )
% 8.83/4.99 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 8.83/4.99 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 8.83/4.99 @ ( if_option_nat
% 8.83/4.99 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 != none_nat )
% 8.83/4.99 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 8.83/4.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 @ ( if_option_nat
% 8.83/4.99 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.83/4.99 = none_nat )
% 8.83/4.99 @ none_nat
% 8.83/4.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 8.83/4.99 @ none_nat ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1214_vebt__pred_Osimps_I7_J,axiom,
% 8.83/4.99 ! [Ma: nat,X2: nat,Mi: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 8.83/4.99 ( ( ( ord_less_nat @ Ma @ X2 )
% 8.83/4.99 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 8.83/4.99 = ( some_nat @ Ma ) ) )
% 8.83/4.99 & ( ~ ( ord_less_nat @ Ma @ X2 )
% 8.83/4.99 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 8.83/4.99 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 8.83/4.99 @ ( if_option_nat
% 8.83/4.99 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 != none_nat )
% 8.83/4.99 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 8.83/4.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 @ ( if_option_nat
% 8.83/4.99 @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.83/4.99 = none_nat )
% 8.83/4.99 @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( some_nat @ Mi ) @ none_nat )
% 8.83/4.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 8.83/4.99 @ none_nat ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1215_vebt__member_Osimps_I5_J,axiom,
% 8.83/4.99 ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 8.83/4.99 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 8.83/4.99 = ( ( X2 != Mi )
% 8.83/4.99 => ( ( X2 != Ma )
% 8.83/4.99 => ( ~ ( ord_less_nat @ X2 @ Mi )
% 8.83/4.99 & ( ~ ( ord_less_nat @ X2 @ Mi )
% 8.83/4.99 => ( ~ ( ord_less_nat @ Ma @ X2 )
% 8.83/4.99 & ( ~ ( ord_less_nat @ Ma @ X2 )
% 8.83/4.99 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 8.83/4.99 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1216_pred__empty,axiom,
% 8.83/4.99 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 8.83/4.99 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.99 => ( ( ( vEBT_vebt_pred @ T @ X2 )
% 8.83/4.99 = none_nat )
% 8.83/4.99 = ( ( collect_nat
% 8.83/4.99 @ ^ [Y: nat] :
% 8.83/4.99 ( ( vEBT_vebt_member @ T @ Y )
% 8.83/4.99 & ( ord_less_nat @ Y @ X2 ) ) )
% 8.83/4.99 = bot_bot_set_nat ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1217_succ__empty,axiom,
% 8.83/4.99 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 8.83/4.99 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.99 => ( ( ( vEBT_vebt_succ @ T @ X2 )
% 8.83/4.99 = none_nat )
% 8.83/4.99 = ( ( collect_nat
% 8.83/4.99 @ ^ [Y: nat] :
% 8.83/4.99 ( ( vEBT_vebt_member @ T @ Y )
% 8.83/4.99 & ( ord_less_nat @ X2 @ Y ) ) )
% 8.83/4.99 = bot_bot_set_nat ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1218_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 8.83/4.99 ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
% 8.83/4.99 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc ) @ X2 )
% 8.83/4.99 = ( ( X2 = Mi )
% 8.83/4.99 | ( X2 = Ma )
% 8.83/4.99 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 8.83/4.99 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1219_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 8.83/4.99 ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
% 8.83/4.99 ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) @ X2 )
% 8.83/4.99 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 8.83/4.99 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1220_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 8.83/4.99 ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X2: nat] :
% 8.83/4.99 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) @ X2 )
% 8.83/4.99 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 8.83/4.99 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1221_vebt__succ_Osimps_I3_J,axiom,
% 8.83/4.99 ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
% 8.83/4.99 ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va )
% 8.83/4.99 = none_nat ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1222_vebt__pred_Osimps_I4_J,axiom,
% 8.83/4.99 ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
% 8.83/4.99 ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
% 8.83/4.99 = none_nat ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1223_both__member__options__def,axiom,
% 8.83/4.99 ( vEBT_V8194947554948674370ptions
% 8.83/4.99 = ( ^ [T2: vEBT_VEBT,X: nat] :
% 8.83/4.99 ( ( vEBT_V5719532721284313246member @ T2 @ X )
% 8.83/4.99 | ( vEBT_VEBT_membermima @ T2 @ X ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1224_member__valid__both__member__options,axiom,
% 8.83/4.99 ! [Tree: vEBT_VEBT,N: nat,X2: nat] :
% 8.83/4.99 ( ( vEBT_invar_vebt @ Tree @ N )
% 8.83/4.99 => ( ( vEBT_vebt_member @ Tree @ X2 )
% 8.83/4.99 => ( ( vEBT_V5719532721284313246member @ Tree @ X2 )
% 8.83/4.99 | ( vEBT_VEBT_membermima @ Tree @ X2 ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1225_mint__corr__help__empty,axiom,
% 8.83/4.99 ! [T: vEBT_VEBT,N: nat] :
% 8.83/4.99 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.99 => ( ( ( vEBT_vebt_mint @ T )
% 8.83/4.99 = none_nat )
% 8.83/4.99 => ( ( vEBT_VEBT_set_vebt @ T )
% 8.83/4.99 = bot_bot_set_nat ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1226_maxt__corr__help__empty,axiom,
% 8.83/4.99 ! [T: vEBT_VEBT,N: nat] :
% 8.83/4.99 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.99 => ( ( ( vEBT_vebt_maxt @ T )
% 8.83/4.99 = none_nat )
% 8.83/4.99 => ( ( vEBT_VEBT_set_vebt @ T )
% 8.83/4.99 = bot_bot_set_nat ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1227_VEBT__internal_OminNull_Osimps_I5_J,axiom,
% 8.83/4.99 ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 8.83/4.99 ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1228_vebt__member_Osimps_I2_J,axiom,
% 8.83/4.99 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
% 8.83/4.99 ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X2 ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1229_VEBT__internal_OminNull_Osimps_I4_J,axiom,
% 8.83/4.99 ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1230_is__succ__in__set__def,axiom,
% 8.83/4.99 ( vEBT_is_succ_in_set
% 8.83/4.99 = ( ^ [Xs: set_nat,X: nat,Y: nat] :
% 8.83/4.99 ( ( member_nat @ Y @ Xs )
% 8.83/4.99 & ( ord_less_nat @ X @ Y )
% 8.83/4.99 & ! [Z3: nat] :
% 8.83/4.99 ( ( member_nat @ Z3 @ Xs )
% 8.83/4.99 => ( ( ord_less_nat @ X @ Z3 )
% 8.83/4.99 => ( ord_less_eq_nat @ Y @ Z3 ) ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1231_is__pred__in__set__def,axiom,
% 8.83/4.99 ( vEBT_is_pred_in_set
% 8.83/4.99 = ( ^ [Xs: set_nat,X: nat,Y: nat] :
% 8.83/4.99 ( ( member_nat @ Y @ Xs )
% 8.83/4.99 & ( ord_less_nat @ Y @ X )
% 8.83/4.99 & ! [Z3: nat] :
% 8.83/4.99 ( ( member_nat @ Z3 @ Xs )
% 8.83/4.99 => ( ( ord_less_nat @ Z3 @ X )
% 8.83/4.99 => ( ord_less_eq_nat @ Z3 @ Y ) ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1232_buildup__gives__empty,axiom,
% 8.83/4.99 ! [N: nat] :
% 8.83/4.99 ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
% 8.83/4.99 = bot_bot_set_nat ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1233_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 8.83/4.99 ! [X2: vEBT_VEBT,Xa2: nat] :
% 8.83/4.99 ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 8.83/4.99 => ( ! [Mi2: nat,Ma2: nat] :
% 8.83/4.99 ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 8.83/4.99 ( X2
% 8.83/4.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 8.83/4.99 => ~ ( ( Xa2 = Mi2 )
% 8.83/4.99 | ( Xa2 = Ma2 ) ) )
% 8.83/4.99 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
% 8.83/4.99 ( ? [Vc2: vEBT_VEBT] :
% 8.83/4.99 ( X2
% 8.83/4.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
% 8.83/4.99 => ~ ( ( Xa2 = Mi2 )
% 8.83/4.99 | ( Xa2 = Ma2 )
% 8.83/4.99 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/4.99 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 8.83/4.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
% 8.83/4.99 => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
% 8.83/4.99 ( ? [Vd2: vEBT_VEBT] :
% 8.83/4.99 ( X2
% 8.83/4.99 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
% 8.83/4.99 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/4.99 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 8.83/4.99 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1234_divmod__step__eq,axiom,
% 8.83/4.99 ! [L: num,R2: nat,Q3: nat] :
% 8.83/4.99 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 8.83/4.99 => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
% 8.83/4.99 = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
% 8.83/4.99 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 8.83/4.99 => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
% 8.83/4.99 = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1235_divmod__step__eq,axiom,
% 8.83/4.99 ! [L: num,R2: int,Q3: int] :
% 8.83/4.99 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 8.83/4.99 => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 8.83/4.99 = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
% 8.83/4.99 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 8.83/4.99 => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 8.83/4.99 = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1236_divmod__step__eq,axiom,
% 8.83/4.99 ! [L: num,R2: code_integer,Q3: code_integer] :
% 8.83/4.99 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 8.83/4.99 => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
% 8.83/4.99 = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
% 8.83/4.99 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 8.83/4.99 => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
% 8.83/4.99 = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1237_nth__mem,axiom,
% 8.83/4.99 ! [N: nat,Xs2: list_P6011104703257516679at_nat] :
% 8.83/4.99 ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
% 8.83/4.99 => ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ N ) @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1238_nth__mem,axiom,
% 8.83/4.99 ! [N: nat,Xs2: list_complex] :
% 8.83/4.99 ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 8.83/4.99 => ( member_complex @ ( nth_complex @ Xs2 @ N ) @ ( set_complex2 @ Xs2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1239_nth__mem,axiom,
% 8.83/4.99 ! [N: nat,Xs2: list_real] :
% 8.83/4.99 ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 8.83/4.99 => ( member_real @ ( nth_real @ Xs2 @ N ) @ ( set_real2 @ Xs2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1240_nth__mem,axiom,
% 8.83/4.99 ! [N: nat,Xs2: list_VEBT_VEBT] :
% 8.83/4.99 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 8.83/4.99 => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1241_nth__mem,axiom,
% 8.83/4.99 ! [N: nat,Xs2: list_o] :
% 8.83/4.99 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 8.83/4.99 => ( member_o @ ( nth_o @ Xs2 @ N ) @ ( set_o2 @ Xs2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1242_nth__mem,axiom,
% 8.83/4.99 ! [N: nat,Xs2: list_nat] :
% 8.83/4.99 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 8.83/4.99 => ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1243_nth__mem,axiom,
% 8.83/4.99 ! [N: nat,Xs2: list_int] :
% 8.83/4.99 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 8.83/4.99 => ( member_int @ ( nth_int @ Xs2 @ N ) @ ( set_int2 @ Xs2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1244_list__ball__nth,axiom,
% 8.83/4.99 ! [N: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 8.83/4.99 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 8.83/4.99 => ( ! [X5: vEBT_VEBT] :
% 8.83/4.99 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 8.83/4.99 => ( P @ X5 ) )
% 8.83/4.99 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1245_list__ball__nth,axiom,
% 8.83/4.99 ! [N: nat,Xs2: list_o,P: $o > $o] :
% 8.83/4.99 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 8.83/4.99 => ( ! [X5: $o] :
% 8.83/4.99 ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
% 8.83/4.99 => ( P @ X5 ) )
% 8.83/4.99 => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1246_list__ball__nth,axiom,
% 8.83/4.99 ! [N: nat,Xs2: list_nat,P: nat > $o] :
% 8.83/4.99 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 8.83/4.99 => ( ! [X5: nat] :
% 8.83/4.99 ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
% 8.83/4.99 => ( P @ X5 ) )
% 8.83/4.99 => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1247_list__ball__nth,axiom,
% 8.83/4.99 ! [N: nat,Xs2: list_int,P: int > $o] :
% 8.83/4.99 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 8.83/4.99 => ( ! [X5: int] :
% 8.83/4.99 ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
% 8.83/4.99 => ( P @ X5 ) )
% 8.83/4.99 => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1248_in__set__conv__nth,axiom,
% 8.83/4.99 ! [X2: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
% 8.83/4.99 ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
% 8.83/4.99 = ( ? [I4: nat] :
% 8.83/4.99 ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
% 8.83/4.99 & ( ( nth_Pr7617993195940197384at_nat @ Xs2 @ I4 )
% 8.83/4.99 = X2 ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1249_in__set__conv__nth,axiom,
% 8.83/4.99 ! [X2: complex,Xs2: list_complex] :
% 8.83/4.99 ( ( member_complex @ X2 @ ( set_complex2 @ Xs2 ) )
% 8.83/4.99 = ( ? [I4: nat] :
% 8.83/4.99 ( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 8.83/4.99 & ( ( nth_complex @ Xs2 @ I4 )
% 8.83/4.99 = X2 ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1250_in__set__conv__nth,axiom,
% 8.83/4.99 ! [X2: real,Xs2: list_real] :
% 8.83/4.99 ( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
% 8.83/4.99 = ( ? [I4: nat] :
% 8.83/4.99 ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs2 ) )
% 8.83/4.99 & ( ( nth_real @ Xs2 @ I4 )
% 8.83/4.99 = X2 ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1251_in__set__conv__nth,axiom,
% 8.83/4.99 ! [X2: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
% 8.83/4.99 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 8.83/4.99 = ( ? [I4: nat] :
% 8.83/4.99 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 8.83/4.99 & ( ( nth_VEBT_VEBT @ Xs2 @ I4 )
% 8.83/4.99 = X2 ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1252_in__set__conv__nth,axiom,
% 8.83/4.99 ! [X2: $o,Xs2: list_o] :
% 8.83/4.99 ( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
% 8.83/4.99 = ( ? [I4: nat] :
% 8.83/4.99 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
% 8.83/4.99 & ( ( nth_o @ Xs2 @ I4 )
% 8.83/4.99 = X2 ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1253_in__set__conv__nth,axiom,
% 8.83/4.99 ! [X2: nat,Xs2: list_nat] :
% 8.83/4.99 ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
% 8.83/4.99 = ( ? [I4: nat] :
% 8.83/4.99 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
% 8.83/4.99 & ( ( nth_nat @ Xs2 @ I4 )
% 8.83/4.99 = X2 ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1254_in__set__conv__nth,axiom,
% 8.83/4.99 ! [X2: int,Xs2: list_int] :
% 8.83/4.99 ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
% 8.83/4.99 = ( ? [I4: nat] :
% 8.83/4.99 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
% 8.83/4.99 & ( ( nth_int @ Xs2 @ I4 )
% 8.83/4.99 = X2 ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1255_all__nth__imp__all__set,axiom,
% 8.83/4.99 ! [Xs2: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,X2: product_prod_nat_nat] :
% 8.83/4.99 ( ! [I3: nat] :
% 8.83/4.99 ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
% 8.83/4.99 => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ I3 ) ) )
% 8.83/4.99 => ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
% 8.83/4.99 => ( P @ X2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1256_all__nth__imp__all__set,axiom,
% 8.83/4.99 ! [Xs2: list_complex,P: complex > $o,X2: complex] :
% 8.83/4.99 ( ! [I3: nat] :
% 8.83/4.99 ( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 8.83/4.99 => ( P @ ( nth_complex @ Xs2 @ I3 ) ) )
% 8.83/4.99 => ( ( member_complex @ X2 @ ( set_complex2 @ Xs2 ) )
% 8.83/4.99 => ( P @ X2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1257_all__nth__imp__all__set,axiom,
% 8.83/4.99 ! [Xs2: list_real,P: real > $o,X2: real] :
% 8.83/4.99 ( ! [I3: nat] :
% 8.83/4.99 ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
% 8.83/4.99 => ( P @ ( nth_real @ Xs2 @ I3 ) ) )
% 8.83/4.99 => ( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
% 8.83/4.99 => ( P @ X2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1258_all__nth__imp__all__set,axiom,
% 8.83/4.99 ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X2: vEBT_VEBT] :
% 8.83/4.99 ( ! [I3: nat] :
% 8.83/4.99 ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 8.83/4.99 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) )
% 8.83/4.99 => ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 8.83/4.99 => ( P @ X2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1259_all__nth__imp__all__set,axiom,
% 8.83/4.99 ! [Xs2: list_o,P: $o > $o,X2: $o] :
% 8.83/4.99 ( ! [I3: nat] :
% 8.83/4.99 ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
% 8.83/4.99 => ( P @ ( nth_o @ Xs2 @ I3 ) ) )
% 8.83/4.99 => ( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
% 8.83/4.99 => ( P @ X2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1260_all__nth__imp__all__set,axiom,
% 8.83/4.99 ! [Xs2: list_nat,P: nat > $o,X2: nat] :
% 8.83/4.99 ( ! [I3: nat] :
% 8.83/4.99 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
% 8.83/4.99 => ( P @ ( nth_nat @ Xs2 @ I3 ) ) )
% 8.83/4.99 => ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
% 8.83/4.99 => ( P @ X2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1261_all__nth__imp__all__set,axiom,
% 8.83/4.99 ! [Xs2: list_int,P: int > $o,X2: int] :
% 8.83/4.99 ( ! [I3: nat] :
% 8.83/4.99 ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
% 8.83/4.99 => ( P @ ( nth_int @ Xs2 @ I3 ) ) )
% 8.83/4.99 => ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
% 8.83/4.99 => ( P @ X2 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1262_all__set__conv__all__nth,axiom,
% 8.83/4.99 ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 8.83/4.99 ( ( ! [X: vEBT_VEBT] :
% 8.83/4.99 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 8.83/4.99 => ( P @ X ) ) )
% 8.83/4.99 = ( ! [I4: nat] :
% 8.83/4.99 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 8.83/4.99 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I4 ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1263_all__set__conv__all__nth,axiom,
% 8.83/4.99 ! [Xs2: list_o,P: $o > $o] :
% 8.83/4.99 ( ( ! [X: $o] :
% 8.83/4.99 ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
% 8.83/4.99 => ( P @ X ) ) )
% 8.83/4.99 = ( ! [I4: nat] :
% 8.83/4.99 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
% 8.83/4.99 => ( P @ ( nth_o @ Xs2 @ I4 ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1264_all__set__conv__all__nth,axiom,
% 8.83/4.99 ! [Xs2: list_nat,P: nat > $o] :
% 8.83/4.99 ( ( ! [X: nat] :
% 8.83/4.99 ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
% 8.83/4.99 => ( P @ X ) ) )
% 8.83/4.99 = ( ! [I4: nat] :
% 8.83/4.99 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
% 8.83/4.99 => ( P @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1265_all__set__conv__all__nth,axiom,
% 8.83/4.99 ! [Xs2: list_int,P: int > $o] :
% 8.83/4.99 ( ( ! [X: int] :
% 8.83/4.99 ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
% 8.83/4.99 => ( P @ X ) ) )
% 8.83/4.99 = ( ! [I4: nat] :
% 8.83/4.99 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
% 8.83/4.99 => ( P @ ( nth_int @ Xs2 @ I4 ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1266_vebt__pred_Oelims,axiom,
% 8.83/4.99 ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 8.83/4.99 ( ( ( vEBT_vebt_pred @ X2 @ Xa2 )
% 8.83/4.99 = Y3 )
% 8.83/4.99 => ( ( ? [Uu2: $o,Uv2: $o] :
% 8.83/4.99 ( X2
% 8.83/4.99 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.83/4.99 => ( ( Xa2 = zero_zero_nat )
% 8.83/4.99 => ( Y3 != none_nat ) ) )
% 8.83/4.99 => ( ! [A4: $o] :
% 8.83/4.99 ( ? [Uw2: $o] :
% 8.83/4.99 ( X2
% 8.83/4.99 = ( vEBT_Leaf @ A4 @ Uw2 ) )
% 8.83/4.99 => ( ( Xa2
% 8.83/4.99 = ( suc @ zero_zero_nat ) )
% 8.83/4.99 => ~ ( ( A4
% 8.83/4.99 => ( Y3
% 8.83/4.99 = ( some_nat @ zero_zero_nat ) ) )
% 8.83/4.99 & ( ~ A4
% 8.83/4.99 => ( Y3 = none_nat ) ) ) ) )
% 8.83/4.99 => ( ! [A4: $o,B4: $o] :
% 8.83/4.99 ( ( X2
% 8.83/4.99 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/4.99 => ( ? [Va3: nat] :
% 8.83/4.99 ( Xa2
% 8.83/4.99 = ( suc @ ( suc @ Va3 ) ) )
% 8.83/4.99 => ~ ( ( B4
% 8.83/4.99 => ( Y3
% 8.83/4.99 = ( some_nat @ one_one_nat ) ) )
% 8.83/4.99 & ( ~ B4
% 8.83/4.99 => ( ( A4
% 8.83/4.99 => ( Y3
% 8.83/4.99 = ( some_nat @ zero_zero_nat ) ) )
% 8.83/4.99 & ( ~ A4
% 8.83/4.99 => ( Y3 = none_nat ) ) ) ) ) ) )
% 8.83/4.99 => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
% 8.83/4.99 ( X2
% 8.83/4.99 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
% 8.83/4.99 => ( Y3 != none_nat ) )
% 8.83/4.99 => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
% 8.83/4.99 ( X2
% 8.83/4.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) )
% 8.83/4.99 => ( Y3 != none_nat ) )
% 8.83/4.99 => ( ( ? [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
% 8.83/4.99 ( X2
% 8.83/4.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
% 8.83/4.99 => ( Y3 != none_nat ) )
% 8.83/4.99 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.83/4.99 ( ( X2
% 8.83/4.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.83/4.99 => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
% 8.83/4.99 => ( Y3
% 8.83/4.99 = ( some_nat @ Ma2 ) ) )
% 8.83/4.99 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 8.83/4.99 => ( Y3
% 8.83/4.99 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/4.99 @ ( if_option_nat
% 8.83/4.99 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 != none_nat )
% 8.83/4.99 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 8.83/4.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 @ ( if_option_nat
% 8.83/4.99 @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.83/4.99 = none_nat )
% 8.83/4.99 @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
% 8.83/4.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 8.83/4.99 @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1267_vebt__succ_Oelims,axiom,
% 8.83/4.99 ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 8.83/4.99 ( ( ( vEBT_vebt_succ @ X2 @ Xa2 )
% 8.83/4.99 = Y3 )
% 8.83/4.99 => ( ! [Uu2: $o,B4: $o] :
% 8.83/4.99 ( ( X2
% 8.83/4.99 = ( vEBT_Leaf @ Uu2 @ B4 ) )
% 8.83/4.99 => ( ( Xa2 = zero_zero_nat )
% 8.83/4.99 => ~ ( ( B4
% 8.83/4.99 => ( Y3
% 8.83/4.99 = ( some_nat @ one_one_nat ) ) )
% 8.83/4.99 & ( ~ B4
% 8.83/4.99 => ( Y3 = none_nat ) ) ) ) )
% 8.83/4.99 => ( ( ? [Uv2: $o,Uw2: $o] :
% 8.83/4.99 ( X2
% 8.83/4.99 = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 8.83/4.99 => ( ? [N2: nat] :
% 8.83/4.99 ( Xa2
% 8.83/4.99 = ( suc @ N2 ) )
% 8.83/4.99 => ( Y3 != none_nat ) ) )
% 8.83/4.99 => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 8.83/4.99 ( X2
% 8.83/4.99 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 8.83/4.99 => ( Y3 != none_nat ) )
% 8.83/4.99 => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 8.83/4.99 ( X2
% 8.83/4.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 8.83/4.99 => ( Y3 != none_nat ) )
% 8.83/4.99 => ( ( ? [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
% 8.83/4.99 ( X2
% 8.83/4.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
% 8.83/4.99 => ( Y3 != none_nat ) )
% 8.83/4.99 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.83/4.99 ( ( X2
% 8.83/4.99 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.83/4.99 => ~ ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 8.83/4.99 => ( Y3
% 8.83/4.99 = ( some_nat @ Mi2 ) ) )
% 8.83/4.99 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 8.83/4.99 => ( Y3
% 8.83/4.99 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/4.99 @ ( if_option_nat
% 8.83/4.99 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 != none_nat )
% 8.83/4.99 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 8.83/4.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/4.99 @ ( if_option_nat
% 8.83/4.99 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.83/4.99 = none_nat )
% 8.83/4.99 @ none_nat
% 8.83/4.99 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 8.83/4.99 @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1268_valid__tree__deg__neq__0,axiom,
% 8.83/4.99 ! [T: vEBT_VEBT] :
% 8.83/4.99 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1269_valid__0__not,axiom,
% 8.83/4.99 ! [T: vEBT_VEBT] :
% 8.83/4.99 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1270_buildup__nothing__in__min__max,axiom,
% 8.83/4.99 ! [N: nat,X2: nat] :
% 8.83/4.99 ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X2 ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1271_buildup__nothing__in__leaf,axiom,
% 8.83/4.99 ! [N: nat,X2: nat] :
% 8.83/4.99 ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X2 ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1272_deg__not__0,axiom,
% 8.83/4.99 ! [T: vEBT_VEBT,N: nat] :
% 8.83/4.99 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.99 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1273_Leaf__0__not,axiom,
% 8.83/4.99 ! [A: $o,B: $o] :
% 8.83/4.99 ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1274_deg1Leaf,axiom,
% 8.83/4.99 ! [T: vEBT_VEBT] :
% 8.83/4.99 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 8.83/4.99 = ( ? [A3: $o,B2: $o] :
% 8.83/4.99 ( T
% 8.83/4.99 = ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1275_deg__1__Leaf,axiom,
% 8.83/4.99 ! [T: vEBT_VEBT] :
% 8.83/4.99 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 8.83/4.99 => ? [A4: $o,B4: $o] :
% 8.83/4.99 ( T
% 8.83/4.99 = ( vEBT_Leaf @ A4 @ B4 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1276_deg__1__Leafy,axiom,
% 8.83/4.99 ! [T: vEBT_VEBT,N: nat] :
% 8.83/4.99 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/4.99 => ( ( N = one_one_nat )
% 8.83/4.99 => ? [A4: $o,B4: $o] :
% 8.83/4.99 ( T
% 8.83/4.99 = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1277_buildup__gives__valid,axiom,
% 8.83/4.99 ! [N: nat] :
% 8.83/4.99 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/4.99 => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1278_zdiv__numeral__Bit0,axiom,
% 8.83/4.99 ! [V: num,W: num] :
% 8.83/4.99 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 8.83/4.99 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1279_VEBT_Oinject_I2_J,axiom,
% 8.83/4.99 ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
% 8.83/4.99 ( ( ( vEBT_Leaf @ X21 @ X222 )
% 8.83/4.99 = ( vEBT_Leaf @ Y21 @ Y22 ) )
% 8.83/4.99 = ( ( X21 = Y21 )
% 8.83/4.99 & ( X222 = Y22 ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1280_le__zero__eq,axiom,
% 8.83/4.99 ! [N: nat] :
% 8.83/4.99 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 8.83/4.99 = ( N = zero_zero_nat ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1281_not__gr__zero,axiom,
% 8.83/4.99 ! [N: nat] :
% 8.83/4.99 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 8.83/4.99 = ( N = zero_zero_nat ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1282_mult__cancel__right,axiom,
% 8.83/4.99 ! [A: complex,C: complex,B: complex] :
% 8.83/4.99 ( ( ( times_times_complex @ A @ C )
% 8.83/4.99 = ( times_times_complex @ B @ C ) )
% 8.83/4.99 = ( ( C = zero_zero_complex )
% 8.83/4.99 | ( A = B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1283_mult__cancel__right,axiom,
% 8.83/4.99 ! [A: real,C: real,B: real] :
% 8.83/4.99 ( ( ( times_times_real @ A @ C )
% 8.83/4.99 = ( times_times_real @ B @ C ) )
% 8.83/4.99 = ( ( C = zero_zero_real )
% 8.83/4.99 | ( A = B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1284_mult__cancel__right,axiom,
% 8.83/4.99 ! [A: rat,C: rat,B: rat] :
% 8.83/4.99 ( ( ( times_times_rat @ A @ C )
% 8.83/4.99 = ( times_times_rat @ B @ C ) )
% 8.83/4.99 = ( ( C = zero_zero_rat )
% 8.83/4.99 | ( A = B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1285_mult__cancel__right,axiom,
% 8.83/4.99 ! [A: nat,C: nat,B: nat] :
% 8.83/4.99 ( ( ( times_times_nat @ A @ C )
% 8.83/4.99 = ( times_times_nat @ B @ C ) )
% 8.83/4.99 = ( ( C = zero_zero_nat )
% 8.83/4.99 | ( A = B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1286_mult__cancel__right,axiom,
% 8.83/4.99 ! [A: int,C: int,B: int] :
% 8.83/4.99 ( ( ( times_times_int @ A @ C )
% 8.83/4.99 = ( times_times_int @ B @ C ) )
% 8.83/4.99 = ( ( C = zero_zero_int )
% 8.83/4.99 | ( A = B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1287_mult__cancel__left,axiom,
% 8.83/4.99 ! [C: complex,A: complex,B: complex] :
% 8.83/4.99 ( ( ( times_times_complex @ C @ A )
% 8.83/4.99 = ( times_times_complex @ C @ B ) )
% 8.83/4.99 = ( ( C = zero_zero_complex )
% 8.83/4.99 | ( A = B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1288_mult__cancel__left,axiom,
% 8.83/4.99 ! [C: real,A: real,B: real] :
% 8.83/4.99 ( ( ( times_times_real @ C @ A )
% 8.83/4.99 = ( times_times_real @ C @ B ) )
% 8.83/4.99 = ( ( C = zero_zero_real )
% 8.83/4.99 | ( A = B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1289_mult__cancel__left,axiom,
% 8.83/4.99 ! [C: rat,A: rat,B: rat] :
% 8.83/4.99 ( ( ( times_times_rat @ C @ A )
% 8.83/4.99 = ( times_times_rat @ C @ B ) )
% 8.83/4.99 = ( ( C = zero_zero_rat )
% 8.83/4.99 | ( A = B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1290_mult__cancel__left,axiom,
% 8.83/4.99 ! [C: nat,A: nat,B: nat] :
% 8.83/4.99 ( ( ( times_times_nat @ C @ A )
% 8.83/4.99 = ( times_times_nat @ C @ B ) )
% 8.83/4.99 = ( ( C = zero_zero_nat )
% 8.83/4.99 | ( A = B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1291_mult__cancel__left,axiom,
% 8.83/4.99 ! [C: int,A: int,B: int] :
% 8.83/4.99 ( ( ( times_times_int @ C @ A )
% 8.83/4.99 = ( times_times_int @ C @ B ) )
% 8.83/4.99 = ( ( C = zero_zero_int )
% 8.83/4.99 | ( A = B ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1292_mult__eq__0__iff,axiom,
% 8.83/4.99 ! [A: complex,B: complex] :
% 8.83/4.99 ( ( ( times_times_complex @ A @ B )
% 8.83/4.99 = zero_zero_complex )
% 8.83/4.99 = ( ( A = zero_zero_complex )
% 8.83/4.99 | ( B = zero_zero_complex ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1293_mult__eq__0__iff,axiom,
% 8.83/4.99 ! [A: real,B: real] :
% 8.83/4.99 ( ( ( times_times_real @ A @ B )
% 8.83/4.99 = zero_zero_real )
% 8.83/4.99 = ( ( A = zero_zero_real )
% 8.83/4.99 | ( B = zero_zero_real ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1294_mult__eq__0__iff,axiom,
% 8.83/4.99 ! [A: rat,B: rat] :
% 8.83/4.99 ( ( ( times_times_rat @ A @ B )
% 8.83/4.99 = zero_zero_rat )
% 8.83/4.99 = ( ( A = zero_zero_rat )
% 8.83/4.99 | ( B = zero_zero_rat ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1295_mult__eq__0__iff,axiom,
% 8.83/4.99 ! [A: nat,B: nat] :
% 8.83/4.99 ( ( ( times_times_nat @ A @ B )
% 8.83/4.99 = zero_zero_nat )
% 8.83/4.99 = ( ( A = zero_zero_nat )
% 8.83/4.99 | ( B = zero_zero_nat ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1296_mult__eq__0__iff,axiom,
% 8.83/4.99 ! [A: int,B: int] :
% 8.83/4.99 ( ( ( times_times_int @ A @ B )
% 8.83/4.99 = zero_zero_int )
% 8.83/4.99 = ( ( A = zero_zero_int )
% 8.83/4.99 | ( B = zero_zero_int ) ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1297_mult__zero__right,axiom,
% 8.83/4.99 ! [A: complex] :
% 8.83/4.99 ( ( times_times_complex @ A @ zero_zero_complex )
% 8.83/4.99 = zero_zero_complex ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1298_mult__zero__right,axiom,
% 8.83/4.99 ! [A: real] :
% 8.83/4.99 ( ( times_times_real @ A @ zero_zero_real )
% 8.83/4.99 = zero_zero_real ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1299_mult__zero__right,axiom,
% 8.83/4.99 ! [A: rat] :
% 8.83/4.99 ( ( times_times_rat @ A @ zero_zero_rat )
% 8.83/4.99 = zero_zero_rat ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1300_mult__zero__right,axiom,
% 8.83/4.99 ! [A: nat] :
% 8.83/4.99 ( ( times_times_nat @ A @ zero_zero_nat )
% 8.83/4.99 = zero_zero_nat ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1301_mult__zero__right,axiom,
% 8.83/4.99 ! [A: int] :
% 8.83/4.99 ( ( times_times_int @ A @ zero_zero_int )
% 8.83/4.99 = zero_zero_int ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1302_mult__zero__left,axiom,
% 8.83/4.99 ! [A: complex] :
% 8.83/4.99 ( ( times_times_complex @ zero_zero_complex @ A )
% 8.83/4.99 = zero_zero_complex ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1303_mult__zero__left,axiom,
% 8.83/4.99 ! [A: real] :
% 8.83/4.99 ( ( times_times_real @ zero_zero_real @ A )
% 8.83/4.99 = zero_zero_real ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1304_mult__zero__left,axiom,
% 8.83/4.99 ! [A: rat] :
% 8.83/4.99 ( ( times_times_rat @ zero_zero_rat @ A )
% 8.83/4.99 = zero_zero_rat ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1305_mult__zero__left,axiom,
% 8.83/4.99 ! [A: nat] :
% 8.83/4.99 ( ( times_times_nat @ zero_zero_nat @ A )
% 8.83/4.99 = zero_zero_nat ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1306_mult__zero__left,axiom,
% 8.83/4.99 ! [A: int] :
% 8.83/4.99 ( ( times_times_int @ zero_zero_int @ A )
% 8.83/4.99 = zero_zero_int ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1307_add_Oright__neutral,axiom,
% 8.83/4.99 ! [A: complex] :
% 8.83/4.99 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1308_add_Oright__neutral,axiom,
% 8.83/4.99 ! [A: real] :
% 8.83/4.99 ( ( plus_plus_real @ A @ zero_zero_real )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1309_add_Oright__neutral,axiom,
% 8.83/4.99 ! [A: rat] :
% 8.83/4.99 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1310_add_Oright__neutral,axiom,
% 8.83/4.99 ! [A: nat] :
% 8.83/4.99 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1311_add_Oright__neutral,axiom,
% 8.83/4.99 ! [A: int] :
% 8.83/4.99 ( ( plus_plus_int @ A @ zero_zero_int )
% 8.83/4.99 = A ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1312_double__zero__sym,axiom,
% 8.83/4.99 ! [A: real] :
% 8.83/4.99 ( ( zero_zero_real
% 8.83/4.99 = ( plus_plus_real @ A @ A ) )
% 8.83/4.99 = ( A = zero_zero_real ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1313_double__zero__sym,axiom,
% 8.83/4.99 ! [A: rat] :
% 8.83/4.99 ( ( zero_zero_rat
% 8.83/4.99 = ( plus_plus_rat @ A @ A ) )
% 8.83/4.99 = ( A = zero_zero_rat ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1314_double__zero__sym,axiom,
% 8.83/4.99 ! [A: int] :
% 8.83/4.99 ( ( zero_zero_int
% 8.83/4.99 = ( plus_plus_int @ A @ A ) )
% 8.83/4.99 = ( A = zero_zero_int ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1315_add__cancel__left__left,axiom,
% 8.83/4.99 ! [B: complex,A: complex] :
% 8.83/4.99 ( ( ( plus_plus_complex @ B @ A )
% 8.83/4.99 = A )
% 8.83/4.99 = ( B = zero_zero_complex ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1316_add__cancel__left__left,axiom,
% 8.83/4.99 ! [B: real,A: real] :
% 8.83/4.99 ( ( ( plus_plus_real @ B @ A )
% 8.83/4.99 = A )
% 8.83/4.99 = ( B = zero_zero_real ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1317_add__cancel__left__left,axiom,
% 8.83/4.99 ! [B: rat,A: rat] :
% 8.83/4.99 ( ( ( plus_plus_rat @ B @ A )
% 8.83/4.99 = A )
% 8.83/4.99 = ( B = zero_zero_rat ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1318_add__cancel__left__left,axiom,
% 8.83/4.99 ! [B: nat,A: nat] :
% 8.83/4.99 ( ( ( plus_plus_nat @ B @ A )
% 8.83/4.99 = A )
% 8.83/4.99 = ( B = zero_zero_nat ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1319_add__cancel__left__left,axiom,
% 8.83/4.99 ! [B: int,A: int] :
% 8.83/4.99 ( ( ( plus_plus_int @ B @ A )
% 8.83/4.99 = A )
% 8.83/4.99 = ( B = zero_zero_int ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1320_add__cancel__left__right,axiom,
% 8.83/4.99 ! [A: complex,B: complex] :
% 8.83/4.99 ( ( ( plus_plus_complex @ A @ B )
% 8.83/4.99 = A )
% 8.83/4.99 = ( B = zero_zero_complex ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1321_add__cancel__left__right,axiom,
% 8.83/4.99 ! [A: real,B: real] :
% 8.83/4.99 ( ( ( plus_plus_real @ A @ B )
% 8.83/4.99 = A )
% 8.83/4.99 = ( B = zero_zero_real ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1322_add__cancel__left__right,axiom,
% 8.83/4.99 ! [A: rat,B: rat] :
% 8.83/4.99 ( ( ( plus_plus_rat @ A @ B )
% 8.83/4.99 = A )
% 8.83/4.99 = ( B = zero_zero_rat ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1323_add__cancel__left__right,axiom,
% 8.83/4.99 ! [A: nat,B: nat] :
% 8.83/4.99 ( ( ( plus_plus_nat @ A @ B )
% 8.83/4.99 = A )
% 8.83/4.99 = ( B = zero_zero_nat ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1324_add__cancel__left__right,axiom,
% 8.83/4.99 ! [A: int,B: int] :
% 8.83/4.99 ( ( ( plus_plus_int @ A @ B )
% 8.83/4.99 = A )
% 8.83/4.99 = ( B = zero_zero_int ) ) ).
% 8.83/4.99
% 8.83/4.99 thf(fact_1325_add__cancel__right__left,axiom,
% 8.83/5.00 ! [A: complex,B: complex] :
% 8.83/5.00 ( ( A
% 8.83/5.00 = ( plus_plus_complex @ B @ A ) )
% 8.83/5.00 = ( B = zero_zero_complex ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1326_add__cancel__right__left,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( A
% 8.83/5.00 = ( plus_plus_real @ B @ A ) )
% 8.83/5.00 = ( B = zero_zero_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1327_add__cancel__right__left,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( A
% 8.83/5.00 = ( plus_plus_rat @ B @ A ) )
% 8.83/5.00 = ( B = zero_zero_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1328_add__cancel__right__left,axiom,
% 8.83/5.00 ! [A: nat,B: nat] :
% 8.83/5.00 ( ( A
% 8.83/5.00 = ( plus_plus_nat @ B @ A ) )
% 8.83/5.00 = ( B = zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1329_add__cancel__right__left,axiom,
% 8.83/5.00 ! [A: int,B: int] :
% 8.83/5.00 ( ( A
% 8.83/5.00 = ( plus_plus_int @ B @ A ) )
% 8.83/5.00 = ( B = zero_zero_int ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1330_add__cancel__right__right,axiom,
% 8.83/5.00 ! [A: complex,B: complex] :
% 8.83/5.00 ( ( A
% 8.83/5.00 = ( plus_plus_complex @ A @ B ) )
% 8.83/5.00 = ( B = zero_zero_complex ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1331_add__cancel__right__right,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( A
% 8.83/5.00 = ( plus_plus_real @ A @ B ) )
% 8.83/5.00 = ( B = zero_zero_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1332_add__cancel__right__right,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( A
% 8.83/5.00 = ( plus_plus_rat @ A @ B ) )
% 8.83/5.00 = ( B = zero_zero_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1333_add__cancel__right__right,axiom,
% 8.83/5.00 ! [A: nat,B: nat] :
% 8.83/5.00 ( ( A
% 8.83/5.00 = ( plus_plus_nat @ A @ B ) )
% 8.83/5.00 = ( B = zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1334_add__cancel__right__right,axiom,
% 8.83/5.00 ! [A: int,B: int] :
% 8.83/5.00 ( ( A
% 8.83/5.00 = ( plus_plus_int @ A @ B ) )
% 8.83/5.00 = ( B = zero_zero_int ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1335_add__eq__0__iff__both__eq__0,axiom,
% 8.83/5.00 ! [X2: nat,Y3: nat] :
% 8.83/5.00 ( ( ( plus_plus_nat @ X2 @ Y3 )
% 8.83/5.00 = zero_zero_nat )
% 8.83/5.00 = ( ( X2 = zero_zero_nat )
% 8.83/5.00 & ( Y3 = zero_zero_nat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1336_zero__eq__add__iff__both__eq__0,axiom,
% 8.83/5.00 ! [X2: nat,Y3: nat] :
% 8.83/5.00 ( ( zero_zero_nat
% 8.83/5.00 = ( plus_plus_nat @ X2 @ Y3 ) )
% 8.83/5.00 = ( ( X2 = zero_zero_nat )
% 8.83/5.00 & ( Y3 = zero_zero_nat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1337_add__0,axiom,
% 8.83/5.00 ! [A: complex] :
% 8.83/5.00 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1338_add__0,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( plus_plus_real @ zero_zero_real @ A )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1339_add__0,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1340_add__0,axiom,
% 8.83/5.00 ! [A: nat] :
% 8.83/5.00 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1341_add__0,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( plus_plus_int @ zero_zero_int @ A )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1342_diff__self,axiom,
% 8.83/5.00 ! [A: complex] :
% 8.83/5.00 ( ( minus_minus_complex @ A @ A )
% 8.83/5.00 = zero_zero_complex ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1343_diff__self,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( minus_minus_real @ A @ A )
% 8.83/5.00 = zero_zero_real ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1344_diff__self,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( minus_minus_rat @ A @ A )
% 8.83/5.00 = zero_zero_rat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1345_diff__self,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( minus_minus_int @ A @ A )
% 8.83/5.00 = zero_zero_int ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1346_diff__0__right,axiom,
% 8.83/5.00 ! [A: complex] :
% 8.83/5.00 ( ( minus_minus_complex @ A @ zero_zero_complex )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1347_diff__0__right,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( minus_minus_real @ A @ zero_zero_real )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1348_diff__0__right,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1349_diff__0__right,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( minus_minus_int @ A @ zero_zero_int )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1350_zero__diff,axiom,
% 8.83/5.00 ! [A: nat] :
% 8.83/5.00 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 8.83/5.00 = zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1351_diff__zero,axiom,
% 8.83/5.00 ! [A: complex] :
% 8.83/5.00 ( ( minus_minus_complex @ A @ zero_zero_complex )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1352_diff__zero,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( minus_minus_real @ A @ zero_zero_real )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1353_diff__zero,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1354_diff__zero,axiom,
% 8.83/5.00 ! [A: nat] :
% 8.83/5.00 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1355_diff__zero,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( minus_minus_int @ A @ zero_zero_int )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1356_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 8.83/5.00 ! [A: complex] :
% 8.83/5.00 ( ( minus_minus_complex @ A @ A )
% 8.83/5.00 = zero_zero_complex ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1357_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( minus_minus_real @ A @ A )
% 8.83/5.00 = zero_zero_real ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1358_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( minus_minus_rat @ A @ A )
% 8.83/5.00 = zero_zero_rat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1359_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 8.83/5.00 ! [A: nat] :
% 8.83/5.00 ( ( minus_minus_nat @ A @ A )
% 8.83/5.00 = zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1360_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( minus_minus_int @ A @ A )
% 8.83/5.00 = zero_zero_int ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1361_division__ring__divide__zero,axiom,
% 8.83/5.00 ! [A: complex] :
% 8.83/5.00 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 8.83/5.00 = zero_zero_complex ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1362_division__ring__divide__zero,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( divide_divide_real @ A @ zero_zero_real )
% 8.83/5.00 = zero_zero_real ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1363_division__ring__divide__zero,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 8.83/5.00 = zero_zero_rat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1364_divide__cancel__right,axiom,
% 8.83/5.00 ! [A: complex,C: complex,B: complex] :
% 8.83/5.00 ( ( ( divide1717551699836669952omplex @ A @ C )
% 8.83/5.00 = ( divide1717551699836669952omplex @ B @ C ) )
% 8.83/5.00 = ( ( C = zero_zero_complex )
% 8.83/5.00 | ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1365_divide__cancel__right,axiom,
% 8.83/5.00 ! [A: real,C: real,B: real] :
% 8.83/5.00 ( ( ( divide_divide_real @ A @ C )
% 8.83/5.00 = ( divide_divide_real @ B @ C ) )
% 8.83/5.00 = ( ( C = zero_zero_real )
% 8.83/5.00 | ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1366_divide__cancel__right,axiom,
% 8.83/5.00 ! [A: rat,C: rat,B: rat] :
% 8.83/5.00 ( ( ( divide_divide_rat @ A @ C )
% 8.83/5.00 = ( divide_divide_rat @ B @ C ) )
% 8.83/5.00 = ( ( C = zero_zero_rat )
% 8.83/5.00 | ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1367_divide__cancel__left,axiom,
% 8.83/5.00 ! [C: complex,A: complex,B: complex] :
% 8.83/5.00 ( ( ( divide1717551699836669952omplex @ C @ A )
% 8.83/5.00 = ( divide1717551699836669952omplex @ C @ B ) )
% 8.83/5.00 = ( ( C = zero_zero_complex )
% 8.83/5.00 | ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1368_divide__cancel__left,axiom,
% 8.83/5.00 ! [C: real,A: real,B: real] :
% 8.83/5.00 ( ( ( divide_divide_real @ C @ A )
% 8.83/5.00 = ( divide_divide_real @ C @ B ) )
% 8.83/5.00 = ( ( C = zero_zero_real )
% 8.83/5.00 | ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1369_divide__cancel__left,axiom,
% 8.83/5.00 ! [C: rat,A: rat,B: rat] :
% 8.83/5.00 ( ( ( divide_divide_rat @ C @ A )
% 8.83/5.00 = ( divide_divide_rat @ C @ B ) )
% 8.83/5.00 = ( ( C = zero_zero_rat )
% 8.83/5.00 | ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1370_divide__eq__0__iff,axiom,
% 8.83/5.00 ! [A: complex,B: complex] :
% 8.83/5.00 ( ( ( divide1717551699836669952omplex @ A @ B )
% 8.83/5.00 = zero_zero_complex )
% 8.83/5.00 = ( ( A = zero_zero_complex )
% 8.83/5.00 | ( B = zero_zero_complex ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1371_divide__eq__0__iff,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ( divide_divide_real @ A @ B )
% 8.83/5.00 = zero_zero_real )
% 8.83/5.00 = ( ( A = zero_zero_real )
% 8.83/5.00 | ( B = zero_zero_real ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1372_divide__eq__0__iff,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ( divide_divide_rat @ A @ B )
% 8.83/5.00 = zero_zero_rat )
% 8.83/5.00 = ( ( A = zero_zero_rat )
% 8.83/5.00 | ( B = zero_zero_rat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1373_bits__div__by__0,axiom,
% 8.83/5.00 ! [A: nat] :
% 8.83/5.00 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 8.83/5.00 = zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1374_bits__div__by__0,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( divide_divide_int @ A @ zero_zero_int )
% 8.83/5.00 = zero_zero_int ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1375_bits__div__0,axiom,
% 8.83/5.00 ! [A: nat] :
% 8.83/5.00 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 8.83/5.00 = zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1376_bits__div__0,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( divide_divide_int @ zero_zero_int @ A )
% 8.83/5.00 = zero_zero_int ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1377_div__by__0,axiom,
% 8.83/5.00 ! [A: complex] :
% 8.83/5.00 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 8.83/5.00 = zero_zero_complex ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1378_div__by__0,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( divide_divide_real @ A @ zero_zero_real )
% 8.83/5.00 = zero_zero_real ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1379_div__by__0,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 8.83/5.00 = zero_zero_rat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1380_div__by__0,axiom,
% 8.83/5.00 ! [A: nat] :
% 8.83/5.00 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 8.83/5.00 = zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1381_div__by__0,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( divide_divide_int @ A @ zero_zero_int )
% 8.83/5.00 = zero_zero_int ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1382_div__0,axiom,
% 8.83/5.00 ! [A: complex] :
% 8.83/5.00 ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 8.83/5.00 = zero_zero_complex ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1383_div__0,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( divide_divide_real @ zero_zero_real @ A )
% 8.83/5.00 = zero_zero_real ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1384_div__0,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( divide_divide_rat @ zero_zero_rat @ A )
% 8.83/5.00 = zero_zero_rat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1385_div__0,axiom,
% 8.83/5.00 ! [A: nat] :
% 8.83/5.00 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 8.83/5.00 = zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1386_div__0,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( divide_divide_int @ zero_zero_int @ A )
% 8.83/5.00 = zero_zero_int ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1387_less__nat__zero__code,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1388_neq0__conv,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( N != zero_zero_nat )
% 8.83/5.00 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1389_bot__nat__0_Onot__eq__extremum,axiom,
% 8.83/5.00 ! [A: nat] :
% 8.83/5.00 ( ( A != zero_zero_nat )
% 8.83/5.00 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1390_bot__nat__0_Oextremum,axiom,
% 8.83/5.00 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1391_le0,axiom,
% 8.83/5.00 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1392_add__is__0,axiom,
% 8.83/5.00 ! [M: nat,N: nat] :
% 8.83/5.00 ( ( ( plus_plus_nat @ M @ N )
% 8.83/5.00 = zero_zero_nat )
% 8.83/5.00 = ( ( M = zero_zero_nat )
% 8.83/5.00 & ( N = zero_zero_nat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1393_Nat_Oadd__0__right,axiom,
% 8.83/5.00 ! [M: nat] :
% 8.83/5.00 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 8.83/5.00 = M ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1394_diff__0__eq__0,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 8.83/5.00 = zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1395_diff__self__eq__0,axiom,
% 8.83/5.00 ! [M: nat] :
% 8.83/5.00 ( ( minus_minus_nat @ M @ M )
% 8.83/5.00 = zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1396_mult__is__0,axiom,
% 8.83/5.00 ! [M: nat,N: nat] :
% 8.83/5.00 ( ( ( times_times_nat @ M @ N )
% 8.83/5.00 = zero_zero_nat )
% 8.83/5.00 = ( ( M = zero_zero_nat )
% 8.83/5.00 | ( N = zero_zero_nat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1397_mult__0__right,axiom,
% 8.83/5.00 ! [M: nat] :
% 8.83/5.00 ( ( times_times_nat @ M @ zero_zero_nat )
% 8.83/5.00 = zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1398_mult__cancel1,axiom,
% 8.83/5.00 ! [K: nat,M: nat,N: nat] :
% 8.83/5.00 ( ( ( times_times_nat @ K @ M )
% 8.83/5.00 = ( times_times_nat @ K @ N ) )
% 8.83/5.00 = ( ( M = N )
% 8.83/5.00 | ( K = zero_zero_nat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1399_mult__cancel2,axiom,
% 8.83/5.00 ! [M: nat,K: nat,N: nat] :
% 8.83/5.00 ( ( ( times_times_nat @ M @ K )
% 8.83/5.00 = ( times_times_nat @ N @ K ) )
% 8.83/5.00 = ( ( M = N )
% 8.83/5.00 | ( K = zero_zero_nat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1400_add__le__same__cancel1,axiom,
% 8.83/5.00 ! [B: real,A: real] :
% 8.83/5.00 ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 8.83/5.00 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1401_add__le__same__cancel1,axiom,
% 8.83/5.00 ! [B: rat,A: rat] :
% 8.83/5.00 ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 8.83/5.00 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1402_add__le__same__cancel1,axiom,
% 8.83/5.00 ! [B: nat,A: nat] :
% 8.83/5.00 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 8.83/5.00 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1403_add__le__same__cancel1,axiom,
% 8.83/5.00 ! [B: int,A: int] :
% 8.83/5.00 ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
% 8.83/5.00 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1404_add__le__same__cancel2,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 8.83/5.00 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1405_add__le__same__cancel2,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 8.83/5.00 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1406_add__le__same__cancel2,axiom,
% 8.83/5.00 ! [A: nat,B: nat] :
% 8.83/5.00 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 8.83/5.00 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1407_add__le__same__cancel2,axiom,
% 8.83/5.00 ! [A: int,B: int] :
% 8.83/5.00 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
% 8.83/5.00 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1408_le__add__same__cancel1,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 8.83/5.00 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1409_le__add__same__cancel1,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 8.83/5.00 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1410_le__add__same__cancel1,axiom,
% 8.83/5.00 ! [A: nat,B: nat] :
% 8.83/5.00 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 8.83/5.00 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1411_le__add__same__cancel1,axiom,
% 8.83/5.00 ! [A: int,B: int] :
% 8.83/5.00 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
% 8.83/5.00 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1412_le__add__same__cancel2,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 8.83/5.00 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1413_le__add__same__cancel2,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 8.83/5.00 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1414_le__add__same__cancel2,axiom,
% 8.83/5.00 ! [A: nat,B: nat] :
% 8.83/5.00 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 8.83/5.00 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1415_le__add__same__cancel2,axiom,
% 8.83/5.00 ! [A: int,B: int] :
% 8.83/5.00 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
% 8.83/5.00 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1416_double__add__le__zero__iff__single__add__le__zero,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 8.83/5.00 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1417_double__add__le__zero__iff__single__add__le__zero,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 8.83/5.00 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1418_double__add__le__zero__iff__single__add__le__zero,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 8.83/5.00 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1419_zero__le__double__add__iff__zero__le__single__add,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 8.83/5.00 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1420_zero__le__double__add__iff__zero__le__single__add,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 8.83/5.00 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1421_zero__le__double__add__iff__zero__le__single__add,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 8.83/5.00 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1422_add__less__same__cancel1,axiom,
% 8.83/5.00 ! [B: real,A: real] :
% 8.83/5.00 ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 8.83/5.00 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1423_add__less__same__cancel1,axiom,
% 8.83/5.00 ! [B: rat,A: rat] :
% 8.83/5.00 ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 8.83/5.00 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1424_add__less__same__cancel1,axiom,
% 8.83/5.00 ! [B: nat,A: nat] :
% 8.83/5.00 ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 8.83/5.00 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1425_add__less__same__cancel1,axiom,
% 8.83/5.00 ! [B: int,A: int] :
% 8.83/5.00 ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
% 8.83/5.00 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1426_add__less__same__cancel2,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 8.83/5.00 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1427_add__less__same__cancel2,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 8.83/5.00 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1428_add__less__same__cancel2,axiom,
% 8.83/5.00 ! [A: nat,B: nat] :
% 8.83/5.00 ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 8.83/5.00 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1429_add__less__same__cancel2,axiom,
% 8.83/5.00 ! [A: int,B: int] :
% 8.83/5.00 ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
% 8.83/5.00 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1430_less__add__same__cancel1,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 8.83/5.00 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1431_less__add__same__cancel1,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 8.83/5.00 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1432_less__add__same__cancel1,axiom,
% 8.83/5.00 ! [A: nat,B: nat] :
% 8.83/5.00 ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 8.83/5.00 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1433_less__add__same__cancel1,axiom,
% 8.83/5.00 ! [A: int,B: int] :
% 8.83/5.00 ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
% 8.83/5.00 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1434_less__add__same__cancel2,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 8.83/5.00 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1435_less__add__same__cancel2,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 8.83/5.00 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1436_less__add__same__cancel2,axiom,
% 8.83/5.00 ! [A: nat,B: nat] :
% 8.83/5.00 ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 8.83/5.00 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1437_less__add__same__cancel2,axiom,
% 8.83/5.00 ! [A: int,B: int] :
% 8.83/5.00 ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
% 8.83/5.00 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1438_double__add__less__zero__iff__single__add__less__zero,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 8.83/5.00 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1439_double__add__less__zero__iff__single__add__less__zero,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 8.83/5.00 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1440_double__add__less__zero__iff__single__add__less__zero,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 8.83/5.00 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1441_zero__less__double__add__iff__zero__less__single__add,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 8.83/5.00 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1442_zero__less__double__add__iff__zero__less__single__add,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 8.83/5.00 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1443_zero__less__double__add__iff__zero__less__single__add,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 8.83/5.00 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1444_diff__ge__0__iff__ge,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 8.83/5.00 = ( ord_less_eq_real @ B @ A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1445_diff__ge__0__iff__ge,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 8.83/5.00 = ( ord_less_eq_rat @ B @ A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1446_diff__ge__0__iff__ge,axiom,
% 8.83/5.00 ! [A: int,B: int] :
% 8.83/5.00 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 8.83/5.00 = ( ord_less_eq_int @ B @ A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1447_diff__gt__0__iff__gt,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 8.83/5.00 = ( ord_less_real @ B @ A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1448_diff__gt__0__iff__gt,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 8.83/5.00 = ( ord_less_rat @ B @ A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1449_diff__gt__0__iff__gt,axiom,
% 8.83/5.00 ! [A: int,B: int] :
% 8.83/5.00 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 8.83/5.00 = ( ord_less_int @ B @ A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1450_mult__cancel__left1,axiom,
% 8.83/5.00 ! [C: complex,B: complex] :
% 8.83/5.00 ( ( C
% 8.83/5.00 = ( times_times_complex @ C @ B ) )
% 8.83/5.00 = ( ( C = zero_zero_complex )
% 8.83/5.00 | ( B = one_one_complex ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1451_mult__cancel__left1,axiom,
% 8.83/5.00 ! [C: real,B: real] :
% 8.83/5.00 ( ( C
% 8.83/5.00 = ( times_times_real @ C @ B ) )
% 8.83/5.00 = ( ( C = zero_zero_real )
% 8.83/5.00 | ( B = one_one_real ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1452_mult__cancel__left1,axiom,
% 8.83/5.00 ! [C: rat,B: rat] :
% 8.83/5.00 ( ( C
% 8.83/5.00 = ( times_times_rat @ C @ B ) )
% 8.83/5.00 = ( ( C = zero_zero_rat )
% 8.83/5.00 | ( B = one_one_rat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1453_mult__cancel__left1,axiom,
% 8.83/5.00 ! [C: int,B: int] :
% 8.83/5.00 ( ( C
% 8.83/5.00 = ( times_times_int @ C @ B ) )
% 8.83/5.00 = ( ( C = zero_zero_int )
% 8.83/5.00 | ( B = one_one_int ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1454_mult__cancel__left2,axiom,
% 8.83/5.00 ! [C: complex,A: complex] :
% 8.83/5.00 ( ( ( times_times_complex @ C @ A )
% 8.83/5.00 = C )
% 8.83/5.00 = ( ( C = zero_zero_complex )
% 8.83/5.00 | ( A = one_one_complex ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1455_mult__cancel__left2,axiom,
% 8.83/5.00 ! [C: real,A: real] :
% 8.83/5.00 ( ( ( times_times_real @ C @ A )
% 8.83/5.00 = C )
% 8.83/5.00 = ( ( C = zero_zero_real )
% 8.83/5.00 | ( A = one_one_real ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1456_mult__cancel__left2,axiom,
% 8.83/5.00 ! [C: rat,A: rat] :
% 8.83/5.00 ( ( ( times_times_rat @ C @ A )
% 8.83/5.00 = C )
% 8.83/5.00 = ( ( C = zero_zero_rat )
% 8.83/5.00 | ( A = one_one_rat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1457_mult__cancel__left2,axiom,
% 8.83/5.00 ! [C: int,A: int] :
% 8.83/5.00 ( ( ( times_times_int @ C @ A )
% 8.83/5.00 = C )
% 8.83/5.00 = ( ( C = zero_zero_int )
% 8.83/5.00 | ( A = one_one_int ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1458_mult__cancel__right1,axiom,
% 8.83/5.00 ! [C: complex,B: complex] :
% 8.83/5.00 ( ( C
% 8.83/5.00 = ( times_times_complex @ B @ C ) )
% 8.83/5.00 = ( ( C = zero_zero_complex )
% 8.83/5.00 | ( B = one_one_complex ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1459_mult__cancel__right1,axiom,
% 8.83/5.00 ! [C: real,B: real] :
% 8.83/5.00 ( ( C
% 8.83/5.00 = ( times_times_real @ B @ C ) )
% 8.83/5.00 = ( ( C = zero_zero_real )
% 8.83/5.00 | ( B = one_one_real ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1460_mult__cancel__right1,axiom,
% 8.83/5.00 ! [C: rat,B: rat] :
% 8.83/5.00 ( ( C
% 8.83/5.00 = ( times_times_rat @ B @ C ) )
% 8.83/5.00 = ( ( C = zero_zero_rat )
% 8.83/5.00 | ( B = one_one_rat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1461_mult__cancel__right1,axiom,
% 8.83/5.00 ! [C: int,B: int] :
% 8.83/5.00 ( ( C
% 8.83/5.00 = ( times_times_int @ B @ C ) )
% 8.83/5.00 = ( ( C = zero_zero_int )
% 8.83/5.00 | ( B = one_one_int ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1462_mult__cancel__right2,axiom,
% 8.83/5.00 ! [A: complex,C: complex] :
% 8.83/5.00 ( ( ( times_times_complex @ A @ C )
% 8.83/5.00 = C )
% 8.83/5.00 = ( ( C = zero_zero_complex )
% 8.83/5.00 | ( A = one_one_complex ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1463_mult__cancel__right2,axiom,
% 8.83/5.00 ! [A: real,C: real] :
% 8.83/5.00 ( ( ( times_times_real @ A @ C )
% 8.83/5.00 = C )
% 8.83/5.00 = ( ( C = zero_zero_real )
% 8.83/5.00 | ( A = one_one_real ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1464_mult__cancel__right2,axiom,
% 8.83/5.00 ! [A: rat,C: rat] :
% 8.83/5.00 ( ( ( times_times_rat @ A @ C )
% 8.83/5.00 = C )
% 8.83/5.00 = ( ( C = zero_zero_rat )
% 8.83/5.00 | ( A = one_one_rat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1465_mult__cancel__right2,axiom,
% 8.83/5.00 ! [A: int,C: int] :
% 8.83/5.00 ( ( ( times_times_int @ A @ C )
% 8.83/5.00 = C )
% 8.83/5.00 = ( ( C = zero_zero_int )
% 8.83/5.00 | ( A = one_one_int ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1466_sum__squares__eq__zero__iff,axiom,
% 8.83/5.00 ! [X2: real,Y3: real] :
% 8.83/5.00 ( ( ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) )
% 8.83/5.00 = zero_zero_real )
% 8.83/5.00 = ( ( X2 = zero_zero_real )
% 8.83/5.00 & ( Y3 = zero_zero_real ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1467_sum__squares__eq__zero__iff,axiom,
% 8.83/5.00 ! [X2: rat,Y3: rat] :
% 8.83/5.00 ( ( ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 8.83/5.00 = zero_zero_rat )
% 8.83/5.00 = ( ( X2 = zero_zero_rat )
% 8.83/5.00 & ( Y3 = zero_zero_rat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1468_sum__squares__eq__zero__iff,axiom,
% 8.83/5.00 ! [X2: int,Y3: int] :
% 8.83/5.00 ( ( ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) )
% 8.83/5.00 = zero_zero_int )
% 8.83/5.00 = ( ( X2 = zero_zero_int )
% 8.83/5.00 & ( Y3 = zero_zero_int ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1469_diff__numeral__special_I9_J,axiom,
% 8.83/5.00 ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 8.83/5.00 = zero_zero_complex ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1470_diff__numeral__special_I9_J,axiom,
% 8.83/5.00 ( ( minus_minus_real @ one_one_real @ one_one_real )
% 8.83/5.00 = zero_zero_real ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1471_diff__numeral__special_I9_J,axiom,
% 8.83/5.00 ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 8.83/5.00 = zero_zero_rat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1472_diff__numeral__special_I9_J,axiom,
% 8.83/5.00 ( ( minus_minus_int @ one_one_int @ one_one_int )
% 8.83/5.00 = zero_zero_int ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1473_nonzero__mult__divide__mult__cancel__right2,axiom,
% 8.83/5.00 ! [C: complex,A: complex,B: complex] :
% 8.83/5.00 ( ( C != zero_zero_complex )
% 8.83/5.00 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
% 8.83/5.00 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1474_nonzero__mult__divide__mult__cancel__right2,axiom,
% 8.83/5.00 ! [C: real,A: real,B: real] :
% 8.83/5.00 ( ( C != zero_zero_real )
% 8.83/5.00 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
% 8.83/5.00 = ( divide_divide_real @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1475_nonzero__mult__divide__mult__cancel__right2,axiom,
% 8.83/5.00 ! [C: rat,A: rat,B: rat] :
% 8.83/5.00 ( ( C != zero_zero_rat )
% 8.83/5.00 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
% 8.83/5.00 = ( divide_divide_rat @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1476_nonzero__mult__divide__mult__cancel__right,axiom,
% 8.83/5.00 ! [C: complex,A: complex,B: complex] :
% 8.83/5.00 ( ( C != zero_zero_complex )
% 8.83/5.00 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 8.83/5.00 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1477_nonzero__mult__divide__mult__cancel__right,axiom,
% 8.83/5.00 ! [C: real,A: real,B: real] :
% 8.83/5.00 ( ( C != zero_zero_real )
% 8.83/5.00 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 8.83/5.00 = ( divide_divide_real @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1478_nonzero__mult__divide__mult__cancel__right,axiom,
% 8.83/5.00 ! [C: rat,A: rat,B: rat] :
% 8.83/5.00 ( ( C != zero_zero_rat )
% 8.83/5.00 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 8.83/5.00 = ( divide_divide_rat @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1479_nonzero__mult__divide__mult__cancel__left2,axiom,
% 8.83/5.00 ! [C: complex,A: complex,B: complex] :
% 8.83/5.00 ( ( C != zero_zero_complex )
% 8.83/5.00 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
% 8.83/5.00 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1480_nonzero__mult__divide__mult__cancel__left2,axiom,
% 8.83/5.00 ! [C: real,A: real,B: real] :
% 8.83/5.00 ( ( C != zero_zero_real )
% 8.83/5.00 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
% 8.83/5.00 = ( divide_divide_real @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1481_nonzero__mult__divide__mult__cancel__left2,axiom,
% 8.83/5.00 ! [C: rat,A: rat,B: rat] :
% 8.83/5.00 ( ( C != zero_zero_rat )
% 8.83/5.00 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
% 8.83/5.00 = ( divide_divide_rat @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1482_nonzero__mult__divide__mult__cancel__left,axiom,
% 8.83/5.00 ! [C: complex,A: complex,B: complex] :
% 8.83/5.00 ( ( C != zero_zero_complex )
% 8.83/5.00 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 8.83/5.00 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1483_nonzero__mult__divide__mult__cancel__left,axiom,
% 8.83/5.00 ! [C: real,A: real,B: real] :
% 8.83/5.00 ( ( C != zero_zero_real )
% 8.83/5.00 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.83/5.00 = ( divide_divide_real @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1484_nonzero__mult__divide__mult__cancel__left,axiom,
% 8.83/5.00 ! [C: rat,A: rat,B: rat] :
% 8.83/5.00 ( ( C != zero_zero_rat )
% 8.83/5.00 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.83/5.00 = ( divide_divide_rat @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1485_mult__divide__mult__cancel__left__if,axiom,
% 8.83/5.00 ! [C: complex,A: complex,B: complex] :
% 8.83/5.00 ( ( ( C = zero_zero_complex )
% 8.83/5.00 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 8.83/5.00 = zero_zero_complex ) )
% 8.83/5.00 & ( ( C != zero_zero_complex )
% 8.83/5.00 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 8.83/5.00 = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1486_mult__divide__mult__cancel__left__if,axiom,
% 8.83/5.00 ! [C: real,A: real,B: real] :
% 8.83/5.00 ( ( ( C = zero_zero_real )
% 8.83/5.00 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.83/5.00 = zero_zero_real ) )
% 8.83/5.00 & ( ( C != zero_zero_real )
% 8.83/5.00 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.83/5.00 = ( divide_divide_real @ A @ B ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1487_mult__divide__mult__cancel__left__if,axiom,
% 8.83/5.00 ! [C: rat,A: rat,B: rat] :
% 8.83/5.00 ( ( ( C = zero_zero_rat )
% 8.83/5.00 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.83/5.00 = zero_zero_rat ) )
% 8.83/5.00 & ( ( C != zero_zero_rat )
% 8.83/5.00 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.83/5.00 = ( divide_divide_rat @ A @ B ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1488_div__mult__mult1,axiom,
% 8.83/5.00 ! [C: nat,A: nat,B: nat] :
% 8.83/5.00 ( ( C != zero_zero_nat )
% 8.83/5.00 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 8.83/5.00 = ( divide_divide_nat @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1489_div__mult__mult1,axiom,
% 8.83/5.00 ! [C: int,A: int,B: int] :
% 8.83/5.00 ( ( C != zero_zero_int )
% 8.83/5.00 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.83/5.00 = ( divide_divide_int @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1490_div__mult__mult2,axiom,
% 8.83/5.00 ! [C: nat,A: nat,B: nat] :
% 8.83/5.00 ( ( C != zero_zero_nat )
% 8.83/5.00 => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 8.83/5.00 = ( divide_divide_nat @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1491_div__mult__mult2,axiom,
% 8.83/5.00 ! [C: int,A: int,B: int] :
% 8.83/5.00 ( ( C != zero_zero_int )
% 8.83/5.00 => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 8.83/5.00 = ( divide_divide_int @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1492_div__mult__mult1__if,axiom,
% 8.83/5.00 ! [C: nat,A: nat,B: nat] :
% 8.83/5.00 ( ( ( C = zero_zero_nat )
% 8.83/5.00 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 8.83/5.00 = zero_zero_nat ) )
% 8.83/5.00 & ( ( C != zero_zero_nat )
% 8.83/5.00 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 8.83/5.00 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1493_div__mult__mult1__if,axiom,
% 8.83/5.00 ! [C: int,A: int,B: int] :
% 8.83/5.00 ( ( ( C = zero_zero_int )
% 8.83/5.00 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.83/5.00 = zero_zero_int ) )
% 8.83/5.00 & ( ( C != zero_zero_int )
% 8.83/5.00 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.83/5.00 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1494_nonzero__mult__div__cancel__right,axiom,
% 8.83/5.00 ! [B: complex,A: complex] :
% 8.83/5.00 ( ( B != zero_zero_complex )
% 8.83/5.00 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
% 8.83/5.00 = A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1495_nonzero__mult__div__cancel__right,axiom,
% 8.83/5.00 ! [B: real,A: real] :
% 8.83/5.00 ( ( B != zero_zero_real )
% 8.83/5.00 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
% 8.83/5.00 = A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1496_nonzero__mult__div__cancel__right,axiom,
% 8.83/5.00 ! [B: rat,A: rat] :
% 8.83/5.00 ( ( B != zero_zero_rat )
% 8.83/5.00 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
% 8.83/5.00 = A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1497_nonzero__mult__div__cancel__right,axiom,
% 8.83/5.00 ! [B: nat,A: nat] :
% 8.83/5.00 ( ( B != zero_zero_nat )
% 8.83/5.00 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
% 8.83/5.00 = A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1498_nonzero__mult__div__cancel__right,axiom,
% 8.83/5.00 ! [B: int,A: int] :
% 8.83/5.00 ( ( B != zero_zero_int )
% 8.83/5.00 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
% 8.83/5.00 = A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1499_nonzero__mult__div__cancel__left,axiom,
% 8.83/5.00 ! [A: complex,B: complex] :
% 8.83/5.00 ( ( A != zero_zero_complex )
% 8.83/5.00 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
% 8.83/5.00 = B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1500_nonzero__mult__div__cancel__left,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( A != zero_zero_real )
% 8.83/5.00 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
% 8.83/5.00 = B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1501_nonzero__mult__div__cancel__left,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( A != zero_zero_rat )
% 8.83/5.00 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
% 8.83/5.00 = B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1502_nonzero__mult__div__cancel__left,axiom,
% 8.83/5.00 ! [A: nat,B: nat] :
% 8.83/5.00 ( ( A != zero_zero_nat )
% 8.83/5.00 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
% 8.83/5.00 = B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1503_nonzero__mult__div__cancel__left,axiom,
% 8.83/5.00 ! [A: int,B: int] :
% 8.83/5.00 ( ( A != zero_zero_int )
% 8.83/5.00 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
% 8.83/5.00 = B ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1504_diff__add__zero,axiom,
% 8.83/5.00 ! [A: nat,B: nat] :
% 8.83/5.00 ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 8.83/5.00 = zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1505_divide__eq__1__iff,axiom,
% 8.83/5.00 ! [A: complex,B: complex] :
% 8.83/5.00 ( ( ( divide1717551699836669952omplex @ A @ B )
% 8.83/5.00 = one_one_complex )
% 8.83/5.00 = ( ( B != zero_zero_complex )
% 8.83/5.00 & ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1506_divide__eq__1__iff,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ( divide_divide_real @ A @ B )
% 8.83/5.00 = one_one_real )
% 8.83/5.00 = ( ( B != zero_zero_real )
% 8.83/5.00 & ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1507_divide__eq__1__iff,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ( divide_divide_rat @ A @ B )
% 8.83/5.00 = one_one_rat )
% 8.83/5.00 = ( ( B != zero_zero_rat )
% 8.83/5.00 & ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1508_one__eq__divide__iff,axiom,
% 8.83/5.00 ! [A: complex,B: complex] :
% 8.83/5.00 ( ( one_one_complex
% 8.83/5.00 = ( divide1717551699836669952omplex @ A @ B ) )
% 8.83/5.00 = ( ( B != zero_zero_complex )
% 8.83/5.00 & ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1509_one__eq__divide__iff,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( one_one_real
% 8.83/5.00 = ( divide_divide_real @ A @ B ) )
% 8.83/5.00 = ( ( B != zero_zero_real )
% 8.83/5.00 & ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1510_one__eq__divide__iff,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( one_one_rat
% 8.83/5.00 = ( divide_divide_rat @ A @ B ) )
% 8.83/5.00 = ( ( B != zero_zero_rat )
% 8.83/5.00 & ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1511_divide__self,axiom,
% 8.83/5.00 ! [A: complex] :
% 8.83/5.00 ( ( A != zero_zero_complex )
% 8.83/5.00 => ( ( divide1717551699836669952omplex @ A @ A )
% 8.83/5.00 = one_one_complex ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1512_divide__self,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( A != zero_zero_real )
% 8.83/5.00 => ( ( divide_divide_real @ A @ A )
% 8.83/5.00 = one_one_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1513_divide__self,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( A != zero_zero_rat )
% 8.83/5.00 => ( ( divide_divide_rat @ A @ A )
% 8.83/5.00 = one_one_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1514_divide__self__if,axiom,
% 8.83/5.00 ! [A: complex] :
% 8.83/5.00 ( ( ( A = zero_zero_complex )
% 8.83/5.00 => ( ( divide1717551699836669952omplex @ A @ A )
% 8.83/5.00 = zero_zero_complex ) )
% 8.83/5.00 & ( ( A != zero_zero_complex )
% 8.83/5.00 => ( ( divide1717551699836669952omplex @ A @ A )
% 8.83/5.00 = one_one_complex ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1515_divide__self__if,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( ( A = zero_zero_real )
% 8.83/5.00 => ( ( divide_divide_real @ A @ A )
% 8.83/5.00 = zero_zero_real ) )
% 8.83/5.00 & ( ( A != zero_zero_real )
% 8.83/5.00 => ( ( divide_divide_real @ A @ A )
% 8.83/5.00 = one_one_real ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1516_divide__self__if,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( ( A = zero_zero_rat )
% 8.83/5.00 => ( ( divide_divide_rat @ A @ A )
% 8.83/5.00 = zero_zero_rat ) )
% 8.83/5.00 & ( ( A != zero_zero_rat )
% 8.83/5.00 => ( ( divide_divide_rat @ A @ A )
% 8.83/5.00 = one_one_rat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1517_divide__eq__eq__1,axiom,
% 8.83/5.00 ! [B: real,A: real] :
% 8.83/5.00 ( ( ( divide_divide_real @ B @ A )
% 8.83/5.00 = one_one_real )
% 8.83/5.00 = ( ( A != zero_zero_real )
% 8.83/5.00 & ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1518_divide__eq__eq__1,axiom,
% 8.83/5.00 ! [B: rat,A: rat] :
% 8.83/5.00 ( ( ( divide_divide_rat @ B @ A )
% 8.83/5.00 = one_one_rat )
% 8.83/5.00 = ( ( A != zero_zero_rat )
% 8.83/5.00 & ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1519_eq__divide__eq__1,axiom,
% 8.83/5.00 ! [B: real,A: real] :
% 8.83/5.00 ( ( one_one_real
% 8.83/5.00 = ( divide_divide_real @ B @ A ) )
% 8.83/5.00 = ( ( A != zero_zero_real )
% 8.83/5.00 & ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1520_eq__divide__eq__1,axiom,
% 8.83/5.00 ! [B: rat,A: rat] :
% 8.83/5.00 ( ( one_one_rat
% 8.83/5.00 = ( divide_divide_rat @ B @ A ) )
% 8.83/5.00 = ( ( A != zero_zero_rat )
% 8.83/5.00 & ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1521_one__divide__eq__0__iff,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( ( divide_divide_real @ one_one_real @ A )
% 8.83/5.00 = zero_zero_real )
% 8.83/5.00 = ( A = zero_zero_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1522_one__divide__eq__0__iff,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( ( divide_divide_rat @ one_one_rat @ A )
% 8.83/5.00 = zero_zero_rat )
% 8.83/5.00 = ( A = zero_zero_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1523_zero__eq__1__divide__iff,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( zero_zero_real
% 8.83/5.00 = ( divide_divide_real @ one_one_real @ A ) )
% 8.83/5.00 = ( A = zero_zero_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1524_zero__eq__1__divide__iff,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( zero_zero_rat
% 8.83/5.00 = ( divide_divide_rat @ one_one_rat @ A ) )
% 8.83/5.00 = ( A = zero_zero_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1525_div__self,axiom,
% 8.83/5.00 ! [A: complex] :
% 8.83/5.00 ( ( A != zero_zero_complex )
% 8.83/5.00 => ( ( divide1717551699836669952omplex @ A @ A )
% 8.83/5.00 = one_one_complex ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1526_div__self,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( A != zero_zero_real )
% 8.83/5.00 => ( ( divide_divide_real @ A @ A )
% 8.83/5.00 = one_one_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1527_div__self,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( A != zero_zero_rat )
% 8.83/5.00 => ( ( divide_divide_rat @ A @ A )
% 8.83/5.00 = one_one_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1528_div__self,axiom,
% 8.83/5.00 ! [A: nat] :
% 8.83/5.00 ( ( A != zero_zero_nat )
% 8.83/5.00 => ( ( divide_divide_nat @ A @ A )
% 8.83/5.00 = one_one_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1529_div__self,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( A != zero_zero_int )
% 8.83/5.00 => ( ( divide_divide_int @ A @ A )
% 8.83/5.00 = one_one_int ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1530_power__0__Suc,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
% 8.83/5.00 = zero_zero_rat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1531_power__0__Suc,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 8.83/5.00 = zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1532_power__0__Suc,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 8.83/5.00 = zero_zero_real ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1533_power__0__Suc,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 8.83/5.00 = zero_zero_int ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1534_power__0__Suc,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
% 8.83/5.00 = zero_zero_complex ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1535_power__zero__numeral,axiom,
% 8.83/5.00 ! [K: num] :
% 8.83/5.00 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 8.83/5.00 = zero_zero_rat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1536_power__zero__numeral,axiom,
% 8.83/5.00 ! [K: num] :
% 8.83/5.00 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 8.83/5.00 = zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1537_power__zero__numeral,axiom,
% 8.83/5.00 ! [K: num] :
% 8.83/5.00 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 8.83/5.00 = zero_zero_real ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1538_power__zero__numeral,axiom,
% 8.83/5.00 ! [K: num] :
% 8.83/5.00 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 8.83/5.00 = zero_zero_int ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1539_power__zero__numeral,axiom,
% 8.83/5.00 ! [K: num] :
% 8.83/5.00 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 8.83/5.00 = zero_zero_complex ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1540_power__Suc0__right,axiom,
% 8.83/5.00 ! [A: nat] :
% 8.83/5.00 ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1541_power__Suc0__right,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1542_power__Suc0__right,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1543_power__Suc0__right,axiom,
% 8.83/5.00 ! [A: complex] :
% 8.83/5.00 ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1544_zero__less__Suc,axiom,
% 8.83/5.00 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1545_less__Suc0,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 8.83/5.00 = ( N = zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1546_add__gr__0,axiom,
% 8.83/5.00 ! [M: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 8.83/5.00 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.83/5.00 | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1547_mult__eq__1__iff,axiom,
% 8.83/5.00 ! [M: nat,N: nat] :
% 8.83/5.00 ( ( ( times_times_nat @ M @ N )
% 8.83/5.00 = ( suc @ zero_zero_nat ) )
% 8.83/5.00 = ( ( M
% 8.83/5.00 = ( suc @ zero_zero_nat ) )
% 8.83/5.00 & ( N
% 8.83/5.00 = ( suc @ zero_zero_nat ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1548_one__eq__mult__iff,axiom,
% 8.83/5.00 ! [M: nat,N: nat] :
% 8.83/5.00 ( ( ( suc @ zero_zero_nat )
% 8.83/5.00 = ( times_times_nat @ M @ N ) )
% 8.83/5.00 = ( ( M
% 8.83/5.00 = ( suc @ zero_zero_nat ) )
% 8.83/5.00 & ( N
% 8.83/5.00 = ( suc @ zero_zero_nat ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1549_div__by__Suc__0,axiom,
% 8.83/5.00 ! [M: nat] :
% 8.83/5.00 ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 8.83/5.00 = M ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1550_less__one,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ N @ one_one_nat )
% 8.83/5.00 = ( N = zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1551_zero__less__diff,axiom,
% 8.83/5.00 ! [N: nat,M: nat] :
% 8.83/5.00 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 8.83/5.00 = ( ord_less_nat @ M @ N ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1552_nat__0__less__mult__iff,axiom,
% 8.83/5.00 ! [M: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 8.83/5.00 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.83/5.00 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1553_mult__less__cancel2,axiom,
% 8.83/5.00 ! [M: nat,K: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 8.83/5.00 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.83/5.00 & ( ord_less_nat @ M @ N ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1554_nat__mult__less__cancel__disj,axiom,
% 8.83/5.00 ! [K: nat,M: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.83/5.00 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.83/5.00 & ( ord_less_nat @ M @ N ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1555_div__less,axiom,
% 8.83/5.00 ! [M: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ M @ N )
% 8.83/5.00 => ( ( divide_divide_nat @ M @ N )
% 8.83/5.00 = zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1556_diff__is__0__eq_H,axiom,
% 8.83/5.00 ! [M: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_eq_nat @ M @ N )
% 8.83/5.00 => ( ( minus_minus_nat @ M @ N )
% 8.83/5.00 = zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1557_diff__is__0__eq,axiom,
% 8.83/5.00 ! [M: nat,N: nat] :
% 8.83/5.00 ( ( ( minus_minus_nat @ M @ N )
% 8.83/5.00 = zero_zero_nat )
% 8.83/5.00 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1558_nat__power__eq__Suc__0__iff,axiom,
% 8.83/5.00 ! [X2: nat,M: nat] :
% 8.83/5.00 ( ( ( power_power_nat @ X2 @ M )
% 8.83/5.00 = ( suc @ zero_zero_nat ) )
% 8.83/5.00 = ( ( M = zero_zero_nat )
% 8.83/5.00 | ( X2
% 8.83/5.00 = ( suc @ zero_zero_nat ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1559_power__Suc__0,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.83/5.00 = ( suc @ zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1560_nat__zero__less__power__iff,axiom,
% 8.83/5.00 ! [X2: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N ) )
% 8.83/5.00 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 8.83/5.00 | ( N = zero_zero_nat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1561_nat__mult__div__cancel__disj,axiom,
% 8.83/5.00 ! [K: nat,M: nat,N: nat] :
% 8.83/5.00 ( ( ( K = zero_zero_nat )
% 8.83/5.00 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.83/5.00 = zero_zero_nat ) )
% 8.83/5.00 & ( ( K != zero_zero_nat )
% 8.83/5.00 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.83/5.00 = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1562_divide__le__0__1__iff,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 8.83/5.00 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1563_divide__le__0__1__iff,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 8.83/5.00 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1564_zero__le__divide__1__iff,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 8.83/5.00 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1565_zero__le__divide__1__iff,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 8.83/5.00 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1566_divide__less__0__1__iff,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 8.83/5.00 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1567_divide__less__0__1__iff,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 8.83/5.00 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1568_divide__less__eq__1__neg,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ord_less_real @ A @ zero_zero_real )
% 8.83/5.00 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 8.83/5.00 = ( ord_less_real @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1569_divide__less__eq__1__neg,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.83/5.00 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 8.83/5.00 = ( ord_less_rat @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1570_divide__less__eq__1__pos,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.00 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 8.83/5.00 = ( ord_less_real @ B @ A ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1571_divide__less__eq__1__pos,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.00 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 8.83/5.00 = ( ord_less_rat @ B @ A ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1572_less__divide__eq__1__neg,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ord_less_real @ A @ zero_zero_real )
% 8.83/5.00 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 8.83/5.00 = ( ord_less_real @ B @ A ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1573_less__divide__eq__1__neg,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.83/5.00 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 8.83/5.00 = ( ord_less_rat @ B @ A ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1574_less__divide__eq__1__pos,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.00 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 8.83/5.00 = ( ord_less_real @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1575_less__divide__eq__1__pos,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.00 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 8.83/5.00 = ( ord_less_rat @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1576_zero__less__divide__1__iff,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 8.83/5.00 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1577_zero__less__divide__1__iff,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 8.83/5.00 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1578_eq__divide__eq__numeral1_I1_J,axiom,
% 8.83/5.00 ! [A: complex,B: complex,W: num] :
% 8.83/5.00 ( ( A
% 8.83/5.00 = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
% 8.83/5.00 = ( ( ( ( numera6690914467698888265omplex @ W )
% 8.83/5.00 != zero_zero_complex )
% 8.83/5.00 => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
% 8.83/5.00 = B ) )
% 8.83/5.00 & ( ( ( numera6690914467698888265omplex @ W )
% 8.83/5.00 = zero_zero_complex )
% 8.83/5.00 => ( A = zero_zero_complex ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1579_eq__divide__eq__numeral1_I1_J,axiom,
% 8.83/5.00 ! [A: real,B: real,W: num] :
% 8.83/5.00 ( ( A
% 8.83/5.00 = ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 8.83/5.00 = ( ( ( ( numeral_numeral_real @ W )
% 8.83/5.00 != zero_zero_real )
% 8.83/5.00 => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
% 8.83/5.00 = B ) )
% 8.83/5.00 & ( ( ( numeral_numeral_real @ W )
% 8.83/5.00 = zero_zero_real )
% 8.83/5.00 => ( A = zero_zero_real ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1580_eq__divide__eq__numeral1_I1_J,axiom,
% 8.83/5.00 ! [A: rat,B: rat,W: num] :
% 8.83/5.00 ( ( A
% 8.83/5.00 = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 8.83/5.00 = ( ( ( ( numeral_numeral_rat @ W )
% 8.83/5.00 != zero_zero_rat )
% 8.83/5.00 => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
% 8.83/5.00 = B ) )
% 8.83/5.00 & ( ( ( numeral_numeral_rat @ W )
% 8.83/5.00 = zero_zero_rat )
% 8.83/5.00 => ( A = zero_zero_rat ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1581_divide__eq__eq__numeral1_I1_J,axiom,
% 8.83/5.00 ! [B: complex,W: num,A: complex] :
% 8.83/5.00 ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
% 8.83/5.00 = A )
% 8.83/5.00 = ( ( ( ( numera6690914467698888265omplex @ W )
% 8.83/5.00 != zero_zero_complex )
% 8.83/5.00 => ( B
% 8.83/5.00 = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
% 8.83/5.00 & ( ( ( numera6690914467698888265omplex @ W )
% 8.83/5.00 = zero_zero_complex )
% 8.83/5.00 => ( A = zero_zero_complex ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1582_divide__eq__eq__numeral1_I1_J,axiom,
% 8.83/5.00 ! [B: real,W: num,A: real] :
% 8.83/5.00 ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
% 8.83/5.00 = A )
% 8.83/5.00 = ( ( ( ( numeral_numeral_real @ W )
% 8.83/5.00 != zero_zero_real )
% 8.83/5.00 => ( B
% 8.83/5.00 = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
% 8.83/5.00 & ( ( ( numeral_numeral_real @ W )
% 8.83/5.00 = zero_zero_real )
% 8.83/5.00 => ( A = zero_zero_real ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1583_divide__eq__eq__numeral1_I1_J,axiom,
% 8.83/5.00 ! [B: rat,W: num,A: rat] :
% 8.83/5.00 ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
% 8.83/5.00 = A )
% 8.83/5.00 = ( ( ( ( numeral_numeral_rat @ W )
% 8.83/5.00 != zero_zero_rat )
% 8.83/5.00 => ( B
% 8.83/5.00 = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
% 8.83/5.00 & ( ( ( numeral_numeral_rat @ W )
% 8.83/5.00 = zero_zero_rat )
% 8.83/5.00 => ( A = zero_zero_rat ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1584_nonzero__divide__mult__cancel__right,axiom,
% 8.83/5.00 ! [B: complex,A: complex] :
% 8.83/5.00 ( ( B != zero_zero_complex )
% 8.83/5.00 => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
% 8.83/5.00 = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1585_nonzero__divide__mult__cancel__right,axiom,
% 8.83/5.00 ! [B: real,A: real] :
% 8.83/5.00 ( ( B != zero_zero_real )
% 8.83/5.00 => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
% 8.83/5.00 = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1586_nonzero__divide__mult__cancel__right,axiom,
% 8.83/5.00 ! [B: rat,A: rat] :
% 8.83/5.00 ( ( B != zero_zero_rat )
% 8.83/5.00 => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
% 8.83/5.00 = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1587_nonzero__divide__mult__cancel__left,axiom,
% 8.83/5.00 ! [A: complex,B: complex] :
% 8.83/5.00 ( ( A != zero_zero_complex )
% 8.83/5.00 => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
% 8.83/5.00 = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1588_nonzero__divide__mult__cancel__left,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( A != zero_zero_real )
% 8.83/5.00 => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
% 8.83/5.00 = ( divide_divide_real @ one_one_real @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1589_nonzero__divide__mult__cancel__left,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( A != zero_zero_rat )
% 8.83/5.00 => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
% 8.83/5.00 = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1590_div__mult__self4,axiom,
% 8.83/5.00 ! [B: nat,C: nat,A: nat] :
% 8.83/5.00 ( ( B != zero_zero_nat )
% 8.83/5.00 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 8.83/5.00 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1591_div__mult__self4,axiom,
% 8.83/5.00 ! [B: int,C: int,A: int] :
% 8.83/5.00 ( ( B != zero_zero_int )
% 8.83/5.00 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 8.83/5.00 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1592_div__mult__self3,axiom,
% 8.83/5.00 ! [B: nat,C: nat,A: nat] :
% 8.83/5.00 ( ( B != zero_zero_nat )
% 8.83/5.00 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 8.83/5.00 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1593_div__mult__self3,axiom,
% 8.83/5.00 ! [B: int,C: int,A: int] :
% 8.83/5.00 ( ( B != zero_zero_int )
% 8.83/5.00 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 8.83/5.00 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1594_div__mult__self2,axiom,
% 8.83/5.00 ! [B: nat,A: nat,C: nat] :
% 8.83/5.00 ( ( B != zero_zero_nat )
% 8.83/5.00 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 8.83/5.00 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1595_div__mult__self2,axiom,
% 8.83/5.00 ! [B: int,A: int,C: int] :
% 8.83/5.00 ( ( B != zero_zero_int )
% 8.83/5.00 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 8.83/5.00 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1596_div__mult__self1,axiom,
% 8.83/5.00 ! [B: nat,A: nat,C: nat] :
% 8.83/5.00 ( ( B != zero_zero_nat )
% 8.83/5.00 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 8.83/5.00 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1597_div__mult__self1,axiom,
% 8.83/5.00 ! [B: int,A: int,C: int] :
% 8.83/5.00 ( ( B != zero_zero_int )
% 8.83/5.00 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 8.83/5.00 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1598_power__eq__0__iff,axiom,
% 8.83/5.00 ! [A: rat,N: nat] :
% 8.83/5.00 ( ( ( power_power_rat @ A @ N )
% 8.83/5.00 = zero_zero_rat )
% 8.83/5.00 = ( ( A = zero_zero_rat )
% 8.83/5.00 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1599_power__eq__0__iff,axiom,
% 8.83/5.00 ! [A: nat,N: nat] :
% 8.83/5.00 ( ( ( power_power_nat @ A @ N )
% 8.83/5.00 = zero_zero_nat )
% 8.83/5.00 = ( ( A = zero_zero_nat )
% 8.83/5.00 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1600_power__eq__0__iff,axiom,
% 8.83/5.00 ! [A: real,N: nat] :
% 8.83/5.00 ( ( ( power_power_real @ A @ N )
% 8.83/5.00 = zero_zero_real )
% 8.83/5.00 = ( ( A = zero_zero_real )
% 8.83/5.00 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1601_power__eq__0__iff,axiom,
% 8.83/5.00 ! [A: int,N: nat] :
% 8.83/5.00 ( ( ( power_power_int @ A @ N )
% 8.83/5.00 = zero_zero_int )
% 8.83/5.00 = ( ( A = zero_zero_int )
% 8.83/5.00 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1602_power__eq__0__iff,axiom,
% 8.83/5.00 ! [A: complex,N: nat] :
% 8.83/5.00 ( ( ( power_power_complex @ A @ N )
% 8.83/5.00 = zero_zero_complex )
% 8.83/5.00 = ( ( A = zero_zero_complex )
% 8.83/5.00 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1603_Suc__pred,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.00 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 8.83/5.00 = N ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1604_one__le__mult__iff,axiom,
% 8.83/5.00 ! [M: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 8.83/5.00 = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 8.83/5.00 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1605_mult__le__cancel2,axiom,
% 8.83/5.00 ! [M: nat,K: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 8.83/5.00 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.83/5.00 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1606_nat__mult__le__cancel__disj,axiom,
% 8.83/5.00 ! [K: nat,M: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.83/5.00 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.83/5.00 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1607_div__mult__self1__is__m,axiom,
% 8.83/5.00 ! [N: nat,M: nat] :
% 8.83/5.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.00 => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 8.83/5.00 = M ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1608_div__mult__self__is__m,axiom,
% 8.83/5.00 ! [N: nat,M: nat] :
% 8.83/5.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.00 => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 8.83/5.00 = M ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1609_divide__le__eq__1__neg,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ord_less_real @ A @ zero_zero_real )
% 8.83/5.00 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 8.83/5.00 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1610_divide__le__eq__1__neg,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.83/5.00 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 8.83/5.00 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1611_divide__le__eq__1__pos,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.00 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 8.83/5.00 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1612_divide__le__eq__1__pos,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.00 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 8.83/5.00 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1613_le__divide__eq__1__neg,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ord_less_real @ A @ zero_zero_real )
% 8.83/5.00 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 8.83/5.00 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1614_le__divide__eq__1__neg,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.83/5.00 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 8.83/5.00 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1615_le__divide__eq__1__pos,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.00 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 8.83/5.00 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1616_le__divide__eq__1__pos,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.00 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 8.83/5.00 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1617_power__strict__decreasing__iff,axiom,
% 8.83/5.00 ! [B: real,M: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_real @ zero_zero_real @ B )
% 8.83/5.00 => ( ( ord_less_real @ B @ one_one_real )
% 8.83/5.00 => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 8.83/5.00 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1618_power__strict__decreasing__iff,axiom,
% 8.83/5.00 ! [B: rat,M: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_rat @ zero_zero_rat @ B )
% 8.83/5.00 => ( ( ord_less_rat @ B @ one_one_rat )
% 8.83/5.00 => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 8.83/5.00 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1619_power__strict__decreasing__iff,axiom,
% 8.83/5.00 ! [B: nat,M: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.83/5.00 => ( ( ord_less_nat @ B @ one_one_nat )
% 8.83/5.00 => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 8.83/5.00 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1620_power__strict__decreasing__iff,axiom,
% 8.83/5.00 ! [B: int,M: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_int @ zero_zero_int @ B )
% 8.83/5.00 => ( ( ord_less_int @ B @ one_one_int )
% 8.83/5.00 => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 8.83/5.00 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1621_power__mono__iff,axiom,
% 8.83/5.00 ! [A: real,B: real,N: nat] :
% 8.83/5.00 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.00 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.83/5.00 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.00 => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 8.83/5.00 = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1622_power__mono__iff,axiom,
% 8.83/5.00 ! [A: rat,B: rat,N: nat] :
% 8.83/5.00 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.00 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.83/5.00 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.00 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 8.83/5.00 = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1623_power__mono__iff,axiom,
% 8.83/5.00 ! [A: nat,B: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.00 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.83/5.00 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.00 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 8.83/5.00 = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1624_power__mono__iff,axiom,
% 8.83/5.00 ! [A: int,B: int,N: nat] :
% 8.83/5.00 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.00 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.83/5.00 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.00 => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 8.83/5.00 = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1625_zero__eq__power2,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.00 = zero_zero_rat )
% 8.83/5.00 = ( A = zero_zero_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1626_zero__eq__power2,axiom,
% 8.83/5.00 ! [A: nat] :
% 8.83/5.00 ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.00 = zero_zero_nat )
% 8.83/5.00 = ( A = zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1627_zero__eq__power2,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.00 = zero_zero_real )
% 8.83/5.00 = ( A = zero_zero_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1628_zero__eq__power2,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.00 = zero_zero_int )
% 8.83/5.00 = ( A = zero_zero_int ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1629_zero__eq__power2,axiom,
% 8.83/5.00 ! [A: complex] :
% 8.83/5.00 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.00 = zero_zero_complex )
% 8.83/5.00 = ( A = zero_zero_complex ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1630_Suc__diff__1,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.00 => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 8.83/5.00 = N ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1631_bits__1__div__2,axiom,
% 8.83/5.00 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.00 = zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1632_bits__1__div__2,axiom,
% 8.83/5.00 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.83/5.00 = zero_zero_int ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1633_one__div__two__eq__zero,axiom,
% 8.83/5.00 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.00 = zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1634_one__div__two__eq__zero,axiom,
% 8.83/5.00 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.83/5.00 = zero_zero_int ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1635_power__decreasing__iff,axiom,
% 8.83/5.00 ! [B: real,M: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_real @ zero_zero_real @ B )
% 8.83/5.00 => ( ( ord_less_real @ B @ one_one_real )
% 8.83/5.00 => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 8.83/5.00 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1636_power__decreasing__iff,axiom,
% 8.83/5.00 ! [B: rat,M: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_rat @ zero_zero_rat @ B )
% 8.83/5.00 => ( ( ord_less_rat @ B @ one_one_rat )
% 8.83/5.00 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 8.83/5.00 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1637_power__decreasing__iff,axiom,
% 8.83/5.00 ! [B: nat,M: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.83/5.00 => ( ( ord_less_nat @ B @ one_one_nat )
% 8.83/5.00 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 8.83/5.00 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1638_power__decreasing__iff,axiom,
% 8.83/5.00 ! [B: int,M: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_int @ zero_zero_int @ B )
% 8.83/5.00 => ( ( ord_less_int @ B @ one_one_int )
% 8.83/5.00 => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 8.83/5.00 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1639_power2__eq__iff__nonneg,axiom,
% 8.83/5.00 ! [X2: real,Y3: real] :
% 8.83/5.00 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.83/5.00 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.83/5.00 => ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.00 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.00 = ( X2 = Y3 ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1640_power2__eq__iff__nonneg,axiom,
% 8.83/5.00 ! [X2: rat,Y3: rat] :
% 8.83/5.00 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 8.83/5.00 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 8.83/5.00 => ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.00 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.00 = ( X2 = Y3 ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1641_power2__eq__iff__nonneg,axiom,
% 8.83/5.00 ! [X2: nat,Y3: nat] :
% 8.83/5.00 ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
% 8.83/5.00 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 8.83/5.00 => ( ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.00 = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.00 = ( X2 = Y3 ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1642_power2__eq__iff__nonneg,axiom,
% 8.83/5.00 ! [X2: int,Y3: int] :
% 8.83/5.00 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.83/5.00 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.83/5.00 => ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.00 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.00 = ( X2 = Y3 ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1643_power2__less__eq__zero__iff,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 8.83/5.00 = ( A = zero_zero_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1644_power2__less__eq__zero__iff,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 8.83/5.00 = ( A = zero_zero_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1645_power2__less__eq__zero__iff,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 8.83/5.00 = ( A = zero_zero_int ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1646_zero__less__power2,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.00 = ( A != zero_zero_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1647_zero__less__power2,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.00 = ( A != zero_zero_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1648_zero__less__power2,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.00 = ( A != zero_zero_int ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1649_sum__power2__eq__zero__iff,axiom,
% 8.83/5.00 ! [X2: rat,Y3: rat] :
% 8.83/5.00 ( ( ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.00 = zero_zero_rat )
% 8.83/5.00 = ( ( X2 = zero_zero_rat )
% 8.83/5.00 & ( Y3 = zero_zero_rat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1650_sum__power2__eq__zero__iff,axiom,
% 8.83/5.00 ! [X2: real,Y3: real] :
% 8.83/5.00 ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.00 = zero_zero_real )
% 8.83/5.00 = ( ( X2 = zero_zero_real )
% 8.83/5.00 & ( Y3 = zero_zero_real ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1651_sum__power2__eq__zero__iff,axiom,
% 8.83/5.00 ! [X2: int,Y3: int] :
% 8.83/5.00 ( ( ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.00 = zero_zero_int )
% 8.83/5.00 = ( ( X2 = zero_zero_int )
% 8.83/5.00 & ( Y3 = zero_zero_int ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1652_VEBT__internal_Ovalid_H_Ocases,axiom,
% 8.83/5.00 ! [X2: produc9072475918466114483BT_nat] :
% 8.83/5.00 ( ! [Uu2: $o,Uv2: $o,D3: nat] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D3 ) )
% 8.83/5.00 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Deg3 ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1653_VEBT_Osize_I4_J,axiom,
% 8.83/5.00 ! [X21: $o,X222: $o] :
% 8.83/5.00 ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 8.83/5.00 = zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1654_vebt__buildup_Osimps_I1_J,axiom,
% 8.83/5.00 ( ( vEBT_vebt_buildup @ zero_zero_nat )
% 8.83/5.00 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1655_zero__reorient,axiom,
% 8.83/5.00 ! [X2: complex] :
% 8.83/5.00 ( ( zero_zero_complex = X2 )
% 8.83/5.00 = ( X2 = zero_zero_complex ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1656_zero__reorient,axiom,
% 8.83/5.00 ! [X2: real] :
% 8.83/5.00 ( ( zero_zero_real = X2 )
% 8.83/5.00 = ( X2 = zero_zero_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1657_zero__reorient,axiom,
% 8.83/5.00 ! [X2: rat] :
% 8.83/5.00 ( ( zero_zero_rat = X2 )
% 8.83/5.00 = ( X2 = zero_zero_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1658_zero__reorient,axiom,
% 8.83/5.00 ! [X2: nat] :
% 8.83/5.00 ( ( zero_zero_nat = X2 )
% 8.83/5.00 = ( X2 = zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1659_zero__reorient,axiom,
% 8.83/5.00 ! [X2: int] :
% 8.83/5.00 ( ( zero_zero_int = X2 )
% 8.83/5.00 = ( X2 = zero_zero_int ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1660_vebt__buildup_Osimps_I2_J,axiom,
% 8.83/5.00 ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
% 8.83/5.00 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1661_VEBT__internal_Onaive__member_Ocases,axiom,
% 8.83/5.00 ! [X2: produc9072475918466114483BT_nat] :
% 8.83/5.00 ( ! [A4: $o,B4: $o,X5: nat] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X5 ) )
% 8.83/5.00 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
% 8.83/5.00 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT,X5: nat] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) @ X5 ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1662_VEBT__internal_Ooption__shift_Ocases,axiom,
% 8.83/5.00 ! [X2: produc5542196010084753463at_nat] :
% 8.83/5.00 ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 8.83/5.00 => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 8.83/5.00 => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A4 ) @ ( some_P7363390416028606310at_nat @ B4 ) ) ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1663_VEBT__internal_Ooption__shift_Ocases,axiom,
% 8.83/5.00 ! [X2: produc8306885398267862888on_nat] :
% 8.83/5.00 ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 8.83/5.00 => ( ! [Uw2: nat > nat > nat,V2: nat] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 8.83/5.00 => ~ ! [F2: nat > nat > nat,A4: nat,B4: nat] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A4 ) @ ( some_nat @ B4 ) ) ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1664_VEBT__internal_Ooption__shift_Ocases,axiom,
% 8.83/5.00 ! [X2: produc1193250871479095198on_num] :
% 8.83/5.00 ( ! [Uu2: num > num > num,Uv2: option_num] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 8.83/5.00 => ( ! [Uw2: num > num > num,V2: num] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 8.83/5.00 => ~ ! [F2: num > num > num,A4: num,B4: num] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A4 ) @ ( some_num @ B4 ) ) ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1665_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 8.83/5.00 ! [X2: produc5491161045314408544at_nat] :
% 8.83/5.00 ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 8.83/5.00 => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 8.83/5.00 => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat,Y4: product_prod_nat_nat] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X5 ) @ ( some_P7363390416028606310at_nat @ Y4 ) ) ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1666_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 8.83/5.00 ! [X2: produc2233624965454879586on_nat] :
% 8.83/5.00 ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 8.83/5.00 => ( ! [Uw2: nat > nat > $o,V2: nat] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 8.83/5.00 => ~ ! [F2: nat > nat > $o,X5: nat,Y4: nat] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X5 ) @ ( some_nat @ Y4 ) ) ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1667_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 8.83/5.00 ! [X2: produc7036089656553540234on_num] :
% 8.83/5.00 ( ! [Uu2: num > num > $o,Uv2: option_num] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 8.83/5.00 => ( ! [Uw2: num > num > $o,V2: num] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 8.83/5.00 => ~ ! [F2: num > num > $o,X5: num,Y4: num] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X5 ) @ ( some_num @ Y4 ) ) ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1668_invar__vebt_Ointros_I1_J,axiom,
% 8.83/5.00 ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1669_power__0__left,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( ( N = zero_zero_nat )
% 8.83/5.00 => ( ( power_power_rat @ zero_zero_rat @ N )
% 8.83/5.00 = one_one_rat ) )
% 8.83/5.00 & ( ( N != zero_zero_nat )
% 8.83/5.00 => ( ( power_power_rat @ zero_zero_rat @ N )
% 8.83/5.00 = zero_zero_rat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1670_power__0__left,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( ( N = zero_zero_nat )
% 8.83/5.00 => ( ( power_power_nat @ zero_zero_nat @ N )
% 8.83/5.00 = one_one_nat ) )
% 8.83/5.00 & ( ( N != zero_zero_nat )
% 8.83/5.00 => ( ( power_power_nat @ zero_zero_nat @ N )
% 8.83/5.00 = zero_zero_nat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1671_power__0__left,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( ( N = zero_zero_nat )
% 8.83/5.00 => ( ( power_power_real @ zero_zero_real @ N )
% 8.83/5.00 = one_one_real ) )
% 8.83/5.00 & ( ( N != zero_zero_nat )
% 8.83/5.00 => ( ( power_power_real @ zero_zero_real @ N )
% 8.83/5.00 = zero_zero_real ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1672_power__0__left,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( ( N = zero_zero_nat )
% 8.83/5.00 => ( ( power_power_int @ zero_zero_int @ N )
% 8.83/5.00 = one_one_int ) )
% 8.83/5.00 & ( ( N != zero_zero_nat )
% 8.83/5.00 => ( ( power_power_int @ zero_zero_int @ N )
% 8.83/5.00 = zero_zero_int ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1673_power__0__left,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( ( N = zero_zero_nat )
% 8.83/5.00 => ( ( power_power_complex @ zero_zero_complex @ N )
% 8.83/5.00 = one_one_complex ) )
% 8.83/5.00 & ( ( N != zero_zero_nat )
% 8.83/5.00 => ( ( power_power_complex @ zero_zero_complex @ N )
% 8.83/5.00 = zero_zero_complex ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1674_zero__power,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.00 => ( ( power_power_rat @ zero_zero_rat @ N )
% 8.83/5.00 = zero_zero_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1675_zero__power,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.00 => ( ( power_power_nat @ zero_zero_nat @ N )
% 8.83/5.00 = zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1676_zero__power,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.00 => ( ( power_power_real @ zero_zero_real @ N )
% 8.83/5.00 = zero_zero_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1677_zero__power,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.00 => ( ( power_power_int @ zero_zero_int @ N )
% 8.83/5.00 = zero_zero_int ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1678_zero__power,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.00 => ( ( power_power_complex @ zero_zero_complex @ N )
% 8.83/5.00 = zero_zero_complex ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1679_vebt__member_Osimps_I1_J,axiom,
% 8.83/5.00 ! [A: $o,B: $o,X2: nat] :
% 8.83/5.00 ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X2 )
% 8.83/5.00 = ( ( ( X2 = zero_zero_nat )
% 8.83/5.00 => A )
% 8.83/5.00 & ( ( X2 != zero_zero_nat )
% 8.83/5.00 => ( ( ( X2 = one_one_nat )
% 8.83/5.00 => B )
% 8.83/5.00 & ( X2 = one_one_nat ) ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1680_VEBT_Odistinct_I1_J,axiom,
% 8.83/5.00 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
% 8.83/5.00 ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 8.83/5.00 != ( vEBT_Leaf @ X21 @ X222 ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1681_VEBT_Oexhaust,axiom,
% 8.83/5.00 ! [Y3: vEBT_VEBT] :
% 8.83/5.00 ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
% 8.83/5.00 ( Y3
% 8.83/5.00 != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
% 8.83/5.00 => ~ ! [X212: $o,X223: $o] :
% 8.83/5.00 ( Y3
% 8.83/5.00 != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1682_vebt__insert_Osimps_I1_J,axiom,
% 8.83/5.00 ! [X2: nat,A: $o,B: $o] :
% 8.83/5.00 ( ( ( X2 = zero_zero_nat )
% 8.83/5.00 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X2 )
% 8.83/5.00 = ( vEBT_Leaf @ $true @ B ) ) )
% 8.83/5.00 & ( ( X2 != zero_zero_nat )
% 8.83/5.00 => ( ( ( X2 = one_one_nat )
% 8.83/5.00 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X2 )
% 8.83/5.00 = ( vEBT_Leaf @ A @ $true ) ) )
% 8.83/5.00 & ( ( X2 != one_one_nat )
% 8.83/5.00 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X2 )
% 8.83/5.00 = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1683_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 8.83/5.00 ! [A: $o,B: $o,X2: nat] :
% 8.83/5.00 ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X2 )
% 8.83/5.00 = ( ( ( X2 = zero_zero_nat )
% 8.83/5.00 => A )
% 8.83/5.00 & ( ( X2 != zero_zero_nat )
% 8.83/5.00 => ( ( ( X2 = one_one_nat )
% 8.83/5.00 => B )
% 8.83/5.00 & ( X2 = one_one_nat ) ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1684_vebt__pred_Osimps_I1_J,axiom,
% 8.83/5.00 ! [Uu: $o,Uv: $o] :
% 8.83/5.00 ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
% 8.83/5.00 = none_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1685_le__numeral__extra_I3_J,axiom,
% 8.83/5.00 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1686_le__numeral__extra_I3_J,axiom,
% 8.83/5.00 ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1687_le__numeral__extra_I3_J,axiom,
% 8.83/5.00 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1688_le__numeral__extra_I3_J,axiom,
% 8.83/5.00 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1689_zero__le,axiom,
% 8.83/5.00 ! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1690_field__lbound__gt__zero,axiom,
% 8.83/5.00 ! [D1: real,D22: real] :
% 8.83/5.00 ( ( ord_less_real @ zero_zero_real @ D1 )
% 8.83/5.00 => ( ( ord_less_real @ zero_zero_real @ D22 )
% 8.83/5.00 => ? [E2: real] :
% 8.83/5.00 ( ( ord_less_real @ zero_zero_real @ E2 )
% 8.83/5.00 & ( ord_less_real @ E2 @ D1 )
% 8.83/5.00 & ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1691_field__lbound__gt__zero,axiom,
% 8.83/5.00 ! [D1: rat,D22: rat] :
% 8.83/5.00 ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 8.83/5.00 => ( ( ord_less_rat @ zero_zero_rat @ D22 )
% 8.83/5.00 => ? [E2: rat] :
% 8.83/5.00 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 8.83/5.00 & ( ord_less_rat @ E2 @ D1 )
% 8.83/5.00 & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1692_less__numeral__extra_I3_J,axiom,
% 8.83/5.00 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1693_less__numeral__extra_I3_J,axiom,
% 8.83/5.00 ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1694_less__numeral__extra_I3_J,axiom,
% 8.83/5.00 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1695_less__numeral__extra_I3_J,axiom,
% 8.83/5.00 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1696_zero__less__iff__neq__zero,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.00 = ( N != zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1697_gr__implies__not__zero,axiom,
% 8.83/5.00 ! [M: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ M @ N )
% 8.83/5.00 => ( N != zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1698_not__less__zero,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1699_gr__zeroI,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( N != zero_zero_nat )
% 8.83/5.00 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1700_zero__neq__numeral,axiom,
% 8.83/5.00 ! [N: num] :
% 8.83/5.00 ( zero_zero_complex
% 8.83/5.00 != ( numera6690914467698888265omplex @ N ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1701_zero__neq__numeral,axiom,
% 8.83/5.00 ! [N: num] :
% 8.83/5.00 ( zero_zero_real
% 8.83/5.00 != ( numeral_numeral_real @ N ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1702_zero__neq__numeral,axiom,
% 8.83/5.00 ! [N: num] :
% 8.83/5.00 ( zero_zero_rat
% 8.83/5.00 != ( numeral_numeral_rat @ N ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1703_zero__neq__numeral,axiom,
% 8.83/5.00 ! [N: num] :
% 8.83/5.00 ( zero_zero_nat
% 8.83/5.00 != ( numeral_numeral_nat @ N ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1704_zero__neq__numeral,axiom,
% 8.83/5.00 ! [N: num] :
% 8.83/5.00 ( zero_zero_int
% 8.83/5.00 != ( numeral_numeral_int @ N ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1705_mult__right__cancel,axiom,
% 8.83/5.00 ! [C: complex,A: complex,B: complex] :
% 8.83/5.00 ( ( C != zero_zero_complex )
% 8.83/5.00 => ( ( ( times_times_complex @ A @ C )
% 8.83/5.00 = ( times_times_complex @ B @ C ) )
% 8.83/5.00 = ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1706_mult__right__cancel,axiom,
% 8.83/5.00 ! [C: real,A: real,B: real] :
% 8.83/5.00 ( ( C != zero_zero_real )
% 8.83/5.00 => ( ( ( times_times_real @ A @ C )
% 8.83/5.00 = ( times_times_real @ B @ C ) )
% 8.83/5.00 = ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1707_mult__right__cancel,axiom,
% 8.83/5.00 ! [C: rat,A: rat,B: rat] :
% 8.83/5.00 ( ( C != zero_zero_rat )
% 8.83/5.00 => ( ( ( times_times_rat @ A @ C )
% 8.83/5.00 = ( times_times_rat @ B @ C ) )
% 8.83/5.00 = ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1708_mult__right__cancel,axiom,
% 8.83/5.00 ! [C: nat,A: nat,B: nat] :
% 8.83/5.00 ( ( C != zero_zero_nat )
% 8.83/5.00 => ( ( ( times_times_nat @ A @ C )
% 8.83/5.00 = ( times_times_nat @ B @ C ) )
% 8.83/5.00 = ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1709_mult__right__cancel,axiom,
% 8.83/5.00 ! [C: int,A: int,B: int] :
% 8.83/5.00 ( ( C != zero_zero_int )
% 8.83/5.00 => ( ( ( times_times_int @ A @ C )
% 8.83/5.00 = ( times_times_int @ B @ C ) )
% 8.83/5.00 = ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1710_mult__left__cancel,axiom,
% 8.83/5.00 ! [C: complex,A: complex,B: complex] :
% 8.83/5.00 ( ( C != zero_zero_complex )
% 8.83/5.00 => ( ( ( times_times_complex @ C @ A )
% 8.83/5.00 = ( times_times_complex @ C @ B ) )
% 8.83/5.00 = ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1711_mult__left__cancel,axiom,
% 8.83/5.00 ! [C: real,A: real,B: real] :
% 8.83/5.00 ( ( C != zero_zero_real )
% 8.83/5.00 => ( ( ( times_times_real @ C @ A )
% 8.83/5.00 = ( times_times_real @ C @ B ) )
% 8.83/5.00 = ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1712_mult__left__cancel,axiom,
% 8.83/5.00 ! [C: rat,A: rat,B: rat] :
% 8.83/5.00 ( ( C != zero_zero_rat )
% 8.83/5.00 => ( ( ( times_times_rat @ C @ A )
% 8.83/5.00 = ( times_times_rat @ C @ B ) )
% 8.83/5.00 = ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1713_mult__left__cancel,axiom,
% 8.83/5.00 ! [C: nat,A: nat,B: nat] :
% 8.83/5.00 ( ( C != zero_zero_nat )
% 8.83/5.00 => ( ( ( times_times_nat @ C @ A )
% 8.83/5.00 = ( times_times_nat @ C @ B ) )
% 8.83/5.00 = ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1714_mult__left__cancel,axiom,
% 8.83/5.00 ! [C: int,A: int,B: int] :
% 8.83/5.00 ( ( C != zero_zero_int )
% 8.83/5.00 => ( ( ( times_times_int @ C @ A )
% 8.83/5.00 = ( times_times_int @ C @ B ) )
% 8.83/5.00 = ( A = B ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1715_no__zero__divisors,axiom,
% 8.83/5.00 ! [A: complex,B: complex] :
% 8.83/5.00 ( ( A != zero_zero_complex )
% 8.83/5.00 => ( ( B != zero_zero_complex )
% 8.83/5.00 => ( ( times_times_complex @ A @ B )
% 8.83/5.00 != zero_zero_complex ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1716_no__zero__divisors,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( A != zero_zero_real )
% 8.83/5.00 => ( ( B != zero_zero_real )
% 8.83/5.00 => ( ( times_times_real @ A @ B )
% 8.83/5.00 != zero_zero_real ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1717_no__zero__divisors,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( A != zero_zero_rat )
% 8.83/5.00 => ( ( B != zero_zero_rat )
% 8.83/5.00 => ( ( times_times_rat @ A @ B )
% 8.83/5.00 != zero_zero_rat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1718_no__zero__divisors,axiom,
% 8.83/5.00 ! [A: nat,B: nat] :
% 8.83/5.00 ( ( A != zero_zero_nat )
% 8.83/5.00 => ( ( B != zero_zero_nat )
% 8.83/5.00 => ( ( times_times_nat @ A @ B )
% 8.83/5.00 != zero_zero_nat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1719_no__zero__divisors,axiom,
% 8.83/5.00 ! [A: int,B: int] :
% 8.83/5.00 ( ( A != zero_zero_int )
% 8.83/5.00 => ( ( B != zero_zero_int )
% 8.83/5.00 => ( ( times_times_int @ A @ B )
% 8.83/5.00 != zero_zero_int ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1720_divisors__zero,axiom,
% 8.83/5.00 ! [A: complex,B: complex] :
% 8.83/5.00 ( ( ( times_times_complex @ A @ B )
% 8.83/5.00 = zero_zero_complex )
% 8.83/5.00 => ( ( A = zero_zero_complex )
% 8.83/5.00 | ( B = zero_zero_complex ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1721_divisors__zero,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ( times_times_real @ A @ B )
% 8.83/5.00 = zero_zero_real )
% 8.83/5.00 => ( ( A = zero_zero_real )
% 8.83/5.00 | ( B = zero_zero_real ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1722_divisors__zero,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ( times_times_rat @ A @ B )
% 8.83/5.00 = zero_zero_rat )
% 8.83/5.00 => ( ( A = zero_zero_rat )
% 8.83/5.00 | ( B = zero_zero_rat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1723_divisors__zero,axiom,
% 8.83/5.00 ! [A: nat,B: nat] :
% 8.83/5.00 ( ( ( times_times_nat @ A @ B )
% 8.83/5.00 = zero_zero_nat )
% 8.83/5.00 => ( ( A = zero_zero_nat )
% 8.83/5.00 | ( B = zero_zero_nat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1724_divisors__zero,axiom,
% 8.83/5.00 ! [A: int,B: int] :
% 8.83/5.00 ( ( ( times_times_int @ A @ B )
% 8.83/5.00 = zero_zero_int )
% 8.83/5.00 => ( ( A = zero_zero_int )
% 8.83/5.00 | ( B = zero_zero_int ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1725_mult__not__zero,axiom,
% 8.83/5.00 ! [A: complex,B: complex] :
% 8.83/5.00 ( ( ( times_times_complex @ A @ B )
% 8.83/5.00 != zero_zero_complex )
% 8.83/5.00 => ( ( A != zero_zero_complex )
% 8.83/5.00 & ( B != zero_zero_complex ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1726_mult__not__zero,axiom,
% 8.83/5.00 ! [A: real,B: real] :
% 8.83/5.00 ( ( ( times_times_real @ A @ B )
% 8.83/5.00 != zero_zero_real )
% 8.83/5.00 => ( ( A != zero_zero_real )
% 8.83/5.00 & ( B != zero_zero_real ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1727_mult__not__zero,axiom,
% 8.83/5.00 ! [A: rat,B: rat] :
% 8.83/5.00 ( ( ( times_times_rat @ A @ B )
% 8.83/5.00 != zero_zero_rat )
% 8.83/5.00 => ( ( A != zero_zero_rat )
% 8.83/5.00 & ( B != zero_zero_rat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1728_mult__not__zero,axiom,
% 8.83/5.00 ! [A: nat,B: nat] :
% 8.83/5.00 ( ( ( times_times_nat @ A @ B )
% 8.83/5.00 != zero_zero_nat )
% 8.83/5.00 => ( ( A != zero_zero_nat )
% 8.83/5.00 & ( B != zero_zero_nat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1729_mult__not__zero,axiom,
% 8.83/5.00 ! [A: int,B: int] :
% 8.83/5.00 ( ( ( times_times_int @ A @ B )
% 8.83/5.00 != zero_zero_int )
% 8.83/5.00 => ( ( A != zero_zero_int )
% 8.83/5.00 & ( B != zero_zero_int ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1730_zero__neq__one,axiom,
% 8.83/5.00 zero_zero_complex != one_one_complex ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1731_zero__neq__one,axiom,
% 8.83/5.00 zero_zero_real != one_one_real ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1732_zero__neq__one,axiom,
% 8.83/5.00 zero_zero_rat != one_one_rat ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1733_zero__neq__one,axiom,
% 8.83/5.00 zero_zero_nat != one_one_nat ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1734_zero__neq__one,axiom,
% 8.83/5.00 zero_zero_int != one_one_int ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1735_comm__monoid__add__class_Oadd__0,axiom,
% 8.83/5.00 ! [A: complex] :
% 8.83/5.00 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1736_comm__monoid__add__class_Oadd__0,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( plus_plus_real @ zero_zero_real @ A )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1737_comm__monoid__add__class_Oadd__0,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1738_comm__monoid__add__class_Oadd__0,axiom,
% 8.83/5.00 ! [A: nat] :
% 8.83/5.00 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1739_comm__monoid__add__class_Oadd__0,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( plus_plus_int @ zero_zero_int @ A )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1740_add_Ocomm__neutral,axiom,
% 8.83/5.00 ! [A: complex] :
% 8.83/5.00 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1741_add_Ocomm__neutral,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( plus_plus_real @ A @ zero_zero_real )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1742_add_Ocomm__neutral,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1743_add_Ocomm__neutral,axiom,
% 8.83/5.00 ! [A: nat] :
% 8.83/5.00 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1744_add_Ocomm__neutral,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( plus_plus_int @ A @ zero_zero_int )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1745_add_Ogroup__left__neutral,axiom,
% 8.83/5.00 ! [A: complex] :
% 8.83/5.00 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1746_add_Ogroup__left__neutral,axiom,
% 8.83/5.00 ! [A: real] :
% 8.83/5.00 ( ( plus_plus_real @ zero_zero_real @ A )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1747_add_Ogroup__left__neutral,axiom,
% 8.83/5.00 ! [A: rat] :
% 8.83/5.00 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1748_add_Ogroup__left__neutral,axiom,
% 8.83/5.00 ! [A: int] :
% 8.83/5.00 ( ( plus_plus_int @ zero_zero_int @ A )
% 8.83/5.00 = A ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1749_eq__iff__diff__eq__0,axiom,
% 8.83/5.00 ( ( ^ [Y6: complex,Z5: complex] : ( Y6 = Z5 ) )
% 8.83/5.00 = ( ^ [A3: complex,B2: complex] :
% 8.83/5.00 ( ( minus_minus_complex @ A3 @ B2 )
% 8.83/5.00 = zero_zero_complex ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1750_eq__iff__diff__eq__0,axiom,
% 8.83/5.00 ( ( ^ [Y6: real,Z5: real] : ( Y6 = Z5 ) )
% 8.83/5.00 = ( ^ [A3: real,B2: real] :
% 8.83/5.00 ( ( minus_minus_real @ A3 @ B2 )
% 8.83/5.00 = zero_zero_real ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1751_eq__iff__diff__eq__0,axiom,
% 8.83/5.00 ( ( ^ [Y6: rat,Z5: rat] : ( Y6 = Z5 ) )
% 8.83/5.00 = ( ^ [A3: rat,B2: rat] :
% 8.83/5.00 ( ( minus_minus_rat @ A3 @ B2 )
% 8.83/5.00 = zero_zero_rat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1752_eq__iff__diff__eq__0,axiom,
% 8.83/5.00 ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 8.83/5.00 = ( ^ [A3: int,B2: int] :
% 8.83/5.00 ( ( minus_minus_int @ A3 @ B2 )
% 8.83/5.00 = zero_zero_int ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1753_power__not__zero,axiom,
% 8.83/5.00 ! [A: rat,N: nat] :
% 8.83/5.00 ( ( A != zero_zero_rat )
% 8.83/5.00 => ( ( power_power_rat @ A @ N )
% 8.83/5.00 != zero_zero_rat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1754_power__not__zero,axiom,
% 8.83/5.00 ! [A: nat,N: nat] :
% 8.83/5.00 ( ( A != zero_zero_nat )
% 8.83/5.00 => ( ( power_power_nat @ A @ N )
% 8.83/5.00 != zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1755_power__not__zero,axiom,
% 8.83/5.00 ! [A: real,N: nat] :
% 8.83/5.00 ( ( A != zero_zero_real )
% 8.83/5.00 => ( ( power_power_real @ A @ N )
% 8.83/5.00 != zero_zero_real ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1756_power__not__zero,axiom,
% 8.83/5.00 ! [A: int,N: nat] :
% 8.83/5.00 ( ( A != zero_zero_int )
% 8.83/5.00 => ( ( power_power_int @ A @ N )
% 8.83/5.00 != zero_zero_int ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1757_power__not__zero,axiom,
% 8.83/5.00 ! [A: complex,N: nat] :
% 8.83/5.00 ( ( A != zero_zero_complex )
% 8.83/5.00 => ( ( power_power_complex @ A @ N )
% 8.83/5.00 != zero_zero_complex ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1758_num_Osize_I4_J,axiom,
% 8.83/5.00 ( ( size_size_num @ one )
% 8.83/5.00 = zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1759_nat_Odistinct_I1_J,axiom,
% 8.83/5.00 ! [X22: nat] :
% 8.83/5.00 ( zero_zero_nat
% 8.83/5.00 != ( suc @ X22 ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1760_old_Onat_Odistinct_I2_J,axiom,
% 8.83/5.00 ! [Nat2: nat] :
% 8.83/5.00 ( ( suc @ Nat2 )
% 8.83/5.00 != zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1761_old_Onat_Odistinct_I1_J,axiom,
% 8.83/5.00 ! [Nat2: nat] :
% 8.83/5.00 ( zero_zero_nat
% 8.83/5.00 != ( suc @ Nat2 ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1762_nat_OdiscI,axiom,
% 8.83/5.00 ! [Nat: nat,X22: nat] :
% 8.83/5.00 ( ( Nat
% 8.83/5.00 = ( suc @ X22 ) )
% 8.83/5.00 => ( Nat != zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1763_old_Onat_Oexhaust,axiom,
% 8.83/5.00 ! [Y3: nat] :
% 8.83/5.00 ( ( Y3 != zero_zero_nat )
% 8.83/5.00 => ~ ! [Nat3: nat] :
% 8.83/5.00 ( Y3
% 8.83/5.00 != ( suc @ Nat3 ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1764_vebt__buildup_Ocases,axiom,
% 8.83/5.00 ! [X2: nat] :
% 8.83/5.00 ( ( X2 != zero_zero_nat )
% 8.83/5.00 => ( ( X2
% 8.83/5.00 != ( suc @ zero_zero_nat ) )
% 8.83/5.00 => ~ ! [Va3: nat] :
% 8.83/5.00 ( X2
% 8.83/5.00 != ( suc @ ( suc @ Va3 ) ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1765_nat__induct,axiom,
% 8.83/5.00 ! [P: nat > $o,N: nat] :
% 8.83/5.00 ( ( P @ zero_zero_nat )
% 8.83/5.00 => ( ! [N2: nat] :
% 8.83/5.00 ( ( P @ N2 )
% 8.83/5.00 => ( P @ ( suc @ N2 ) ) )
% 8.83/5.00 => ( P @ N ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1766_diff__induct,axiom,
% 8.83/5.00 ! [P: nat > nat > $o,M: nat,N: nat] :
% 8.83/5.00 ( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
% 8.83/5.00 => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
% 8.83/5.00 => ( ! [X5: nat,Y4: nat] :
% 8.83/5.00 ( ( P @ X5 @ Y4 )
% 8.83/5.00 => ( P @ ( suc @ X5 ) @ ( suc @ Y4 ) ) )
% 8.83/5.00 => ( P @ M @ N ) ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1767_zero__induct,axiom,
% 8.83/5.00 ! [P: nat > $o,K: nat] :
% 8.83/5.00 ( ( P @ K )
% 8.83/5.00 => ( ! [N2: nat] :
% 8.83/5.00 ( ( P @ ( suc @ N2 ) )
% 8.83/5.00 => ( P @ N2 ) )
% 8.83/5.00 => ( P @ zero_zero_nat ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1768_Suc__neq__Zero,axiom,
% 8.83/5.00 ! [M: nat] :
% 8.83/5.00 ( ( suc @ M )
% 8.83/5.00 != zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1769_Zero__neq__Suc,axiom,
% 8.83/5.00 ! [M: nat] :
% 8.83/5.00 ( zero_zero_nat
% 8.83/5.00 != ( suc @ M ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1770_Zero__not__Suc,axiom,
% 8.83/5.00 ! [M: nat] :
% 8.83/5.00 ( zero_zero_nat
% 8.83/5.00 != ( suc @ M ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1771_not0__implies__Suc,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( N != zero_zero_nat )
% 8.83/5.00 => ? [M4: nat] :
% 8.83/5.00 ( N
% 8.83/5.00 = ( suc @ M4 ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1772_infinite__descent0,axiom,
% 8.83/5.00 ! [P: nat > $o,N: nat] :
% 8.83/5.00 ( ( P @ zero_zero_nat )
% 8.83/5.00 => ( ! [N2: nat] :
% 8.83/5.00 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 8.83/5.00 => ( ~ ( P @ N2 )
% 8.83/5.00 => ? [M2: nat] :
% 8.83/5.00 ( ( ord_less_nat @ M2 @ N2 )
% 8.83/5.00 & ~ ( P @ M2 ) ) ) )
% 8.83/5.00 => ( P @ N ) ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1773_gr__implies__not0,axiom,
% 8.83/5.00 ! [M: nat,N: nat] :
% 8.83/5.00 ( ( ord_less_nat @ M @ N )
% 8.83/5.00 => ( N != zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1774_less__zeroE,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1775_not__less0,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1776_not__gr0,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 8.83/5.00 = ( N = zero_zero_nat ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1777_gr0I,axiom,
% 8.83/5.00 ! [N: nat] :
% 8.83/5.00 ( ( N != zero_zero_nat )
% 8.83/5.00 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1778_bot__nat__0_Oextremum__strict,axiom,
% 8.83/5.00 ! [A: nat] :
% 8.83/5.00 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1779_VEBT__internal_OminNull_Osimps_I1_J,axiom,
% 8.83/5.00 vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1780_VEBT__internal_OminNull_Osimps_I2_J,axiom,
% 8.83/5.00 ! [Uv: $o] :
% 8.83/5.00 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1781_VEBT__internal_OminNull_Osimps_I3_J,axiom,
% 8.83/5.00 ! [Uu: $o] :
% 8.83/5.00 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% 8.83/5.00
% 8.83/5.00 thf(fact_1782_less__eq__nat_Osimps_I1_J,axiom,
% 8.83/5.01 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1783_bot__nat__0_Oextremum__unique,axiom,
% 8.83/5.01 ! [A: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 8.83/5.01 = ( A = zero_zero_nat ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1784_bot__nat__0_Oextremum__uniqueI,axiom,
% 8.83/5.01 ! [A: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 8.83/5.01 => ( A = zero_zero_nat ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1785_le__0__eq,axiom,
% 8.83/5.01 ! [N: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 8.83/5.01 = ( N = zero_zero_nat ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1786_plus__nat_Oadd__0,axiom,
% 8.83/5.01 ! [N: nat] :
% 8.83/5.01 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 8.83/5.01 = N ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1787_add__eq__self__zero,axiom,
% 8.83/5.01 ! [M: nat,N: nat] :
% 8.83/5.01 ( ( ( plus_plus_nat @ M @ N )
% 8.83/5.01 = M )
% 8.83/5.01 => ( N = zero_zero_nat ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1788_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
% 8.83/5.01 ! [Uu: $o,Uv: $o,Uw: nat] :
% 8.83/5.01 ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1789_minus__nat_Odiff__0,axiom,
% 8.83/5.01 ! [M: nat] :
% 8.83/5.01 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 8.83/5.01 = M ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1790_diffs0__imp__equal,axiom,
% 8.83/5.01 ! [M: nat,N: nat] :
% 8.83/5.01 ( ( ( minus_minus_nat @ M @ N )
% 8.83/5.01 = zero_zero_nat )
% 8.83/5.01 => ( ( ( minus_minus_nat @ N @ M )
% 8.83/5.01 = zero_zero_nat )
% 8.83/5.01 => ( M = N ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1791_nat__mult__eq__cancel__disj,axiom,
% 8.83/5.01 ! [K: nat,M: nat,N: nat] :
% 8.83/5.01 ( ( ( times_times_nat @ K @ M )
% 8.83/5.01 = ( times_times_nat @ K @ N ) )
% 8.83/5.01 = ( ( K = zero_zero_nat )
% 8.83/5.01 | ( M = N ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1792_mult__0,axiom,
% 8.83/5.01 ! [N: nat] :
% 8.83/5.01 ( ( times_times_nat @ zero_zero_nat @ N )
% 8.83/5.01 = zero_zero_nat ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1793_power__eq__imp__eq__base,axiom,
% 8.83/5.01 ! [A: real,N: nat,B: real] :
% 8.83/5.01 ( ( ( power_power_real @ A @ N )
% 8.83/5.01 = ( power_power_real @ B @ N ) )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.01 => ( A = B ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1794_power__eq__imp__eq__base,axiom,
% 8.83/5.01 ! [A: rat,N: nat,B: rat] :
% 8.83/5.01 ( ( ( power_power_rat @ A @ N )
% 8.83/5.01 = ( power_power_rat @ B @ N ) )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.01 => ( A = B ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1795_power__eq__imp__eq__base,axiom,
% 8.83/5.01 ! [A: nat,N: nat,B: nat] :
% 8.83/5.01 ( ( ( power_power_nat @ A @ N )
% 8.83/5.01 = ( power_power_nat @ B @ N ) )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.01 => ( A = B ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1796_power__eq__imp__eq__base,axiom,
% 8.83/5.01 ! [A: int,N: nat,B: int] :
% 8.83/5.01 ( ( ( power_power_int @ A @ N )
% 8.83/5.01 = ( power_power_int @ B @ N ) )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.01 => ( A = B ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1797_power__eq__iff__eq__base,axiom,
% 8.83/5.01 ! [N: nat,A: real,B: real] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.83/5.01 => ( ( ( power_power_real @ A @ N )
% 8.83/5.01 = ( power_power_real @ B @ N ) )
% 8.83/5.01 = ( A = B ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1798_power__eq__iff__eq__base,axiom,
% 8.83/5.01 ! [N: nat,A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.83/5.01 => ( ( ( power_power_rat @ A @ N )
% 8.83/5.01 = ( power_power_rat @ B @ N ) )
% 8.83/5.01 = ( A = B ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1799_power__eq__iff__eq__base,axiom,
% 8.83/5.01 ! [N: nat,A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.83/5.01 => ( ( ( power_power_nat @ A @ N )
% 8.83/5.01 = ( power_power_nat @ B @ N ) )
% 8.83/5.01 = ( A = B ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1800_power__eq__iff__eq__base,axiom,
% 8.83/5.01 ! [N: nat,A: int,B: int] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.83/5.01 => ( ( ( power_power_int @ A @ N )
% 8.83/5.01 = ( power_power_int @ B @ N ) )
% 8.83/5.01 = ( A = B ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1801_lambda__zero,axiom,
% 8.83/5.01 ( ( ^ [H: complex] : zero_zero_complex )
% 8.83/5.01 = ( times_times_complex @ zero_zero_complex ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1802_lambda__zero,axiom,
% 8.83/5.01 ( ( ^ [H: real] : zero_zero_real )
% 8.83/5.01 = ( times_times_real @ zero_zero_real ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1803_lambda__zero,axiom,
% 8.83/5.01 ( ( ^ [H: rat] : zero_zero_rat )
% 8.83/5.01 = ( times_times_rat @ zero_zero_rat ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1804_lambda__zero,axiom,
% 8.83/5.01 ( ( ^ [H: nat] : zero_zero_nat )
% 8.83/5.01 = ( times_times_nat @ zero_zero_nat ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1805_lambda__zero,axiom,
% 8.83/5.01 ( ( ^ [H: int] : zero_zero_int )
% 8.83/5.01 = ( times_times_int @ zero_zero_int ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1806_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
% 8.83/5.01 ! [X2: produc9072475918466114483BT_nat] :
% 8.83/5.01 ( ! [A4: $o,B4: $o,X5: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X5 ) )
% 8.83/5.01 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X5: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) @ X5 ) )
% 8.83/5.01 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X5: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) @ X5 ) )
% 8.83/5.01 => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ X5 ) )
% 8.83/5.01 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X5 ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1807_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
% 8.83/5.01 ! [X2: produc9072475918466114483BT_nat] :
% 8.83/5.01 ( ! [A4: $o,B4: $o,X5: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X5 ) )
% 8.83/5.01 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X5: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X5 ) )
% 8.83/5.01 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X5: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X5 ) )
% 8.83/5.01 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X5 ) )
% 8.83/5.01 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X5 ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1808_VEBT__internal_Omembermima_Ocases,axiom,
% 8.83/5.01 ! [X2: produc9072475918466114483BT_nat] :
% 8.83/5.01 ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
% 8.83/5.01 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
% 8.83/5.01 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT,X5: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ X5 ) )
% 8.83/5.01 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ X5 ) )
% 8.83/5.01 => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X5: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ X5 ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1809_vebt__pred_Ocases,axiom,
% 8.83/5.01 ! [X2: produc9072475918466114483BT_nat] :
% 8.83/5.01 ( ! [Uu2: $o,Uv2: $o] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
% 8.83/5.01 => ( ! [A4: $o,Uw2: $o] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
% 8.83/5.01 => ( ! [A4: $o,B4: $o,Va3: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ Va3 ) ) ) )
% 8.83/5.01 => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb2: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Vb2 ) )
% 8.83/5.01 => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) @ Vf ) )
% 8.83/5.01 => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj ) )
% 8.83/5.01 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X5 ) ) ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1810_vebt__succ_Ocases,axiom,
% 8.83/5.01 ! [X2: produc9072475918466114483BT_nat] :
% 8.83/5.01 ( ! [Uu2: $o,B4: $o] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B4 ) @ zero_zero_nat ) )
% 8.83/5.01 => ( ! [Uv2: $o,Uw2: $o,N2: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N2 ) ) )
% 8.83/5.01 => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va2 ) )
% 8.83/5.01 => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve ) )
% 8.83/5.01 => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi ) )
% 8.83/5.01 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X5 ) ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1811_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
% 8.83/5.01 ! [A: $o,B: $o] :
% 8.83/5.01 ( ( vEBT_T_m_i_n_t @ ( vEBT_Leaf @ A @ B ) )
% 8.83/5.01 = ( plus_plus_nat @ one_one_nat @ ( if_nat @ A @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1812_power__strict__mono,axiom,
% 8.83/5.01 ! [A: real,B: real,N: nat] :
% 8.83/5.01 ( ( ord_less_real @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.01 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1813_power__strict__mono,axiom,
% 8.83/5.01 ! [A: rat,B: rat,N: nat] :
% 8.83/5.01 ( ( ord_less_rat @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.01 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1814_power__strict__mono,axiom,
% 8.83/5.01 ! [A: nat,B: nat,N: nat] :
% 8.83/5.01 ( ( ord_less_nat @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.01 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1815_power__strict__mono,axiom,
% 8.83/5.01 ! [A: int,B: int,N: nat] :
% 8.83/5.01 ( ( ord_less_int @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.01 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1816_div__positive,axiom,
% 8.83/5.01 ! [B: nat,A: nat] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.83/5.01 => ( ( ord_less_eq_nat @ B @ A )
% 8.83/5.01 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1817_div__positive,axiom,
% 8.83/5.01 ! [B: int,A: int] :
% 8.83/5.01 ( ( ord_less_int @ zero_zero_int @ B )
% 8.83/5.01 => ( ( ord_less_eq_int @ B @ A )
% 8.83/5.01 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1818_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 8.83/5.01 ! [A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ( ord_less_nat @ A @ B )
% 8.83/5.01 => ( ( divide_divide_nat @ A @ B )
% 8.83/5.01 = zero_zero_nat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1819_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ( ord_less_int @ A @ B )
% 8.83/5.01 => ( ( divide_divide_int @ A @ B )
% 8.83/5.01 = zero_zero_int ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1820_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 8.83/5.01 ! [C: nat,A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.83/5.01 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 8.83/5.01 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1821_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 8.83/5.01 ! [C: int,A: int,B: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 8.83/5.01 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1822_vebt__pred_Osimps_I2_J,axiom,
% 8.83/5.01 ! [A: $o,Uw: $o] :
% 8.83/5.01 ( ( A
% 8.83/5.01 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
% 8.83/5.01 = ( some_nat @ zero_zero_nat ) ) )
% 8.83/5.01 & ( ~ A
% 8.83/5.01 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
% 8.83/5.01 = none_nat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1823_vebt__mint_Osimps_I1_J,axiom,
% 8.83/5.01 ! [A: $o,B: $o] :
% 8.83/5.01 ( ( A
% 8.83/5.01 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 8.83/5.01 = ( some_nat @ zero_zero_nat ) ) )
% 8.83/5.01 & ( ~ A
% 8.83/5.01 => ( ( B
% 8.83/5.01 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 8.83/5.01 = ( some_nat @ one_one_nat ) ) )
% 8.83/5.01 & ( ~ B
% 8.83/5.01 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 8.83/5.01 = none_nat ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1824_vebt__succ_Osimps_I1_J,axiom,
% 8.83/5.01 ! [B: $o,Uu: $o] :
% 8.83/5.01 ( ( B
% 8.83/5.01 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
% 8.83/5.01 = ( some_nat @ one_one_nat ) ) )
% 8.83/5.01 & ( ~ B
% 8.83/5.01 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
% 8.83/5.01 = none_nat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1825_vebt__maxt_Osimps_I1_J,axiom,
% 8.83/5.01 ! [B: $o,A: $o] :
% 8.83/5.01 ( ( B
% 8.83/5.01 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 8.83/5.01 = ( some_nat @ one_one_nat ) ) )
% 8.83/5.01 & ( ~ B
% 8.83/5.01 => ( ( A
% 8.83/5.01 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 8.83/5.01 = ( some_nat @ zero_zero_nat ) ) )
% 8.83/5.01 & ( ~ A
% 8.83/5.01 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 8.83/5.01 = none_nat ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1826_zero__le__numeral,axiom,
% 8.83/5.01 ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1827_zero__le__numeral,axiom,
% 8.83/5.01 ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1828_zero__le__numeral,axiom,
% 8.83/5.01 ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1829_zero__le__numeral,axiom,
% 8.83/5.01 ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1830_not__numeral__le__zero,axiom,
% 8.83/5.01 ! [N: num] :
% 8.83/5.01 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1831_not__numeral__le__zero,axiom,
% 8.83/5.01 ! [N: num] :
% 8.83/5.01 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1832_not__numeral__le__zero,axiom,
% 8.83/5.01 ! [N: num] :
% 8.83/5.01 ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1833_not__numeral__le__zero,axiom,
% 8.83/5.01 ! [N: num] :
% 8.83/5.01 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1834_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 8.83/5.01 ! [A: real,B: real,C: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1835_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 8.83/5.01 ! [A: rat,B: rat,C: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1836_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 8.83/5.01 ! [A: nat,B: nat,C: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.83/5.01 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1837_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 8.83/5.01 ! [A: int,B: int,C: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1838_zero__le__mult__iff,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 8.83/5.01 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.01 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 8.83/5.01 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.83/5.01 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1839_zero__le__mult__iff,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 8.83/5.01 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.01 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 8.83/5.01 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.83/5.01 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1840_zero__le__mult__iff,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 8.83/5.01 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.01 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 8.83/5.01 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.83/5.01 & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1841_mult__nonneg__nonpos2,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 8.83/5.01 => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1842_mult__nonneg__nonpos2,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1843_mult__nonneg__nonpos2,axiom,
% 8.83/5.01 ! [A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 8.83/5.01 => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1844_mult__nonneg__nonpos2,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 8.83/5.01 => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1845_mult__nonpos__nonneg,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.83/5.01 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1846_mult__nonpos__nonneg,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.83/5.01 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1847_mult__nonpos__nonneg,axiom,
% 8.83/5.01 ! [A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.83/5.01 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1848_mult__nonpos__nonneg,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.83/5.01 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1849_mult__nonneg__nonpos,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 8.83/5.01 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1850_mult__nonneg__nonpos,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1851_mult__nonneg__nonpos,axiom,
% 8.83/5.01 ! [A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 8.83/5.01 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1852_mult__nonneg__nonpos,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 8.83/5.01 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1853_mult__nonneg__nonneg,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.83/5.01 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1854_mult__nonneg__nonneg,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.83/5.01 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1855_mult__nonneg__nonneg,axiom,
% 8.83/5.01 ! [A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.83/5.01 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1856_mult__nonneg__nonneg,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.83/5.01 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1857_split__mult__neg__le,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.01 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 8.83/5.01 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.83/5.01 & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
% 8.83/5.01 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1858_split__mult__neg__le,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.01 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 8.83/5.01 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.83/5.01 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
% 8.83/5.01 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1859_split__mult__neg__le,axiom,
% 8.83/5.01 ! [A: nat,B: nat] :
% 8.83/5.01 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.01 & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 8.83/5.01 | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 8.83/5.01 & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 8.83/5.01 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1860_split__mult__neg__le,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.01 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 8.83/5.01 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.83/5.01 & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
% 8.83/5.01 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1861_mult__le__0__iff,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 8.83/5.01 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.01 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 8.83/5.01 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.83/5.01 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1862_mult__le__0__iff,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 8.83/5.01 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.01 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 8.83/5.01 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.83/5.01 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1863_mult__le__0__iff,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 8.83/5.01 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.01 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 8.83/5.01 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.83/5.01 & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1864_mult__right__mono,axiom,
% 8.83/5.01 ! [A: real,B: real,C: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1865_mult__right__mono,axiom,
% 8.83/5.01 ! [A: rat,B: rat,C: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1866_mult__right__mono,axiom,
% 8.83/5.01 ! [A: nat,B: nat,C: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.83/5.01 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1867_mult__right__mono,axiom,
% 8.83/5.01 ! [A: int,B: int,C: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1868_mult__right__mono__neg,axiom,
% 8.83/5.01 ! [B: real,A: real,C: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ B @ A )
% 8.83/5.01 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.83/5.01 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1869_mult__right__mono__neg,axiom,
% 8.83/5.01 ! [B: rat,A: rat,C: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ B @ A )
% 8.83/5.01 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1870_mult__right__mono__neg,axiom,
% 8.83/5.01 ! [B: int,A: int,C: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ B @ A )
% 8.83/5.01 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 8.83/5.01 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1871_mult__left__mono,axiom,
% 8.83/5.01 ! [A: real,B: real,C: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1872_mult__left__mono,axiom,
% 8.83/5.01 ! [A: rat,B: rat,C: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1873_mult__left__mono,axiom,
% 8.83/5.01 ! [A: nat,B: nat,C: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.83/5.01 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1874_mult__left__mono,axiom,
% 8.83/5.01 ! [A: int,B: int,C: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1875_mult__nonpos__nonpos,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.83/5.01 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 8.83/5.01 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1876_mult__nonpos__nonpos,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.83/5.01 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1877_mult__nonpos__nonpos,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.83/5.01 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 8.83/5.01 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1878_mult__left__mono__neg,axiom,
% 8.83/5.01 ! [B: real,A: real,C: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ B @ A )
% 8.83/5.01 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.83/5.01 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1879_mult__left__mono__neg,axiom,
% 8.83/5.01 ! [B: rat,A: rat,C: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ B @ A )
% 8.83/5.01 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1880_mult__left__mono__neg,axiom,
% 8.83/5.01 ! [B: int,A: int,C: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ B @ A )
% 8.83/5.01 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 8.83/5.01 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1881_split__mult__pos__le,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.01 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 8.83/5.01 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.83/5.01 & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 8.83/5.01 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1882_split__mult__pos__le,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.01 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 8.83/5.01 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.83/5.01 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 8.83/5.01 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1883_split__mult__pos__le,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.01 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 8.83/5.01 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.83/5.01 & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 8.83/5.01 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1884_zero__le__square,axiom,
% 8.83/5.01 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1885_zero__le__square,axiom,
% 8.83/5.01 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1886_zero__le__square,axiom,
% 8.83/5.01 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1887_mult__mono_H,axiom,
% 8.83/5.01 ! [A: real,B: real,C: real,D: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_real @ C @ D )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1888_mult__mono_H,axiom,
% 8.83/5.01 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_rat @ C @ D )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1889_mult__mono_H,axiom,
% 8.83/5.01 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_nat @ C @ D )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.83/5.01 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1890_mult__mono_H,axiom,
% 8.83/5.01 ! [A: int,B: int,C: int,D: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_int @ C @ D )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1891_mult__mono,axiom,
% 8.83/5.01 ! [A: real,B: real,C: real,D: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_real @ C @ D )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1892_mult__mono,axiom,
% 8.83/5.01 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_rat @ C @ D )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1893_mult__mono,axiom,
% 8.83/5.01 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_nat @ C @ D )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.83/5.01 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1894_mult__mono,axiom,
% 8.83/5.01 ! [A: int,B: int,C: int,D: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_int @ C @ D )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1895_not__numeral__less__zero,axiom,
% 8.83/5.01 ! [N: num] :
% 8.83/5.01 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1896_not__numeral__less__zero,axiom,
% 8.83/5.01 ! [N: num] :
% 8.83/5.01 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1897_not__numeral__less__zero,axiom,
% 8.83/5.01 ! [N: num] :
% 8.83/5.01 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1898_not__numeral__less__zero,axiom,
% 8.83/5.01 ! [N: num] :
% 8.83/5.01 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1899_zero__less__numeral,axiom,
% 8.83/5.01 ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1900_zero__less__numeral,axiom,
% 8.83/5.01 ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1901_zero__less__numeral,axiom,
% 8.83/5.01 ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1902_zero__less__numeral,axiom,
% 8.83/5.01 ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1903_zero__less__one__class_Ozero__le__one,axiom,
% 8.83/5.01 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1904_zero__less__one__class_Ozero__le__one,axiom,
% 8.83/5.01 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1905_zero__less__one__class_Ozero__le__one,axiom,
% 8.83/5.01 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1906_zero__less__one__class_Ozero__le__one,axiom,
% 8.83/5.01 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1907_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 8.83/5.01 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1908_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 8.83/5.01 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1909_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 8.83/5.01 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1910_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 8.83/5.01 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1911_not__one__le__zero,axiom,
% 8.83/5.01 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1912_not__one__le__zero,axiom,
% 8.83/5.01 ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1913_not__one__le__zero,axiom,
% 8.83/5.01 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1914_not__one__le__zero,axiom,
% 8.83/5.01 ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1915_add__decreasing,axiom,
% 8.83/5.01 ! [A: real,C: real,B: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.83/5.01 => ( ( ord_less_eq_real @ C @ B )
% 8.83/5.01 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1916_add__decreasing,axiom,
% 8.83/5.01 ! [A: rat,C: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.83/5.01 => ( ( ord_less_eq_rat @ C @ B )
% 8.83/5.01 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1917_add__decreasing,axiom,
% 8.83/5.01 ! [A: nat,C: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 8.83/5.01 => ( ( ord_less_eq_nat @ C @ B )
% 8.83/5.01 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1918_add__decreasing,axiom,
% 8.83/5.01 ! [A: int,C: int,B: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.83/5.01 => ( ( ord_less_eq_int @ C @ B )
% 8.83/5.01 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1919_add__increasing,axiom,
% 8.83/5.01 ! [A: real,B: real,C: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ( ord_less_eq_real @ B @ C )
% 8.83/5.01 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1920_add__increasing,axiom,
% 8.83/5.01 ! [A: rat,B: rat,C: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ( ord_less_eq_rat @ B @ C )
% 8.83/5.01 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1921_add__increasing,axiom,
% 8.83/5.01 ! [A: nat,B: nat,C: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ( ord_less_eq_nat @ B @ C )
% 8.83/5.01 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1922_add__increasing,axiom,
% 8.83/5.01 ! [A: int,B: int,C: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ( ord_less_eq_int @ B @ C )
% 8.83/5.01 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1923_add__decreasing2,axiom,
% 8.83/5.01 ! [C: real,A: real,B: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.83/5.01 => ( ( ord_less_eq_real @ A @ B )
% 8.83/5.01 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1924_add__decreasing2,axiom,
% 8.83/5.01 ! [C: rat,A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.83/5.01 => ( ( ord_less_eq_rat @ A @ B )
% 8.83/5.01 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1925_add__decreasing2,axiom,
% 8.83/5.01 ! [C: nat,A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 8.83/5.01 => ( ( ord_less_eq_nat @ A @ B )
% 8.83/5.01 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1926_add__decreasing2,axiom,
% 8.83/5.01 ! [C: int,A: int,B: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ C @ zero_zero_int )
% 8.83/5.01 => ( ( ord_less_eq_int @ A @ B )
% 8.83/5.01 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1927_add__increasing2,axiom,
% 8.83/5.01 ! [C: real,B: real,A: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ( ord_less_eq_real @ B @ A )
% 8.83/5.01 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1928_add__increasing2,axiom,
% 8.83/5.01 ! [C: rat,B: rat,A: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ( ord_less_eq_rat @ B @ A )
% 8.83/5.01 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1929_add__increasing2,axiom,
% 8.83/5.01 ! [C: nat,B: nat,A: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.83/5.01 => ( ( ord_less_eq_nat @ B @ A )
% 8.83/5.01 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1930_add__increasing2,axiom,
% 8.83/5.01 ! [C: int,B: int,A: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ( ord_less_eq_int @ B @ A )
% 8.83/5.01 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1931_add__nonneg__nonneg,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.83/5.01 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1932_add__nonneg__nonneg,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.83/5.01 => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1933_add__nonneg__nonneg,axiom,
% 8.83/5.01 ! [A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.83/5.01 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1934_add__nonneg__nonneg,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.83/5.01 => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1935_add__nonpos__nonpos,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.83/5.01 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 8.83/5.01 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1936_add__nonpos__nonpos,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.83/5.01 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1937_add__nonpos__nonpos,axiom,
% 8.83/5.01 ! [A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 8.83/5.01 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 8.83/5.01 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1938_add__nonpos__nonpos,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.83/5.01 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 8.83/5.01 => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1939_add__nonneg__eq__0__iff,axiom,
% 8.83/5.01 ! [X2: real,Y3: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.83/5.01 => ( ( ( plus_plus_real @ X2 @ Y3 )
% 8.83/5.01 = zero_zero_real )
% 8.83/5.01 = ( ( X2 = zero_zero_real )
% 8.83/5.01 & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1940_add__nonneg__eq__0__iff,axiom,
% 8.83/5.01 ! [X2: rat,Y3: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 8.83/5.01 => ( ( ( plus_plus_rat @ X2 @ Y3 )
% 8.83/5.01 = zero_zero_rat )
% 8.83/5.01 = ( ( X2 = zero_zero_rat )
% 8.83/5.01 & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1941_add__nonneg__eq__0__iff,axiom,
% 8.83/5.01 ! [X2: nat,Y3: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 8.83/5.01 => ( ( ( plus_plus_nat @ X2 @ Y3 )
% 8.83/5.01 = zero_zero_nat )
% 8.83/5.01 = ( ( X2 = zero_zero_nat )
% 8.83/5.01 & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1942_add__nonneg__eq__0__iff,axiom,
% 8.83/5.01 ! [X2: int,Y3: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.83/5.01 => ( ( ( plus_plus_int @ X2 @ Y3 )
% 8.83/5.01 = zero_zero_int )
% 8.83/5.01 = ( ( X2 = zero_zero_int )
% 8.83/5.01 & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1943_add__nonpos__eq__0__iff,axiom,
% 8.83/5.01 ! [X2: real,Y3: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 8.83/5.01 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 8.83/5.01 => ( ( ( plus_plus_real @ X2 @ Y3 )
% 8.83/5.01 = zero_zero_real )
% 8.83/5.01 = ( ( X2 = zero_zero_real )
% 8.83/5.01 & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1944_add__nonpos__eq__0__iff,axiom,
% 8.83/5.01 ! [X2: rat,Y3: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 8.83/5.01 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 8.83/5.01 => ( ( ( plus_plus_rat @ X2 @ Y3 )
% 8.83/5.01 = zero_zero_rat )
% 8.83/5.01 = ( ( X2 = zero_zero_rat )
% 8.83/5.01 & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1945_add__nonpos__eq__0__iff,axiom,
% 8.83/5.01 ! [X2: nat,Y3: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
% 8.83/5.01 => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
% 8.83/5.01 => ( ( ( plus_plus_nat @ X2 @ Y3 )
% 8.83/5.01 = zero_zero_nat )
% 8.83/5.01 = ( ( X2 = zero_zero_nat )
% 8.83/5.01 & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1946_add__nonpos__eq__0__iff,axiom,
% 8.83/5.01 ! [X2: int,Y3: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ X2 @ zero_zero_int )
% 8.83/5.01 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 8.83/5.01 => ( ( ( plus_plus_int @ X2 @ Y3 )
% 8.83/5.01 = zero_zero_int )
% 8.83/5.01 = ( ( X2 = zero_zero_int )
% 8.83/5.01 & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1947_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 8.83/5.01 ! [A: real,B: real,C: real] :
% 8.83/5.01 ( ( ord_less_real @ A @ B )
% 8.83/5.01 => ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1948_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 8.83/5.01 ! [A: rat,B: rat,C: rat] :
% 8.83/5.01 ( ( ord_less_rat @ A @ B )
% 8.83/5.01 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1949_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 8.83/5.01 ! [A: nat,B: nat,C: nat] :
% 8.83/5.01 ( ( ord_less_nat @ A @ B )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 8.83/5.01 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1950_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 8.83/5.01 ! [A: int,B: int,C: int] :
% 8.83/5.01 ( ( ord_less_int @ A @ B )
% 8.83/5.01 => ( ( ord_less_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1951_mult__less__cancel__right__disj,axiom,
% 8.83/5.01 ! [A: real,C: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 8.83/5.01 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.01 & ( ord_less_real @ A @ B ) )
% 8.83/5.01 | ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.01 & ( ord_less_real @ B @ A ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1952_mult__less__cancel__right__disj,axiom,
% 8.83/5.01 ! [A: rat,C: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 8.83/5.01 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.01 & ( ord_less_rat @ A @ B ) )
% 8.83/5.01 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.01 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1953_mult__less__cancel__right__disj,axiom,
% 8.83/5.01 ! [A: int,C: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 8.83/5.01 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 8.83/5.01 & ( ord_less_int @ A @ B ) )
% 8.83/5.01 | ( ( ord_less_int @ C @ zero_zero_int )
% 8.83/5.01 & ( ord_less_int @ B @ A ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1954_mult__strict__right__mono,axiom,
% 8.83/5.01 ! [A: real,B: real,C: real] :
% 8.83/5.01 ( ( ord_less_real @ A @ B )
% 8.83/5.01 => ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1955_mult__strict__right__mono,axiom,
% 8.83/5.01 ! [A: rat,B: rat,C: rat] :
% 8.83/5.01 ( ( ord_less_rat @ A @ B )
% 8.83/5.01 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1956_mult__strict__right__mono,axiom,
% 8.83/5.01 ! [A: nat,B: nat,C: nat] :
% 8.83/5.01 ( ( ord_less_nat @ A @ B )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 8.83/5.01 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1957_mult__strict__right__mono,axiom,
% 8.83/5.01 ! [A: int,B: int,C: int] :
% 8.83/5.01 ( ( ord_less_int @ A @ B )
% 8.83/5.01 => ( ( ord_less_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1958_mult__strict__right__mono__neg,axiom,
% 8.83/5.01 ! [B: real,A: real,C: real] :
% 8.83/5.01 ( ( ord_less_real @ B @ A )
% 8.83/5.01 => ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.01 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1959_mult__strict__right__mono__neg,axiom,
% 8.83/5.01 ! [B: rat,A: rat,C: rat] :
% 8.83/5.01 ( ( ord_less_rat @ B @ A )
% 8.83/5.01 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1960_mult__strict__right__mono__neg,axiom,
% 8.83/5.01 ! [B: int,A: int,C: int] :
% 8.83/5.01 ( ( ord_less_int @ B @ A )
% 8.83/5.01 => ( ( ord_less_int @ C @ zero_zero_int )
% 8.83/5.01 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1961_mult__less__cancel__left__disj,axiom,
% 8.83/5.01 ! [C: real,A: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.83/5.01 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.01 & ( ord_less_real @ A @ B ) )
% 8.83/5.01 | ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.01 & ( ord_less_real @ B @ A ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1962_mult__less__cancel__left__disj,axiom,
% 8.83/5.01 ! [C: rat,A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.83/5.01 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.01 & ( ord_less_rat @ A @ B ) )
% 8.83/5.01 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.01 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1963_mult__less__cancel__left__disj,axiom,
% 8.83/5.01 ! [C: int,A: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.83/5.01 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 8.83/5.01 & ( ord_less_int @ A @ B ) )
% 8.83/5.01 | ( ( ord_less_int @ C @ zero_zero_int )
% 8.83/5.01 & ( ord_less_int @ B @ A ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1964_mult__strict__left__mono,axiom,
% 8.83/5.01 ! [A: real,B: real,C: real] :
% 8.83/5.01 ( ( ord_less_real @ A @ B )
% 8.83/5.01 => ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1965_mult__strict__left__mono,axiom,
% 8.83/5.01 ! [A: rat,B: rat,C: rat] :
% 8.83/5.01 ( ( ord_less_rat @ A @ B )
% 8.83/5.01 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1966_mult__strict__left__mono,axiom,
% 8.83/5.01 ! [A: nat,B: nat,C: nat] :
% 8.83/5.01 ( ( ord_less_nat @ A @ B )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 8.83/5.01 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1967_mult__strict__left__mono,axiom,
% 8.83/5.01 ! [A: int,B: int,C: int] :
% 8.83/5.01 ( ( ord_less_int @ A @ B )
% 8.83/5.01 => ( ( ord_less_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1968_mult__strict__left__mono__neg,axiom,
% 8.83/5.01 ! [B: real,A: real,C: real] :
% 8.83/5.01 ( ( ord_less_real @ B @ A )
% 8.83/5.01 => ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.01 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1969_mult__strict__left__mono__neg,axiom,
% 8.83/5.01 ! [B: rat,A: rat,C: rat] :
% 8.83/5.01 ( ( ord_less_rat @ B @ A )
% 8.83/5.01 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1970_mult__strict__left__mono__neg,axiom,
% 8.83/5.01 ! [B: int,A: int,C: int] :
% 8.83/5.01 ( ( ord_less_int @ B @ A )
% 8.83/5.01 => ( ( ord_less_int @ C @ zero_zero_int )
% 8.83/5.01 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1971_mult__less__cancel__left__pos,axiom,
% 8.83/5.01 ! [C: real,A: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.83/5.01 = ( ord_less_real @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1972_mult__less__cancel__left__pos,axiom,
% 8.83/5.01 ! [C: rat,A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.83/5.01 = ( ord_less_rat @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1973_mult__less__cancel__left__pos,axiom,
% 8.83/5.01 ! [C: int,A: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.83/5.01 = ( ord_less_int @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1974_mult__less__cancel__left__neg,axiom,
% 8.83/5.01 ! [C: real,A: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.01 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.83/5.01 = ( ord_less_real @ B @ A ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1975_mult__less__cancel__left__neg,axiom,
% 8.83/5.01 ! [C: rat,A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.01 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.83/5.01 = ( ord_less_rat @ B @ A ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1976_mult__less__cancel__left__neg,axiom,
% 8.83/5.01 ! [C: int,A: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ C @ zero_zero_int )
% 8.83/5.01 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.83/5.01 = ( ord_less_int @ B @ A ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1977_zero__less__mult__pos2,axiom,
% 8.83/5.01 ! [B: real,A: real] :
% 8.83/5.01 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
% 8.83/5.01 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1978_zero__less__mult__pos2,axiom,
% 8.83/5.01 ! [B: rat,A: rat] :
% 8.83/5.01 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
% 8.83/5.01 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1979_zero__less__mult__pos2,axiom,
% 8.83/5.01 ! [B: nat,A: nat] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1980_zero__less__mult__pos2,axiom,
% 8.83/5.01 ! [B: int,A: int] :
% 8.83/5.01 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
% 8.83/5.01 => ( ( ord_less_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1981_zero__less__mult__pos,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 8.83/5.01 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1982_zero__less__mult__pos,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 8.83/5.01 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1983_zero__less__mult__pos,axiom,
% 8.83/5.01 ! [A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1984_zero__less__mult__pos,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 8.83/5.01 => ( ( ord_less_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1985_zero__less__mult__iff,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 8.83/5.01 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.01 & ( ord_less_real @ zero_zero_real @ B ) )
% 8.83/5.01 | ( ( ord_less_real @ A @ zero_zero_real )
% 8.83/5.01 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1986_zero__less__mult__iff,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 8.83/5.01 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.01 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 8.83/5.01 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.83/5.01 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1987_zero__less__mult__iff,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 8.83/5.01 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 8.83/5.01 & ( ord_less_int @ zero_zero_int @ B ) )
% 8.83/5.01 | ( ( ord_less_int @ A @ zero_zero_int )
% 8.83/5.01 & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1988_mult__pos__neg2,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ( ord_less_real @ B @ zero_zero_real )
% 8.83/5.01 => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1989_mult__pos__neg2,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1990_mult__pos__neg2,axiom,
% 8.83/5.01 ! [A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 8.83/5.01 => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1991_mult__pos__neg2,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ( ord_less_int @ B @ zero_zero_int )
% 8.83/5.01 => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1992_mult__pos__pos,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ( ord_less_real @ zero_zero_real @ B )
% 8.83/5.01 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1993_mult__pos__pos,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 8.83/5.01 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1994_mult__pos__pos,axiom,
% 8.83/5.01 ! [A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.83/5.01 => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1995_mult__pos__pos,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ( ord_less_int @ zero_zero_int @ B )
% 8.83/5.01 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1996_mult__pos__neg,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ( ord_less_real @ B @ zero_zero_real )
% 8.83/5.01 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1997_mult__pos__neg,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1998_mult__pos__neg,axiom,
% 8.83/5.01 ! [A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 8.83/5.01 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_1999_mult__pos__neg,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ( ord_less_int @ B @ zero_zero_int )
% 8.83/5.01 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2000_mult__neg__pos,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ A @ zero_zero_real )
% 8.83/5.01 => ( ( ord_less_real @ zero_zero_real @ B )
% 8.83/5.01 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2001_mult__neg__pos,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.83/5.01 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 8.83/5.01 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2002_mult__neg__pos,axiom,
% 8.83/5.01 ! [A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_nat @ A @ zero_zero_nat )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.83/5.01 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2003_mult__neg__pos,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ A @ zero_zero_int )
% 8.83/5.01 => ( ( ord_less_int @ zero_zero_int @ B )
% 8.83/5.01 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2004_mult__less__0__iff,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 8.83/5.01 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.01 & ( ord_less_real @ B @ zero_zero_real ) )
% 8.83/5.01 | ( ( ord_less_real @ A @ zero_zero_real )
% 8.83/5.01 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2005_mult__less__0__iff,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 8.83/5.01 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.01 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 8.83/5.01 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.83/5.01 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2006_mult__less__0__iff,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 8.83/5.01 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 8.83/5.01 & ( ord_less_int @ B @ zero_zero_int ) )
% 8.83/5.01 | ( ( ord_less_int @ A @ zero_zero_int )
% 8.83/5.01 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2007_not__square__less__zero,axiom,
% 8.83/5.01 ! [A: real] :
% 8.83/5.01 ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2008_not__square__less__zero,axiom,
% 8.83/5.01 ! [A: rat] :
% 8.83/5.01 ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2009_not__square__less__zero,axiom,
% 8.83/5.01 ! [A: int] :
% 8.83/5.01 ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2010_mult__neg__neg,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ A @ zero_zero_real )
% 8.83/5.01 => ( ( ord_less_real @ B @ zero_zero_real )
% 8.83/5.01 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2011_mult__neg__neg,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.83/5.01 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2012_mult__neg__neg,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ A @ zero_zero_int )
% 8.83/5.01 => ( ( ord_less_int @ B @ zero_zero_int )
% 8.83/5.01 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2013_less__numeral__extra_I1_J,axiom,
% 8.83/5.01 ord_less_real @ zero_zero_real @ one_one_real ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2014_less__numeral__extra_I1_J,axiom,
% 8.83/5.01 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2015_less__numeral__extra_I1_J,axiom,
% 8.83/5.01 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2016_less__numeral__extra_I1_J,axiom,
% 8.83/5.01 ord_less_int @ zero_zero_int @ one_one_int ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2017_zero__less__one,axiom,
% 8.83/5.01 ord_less_real @ zero_zero_real @ one_one_real ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2018_zero__less__one,axiom,
% 8.83/5.01 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2019_zero__less__one,axiom,
% 8.83/5.01 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2020_zero__less__one,axiom,
% 8.83/5.01 ord_less_int @ zero_zero_int @ one_one_int ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2021_not__one__less__zero,axiom,
% 8.83/5.01 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2022_not__one__less__zero,axiom,
% 8.83/5.01 ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2023_not__one__less__zero,axiom,
% 8.83/5.01 ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2024_not__one__less__zero,axiom,
% 8.83/5.01 ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2025_add__neg__neg,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ A @ zero_zero_real )
% 8.83/5.01 => ( ( ord_less_real @ B @ zero_zero_real )
% 8.83/5.01 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2026_add__neg__neg,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.83/5.01 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2027_add__neg__neg,axiom,
% 8.83/5.01 ! [A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_nat @ A @ zero_zero_nat )
% 8.83/5.01 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 8.83/5.01 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2028_add__neg__neg,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ A @ zero_zero_int )
% 8.83/5.01 => ( ( ord_less_int @ B @ zero_zero_int )
% 8.83/5.01 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2029_add__pos__pos,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ( ord_less_real @ zero_zero_real @ B )
% 8.83/5.01 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2030_add__pos__pos,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 8.83/5.01 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2031_add__pos__pos,axiom,
% 8.83/5.01 ! [A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.83/5.01 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2032_add__pos__pos,axiom,
% 8.83/5.01 ! [A: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ( ord_less_int @ zero_zero_int @ B )
% 8.83/5.01 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2033_canonically__ordered__monoid__add__class_OlessE,axiom,
% 8.83/5.01 ! [A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_nat @ A @ B )
% 8.83/5.01 => ~ ! [C3: nat] :
% 8.83/5.01 ( ( B
% 8.83/5.01 = ( plus_plus_nat @ A @ C3 ) )
% 8.83/5.01 => ( C3 = zero_zero_nat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2034_pos__add__strict,axiom,
% 8.83/5.01 ! [A: real,B: real,C: real] :
% 8.83/5.01 ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ( ord_less_real @ B @ C )
% 8.83/5.01 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2035_pos__add__strict,axiom,
% 8.83/5.01 ! [A: rat,B: rat,C: rat] :
% 8.83/5.01 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ( ord_less_rat @ B @ C )
% 8.83/5.01 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2036_pos__add__strict,axiom,
% 8.83/5.01 ! [A: nat,B: nat,C: nat] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ( ord_less_nat @ B @ C )
% 8.83/5.01 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2037_pos__add__strict,axiom,
% 8.83/5.01 ! [A: int,B: int,C: int] :
% 8.83/5.01 ( ( ord_less_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ( ord_less_int @ B @ C )
% 8.83/5.01 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2038_add__less__zeroD,axiom,
% 8.83/5.01 ! [X2: real,Y3: real] :
% 8.83/5.01 ( ( ord_less_real @ ( plus_plus_real @ X2 @ Y3 ) @ zero_zero_real )
% 8.83/5.01 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 8.83/5.01 | ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2039_add__less__zeroD,axiom,
% 8.83/5.01 ! [X2: rat,Y3: rat] :
% 8.83/5.01 ( ( ord_less_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ zero_zero_rat )
% 8.83/5.01 => ( ( ord_less_rat @ X2 @ zero_zero_rat )
% 8.83/5.01 | ( ord_less_rat @ Y3 @ zero_zero_rat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2040_add__less__zeroD,axiom,
% 8.83/5.01 ! [X2: int,Y3: int] :
% 8.83/5.01 ( ( ord_less_int @ ( plus_plus_int @ X2 @ Y3 ) @ zero_zero_int )
% 8.83/5.01 => ( ( ord_less_int @ X2 @ zero_zero_int )
% 8.83/5.01 | ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2041_le__iff__diff__le__0,axiom,
% 8.83/5.01 ( ord_less_eq_real
% 8.83/5.01 = ( ^ [A3: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2042_le__iff__diff__le__0,axiom,
% 8.83/5.01 ( ord_less_eq_rat
% 8.83/5.01 = ( ^ [A3: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B2 ) @ zero_zero_rat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2043_le__iff__diff__le__0,axiom,
% 8.83/5.01 ( ord_less_eq_int
% 8.83/5.01 = ( ^ [A3: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2044_divide__le__0__iff,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 8.83/5.01 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.01 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 8.83/5.01 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.83/5.01 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2045_divide__le__0__iff,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 8.83/5.01 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.01 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 8.83/5.01 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.83/5.01 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2046_divide__right__mono,axiom,
% 8.83/5.01 ! [A: real,B: real,C: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2047_divide__right__mono,axiom,
% 8.83/5.01 ! [A: rat,B: rat,C: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2048_zero__le__divide__iff,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 8.83/5.01 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.01 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 8.83/5.01 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.83/5.01 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2049_zero__le__divide__iff,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 8.83/5.01 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.01 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 8.83/5.01 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.83/5.01 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2050_divide__nonneg__nonneg,axiom,
% 8.83/5.01 ! [X2: real,Y3: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.83/5.01 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2051_divide__nonneg__nonneg,axiom,
% 8.83/5.01 ! [X2: rat,Y3: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 8.83/5.01 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2052_divide__nonneg__nonpos,axiom,
% 8.83/5.01 ! [X2: real,Y3: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.83/5.01 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 8.83/5.01 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2053_divide__nonneg__nonpos,axiom,
% 8.83/5.01 ! [X2: rat,Y3: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 8.83/5.01 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2054_divide__nonpos__nonneg,axiom,
% 8.83/5.01 ! [X2: real,Y3: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.83/5.01 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2055_divide__nonpos__nonneg,axiom,
% 8.83/5.01 ! [X2: rat,Y3: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 8.83/5.01 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2056_divide__nonpos__nonpos,axiom,
% 8.83/5.01 ! [X2: real,Y3: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 8.83/5.01 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 8.83/5.01 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2057_divide__nonpos__nonpos,axiom,
% 8.83/5.01 ! [X2: rat,Y3: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 8.83/5.01 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2058_divide__right__mono__neg,axiom,
% 8.83/5.01 ! [A: real,B: real,C: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.83/5.01 => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2059_divide__right__mono__neg,axiom,
% 8.83/5.01 ! [A: rat,B: rat,C: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2060_less__iff__diff__less__0,axiom,
% 8.83/5.01 ( ord_less_real
% 8.83/5.01 = ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2061_less__iff__diff__less__0,axiom,
% 8.83/5.01 ( ord_less_rat
% 8.83/5.01 = ( ^ [A3: rat,B2: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B2 ) @ zero_zero_rat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2062_less__iff__diff__less__0,axiom,
% 8.83/5.01 ( ord_less_int
% 8.83/5.01 = ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2063_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I1_J,axiom,
% 8.83/5.01 ! [A: $o,B: $o,X2: nat] :
% 8.83/5.01 ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Leaf @ A @ B ) @ X2 )
% 8.83/5.01 = one_one_nat ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2064_divide__strict__right__mono__neg,axiom,
% 8.83/5.01 ! [B: real,A: real,C: real] :
% 8.83/5.01 ( ( ord_less_real @ B @ A )
% 8.83/5.01 => ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.01 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2065_divide__strict__right__mono__neg,axiom,
% 8.83/5.01 ! [B: rat,A: rat,C: rat] :
% 8.83/5.01 ( ( ord_less_rat @ B @ A )
% 8.83/5.01 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2066_divide__strict__right__mono,axiom,
% 8.83/5.01 ! [A: real,B: real,C: real] :
% 8.83/5.01 ( ( ord_less_real @ A @ B )
% 8.83/5.01 => ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2067_divide__strict__right__mono,axiom,
% 8.83/5.01 ! [A: rat,B: rat,C: rat] :
% 8.83/5.01 ( ( ord_less_rat @ A @ B )
% 8.83/5.01 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2068_zero__less__divide__iff,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 8.83/5.01 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.01 & ( ord_less_real @ zero_zero_real @ B ) )
% 8.83/5.01 | ( ( ord_less_real @ A @ zero_zero_real )
% 8.83/5.01 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2069_zero__less__divide__iff,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 8.83/5.01 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.01 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 8.83/5.01 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.83/5.01 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2070_divide__less__cancel,axiom,
% 8.83/5.01 ! [A: real,C: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 8.83/5.01 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_real @ A @ B ) )
% 8.83/5.01 & ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.01 => ( ord_less_real @ B @ A ) )
% 8.83/5.01 & ( C != zero_zero_real ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2071_divide__less__cancel,axiom,
% 8.83/5.01 ! [A: rat,C: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 8.83/5.01 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_rat @ A @ B ) )
% 8.83/5.01 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_rat @ B @ A ) )
% 8.83/5.01 & ( C != zero_zero_rat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2072_divide__less__0__iff,axiom,
% 8.83/5.01 ! [A: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 8.83/5.01 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.01 & ( ord_less_real @ B @ zero_zero_real ) )
% 8.83/5.01 | ( ( ord_less_real @ A @ zero_zero_real )
% 8.83/5.01 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2073_divide__less__0__iff,axiom,
% 8.83/5.01 ! [A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 8.83/5.01 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.01 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 8.83/5.01 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.83/5.01 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2074_divide__pos__pos,axiom,
% 8.83/5.01 ! [X2: real,Y3: real] :
% 8.83/5.01 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.83/5.01 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.83/5.01 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2075_divide__pos__pos,axiom,
% 8.83/5.01 ! [X2: rat,Y3: rat] :
% 8.83/5.01 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 8.83/5.01 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 8.83/5.01 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2076_divide__pos__neg,axiom,
% 8.83/5.01 ! [X2: real,Y3: real] :
% 8.83/5.01 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.83/5.01 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 8.83/5.01 => ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2077_divide__pos__neg,axiom,
% 8.83/5.01 ! [X2: rat,Y3: rat] :
% 8.83/5.01 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 8.83/5.01 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2078_divide__neg__pos,axiom,
% 8.83/5.01 ! [X2: real,Y3: real] :
% 8.83/5.01 ( ( ord_less_real @ X2 @ zero_zero_real )
% 8.83/5.01 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.83/5.01 => ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2079_divide__neg__pos,axiom,
% 8.83/5.01 ! [X2: rat,Y3: rat] :
% 8.83/5.01 ( ( ord_less_rat @ X2 @ zero_zero_rat )
% 8.83/5.01 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 8.83/5.01 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2080_divide__neg__neg,axiom,
% 8.83/5.01 ! [X2: real,Y3: real] :
% 8.83/5.01 ( ( ord_less_real @ X2 @ zero_zero_real )
% 8.83/5.01 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 8.83/5.01 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2081_divide__neg__neg,axiom,
% 8.83/5.01 ! [X2: rat,Y3: rat] :
% 8.83/5.01 ( ( ord_less_rat @ X2 @ zero_zero_rat )
% 8.83/5.01 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2082_power__mono,axiom,
% 8.83/5.01 ! [A: real,B: real,N: nat] :
% 8.83/5.01 ( ( ord_less_eq_real @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2083_power__mono,axiom,
% 8.83/5.01 ! [A: rat,B: rat,N: nat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2084_power__mono,axiom,
% 8.83/5.01 ! [A: nat,B: nat,N: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2085_power__mono,axiom,
% 8.83/5.01 ! [A: int,B: int,N: nat] :
% 8.83/5.01 ( ( ord_less_eq_int @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2086_zero__le__power,axiom,
% 8.83/5.01 ! [A: real,N: nat] :
% 8.83/5.01 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2087_zero__le__power,axiom,
% 8.83/5.01 ! [A: rat,N: nat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2088_zero__le__power,axiom,
% 8.83/5.01 ! [A: nat,N: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2089_zero__le__power,axiom,
% 8.83/5.01 ! [A: int,N: nat] :
% 8.83/5.01 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2090_zero__less__power,axiom,
% 8.83/5.01 ! [A: real,N: nat] :
% 8.83/5.01 ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2091_zero__less__power,axiom,
% 8.83/5.01 ! [A: rat,N: nat] :
% 8.83/5.01 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2092_zero__less__power,axiom,
% 8.83/5.01 ! [A: nat,N: nat] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2093_zero__less__power,axiom,
% 8.83/5.01 ! [A: int,N: nat] :
% 8.83/5.01 ( ( ord_less_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2094_nonzero__eq__divide__eq,axiom,
% 8.83/5.01 ! [C: complex,A: complex,B: complex] :
% 8.83/5.01 ( ( C != zero_zero_complex )
% 8.83/5.01 => ( ( A
% 8.83/5.01 = ( divide1717551699836669952omplex @ B @ C ) )
% 8.83/5.01 = ( ( times_times_complex @ A @ C )
% 8.83/5.01 = B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2095_nonzero__eq__divide__eq,axiom,
% 8.83/5.01 ! [C: real,A: real,B: real] :
% 8.83/5.01 ( ( C != zero_zero_real )
% 8.83/5.01 => ( ( A
% 8.83/5.01 = ( divide_divide_real @ B @ C ) )
% 8.83/5.01 = ( ( times_times_real @ A @ C )
% 8.83/5.01 = B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2096_nonzero__eq__divide__eq,axiom,
% 8.83/5.01 ! [C: rat,A: rat,B: rat] :
% 8.83/5.01 ( ( C != zero_zero_rat )
% 8.83/5.01 => ( ( A
% 8.83/5.01 = ( divide_divide_rat @ B @ C ) )
% 8.83/5.01 = ( ( times_times_rat @ A @ C )
% 8.83/5.01 = B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2097_nonzero__divide__eq__eq,axiom,
% 8.83/5.01 ! [C: complex,B: complex,A: complex] :
% 8.83/5.01 ( ( C != zero_zero_complex )
% 8.83/5.01 => ( ( ( divide1717551699836669952omplex @ B @ C )
% 8.83/5.01 = A )
% 8.83/5.01 = ( B
% 8.83/5.01 = ( times_times_complex @ A @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2098_nonzero__divide__eq__eq,axiom,
% 8.83/5.01 ! [C: real,B: real,A: real] :
% 8.83/5.01 ( ( C != zero_zero_real )
% 8.83/5.01 => ( ( ( divide_divide_real @ B @ C )
% 8.83/5.01 = A )
% 8.83/5.01 = ( B
% 8.83/5.01 = ( times_times_real @ A @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2099_nonzero__divide__eq__eq,axiom,
% 8.83/5.01 ! [C: rat,B: rat,A: rat] :
% 8.83/5.01 ( ( C != zero_zero_rat )
% 8.83/5.01 => ( ( ( divide_divide_rat @ B @ C )
% 8.83/5.01 = A )
% 8.83/5.01 = ( B
% 8.83/5.01 = ( times_times_rat @ A @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2100_eq__divide__imp,axiom,
% 8.83/5.01 ! [C: complex,A: complex,B: complex] :
% 8.83/5.01 ( ( C != zero_zero_complex )
% 8.83/5.01 => ( ( ( times_times_complex @ A @ C )
% 8.83/5.01 = B )
% 8.83/5.01 => ( A
% 8.83/5.01 = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2101_eq__divide__imp,axiom,
% 8.83/5.01 ! [C: real,A: real,B: real] :
% 8.83/5.01 ( ( C != zero_zero_real )
% 8.83/5.01 => ( ( ( times_times_real @ A @ C )
% 8.83/5.01 = B )
% 8.83/5.01 => ( A
% 8.83/5.01 = ( divide_divide_real @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2102_eq__divide__imp,axiom,
% 8.83/5.01 ! [C: rat,A: rat,B: rat] :
% 8.83/5.01 ( ( C != zero_zero_rat )
% 8.83/5.01 => ( ( ( times_times_rat @ A @ C )
% 8.83/5.01 = B )
% 8.83/5.01 => ( A
% 8.83/5.01 = ( divide_divide_rat @ B @ C ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2103_divide__eq__imp,axiom,
% 8.83/5.01 ! [C: complex,B: complex,A: complex] :
% 8.83/5.01 ( ( C != zero_zero_complex )
% 8.83/5.01 => ( ( B
% 8.83/5.01 = ( times_times_complex @ A @ C ) )
% 8.83/5.01 => ( ( divide1717551699836669952omplex @ B @ C )
% 8.83/5.01 = A ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2104_divide__eq__imp,axiom,
% 8.83/5.01 ! [C: real,B: real,A: real] :
% 8.83/5.01 ( ( C != zero_zero_real )
% 8.83/5.01 => ( ( B
% 8.83/5.01 = ( times_times_real @ A @ C ) )
% 8.83/5.01 => ( ( divide_divide_real @ B @ C )
% 8.83/5.01 = A ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2105_divide__eq__imp,axiom,
% 8.83/5.01 ! [C: rat,B: rat,A: rat] :
% 8.83/5.01 ( ( C != zero_zero_rat )
% 8.83/5.01 => ( ( B
% 8.83/5.01 = ( times_times_rat @ A @ C ) )
% 8.83/5.01 => ( ( divide_divide_rat @ B @ C )
% 8.83/5.01 = A ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2106_eq__divide__eq,axiom,
% 8.83/5.01 ! [A: complex,B: complex,C: complex] :
% 8.83/5.01 ( ( A
% 8.83/5.01 = ( divide1717551699836669952omplex @ B @ C ) )
% 8.83/5.01 = ( ( ( C != zero_zero_complex )
% 8.83/5.01 => ( ( times_times_complex @ A @ C )
% 8.83/5.01 = B ) )
% 8.83/5.01 & ( ( C = zero_zero_complex )
% 8.83/5.01 => ( A = zero_zero_complex ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2107_eq__divide__eq,axiom,
% 8.83/5.01 ! [A: real,B: real,C: real] :
% 8.83/5.01 ( ( A
% 8.83/5.01 = ( divide_divide_real @ B @ C ) )
% 8.83/5.01 = ( ( ( C != zero_zero_real )
% 8.83/5.01 => ( ( times_times_real @ A @ C )
% 8.83/5.01 = B ) )
% 8.83/5.01 & ( ( C = zero_zero_real )
% 8.83/5.01 => ( A = zero_zero_real ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2108_eq__divide__eq,axiom,
% 8.83/5.01 ! [A: rat,B: rat,C: rat] :
% 8.83/5.01 ( ( A
% 8.83/5.01 = ( divide_divide_rat @ B @ C ) )
% 8.83/5.01 = ( ( ( C != zero_zero_rat )
% 8.83/5.01 => ( ( times_times_rat @ A @ C )
% 8.83/5.01 = B ) )
% 8.83/5.01 & ( ( C = zero_zero_rat )
% 8.83/5.01 => ( A = zero_zero_rat ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2109_divide__eq__eq,axiom,
% 8.83/5.01 ! [B: complex,C: complex,A: complex] :
% 8.83/5.01 ( ( ( divide1717551699836669952omplex @ B @ C )
% 8.83/5.01 = A )
% 8.83/5.01 = ( ( ( C != zero_zero_complex )
% 8.83/5.01 => ( B
% 8.83/5.01 = ( times_times_complex @ A @ C ) ) )
% 8.83/5.01 & ( ( C = zero_zero_complex )
% 8.83/5.01 => ( A = zero_zero_complex ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2110_divide__eq__eq,axiom,
% 8.83/5.01 ! [B: real,C: real,A: real] :
% 8.83/5.01 ( ( ( divide_divide_real @ B @ C )
% 8.83/5.01 = A )
% 8.83/5.01 = ( ( ( C != zero_zero_real )
% 8.83/5.01 => ( B
% 8.83/5.01 = ( times_times_real @ A @ C ) ) )
% 8.83/5.01 & ( ( C = zero_zero_real )
% 8.83/5.01 => ( A = zero_zero_real ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2111_divide__eq__eq,axiom,
% 8.83/5.01 ! [B: rat,C: rat,A: rat] :
% 8.83/5.01 ( ( ( divide_divide_rat @ B @ C )
% 8.83/5.01 = A )
% 8.83/5.01 = ( ( ( C != zero_zero_rat )
% 8.83/5.01 => ( B
% 8.83/5.01 = ( times_times_rat @ A @ C ) ) )
% 8.83/5.01 & ( ( C = zero_zero_rat )
% 8.83/5.01 => ( A = zero_zero_rat ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2112_frac__eq__eq,axiom,
% 8.83/5.01 ! [Y3: complex,Z: complex,X2: complex,W: complex] :
% 8.83/5.01 ( ( Y3 != zero_zero_complex )
% 8.83/5.01 => ( ( Z != zero_zero_complex )
% 8.83/5.01 => ( ( ( divide1717551699836669952omplex @ X2 @ Y3 )
% 8.83/5.01 = ( divide1717551699836669952omplex @ W @ Z ) )
% 8.83/5.01 = ( ( times_times_complex @ X2 @ Z )
% 8.83/5.01 = ( times_times_complex @ W @ Y3 ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2113_frac__eq__eq,axiom,
% 8.83/5.01 ! [Y3: real,Z: real,X2: real,W: real] :
% 8.83/5.01 ( ( Y3 != zero_zero_real )
% 8.83/5.01 => ( ( Z != zero_zero_real )
% 8.83/5.01 => ( ( ( divide_divide_real @ X2 @ Y3 )
% 8.83/5.01 = ( divide_divide_real @ W @ Z ) )
% 8.83/5.01 = ( ( times_times_real @ X2 @ Z )
% 8.83/5.01 = ( times_times_real @ W @ Y3 ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2114_frac__eq__eq,axiom,
% 8.83/5.01 ! [Y3: rat,Z: rat,X2: rat,W: rat] :
% 8.83/5.01 ( ( Y3 != zero_zero_rat )
% 8.83/5.01 => ( ( Z != zero_zero_rat )
% 8.83/5.01 => ( ( ( divide_divide_rat @ X2 @ Y3 )
% 8.83/5.01 = ( divide_divide_rat @ W @ Z ) )
% 8.83/5.01 = ( ( times_times_rat @ X2 @ Z )
% 8.83/5.01 = ( times_times_rat @ W @ Y3 ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2115_right__inverse__eq,axiom,
% 8.83/5.01 ! [B: complex,A: complex] :
% 8.83/5.01 ( ( B != zero_zero_complex )
% 8.83/5.01 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 8.83/5.01 = one_one_complex )
% 8.83/5.01 = ( A = B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2116_right__inverse__eq,axiom,
% 8.83/5.01 ! [B: real,A: real] :
% 8.83/5.01 ( ( B != zero_zero_real )
% 8.83/5.01 => ( ( ( divide_divide_real @ A @ B )
% 8.83/5.01 = one_one_real )
% 8.83/5.01 = ( A = B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2117_right__inverse__eq,axiom,
% 8.83/5.01 ! [B: rat,A: rat] :
% 8.83/5.01 ( ( B != zero_zero_rat )
% 8.83/5.01 => ( ( ( divide_divide_rat @ A @ B )
% 8.83/5.01 = one_one_rat )
% 8.83/5.01 = ( A = B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2118_length__pos__if__in__set,axiom,
% 8.83/5.01 ! [X2: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
% 8.83/5.01 ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
% 8.83/5.01 => ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2119_length__pos__if__in__set,axiom,
% 8.83/5.01 ! [X2: complex,Xs2: list_complex] :
% 8.83/5.01 ( ( member_complex @ X2 @ ( set_complex2 @ Xs2 ) )
% 8.83/5.01 => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2120_length__pos__if__in__set,axiom,
% 8.83/5.01 ! [X2: real,Xs2: list_real] :
% 8.83/5.01 ( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
% 8.83/5.01 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2121_length__pos__if__in__set,axiom,
% 8.83/5.01 ! [X2: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
% 8.83/5.01 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 8.83/5.01 => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2122_length__pos__if__in__set,axiom,
% 8.83/5.01 ! [X2: $o,Xs2: list_o] :
% 8.83/5.01 ( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
% 8.83/5.01 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2123_length__pos__if__in__set,axiom,
% 8.83/5.01 ! [X2: nat,Xs2: list_nat] :
% 8.83/5.01 ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
% 8.83/5.01 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2124_length__pos__if__in__set,axiom,
% 8.83/5.01 ! [X2: int,Xs2: list_int] :
% 8.83/5.01 ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
% 8.83/5.01 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2125_power__0,axiom,
% 8.83/5.01 ! [A: rat] :
% 8.83/5.01 ( ( power_power_rat @ A @ zero_zero_nat )
% 8.83/5.01 = one_one_rat ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2126_power__0,axiom,
% 8.83/5.01 ! [A: nat] :
% 8.83/5.01 ( ( power_power_nat @ A @ zero_zero_nat )
% 8.83/5.01 = one_one_nat ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2127_power__0,axiom,
% 8.83/5.01 ! [A: real] :
% 8.83/5.01 ( ( power_power_real @ A @ zero_zero_nat )
% 8.83/5.01 = one_one_real ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2128_power__0,axiom,
% 8.83/5.01 ! [A: int] :
% 8.83/5.01 ( ( power_power_int @ A @ zero_zero_nat )
% 8.83/5.01 = one_one_int ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2129_power__0,axiom,
% 8.83/5.01 ! [A: complex] :
% 8.83/5.01 ( ( power_power_complex @ A @ zero_zero_nat )
% 8.83/5.01 = one_one_complex ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2130_less__Suc__eq__0__disj,axiom,
% 8.83/5.01 ! [M: nat,N: nat] :
% 8.83/5.01 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 8.83/5.01 = ( ( M = zero_zero_nat )
% 8.83/5.01 | ? [J3: nat] :
% 8.83/5.01 ( ( M
% 8.83/5.01 = ( suc @ J3 ) )
% 8.83/5.01 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2131_gr0__implies__Suc,axiom,
% 8.83/5.01 ! [N: nat] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.01 => ? [M4: nat] :
% 8.83/5.01 ( N
% 8.83/5.01 = ( suc @ M4 ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2132_All__less__Suc2,axiom,
% 8.83/5.01 ! [N: nat,P: nat > $o] :
% 8.83/5.01 ( ( ! [I4: nat] :
% 8.83/5.01 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 8.83/5.01 => ( P @ I4 ) ) )
% 8.83/5.01 = ( ( P @ zero_zero_nat )
% 8.83/5.01 & ! [I4: nat] :
% 8.83/5.01 ( ( ord_less_nat @ I4 @ N )
% 8.83/5.01 => ( P @ ( suc @ I4 ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2133_gr0__conv__Suc,axiom,
% 8.83/5.01 ! [N: nat] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.01 = ( ? [M3: nat] :
% 8.83/5.01 ( N
% 8.83/5.01 = ( suc @ M3 ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2134_Ex__less__Suc2,axiom,
% 8.83/5.01 ! [N: nat,P: nat > $o] :
% 8.83/5.01 ( ( ? [I4: nat] :
% 8.83/5.01 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 8.83/5.01 & ( P @ I4 ) ) )
% 8.83/5.01 = ( ( P @ zero_zero_nat )
% 8.83/5.01 | ? [I4: nat] :
% 8.83/5.01 ( ( ord_less_nat @ I4 @ N )
% 8.83/5.01 & ( P @ ( suc @ I4 ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2135_one__is__add,axiom,
% 8.83/5.01 ! [M: nat,N: nat] :
% 8.83/5.01 ( ( ( suc @ zero_zero_nat )
% 8.83/5.01 = ( plus_plus_nat @ M @ N ) )
% 8.83/5.01 = ( ( ( M
% 8.83/5.01 = ( suc @ zero_zero_nat ) )
% 8.83/5.01 & ( N = zero_zero_nat ) )
% 8.83/5.01 | ( ( M = zero_zero_nat )
% 8.83/5.01 & ( N
% 8.83/5.01 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2136_add__is__1,axiom,
% 8.83/5.01 ! [M: nat,N: nat] :
% 8.83/5.01 ( ( ( plus_plus_nat @ M @ N )
% 8.83/5.01 = ( suc @ zero_zero_nat ) )
% 8.83/5.01 = ( ( ( M
% 8.83/5.01 = ( suc @ zero_zero_nat ) )
% 8.83/5.01 & ( N = zero_zero_nat ) )
% 8.83/5.01 | ( ( M = zero_zero_nat )
% 8.83/5.01 & ( N
% 8.83/5.01 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2137_option_Osize_I4_J,axiom,
% 8.83/5.01 ! [X22: product_prod_nat_nat] :
% 8.83/5.01 ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 8.83/5.01 = ( suc @ zero_zero_nat ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2138_option_Osize_I4_J,axiom,
% 8.83/5.01 ! [X22: nat] :
% 8.83/5.01 ( ( size_size_option_nat @ ( some_nat @ X22 ) )
% 8.83/5.01 = ( suc @ zero_zero_nat ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2139_option_Osize_I4_J,axiom,
% 8.83/5.01 ! [X22: num] :
% 8.83/5.01 ( ( size_size_option_num @ ( some_num @ X22 ) )
% 8.83/5.01 = ( suc @ zero_zero_nat ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2140_ex__least__nat__le,axiom,
% 8.83/5.01 ! [P: nat > $o,N: nat] :
% 8.83/5.01 ( ( P @ N )
% 8.83/5.01 => ( ~ ( P @ zero_zero_nat )
% 8.83/5.01 => ? [K3: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ K3 @ N )
% 8.83/5.01 & ! [I: nat] :
% 8.83/5.01 ( ( ord_less_nat @ I @ K3 )
% 8.83/5.01 => ~ ( P @ I ) )
% 8.83/5.01 & ( P @ K3 ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2141_less__imp__add__positive,axiom,
% 8.83/5.01 ! [I2: nat,J: nat] :
% 8.83/5.01 ( ( ord_less_nat @ I2 @ J )
% 8.83/5.01 => ? [K3: nat] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 8.83/5.01 & ( ( plus_plus_nat @ I2 @ K3 )
% 8.83/5.01 = J ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2142_One__nat__def,axiom,
% 8.83/5.01 ( one_one_nat
% 8.83/5.01 = ( suc @ zero_zero_nat ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2143_option_Osize_I3_J,axiom,
% 8.83/5.01 ( ( size_size_option_nat @ none_nat )
% 8.83/5.01 = ( suc @ zero_zero_nat ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2144_option_Osize_I3_J,axiom,
% 8.83/5.01 ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
% 8.83/5.01 = ( suc @ zero_zero_nat ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2145_option_Osize_I3_J,axiom,
% 8.83/5.01 ( ( size_size_option_num @ none_num )
% 8.83/5.01 = ( suc @ zero_zero_nat ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2146_diff__less,axiom,
% 8.83/5.01 ! [N: nat,M: nat] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.83/5.01 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2147_mult__less__mono2,axiom,
% 8.83/5.01 ! [I2: nat,J: nat,K: nat] :
% 8.83/5.01 ( ( ord_less_nat @ I2 @ J )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.83/5.01 => ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2148_mult__less__mono1,axiom,
% 8.83/5.01 ! [I2: nat,J: nat,K: nat] :
% 8.83/5.01 ( ( ord_less_nat @ I2 @ J )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.83/5.01 => ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2149_nat__mult__less__cancel1,axiom,
% 8.83/5.01 ! [K: nat,M: nat,N: nat] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.83/5.01 => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.83/5.01 = ( ord_less_nat @ M @ N ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2150_nat__mult__eq__cancel1,axiom,
% 8.83/5.01 ! [K: nat,M: nat,N: nat] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.83/5.01 => ( ( ( times_times_nat @ K @ M )
% 8.83/5.01 = ( times_times_nat @ K @ N ) )
% 8.83/5.01 = ( M = N ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2151_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 8.83/5.01 ! [M: nat,N: nat] :
% 8.83/5.01 ( ( ( divide_divide_nat @ M @ N )
% 8.83/5.01 = zero_zero_nat )
% 8.83/5.01 = ( ( ord_less_nat @ M @ N )
% 8.83/5.01 | ( N = zero_zero_nat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2152_diff__add__0,axiom,
% 8.83/5.01 ! [N: nat,M: nat] :
% 8.83/5.01 ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 8.83/5.01 = zero_zero_nat ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2153_nat__power__less__imp__less,axiom,
% 8.83/5.01 ! [I2: nat,M: nat,N: nat] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ I2 )
% 8.83/5.01 => ( ( ord_less_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N ) )
% 8.83/5.01 => ( ord_less_nat @ M @ N ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2154_mult__eq__self__implies__10,axiom,
% 8.83/5.01 ! [M: nat,N: nat] :
% 8.83/5.01 ( ( M
% 8.83/5.01 = ( times_times_nat @ M @ N ) )
% 8.83/5.01 => ( ( N = one_one_nat )
% 8.83/5.01 | ( M = zero_zero_nat ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2155_vebt__insert_Osimps_I2_J,axiom,
% 8.83/5.01 ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
% 8.83/5.01 ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S ) @ X2 )
% 8.83/5.01 = ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2156_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
% 8.83/5.01 ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
% 8.83/5.01 ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2157_vebt__pred_Osimps_I3_J,axiom,
% 8.83/5.01 ! [B: $o,A: $o,Va: nat] :
% 8.83/5.01 ( ( B
% 8.83/5.01 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
% 8.83/5.01 = ( some_nat @ one_one_nat ) ) )
% 8.83/5.01 & ( ~ B
% 8.83/5.01 => ( ( A
% 8.83/5.01 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
% 8.83/5.01 = ( some_nat @ zero_zero_nat ) ) )
% 8.83/5.01 & ( ~ A
% 8.83/5.01 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
% 8.83/5.01 = none_nat ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2158_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
% 8.83/5.01 ! [X2: vEBT_VEBT] :
% 8.83/5.01 ( ( X2
% 8.83/5.01 != ( vEBT_Leaf @ $false @ $false ) )
% 8.83/5.01 => ( ! [Uv2: $o] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( vEBT_Leaf @ $true @ Uv2 ) )
% 8.83/5.01 => ( ! [Uu2: $o] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( vEBT_Leaf @ Uu2 @ $true ) )
% 8.83/5.01 => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 8.83/5.01 => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2159_div__geq,axiom,
% 8.83/5.01 ! [N: nat,M: nat] :
% 8.83/5.01 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.01 => ( ~ ( ord_less_nat @ M @ N )
% 8.83/5.01 => ( ( divide_divide_nat @ M @ N )
% 8.83/5.01 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2160_VEBT__internal_OminNull_Oelims_I3_J,axiom,
% 8.83/5.01 ! [X2: vEBT_VEBT] :
% 8.83/5.01 ( ~ ( vEBT_VEBT_minNull @ X2 )
% 8.83/5.01 => ( ! [Uv2: $o] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( vEBT_Leaf @ $true @ Uv2 ) )
% 8.83/5.01 => ( ! [Uu2: $o] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( vEBT_Leaf @ Uu2 @ $true ) )
% 8.83/5.01 => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.83/5.01 ( X2
% 8.83/5.01 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2161_vebt__succ_Osimps_I2_J,axiom,
% 8.83/5.01 ! [Uv: $o,Uw: $o,N: nat] :
% 8.83/5.01 ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
% 8.83/5.01 = none_nat ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2162_mult__less__le__imp__less,axiom,
% 8.83/5.01 ! [A: real,B: real,C: real,D: real] :
% 8.83/5.01 ( ( ord_less_real @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_real @ C @ D )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2163_mult__less__le__imp__less,axiom,
% 8.83/5.01 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.83/5.01 ( ( ord_less_rat @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_rat @ C @ D )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2164_mult__less__le__imp__less,axiom,
% 8.83/5.01 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.83/5.01 ( ( ord_less_nat @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_nat @ C @ D )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 8.83/5.01 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2165_mult__less__le__imp__less,axiom,
% 8.83/5.01 ! [A: int,B: int,C: int,D: int] :
% 8.83/5.01 ( ( ord_less_int @ A @ B )
% 8.83/5.01 => ( ( ord_less_eq_int @ C @ D )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ( ord_less_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2166_mult__le__less__imp__less,axiom,
% 8.83/5.01 ! [A: real,B: real,C: real,D: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ A @ B )
% 8.83/5.01 => ( ( ord_less_real @ C @ D )
% 8.83/5.01 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2167_mult__le__less__imp__less,axiom,
% 8.83/5.01 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ A @ B )
% 8.83/5.01 => ( ( ord_less_rat @ C @ D )
% 8.83/5.01 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2168_mult__le__less__imp__less,axiom,
% 8.83/5.01 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ A @ B )
% 8.83/5.01 => ( ( ord_less_nat @ C @ D )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.83/5.01 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2169_mult__le__less__imp__less,axiom,
% 8.83/5.01 ! [A: int,B: int,C: int,D: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ A @ B )
% 8.83/5.01 => ( ( ord_less_int @ C @ D )
% 8.83/5.01 => ( ( ord_less_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2170_mult__right__le__imp__le,axiom,
% 8.83/5.01 ! [A: real,C: real,B: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 8.83/5.01 => ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2171_mult__right__le__imp__le,axiom,
% 8.83/5.01 ! [A: rat,C: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 8.83/5.01 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2172_mult__right__le__imp__le,axiom,
% 8.83/5.01 ! [A: nat,C: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 8.83/5.01 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2173_mult__right__le__imp__le,axiom,
% 8.83/5.01 ! [A: int,C: int,B: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 8.83/5.01 => ( ( ord_less_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2174_mult__left__le__imp__le,axiom,
% 8.83/5.01 ! [C: real,A: real,B: real] :
% 8.83/5.01 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.83/5.01 => ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2175_mult__left__le__imp__le,axiom,
% 8.83/5.01 ! [C: rat,A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.83/5.01 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2176_mult__left__le__imp__le,axiom,
% 8.83/5.01 ! [C: nat,A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 8.83/5.01 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2177_mult__left__le__imp__le,axiom,
% 8.83/5.01 ! [C: int,A: int,B: int] :
% 8.83/5.01 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.83/5.01 => ( ( ord_less_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2178_mult__le__cancel__left__pos,axiom,
% 8.83/5.01 ! [C: real,A: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.83/5.01 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2179_mult__le__cancel__left__pos,axiom,
% 8.83/5.01 ! [C: rat,A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.83/5.01 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2180_mult__le__cancel__left__pos,axiom,
% 8.83/5.01 ! [C: int,A: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.83/5.01 = ( ord_less_eq_int @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2181_mult__le__cancel__left__neg,axiom,
% 8.83/5.01 ! [C: real,A: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.01 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.83/5.01 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2182_mult__le__cancel__left__neg,axiom,
% 8.83/5.01 ! [C: rat,A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.01 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.83/5.01 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2183_mult__le__cancel__left__neg,axiom,
% 8.83/5.01 ! [C: int,A: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ C @ zero_zero_int )
% 8.83/5.01 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.83/5.01 = ( ord_less_eq_int @ B @ A ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2184_mult__less__cancel__right,axiom,
% 8.83/5.01 ! [A: real,C: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 8.83/5.01 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_real @ A @ B ) )
% 8.83/5.01 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.83/5.01 => ( ord_less_real @ B @ A ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2185_mult__less__cancel__right,axiom,
% 8.83/5.01 ! [A: rat,C: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 8.83/5.01 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_rat @ A @ B ) )
% 8.83/5.01 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_rat @ B @ A ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2186_mult__less__cancel__right,axiom,
% 8.83/5.01 ! [A: int,C: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 8.83/5.01 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ord_less_int @ A @ B ) )
% 8.83/5.01 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 8.83/5.01 => ( ord_less_int @ B @ A ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2187_mult__strict__mono_H,axiom,
% 8.83/5.01 ! [A: real,B: real,C: real,D: real] :
% 8.83/5.01 ( ( ord_less_real @ A @ B )
% 8.83/5.01 => ( ( ord_less_real @ C @ D )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2188_mult__strict__mono_H,axiom,
% 8.83/5.01 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.83/5.01 ( ( ord_less_rat @ A @ B )
% 8.83/5.01 => ( ( ord_less_rat @ C @ D )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2189_mult__strict__mono_H,axiom,
% 8.83/5.01 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.83/5.01 ( ( ord_less_nat @ A @ B )
% 8.83/5.01 => ( ( ord_less_nat @ C @ D )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.83/5.01 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2190_mult__strict__mono_H,axiom,
% 8.83/5.01 ! [A: int,B: int,C: int,D: int] :
% 8.83/5.01 ( ( ord_less_int @ A @ B )
% 8.83/5.01 => ( ( ord_less_int @ C @ D )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2191_mult__right__less__imp__less,axiom,
% 8.83/5.01 ! [A: real,C: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_real @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2192_mult__right__less__imp__less,axiom,
% 8.83/5.01 ! [A: rat,C: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_rat @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2193_mult__right__less__imp__less,axiom,
% 8.83/5.01 ! [A: nat,C: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.83/5.01 => ( ord_less_nat @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2194_mult__right__less__imp__less,axiom,
% 8.83/5.01 ! [A: int,C: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ord_less_int @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2195_mult__less__cancel__left,axiom,
% 8.83/5.01 ! [C: real,A: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.83/5.01 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_real @ A @ B ) )
% 8.83/5.01 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.83/5.01 => ( ord_less_real @ B @ A ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2196_mult__less__cancel__left,axiom,
% 8.83/5.01 ! [C: rat,A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.83/5.01 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_rat @ A @ B ) )
% 8.83/5.01 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.83/5.01 => ( ord_less_rat @ B @ A ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2197_mult__less__cancel__left,axiom,
% 8.83/5.01 ! [C: int,A: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.83/5.01 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ord_less_int @ A @ B ) )
% 8.83/5.01 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 8.83/5.01 => ( ord_less_int @ B @ A ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2198_mult__strict__mono,axiom,
% 8.83/5.01 ! [A: real,B: real,C: real,D: real] :
% 8.83/5.01 ( ( ord_less_real @ A @ B )
% 8.83/5.01 => ( ( ord_less_real @ C @ D )
% 8.83/5.01 => ( ( ord_less_real @ zero_zero_real @ B )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2199_mult__strict__mono,axiom,
% 8.83/5.01 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.83/5.01 ( ( ord_less_rat @ A @ B )
% 8.83/5.01 => ( ( ord_less_rat @ C @ D )
% 8.83/5.01 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2200_mult__strict__mono,axiom,
% 8.83/5.01 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.83/5.01 ( ( ord_less_nat @ A @ B )
% 8.83/5.01 => ( ( ord_less_nat @ C @ D )
% 8.83/5.01 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.83/5.01 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2201_mult__strict__mono,axiom,
% 8.83/5.01 ! [A: int,B: int,C: int,D: int] :
% 8.83/5.01 ( ( ord_less_int @ A @ B )
% 8.83/5.01 => ( ( ord_less_int @ C @ D )
% 8.83/5.01 => ( ( ord_less_int @ zero_zero_int @ B )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2202_mult__left__less__imp__less,axiom,
% 8.83/5.01 ! [C: real,A: real,B: real] :
% 8.83/5.01 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.83/5.01 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.83/5.01 => ( ord_less_real @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2203_mult__left__less__imp__less,axiom,
% 8.83/5.01 ! [C: rat,A: rat,B: rat] :
% 8.83/5.01 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.83/5.01 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.83/5.01 => ( ord_less_rat @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2204_mult__left__less__imp__less,axiom,
% 8.83/5.01 ! [C: nat,A: nat,B: nat] :
% 8.83/5.01 ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 8.83/5.01 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.83/5.01 => ( ord_less_nat @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2205_mult__left__less__imp__less,axiom,
% 8.83/5.01 ! [C: int,A: int,B: int] :
% 8.83/5.01 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.83/5.01 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.83/5.01 => ( ord_less_int @ A @ B ) ) ) ).
% 8.83/5.01
% 8.83/5.01 thf(fact_2206_mult__le__cancel__right,axiom,
% 8.83/5.02 ! [A: real,C: real,B: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 8.83/5.02 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ord_less_eq_real @ A @ B ) )
% 8.83/5.02 & ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2207_mult__le__cancel__right,axiom,
% 8.83/5.02 ! [A: rat,C: rat,B: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 8.83/5.02 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ord_less_eq_rat @ A @ B ) )
% 8.83/5.02 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2208_mult__le__cancel__right,axiom,
% 8.83/5.02 ! [A: int,C: int,B: int] :
% 8.83/5.02 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 8.83/5.02 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 8.83/5.02 => ( ord_less_eq_int @ A @ B ) )
% 8.83/5.02 & ( ( ord_less_int @ C @ zero_zero_int )
% 8.83/5.02 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2209_mult__le__cancel__left,axiom,
% 8.83/5.02 ! [C: real,A: real,B: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.83/5.02 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ord_less_eq_real @ A @ B ) )
% 8.83/5.02 & ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2210_mult__le__cancel__left,axiom,
% 8.83/5.02 ! [C: rat,A: rat,B: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.83/5.02 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ord_less_eq_rat @ A @ B ) )
% 8.83/5.02 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2211_mult__le__cancel__left,axiom,
% 8.83/5.02 ! [C: int,A: int,B: int] :
% 8.83/5.02 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.83/5.02 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 8.83/5.02 => ( ord_less_eq_int @ A @ B ) )
% 8.83/5.02 & ( ( ord_less_int @ C @ zero_zero_int )
% 8.83/5.02 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2212_VEBT__internal_OminNull_Oelims_I2_J,axiom,
% 8.83/5.02 ! [X2: vEBT_VEBT] :
% 8.83/5.02 ( ( vEBT_VEBT_minNull @ X2 )
% 8.83/5.02 => ( ( X2
% 8.83/5.02 != ( vEBT_Leaf @ $false @ $false ) )
% 8.83/5.02 => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 8.83/5.02 ( X2
% 8.83/5.02 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2213_field__le__epsilon,axiom,
% 8.83/5.02 ! [X2: real,Y3: real] :
% 8.83/5.02 ( ! [E2: real] :
% 8.83/5.02 ( ( ord_less_real @ zero_zero_real @ E2 )
% 8.83/5.02 => ( ord_less_eq_real @ X2 @ ( plus_plus_real @ Y3 @ E2 ) ) )
% 8.83/5.02 => ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2214_field__le__epsilon,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat] :
% 8.83/5.02 ( ! [E2: rat] :
% 8.83/5.02 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 8.83/5.02 => ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ Y3 @ E2 ) ) )
% 8.83/5.02 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2215_add__neg__nonpos,axiom,
% 8.83/5.02 ! [A: real,B: real] :
% 8.83/5.02 ( ( ord_less_real @ A @ zero_zero_real )
% 8.83/5.02 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 8.83/5.02 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2216_add__neg__nonpos,axiom,
% 8.83/5.02 ! [A: rat,B: rat] :
% 8.83/5.02 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.83/5.02 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2217_add__neg__nonpos,axiom,
% 8.83/5.02 ! [A: nat,B: nat] :
% 8.83/5.02 ( ( ord_less_nat @ A @ zero_zero_nat )
% 8.83/5.02 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 8.83/5.02 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2218_add__neg__nonpos,axiom,
% 8.83/5.02 ! [A: int,B: int] :
% 8.83/5.02 ( ( ord_less_int @ A @ zero_zero_int )
% 8.83/5.02 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 8.83/5.02 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2219_add__nonneg__pos,axiom,
% 8.83/5.02 ! [A: real,B: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.02 => ( ( ord_less_real @ zero_zero_real @ B )
% 8.83/5.02 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2220_add__nonneg__pos,axiom,
% 8.83/5.02 ! [A: rat,B: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.02 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 8.83/5.02 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2221_add__nonneg__pos,axiom,
% 8.83/5.02 ! [A: nat,B: nat] :
% 8.83/5.02 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.02 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.83/5.02 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2222_add__nonneg__pos,axiom,
% 8.83/5.02 ! [A: int,B: int] :
% 8.83/5.02 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.02 => ( ( ord_less_int @ zero_zero_int @ B )
% 8.83/5.02 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2223_add__nonpos__neg,axiom,
% 8.83/5.02 ! [A: real,B: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.83/5.02 => ( ( ord_less_real @ B @ zero_zero_real )
% 8.83/5.02 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2224_add__nonpos__neg,axiom,
% 8.83/5.02 ! [A: rat,B: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.83/5.02 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2225_add__nonpos__neg,axiom,
% 8.83/5.02 ! [A: nat,B: nat] :
% 8.83/5.02 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 8.83/5.02 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 8.83/5.02 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2226_add__nonpos__neg,axiom,
% 8.83/5.02 ! [A: int,B: int] :
% 8.83/5.02 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.83/5.02 => ( ( ord_less_int @ B @ zero_zero_int )
% 8.83/5.02 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2227_add__pos__nonneg,axiom,
% 8.83/5.02 ! [A: real,B: real] :
% 8.83/5.02 ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.83/5.02 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2228_add__pos__nonneg,axiom,
% 8.83/5.02 ! [A: rat,B: rat] :
% 8.83/5.02 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.83/5.02 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2229_add__pos__nonneg,axiom,
% 8.83/5.02 ! [A: nat,B: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.83/5.02 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.83/5.02 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2230_add__pos__nonneg,axiom,
% 8.83/5.02 ! [A: int,B: int] :
% 8.83/5.02 ( ( ord_less_int @ zero_zero_int @ A )
% 8.83/5.02 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.83/5.02 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2231_add__strict__increasing,axiom,
% 8.83/5.02 ! [A: real,B: real,C: real] :
% 8.83/5.02 ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.02 => ( ( ord_less_eq_real @ B @ C )
% 8.83/5.02 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2232_add__strict__increasing,axiom,
% 8.83/5.02 ! [A: rat,B: rat,C: rat] :
% 8.83/5.02 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.02 => ( ( ord_less_eq_rat @ B @ C )
% 8.83/5.02 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2233_add__strict__increasing,axiom,
% 8.83/5.02 ! [A: nat,B: nat,C: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.83/5.02 => ( ( ord_less_eq_nat @ B @ C )
% 8.83/5.02 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2234_add__strict__increasing,axiom,
% 8.83/5.02 ! [A: int,B: int,C: int] :
% 8.83/5.02 ( ( ord_less_int @ zero_zero_int @ A )
% 8.83/5.02 => ( ( ord_less_eq_int @ B @ C )
% 8.83/5.02 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2235_add__strict__increasing2,axiom,
% 8.83/5.02 ! [A: real,B: real,C: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.02 => ( ( ord_less_real @ B @ C )
% 8.83/5.02 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2236_add__strict__increasing2,axiom,
% 8.83/5.02 ! [A: rat,B: rat,C: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.02 => ( ( ord_less_rat @ B @ C )
% 8.83/5.02 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2237_add__strict__increasing2,axiom,
% 8.83/5.02 ! [A: nat,B: nat,C: nat] :
% 8.83/5.02 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.02 => ( ( ord_less_nat @ B @ C )
% 8.83/5.02 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2238_add__strict__increasing2,axiom,
% 8.83/5.02 ! [A: int,B: int,C: int] :
% 8.83/5.02 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.02 => ( ( ord_less_int @ B @ C )
% 8.83/5.02 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2239_mult__left__le__one__le,axiom,
% 8.83/5.02 ! [X2: real,Y3: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.83/5.02 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.83/5.02 => ( ord_less_eq_real @ ( times_times_real @ Y3 @ X2 ) @ X2 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2240_mult__left__le__one__le,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 8.83/5.02 => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ ( times_times_rat @ Y3 @ X2 ) @ X2 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2241_mult__left__le__one__le,axiom,
% 8.83/5.02 ! [X2: int,Y3: int] :
% 8.83/5.02 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.83/5.02 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.83/5.02 => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 8.83/5.02 => ( ord_less_eq_int @ ( times_times_int @ Y3 @ X2 ) @ X2 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2242_mult__right__le__one__le,axiom,
% 8.83/5.02 ! [X2: real,Y3: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.83/5.02 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.83/5.02 => ( ord_less_eq_real @ ( times_times_real @ X2 @ Y3 ) @ X2 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2243_mult__right__le__one__le,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 8.83/5.02 => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ ( times_times_rat @ X2 @ Y3 ) @ X2 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2244_mult__right__le__one__le,axiom,
% 8.83/5.02 ! [X2: int,Y3: int] :
% 8.83/5.02 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.83/5.02 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.83/5.02 => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 8.83/5.02 => ( ord_less_eq_int @ ( times_times_int @ X2 @ Y3 ) @ X2 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2245_mult__le__one,axiom,
% 8.83/5.02 ! [A: real,B: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ A @ one_one_real )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.83/5.02 => ( ( ord_less_eq_real @ B @ one_one_real )
% 8.83/5.02 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2246_mult__le__one,axiom,
% 8.83/5.02 ! [A: rat,B: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ A @ one_one_rat )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.83/5.02 => ( ( ord_less_eq_rat @ B @ one_one_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2247_mult__le__one,axiom,
% 8.83/5.02 ! [A: nat,B: nat] :
% 8.83/5.02 ( ( ord_less_eq_nat @ A @ one_one_nat )
% 8.83/5.02 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.83/5.02 => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 8.83/5.02 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2248_mult__le__one,axiom,
% 8.83/5.02 ! [A: int,B: int] :
% 8.83/5.02 ( ( ord_less_eq_int @ A @ one_one_int )
% 8.83/5.02 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.83/5.02 => ( ( ord_less_eq_int @ B @ one_one_int )
% 8.83/5.02 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2249_mult__left__le,axiom,
% 8.83/5.02 ! [C: real,A: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ C @ one_one_real )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.02 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2250_mult__left__le,axiom,
% 8.83/5.02 ! [C: rat,A: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ C @ one_one_rat )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.02 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2251_mult__left__le,axiom,
% 8.83/5.02 ! [C: nat,A: nat] :
% 8.83/5.02 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 8.83/5.02 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.02 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2252_mult__left__le,axiom,
% 8.83/5.02 ! [C: int,A: int] :
% 8.83/5.02 ( ( ord_less_eq_int @ C @ one_one_int )
% 8.83/5.02 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.02 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2253_divide__nonpos__pos,axiom,
% 8.83/5.02 ! [X2: real,Y3: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 8.83/5.02 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.83/5.02 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2254_divide__nonpos__pos,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 8.83/5.02 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 8.83/5.02 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2255_divide__nonpos__neg,axiom,
% 8.83/5.02 ! [X2: real,Y3: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 8.83/5.02 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 8.83/5.02 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2256_divide__nonpos__neg,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 8.83/5.02 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2257_divide__nonneg__pos,axiom,
% 8.83/5.02 ! [X2: real,Y3: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.83/5.02 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.83/5.02 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2258_divide__nonneg__pos,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 8.83/5.02 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 8.83/5.02 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2259_divide__nonneg__neg,axiom,
% 8.83/5.02 ! [X2: real,Y3: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.83/5.02 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 8.83/5.02 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2260_divide__nonneg__neg,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 8.83/5.02 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2261_divide__le__cancel,axiom,
% 8.83/5.02 ! [A: real,C: real,B: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 8.83/5.02 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ord_less_eq_real @ A @ B ) )
% 8.83/5.02 & ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2262_divide__le__cancel,axiom,
% 8.83/5.02 ! [A: rat,C: rat,B: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 8.83/5.02 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ord_less_eq_rat @ A @ B ) )
% 8.83/5.02 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2263_frac__less2,axiom,
% 8.83/5.02 ! [X2: real,Y3: real,W: real,Z: real] :
% 8.83/5.02 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.83/5.02 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 8.83/5.02 => ( ( ord_less_real @ zero_zero_real @ W )
% 8.83/5.02 => ( ( ord_less_real @ W @ Z )
% 8.83/5.02 => ( ord_less_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2264_frac__less2,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat,W: rat,Z: rat] :
% 8.83/5.02 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 8.83/5.02 => ( ( ord_less_eq_rat @ X2 @ Y3 )
% 8.83/5.02 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 8.83/5.02 => ( ( ord_less_rat @ W @ Z )
% 8.83/5.02 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2265_frac__less,axiom,
% 8.83/5.02 ! [X2: real,Y3: real,W: real,Z: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.83/5.02 => ( ( ord_less_real @ X2 @ Y3 )
% 8.83/5.02 => ( ( ord_less_real @ zero_zero_real @ W )
% 8.83/5.02 => ( ( ord_less_eq_real @ W @ Z )
% 8.83/5.02 => ( ord_less_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2266_frac__less,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat,W: rat,Z: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 8.83/5.02 => ( ( ord_less_rat @ X2 @ Y3 )
% 8.83/5.02 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 8.83/5.02 => ( ( ord_less_eq_rat @ W @ Z )
% 8.83/5.02 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2267_frac__le,axiom,
% 8.83/5.02 ! [Y3: real,X2: real,W: real,Z: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.83/5.02 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 8.83/5.02 => ( ( ord_less_real @ zero_zero_real @ W )
% 8.83/5.02 => ( ( ord_less_eq_real @ W @ Z )
% 8.83/5.02 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2268_frac__le,axiom,
% 8.83/5.02 ! [Y3: rat,X2: rat,W: rat,Z: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 8.83/5.02 => ( ( ord_less_eq_rat @ X2 @ Y3 )
% 8.83/5.02 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 8.83/5.02 => ( ( ord_less_eq_rat @ W @ Z )
% 8.83/5.02 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2269_sum__squares__le__zero__iff,axiom,
% 8.83/5.02 ! [X2: real,Y3: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real )
% 8.83/5.02 = ( ( X2 = zero_zero_real )
% 8.83/5.02 & ( Y3 = zero_zero_real ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2270_sum__squares__le__zero__iff,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat )
% 8.83/5.02 = ( ( X2 = zero_zero_rat )
% 8.83/5.02 & ( Y3 = zero_zero_rat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2271_sum__squares__le__zero__iff,axiom,
% 8.83/5.02 ! [X2: int,Y3: int] :
% 8.83/5.02 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int )
% 8.83/5.02 = ( ( X2 = zero_zero_int )
% 8.83/5.02 & ( Y3 = zero_zero_int ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2272_sum__squares__ge__zero,axiom,
% 8.83/5.02 ! [X2: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2273_sum__squares__ge__zero,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2274_sum__squares__ge__zero,axiom,
% 8.83/5.02 ! [X2: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2275_sum__squares__gt__zero__iff,axiom,
% 8.83/5.02 ! [X2: real,Y3: real] :
% 8.83/5.02 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) )
% 8.83/5.02 = ( ( X2 != zero_zero_real )
% 8.83/5.02 | ( Y3 != zero_zero_real ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2276_sum__squares__gt__zero__iff,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat] :
% 8.83/5.02 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) )
% 8.83/5.02 = ( ( X2 != zero_zero_rat )
% 8.83/5.02 | ( Y3 != zero_zero_rat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2277_sum__squares__gt__zero__iff,axiom,
% 8.83/5.02 ! [X2: int,Y3: int] :
% 8.83/5.02 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) )
% 8.83/5.02 = ( ( X2 != zero_zero_int )
% 8.83/5.02 | ( Y3 != zero_zero_int ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2278_not__sum__squares__lt__zero,axiom,
% 8.83/5.02 ! [X2: real,Y3: real] :
% 8.83/5.02 ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2279_not__sum__squares__lt__zero,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat] :
% 8.83/5.02 ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2280_not__sum__squares__lt__zero,axiom,
% 8.83/5.02 ! [X2: int,Y3: int] :
% 8.83/5.02 ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2281_power__less__imp__less__base,axiom,
% 8.83/5.02 ! [A: real,N: nat,B: real] :
% 8.83/5.02 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.83/5.02 => ( ord_less_real @ A @ B ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2282_power__less__imp__less__base,axiom,
% 8.83/5.02 ! [A: rat,N: nat,B: rat] :
% 8.83/5.02 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.83/5.02 => ( ord_less_rat @ A @ B ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2283_power__less__imp__less__base,axiom,
% 8.83/5.02 ! [A: nat,N: nat,B: nat] :
% 8.83/5.02 ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 8.83/5.02 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.83/5.02 => ( ord_less_nat @ A @ B ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2284_power__less__imp__less__base,axiom,
% 8.83/5.02 ! [A: int,N: nat,B: int] :
% 8.83/5.02 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 8.83/5.02 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.83/5.02 => ( ord_less_int @ A @ B ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2285_zero__less__two,axiom,
% 8.83/5.02 ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2286_zero__less__two,axiom,
% 8.83/5.02 ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2287_zero__less__two,axiom,
% 8.83/5.02 ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2288_zero__less__two,axiom,
% 8.83/5.02 ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2289_divide__strict__left__mono__neg,axiom,
% 8.83/5.02 ! [A: real,B: real,C: real] :
% 8.83/5.02 ( ( ord_less_real @ A @ B )
% 8.83/5.02 => ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 8.83/5.02 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2290_divide__strict__left__mono__neg,axiom,
% 8.83/5.02 ! [A: rat,B: rat,C: rat] :
% 8.83/5.02 ( ( ord_less_rat @ A @ B )
% 8.83/5.02 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 8.83/5.02 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2291_divide__strict__left__mono,axiom,
% 8.83/5.02 ! [B: real,A: real,C: real] :
% 8.83/5.02 ( ( ord_less_real @ B @ A )
% 8.83/5.02 => ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 8.83/5.02 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2292_divide__strict__left__mono,axiom,
% 8.83/5.02 ! [B: rat,A: rat,C: rat] :
% 8.83/5.02 ( ( ord_less_rat @ B @ A )
% 8.83/5.02 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 8.83/5.02 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2293_mult__imp__less__div__pos,axiom,
% 8.83/5.02 ! [Y3: real,Z: real,X2: real] :
% 8.83/5.02 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.83/5.02 => ( ( ord_less_real @ ( times_times_real @ Z @ Y3 ) @ X2 )
% 8.83/5.02 => ( ord_less_real @ Z @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2294_mult__imp__less__div__pos,axiom,
% 8.83/5.02 ! [Y3: rat,Z: rat,X2: rat] :
% 8.83/5.02 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 8.83/5.02 => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y3 ) @ X2 )
% 8.83/5.02 => ( ord_less_rat @ Z @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2295_mult__imp__div__pos__less,axiom,
% 8.83/5.02 ! [Y3: real,X2: real,Z: real] :
% 8.83/5.02 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.83/5.02 => ( ( ord_less_real @ X2 @ ( times_times_real @ Z @ Y3 ) )
% 8.83/5.02 => ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ Z ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2296_mult__imp__div__pos__less,axiom,
% 8.83/5.02 ! [Y3: rat,X2: rat,Z: rat] :
% 8.83/5.02 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 8.83/5.02 => ( ( ord_less_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) )
% 8.83/5.02 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ Z ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2297_pos__less__divide__eq,axiom,
% 8.83/5.02 ! [C: real,A: real,B: real] :
% 8.83/5.02 ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 8.83/5.02 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2298_pos__less__divide__eq,axiom,
% 8.83/5.02 ! [C: rat,A: rat,B: rat] :
% 8.83/5.02 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 8.83/5.02 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2299_pos__divide__less__eq,axiom,
% 8.83/5.02 ! [C: real,B: real,A: real] :
% 8.83/5.02 ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 8.83/5.02 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2300_pos__divide__less__eq,axiom,
% 8.83/5.02 ! [C: rat,B: rat,A: rat] :
% 8.83/5.02 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 8.83/5.02 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2301_neg__less__divide__eq,axiom,
% 8.83/5.02 ! [C: real,A: real,B: real] :
% 8.83/5.02 ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 8.83/5.02 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2302_neg__less__divide__eq,axiom,
% 8.83/5.02 ! [C: rat,A: rat,B: rat] :
% 8.83/5.02 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 8.83/5.02 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2303_neg__divide__less__eq,axiom,
% 8.83/5.02 ! [C: real,B: real,A: real] :
% 8.83/5.02 ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 8.83/5.02 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2304_neg__divide__less__eq,axiom,
% 8.83/5.02 ! [C: rat,B: rat,A: rat] :
% 8.83/5.02 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 8.83/5.02 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2305_less__divide__eq,axiom,
% 8.83/5.02 ! [A: real,B: real,C: real] :
% 8.83/5.02 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 8.83/5.02 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 8.83/5.02 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 8.83/5.02 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2306_less__divide__eq,axiom,
% 8.83/5.02 ! [A: rat,B: rat,C: rat] :
% 8.83/5.02 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 8.83/5.02 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 8.83/5.02 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 8.83/5.02 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2307_divide__less__eq,axiom,
% 8.83/5.02 ! [B: real,C: real,A: real] :
% 8.83/5.02 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 8.83/5.02 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 8.83/5.02 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 8.83/5.02 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2308_divide__less__eq,axiom,
% 8.83/5.02 ! [B: rat,C: rat,A: rat] :
% 8.83/5.02 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 8.83/5.02 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 8.83/5.02 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 8.83/5.02 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2309_less__divide__eq__1,axiom,
% 8.83/5.02 ! [B: real,A: real] :
% 8.83/5.02 ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 8.83/5.02 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.02 & ( ord_less_real @ A @ B ) )
% 8.83/5.02 | ( ( ord_less_real @ A @ zero_zero_real )
% 8.83/5.02 & ( ord_less_real @ B @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2310_less__divide__eq__1,axiom,
% 8.83/5.02 ! [B: rat,A: rat] :
% 8.83/5.02 ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 8.83/5.02 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.02 & ( ord_less_rat @ A @ B ) )
% 8.83/5.02 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.83/5.02 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2311_divide__less__eq__1,axiom,
% 8.83/5.02 ! [B: real,A: real] :
% 8.83/5.02 ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 8.83/5.02 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.02 & ( ord_less_real @ B @ A ) )
% 8.83/5.02 | ( ( ord_less_real @ A @ zero_zero_real )
% 8.83/5.02 & ( ord_less_real @ A @ B ) )
% 8.83/5.02 | ( A = zero_zero_real ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2312_divide__less__eq__1,axiom,
% 8.83/5.02 ! [B: rat,A: rat] :
% 8.83/5.02 ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 8.83/5.02 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.02 & ( ord_less_rat @ B @ A ) )
% 8.83/5.02 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.83/5.02 & ( ord_less_rat @ A @ B ) )
% 8.83/5.02 | ( A = zero_zero_rat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2313_power__le__one,axiom,
% 8.83/5.02 ! [A: real,N: nat] :
% 8.83/5.02 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.02 => ( ( ord_less_eq_real @ A @ one_one_real )
% 8.83/5.02 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2314_power__le__one,axiom,
% 8.83/5.02 ! [A: rat,N: nat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.02 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2315_power__le__one,axiom,
% 8.83/5.02 ! [A: nat,N: nat] :
% 8.83/5.02 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.02 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 8.83/5.02 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2316_power__le__one,axiom,
% 8.83/5.02 ! [A: int,N: nat] :
% 8.83/5.02 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.02 => ( ( ord_less_eq_int @ A @ one_one_int )
% 8.83/5.02 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2317_eq__divide__eq__numeral_I1_J,axiom,
% 8.83/5.02 ! [W: num,B: complex,C: complex] :
% 8.83/5.02 ( ( ( numera6690914467698888265omplex @ W )
% 8.83/5.02 = ( divide1717551699836669952omplex @ B @ C ) )
% 8.83/5.02 = ( ( ( C != zero_zero_complex )
% 8.83/5.02 => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
% 8.83/5.02 = B ) )
% 8.83/5.02 & ( ( C = zero_zero_complex )
% 8.83/5.02 => ( ( numera6690914467698888265omplex @ W )
% 8.83/5.02 = zero_zero_complex ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2318_eq__divide__eq__numeral_I1_J,axiom,
% 8.83/5.02 ! [W: num,B: real,C: real] :
% 8.83/5.02 ( ( ( numeral_numeral_real @ W )
% 8.83/5.02 = ( divide_divide_real @ B @ C ) )
% 8.83/5.02 = ( ( ( C != zero_zero_real )
% 8.83/5.02 => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
% 8.83/5.02 = B ) )
% 8.83/5.02 & ( ( C = zero_zero_real )
% 8.83/5.02 => ( ( numeral_numeral_real @ W )
% 8.83/5.02 = zero_zero_real ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2319_eq__divide__eq__numeral_I1_J,axiom,
% 8.83/5.02 ! [W: num,B: rat,C: rat] :
% 8.83/5.02 ( ( ( numeral_numeral_rat @ W )
% 8.83/5.02 = ( divide_divide_rat @ B @ C ) )
% 8.83/5.02 = ( ( ( C != zero_zero_rat )
% 8.83/5.02 => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
% 8.83/5.02 = B ) )
% 8.83/5.02 & ( ( C = zero_zero_rat )
% 8.83/5.02 => ( ( numeral_numeral_rat @ W )
% 8.83/5.02 = zero_zero_rat ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2320_divide__eq__eq__numeral_I1_J,axiom,
% 8.83/5.02 ! [B: complex,C: complex,W: num] :
% 8.83/5.02 ( ( ( divide1717551699836669952omplex @ B @ C )
% 8.83/5.02 = ( numera6690914467698888265omplex @ W ) )
% 8.83/5.02 = ( ( ( C != zero_zero_complex )
% 8.83/5.02 => ( B
% 8.83/5.02 = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
% 8.83/5.02 & ( ( C = zero_zero_complex )
% 8.83/5.02 => ( ( numera6690914467698888265omplex @ W )
% 8.83/5.02 = zero_zero_complex ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2321_divide__eq__eq__numeral_I1_J,axiom,
% 8.83/5.02 ! [B: real,C: real,W: num] :
% 8.83/5.02 ( ( ( divide_divide_real @ B @ C )
% 8.83/5.02 = ( numeral_numeral_real @ W ) )
% 8.83/5.02 = ( ( ( C != zero_zero_real )
% 8.83/5.02 => ( B
% 8.83/5.02 = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 8.83/5.02 & ( ( C = zero_zero_real )
% 8.83/5.02 => ( ( numeral_numeral_real @ W )
% 8.83/5.02 = zero_zero_real ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2322_divide__eq__eq__numeral_I1_J,axiom,
% 8.83/5.02 ! [B: rat,C: rat,W: num] :
% 8.83/5.02 ( ( ( divide_divide_rat @ B @ C )
% 8.83/5.02 = ( numeral_numeral_rat @ W ) )
% 8.83/5.02 = ( ( ( C != zero_zero_rat )
% 8.83/5.02 => ( B
% 8.83/5.02 = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 8.83/5.02 & ( ( C = zero_zero_rat )
% 8.83/5.02 => ( ( numeral_numeral_rat @ W )
% 8.83/5.02 = zero_zero_rat ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2323_divide__add__eq__iff,axiom,
% 8.83/5.02 ! [Z: complex,X2: complex,Y3: complex] :
% 8.83/5.02 ( ( Z != zero_zero_complex )
% 8.83/5.02 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Z ) @ Y3 )
% 8.83/5.02 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2324_divide__add__eq__iff,axiom,
% 8.83/5.02 ! [Z: real,X2: real,Y3: real] :
% 8.83/5.02 ( ( Z != zero_zero_real )
% 8.83/5.02 => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Z ) @ Y3 )
% 8.83/5.02 = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2325_divide__add__eq__iff,axiom,
% 8.83/5.02 ! [Z: rat,X2: rat,Y3: rat] :
% 8.83/5.02 ( ( Z != zero_zero_rat )
% 8.83/5.02 => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Z ) @ Y3 )
% 8.83/5.02 = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2326_add__divide__eq__iff,axiom,
% 8.83/5.02 ! [Z: complex,X2: complex,Y3: complex] :
% 8.83/5.02 ( ( Z != zero_zero_complex )
% 8.83/5.02 => ( ( plus_plus_complex @ X2 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 8.83/5.02 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2327_add__divide__eq__iff,axiom,
% 8.83/5.02 ! [Z: real,X2: real,Y3: real] :
% 8.83/5.02 ( ( Z != zero_zero_real )
% 8.83/5.02 => ( ( plus_plus_real @ X2 @ ( divide_divide_real @ Y3 @ Z ) )
% 8.83/5.02 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2328_add__divide__eq__iff,axiom,
% 8.83/5.02 ! [Z: rat,X2: rat,Y3: rat] :
% 8.83/5.02 ( ( Z != zero_zero_rat )
% 8.83/5.02 => ( ( plus_plus_rat @ X2 @ ( divide_divide_rat @ Y3 @ Z ) )
% 8.83/5.02 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2329_add__num__frac,axiom,
% 8.83/5.02 ! [Y3: complex,Z: complex,X2: complex] :
% 8.83/5.02 ( ( Y3 != zero_zero_complex )
% 8.83/5.02 => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X2 @ Y3 ) )
% 8.83/5.02 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2330_add__num__frac,axiom,
% 8.83/5.02 ! [Y3: real,Z: real,X2: real] :
% 8.83/5.02 ( ( Y3 != zero_zero_real )
% 8.83/5.02 => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X2 @ Y3 ) )
% 8.83/5.02 = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2331_add__num__frac,axiom,
% 8.83/5.02 ! [Y3: rat,Z: rat,X2: rat] :
% 8.83/5.02 ( ( Y3 != zero_zero_rat )
% 8.83/5.02 => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X2 @ Y3 ) )
% 8.83/5.02 = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2332_add__frac__num,axiom,
% 8.83/5.02 ! [Y3: complex,X2: complex,Z: complex] :
% 8.83/5.02 ( ( Y3 != zero_zero_complex )
% 8.83/5.02 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Y3 ) @ Z )
% 8.83/5.02 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2333_add__frac__num,axiom,
% 8.83/5.02 ! [Y3: real,X2: real,Z: real] :
% 8.83/5.02 ( ( Y3 != zero_zero_real )
% 8.83/5.02 => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y3 ) @ Z )
% 8.83/5.02 = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2334_add__frac__num,axiom,
% 8.83/5.02 ! [Y3: rat,X2: rat,Z: rat] :
% 8.83/5.02 ( ( Y3 != zero_zero_rat )
% 8.83/5.02 => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ Z )
% 8.83/5.02 = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2335_add__frac__eq,axiom,
% 8.83/5.02 ! [Y3: complex,Z: complex,X2: complex,W: complex] :
% 8.83/5.02 ( ( Y3 != zero_zero_complex )
% 8.83/5.02 => ( ( Z != zero_zero_complex )
% 8.83/5.02 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 8.83/5.02 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2336_add__frac__eq,axiom,
% 8.83/5.02 ! [Y3: real,Z: real,X2: real,W: real] :
% 8.83/5.02 ( ( Y3 != zero_zero_real )
% 8.83/5.02 => ( ( Z != zero_zero_real )
% 8.83/5.02 => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 8.83/5.02 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2337_add__frac__eq,axiom,
% 8.83/5.02 ! [Y3: rat,Z: rat,X2: rat,W: rat] :
% 8.83/5.02 ( ( Y3 != zero_zero_rat )
% 8.83/5.02 => ( ( Z != zero_zero_rat )
% 8.83/5.02 => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 8.83/5.02 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2338_add__divide__eq__if__simps_I1_J,axiom,
% 8.83/5.02 ! [Z: complex,A: complex,B: complex] :
% 8.83/5.02 ( ( ( Z = zero_zero_complex )
% 8.83/5.02 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 8.83/5.02 = A ) )
% 8.83/5.02 & ( ( Z != zero_zero_complex )
% 8.83/5.02 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 8.83/5.02 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2339_add__divide__eq__if__simps_I1_J,axiom,
% 8.83/5.02 ! [Z: real,A: real,B: real] :
% 8.83/5.02 ( ( ( Z = zero_zero_real )
% 8.83/5.02 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 8.83/5.02 = A ) )
% 8.83/5.02 & ( ( Z != zero_zero_real )
% 8.83/5.02 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 8.83/5.02 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2340_add__divide__eq__if__simps_I1_J,axiom,
% 8.83/5.02 ! [Z: rat,A: rat,B: rat] :
% 8.83/5.02 ( ( ( Z = zero_zero_rat )
% 8.83/5.02 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 8.83/5.02 = A ) )
% 8.83/5.02 & ( ( Z != zero_zero_rat )
% 8.83/5.02 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 8.83/5.02 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2341_add__divide__eq__if__simps_I2_J,axiom,
% 8.83/5.02 ! [Z: complex,A: complex,B: complex] :
% 8.83/5.02 ( ( ( Z = zero_zero_complex )
% 8.83/5.02 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 8.83/5.02 = B ) )
% 8.83/5.02 & ( ( Z != zero_zero_complex )
% 8.83/5.02 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 8.83/5.02 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2342_add__divide__eq__if__simps_I2_J,axiom,
% 8.83/5.02 ! [Z: real,A: real,B: real] :
% 8.83/5.02 ( ( ( Z = zero_zero_real )
% 8.83/5.02 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 8.83/5.02 = B ) )
% 8.83/5.02 & ( ( Z != zero_zero_real )
% 8.83/5.02 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 8.83/5.02 = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2343_add__divide__eq__if__simps_I2_J,axiom,
% 8.83/5.02 ! [Z: rat,A: rat,B: rat] :
% 8.83/5.02 ( ( ( Z = zero_zero_rat )
% 8.83/5.02 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 8.83/5.02 = B ) )
% 8.83/5.02 & ( ( Z != zero_zero_rat )
% 8.83/5.02 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 8.83/5.02 = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2344_power__inject__base,axiom,
% 8.83/5.02 ! [A: real,N: nat,B: real] :
% 8.83/5.02 ( ( ( power_power_real @ A @ ( suc @ N ) )
% 8.83/5.02 = ( power_power_real @ B @ ( suc @ N ) ) )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.83/5.02 => ( A = B ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2345_power__inject__base,axiom,
% 8.83/5.02 ! [A: rat,N: nat,B: rat] :
% 8.83/5.02 ( ( ( power_power_rat @ A @ ( suc @ N ) )
% 8.83/5.02 = ( power_power_rat @ B @ ( suc @ N ) ) )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.83/5.02 => ( A = B ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2346_power__inject__base,axiom,
% 8.83/5.02 ! [A: nat,N: nat,B: nat] :
% 8.83/5.02 ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 8.83/5.02 = ( power_power_nat @ B @ ( suc @ N ) ) )
% 8.83/5.02 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.02 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.83/5.02 => ( A = B ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2347_power__inject__base,axiom,
% 8.83/5.02 ! [A: int,N: nat,B: int] :
% 8.83/5.02 ( ( ( power_power_int @ A @ ( suc @ N ) )
% 8.83/5.02 = ( power_power_int @ B @ ( suc @ N ) ) )
% 8.83/5.02 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.02 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.83/5.02 => ( A = B ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2348_power__le__imp__le__base,axiom,
% 8.83/5.02 ! [A: real,N: nat,B: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.83/5.02 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2349_power__le__imp__le__base,axiom,
% 8.83/5.02 ! [A: rat,N: nat,B: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.83/5.02 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2350_power__le__imp__le__base,axiom,
% 8.83/5.02 ! [A: nat,N: nat,B: nat] :
% 8.83/5.02 ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
% 8.83/5.02 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 8.83/5.02 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2351_power__le__imp__le__base,axiom,
% 8.83/5.02 ! [A: int,N: nat,B: int] :
% 8.83/5.02 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
% 8.83/5.02 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.83/5.02 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2352_div__add__self1,axiom,
% 8.83/5.02 ! [B: nat,A: nat] :
% 8.83/5.02 ( ( B != zero_zero_nat )
% 8.83/5.02 => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 8.83/5.02 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2353_div__add__self1,axiom,
% 8.83/5.02 ! [B: int,A: int] :
% 8.83/5.02 ( ( B != zero_zero_int )
% 8.83/5.02 => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
% 8.83/5.02 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2354_div__add__self2,axiom,
% 8.83/5.02 ! [B: nat,A: nat] :
% 8.83/5.02 ( ( B != zero_zero_nat )
% 8.83/5.02 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 8.83/5.02 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2355_div__add__self2,axiom,
% 8.83/5.02 ! [B: int,A: int] :
% 8.83/5.02 ( ( B != zero_zero_int )
% 8.83/5.02 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
% 8.83/5.02 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2356_divide__diff__eq__iff,axiom,
% 8.83/5.02 ! [Z: complex,X2: complex,Y3: complex] :
% 8.83/5.02 ( ( Z != zero_zero_complex )
% 8.83/5.02 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X2 @ Z ) @ Y3 )
% 8.83/5.02 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2357_divide__diff__eq__iff,axiom,
% 8.83/5.02 ! [Z: real,X2: real,Y3: real] :
% 8.83/5.02 ( ( Z != zero_zero_real )
% 8.83/5.02 => ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Z ) @ Y3 )
% 8.83/5.02 = ( divide_divide_real @ ( minus_minus_real @ X2 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2358_divide__diff__eq__iff,axiom,
% 8.83/5.02 ! [Z: rat,X2: rat,Y3: rat] :
% 8.83/5.02 ( ( Z != zero_zero_rat )
% 8.83/5.02 => ( ( minus_minus_rat @ ( divide_divide_rat @ X2 @ Z ) @ Y3 )
% 8.83/5.02 = ( divide_divide_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2359_diff__divide__eq__iff,axiom,
% 8.83/5.02 ! [Z: complex,X2: complex,Y3: complex] :
% 8.83/5.02 ( ( Z != zero_zero_complex )
% 8.83/5.02 => ( ( minus_minus_complex @ X2 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 8.83/5.02 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2360_diff__divide__eq__iff,axiom,
% 8.83/5.02 ! [Z: real,X2: real,Y3: real] :
% 8.83/5.02 ( ( Z != zero_zero_real )
% 8.83/5.02 => ( ( minus_minus_real @ X2 @ ( divide_divide_real @ Y3 @ Z ) )
% 8.83/5.02 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2361_diff__divide__eq__iff,axiom,
% 8.83/5.02 ! [Z: rat,X2: rat,Y3: rat] :
% 8.83/5.02 ( ( Z != zero_zero_rat )
% 8.83/5.02 => ( ( minus_minus_rat @ X2 @ ( divide_divide_rat @ Y3 @ Z ) )
% 8.83/5.02 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2362_diff__frac__eq,axiom,
% 8.83/5.02 ! [Y3: complex,Z: complex,X2: complex,W: complex] :
% 8.83/5.02 ( ( Y3 != zero_zero_complex )
% 8.83/5.02 => ( ( Z != zero_zero_complex )
% 8.83/5.02 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X2 @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 8.83/5.02 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2363_diff__frac__eq,axiom,
% 8.83/5.02 ! [Y3: real,Z: real,X2: real,W: real] :
% 8.83/5.02 ( ( Y3 != zero_zero_real )
% 8.83/5.02 => ( ( Z != zero_zero_real )
% 8.83/5.02 => ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 8.83/5.02 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2364_diff__frac__eq,axiom,
% 8.83/5.02 ! [Y3: rat,Z: rat,X2: rat,W: rat] :
% 8.83/5.02 ( ( Y3 != zero_zero_rat )
% 8.83/5.02 => ( ( Z != zero_zero_rat )
% 8.83/5.02 => ( ( minus_minus_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 8.83/5.02 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2365_add__divide__eq__if__simps_I4_J,axiom,
% 8.83/5.02 ! [Z: complex,A: complex,B: complex] :
% 8.83/5.02 ( ( ( Z = zero_zero_complex )
% 8.83/5.02 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 8.83/5.02 = A ) )
% 8.83/5.02 & ( ( Z != zero_zero_complex )
% 8.83/5.02 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 8.83/5.02 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2366_add__divide__eq__if__simps_I4_J,axiom,
% 8.83/5.02 ! [Z: real,A: real,B: real] :
% 8.83/5.02 ( ( ( Z = zero_zero_real )
% 8.83/5.02 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 8.83/5.02 = A ) )
% 8.83/5.02 & ( ( Z != zero_zero_real )
% 8.83/5.02 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 8.83/5.02 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2367_add__divide__eq__if__simps_I4_J,axiom,
% 8.83/5.02 ! [Z: rat,A: rat,B: rat] :
% 8.83/5.02 ( ( ( Z = zero_zero_rat )
% 8.83/5.02 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 8.83/5.02 = A ) )
% 8.83/5.02 & ( ( Z != zero_zero_rat )
% 8.83/5.02 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 8.83/5.02 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2368_numeral__1__eq__Suc__0,axiom,
% 8.83/5.02 ( ( numeral_numeral_nat @ one )
% 8.83/5.02 = ( suc @ zero_zero_nat ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2369_subset__code_I1_J,axiom,
% 8.83/5.02 ! [Xs2: list_P6011104703257516679at_nat,B3: set_Pr1261947904930325089at_nat] :
% 8.83/5.02 ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ B3 )
% 8.83/5.02 = ( ! [X: product_prod_nat_nat] :
% 8.83/5.02 ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
% 8.83/5.02 => ( member8440522571783428010at_nat @ X @ B3 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2370_subset__code_I1_J,axiom,
% 8.83/5.02 ! [Xs2: list_complex,B3: set_complex] :
% 8.83/5.02 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ B3 )
% 8.83/5.02 = ( ! [X: complex] :
% 8.83/5.02 ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
% 8.83/5.02 => ( member_complex @ X @ B3 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2371_subset__code_I1_J,axiom,
% 8.83/5.02 ! [Xs2: list_real,B3: set_real] :
% 8.83/5.02 ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B3 )
% 8.83/5.02 = ( ! [X: real] :
% 8.83/5.02 ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
% 8.83/5.02 => ( member_real @ X @ B3 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2372_subset__code_I1_J,axiom,
% 8.83/5.02 ! [Xs2: list_VEBT_VEBT,B3: set_VEBT_VEBT] :
% 8.83/5.02 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ B3 )
% 8.83/5.02 = ( ! [X: vEBT_VEBT] :
% 8.83/5.02 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 8.83/5.02 => ( member_VEBT_VEBT @ X @ B3 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2373_subset__code_I1_J,axiom,
% 8.83/5.02 ! [Xs2: list_nat,B3: set_nat] :
% 8.83/5.02 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B3 )
% 8.83/5.02 = ( ! [X: nat] :
% 8.83/5.02 ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
% 8.83/5.02 => ( member_nat @ X @ B3 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2374_subset__code_I1_J,axiom,
% 8.83/5.02 ! [Xs2: list_int,B3: set_int] :
% 8.83/5.02 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B3 )
% 8.83/5.02 = ( ! [X: int] :
% 8.83/5.02 ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
% 8.83/5.02 => ( member_int @ X @ B3 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2375_num_Osize_I5_J,axiom,
% 8.83/5.02 ! [X22: num] :
% 8.83/5.02 ( ( size_size_num @ ( bit0 @ X22 ) )
% 8.83/5.02 = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2376_neq__if__length__neq,axiom,
% 8.83/5.02 ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 8.83/5.02 ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 8.83/5.02 != ( size_s6755466524823107622T_VEBT @ Ys ) )
% 8.83/5.02 => ( Xs2 != Ys ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2377_neq__if__length__neq,axiom,
% 8.83/5.02 ! [Xs2: list_o,Ys: list_o] :
% 8.83/5.02 ( ( ( size_size_list_o @ Xs2 )
% 8.83/5.02 != ( size_size_list_o @ Ys ) )
% 8.83/5.02 => ( Xs2 != Ys ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2378_neq__if__length__neq,axiom,
% 8.83/5.02 ! [Xs2: list_nat,Ys: list_nat] :
% 8.83/5.02 ( ( ( size_size_list_nat @ Xs2 )
% 8.83/5.02 != ( size_size_list_nat @ Ys ) )
% 8.83/5.02 => ( Xs2 != Ys ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2379_neq__if__length__neq,axiom,
% 8.83/5.02 ! [Xs2: list_int,Ys: list_int] :
% 8.83/5.02 ( ( ( size_size_list_int @ Xs2 )
% 8.83/5.02 != ( size_size_list_int @ Ys ) )
% 8.83/5.02 => ( Xs2 != Ys ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2380_Ex__list__of__length,axiom,
% 8.83/5.02 ! [N: nat] :
% 8.83/5.02 ? [Xs3: list_VEBT_VEBT] :
% 8.83/5.02 ( ( size_s6755466524823107622T_VEBT @ Xs3 )
% 8.83/5.02 = N ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2381_Ex__list__of__length,axiom,
% 8.83/5.02 ! [N: nat] :
% 8.83/5.02 ? [Xs3: list_o] :
% 8.83/5.02 ( ( size_size_list_o @ Xs3 )
% 8.83/5.02 = N ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2382_Ex__list__of__length,axiom,
% 8.83/5.02 ! [N: nat] :
% 8.83/5.02 ? [Xs3: list_nat] :
% 8.83/5.02 ( ( size_size_list_nat @ Xs3 )
% 8.83/5.02 = N ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2383_Ex__list__of__length,axiom,
% 8.83/5.02 ! [N: nat] :
% 8.83/5.02 ? [Xs3: list_int] :
% 8.83/5.02 ( ( size_size_list_int @ Xs3 )
% 8.83/5.02 = N ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2384_ex__least__nat__less,axiom,
% 8.83/5.02 ! [P: nat > $o,N: nat] :
% 8.83/5.02 ( ( P @ N )
% 8.83/5.02 => ( ~ ( P @ zero_zero_nat )
% 8.83/5.02 => ? [K3: nat] :
% 8.83/5.02 ( ( ord_less_nat @ K3 @ N )
% 8.83/5.02 & ! [I: nat] :
% 8.83/5.02 ( ( ord_less_eq_nat @ I @ K3 )
% 8.83/5.02 => ~ ( P @ I ) )
% 8.83/5.02 & ( P @ ( suc @ K3 ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2385_nat__induct__non__zero,axiom,
% 8.83/5.02 ! [N: nat,P: nat > $o] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ( P @ one_one_nat )
% 8.83/5.02 => ( ! [N2: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 8.83/5.02 => ( ( P @ N2 )
% 8.83/5.02 => ( P @ ( suc @ N2 ) ) ) )
% 8.83/5.02 => ( P @ N ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2386_diff__Suc__less,axiom,
% 8.83/5.02 ! [N: nat,I2: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2387_one__less__mult,axiom,
% 8.83/5.02 ! [N: nat,M: nat] :
% 8.83/5.02 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.83/5.02 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 8.83/5.02 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2388_n__less__m__mult__n,axiom,
% 8.83/5.02 ! [N: nat,M: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 8.83/5.02 => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2389_n__less__n__mult__m,axiom,
% 8.83/5.02 ! [N: nat,M: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 8.83/5.02 => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2390_nat__mult__le__cancel1,axiom,
% 8.83/5.02 ! [K: nat,M: nat,N: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.83/5.02 => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.83/5.02 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2391_power__gt__expt,axiom,
% 8.83/5.02 ! [N: nat,K: nat] :
% 8.83/5.02 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.83/5.02 => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2392_nat__diff__split__asm,axiom,
% 8.83/5.02 ! [P: nat > $o,A: nat,B: nat] :
% 8.83/5.02 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 8.83/5.02 = ( ~ ( ( ( ord_less_nat @ A @ B )
% 8.83/5.02 & ~ ( P @ zero_zero_nat ) )
% 8.83/5.02 | ? [D2: nat] :
% 8.83/5.02 ( ( A
% 8.83/5.02 = ( plus_plus_nat @ B @ D2 ) )
% 8.83/5.02 & ~ ( P @ D2 ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2393_nat__diff__split,axiom,
% 8.83/5.02 ! [P: nat > $o,A: nat,B: nat] :
% 8.83/5.02 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 8.83/5.02 = ( ( ( ord_less_nat @ A @ B )
% 8.83/5.02 => ( P @ zero_zero_nat ) )
% 8.83/5.02 & ! [D2: nat] :
% 8.83/5.02 ( ( A
% 8.83/5.02 = ( plus_plus_nat @ B @ D2 ) )
% 8.83/5.02 => ( P @ D2 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2394_div__le__mono2,axiom,
% 8.83/5.02 ! [M: nat,N: nat,K: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.83/5.02 => ( ( ord_less_eq_nat @ M @ N )
% 8.83/5.02 => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2395_div__greater__zero__iff,axiom,
% 8.83/5.02 ! [M: nat,N: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 8.83/5.02 = ( ( ord_less_eq_nat @ N @ M )
% 8.83/5.02 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2396_nat__one__le__power,axiom,
% 8.83/5.02 ! [I2: nat,N: nat] :
% 8.83/5.02 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I2 )
% 8.83/5.02 => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I2 @ N ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2397_div__less__dividend,axiom,
% 8.83/5.02 ! [N: nat,M: nat] :
% 8.83/5.02 ( ( ord_less_nat @ one_one_nat @ N )
% 8.83/5.02 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.83/5.02 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2398_div__eq__dividend__iff,axiom,
% 8.83/5.02 ! [M: nat,N: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.83/5.02 => ( ( ( divide_divide_nat @ M @ N )
% 8.83/5.02 = M )
% 8.83/5.02 = ( N = one_one_nat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2399_div__less__iff__less__mult,axiom,
% 8.83/5.02 ! [Q3: nat,M: nat,N: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ Q3 )
% 8.83/5.02 => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N )
% 8.83/5.02 = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2400_nat__mult__div__cancel1,axiom,
% 8.83/5.02 ! [K: nat,M: nat,N: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.83/5.02 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.83/5.02 = ( divide_divide_nat @ M @ N ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2401_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
% 8.83/5.02 ! [A: $o,B: $o] :
% 8.83/5.02 ( ( vEBT_T_m_a_x_t @ ( vEBT_Leaf @ A @ B ) )
% 8.83/5.02 = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2402_vebt__insert_Osimps_I3_J,axiom,
% 8.83/5.02 ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
% 8.83/5.02 ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ X2 )
% 8.83/5.02 = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2403_vebt__member_Osimps_I3_J,axiom,
% 8.83/5.02 ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
% 8.83/5.02 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X2 ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2404_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
% 8.83/5.02 ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
% 8.83/5.02 ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S ) @ X2 )
% 8.83/5.02 = one_one_nat ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2405_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
% 8.83/5.02 ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
% 8.83/5.02 ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2406_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
% 8.83/5.02 ! [X2: vEBT_VEBT,Y3: nat] :
% 8.83/5.02 ( ( ( vEBT_T_m_i_n_t @ X2 )
% 8.83/5.02 = Y3 )
% 8.83/5.02 => ( ! [A4: $o] :
% 8.83/5.02 ( ? [B4: $o] :
% 8.83/5.02 ( X2
% 8.83/5.02 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.02 => ( Y3
% 8.83/5.02 != ( plus_plus_nat @ one_one_nat @ ( if_nat @ A4 @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
% 8.83/5.02 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.83/5.02 ( X2
% 8.83/5.02 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 8.83/5.02 => ( Y3 != one_one_nat ) )
% 8.83/5.02 => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 8.83/5.02 ( X2
% 8.83/5.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 8.83/5.02 => ( Y3 != one_one_nat ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2407_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
% 8.83/5.02 ! [X2: vEBT_VEBT] :
% 8.83/5.02 ( ! [A4: $o,B4: $o] :
% 8.83/5.02 ( X2
% 8.83/5.02 != ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.02 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.83/5.02 ( X2
% 8.83/5.02 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 8.83/5.02 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 8.83/5.02 ( X2
% 8.83/5.02 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2408_VEBT__internal_OminNull_Oelims_I1_J,axiom,
% 8.83/5.02 ! [X2: vEBT_VEBT,Y3: $o] :
% 8.83/5.02 ( ( ( vEBT_VEBT_minNull @ X2 )
% 8.83/5.02 = Y3 )
% 8.83/5.02 => ( ( ( X2
% 8.83/5.02 = ( vEBT_Leaf @ $false @ $false ) )
% 8.83/5.02 => ~ Y3 )
% 8.83/5.02 => ( ( ? [Uv2: $o] :
% 8.83/5.02 ( X2
% 8.83/5.02 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 8.83/5.02 => Y3 )
% 8.83/5.02 => ( ( ? [Uu2: $o] :
% 8.83/5.02 ( X2
% 8.83/5.02 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 8.83/5.02 => Y3 )
% 8.83/5.02 => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 8.83/5.02 ( X2
% 8.83/5.02 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 8.83/5.02 => ~ Y3 )
% 8.83/5.02 => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.83/5.02 ( X2
% 8.83/5.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
% 8.83/5.02 => Y3 ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2409_field__le__mult__one__interval,axiom,
% 8.83/5.02 ! [X2: real,Y3: real] :
% 8.83/5.02 ( ! [Z2: real] :
% 8.83/5.02 ( ( ord_less_real @ zero_zero_real @ Z2 )
% 8.83/5.02 => ( ( ord_less_real @ Z2 @ one_one_real )
% 8.83/5.02 => ( ord_less_eq_real @ ( times_times_real @ Z2 @ X2 ) @ Y3 ) ) )
% 8.83/5.02 => ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2410_field__le__mult__one__interval,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat] :
% 8.83/5.02 ( ! [Z2: rat] :
% 8.83/5.02 ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 8.83/5.02 => ( ( ord_less_rat @ Z2 @ one_one_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X2 ) @ Y3 ) ) )
% 8.83/5.02 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2411_mult__le__cancel__left1,axiom,
% 8.83/5.02 ! [C: real,B: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
% 8.83/5.02 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ord_less_eq_real @ one_one_real @ B ) )
% 8.83/5.02 & ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2412_mult__le__cancel__left1,axiom,
% 8.83/5.02 ! [C: rat,B: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
% 8.83/5.02 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 8.83/5.02 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2413_mult__le__cancel__left1,axiom,
% 8.83/5.02 ! [C: int,B: int] :
% 8.83/5.02 ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
% 8.83/5.02 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 8.83/5.02 => ( ord_less_eq_int @ one_one_int @ B ) )
% 8.83/5.02 & ( ( ord_less_int @ C @ zero_zero_int )
% 8.83/5.02 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2414_mult__le__cancel__left2,axiom,
% 8.83/5.02 ! [C: real,A: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 8.83/5.02 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ord_less_eq_real @ A @ one_one_real ) )
% 8.83/5.02 & ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2415_mult__le__cancel__left2,axiom,
% 8.83/5.02 ! [C: rat,A: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 8.83/5.02 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 8.83/5.02 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2416_mult__le__cancel__left2,axiom,
% 8.83/5.02 ! [C: int,A: int] :
% 8.83/5.02 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 8.83/5.02 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 8.83/5.02 => ( ord_less_eq_int @ A @ one_one_int ) )
% 8.83/5.02 & ( ( ord_less_int @ C @ zero_zero_int )
% 8.83/5.02 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2417_mult__le__cancel__right1,axiom,
% 8.83/5.02 ! [C: real,B: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
% 8.83/5.02 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ord_less_eq_real @ one_one_real @ B ) )
% 8.83/5.02 & ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2418_mult__le__cancel__right1,axiom,
% 8.83/5.02 ! [C: rat,B: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
% 8.83/5.02 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 8.83/5.02 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2419_mult__le__cancel__right1,axiom,
% 8.83/5.02 ! [C: int,B: int] :
% 8.83/5.02 ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
% 8.83/5.02 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 8.83/5.02 => ( ord_less_eq_int @ one_one_int @ B ) )
% 8.83/5.02 & ( ( ord_less_int @ C @ zero_zero_int )
% 8.83/5.02 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2420_mult__le__cancel__right2,axiom,
% 8.83/5.02 ! [A: real,C: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 8.83/5.02 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ord_less_eq_real @ A @ one_one_real ) )
% 8.83/5.02 & ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2421_mult__le__cancel__right2,axiom,
% 8.83/5.02 ! [A: rat,C: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 8.83/5.02 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 8.83/5.02 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2422_mult__le__cancel__right2,axiom,
% 8.83/5.02 ! [A: int,C: int] :
% 8.83/5.02 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 8.83/5.02 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 8.83/5.02 => ( ord_less_eq_int @ A @ one_one_int ) )
% 8.83/5.02 & ( ( ord_less_int @ C @ zero_zero_int )
% 8.83/5.02 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2423_mult__less__cancel__left1,axiom,
% 8.83/5.02 ! [C: real,B: real] :
% 8.83/5.02 ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
% 8.83/5.02 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ord_less_real @ one_one_real @ B ) )
% 8.83/5.02 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2424_mult__less__cancel__left1,axiom,
% 8.83/5.02 ! [C: rat,B: rat] :
% 8.83/5.02 ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
% 8.83/5.02 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ord_less_rat @ one_one_rat @ B ) )
% 8.83/5.02 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2425_mult__less__cancel__left1,axiom,
% 8.83/5.02 ! [C: int,B: int] :
% 8.83/5.02 ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
% 8.83/5.02 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.83/5.02 => ( ord_less_int @ one_one_int @ B ) )
% 8.83/5.02 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 8.83/5.02 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2426_mult__less__cancel__left2,axiom,
% 8.83/5.02 ! [C: real,A: real] :
% 8.83/5.02 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 8.83/5.02 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ord_less_real @ A @ one_one_real ) )
% 8.83/5.02 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2427_mult__less__cancel__left2,axiom,
% 8.83/5.02 ! [C: rat,A: rat] :
% 8.83/5.02 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 8.83/5.02 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ord_less_rat @ A @ one_one_rat ) )
% 8.83/5.02 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2428_mult__less__cancel__left2,axiom,
% 8.83/5.02 ! [C: int,A: int] :
% 8.83/5.02 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 8.83/5.02 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.83/5.02 => ( ord_less_int @ A @ one_one_int ) )
% 8.83/5.02 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 8.83/5.02 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2429_mult__less__cancel__right1,axiom,
% 8.83/5.02 ! [C: real,B: real] :
% 8.83/5.02 ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
% 8.83/5.02 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ord_less_real @ one_one_real @ B ) )
% 8.83/5.02 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2430_mult__less__cancel__right1,axiom,
% 8.83/5.02 ! [C: rat,B: rat] :
% 8.83/5.02 ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
% 8.83/5.02 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ord_less_rat @ one_one_rat @ B ) )
% 8.83/5.02 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2431_mult__less__cancel__right1,axiom,
% 8.83/5.02 ! [C: int,B: int] :
% 8.83/5.02 ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
% 8.83/5.02 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.83/5.02 => ( ord_less_int @ one_one_int @ B ) )
% 8.83/5.02 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 8.83/5.02 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2432_mult__less__cancel__right2,axiom,
% 8.83/5.02 ! [A: real,C: real] :
% 8.83/5.02 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 8.83/5.02 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ord_less_real @ A @ one_one_real ) )
% 8.83/5.02 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2433_mult__less__cancel__right2,axiom,
% 8.83/5.02 ! [A: rat,C: rat] :
% 8.83/5.02 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 8.83/5.02 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ord_less_rat @ A @ one_one_rat ) )
% 8.83/5.02 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2434_mult__less__cancel__right2,axiom,
% 8.83/5.02 ! [A: int,C: int] :
% 8.83/5.02 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 8.83/5.02 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.83/5.02 => ( ord_less_int @ A @ one_one_int ) )
% 8.83/5.02 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 8.83/5.02 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2435_divide__left__mono__neg,axiom,
% 8.83/5.02 ! [A: real,B: real,C: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ A @ B )
% 8.83/5.02 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 8.83/5.02 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2436_divide__left__mono__neg,axiom,
% 8.83/5.02 ! [A: rat,B: rat,C: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ A @ B )
% 8.83/5.02 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 8.83/5.02 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2437_mult__imp__le__div__pos,axiom,
% 8.83/5.02 ! [Y3: real,Z: real,X2: real] :
% 8.83/5.02 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.83/5.02 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y3 ) @ X2 )
% 8.83/5.02 => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2438_mult__imp__le__div__pos,axiom,
% 8.83/5.02 ! [Y3: rat,Z: rat,X2: rat] :
% 8.83/5.02 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 8.83/5.02 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y3 ) @ X2 )
% 8.83/5.02 => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2439_mult__imp__div__pos__le,axiom,
% 8.83/5.02 ! [Y3: real,X2: real,Z: real] :
% 8.83/5.02 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.83/5.02 => ( ( ord_less_eq_real @ X2 @ ( times_times_real @ Z @ Y3 ) )
% 8.83/5.02 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ Z ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2440_mult__imp__div__pos__le,axiom,
% 8.83/5.02 ! [Y3: rat,X2: rat,Z: rat] :
% 8.83/5.02 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 8.83/5.02 => ( ( ord_less_eq_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) )
% 8.83/5.02 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ Z ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2441_pos__le__divide__eq,axiom,
% 8.83/5.02 ! [C: real,A: real,B: real] :
% 8.83/5.02 ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 8.83/5.02 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2442_pos__le__divide__eq,axiom,
% 8.83/5.02 ! [C: rat,A: rat,B: rat] :
% 8.83/5.02 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 8.83/5.02 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2443_pos__divide__le__eq,axiom,
% 8.83/5.02 ! [C: real,B: real,A: real] :
% 8.83/5.02 ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 8.83/5.02 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2444_pos__divide__le__eq,axiom,
% 8.83/5.02 ! [C: rat,B: rat,A: rat] :
% 8.83/5.02 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 8.83/5.02 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2445_neg__le__divide__eq,axiom,
% 8.83/5.02 ! [C: real,A: real,B: real] :
% 8.83/5.02 ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 8.83/5.02 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2446_neg__le__divide__eq,axiom,
% 8.83/5.02 ! [C: rat,A: rat,B: rat] :
% 8.83/5.02 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 8.83/5.02 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2447_neg__divide__le__eq,axiom,
% 8.83/5.02 ! [C: real,B: real,A: real] :
% 8.83/5.02 ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 8.83/5.02 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2448_neg__divide__le__eq,axiom,
% 8.83/5.02 ! [C: rat,B: rat,A: rat] :
% 8.83/5.02 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 8.83/5.02 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2449_divide__left__mono,axiom,
% 8.83/5.02 ! [B: real,A: real,C: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ B @ A )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 8.83/5.02 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2450_divide__left__mono,axiom,
% 8.83/5.02 ! [B: rat,A: rat,C: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ B @ A )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 8.83/5.02 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2451_le__divide__eq,axiom,
% 8.83/5.02 ! [A: real,B: real,C: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 8.83/5.02 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 8.83/5.02 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 8.83/5.02 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2452_le__divide__eq,axiom,
% 8.83/5.02 ! [A: rat,B: rat,C: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 8.83/5.02 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 8.83/5.02 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 8.83/5.02 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2453_divide__le__eq,axiom,
% 8.83/5.02 ! [B: real,C: real,A: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 8.83/5.02 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 8.83/5.02 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 8.83/5.02 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2454_divide__le__eq,axiom,
% 8.83/5.02 ! [B: rat,C: rat,A: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 8.83/5.02 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 8.83/5.02 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 8.83/5.02 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2455_divide__le__eq__1,axiom,
% 8.83/5.02 ! [B: real,A: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 8.83/5.02 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.02 & ( ord_less_eq_real @ B @ A ) )
% 8.83/5.02 | ( ( ord_less_real @ A @ zero_zero_real )
% 8.83/5.02 & ( ord_less_eq_real @ A @ B ) )
% 8.83/5.02 | ( A = zero_zero_real ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2456_divide__le__eq__1,axiom,
% 8.83/5.02 ! [B: rat,A: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 8.83/5.02 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.02 & ( ord_less_eq_rat @ B @ A ) )
% 8.83/5.02 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.83/5.02 & ( ord_less_eq_rat @ A @ B ) )
% 8.83/5.02 | ( A = zero_zero_rat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2457_le__divide__eq__1,axiom,
% 8.83/5.02 ! [B: real,A: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 8.83/5.02 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.02 & ( ord_less_eq_real @ A @ B ) )
% 8.83/5.02 | ( ( ord_less_real @ A @ zero_zero_real )
% 8.83/5.02 & ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2458_le__divide__eq__1,axiom,
% 8.83/5.02 ! [B: rat,A: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 8.83/5.02 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.02 & ( ord_less_eq_rat @ A @ B ) )
% 8.83/5.02 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.83/5.02 & ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2459_convex__bound__le,axiom,
% 8.83/5.02 ! [X2: real,A: real,Y3: real,U: real,V: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ X2 @ A )
% 8.83/5.02 => ( ( ord_less_eq_real @ Y3 @ A )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 8.83/5.02 => ( ( ( plus_plus_real @ U @ V )
% 8.83/5.02 = one_one_real )
% 8.83/5.02 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X2 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2460_convex__bound__le,axiom,
% 8.83/5.02 ! [X2: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ X2 @ A )
% 8.83/5.02 => ( ( ord_less_eq_rat @ Y3 @ A )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 8.83/5.02 => ( ( ( plus_plus_rat @ U @ V )
% 8.83/5.02 = one_one_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X2 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2461_convex__bound__le,axiom,
% 8.83/5.02 ! [X2: int,A: int,Y3: int,U: int,V: int] :
% 8.83/5.02 ( ( ord_less_eq_int @ X2 @ A )
% 8.83/5.02 => ( ( ord_less_eq_int @ Y3 @ A )
% 8.83/5.02 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 8.83/5.02 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 8.83/5.02 => ( ( ( plus_plus_int @ U @ V )
% 8.83/5.02 = one_one_int )
% 8.83/5.02 => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X2 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2462_less__divide__eq__numeral_I1_J,axiom,
% 8.83/5.02 ! [W: num,B: real,C: real] :
% 8.83/5.02 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 8.83/5.02 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 8.83/5.02 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 8.83/5.02 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2463_less__divide__eq__numeral_I1_J,axiom,
% 8.83/5.02 ! [W: num,B: rat,C: rat] :
% 8.83/5.02 ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 8.83/5.02 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 8.83/5.02 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 8.83/5.02 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2464_divide__less__eq__numeral_I1_J,axiom,
% 8.83/5.02 ! [B: real,C: real,W: num] :
% 8.83/5.02 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 8.83/5.02 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 8.83/5.02 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 8.83/5.02 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2465_divide__less__eq__numeral_I1_J,axiom,
% 8.83/5.02 ! [B: rat,C: rat,W: num] :
% 8.83/5.02 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 8.83/5.02 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 8.83/5.02 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 8.83/5.02 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2466_frac__le__eq,axiom,
% 8.83/5.02 ! [Y3: real,Z: real,X2: real,W: real] :
% 8.83/5.02 ( ( Y3 != zero_zero_real )
% 8.83/5.02 => ( ( Z != zero_zero_real )
% 8.83/5.02 => ( ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 8.83/5.02 = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2467_frac__le__eq,axiom,
% 8.83/5.02 ! [Y3: rat,Z: rat,X2: rat,W: rat] :
% 8.83/5.02 ( ( Y3 != zero_zero_rat )
% 8.83/5.02 => ( ( Z != zero_zero_rat )
% 8.83/5.02 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 8.83/5.02 = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2468_power__Suc__less,axiom,
% 8.83/5.02 ! [A: real,N: nat] :
% 8.83/5.02 ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.02 => ( ( ord_less_real @ A @ one_one_real )
% 8.83/5.02 => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2469_power__Suc__less,axiom,
% 8.83/5.02 ! [A: rat,N: nat] :
% 8.83/5.02 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.02 => ( ( ord_less_rat @ A @ one_one_rat )
% 8.83/5.02 => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2470_power__Suc__less,axiom,
% 8.83/5.02 ! [A: nat,N: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.83/5.02 => ( ( ord_less_nat @ A @ one_one_nat )
% 8.83/5.02 => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2471_power__Suc__less,axiom,
% 8.83/5.02 ! [A: int,N: nat] :
% 8.83/5.02 ( ( ord_less_int @ zero_zero_int @ A )
% 8.83/5.02 => ( ( ord_less_int @ A @ one_one_int )
% 8.83/5.02 => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2472_frac__less__eq,axiom,
% 8.83/5.02 ! [Y3: real,Z: real,X2: real,W: real] :
% 8.83/5.02 ( ( Y3 != zero_zero_real )
% 8.83/5.02 => ( ( Z != zero_zero_real )
% 8.83/5.02 => ( ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 8.83/5.02 = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2473_frac__less__eq,axiom,
% 8.83/5.02 ! [Y3: rat,Z: rat,X2: rat,W: rat] :
% 8.83/5.02 ( ( Y3 != zero_zero_rat )
% 8.83/5.02 => ( ( Z != zero_zero_rat )
% 8.83/5.02 => ( ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 8.83/5.02 = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2474_vebt__mint_Oelims,axiom,
% 8.83/5.02 ! [X2: vEBT_VEBT,Y3: option_nat] :
% 8.83/5.02 ( ( ( vEBT_vebt_mint @ X2 )
% 8.83/5.02 = Y3 )
% 8.83/5.02 => ( ! [A4: $o,B4: $o] :
% 8.83/5.02 ( ( X2
% 8.83/5.02 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.02 => ~ ( ( A4
% 8.83/5.02 => ( Y3
% 8.83/5.02 = ( some_nat @ zero_zero_nat ) ) )
% 8.83/5.02 & ( ~ A4
% 8.83/5.02 => ( ( B4
% 8.83/5.02 => ( Y3
% 8.83/5.02 = ( some_nat @ one_one_nat ) ) )
% 8.83/5.02 & ( ~ B4
% 8.83/5.02 => ( Y3 = none_nat ) ) ) ) ) )
% 8.83/5.02 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.83/5.02 ( X2
% 8.83/5.02 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 8.83/5.02 => ( Y3 != none_nat ) )
% 8.83/5.02 => ~ ! [Mi2: nat] :
% 8.83/5.02 ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 8.83/5.02 ( X2
% 8.83/5.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 8.83/5.02 => ( Y3
% 8.83/5.02 != ( some_nat @ Mi2 ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2475_power__Suc__le__self,axiom,
% 8.83/5.02 ! [A: real,N: nat] :
% 8.83/5.02 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.02 => ( ( ord_less_eq_real @ A @ one_one_real )
% 8.83/5.02 => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2476_power__Suc__le__self,axiom,
% 8.83/5.02 ! [A: rat,N: nat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.02 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2477_power__Suc__le__self,axiom,
% 8.83/5.02 ! [A: nat,N: nat] :
% 8.83/5.02 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.02 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 8.83/5.02 => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2478_power__Suc__le__self,axiom,
% 8.83/5.02 ! [A: int,N: nat] :
% 8.83/5.02 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.02 => ( ( ord_less_eq_int @ A @ one_one_int )
% 8.83/5.02 => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2479_power__Suc__less__one,axiom,
% 8.83/5.02 ! [A: real,N: nat] :
% 8.83/5.02 ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.02 => ( ( ord_less_real @ A @ one_one_real )
% 8.83/5.02 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2480_power__Suc__less__one,axiom,
% 8.83/5.02 ! [A: rat,N: nat] :
% 8.83/5.02 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.02 => ( ( ord_less_rat @ A @ one_one_rat )
% 8.83/5.02 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2481_power__Suc__less__one,axiom,
% 8.83/5.02 ! [A: nat,N: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.83/5.02 => ( ( ord_less_nat @ A @ one_one_nat )
% 8.83/5.02 => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2482_power__Suc__less__one,axiom,
% 8.83/5.02 ! [A: int,N: nat] :
% 8.83/5.02 ( ( ord_less_int @ zero_zero_int @ A )
% 8.83/5.02 => ( ( ord_less_int @ A @ one_one_int )
% 8.83/5.02 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2483_vebt__maxt_Oelims,axiom,
% 8.83/5.02 ! [X2: vEBT_VEBT,Y3: option_nat] :
% 8.83/5.02 ( ( ( vEBT_vebt_maxt @ X2 )
% 8.83/5.02 = Y3 )
% 8.83/5.02 => ( ! [A4: $o,B4: $o] :
% 8.83/5.02 ( ( X2
% 8.83/5.02 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.02 => ~ ( ( B4
% 8.83/5.02 => ( Y3
% 8.83/5.02 = ( some_nat @ one_one_nat ) ) )
% 8.83/5.02 & ( ~ B4
% 8.83/5.02 => ( ( A4
% 8.83/5.02 => ( Y3
% 8.83/5.02 = ( some_nat @ zero_zero_nat ) ) )
% 8.83/5.02 & ( ~ A4
% 8.83/5.02 => ( Y3 = none_nat ) ) ) ) ) )
% 8.83/5.02 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.83/5.02 ( X2
% 8.83/5.02 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 8.83/5.02 => ( Y3 != none_nat ) )
% 8.83/5.02 => ~ ! [Mi2: nat,Ma2: nat] :
% 8.83/5.02 ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 8.83/5.02 ( X2
% 8.83/5.02 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 8.83/5.02 => ( Y3
% 8.83/5.02 != ( some_nat @ Ma2 ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2484_power__strict__decreasing,axiom,
% 8.83/5.02 ! [N: nat,N5: nat,A: real] :
% 8.83/5.02 ( ( ord_less_nat @ N @ N5 )
% 8.83/5.02 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.02 => ( ( ord_less_real @ A @ one_one_real )
% 8.83/5.02 => ( ord_less_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2485_power__strict__decreasing,axiom,
% 8.83/5.02 ! [N: nat,N5: nat,A: rat] :
% 8.83/5.02 ( ( ord_less_nat @ N @ N5 )
% 8.83/5.02 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.02 => ( ( ord_less_rat @ A @ one_one_rat )
% 8.83/5.02 => ( ord_less_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2486_power__strict__decreasing,axiom,
% 8.83/5.02 ! [N: nat,N5: nat,A: nat] :
% 8.83/5.02 ( ( ord_less_nat @ N @ N5 )
% 8.83/5.02 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 8.83/5.02 => ( ( ord_less_nat @ A @ one_one_nat )
% 8.83/5.02 => ( ord_less_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2487_power__strict__decreasing,axiom,
% 8.83/5.02 ! [N: nat,N5: nat,A: int] :
% 8.83/5.02 ( ( ord_less_nat @ N @ N5 )
% 8.83/5.02 => ( ( ord_less_int @ zero_zero_int @ A )
% 8.83/5.02 => ( ( ord_less_int @ A @ one_one_int )
% 8.83/5.02 => ( ord_less_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2488_power__decreasing,axiom,
% 8.83/5.02 ! [N: nat,N5: nat,A: real] :
% 8.83/5.02 ( ( ord_less_eq_nat @ N @ N5 )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.83/5.02 => ( ( ord_less_eq_real @ A @ one_one_real )
% 8.83/5.02 => ( ord_less_eq_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2489_power__decreasing,axiom,
% 8.83/5.02 ! [N: nat,N5: nat,A: rat] :
% 8.83/5.02 ( ( ord_less_eq_nat @ N @ N5 )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.83/5.02 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2490_power__decreasing,axiom,
% 8.83/5.02 ! [N: nat,N5: nat,A: nat] :
% 8.83/5.02 ( ( ord_less_eq_nat @ N @ N5 )
% 8.83/5.02 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.83/5.02 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 8.83/5.02 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2491_power__decreasing,axiom,
% 8.83/5.02 ! [N: nat,N5: nat,A: int] :
% 8.83/5.02 ( ( ord_less_eq_nat @ N @ N5 )
% 8.83/5.02 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.02 => ( ( ord_less_eq_int @ A @ one_one_int )
% 8.83/5.02 => ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2492_zero__power2,axiom,
% 8.83/5.02 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.02 = zero_zero_rat ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2493_zero__power2,axiom,
% 8.83/5.02 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.02 = zero_zero_nat ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2494_zero__power2,axiom,
% 8.83/5.02 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.02 = zero_zero_real ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2495_zero__power2,axiom,
% 8.83/5.02 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.02 = zero_zero_int ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2496_zero__power2,axiom,
% 8.83/5.02 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.02 = zero_zero_complex ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2497_self__le__power,axiom,
% 8.83/5.02 ! [A: real,N: nat] :
% 8.83/5.02 ( ( ord_less_eq_real @ one_one_real @ A )
% 8.83/5.02 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2498_self__le__power,axiom,
% 8.83/5.02 ! [A: rat,N: nat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 8.83/5.02 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2499_self__le__power,axiom,
% 8.83/5.02 ! [A: nat,N: nat] :
% 8.83/5.02 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 8.83/5.02 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2500_self__le__power,axiom,
% 8.83/5.02 ! [A: int,N: nat] :
% 8.83/5.02 ( ( ord_less_eq_int @ one_one_int @ A )
% 8.83/5.02 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2501_one__less__power,axiom,
% 8.83/5.02 ! [A: real,N: nat] :
% 8.83/5.02 ( ( ord_less_real @ one_one_real @ A )
% 8.83/5.02 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2502_one__less__power,axiom,
% 8.83/5.02 ! [A: rat,N: nat] :
% 8.83/5.02 ( ( ord_less_rat @ one_one_rat @ A )
% 8.83/5.02 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2503_one__less__power,axiom,
% 8.83/5.02 ! [A: nat,N: nat] :
% 8.83/5.02 ( ( ord_less_nat @ one_one_nat @ A )
% 8.83/5.02 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2504_one__less__power,axiom,
% 8.83/5.02 ! [A: int,N: nat] :
% 8.83/5.02 ( ( ord_less_int @ one_one_int @ A )
% 8.83/5.02 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2505_numeral__2__eq__2,axiom,
% 8.83/5.02 ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 8.83/5.02 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2506_pos2,axiom,
% 8.83/5.02 ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2507_power__diff,axiom,
% 8.83/5.02 ! [A: complex,N: nat,M: nat] :
% 8.83/5.02 ( ( A != zero_zero_complex )
% 8.83/5.02 => ( ( ord_less_eq_nat @ N @ M )
% 8.83/5.02 => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
% 8.83/5.02 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2508_power__diff,axiom,
% 8.83/5.02 ! [A: real,N: nat,M: nat] :
% 8.83/5.02 ( ( A != zero_zero_real )
% 8.83/5.02 => ( ( ord_less_eq_nat @ N @ M )
% 8.83/5.02 => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
% 8.83/5.02 = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2509_power__diff,axiom,
% 8.83/5.02 ! [A: rat,N: nat,M: nat] :
% 8.83/5.02 ( ( A != zero_zero_rat )
% 8.83/5.02 => ( ( ord_less_eq_nat @ N @ M )
% 8.83/5.02 => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
% 8.83/5.02 = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2510_power__diff,axiom,
% 8.83/5.02 ! [A: nat,N: nat,M: nat] :
% 8.83/5.02 ( ( A != zero_zero_nat )
% 8.83/5.02 => ( ( ord_less_eq_nat @ N @ M )
% 8.83/5.02 => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
% 8.83/5.02 = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2511_power__diff,axiom,
% 8.83/5.02 ! [A: int,N: nat,M: nat] :
% 8.83/5.02 ( ( A != zero_zero_int )
% 8.83/5.02 => ( ( ord_less_eq_nat @ N @ M )
% 8.83/5.02 => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
% 8.83/5.02 = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2512_Suc__diff__eq__diff__pred,axiom,
% 8.83/5.02 ! [N: nat,M: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 8.83/5.02 = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2513_Suc__pred_H,axiom,
% 8.83/5.02 ! [N: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( N
% 8.83/5.02 = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2514_div__if,axiom,
% 8.83/5.02 ( divide_divide_nat
% 8.83/5.02 = ( ^ [M3: nat,N3: nat] :
% 8.83/5.02 ( if_nat
% 8.83/5.02 @ ( ( ord_less_nat @ M3 @ N3 )
% 8.83/5.02 | ( N3 = zero_zero_nat ) )
% 8.83/5.02 @ zero_zero_nat
% 8.83/5.02 @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N3 ) @ N3 ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2515_add__eq__if,axiom,
% 8.83/5.02 ( plus_plus_nat
% 8.83/5.02 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2516_less__eq__div__iff__mult__less__eq,axiom,
% 8.83/5.02 ! [Q3: nat,M: nat,N: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ Q3 )
% 8.83/5.02 => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q3 ) )
% 8.83/5.02 = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q3 ) @ N ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2517_dividend__less__times__div,axiom,
% 8.83/5.02 ! [N: nat,M: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2518_dividend__less__div__times,axiom,
% 8.83/5.02 ! [N: nat,M: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2519_split__div,axiom,
% 8.83/5.02 ! [P: nat > $o,M: nat,N: nat] :
% 8.83/5.02 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 8.83/5.02 = ( ( ( N = zero_zero_nat )
% 8.83/5.02 => ( P @ zero_zero_nat ) )
% 8.83/5.02 & ( ( N != zero_zero_nat )
% 8.83/5.02 => ! [I4: nat,J3: nat] :
% 8.83/5.02 ( ( ord_less_nat @ J3 @ N )
% 8.83/5.02 => ( ( M
% 8.83/5.02 = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 8.83/5.02 => ( P @ I4 ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2520_mult__eq__if,axiom,
% 8.83/5.02 ( times_times_nat
% 8.83/5.02 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2521_vebt__member_Osimps_I4_J,axiom,
% 8.83/5.02 ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
% 8.83/5.02 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X2 ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2522_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
% 8.83/5.02 ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
% 8.83/5.02 ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ X2 )
% 8.83/5.02 = one_one_nat ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2523_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
% 8.83/5.02 ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X2: nat] :
% 8.83/5.02 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X2 )
% 8.83/5.02 = ( ( X2 = Mi )
% 8.83/5.02 | ( X2 = Ma ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2524_vebt__pred_Osimps_I5_J,axiom,
% 8.83/5.02 ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
% 8.83/5.02 ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve2 ) @ Vf2 )
% 8.83/5.02 = none_nat ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2525_vebt__succ_Osimps_I4_J,axiom,
% 8.83/5.02 ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve2: nat] :
% 8.83/5.02 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve2 )
% 8.83/5.02 = none_nat ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2526_convex__bound__lt,axiom,
% 8.83/5.02 ! [X2: real,A: real,Y3: real,U: real,V: real] :
% 8.83/5.02 ( ( ord_less_real @ X2 @ A )
% 8.83/5.02 => ( ( ord_less_real @ Y3 @ A )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 8.83/5.02 => ( ( ( plus_plus_real @ U @ V )
% 8.83/5.02 = one_one_real )
% 8.83/5.02 => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X2 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2527_convex__bound__lt,axiom,
% 8.83/5.02 ! [X2: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 8.83/5.02 ( ( ord_less_rat @ X2 @ A )
% 8.83/5.02 => ( ( ord_less_rat @ Y3 @ A )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 8.83/5.02 => ( ( ( plus_plus_rat @ U @ V )
% 8.83/5.02 = one_one_rat )
% 8.83/5.02 => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X2 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2528_convex__bound__lt,axiom,
% 8.83/5.02 ! [X2: int,A: int,Y3: int,U: int,V: int] :
% 8.83/5.02 ( ( ord_less_int @ X2 @ A )
% 8.83/5.02 => ( ( ord_less_int @ Y3 @ A )
% 8.83/5.02 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 8.83/5.02 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 8.83/5.02 => ( ( ( plus_plus_int @ U @ V )
% 8.83/5.02 = one_one_int )
% 8.83/5.02 => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X2 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2529_le__divide__eq__numeral_I1_J,axiom,
% 8.83/5.02 ! [W: num,B: real,C: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 8.83/5.02 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 8.83/5.02 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 8.83/5.02 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2530_le__divide__eq__numeral_I1_J,axiom,
% 8.83/5.02 ! [W: num,B: rat,C: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 8.83/5.02 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 8.83/5.02 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 8.83/5.02 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2531_divide__le__eq__numeral_I1_J,axiom,
% 8.83/5.02 ! [B: real,C: real,W: num] :
% 8.83/5.02 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 8.83/5.02 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 8.83/5.02 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.83/5.02 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 8.83/5.02 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.83/5.02 => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2532_divide__le__eq__numeral_I1_J,axiom,
% 8.83/5.02 ! [B: rat,C: rat,W: num] :
% 8.83/5.02 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 8.83/5.02 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 8.83/5.02 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.83/5.02 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 8.83/5.02 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.83/5.02 => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2533_half__gt__zero__iff,axiom,
% 8.83/5.02 ! [A: real] :
% 8.83/5.02 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.83/5.02 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2534_half__gt__zero__iff,axiom,
% 8.83/5.02 ! [A: rat] :
% 8.83/5.02 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 8.83/5.02 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2535_half__gt__zero,axiom,
% 8.83/5.02 ! [A: real] :
% 8.83/5.02 ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.02 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2536_half__gt__zero,axiom,
% 8.83/5.02 ! [A: rat] :
% 8.83/5.02 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.83/5.02 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2537_scaling__mono,axiom,
% 8.83/5.02 ! [U: real,V: real,R2: real,S: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ U @ V )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 8.83/5.02 => ( ( ord_less_eq_real @ R2 @ S )
% 8.83/5.02 => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2538_scaling__mono,axiom,
% 8.83/5.02 ! [U: rat,V: rat,R2: rat,S: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ U @ V )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 8.83/5.02 => ( ( ord_less_eq_rat @ R2 @ S )
% 8.83/5.02 => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2539_power2__le__imp__le,axiom,
% 8.83/5.02 ! [X2: real,Y3: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.83/5.02 => ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2540_power2__le__imp__le,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 8.83/5.02 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2541_power2__le__imp__le,axiom,
% 8.83/5.02 ! [X2: nat,Y3: nat] :
% 8.83/5.02 ( ( ord_less_eq_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.02 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 8.83/5.02 => ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2542_power2__le__imp__le,axiom,
% 8.83/5.02 ! [X2: int,Y3: int] :
% 8.83/5.02 ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.02 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.83/5.02 => ( ord_less_eq_int @ X2 @ Y3 ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2543_power2__eq__imp__eq,axiom,
% 8.83/5.02 ! [X2: real,Y3: real] :
% 8.83/5.02 ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.02 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.83/5.02 => ( X2 = Y3 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2544_power2__eq__imp__eq,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat] :
% 8.83/5.02 ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.02 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 8.83/5.02 => ( X2 = Y3 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2545_power2__eq__imp__eq,axiom,
% 8.83/5.02 ! [X2: nat,Y3: nat] :
% 8.83/5.02 ( ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.02 = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.02 => ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
% 8.83/5.02 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 8.83/5.02 => ( X2 = Y3 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2546_power2__eq__imp__eq,axiom,
% 8.83/5.02 ! [X2: int,Y3: int] :
% 8.83/5.02 ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.02 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.02 => ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.83/5.02 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.83/5.02 => ( X2 = Y3 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2547_zero__le__power2,axiom,
% 8.83/5.02 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2548_zero__le__power2,axiom,
% 8.83/5.02 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2549_zero__le__power2,axiom,
% 8.83/5.02 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2550_power2__less__0,axiom,
% 8.83/5.02 ! [A: real] :
% 8.83/5.02 ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2551_power2__less__0,axiom,
% 8.83/5.02 ! [A: rat] :
% 8.83/5.02 ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2552_power2__less__0,axiom,
% 8.83/5.02 ! [A: int] :
% 8.83/5.02 ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2553_exp__add__not__zero__imp__right,axiom,
% 8.83/5.02 ! [M: nat,N: nat] :
% 8.83/5.02 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 8.83/5.02 != zero_zero_nat )
% 8.83/5.02 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.83/5.02 != zero_zero_nat ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2554_exp__add__not__zero__imp__right,axiom,
% 8.83/5.02 ! [M: nat,N: nat] :
% 8.83/5.02 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 8.83/5.02 != zero_zero_int )
% 8.83/5.02 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 8.83/5.02 != zero_zero_int ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2555_exp__add__not__zero__imp__left,axiom,
% 8.83/5.02 ! [M: nat,N: nat] :
% 8.83/5.02 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 8.83/5.02 != zero_zero_nat )
% 8.83/5.02 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 8.83/5.02 != zero_zero_nat ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2556_exp__add__not__zero__imp__left,axiom,
% 8.83/5.02 ! [M: nat,N: nat] :
% 8.83/5.02 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 8.83/5.02 != zero_zero_int )
% 8.83/5.02 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 8.83/5.02 != zero_zero_int ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2557_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 8.83/5.02 ! [N: nat,M: nat] :
% 8.83/5.02 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.83/5.02 != zero_zero_nat )
% 8.83/5.02 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 8.83/5.02 != zero_zero_nat ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2558_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 8.83/5.02 ! [N: nat,M: nat] :
% 8.83/5.02 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 8.83/5.02 != zero_zero_int )
% 8.83/5.02 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 8.83/5.02 != zero_zero_int ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2559_power__diff__power__eq,axiom,
% 8.83/5.02 ! [A: nat,N: nat,M: nat] :
% 8.83/5.02 ( ( A != zero_zero_nat )
% 8.83/5.02 => ( ( ( ord_less_eq_nat @ N @ M )
% 8.83/5.02 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 8.83/5.02 = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 8.83/5.02 & ( ~ ( ord_less_eq_nat @ N @ M )
% 8.83/5.02 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 8.83/5.02 = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2560_power__diff__power__eq,axiom,
% 8.83/5.02 ! [A: int,N: nat,M: nat] :
% 8.83/5.02 ( ( A != zero_zero_int )
% 8.83/5.02 => ( ( ( ord_less_eq_nat @ N @ M )
% 8.83/5.02 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 8.83/5.02 = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 8.83/5.02 & ( ~ ( ord_less_eq_nat @ N @ M )
% 8.83/5.02 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 8.83/5.02 = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2561_less__2__cases__iff,axiom,
% 8.83/5.02 ! [N: nat] :
% 8.83/5.02 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.02 = ( ( N = zero_zero_nat )
% 8.83/5.02 | ( N
% 8.83/5.02 = ( suc @ zero_zero_nat ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2562_less__2__cases,axiom,
% 8.83/5.02 ! [N: nat] :
% 8.83/5.02 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.02 => ( ( N = zero_zero_nat )
% 8.83/5.02 | ( N
% 8.83/5.02 = ( suc @ zero_zero_nat ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2563_nat__induct2,axiom,
% 8.83/5.02 ! [P: nat > $o,N: nat] :
% 8.83/5.02 ( ( P @ zero_zero_nat )
% 8.83/5.02 => ( ( P @ one_one_nat )
% 8.83/5.02 => ( ! [N2: nat] :
% 8.83/5.02 ( ( P @ N2 )
% 8.83/5.02 => ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.83/5.02 => ( P @ N ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2564_power__eq__if,axiom,
% 8.83/5.02 ( power_power_complex
% 8.83/5.02 = ( ^ [P4: complex,M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P4 @ ( power_power_complex @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2565_power__eq__if,axiom,
% 8.83/5.02 ( power_power_real
% 8.83/5.02 = ( ^ [P4: real,M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P4 @ ( power_power_real @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2566_power__eq__if,axiom,
% 8.83/5.02 ( power_power_rat
% 8.83/5.02 = ( ^ [P4: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P4 @ ( power_power_rat @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2567_power__eq__if,axiom,
% 8.83/5.02 ( power_power_nat
% 8.83/5.02 = ( ^ [P4: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P4 @ ( power_power_nat @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2568_power__eq__if,axiom,
% 8.83/5.02 ( power_power_int
% 8.83/5.02 = ( ^ [P4: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P4 @ ( power_power_int @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2569_power__minus__mult,axiom,
% 8.83/5.02 ! [N: nat,A: complex] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 8.83/5.02 = ( power_power_complex @ A @ N ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2570_power__minus__mult,axiom,
% 8.83/5.02 ! [N: nat,A: real] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 8.83/5.02 = ( power_power_real @ A @ N ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2571_power__minus__mult,axiom,
% 8.83/5.02 ! [N: nat,A: rat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 8.83/5.02 = ( power_power_rat @ A @ N ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2572_power__minus__mult,axiom,
% 8.83/5.02 ! [N: nat,A: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 8.83/5.02 = ( power_power_nat @ A @ N ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2573_power__minus__mult,axiom,
% 8.83/5.02 ! [N: nat,A: int] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 8.83/5.02 = ( power_power_int @ A @ N ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2574_le__div__geq,axiom,
% 8.83/5.02 ! [N: nat,M: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ( ord_less_eq_nat @ N @ M )
% 8.83/5.02 => ( ( divide_divide_nat @ M @ N )
% 8.83/5.02 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2575_split__div_H,axiom,
% 8.83/5.02 ! [P: nat > $o,M: nat,N: nat] :
% 8.83/5.02 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 8.83/5.02 = ( ( ( N = zero_zero_nat )
% 8.83/5.02 & ( P @ zero_zero_nat ) )
% 8.83/5.02 | ? [Q4: nat] :
% 8.83/5.02 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
% 8.83/5.02 & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
% 8.83/5.02 & ( P @ Q4 ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2576_vebt__pred_Osimps_I6_J,axiom,
% 8.83/5.02 ! [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
% 8.83/5.02 ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 )
% 8.83/5.02 = none_nat ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2577_vebt__succ_Osimps_I5_J,axiom,
% 8.83/5.02 ! [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
% 8.83/5.02 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 )
% 8.83/5.02 = none_nat ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2578_power2__less__imp__less,axiom,
% 8.83/5.02 ! [X2: real,Y3: real] :
% 8.83/5.02 ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.02 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.83/5.02 => ( ord_less_real @ X2 @ Y3 ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2579_power2__less__imp__less,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat] :
% 8.83/5.02 ( ( ord_less_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 8.83/5.02 => ( ord_less_rat @ X2 @ Y3 ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2580_power2__less__imp__less,axiom,
% 8.83/5.02 ! [X2: nat,Y3: nat] :
% 8.83/5.02 ( ( ord_less_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.02 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 8.83/5.02 => ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2581_power2__less__imp__less,axiom,
% 8.83/5.02 ! [X2: int,Y3: int] :
% 8.83/5.02 ( ( ord_less_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.83/5.02 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.83/5.02 => ( ord_less_int @ X2 @ Y3 ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2582_sum__power2__ge__zero,axiom,
% 8.83/5.02 ! [X2: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2583_sum__power2__ge__zero,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2584_sum__power2__ge__zero,axiom,
% 8.83/5.02 ! [X2: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2585_sum__power2__le__zero__iff,axiom,
% 8.83/5.02 ! [X2: real,Y3: real] :
% 8.83/5.02 ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 8.83/5.02 = ( ( X2 = zero_zero_real )
% 8.83/5.02 & ( Y3 = zero_zero_real ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2586_sum__power2__le__zero__iff,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat] :
% 8.83/5.02 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 8.83/5.02 = ( ( X2 = zero_zero_rat )
% 8.83/5.02 & ( Y3 = zero_zero_rat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2587_sum__power2__le__zero__iff,axiom,
% 8.83/5.02 ! [X2: int,Y3: int] :
% 8.83/5.02 ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 8.83/5.02 = ( ( X2 = zero_zero_int )
% 8.83/5.02 & ( Y3 = zero_zero_int ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2588_sum__power2__gt__zero__iff,axiom,
% 8.83/5.02 ! [X2: real,Y3: real] :
% 8.83/5.02 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.83/5.02 = ( ( X2 != zero_zero_real )
% 8.83/5.02 | ( Y3 != zero_zero_real ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2589_sum__power2__gt__zero__iff,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat] :
% 8.83/5.02 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.83/5.02 = ( ( X2 != zero_zero_rat )
% 8.83/5.02 | ( Y3 != zero_zero_rat ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2590_sum__power2__gt__zero__iff,axiom,
% 8.83/5.02 ! [X2: int,Y3: int] :
% 8.83/5.02 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.83/5.02 = ( ( X2 != zero_zero_int )
% 8.83/5.02 | ( Y3 != zero_zero_int ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2591_not__sum__power2__lt__zero,axiom,
% 8.83/5.02 ! [X2: real,Y3: real] :
% 8.83/5.02 ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2592_not__sum__power2__lt__zero,axiom,
% 8.83/5.02 ! [X2: rat,Y3: rat] :
% 8.83/5.02 ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2593_not__sum__power2__lt__zero,axiom,
% 8.83/5.02 ! [X2: int,Y3: int] :
% 8.83/5.02 ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2594_zero__le__even__power_H,axiom,
% 8.83/5.02 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2595_zero__le__even__power_H,axiom,
% 8.83/5.02 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2596_zero__le__even__power_H,axiom,
% 8.83/5.02 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2597_nat__bit__induct,axiom,
% 8.83/5.02 ! [P: nat > $o,N: nat] :
% 8.83/5.02 ( ( P @ zero_zero_nat )
% 8.83/5.02 => ( ! [N2: nat] :
% 8.83/5.02 ( ( P @ N2 )
% 8.83/5.02 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 8.83/5.02 => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 8.83/5.02 => ( ! [N2: nat] :
% 8.83/5.02 ( ( P @ N2 )
% 8.83/5.02 => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 8.83/5.02 => ( P @ N ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2598_Suc__n__div__2__gt__zero,axiom,
% 8.83/5.02 ! [N: nat] :
% 8.83/5.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.02 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2599_div__2__gt__zero,axiom,
% 8.83/5.02 ! [N: nat] :
% 8.83/5.02 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.83/5.02 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2600_length__induct,axiom,
% 8.83/5.02 ! [P: list_VEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
% 8.83/5.02 ( ! [Xs3: list_VEBT_VEBT] :
% 8.83/5.02 ( ! [Ys2: list_VEBT_VEBT] :
% 8.83/5.02 ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
% 8.83/5.02 => ( P @ Ys2 ) )
% 8.83/5.02 => ( P @ Xs3 ) )
% 8.83/5.02 => ( P @ Xs2 ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2601_length__induct,axiom,
% 8.83/5.02 ! [P: list_o > $o,Xs2: list_o] :
% 8.83/5.02 ( ! [Xs3: list_o] :
% 8.83/5.02 ( ! [Ys2: list_o] :
% 8.83/5.02 ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
% 8.83/5.02 => ( P @ Ys2 ) )
% 8.83/5.02 => ( P @ Xs3 ) )
% 8.83/5.02 => ( P @ Xs2 ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2602_length__induct,axiom,
% 8.83/5.02 ! [P: list_nat > $o,Xs2: list_nat] :
% 8.83/5.02 ( ! [Xs3: list_nat] :
% 8.83/5.02 ( ! [Ys2: list_nat] :
% 8.83/5.02 ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
% 8.83/5.02 => ( P @ Ys2 ) )
% 8.83/5.02 => ( P @ Xs3 ) )
% 8.83/5.02 => ( P @ Xs2 ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2603_length__induct,axiom,
% 8.83/5.02 ! [P: list_int > $o,Xs2: list_int] :
% 8.83/5.02 ( ! [Xs3: list_int] :
% 8.83/5.02 ( ! [Ys2: list_int] :
% 8.83/5.02 ( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs3 ) )
% 8.83/5.02 => ( P @ Ys2 ) )
% 8.83/5.02 => ( P @ Xs3 ) )
% 8.83/5.02 => ( P @ Xs2 ) ) ).
% 8.83/5.02
% 8.83/5.02 thf(fact_2604_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 8.83/5.02 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 8.83/5.02 ( ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 8.83/5.02 = Y3 )
% 8.83/5.02 => ( ! [A4: $o,B4: $o] :
% 8.83/5.02 ( ( X2
% 8.83/5.02 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.02 => ( Y3
% 8.83/5.02 = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 8.83/5.02 => A4 )
% 8.83/5.02 & ( ( Xa2 != zero_zero_nat )
% 8.83/5.02 => ( ( ( Xa2 = one_one_nat )
% 8.83/5.02 => B4 )
% 8.83/5.02 & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 8.83/5.02 => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 8.83/5.03 => Y3 )
% 8.83/5.03 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
% 8.83/5.03 ( ? [S2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) )
% 8.83/5.03 => ( Y3
% 8.83/5.03 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2605_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat] :
% 8.83/5.03 ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 8.83/5.03 => ( ! [A4: $o,B4: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.03 => ~ ( ( ( Xa2 = zero_zero_nat )
% 8.83/5.03 => A4 )
% 8.83/5.03 & ( ( Xa2 != zero_zero_nat )
% 8.83/5.03 => ( ( ( Xa2 = one_one_nat )
% 8.83/5.03 => B4 )
% 8.83/5.03 & ( Xa2 = one_one_nat ) ) ) ) )
% 8.83/5.03 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
% 8.83/5.03 ( ? [S2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) )
% 8.83/5.03 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2606_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat] :
% 8.83/5.03 ( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 8.83/5.03 => ( ! [A4: $o,B4: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.03 => ( ( ( Xa2 = zero_zero_nat )
% 8.83/5.03 => A4 )
% 8.83/5.03 & ( ( Xa2 != zero_zero_nat )
% 8.83/5.03 => ( ( ( Xa2 = one_one_nat )
% 8.83/5.03 => B4 )
% 8.83/5.03 & ( Xa2 = one_one_nat ) ) ) ) )
% 8.83/5.03 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 8.83/5.03 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
% 8.83/5.03 ( ? [S2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) )
% 8.83/5.03 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2607_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Y3: nat] :
% 8.83/5.03 ( ( ( vEBT_T_m_a_x_t @ X2 )
% 8.83/5.03 = Y3 )
% 8.83/5.03 => ( ! [A4: $o,B4: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.03 => ( Y3
% 8.83/5.03 != ( plus_plus_nat @ one_one_nat @ ( if_nat @ B4 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
% 8.83/5.03 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 8.83/5.03 => ( Y3 != one_one_nat ) )
% 8.83/5.03 => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 8.83/5.03 => ( Y3 != one_one_nat ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2608_odd__0__le__power__imp__0__le,axiom,
% 8.83/5.03 ! [A: real,N: nat] :
% 8.83/5.03 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 8.83/5.03 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2609_odd__0__le__power__imp__0__le,axiom,
% 8.83/5.03 ! [A: rat,N: nat] :
% 8.83/5.03 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 8.83/5.03 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2610_odd__0__le__power__imp__0__le,axiom,
% 8.83/5.03 ! [A: int,N: nat] :
% 8.83/5.03 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 8.83/5.03 => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2611_odd__power__less__zero,axiom,
% 8.83/5.03 ! [A: real,N: nat] :
% 8.83/5.03 ( ( ord_less_real @ A @ zero_zero_real )
% 8.83/5.03 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2612_odd__power__less__zero,axiom,
% 8.83/5.03 ! [A: rat,N: nat] :
% 8.83/5.03 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.83/5.03 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2613_odd__power__less__zero,axiom,
% 8.83/5.03 ! [A: int,N: nat] :
% 8.83/5.03 ( ( ord_less_int @ A @ zero_zero_int )
% 8.83/5.03 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2614_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 8.83/5.03 ! [X2: nat,N: nat,M: nat] :
% 8.83/5.03 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 8.83/5.03 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.03 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.83/5.03 => ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2615_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 8.83/5.03 ! [X2: nat,N: nat,M: nat] :
% 8.83/5.03 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 8.83/5.03 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.03 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.83/5.03 => ( ord_less_nat @ ( vEBT_VEBT_low @ X2 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2616_vebt__member_Oelims_I2_J,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat] :
% 8.83/5.03 ( ( vEBT_vebt_member @ X2 @ Xa2 )
% 8.83/5.03 => ( ! [A4: $o,B4: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.03 => ~ ( ( ( Xa2 = zero_zero_nat )
% 8.83/5.03 => A4 )
% 8.83/5.03 & ( ( Xa2 != zero_zero_nat )
% 8.83/5.03 => ( ( ( Xa2 = one_one_nat )
% 8.83/5.03 => B4 )
% 8.83/5.03 & ( Xa2 = one_one_nat ) ) ) ) )
% 8.83/5.03 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
% 8.83/5.03 ( ? [Summary2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.83/5.03 => ~ ( ( Xa2 != Mi2 )
% 8.83/5.03 => ( ( Xa2 != Ma2 )
% 8.83/5.03 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 8.83/5.03 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 8.83/5.03 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 8.83/5.03 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 8.83/5.03 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2617_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 8.83/5.03 ( ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 8.83/5.03 = Y3 )
% 8.83/5.03 => ( ( ? [Uu2: $o,Uv2: $o] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.83/5.03 => Y3 )
% 8.83/5.03 => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 8.83/5.03 => Y3 )
% 8.83/5.03 => ( ! [Mi2: nat,Ma2: nat] :
% 8.83/5.03 ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 8.83/5.03 => ( Y3
% 8.83/5.03 = ( ~ ( ( Xa2 = Mi2 )
% 8.83/5.03 | ( Xa2 = Ma2 ) ) ) ) )
% 8.83/5.03 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
% 8.83/5.03 ( ? [Vc2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
% 8.83/5.03 => ( Y3
% 8.83/5.03 = ( ~ ( ( Xa2 = Mi2 )
% 8.83/5.03 | ( Xa2 = Ma2 )
% 8.83/5.03 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) )
% 8.83/5.03 => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
% 8.83/5.03 ( ? [Vd2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
% 8.83/5.03 => ( Y3
% 8.83/5.03 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2618_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat] :
% 8.83/5.03 ( ~ ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 8.83/5.03 => ( ! [Uu2: $o,Uv2: $o] :
% 8.83/5.03 ( X2
% 8.83/5.03 != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.83/5.03 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 8.83/5.03 => ( ! [Mi2: nat,Ma2: nat] :
% 8.83/5.03 ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 8.83/5.03 => ( ( Xa2 = Mi2 )
% 8.83/5.03 | ( Xa2 = Ma2 ) ) )
% 8.83/5.03 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
% 8.83/5.03 ( ? [Vc2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
% 8.83/5.03 => ( ( Xa2 = Mi2 )
% 8.83/5.03 | ( Xa2 = Ma2 )
% 8.83/5.03 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
% 8.83/5.03 => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
% 8.83/5.03 ( ? [Vd2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
% 8.83/5.03 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2619_vebt__member_Oelims_I1_J,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 8.83/5.03 ( ( ( vEBT_vebt_member @ X2 @ Xa2 )
% 8.83/5.03 = Y3 )
% 8.83/5.03 => ( ! [A4: $o,B4: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.03 => ( Y3
% 8.83/5.03 = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 8.83/5.03 => A4 )
% 8.83/5.03 & ( ( Xa2 != zero_zero_nat )
% 8.83/5.03 => ( ( ( Xa2 = one_one_nat )
% 8.83/5.03 => B4 )
% 8.83/5.03 & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 8.83/5.03 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 8.83/5.03 => Y3 )
% 8.83/5.03 => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 8.83/5.03 => Y3 )
% 8.83/5.03 => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 8.83/5.03 => Y3 )
% 8.83/5.03 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
% 8.83/5.03 ( ? [Summary2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.83/5.03 => ( Y3
% 8.83/5.03 = ( ~ ( ( Xa2 != Mi2 )
% 8.83/5.03 => ( ( Xa2 != Ma2 )
% 8.83/5.03 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 8.83/5.03 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 8.83/5.03 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 8.83/5.03 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 8.83/5.03 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2620_vebt__member_Oelims_I3_J,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat] :
% 8.83/5.03 ( ~ ( vEBT_vebt_member @ X2 @ Xa2 )
% 8.83/5.03 => ( ! [A4: $o,B4: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.03 => ( ( ( Xa2 = zero_zero_nat )
% 8.83/5.03 => A4 )
% 8.83/5.03 & ( ( Xa2 != zero_zero_nat )
% 8.83/5.03 => ( ( ( Xa2 = one_one_nat )
% 8.83/5.03 => B4 )
% 8.83/5.03 & ( Xa2 = one_one_nat ) ) ) ) )
% 8.83/5.03 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 8.83/5.03 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 8.83/5.03 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 8.83/5.03 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
% 8.83/5.03 ( ? [Summary2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.83/5.03 => ( ( Xa2 != Mi2 )
% 8.83/5.03 => ( ( Xa2 != Ma2 )
% 8.83/5.03 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 8.83/5.03 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 8.83/5.03 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 8.83/5.03 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 8.83/5.03 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2621_arith__geo__mean,axiom,
% 8.83/5.03 ! [U: real,X2: real,Y3: real] :
% 8.83/5.03 ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.03 = ( times_times_real @ X2 @ Y3 ) )
% 8.83/5.03 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.83/5.03 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.83/5.03 => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2622_arith__geo__mean,axiom,
% 8.83/5.03 ! [U: rat,X2: rat,Y3: rat] :
% 8.83/5.03 ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.83/5.03 = ( times_times_rat @ X2 @ Y3 ) )
% 8.83/5.03 => ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 8.83/5.03 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 8.83/5.03 => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2623_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat,Y3: nat] :
% 8.83/5.03 ( ( ( vEBT_T_i_n_s_e_r_t2 @ X2 @ Xa2 )
% 8.83/5.03 = Y3 )
% 8.83/5.03 => ( ( ? [A4: $o,B4: $o] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.03 => ( Y3 != one_one_nat ) )
% 8.83/5.03 => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 8.83/5.03 => ( Y3 != one_one_nat ) )
% 8.83/5.03 => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 8.83/5.03 => ( Y3 != one_one_nat ) )
% 8.83/5.03 => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
% 8.83/5.03 => ( Y3 != one_one_nat ) )
% 8.83/5.03 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.83/5.03 => ( Y3
% 8.83/5.03 != ( if_nat
% 8.83/5.03 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 & ~ ( ( Xa2 = Mi2 )
% 8.83/5.03 | ( Xa2 = Ma2 ) ) )
% 8.83/5.03 @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
% 8.83/5.03 @ one_one_nat ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2624_invar__vebt_Osimps,axiom,
% 8.83/5.03 ( vEBT_invar_vebt
% 8.83/5.03 = ( ^ [A1: vEBT_VEBT,A22: nat] :
% 8.83/5.03 ( ( ? [A3: $o,B2: $o] :
% 8.83/5.03 ( A1
% 8.83/5.03 = ( vEBT_Leaf @ A3 @ B2 ) )
% 8.83/5.03 & ( A22
% 8.83/5.03 = ( suc @ zero_zero_nat ) ) )
% 8.83/5.03 | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
% 8.83/5.03 ( ( A1
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList3 @ Summary3 ) )
% 8.83/5.03 & ! [X: vEBT_VEBT] :
% 8.83/5.03 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.83/5.03 => ( vEBT_invar_vebt @ X @ N3 ) )
% 8.83/5.03 & ( vEBT_invar_vebt @ Summary3 @ N3 )
% 8.83/5.03 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 8.83/5.03 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 8.83/5.03 & ( A22
% 8.83/5.03 = ( plus_plus_nat @ N3 @ N3 ) )
% 8.83/5.03 & ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X4 )
% 8.83/5.03 & ! [X: vEBT_VEBT] :
% 8.83/5.03 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.83/5.03 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 8.83/5.03 | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
% 8.83/5.03 ( ( A1
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList3 @ Summary3 ) )
% 8.83/5.03 & ! [X: vEBT_VEBT] :
% 8.83/5.03 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.83/5.03 => ( vEBT_invar_vebt @ X @ N3 ) )
% 8.83/5.03 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
% 8.83/5.03 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 8.83/5.03 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 8.83/5.03 & ( A22
% 8.83/5.03 = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
% 8.83/5.03 & ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X4 )
% 8.83/5.03 & ! [X: vEBT_VEBT] :
% 8.83/5.03 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.83/5.03 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 8.83/5.03 | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 8.83/5.03 ( ( A1
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList3 @ Summary3 ) )
% 8.83/5.03 & ! [X: vEBT_VEBT] :
% 8.83/5.03 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.83/5.03 => ( vEBT_invar_vebt @ X @ N3 ) )
% 8.83/5.03 & ( vEBT_invar_vebt @ Summary3 @ N3 )
% 8.83/5.03 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 8.83/5.03 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 8.83/5.03 & ( A22
% 8.83/5.03 = ( plus_plus_nat @ N3 @ N3 ) )
% 8.83/5.03 & ! [I4: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 8.83/5.03 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X4 ) )
% 8.83/5.03 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 8.83/5.03 & ( ( Mi3 = Ma3 )
% 8.83/5.03 => ! [X: vEBT_VEBT] :
% 8.83/5.03 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.83/5.03 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 8.83/5.03 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 8.83/5.03 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 8.83/5.03 & ( ( Mi3 != Ma3 )
% 8.83/5.03 => ! [I4: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
% 8.83/5.03 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
% 8.83/5.03 = I4 )
% 8.83/5.03 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
% 8.83/5.03 & ! [X: nat] :
% 8.83/5.03 ( ( ( ( vEBT_VEBT_high @ X @ N3 )
% 8.83/5.03 = I4 )
% 8.83/5.03 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X @ N3 ) ) )
% 8.83/5.03 => ( ( ord_less_nat @ Mi3 @ X )
% 8.83/5.03 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) )
% 8.83/5.03 | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 8.83/5.03 ( ( A1
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList3 @ Summary3 ) )
% 8.83/5.03 & ! [X: vEBT_VEBT] :
% 8.83/5.03 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.83/5.03 => ( vEBT_invar_vebt @ X @ N3 ) )
% 8.83/5.03 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
% 8.83/5.03 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 8.83/5.03 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 8.83/5.03 & ( A22
% 8.83/5.03 = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
% 8.83/5.03 & ! [I4: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 8.83/5.03 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X4 ) )
% 8.83/5.03 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 8.83/5.03 & ( ( Mi3 = Ma3 )
% 8.83/5.03 => ! [X: vEBT_VEBT] :
% 8.83/5.03 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 8.83/5.03 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 8.83/5.03 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 8.83/5.03 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 8.83/5.03 & ( ( Mi3 != Ma3 )
% 8.83/5.03 => ! [I4: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
% 8.83/5.03 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
% 8.83/5.03 = I4 )
% 8.83/5.03 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
% 8.83/5.03 & ! [X: nat] :
% 8.83/5.03 ( ( ( ( vEBT_VEBT_high @ X @ N3 )
% 8.83/5.03 = I4 )
% 8.83/5.03 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X @ N3 ) ) )
% 8.83/5.03 => ( ( ord_less_nat @ Mi3 @ X )
% 8.83/5.03 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2625_invar__vebt_Ocases,axiom,
% 8.83/5.03 ! [A12: vEBT_VEBT,A23: nat] :
% 8.83/5.03 ( ( vEBT_invar_vebt @ A12 @ A23 )
% 8.83/5.03 => ( ( ? [A4: $o,B4: $o] :
% 8.83/5.03 ( A12
% 8.83/5.03 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.03 => ( A23
% 8.83/5.03 != ( suc @ zero_zero_nat ) ) )
% 8.83/5.03 => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 8.83/5.03 ( ( A12
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
% 8.83/5.03 => ( ( A23 = Deg2 )
% 8.83/5.03 => ( ! [X3: vEBT_VEBT] :
% 8.83/5.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_invar_vebt @ X3 @ N2 ) )
% 8.83/5.03 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 8.83/5.03 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 8.83/5.03 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 8.83/5.03 => ( ( M4 = N2 )
% 8.83/5.03 => ( ( Deg2
% 8.83/5.03 = ( plus_plus_nat @ N2 @ M4 ) )
% 8.83/5.03 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
% 8.83/5.03 => ~ ! [X3: vEBT_VEBT] :
% 8.83/5.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.83/5.03 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ) ) ) ) ) ) )
% 8.83/5.03 => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 8.83/5.03 ( ( A12
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
% 8.83/5.03 => ( ( A23 = Deg2 )
% 8.83/5.03 => ( ! [X3: vEBT_VEBT] :
% 8.83/5.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_invar_vebt @ X3 @ N2 ) )
% 8.83/5.03 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 8.83/5.03 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 8.83/5.03 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 8.83/5.03 => ( ( M4
% 8.83/5.03 = ( suc @ N2 ) )
% 8.83/5.03 => ( ( Deg2
% 8.83/5.03 = ( plus_plus_nat @ N2 @ M4 ) )
% 8.83/5.03 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
% 8.83/5.03 => ~ ! [X3: vEBT_VEBT] :
% 8.83/5.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.83/5.03 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ) ) ) ) ) ) )
% 8.83/5.03 => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 8.83/5.03 ( ( A12
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
% 8.83/5.03 => ( ( A23 = Deg2 )
% 8.83/5.03 => ( ! [X3: vEBT_VEBT] :
% 8.83/5.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_invar_vebt @ X3 @ N2 ) )
% 8.83/5.03 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 8.83/5.03 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 8.83/5.03 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 8.83/5.03 => ( ( M4 = N2 )
% 8.83/5.03 => ( ( Deg2
% 8.83/5.03 = ( plus_plus_nat @ N2 @ M4 ) )
% 8.83/5.03 => ( ! [I: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 8.83/5.03 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ X4 ) )
% 8.83/5.03 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
% 8.83/5.03 => ( ( ( Mi2 = Ma2 )
% 8.83/5.03 => ! [X3: vEBT_VEBT] :
% 8.83/5.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.83/5.03 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
% 8.83/5.03 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 8.83/5.03 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 8.83/5.03 => ~ ( ( Mi2 != Ma2 )
% 8.83/5.03 => ! [I: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 8.83/5.03 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 8.83/5.03 = I )
% 8.83/5.03 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 8.83/5.03 & ! [X3: nat] :
% 8.83/5.03 ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
% 8.83/5.03 = I )
% 8.83/5.03 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
% 8.83/5.03 => ( ( ord_less_nat @ Mi2 @ X3 )
% 8.83/5.03 & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 8.83/5.03 => ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 8.83/5.03 ( ( A12
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
% 8.83/5.03 => ( ( A23 = Deg2 )
% 8.83/5.03 => ( ! [X3: vEBT_VEBT] :
% 8.83/5.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_invar_vebt @ X3 @ N2 ) )
% 8.83/5.03 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 8.83/5.03 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 8.83/5.03 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 8.83/5.03 => ( ( M4
% 8.83/5.03 = ( suc @ N2 ) )
% 8.83/5.03 => ( ( Deg2
% 8.83/5.03 = ( plus_plus_nat @ N2 @ M4 ) )
% 8.83/5.03 => ( ! [I: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 8.83/5.03 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ X4 ) )
% 8.83/5.03 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
% 8.83/5.03 => ( ( ( Mi2 = Ma2 )
% 8.83/5.03 => ! [X3: vEBT_VEBT] :
% 8.83/5.03 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.83/5.03 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
% 8.83/5.03 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 8.83/5.03 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 8.83/5.03 => ~ ( ( Mi2 != Ma2 )
% 8.83/5.03 => ! [I: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 8.83/5.03 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
% 8.83/5.03 = I )
% 8.83/5.03 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
% 8.83/5.03 & ! [X3: nat] :
% 8.83/5.03 ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
% 8.83/5.03 = I )
% 8.83/5.03 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
% 8.83/5.03 => ( ( ord_less_nat @ Mi2 @ X3 )
% 8.83/5.03 & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2626_list__eq__iff__nth__eq,axiom,
% 8.83/5.03 ( ( ^ [Y6: list_VEBT_VEBT,Z5: list_VEBT_VEBT] : ( Y6 = Z5 ) )
% 8.83/5.03 = ( ^ [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 8.83/5.03 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 8.83/5.03 = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
% 8.83/5.03 & ! [I4: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 8.83/5.03 => ( ( nth_VEBT_VEBT @ Xs @ I4 )
% 8.83/5.03 = ( nth_VEBT_VEBT @ Ys3 @ I4 ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2627_list__eq__iff__nth__eq,axiom,
% 8.83/5.03 ( ( ^ [Y6: list_o,Z5: list_o] : ( Y6 = Z5 ) )
% 8.83/5.03 = ( ^ [Xs: list_o,Ys3: list_o] :
% 8.83/5.03 ( ( ( size_size_list_o @ Xs )
% 8.83/5.03 = ( size_size_list_o @ Ys3 ) )
% 8.83/5.03 & ! [I4: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 8.83/5.03 => ( ( nth_o @ Xs @ I4 )
% 8.83/5.03 = ( nth_o @ Ys3 @ I4 ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2628_list__eq__iff__nth__eq,axiom,
% 8.83/5.03 ( ( ^ [Y6: list_nat,Z5: list_nat] : ( Y6 = Z5 ) )
% 8.83/5.03 = ( ^ [Xs: list_nat,Ys3: list_nat] :
% 8.83/5.03 ( ( ( size_size_list_nat @ Xs )
% 8.83/5.03 = ( size_size_list_nat @ Ys3 ) )
% 8.83/5.03 & ! [I4: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 8.83/5.03 => ( ( nth_nat @ Xs @ I4 )
% 8.83/5.03 = ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2629_list__eq__iff__nth__eq,axiom,
% 8.83/5.03 ( ( ^ [Y6: list_int,Z5: list_int] : ( Y6 = Z5 ) )
% 8.83/5.03 = ( ^ [Xs: list_int,Ys3: list_int] :
% 8.83/5.03 ( ( ( size_size_list_int @ Xs )
% 8.83/5.03 = ( size_size_list_int @ Ys3 ) )
% 8.83/5.03 & ! [I4: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 8.83/5.03 => ( ( nth_int @ Xs @ I4 )
% 8.83/5.03 = ( nth_int @ Ys3 @ I4 ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2630_Skolem__list__nth,axiom,
% 8.83/5.03 ! [K: nat,P: nat > vEBT_VEBT > $o] :
% 8.83/5.03 ( ( ! [I4: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I4 @ K )
% 8.83/5.03 => ? [X4: vEBT_VEBT] : ( P @ I4 @ X4 ) ) )
% 8.83/5.03 = ( ? [Xs: list_VEBT_VEBT] :
% 8.83/5.03 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 8.83/5.03 = K )
% 8.83/5.03 & ! [I4: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I4 @ K )
% 8.83/5.03 => ( P @ I4 @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2631_Skolem__list__nth,axiom,
% 8.83/5.03 ! [K: nat,P: nat > $o > $o] :
% 8.83/5.03 ( ( ! [I4: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I4 @ K )
% 8.83/5.03 => ? [X4: $o] : ( P @ I4 @ X4 ) ) )
% 8.83/5.03 = ( ? [Xs: list_o] :
% 8.83/5.03 ( ( ( size_size_list_o @ Xs )
% 8.83/5.03 = K )
% 8.83/5.03 & ! [I4: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I4 @ K )
% 8.83/5.03 => ( P @ I4 @ ( nth_o @ Xs @ I4 ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2632_Skolem__list__nth,axiom,
% 8.83/5.03 ! [K: nat,P: nat > nat > $o] :
% 8.83/5.03 ( ( ! [I4: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I4 @ K )
% 8.83/5.03 => ? [X4: nat] : ( P @ I4 @ X4 ) ) )
% 8.83/5.03 = ( ? [Xs: list_nat] :
% 8.83/5.03 ( ( ( size_size_list_nat @ Xs )
% 8.83/5.03 = K )
% 8.83/5.03 & ! [I4: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I4 @ K )
% 8.83/5.03 => ( P @ I4 @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2633_Skolem__list__nth,axiom,
% 8.83/5.03 ! [K: nat,P: nat > int > $o] :
% 8.83/5.03 ( ( ! [I4: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I4 @ K )
% 8.83/5.03 => ? [X4: int] : ( P @ I4 @ X4 ) ) )
% 8.83/5.03 = ( ? [Xs: list_int] :
% 8.83/5.03 ( ( ( size_size_list_int @ Xs )
% 8.83/5.03 = K )
% 8.83/5.03 & ! [I4: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I4 @ K )
% 8.83/5.03 => ( P @ I4 @ ( nth_int @ Xs @ I4 ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2634_nth__equalityI,axiom,
% 8.83/5.03 ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 8.83/5.03 ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 8.83/5.03 = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 8.83/5.03 => ( ! [I3: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 8.83/5.03 => ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
% 8.83/5.03 = ( nth_VEBT_VEBT @ Ys @ I3 ) ) )
% 8.83/5.03 => ( Xs2 = Ys ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2635_nth__equalityI,axiom,
% 8.83/5.03 ! [Xs2: list_o,Ys: list_o] :
% 8.83/5.03 ( ( ( size_size_list_o @ Xs2 )
% 8.83/5.03 = ( size_size_list_o @ Ys ) )
% 8.83/5.03 => ( ! [I3: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
% 8.83/5.03 => ( ( nth_o @ Xs2 @ I3 )
% 8.83/5.03 = ( nth_o @ Ys @ I3 ) ) )
% 8.83/5.03 => ( Xs2 = Ys ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2636_nth__equalityI,axiom,
% 8.83/5.03 ! [Xs2: list_nat,Ys: list_nat] :
% 8.83/5.03 ( ( ( size_size_list_nat @ Xs2 )
% 8.83/5.03 = ( size_size_list_nat @ Ys ) )
% 8.83/5.03 => ( ! [I3: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
% 8.83/5.03 => ( ( nth_nat @ Xs2 @ I3 )
% 8.83/5.03 = ( nth_nat @ Ys @ I3 ) ) )
% 8.83/5.03 => ( Xs2 = Ys ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2637_nth__equalityI,axiom,
% 8.83/5.03 ! [Xs2: list_int,Ys: list_int] :
% 8.83/5.03 ( ( ( size_size_list_int @ Xs2 )
% 8.83/5.03 = ( size_size_list_int @ Ys ) )
% 8.83/5.03 => ( ! [I3: nat] :
% 8.83/5.03 ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
% 8.83/5.03 => ( ( nth_int @ Xs2 @ I3 )
% 8.83/5.03 = ( nth_int @ Ys @ I3 ) ) )
% 8.83/5.03 => ( Xs2 = Ys ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2638_discrete,axiom,
% 8.83/5.03 ( ord_less_nat
% 8.83/5.03 = ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2639_discrete,axiom,
% 8.83/5.03 ( ord_less_int
% 8.83/5.03 = ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2640_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat,Y3: nat] :
% 8.83/5.03 ( ( ( vEBT_T_m_e_m_b_e_r2 @ X2 @ Xa2 )
% 8.83/5.03 = Y3 )
% 8.83/5.03 => ( ( ? [A4: $o,B4: $o] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.03 => ( Y3 != one_one_nat ) )
% 8.83/5.03 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 8.83/5.03 => ( Y3 != one_one_nat ) )
% 8.83/5.03 => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 8.83/5.03 => ( Y3 != one_one_nat ) )
% 8.83/5.03 => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 8.83/5.03 => ( Y3 != one_one_nat ) )
% 8.83/5.03 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
% 8.83/5.03 ( ? [Summary2: vEBT_VEBT] :
% 8.83/5.03 ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.83/5.03 => ( Y3
% 8.83/5.03 != ( plus_plus_nat @ one_one_nat
% 8.83/5.03 @ ( if_nat @ ( Xa2 = Mi2 ) @ zero_zero_nat
% 8.83/5.03 @ ( if_nat @ ( Xa2 = Ma2 ) @ zero_zero_nat
% 8.83/5.03 @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ zero_zero_nat
% 8.83/5.03 @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa2 ) @ zero_zero_nat
% 8.83/5.03 @ ( if_nat
% 8.83/5.03 @ ( ( ord_less_nat @ Mi2 @ Xa2 )
% 8.83/5.03 & ( ord_less_nat @ Xa2 @ Ma2 ) )
% 8.83/5.03 @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
% 8.83/5.03 @ zero_zero_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2641_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
% 8.83/5.03 ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 8.83/5.03 ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 8.83/5.03 = ( plus_plus_nat @ one_one_nat
% 8.83/5.03 @ ( if_nat @ ( X2 = Mi ) @ zero_zero_nat
% 8.83/5.03 @ ( if_nat @ ( X2 = Ma ) @ zero_zero_nat
% 8.83/5.03 @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ zero_zero_nat
% 8.83/5.03 @ ( if_nat @ ( ord_less_nat @ Ma @ X2 ) @ zero_zero_nat
% 8.83/5.03 @ ( if_nat
% 8.83/5.03 @ ( ( ord_less_nat @ Mi @ X2 )
% 8.83/5.03 & ( ord_less_nat @ X2 @ Ma ) )
% 8.83/5.03 @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
% 8.83/5.03 @ zero_zero_nat ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2642_inrange,axiom,
% 8.83/5.03 ! [T: vEBT_VEBT,N: nat] :
% 8.83/5.03 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/5.03 => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2643_set__bit__0,axiom,
% 8.83/5.03 ! [A: int] :
% 8.83/5.03 ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
% 8.83/5.03 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2644_set__bit__0,axiom,
% 8.83/5.03 ! [A: nat] :
% 8.83/5.03 ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
% 8.83/5.03 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2645_vebt__succ_Opelims,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 8.83/5.03 ( ( ( vEBT_vebt_succ @ X2 @ Xa2 )
% 8.83/5.03 = Y3 )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 8.83/5.03 => ( ! [Uu2: $o,B4: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ Uu2 @ B4 ) )
% 8.83/5.03 => ( ( Xa2 = zero_zero_nat )
% 8.83/5.03 => ( ( ( B4
% 8.83/5.03 => ( Y3
% 8.83/5.03 = ( some_nat @ one_one_nat ) ) )
% 8.83/5.03 & ( ~ B4
% 8.83/5.03 => ( Y3 = none_nat ) ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B4 ) @ zero_zero_nat ) ) ) ) )
% 8.83/5.03 => ( ! [Uv2: $o,Uw2: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 8.83/5.03 => ! [N2: nat] :
% 8.83/5.03 ( ( Xa2
% 8.83/5.03 = ( suc @ N2 ) )
% 8.83/5.03 => ( ( Y3 = none_nat )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N2 ) ) ) ) ) )
% 8.83/5.03 => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 8.83/5.03 => ( ( Y3 = none_nat )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 8.83/5.03 => ( ( Y3 = none_nat )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
% 8.83/5.03 => ( ( Y3 = none_nat )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Xa2 ) ) ) )
% 8.83/5.03 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.83/5.03 => ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 8.83/5.03 => ( Y3
% 8.83/5.03 = ( some_nat @ Mi2 ) ) )
% 8.83/5.03 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 8.83/5.03 => ( Y3
% 8.83/5.03 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 @ ( if_option_nat
% 8.83/5.03 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/5.03 != none_nat )
% 8.83/5.03 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 8.83/5.03 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/5.03 @ ( if_option_nat
% 8.83/5.03 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.83/5.03 = none_nat )
% 8.83/5.03 @ none_nat
% 8.83/5.03 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 8.83/5.03 @ none_nat ) ) ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2646_vebt__pred_Opelims,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 8.83/5.03 ( ( ( vEBT_vebt_pred @ X2 @ Xa2 )
% 8.83/5.03 = Y3 )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 8.83/5.03 => ( ! [Uu2: $o,Uv2: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.83/5.03 => ( ( Xa2 = zero_zero_nat )
% 8.83/5.03 => ( ( Y3 = none_nat )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
% 8.83/5.03 => ( ! [A4: $o,Uw2: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ A4 @ Uw2 ) )
% 8.83/5.03 => ( ( Xa2
% 8.83/5.03 = ( suc @ zero_zero_nat ) )
% 8.83/5.03 => ( ( ( A4
% 8.83/5.03 => ( Y3
% 8.83/5.03 = ( some_nat @ zero_zero_nat ) ) )
% 8.83/5.03 & ( ~ A4
% 8.83/5.03 => ( Y3 = none_nat ) ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
% 8.83/5.03 => ( ! [A4: $o,B4: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.03 => ! [Va3: nat] :
% 8.83/5.03 ( ( Xa2
% 8.83/5.03 = ( suc @ ( suc @ Va3 ) ) )
% 8.83/5.03 => ( ( ( B4
% 8.83/5.03 => ( Y3
% 8.83/5.03 = ( some_nat @ one_one_nat ) ) )
% 8.83/5.03 & ( ~ B4
% 8.83/5.03 => ( ( A4
% 8.83/5.03 => ( Y3
% 8.83/5.03 = ( some_nat @ zero_zero_nat ) ) )
% 8.83/5.03 & ( ~ A4
% 8.83/5.03 => ( Y3 = none_nat ) ) ) ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
% 8.83/5.03 => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
% 8.83/5.03 => ( ( Y3 = none_nat )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) )
% 8.83/5.03 => ( ( Y3 = none_nat )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) @ Xa2 ) ) ) )
% 8.83/5.03 => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
% 8.83/5.03 => ( ( Y3 = none_nat )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Xa2 ) ) ) )
% 8.83/5.03 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.83/5.03 => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
% 8.83/5.03 => ( Y3
% 8.83/5.03 = ( some_nat @ Ma2 ) ) )
% 8.83/5.03 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 8.83/5.03 => ( Y3
% 8.83/5.03 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 @ ( if_option_nat
% 8.83/5.03 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/5.03 != none_nat )
% 8.83/5.03 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 8.83/5.03 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/5.03 @ ( if_option_nat
% 8.83/5.03 @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.83/5.03 = none_nat )
% 8.83/5.03 @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
% 8.83/5.03 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 8.83/5.03 @ none_nat ) ) ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2647_double__eq__0__iff,axiom,
% 8.83/5.03 ! [A: real] :
% 8.83/5.03 ( ( ( plus_plus_real @ A @ A )
% 8.83/5.03 = zero_zero_real )
% 8.83/5.03 = ( A = zero_zero_real ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2648_double__eq__0__iff,axiom,
% 8.83/5.03 ! [A: rat] :
% 8.83/5.03 ( ( ( plus_plus_rat @ A @ A )
% 8.83/5.03 = zero_zero_rat )
% 8.83/5.03 = ( A = zero_zero_rat ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2649_double__eq__0__iff,axiom,
% 8.83/5.03 ! [A: int] :
% 8.83/5.03 ( ( ( plus_plus_int @ A @ A )
% 8.83/5.03 = zero_zero_int )
% 8.83/5.03 = ( A = zero_zero_int ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2650_unset__bit__0,axiom,
% 8.83/5.03 ! [A: int] :
% 8.83/5.03 ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
% 8.83/5.03 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2651_unset__bit__0,axiom,
% 8.83/5.03 ! [A: nat] :
% 8.83/5.03 ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
% 8.83/5.03 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2652_mult__le__cancel__iff2,axiom,
% 8.83/5.03 ! [Z: real,X2: real,Y3: real] :
% 8.83/5.03 ( ( ord_less_real @ zero_zero_real @ Z )
% 8.83/5.03 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X2 ) @ ( times_times_real @ Z @ Y3 ) )
% 8.83/5.03 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2653_mult__le__cancel__iff2,axiom,
% 8.83/5.03 ! [Z: rat,X2: rat,Y3: rat] :
% 8.83/5.03 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 8.83/5.03 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X2 ) @ ( times_times_rat @ Z @ Y3 ) )
% 8.83/5.03 = ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2654_mult__le__cancel__iff2,axiom,
% 8.83/5.03 ! [Z: int,X2: int,Y3: int] :
% 8.83/5.03 ( ( ord_less_int @ zero_zero_int @ Z )
% 8.83/5.03 => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X2 ) @ ( times_times_int @ Z @ Y3 ) )
% 8.83/5.03 = ( ord_less_eq_int @ X2 @ Y3 ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2655_div__neg__neg__trivial,axiom,
% 8.83/5.03 ! [K: int,L: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 8.83/5.03 => ( ( ord_less_int @ L @ K )
% 8.83/5.03 => ( ( divide_divide_int @ K @ L )
% 8.83/5.03 = zero_zero_int ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2656_div__pos__pos__trivial,axiom,
% 8.83/5.03 ! [K: int,L: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.83/5.03 => ( ( ord_less_int @ K @ L )
% 8.83/5.03 => ( ( divide_divide_int @ K @ L )
% 8.83/5.03 = zero_zero_int ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2657_i0__less,axiom,
% 8.83/5.03 ! [N: extended_enat] :
% 8.83/5.03 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 8.83/5.03 = ( N != zero_z5237406670263579293d_enat ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2658_idiff__0,axiom,
% 8.83/5.03 ! [N: extended_enat] :
% 8.83/5.03 ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
% 8.83/5.03 = zero_z5237406670263579293d_enat ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2659_idiff__0__right,axiom,
% 8.83/5.03 ! [N: extended_enat] :
% 8.83/5.03 ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
% 8.83/5.03 = N ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2660_not__real__square__gt__zero,axiom,
% 8.83/5.03 ! [X2: real] :
% 8.83/5.03 ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X2 @ X2 ) ) )
% 8.83/5.03 = ( X2 = zero_zero_real ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2661_half__nonnegative__int__iff,axiom,
% 8.83/5.03 ! [K: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 8.83/5.03 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2662_half__negative__int__iff,axiom,
% 8.83/5.03 ! [K: int] :
% 8.83/5.03 ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 8.83/5.03 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2663_zdiv__mono1,axiom,
% 8.83/5.03 ! [A: int,A5: int,B: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ A @ A5 )
% 8.83/5.03 => ( ( ord_less_int @ zero_zero_int @ B )
% 8.83/5.03 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A5 @ B ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2664_zdiv__mono2,axiom,
% 8.83/5.03 ! [A: int,B5: int,B: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.03 => ( ( ord_less_int @ zero_zero_int @ B5 )
% 8.83/5.03 => ( ( ord_less_eq_int @ B5 @ B )
% 8.83/5.03 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B5 ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2665_zdiv__eq__0__iff,axiom,
% 8.83/5.03 ! [I2: int,K: int] :
% 8.83/5.03 ( ( ( divide_divide_int @ I2 @ K )
% 8.83/5.03 = zero_zero_int )
% 8.83/5.03 = ( ( K = zero_zero_int )
% 8.83/5.03 | ( ( ord_less_eq_int @ zero_zero_int @ I2 )
% 8.83/5.03 & ( ord_less_int @ I2 @ K ) )
% 8.83/5.03 | ( ( ord_less_eq_int @ I2 @ zero_zero_int )
% 8.83/5.03 & ( ord_less_int @ K @ I2 ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2666_zdiv__mono1__neg,axiom,
% 8.83/5.03 ! [A: int,A5: int,B: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ A @ A5 )
% 8.83/5.03 => ( ( ord_less_int @ B @ zero_zero_int )
% 8.83/5.03 => ( ord_less_eq_int @ ( divide_divide_int @ A5 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2667_zdiv__mono2__neg,axiom,
% 8.83/5.03 ! [A: int,B5: int,B: int] :
% 8.83/5.03 ( ( ord_less_int @ A @ zero_zero_int )
% 8.83/5.03 => ( ( ord_less_int @ zero_zero_int @ B5 )
% 8.83/5.03 => ( ( ord_less_eq_int @ B5 @ B )
% 8.83/5.03 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B5 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2668_div__int__pos__iff,axiom,
% 8.83/5.03 ! [K: int,L: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
% 8.83/5.03 = ( ( K = zero_zero_int )
% 8.83/5.03 | ( L = zero_zero_int )
% 8.83/5.03 | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.83/5.03 & ( ord_less_eq_int @ zero_zero_int @ L ) )
% 8.83/5.03 | ( ( ord_less_int @ K @ zero_zero_int )
% 8.83/5.03 & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2669_div__positive__int,axiom,
% 8.83/5.03 ! [L: int,K: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ L @ K )
% 8.83/5.03 => ( ( ord_less_int @ zero_zero_int @ L )
% 8.83/5.03 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2670_int__div__less__self,axiom,
% 8.83/5.03 ! [X2: int,K: int] :
% 8.83/5.03 ( ( ord_less_int @ zero_zero_int @ X2 )
% 8.83/5.03 => ( ( ord_less_int @ one_one_int @ K )
% 8.83/5.03 => ( ord_less_int @ ( divide_divide_int @ X2 @ K ) @ X2 ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2671_div__nonneg__neg__le0,axiom,
% 8.83/5.03 ! [A: int,B: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.03 => ( ( ord_less_int @ B @ zero_zero_int )
% 8.83/5.03 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2672_div__nonpos__pos__le0,axiom,
% 8.83/5.03 ! [A: int,B: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.83/5.03 => ( ( ord_less_int @ zero_zero_int @ B )
% 8.83/5.03 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2673_pos__imp__zdiv__pos__iff,axiom,
% 8.83/5.03 ! [K: int,I2: int] :
% 8.83/5.03 ( ( ord_less_int @ zero_zero_int @ K )
% 8.83/5.03 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I2 @ K ) )
% 8.83/5.03 = ( ord_less_eq_int @ K @ I2 ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2674_neg__imp__zdiv__nonneg__iff,axiom,
% 8.83/5.03 ! [B: int,A: int] :
% 8.83/5.03 ( ( ord_less_int @ B @ zero_zero_int )
% 8.83/5.03 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 8.83/5.03 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2675_pos__imp__zdiv__nonneg__iff,axiom,
% 8.83/5.03 ! [B: int,A: int] :
% 8.83/5.03 ( ( ord_less_int @ zero_zero_int @ B )
% 8.83/5.03 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 8.83/5.03 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2676_nonneg1__imp__zdiv__pos__iff,axiom,
% 8.83/5.03 ! [A: int,B: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.03 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 8.83/5.03 = ( ( ord_less_eq_int @ B @ A )
% 8.83/5.03 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2677_zdiv__zmult2__eq,axiom,
% 8.83/5.03 ! [C: int,A: int,B: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.83/5.03 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 8.83/5.03 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2678_int__div__pos__eq,axiom,
% 8.83/5.03 ! [A: int,B: int,Q3: int,R2: int] :
% 8.83/5.03 ( ( A
% 8.83/5.03 = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 8.83/5.03 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 8.83/5.03 => ( ( ord_less_int @ R2 @ B )
% 8.83/5.03 => ( ( divide_divide_int @ A @ B )
% 8.83/5.03 = Q3 ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2679_int__div__neg__eq,axiom,
% 8.83/5.03 ! [A: int,B: int,Q3: int,R2: int] :
% 8.83/5.03 ( ( A
% 8.83/5.03 = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 8.83/5.03 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 8.83/5.03 => ( ( ord_less_int @ B @ R2 )
% 8.83/5.03 => ( ( divide_divide_int @ A @ B )
% 8.83/5.03 = Q3 ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2680_div__pos__geq,axiom,
% 8.83/5.03 ! [L: int,K: int] :
% 8.83/5.03 ( ( ord_less_int @ zero_zero_int @ L )
% 8.83/5.03 => ( ( ord_less_eq_int @ L @ K )
% 8.83/5.03 => ( ( divide_divide_int @ K @ L )
% 8.83/5.03 = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2681_split__zdiv,axiom,
% 8.83/5.03 ! [P: int > $o,N: int,K: int] :
% 8.83/5.03 ( ( P @ ( divide_divide_int @ N @ K ) )
% 8.83/5.03 = ( ( ( K = zero_zero_int )
% 8.83/5.03 => ( P @ zero_zero_int ) )
% 8.83/5.03 & ( ( ord_less_int @ zero_zero_int @ K )
% 8.83/5.03 => ! [I4: int,J3: int] :
% 8.83/5.03 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 8.83/5.03 & ( ord_less_int @ J3 @ K )
% 8.83/5.03 & ( N
% 8.83/5.03 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 8.83/5.03 => ( P @ I4 ) ) )
% 8.83/5.03 & ( ( ord_less_int @ K @ zero_zero_int )
% 8.83/5.03 => ! [I4: int,J3: int] :
% 8.83/5.03 ( ( ( ord_less_int @ K @ J3 )
% 8.83/5.03 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 8.83/5.03 & ( N
% 8.83/5.03 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 8.83/5.03 => ( P @ I4 ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2682_not__iless0,axiom,
% 8.83/5.03 ! [N: extended_enat] :
% 8.83/5.03 ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2683_enat__0__less__mult__iff,axiom,
% 8.83/5.03 ! [M: extended_enat,N: extended_enat] :
% 8.83/5.03 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
% 8.83/5.03 = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 8.83/5.03 & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2684_iadd__is__0,axiom,
% 8.83/5.03 ! [M: extended_enat,N: extended_enat] :
% 8.83/5.03 ( ( ( plus_p3455044024723400733d_enat @ M @ N )
% 8.83/5.03 = zero_z5237406670263579293d_enat )
% 8.83/5.03 = ( ( M = zero_z5237406670263579293d_enat )
% 8.83/5.03 & ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2685_i0__lb,axiom,
% 8.83/5.03 ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2686_ile0__eq,axiom,
% 8.83/5.03 ! [N: extended_enat] :
% 8.83/5.03 ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
% 8.83/5.03 = ( N = zero_z5237406670263579293d_enat ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2687_all__nat__less,axiom,
% 8.83/5.03 ! [N: nat,P: nat > $o] :
% 8.83/5.03 ( ( ! [M3: nat] :
% 8.83/5.03 ( ( ord_less_eq_nat @ M3 @ N )
% 8.83/5.03 => ( P @ M3 ) ) )
% 8.83/5.03 = ( ! [X: nat] :
% 8.83/5.03 ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 8.83/5.03 => ( P @ X ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2688_ex__nat__less,axiom,
% 8.83/5.03 ! [N: nat,P: nat > $o] :
% 8.83/5.03 ( ( ? [M3: nat] :
% 8.83/5.03 ( ( ord_less_eq_nat @ M3 @ N )
% 8.83/5.03 & ( P @ M3 ) ) )
% 8.83/5.03 = ( ? [X: nat] :
% 8.83/5.03 ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 8.83/5.03 & ( P @ X ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2689_not__exp__less__eq__0__int,axiom,
% 8.83/5.03 ! [N: nat] :
% 8.83/5.03 ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2690_pos__zdiv__mult__2,axiom,
% 8.83/5.03 ! [A: int,B: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.83/5.03 => ( ( 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 ) )
% 8.83/5.03 = ( divide_divide_int @ B @ A ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2691_neg__zdiv__mult__2,axiom,
% 8.83/5.03 ! [A: int,B: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.83/5.03 => ( ( 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 ) )
% 8.83/5.03 = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2692_realpow__pos__nth2,axiom,
% 8.83/5.03 ! [A: real,N: nat] :
% 8.83/5.03 ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.03 => ? [R3: real] :
% 8.83/5.03 ( ( ord_less_real @ zero_zero_real @ R3 )
% 8.83/5.03 & ( ( power_power_real @ R3 @ ( suc @ N ) )
% 8.83/5.03 = A ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2693_real__arch__pow__inv,axiom,
% 8.83/5.03 ! [Y3: real,X2: real] :
% 8.83/5.03 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.83/5.03 => ( ( ord_less_real @ X2 @ one_one_real )
% 8.83/5.03 => ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X2 @ N2 ) @ Y3 ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2694_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I1_J,axiom,
% 8.83/5.03 ! [A: $o,B: $o,X2: nat] :
% 8.83/5.03 ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Leaf @ A @ B ) @ X2 )
% 8.83/5.03 = one_one_nat ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2695_int__power__div__base,axiom,
% 8.83/5.03 ! [M: nat,K: int] :
% 8.83/5.03 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.83/5.03 => ( ( ord_less_int @ zero_zero_int @ K )
% 8.83/5.03 => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 8.83/5.03 = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2696_realpow__pos__nth__unique,axiom,
% 8.83/5.03 ! [N: nat,A: real] :
% 8.83/5.03 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.03 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.03 => ? [X5: real] :
% 8.83/5.03 ( ( ord_less_real @ zero_zero_real @ X5 )
% 8.83/5.03 & ( ( power_power_real @ X5 @ N )
% 8.83/5.03 = A )
% 8.83/5.03 & ! [Y5: real] :
% 8.83/5.03 ( ( ( ord_less_real @ zero_zero_real @ Y5 )
% 8.83/5.03 & ( ( power_power_real @ Y5 @ N )
% 8.83/5.03 = A ) )
% 8.83/5.03 => ( Y5 = X5 ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2697_realpow__pos__nth,axiom,
% 8.83/5.03 ! [N: nat,A: real] :
% 8.83/5.03 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.83/5.03 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.83/5.03 => ? [R3: real] :
% 8.83/5.03 ( ( ord_less_real @ zero_zero_real @ R3 )
% 8.83/5.03 & ( ( power_power_real @ R3 @ N )
% 8.83/5.03 = A ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2698_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
% 8.83/5.03 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
% 8.83/5.03 ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X2 )
% 8.83/5.03 = one_one_nat ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2699_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
% 8.83/5.03 ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
% 8.83/5.03 ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X2 )
% 8.83/5.03 = one_one_nat ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2700_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
% 8.83/5.03 ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
% 8.83/5.03 ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X2 )
% 8.83/5.03 = one_one_nat ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2701_member__bound__height_H,axiom,
% 8.83/5.03 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 8.83/5.03 ( ( vEBT_invar_vebt @ T @ N )
% 8.83/5.03 => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r2 @ T @ X2 ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2702_mult__less__iff1,axiom,
% 8.83/5.03 ! [Z: real,X2: real,Y3: real] :
% 8.83/5.03 ( ( ord_less_real @ zero_zero_real @ Z )
% 8.83/5.03 => ( ( ord_less_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 8.83/5.03 = ( ord_less_real @ X2 @ Y3 ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2703_mult__less__iff1,axiom,
% 8.83/5.03 ! [Z: rat,X2: rat,Y3: rat] :
% 8.83/5.03 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 8.83/5.03 => ( ( ord_less_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 8.83/5.03 = ( ord_less_rat @ X2 @ Y3 ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2704_mult__less__iff1,axiom,
% 8.83/5.03 ! [Z: int,X2: int,Y3: int] :
% 8.83/5.03 ( ( ord_less_int @ zero_zero_int @ Z )
% 8.83/5.03 => ( ( ord_less_int @ ( times_times_int @ X2 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 8.83/5.03 = ( ord_less_int @ X2 @ Y3 ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2705_mult__le__cancel__iff1,axiom,
% 8.83/5.03 ! [Z: real,X2: real,Y3: real] :
% 8.83/5.03 ( ( ord_less_real @ zero_zero_real @ Z )
% 8.83/5.03 => ( ( ord_less_eq_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 8.83/5.03 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2706_mult__le__cancel__iff1,axiom,
% 8.83/5.03 ! [Z: rat,X2: rat,Y3: rat] :
% 8.83/5.03 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 8.83/5.03 => ( ( ord_less_eq_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 8.83/5.03 = ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2707_mult__le__cancel__iff1,axiom,
% 8.83/5.03 ! [Z: int,X2: int,Y3: int] :
% 8.83/5.03 ( ( ord_less_int @ zero_zero_int @ Z )
% 8.83/5.03 => ( ( ord_less_eq_int @ ( times_times_int @ X2 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 8.83/5.03 = ( ord_less_eq_int @ X2 @ Y3 ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2708_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat,Y3: nat] :
% 8.83/5.03 ( ( ( vEBT_T_i_n_s_e_r_t2 @ X2 @ Xa2 )
% 8.83/5.03 = Y3 )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 8.83/5.03 => ( ! [A4: $o,B4: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.03 => ( ( Y3 = one_one_nat )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 8.83/5.03 => ( ( Y3 = one_one_nat )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 8.83/5.03 => ( ( Y3 = one_one_nat )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
% 8.83/5.03 => ( ( Y3 = one_one_nat )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.83/5.03 => ( ( Y3
% 8.83/5.03 = ( if_nat
% 8.83/5.03 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 & ~ ( ( Xa2 = Mi2 )
% 8.83/5.03 | ( Xa2 = Ma2 ) ) )
% 8.83/5.03 @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
% 8.83/5.03 @ one_one_nat ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2709_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat,Y3: nat] :
% 8.83/5.03 ( ( ( vEBT_T_m_e_m_b_e_r2 @ X2 @ Xa2 )
% 8.83/5.03 = Y3 )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 8.83/5.03 => ( ! [A4: $o,B4: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.03 => ( ( Y3 = one_one_nat )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 8.83/5.03 => ( ( Y3 = one_one_nat )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 8.83/5.03 => ( ( Y3 = one_one_nat )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 8.83/5.03 => ( ( Y3 = one_one_nat )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.83/5.03 => ( ( Y3
% 8.83/5.03 = ( plus_plus_nat @ one_one_nat
% 8.83/5.03 @ ( if_nat @ ( Xa2 = Mi2 ) @ zero_zero_nat
% 8.83/5.03 @ ( if_nat @ ( Xa2 = Ma2 ) @ zero_zero_nat
% 8.83/5.03 @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ zero_zero_nat
% 8.83/5.03 @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa2 ) @ zero_zero_nat
% 8.83/5.03 @ ( if_nat
% 8.83/5.03 @ ( ( ord_less_nat @ Mi2 @ Xa2 )
% 8.83/5.03 & ( ord_less_nat @ Xa2 @ Ma2 ) )
% 8.83/5.03 @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
% 8.83/5.03 @ zero_zero_nat ) ) ) ) ) ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2710_vebt__member_Opelims_I3_J,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat] :
% 8.83/5.03 ( ~ ( vEBT_vebt_member @ X2 @ Xa2 )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 8.83/5.03 => ( ! [A4: $o,B4: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
% 8.83/5.03 => ( ( ( Xa2 = zero_zero_nat )
% 8.83/5.03 => A4 )
% 8.83/5.03 & ( ( Xa2 != zero_zero_nat )
% 8.83/5.03 => ( ( ( Xa2 = one_one_nat )
% 8.83/5.03 => B4 )
% 8.83/5.03 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 8.83/5.03 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 8.83/5.03 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
% 8.83/5.03 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
% 8.83/5.03 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) )
% 8.83/5.03 => ( ( Xa2 != Mi2 )
% 8.83/5.03 => ( ( Xa2 != Ma2 )
% 8.83/5.03 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 8.83/5.03 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 8.83/5.03 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 8.83/5.03 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 8.83/5.03 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2711_vebt__member_Opelims_I1_J,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 8.83/5.03 ( ( ( vEBT_vebt_member @ X2 @ Xa2 )
% 8.83/5.03 = Y3 )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 8.83/5.03 => ( ! [A4: $o,B4: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.03 => ( ( Y3
% 8.83/5.03 = ( ( ( Xa2 = zero_zero_nat )
% 8.83/5.03 => A4 )
% 8.83/5.03 & ( ( Xa2 != zero_zero_nat )
% 8.83/5.03 => ( ( ( Xa2 = one_one_nat )
% 8.83/5.03 => B4 )
% 8.83/5.03 & ( Xa2 = one_one_nat ) ) ) ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 8.83/5.03 => ( ~ Y3
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 8.83/5.03 => ( ~ Y3
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 8.83/5.03 => ( ~ Y3
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.83/5.03 => ( ( Y3
% 8.83/5.03 = ( ( Xa2 != Mi2 )
% 8.83/5.03 => ( ( Xa2 != Ma2 )
% 8.83/5.03 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 8.83/5.03 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 8.83/5.03 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 8.83/5.03 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 8.83/5.03 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2712_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat] :
% 8.83/5.03 ( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 8.83/5.03 => ( ! [A4: $o,B4: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
% 8.83/5.03 => ( ( ( Xa2 = zero_zero_nat )
% 8.83/5.03 => A4 )
% 8.83/5.03 & ( ( Xa2 != zero_zero_nat )
% 8.83/5.03 => ( ( ( Xa2 = one_one_nat )
% 8.83/5.03 => B4 )
% 8.83/5.03 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 8.83/5.03 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 8.83/5.03 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) @ Xa2 ) )
% 8.83/5.03 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2713_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat] :
% 8.83/5.03 ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 8.83/5.03 => ( ! [A4: $o,B4: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
% 8.83/5.03 => ~ ( ( ( Xa2 = zero_zero_nat )
% 8.83/5.03 => A4 )
% 8.83/5.03 & ( ( Xa2 != zero_zero_nat )
% 8.83/5.03 => ( ( ( Xa2 = one_one_nat )
% 8.83/5.03 => B4 )
% 8.83/5.03 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 8.83/5.03 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) @ Xa2 ) )
% 8.83/5.03 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2714_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 8.83/5.03 ( ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 8.83/5.03 = Y3 )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 8.83/5.03 => ( ! [A4: $o,B4: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.03 => ( ( Y3
% 8.83/5.03 = ( ( ( Xa2 = zero_zero_nat )
% 8.83/5.03 => A4 )
% 8.83/5.03 & ( ( Xa2 != zero_zero_nat )
% 8.83/5.03 => ( ( ( Xa2 = one_one_nat )
% 8.83/5.03 => B4 )
% 8.83/5.03 & ( Xa2 = one_one_nat ) ) ) ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 8.83/5.03 => ( ~ Y3
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) )
% 8.83/5.03 => ( ( Y3
% 8.83/5.03 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) @ Xa2 ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2715_vebt__member_Opelims_I2_J,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat] :
% 8.83/5.03 ( ( vEBT_vebt_member @ X2 @ Xa2 )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 8.83/5.03 => ( ! [A4: $o,B4: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
% 8.83/5.03 => ~ ( ( ( Xa2 = zero_zero_nat )
% 8.83/5.03 => A4 )
% 8.83/5.03 & ( ( Xa2 != zero_zero_nat )
% 8.83/5.03 => ( ( ( Xa2 = one_one_nat )
% 8.83/5.03 => B4 )
% 8.83/5.03 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 8.83/5.03 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) )
% 8.83/5.03 => ~ ( ( Xa2 != Mi2 )
% 8.83/5.03 => ( ( Xa2 != Ma2 )
% 8.83/5.03 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 8.83/5.03 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 8.83/5.03 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 8.83/5.03 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 8.83/5.03 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2716_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat] :
% 8.83/5.03 ( ~ ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 8.83/5.03 => ( ! [Uu2: $o,Uv2: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
% 8.83/5.03 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
% 8.83/5.03 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) )
% 8.83/5.03 => ( ( Xa2 = Mi2 )
% 8.83/5.03 | ( Xa2 = Ma2 ) ) ) )
% 8.83/5.03 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
% 8.83/5.03 => ( ( Xa2 = Mi2 )
% 8.83/5.03 | ( Xa2 = Ma2 )
% 8.83/5.03 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
% 8.83/5.03 => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa2 ) )
% 8.83/5.03 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2717_unset__bit__nonnegative__int__iff,axiom,
% 8.83/5.03 ! [N: nat,K: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
% 8.83/5.03 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2718_set__bit__nonnegative__int__iff,axiom,
% 8.83/5.03 ! [N: nat,K: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
% 8.83/5.03 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2719_unset__bit__negative__int__iff,axiom,
% 8.83/5.03 ! [N: nat,K: int] :
% 8.83/5.03 ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
% 8.83/5.03 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2720_set__bit__negative__int__iff,axiom,
% 8.83/5.03 ! [N: nat,K: int] :
% 8.83/5.03 ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
% 8.83/5.03 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2721_zle__add1__eq__le,axiom,
% 8.83/5.03 ! [W: int,Z: int] :
% 8.83/5.03 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 8.83/5.03 = ( ord_less_eq_int @ W @ Z ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2722_zle__diff1__eq,axiom,
% 8.83/5.03 ! [W: int,Z: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
% 8.83/5.03 = ( ord_less_int @ W @ Z ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2723_less__eq__int__code_I1_J,axiom,
% 8.83/5.03 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2724_zero__one__enat__neq_I1_J,axiom,
% 8.83/5.03 zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2725_int__one__le__iff__zero__less,axiom,
% 8.83/5.03 ! [Z: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ one_one_int @ Z )
% 8.83/5.03 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2726_odd__less__0__iff,axiom,
% 8.83/5.03 ! [Z: int] :
% 8.83/5.03 ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
% 8.83/5.03 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2727_le__imp__0__less,axiom,
% 8.83/5.03 ! [Z: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 8.83/5.03 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2728_odd__nonzero,axiom,
% 8.83/5.03 ! [Z: int] :
% 8.83/5.03 ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
% 8.83/5.03 != zero_zero_int ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2729_q__pos__lemma,axiom,
% 8.83/5.03 ! [B5: int,Q5: int,R4: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) )
% 8.83/5.03 => ( ( ord_less_int @ R4 @ B5 )
% 8.83/5.03 => ( ( ord_less_int @ zero_zero_int @ B5 )
% 8.83/5.03 => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2730_zmult__zless__mono2,axiom,
% 8.83/5.03 ! [I2: int,J: int,K: int] :
% 8.83/5.03 ( ( ord_less_int @ I2 @ J )
% 8.83/5.03 => ( ( ord_less_int @ zero_zero_int @ K )
% 8.83/5.03 => ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2731_pos__zmult__eq__1__iff,axiom,
% 8.83/5.03 ! [M: int,N: int] :
% 8.83/5.03 ( ( ord_less_int @ zero_zero_int @ M )
% 8.83/5.03 => ( ( ( times_times_int @ M @ N )
% 8.83/5.03 = one_one_int )
% 8.83/5.03 = ( ( M = one_one_int )
% 8.83/5.03 & ( N = one_one_int ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2732_zdiv__mono2__lemma,axiom,
% 8.83/5.03 ! [B: int,Q3: int,R2: int,B5: int,Q5: int,R4: int] :
% 8.83/5.03 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 )
% 8.83/5.03 = ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) )
% 8.83/5.03 => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) )
% 8.83/5.03 => ( ( ord_less_int @ R4 @ B5 )
% 8.83/5.03 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 8.83/5.03 => ( ( ord_less_int @ zero_zero_int @ B5 )
% 8.83/5.03 => ( ( ord_less_eq_int @ B5 @ B )
% 8.83/5.03 => ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2733_zdiv__mono2__neg__lemma,axiom,
% 8.83/5.03 ! [B: int,Q3: int,R2: int,B5: int,Q5: int,R4: int] :
% 8.83/5.03 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 )
% 8.83/5.03 = ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) )
% 8.83/5.03 => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B5 @ Q5 ) @ R4 ) @ zero_zero_int )
% 8.83/5.03 => ( ( ord_less_int @ R2 @ B )
% 8.83/5.03 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 8.83/5.03 => ( ( ord_less_int @ zero_zero_int @ B5 )
% 8.83/5.03 => ( ( ord_less_eq_int @ B5 @ B )
% 8.83/5.03 => ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2734_unique__quotient__lemma,axiom,
% 8.83/5.03 ! [B: int,Q5: int,R4: int,Q3: int,R2: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 8.83/5.03 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 8.83/5.03 => ( ( ord_less_int @ R4 @ B )
% 8.83/5.03 => ( ( ord_less_int @ R2 @ B )
% 8.83/5.03 => ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2735_unique__quotient__lemma__neg,axiom,
% 8.83/5.03 ! [B: int,Q5: int,R4: int,Q3: int,R2: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 8.83/5.03 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 8.83/5.03 => ( ( ord_less_int @ B @ R2 )
% 8.83/5.03 => ( ( ord_less_int @ B @ R4 )
% 8.83/5.03 => ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2736_minus__int__code_I1_J,axiom,
% 8.83/5.03 ! [K: int] :
% 8.83/5.03 ( ( minus_minus_int @ K @ zero_zero_int )
% 8.83/5.03 = K ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2737_times__int__code_I2_J,axiom,
% 8.83/5.03 ! [L: int] :
% 8.83/5.03 ( ( times_times_int @ zero_zero_int @ L )
% 8.83/5.03 = zero_zero_int ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2738_times__int__code_I1_J,axiom,
% 8.83/5.03 ! [K: int] :
% 8.83/5.03 ( ( times_times_int @ K @ zero_zero_int )
% 8.83/5.03 = zero_zero_int ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2739_imult__is__0,axiom,
% 8.83/5.03 ! [M: extended_enat,N: extended_enat] :
% 8.83/5.03 ( ( ( times_7803423173614009249d_enat @ M @ N )
% 8.83/5.03 = zero_z5237406670263579293d_enat )
% 8.83/5.03 = ( ( M = zero_z5237406670263579293d_enat )
% 8.83/5.03 | ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2740_int__distrib_I3_J,axiom,
% 8.83/5.03 ! [Z1: int,Z22: int,W: int] :
% 8.83/5.03 ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
% 8.83/5.03 = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2741_int__distrib_I4_J,axiom,
% 8.83/5.03 ! [W: int,Z1: int,Z22: int] :
% 8.83/5.03 ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
% 8.83/5.03 = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2742_int__less__induct,axiom,
% 8.83/5.03 ! [I2: int,K: int,P: int > $o] :
% 8.83/5.03 ( ( ord_less_int @ I2 @ K )
% 8.83/5.03 => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 8.83/5.03 => ( ! [I3: int] :
% 8.83/5.03 ( ( ord_less_int @ I3 @ K )
% 8.83/5.03 => ( ( P @ I3 )
% 8.83/5.03 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 8.83/5.03 => ( P @ I2 ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2743_int__le__induct,axiom,
% 8.83/5.03 ! [I2: int,K: int,P: int > $o] :
% 8.83/5.03 ( ( ord_less_eq_int @ I2 @ K )
% 8.83/5.03 => ( ( P @ K )
% 8.83/5.03 => ( ! [I3: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ I3 @ K )
% 8.83/5.03 => ( ( P @ I3 )
% 8.83/5.03 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 8.83/5.03 => ( P @ I2 ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2744_int__induct,axiom,
% 8.83/5.03 ! [P: int > $o,K: int,I2: int] :
% 8.83/5.03 ( ( P @ K )
% 8.83/5.03 => ( ! [I3: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ K @ I3 )
% 8.83/5.03 => ( ( P @ I3 )
% 8.83/5.03 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 8.83/5.03 => ( ! [I3: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ I3 @ K )
% 8.83/5.03 => ( ( P @ I3 )
% 8.83/5.03 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 8.83/5.03 => ( P @ I2 ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2745_int__distrib_I1_J,axiom,
% 8.83/5.03 ! [Z1: int,Z22: int,W: int] :
% 8.83/5.03 ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
% 8.83/5.03 = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2746_int__distrib_I2_J,axiom,
% 8.83/5.03 ! [W: int,Z1: int,Z22: int] :
% 8.83/5.03 ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
% 8.83/5.03 = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2747_add1__zle__eq,axiom,
% 8.83/5.03 ! [W: int,Z: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
% 8.83/5.03 = ( ord_less_int @ W @ Z ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2748_int__ge__induct,axiom,
% 8.83/5.03 ! [K: int,I2: int,P: int > $o] :
% 8.83/5.03 ( ( ord_less_eq_int @ K @ I2 )
% 8.83/5.03 => ( ( P @ K )
% 8.83/5.03 => ( ! [I3: int] :
% 8.83/5.03 ( ( ord_less_eq_int @ K @ I3 )
% 8.83/5.03 => ( ( P @ I3 )
% 8.83/5.03 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 8.83/5.03 => ( P @ I2 ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2749_int__gr__induct,axiom,
% 8.83/5.03 ! [K: int,I2: int,P: int > $o] :
% 8.83/5.03 ( ( ord_less_int @ K @ I2 )
% 8.83/5.03 => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 8.83/5.03 => ( ! [I3: int] :
% 8.83/5.03 ( ( ord_less_int @ K @ I3 )
% 8.83/5.03 => ( ( P @ I3 )
% 8.83/5.03 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 8.83/5.03 => ( P @ I2 ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2750_zless__add1__eq,axiom,
% 8.83/5.03 ! [W: int,Z: int] :
% 8.83/5.03 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 8.83/5.03 = ( ( ord_less_int @ W @ Z )
% 8.83/5.03 | ( W = Z ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2751_zless__imp__add1__zle,axiom,
% 8.83/5.03 ! [W: int,Z: int] :
% 8.83/5.03 ( ( ord_less_int @ W @ Z )
% 8.83/5.03 => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2752_unset__bit__less__eq,axiom,
% 8.83/5.03 ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2753_set__bit__greater__eq,axiom,
% 8.83/5.03 ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2754_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat] :
% 8.83/5.03 ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 8.83/5.03 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) )
% 8.83/5.03 => ~ ( ( Xa2 = Mi2 )
% 8.83/5.03 | ( Xa2 = Ma2 ) ) ) )
% 8.83/5.03 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
% 8.83/5.03 => ~ ( ( Xa2 = Mi2 )
% 8.83/5.03 | ( Xa2 = Ma2 )
% 8.83/5.03 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
% 8.83/5.03 => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa2 ) )
% 8.83/5.03 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2755_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 8.83/5.03 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 8.83/5.03 ( ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 8.83/5.03 = Y3 )
% 8.83/5.03 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 8.83/5.03 => ( ! [Uu2: $o,Uv2: $o] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.83/5.03 => ( ~ Y3
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 8.83/5.03 => ( ~ Y3
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 8.83/5.03 => ( ( Y3
% 8.83/5.03 = ( ( Xa2 = Mi2 )
% 8.83/5.03 | ( Xa2 = Ma2 ) ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
% 8.83/5.03 => ( ( Y3
% 8.83/5.03 = ( ( Xa2 = Mi2 )
% 8.83/5.03 | ( Xa2 = Ma2 )
% 8.83/5.03 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa2 ) ) ) )
% 8.83/5.03 => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 8.83/5.03 ( ( X2
% 8.83/5.03 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
% 8.83/5.03 => ( ( Y3
% 8.83/5.03 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.83/5.03 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 8.83/5.03 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
% 8.83/5.03 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2756_atLeastatMost__empty,axiom,
% 8.83/5.03 ! [B: rat,A: rat] :
% 8.83/5.03 ( ( ord_less_rat @ B @ A )
% 8.83/5.03 => ( ( set_or633870826150836451st_rat @ A @ B )
% 8.83/5.03 = bot_bot_set_rat ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2757_atLeastatMost__empty,axiom,
% 8.83/5.03 ! [B: num,A: num] :
% 8.83/5.03 ( ( ord_less_num @ B @ A )
% 8.83/5.03 => ( ( set_or7049704709247886629st_num @ A @ B )
% 8.83/5.03 = bot_bot_set_num ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2758_atLeastatMost__empty,axiom,
% 8.83/5.03 ! [B: nat,A: nat] :
% 8.83/5.03 ( ( ord_less_nat @ B @ A )
% 8.83/5.03 => ( ( set_or1269000886237332187st_nat @ A @ B )
% 8.83/5.03 = bot_bot_set_nat ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2759_atLeastatMost__empty,axiom,
% 8.83/5.03 ! [B: int,A: int] :
% 8.83/5.03 ( ( ord_less_int @ B @ A )
% 8.83/5.03 => ( ( set_or1266510415728281911st_int @ A @ B )
% 8.83/5.03 = bot_bot_set_int ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2760_atLeastatMost__empty,axiom,
% 8.83/5.03 ! [B: real,A: real] :
% 8.83/5.03 ( ( ord_less_real @ B @ A )
% 8.83/5.03 => ( ( set_or1222579329274155063t_real @ A @ B )
% 8.83/5.03 = bot_bot_set_real ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2761_atLeastatMost__subset__iff,axiom,
% 8.83/5.03 ! [A: set_int,B: set_int,C: set_int,D: set_int] :
% 8.83/5.03 ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 8.83/5.03 = ( ~ ( ord_less_eq_set_int @ A @ B )
% 8.83/5.03 | ( ( ord_less_eq_set_int @ C @ A )
% 8.83/5.03 & ( ord_less_eq_set_int @ B @ D ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2762_atLeastatMost__subset__iff,axiom,
% 8.83/5.03 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.83/5.03 ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 8.83/5.03 = ( ~ ( ord_less_eq_rat @ A @ B )
% 8.83/5.03 | ( ( ord_less_eq_rat @ C @ A )
% 8.83/5.03 & ( ord_less_eq_rat @ B @ D ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2763_atLeastatMost__subset__iff,axiom,
% 8.83/5.03 ! [A: num,B: num,C: num,D: num] :
% 8.83/5.03 ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 8.83/5.03 = ( ~ ( ord_less_eq_num @ A @ B )
% 8.83/5.03 | ( ( ord_less_eq_num @ C @ A )
% 8.83/5.03 & ( ord_less_eq_num @ B @ D ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2764_atLeastatMost__subset__iff,axiom,
% 8.83/5.03 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.83/5.03 ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 8.83/5.03 = ( ~ ( ord_less_eq_nat @ A @ B )
% 8.83/5.03 | ( ( ord_less_eq_nat @ C @ A )
% 8.83/5.03 & ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2765_atLeastatMost__subset__iff,axiom,
% 8.83/5.03 ! [A: int,B: int,C: int,D: int] :
% 8.83/5.03 ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 8.83/5.03 = ( ~ ( ord_less_eq_int @ A @ B )
% 8.83/5.03 | ( ( ord_less_eq_int @ C @ A )
% 8.83/5.03 & ( ord_less_eq_int @ B @ D ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2766_atLeastatMost__subset__iff,axiom,
% 8.83/5.03 ! [A: real,B: real,C: real,D: real] :
% 8.83/5.03 ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 8.83/5.03 = ( ~ ( ord_less_eq_real @ A @ B )
% 8.83/5.03 | ( ( ord_less_eq_real @ C @ A )
% 8.83/5.03 & ( ord_less_eq_real @ B @ D ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2767_atLeastatMost__empty__iff,axiom,
% 8.83/5.03 ! [A: set_int,B: set_int] :
% 8.83/5.03 ( ( ( set_or370866239135849197et_int @ A @ B )
% 8.83/5.03 = bot_bot_set_set_int )
% 8.83/5.03 = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2768_atLeastatMost__empty__iff,axiom,
% 8.83/5.03 ! [A: rat,B: rat] :
% 8.83/5.03 ( ( ( set_or633870826150836451st_rat @ A @ B )
% 8.83/5.03 = bot_bot_set_rat )
% 8.83/5.03 = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2769_atLeastatMost__empty__iff,axiom,
% 8.83/5.03 ! [A: num,B: num] :
% 8.83/5.03 ( ( ( set_or7049704709247886629st_num @ A @ B )
% 8.83/5.03 = bot_bot_set_num )
% 8.83/5.03 = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2770_atLeastatMost__empty__iff,axiom,
% 8.83/5.03 ! [A: nat,B: nat] :
% 8.83/5.03 ( ( ( set_or1269000886237332187st_nat @ A @ B )
% 8.83/5.03 = bot_bot_set_nat )
% 8.83/5.03 = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2771_atLeastatMost__empty__iff,axiom,
% 8.83/5.03 ! [A: int,B: int] :
% 8.83/5.03 ( ( ( set_or1266510415728281911st_int @ A @ B )
% 8.83/5.03 = bot_bot_set_int )
% 8.83/5.03 = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2772_atLeastatMost__empty__iff,axiom,
% 8.83/5.03 ! [A: real,B: real] :
% 8.83/5.03 ( ( ( set_or1222579329274155063t_real @ A @ B )
% 8.83/5.03 = bot_bot_set_real )
% 8.83/5.03 = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2773_atLeastatMost__empty__iff2,axiom,
% 8.83/5.03 ! [A: set_int,B: set_int] :
% 8.83/5.03 ( ( bot_bot_set_set_int
% 8.83/5.03 = ( set_or370866239135849197et_int @ A @ B ) )
% 8.83/5.03 = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2774_atLeastatMost__empty__iff2,axiom,
% 8.83/5.03 ! [A: rat,B: rat] :
% 8.83/5.03 ( ( bot_bot_set_rat
% 8.83/5.03 = ( set_or633870826150836451st_rat @ A @ B ) )
% 8.83/5.03 = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2775_atLeastatMost__empty__iff2,axiom,
% 8.83/5.03 ! [A: num,B: num] :
% 8.83/5.03 ( ( bot_bot_set_num
% 8.83/5.03 = ( set_or7049704709247886629st_num @ A @ B ) )
% 8.83/5.03 = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2776_atLeastatMost__empty__iff2,axiom,
% 8.83/5.03 ! [A: nat,B: nat] :
% 8.83/5.03 ( ( bot_bot_set_nat
% 8.83/5.03 = ( set_or1269000886237332187st_nat @ A @ B ) )
% 8.83/5.03 = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2777_atLeastatMost__empty__iff2,axiom,
% 8.83/5.03 ! [A: int,B: int] :
% 8.83/5.03 ( ( bot_bot_set_int
% 8.83/5.03 = ( set_or1266510415728281911st_int @ A @ B ) )
% 8.83/5.03 = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2778_atLeastatMost__empty__iff2,axiom,
% 8.83/5.03 ! [A: real,B: real] :
% 8.83/5.03 ( ( bot_bot_set_real
% 8.83/5.03 = ( set_or1222579329274155063t_real @ A @ B ) )
% 8.83/5.03 = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2779_decr__mult__lemma,axiom,
% 8.83/5.03 ! [D: int,P: int > $o,K: int] :
% 8.83/5.03 ( ( ord_less_int @ zero_zero_int @ D )
% 8.83/5.03 => ( ! [X5: int] :
% 8.83/5.03 ( ( P @ X5 )
% 8.83/5.03 => ( P @ ( minus_minus_int @ X5 @ D ) ) )
% 8.83/5.03 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.83/5.03 => ! [X3: int] :
% 8.83/5.03 ( ( P @ X3 )
% 8.83/5.03 => ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2780_incr__mult__lemma,axiom,
% 8.83/5.03 ! [D: int,P: int > $o,K: int] :
% 8.83/5.03 ( ( ord_less_int @ zero_zero_int @ D )
% 8.83/5.03 => ( ! [X5: int] :
% 8.83/5.03 ( ( P @ X5 )
% 8.83/5.03 => ( P @ ( plus_plus_int @ X5 @ D ) ) )
% 8.83/5.03 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.83/5.03 => ! [X3: int] :
% 8.83/5.03 ( ( P @ X3 )
% 8.83/5.03 => ( P @ ( plus_plus_int @ X3 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2781_Icc__eq__Icc,axiom,
% 8.83/5.03 ! [L: set_int,H2: set_int,L3: set_int,H3: set_int] :
% 8.83/5.03 ( ( ( set_or370866239135849197et_int @ L @ H2 )
% 8.83/5.03 = ( set_or370866239135849197et_int @ L3 @ H3 ) )
% 8.83/5.03 = ( ( ( L = L3 )
% 8.83/5.03 & ( H2 = H3 ) )
% 8.83/5.03 | ( ~ ( ord_less_eq_set_int @ L @ H2 )
% 8.83/5.03 & ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2782_Icc__eq__Icc,axiom,
% 8.83/5.03 ! [L: rat,H2: rat,L3: rat,H3: rat] :
% 8.83/5.03 ( ( ( set_or633870826150836451st_rat @ L @ H2 )
% 8.83/5.03 = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
% 8.83/5.03 = ( ( ( L = L3 )
% 8.83/5.03 & ( H2 = H3 ) )
% 8.83/5.03 | ( ~ ( ord_less_eq_rat @ L @ H2 )
% 8.83/5.03 & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2783_Icc__eq__Icc,axiom,
% 8.83/5.03 ! [L: num,H2: num,L3: num,H3: num] :
% 8.83/5.03 ( ( ( set_or7049704709247886629st_num @ L @ H2 )
% 8.83/5.03 = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
% 8.83/5.03 = ( ( ( L = L3 )
% 8.83/5.03 & ( H2 = H3 ) )
% 8.83/5.03 | ( ~ ( ord_less_eq_num @ L @ H2 )
% 8.83/5.03 & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2784_Icc__eq__Icc,axiom,
% 8.83/5.03 ! [L: nat,H2: nat,L3: nat,H3: nat] :
% 8.83/5.03 ( ( ( set_or1269000886237332187st_nat @ L @ H2 )
% 8.83/5.03 = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
% 8.83/5.03 = ( ( ( L = L3 )
% 8.83/5.03 & ( H2 = H3 ) )
% 8.83/5.03 | ( ~ ( ord_less_eq_nat @ L @ H2 )
% 8.83/5.03 & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2785_Icc__eq__Icc,axiom,
% 8.83/5.03 ! [L: int,H2: int,L3: int,H3: int] :
% 8.83/5.03 ( ( ( set_or1266510415728281911st_int @ L @ H2 )
% 8.83/5.03 = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
% 8.83/5.03 = ( ( ( L = L3 )
% 8.83/5.03 & ( H2 = H3 ) )
% 8.83/5.03 | ( ~ ( ord_less_eq_int @ L @ H2 )
% 8.83/5.03 & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2786_Icc__eq__Icc,axiom,
% 8.83/5.03 ! [L: real,H2: real,L3: real,H3: real] :
% 8.83/5.03 ( ( ( set_or1222579329274155063t_real @ L @ H2 )
% 8.83/5.03 = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
% 8.83/5.03 = ( ( ( L = L3 )
% 8.83/5.03 & ( H2 = H3 ) )
% 8.83/5.03 | ( ~ ( ord_less_eq_real @ L @ H2 )
% 8.83/5.03 & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2787_atLeastAtMost__iff,axiom,
% 8.83/5.03 ! [I2: set_int,L: set_int,U: set_int] :
% 8.83/5.03 ( ( member_set_int @ I2 @ ( set_or370866239135849197et_int @ L @ U ) )
% 8.83/5.03 = ( ( ord_less_eq_set_int @ L @ I2 )
% 8.83/5.03 & ( ord_less_eq_set_int @ I2 @ U ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2788_atLeastAtMost__iff,axiom,
% 8.83/5.03 ! [I2: rat,L: rat,U: rat] :
% 8.83/5.03 ( ( member_rat @ I2 @ ( set_or633870826150836451st_rat @ L @ U ) )
% 8.83/5.03 = ( ( ord_less_eq_rat @ L @ I2 )
% 8.83/5.03 & ( ord_less_eq_rat @ I2 @ U ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2789_atLeastAtMost__iff,axiom,
% 8.83/5.03 ! [I2: num,L: num,U: num] :
% 8.83/5.03 ( ( member_num @ I2 @ ( set_or7049704709247886629st_num @ L @ U ) )
% 8.83/5.03 = ( ( ord_less_eq_num @ L @ I2 )
% 8.83/5.03 & ( ord_less_eq_num @ I2 @ U ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2790_atLeastAtMost__iff,axiom,
% 8.83/5.03 ! [I2: nat,L: nat,U: nat] :
% 8.83/5.03 ( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 8.83/5.03 = ( ( ord_less_eq_nat @ L @ I2 )
% 8.83/5.03 & ( ord_less_eq_nat @ I2 @ U ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2791_atLeastAtMost__iff,axiom,
% 8.83/5.03 ! [I2: int,L: int,U: int] :
% 8.83/5.03 ( ( member_int @ I2 @ ( set_or1266510415728281911st_int @ L @ U ) )
% 8.83/5.03 = ( ( ord_less_eq_int @ L @ I2 )
% 8.83/5.03 & ( ord_less_eq_int @ I2 @ U ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2792_atLeastAtMost__iff,axiom,
% 8.83/5.03 ! [I2: real,L: real,U: real] :
% 8.83/5.03 ( ( member_real @ I2 @ ( set_or1222579329274155063t_real @ L @ U ) )
% 8.83/5.03 = ( ( ord_less_eq_real @ L @ I2 )
% 8.83/5.03 & ( ord_less_eq_real @ I2 @ U ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2793_aset_I2_J,axiom,
% 8.83/5.03 ! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
% 8.83/5.03 ( ! [X5: int] :
% 8.83/5.03 ( ! [Xa: int] :
% 8.83/5.03 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.03 => ! [Xb2: int] :
% 8.83/5.03 ( ( member_int @ Xb2 @ A2 )
% 8.83/5.03 => ( X5
% 8.83/5.03 != ( minus_minus_int @ Xb2 @ Xa ) ) ) )
% 8.83/5.03 => ( ( P @ X5 )
% 8.83/5.03 => ( P @ ( plus_plus_int @ X5 @ D4 ) ) ) )
% 8.83/5.03 => ( ! [X5: int] :
% 8.83/5.03 ( ! [Xa: int] :
% 8.83/5.03 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.03 => ! [Xb2: int] :
% 8.83/5.03 ( ( member_int @ Xb2 @ A2 )
% 8.83/5.03 => ( X5
% 8.83/5.03 != ( minus_minus_int @ Xb2 @ Xa ) ) ) )
% 8.83/5.03 => ( ( Q @ X5 )
% 8.83/5.03 => ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) )
% 8.83/5.03 => ! [X3: int] :
% 8.83/5.03 ( ! [Xa3: int] :
% 8.83/5.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.03 => ! [Xb3: int] :
% 8.83/5.03 ( ( member_int @ Xb3 @ A2 )
% 8.83/5.03 => ( X3
% 8.83/5.03 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.83/5.03 => ( ( ( P @ X3 )
% 8.83/5.03 | ( Q @ X3 ) )
% 8.83/5.03 => ( ( P @ ( plus_plus_int @ X3 @ D4 ) )
% 8.83/5.03 | ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2794_aset_I1_J,axiom,
% 8.83/5.03 ! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
% 8.83/5.03 ( ! [X5: int] :
% 8.83/5.03 ( ! [Xa: int] :
% 8.83/5.03 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.03 => ! [Xb2: int] :
% 8.83/5.03 ( ( member_int @ Xb2 @ A2 )
% 8.83/5.03 => ( X5
% 8.83/5.03 != ( minus_minus_int @ Xb2 @ Xa ) ) ) )
% 8.83/5.03 => ( ( P @ X5 )
% 8.83/5.03 => ( P @ ( plus_plus_int @ X5 @ D4 ) ) ) )
% 8.83/5.03 => ( ! [X5: int] :
% 8.83/5.03 ( ! [Xa: int] :
% 8.83/5.03 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.03 => ! [Xb2: int] :
% 8.83/5.03 ( ( member_int @ Xb2 @ A2 )
% 8.83/5.03 => ( X5
% 8.83/5.03 != ( minus_minus_int @ Xb2 @ Xa ) ) ) )
% 8.83/5.03 => ( ( Q @ X5 )
% 8.83/5.03 => ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) )
% 8.83/5.03 => ! [X3: int] :
% 8.83/5.03 ( ! [Xa3: int] :
% 8.83/5.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.03 => ! [Xb3: int] :
% 8.83/5.03 ( ( member_int @ Xb3 @ A2 )
% 8.83/5.03 => ( X3
% 8.83/5.03 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.83/5.03 => ( ( ( P @ X3 )
% 8.83/5.03 & ( Q @ X3 ) )
% 8.83/5.03 => ( ( P @ ( plus_plus_int @ X3 @ D4 ) )
% 8.83/5.03 & ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2795_bset_I2_J,axiom,
% 8.83/5.03 ! [D4: int,B3: set_int,P: int > $o,Q: int > $o] :
% 8.83/5.03 ( ! [X5: int] :
% 8.83/5.03 ( ! [Xa: int] :
% 8.83/5.03 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.03 => ! [Xb2: int] :
% 8.83/5.03 ( ( member_int @ Xb2 @ B3 )
% 8.83/5.03 => ( X5
% 8.83/5.03 != ( plus_plus_int @ Xb2 @ Xa ) ) ) )
% 8.83/5.03 => ( ( P @ X5 )
% 8.83/5.03 => ( P @ ( minus_minus_int @ X5 @ D4 ) ) ) )
% 8.83/5.03 => ( ! [X5: int] :
% 8.83/5.03 ( ! [Xa: int] :
% 8.83/5.03 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.03 => ! [Xb2: int] :
% 8.83/5.03 ( ( member_int @ Xb2 @ B3 )
% 8.83/5.03 => ( X5
% 8.83/5.03 != ( plus_plus_int @ Xb2 @ Xa ) ) ) )
% 8.83/5.03 => ( ( Q @ X5 )
% 8.83/5.03 => ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) )
% 8.83/5.03 => ! [X3: int] :
% 8.83/5.03 ( ! [Xa3: int] :
% 8.83/5.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.03 => ! [Xb3: int] :
% 8.83/5.03 ( ( member_int @ Xb3 @ B3 )
% 8.83/5.03 => ( X3
% 8.83/5.03 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.83/5.03 => ( ( ( P @ X3 )
% 8.83/5.03 | ( Q @ X3 ) )
% 8.83/5.03 => ( ( P @ ( minus_minus_int @ X3 @ D4 ) )
% 8.83/5.03 | ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2796_bset_I1_J,axiom,
% 8.83/5.03 ! [D4: int,B3: set_int,P: int > $o,Q: int > $o] :
% 8.83/5.03 ( ! [X5: int] :
% 8.83/5.03 ( ! [Xa: int] :
% 8.83/5.03 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.03 => ! [Xb2: int] :
% 8.83/5.03 ( ( member_int @ Xb2 @ B3 )
% 8.83/5.03 => ( X5
% 8.83/5.03 != ( plus_plus_int @ Xb2 @ Xa ) ) ) )
% 8.83/5.03 => ( ( P @ X5 )
% 8.83/5.03 => ( P @ ( minus_minus_int @ X5 @ D4 ) ) ) )
% 8.83/5.03 => ( ! [X5: int] :
% 8.83/5.03 ( ! [Xa: int] :
% 8.83/5.03 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.03 => ! [Xb2: int] :
% 8.83/5.03 ( ( member_int @ Xb2 @ B3 )
% 8.83/5.03 => ( X5
% 8.83/5.03 != ( plus_plus_int @ Xb2 @ Xa ) ) ) )
% 8.83/5.03 => ( ( Q @ X5 )
% 8.83/5.03 => ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) )
% 8.83/5.03 => ! [X3: int] :
% 8.83/5.03 ( ! [Xa3: int] :
% 8.83/5.03 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.03 => ! [Xb3: int] :
% 8.83/5.03 ( ( member_int @ Xb3 @ B3 )
% 8.83/5.03 => ( X3
% 8.83/5.03 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.83/5.03 => ( ( ( P @ X3 )
% 8.83/5.03 & ( Q @ X3 ) )
% 8.83/5.03 => ( ( P @ ( minus_minus_int @ X3 @ D4 ) )
% 8.83/5.03 & ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2797_minf_I7_J,axiom,
% 8.83/5.03 ! [T: real] :
% 8.83/5.03 ? [Z2: real] :
% 8.83/5.03 ! [X3: real] :
% 8.83/5.03 ( ( ord_less_real @ X3 @ Z2 )
% 8.83/5.03 => ~ ( ord_less_real @ T @ X3 ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2798_minf_I7_J,axiom,
% 8.83/5.03 ! [T: rat] :
% 8.83/5.03 ? [Z2: rat] :
% 8.83/5.03 ! [X3: rat] :
% 8.83/5.03 ( ( ord_less_rat @ X3 @ Z2 )
% 8.83/5.03 => ~ ( ord_less_rat @ T @ X3 ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2799_minf_I7_J,axiom,
% 8.83/5.03 ! [T: num] :
% 8.83/5.03 ? [Z2: num] :
% 8.83/5.03 ! [X3: num] :
% 8.83/5.03 ( ( ord_less_num @ X3 @ Z2 )
% 8.83/5.03 => ~ ( ord_less_num @ T @ X3 ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2800_minf_I7_J,axiom,
% 8.83/5.03 ! [T: nat] :
% 8.83/5.03 ? [Z2: nat] :
% 8.83/5.03 ! [X3: nat] :
% 8.83/5.03 ( ( ord_less_nat @ X3 @ Z2 )
% 8.83/5.03 => ~ ( ord_less_nat @ T @ X3 ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2801_minf_I7_J,axiom,
% 8.83/5.03 ! [T: int] :
% 8.83/5.03 ? [Z2: int] :
% 8.83/5.03 ! [X3: int] :
% 8.83/5.03 ( ( ord_less_int @ X3 @ Z2 )
% 8.83/5.03 => ~ ( ord_less_int @ T @ X3 ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2802_minf_I5_J,axiom,
% 8.83/5.03 ! [T: real] :
% 8.83/5.03 ? [Z2: real] :
% 8.83/5.03 ! [X3: real] :
% 8.83/5.03 ( ( ord_less_real @ X3 @ Z2 )
% 8.83/5.03 => ( ord_less_real @ X3 @ T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2803_minf_I5_J,axiom,
% 8.83/5.03 ! [T: rat] :
% 8.83/5.03 ? [Z2: rat] :
% 8.83/5.03 ! [X3: rat] :
% 8.83/5.03 ( ( ord_less_rat @ X3 @ Z2 )
% 8.83/5.03 => ( ord_less_rat @ X3 @ T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2804_minf_I5_J,axiom,
% 8.83/5.03 ! [T: num] :
% 8.83/5.03 ? [Z2: num] :
% 8.83/5.03 ! [X3: num] :
% 8.83/5.03 ( ( ord_less_num @ X3 @ Z2 )
% 8.83/5.03 => ( ord_less_num @ X3 @ T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2805_minf_I5_J,axiom,
% 8.83/5.03 ! [T: nat] :
% 8.83/5.03 ? [Z2: nat] :
% 8.83/5.03 ! [X3: nat] :
% 8.83/5.03 ( ( ord_less_nat @ X3 @ Z2 )
% 8.83/5.03 => ( ord_less_nat @ X3 @ T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2806_minf_I5_J,axiom,
% 8.83/5.03 ! [T: int] :
% 8.83/5.03 ? [Z2: int] :
% 8.83/5.03 ! [X3: int] :
% 8.83/5.03 ( ( ord_less_int @ X3 @ Z2 )
% 8.83/5.03 => ( ord_less_int @ X3 @ T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2807_minf_I4_J,axiom,
% 8.83/5.03 ! [T: real] :
% 8.83/5.03 ? [Z2: real] :
% 8.83/5.03 ! [X3: real] :
% 8.83/5.03 ( ( ord_less_real @ X3 @ Z2 )
% 8.83/5.03 => ( X3 != T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2808_minf_I4_J,axiom,
% 8.83/5.03 ! [T: rat] :
% 8.83/5.03 ? [Z2: rat] :
% 8.83/5.03 ! [X3: rat] :
% 8.83/5.03 ( ( ord_less_rat @ X3 @ Z2 )
% 8.83/5.03 => ( X3 != T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2809_minf_I4_J,axiom,
% 8.83/5.03 ! [T: num] :
% 8.83/5.03 ? [Z2: num] :
% 8.83/5.03 ! [X3: num] :
% 8.83/5.03 ( ( ord_less_num @ X3 @ Z2 )
% 8.83/5.03 => ( X3 != T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2810_minf_I4_J,axiom,
% 8.83/5.03 ! [T: nat] :
% 8.83/5.03 ? [Z2: nat] :
% 8.83/5.03 ! [X3: nat] :
% 8.83/5.03 ( ( ord_less_nat @ X3 @ Z2 )
% 8.83/5.03 => ( X3 != T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2811_minf_I4_J,axiom,
% 8.83/5.03 ! [T: int] :
% 8.83/5.03 ? [Z2: int] :
% 8.83/5.03 ! [X3: int] :
% 8.83/5.03 ( ( ord_less_int @ X3 @ Z2 )
% 8.83/5.03 => ( X3 != T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2812_minf_I3_J,axiom,
% 8.83/5.03 ! [T: real] :
% 8.83/5.03 ? [Z2: real] :
% 8.83/5.03 ! [X3: real] :
% 8.83/5.03 ( ( ord_less_real @ X3 @ Z2 )
% 8.83/5.03 => ( X3 != T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2813_minf_I3_J,axiom,
% 8.83/5.03 ! [T: rat] :
% 8.83/5.03 ? [Z2: rat] :
% 8.83/5.03 ! [X3: rat] :
% 8.83/5.03 ( ( ord_less_rat @ X3 @ Z2 )
% 8.83/5.03 => ( X3 != T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2814_minf_I3_J,axiom,
% 8.83/5.03 ! [T: num] :
% 8.83/5.03 ? [Z2: num] :
% 8.83/5.03 ! [X3: num] :
% 8.83/5.03 ( ( ord_less_num @ X3 @ Z2 )
% 8.83/5.03 => ( X3 != T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2815_minf_I3_J,axiom,
% 8.83/5.03 ! [T: nat] :
% 8.83/5.03 ? [Z2: nat] :
% 8.83/5.03 ! [X3: nat] :
% 8.83/5.03 ( ( ord_less_nat @ X3 @ Z2 )
% 8.83/5.03 => ( X3 != T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2816_minf_I3_J,axiom,
% 8.83/5.03 ! [T: int] :
% 8.83/5.03 ? [Z2: int] :
% 8.83/5.03 ! [X3: int] :
% 8.83/5.03 ( ( ord_less_int @ X3 @ Z2 )
% 8.83/5.03 => ( X3 != T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2817_minf_I2_J,axiom,
% 8.83/5.03 ! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
% 8.83/5.03 ( ? [Z4: real] :
% 8.83/5.03 ! [X5: real] :
% 8.83/5.03 ( ( ord_less_real @ X5 @ Z4 )
% 8.83/5.03 => ( ( P @ X5 )
% 8.83/5.03 = ( P5 @ X5 ) ) )
% 8.83/5.03 => ( ? [Z4: real] :
% 8.83/5.03 ! [X5: real] :
% 8.83/5.03 ( ( ord_less_real @ X5 @ Z4 )
% 8.83/5.03 => ( ( Q @ X5 )
% 8.83/5.03 = ( Q6 @ X5 ) ) )
% 8.83/5.03 => ? [Z2: real] :
% 8.83/5.03 ! [X3: real] :
% 8.83/5.03 ( ( ord_less_real @ X3 @ Z2 )
% 8.83/5.03 => ( ( ( P @ X3 )
% 8.83/5.03 | ( Q @ X3 ) )
% 8.83/5.03 = ( ( P5 @ X3 )
% 8.83/5.03 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2818_minf_I2_J,axiom,
% 8.83/5.03 ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 8.83/5.03 ( ? [Z4: rat] :
% 8.83/5.03 ! [X5: rat] :
% 8.83/5.03 ( ( ord_less_rat @ X5 @ Z4 )
% 8.83/5.03 => ( ( P @ X5 )
% 8.83/5.03 = ( P5 @ X5 ) ) )
% 8.83/5.03 => ( ? [Z4: rat] :
% 8.83/5.03 ! [X5: rat] :
% 8.83/5.03 ( ( ord_less_rat @ X5 @ Z4 )
% 8.83/5.03 => ( ( Q @ X5 )
% 8.83/5.03 = ( Q6 @ X5 ) ) )
% 8.83/5.03 => ? [Z2: rat] :
% 8.83/5.03 ! [X3: rat] :
% 8.83/5.03 ( ( ord_less_rat @ X3 @ Z2 )
% 8.83/5.03 => ( ( ( P @ X3 )
% 8.83/5.03 | ( Q @ X3 ) )
% 8.83/5.03 = ( ( P5 @ X3 )
% 8.83/5.03 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2819_minf_I2_J,axiom,
% 8.83/5.03 ! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
% 8.83/5.03 ( ? [Z4: num] :
% 8.83/5.03 ! [X5: num] :
% 8.83/5.03 ( ( ord_less_num @ X5 @ Z4 )
% 8.83/5.03 => ( ( P @ X5 )
% 8.83/5.03 = ( P5 @ X5 ) ) )
% 8.83/5.03 => ( ? [Z4: num] :
% 8.83/5.03 ! [X5: num] :
% 8.83/5.03 ( ( ord_less_num @ X5 @ Z4 )
% 8.83/5.03 => ( ( Q @ X5 )
% 8.83/5.03 = ( Q6 @ X5 ) ) )
% 8.83/5.03 => ? [Z2: num] :
% 8.83/5.03 ! [X3: num] :
% 8.83/5.03 ( ( ord_less_num @ X3 @ Z2 )
% 8.83/5.03 => ( ( ( P @ X3 )
% 8.83/5.03 | ( Q @ X3 ) )
% 8.83/5.03 = ( ( P5 @ X3 )
% 8.83/5.03 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2820_minf_I2_J,axiom,
% 8.83/5.03 ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 8.83/5.03 ( ? [Z4: nat] :
% 8.83/5.03 ! [X5: nat] :
% 8.83/5.03 ( ( ord_less_nat @ X5 @ Z4 )
% 8.83/5.03 => ( ( P @ X5 )
% 8.83/5.03 = ( P5 @ X5 ) ) )
% 8.83/5.03 => ( ? [Z4: nat] :
% 8.83/5.03 ! [X5: nat] :
% 8.83/5.03 ( ( ord_less_nat @ X5 @ Z4 )
% 8.83/5.03 => ( ( Q @ X5 )
% 8.83/5.03 = ( Q6 @ X5 ) ) )
% 8.83/5.03 => ? [Z2: nat] :
% 8.83/5.03 ! [X3: nat] :
% 8.83/5.03 ( ( ord_less_nat @ X3 @ Z2 )
% 8.83/5.03 => ( ( ( P @ X3 )
% 8.83/5.03 | ( Q @ X3 ) )
% 8.83/5.03 = ( ( P5 @ X3 )
% 8.83/5.03 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2821_minf_I2_J,axiom,
% 8.83/5.03 ! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
% 8.83/5.03 ( ? [Z4: int] :
% 8.83/5.03 ! [X5: int] :
% 8.83/5.03 ( ( ord_less_int @ X5 @ Z4 )
% 8.83/5.03 => ( ( P @ X5 )
% 8.83/5.03 = ( P5 @ X5 ) ) )
% 8.83/5.03 => ( ? [Z4: int] :
% 8.83/5.03 ! [X5: int] :
% 8.83/5.03 ( ( ord_less_int @ X5 @ Z4 )
% 8.83/5.03 => ( ( Q @ X5 )
% 8.83/5.03 = ( Q6 @ X5 ) ) )
% 8.83/5.03 => ? [Z2: int] :
% 8.83/5.03 ! [X3: int] :
% 8.83/5.03 ( ( ord_less_int @ X3 @ Z2 )
% 8.83/5.03 => ( ( ( P @ X3 )
% 8.83/5.03 | ( Q @ X3 ) )
% 8.83/5.03 = ( ( P5 @ X3 )
% 8.83/5.03 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2822_minf_I1_J,axiom,
% 8.83/5.03 ! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
% 8.83/5.03 ( ? [Z4: real] :
% 8.83/5.03 ! [X5: real] :
% 8.83/5.03 ( ( ord_less_real @ X5 @ Z4 )
% 8.83/5.03 => ( ( P @ X5 )
% 8.83/5.03 = ( P5 @ X5 ) ) )
% 8.83/5.03 => ( ? [Z4: real] :
% 8.83/5.03 ! [X5: real] :
% 8.83/5.03 ( ( ord_less_real @ X5 @ Z4 )
% 8.83/5.03 => ( ( Q @ X5 )
% 8.83/5.03 = ( Q6 @ X5 ) ) )
% 8.83/5.03 => ? [Z2: real] :
% 8.83/5.03 ! [X3: real] :
% 8.83/5.03 ( ( ord_less_real @ X3 @ Z2 )
% 8.83/5.03 => ( ( ( P @ X3 )
% 8.83/5.03 & ( Q @ X3 ) )
% 8.83/5.03 = ( ( P5 @ X3 )
% 8.83/5.03 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2823_minf_I1_J,axiom,
% 8.83/5.03 ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 8.83/5.03 ( ? [Z4: rat] :
% 8.83/5.03 ! [X5: rat] :
% 8.83/5.03 ( ( ord_less_rat @ X5 @ Z4 )
% 8.83/5.03 => ( ( P @ X5 )
% 8.83/5.03 = ( P5 @ X5 ) ) )
% 8.83/5.03 => ( ? [Z4: rat] :
% 8.83/5.03 ! [X5: rat] :
% 8.83/5.03 ( ( ord_less_rat @ X5 @ Z4 )
% 8.83/5.03 => ( ( Q @ X5 )
% 8.83/5.03 = ( Q6 @ X5 ) ) )
% 8.83/5.03 => ? [Z2: rat] :
% 8.83/5.03 ! [X3: rat] :
% 8.83/5.03 ( ( ord_less_rat @ X3 @ Z2 )
% 8.83/5.03 => ( ( ( P @ X3 )
% 8.83/5.03 & ( Q @ X3 ) )
% 8.83/5.03 = ( ( P5 @ X3 )
% 8.83/5.03 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2824_minf_I1_J,axiom,
% 8.83/5.03 ! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
% 8.83/5.03 ( ? [Z4: num] :
% 8.83/5.03 ! [X5: num] :
% 8.83/5.03 ( ( ord_less_num @ X5 @ Z4 )
% 8.83/5.03 => ( ( P @ X5 )
% 8.83/5.03 = ( P5 @ X5 ) ) )
% 8.83/5.03 => ( ? [Z4: num] :
% 8.83/5.03 ! [X5: num] :
% 8.83/5.03 ( ( ord_less_num @ X5 @ Z4 )
% 8.83/5.03 => ( ( Q @ X5 )
% 8.83/5.03 = ( Q6 @ X5 ) ) )
% 8.83/5.03 => ? [Z2: num] :
% 8.83/5.03 ! [X3: num] :
% 8.83/5.03 ( ( ord_less_num @ X3 @ Z2 )
% 8.83/5.03 => ( ( ( P @ X3 )
% 8.83/5.03 & ( Q @ X3 ) )
% 8.83/5.03 = ( ( P5 @ X3 )
% 8.83/5.03 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2825_minf_I1_J,axiom,
% 8.83/5.03 ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 8.83/5.03 ( ? [Z4: nat] :
% 8.83/5.03 ! [X5: nat] :
% 8.83/5.03 ( ( ord_less_nat @ X5 @ Z4 )
% 8.83/5.03 => ( ( P @ X5 )
% 8.83/5.03 = ( P5 @ X5 ) ) )
% 8.83/5.03 => ( ? [Z4: nat] :
% 8.83/5.03 ! [X5: nat] :
% 8.83/5.03 ( ( ord_less_nat @ X5 @ Z4 )
% 8.83/5.03 => ( ( Q @ X5 )
% 8.83/5.03 = ( Q6 @ X5 ) ) )
% 8.83/5.03 => ? [Z2: nat] :
% 8.83/5.03 ! [X3: nat] :
% 8.83/5.03 ( ( ord_less_nat @ X3 @ Z2 )
% 8.83/5.03 => ( ( ( P @ X3 )
% 8.83/5.03 & ( Q @ X3 ) )
% 8.83/5.03 = ( ( P5 @ X3 )
% 8.83/5.03 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2826_minf_I1_J,axiom,
% 8.83/5.03 ! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
% 8.83/5.03 ( ? [Z4: int] :
% 8.83/5.03 ! [X5: int] :
% 8.83/5.03 ( ( ord_less_int @ X5 @ Z4 )
% 8.83/5.03 => ( ( P @ X5 )
% 8.83/5.03 = ( P5 @ X5 ) ) )
% 8.83/5.03 => ( ? [Z4: int] :
% 8.83/5.03 ! [X5: int] :
% 8.83/5.03 ( ( ord_less_int @ X5 @ Z4 )
% 8.83/5.03 => ( ( Q @ X5 )
% 8.83/5.03 = ( Q6 @ X5 ) ) )
% 8.83/5.03 => ? [Z2: int] :
% 8.83/5.03 ! [X3: int] :
% 8.83/5.03 ( ( ord_less_int @ X3 @ Z2 )
% 8.83/5.03 => ( ( ( P @ X3 )
% 8.83/5.03 & ( Q @ X3 ) )
% 8.83/5.03 = ( ( P5 @ X3 )
% 8.83/5.03 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2827_pinf_I7_J,axiom,
% 8.83/5.03 ! [T: real] :
% 8.83/5.03 ? [Z2: real] :
% 8.83/5.03 ! [X3: real] :
% 8.83/5.03 ( ( ord_less_real @ Z2 @ X3 )
% 8.83/5.03 => ( ord_less_real @ T @ X3 ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2828_pinf_I7_J,axiom,
% 8.83/5.03 ! [T: rat] :
% 8.83/5.03 ? [Z2: rat] :
% 8.83/5.03 ! [X3: rat] :
% 8.83/5.03 ( ( ord_less_rat @ Z2 @ X3 )
% 8.83/5.03 => ( ord_less_rat @ T @ X3 ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2829_pinf_I7_J,axiom,
% 8.83/5.03 ! [T: num] :
% 8.83/5.03 ? [Z2: num] :
% 8.83/5.03 ! [X3: num] :
% 8.83/5.03 ( ( ord_less_num @ Z2 @ X3 )
% 8.83/5.03 => ( ord_less_num @ T @ X3 ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2830_pinf_I7_J,axiom,
% 8.83/5.03 ! [T: nat] :
% 8.83/5.03 ? [Z2: nat] :
% 8.83/5.03 ! [X3: nat] :
% 8.83/5.03 ( ( ord_less_nat @ Z2 @ X3 )
% 8.83/5.03 => ( ord_less_nat @ T @ X3 ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2831_pinf_I7_J,axiom,
% 8.83/5.03 ! [T: int] :
% 8.83/5.03 ? [Z2: int] :
% 8.83/5.03 ! [X3: int] :
% 8.83/5.03 ( ( ord_less_int @ Z2 @ X3 )
% 8.83/5.03 => ( ord_less_int @ T @ X3 ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2832_pinf_I5_J,axiom,
% 8.83/5.03 ! [T: real] :
% 8.83/5.03 ? [Z2: real] :
% 8.83/5.03 ! [X3: real] :
% 8.83/5.03 ( ( ord_less_real @ Z2 @ X3 )
% 8.83/5.03 => ~ ( ord_less_real @ X3 @ T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2833_pinf_I5_J,axiom,
% 8.83/5.03 ! [T: rat] :
% 8.83/5.03 ? [Z2: rat] :
% 8.83/5.03 ! [X3: rat] :
% 8.83/5.03 ( ( ord_less_rat @ Z2 @ X3 )
% 8.83/5.03 => ~ ( ord_less_rat @ X3 @ T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2834_pinf_I5_J,axiom,
% 8.83/5.03 ! [T: num] :
% 8.83/5.03 ? [Z2: num] :
% 8.83/5.03 ! [X3: num] :
% 8.83/5.03 ( ( ord_less_num @ Z2 @ X3 )
% 8.83/5.03 => ~ ( ord_less_num @ X3 @ T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2835_pinf_I5_J,axiom,
% 8.83/5.03 ! [T: nat] :
% 8.83/5.03 ? [Z2: nat] :
% 8.83/5.03 ! [X3: nat] :
% 8.83/5.03 ( ( ord_less_nat @ Z2 @ X3 )
% 8.83/5.03 => ~ ( ord_less_nat @ X3 @ T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2836_pinf_I5_J,axiom,
% 8.83/5.03 ! [T: int] :
% 8.83/5.03 ? [Z2: int] :
% 8.83/5.03 ! [X3: int] :
% 8.83/5.03 ( ( ord_less_int @ Z2 @ X3 )
% 8.83/5.03 => ~ ( ord_less_int @ X3 @ T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2837_pinf_I4_J,axiom,
% 8.83/5.03 ! [T: real] :
% 8.83/5.03 ? [Z2: real] :
% 8.83/5.03 ! [X3: real] :
% 8.83/5.03 ( ( ord_less_real @ Z2 @ X3 )
% 8.83/5.03 => ( X3 != T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2838_pinf_I4_J,axiom,
% 8.83/5.03 ! [T: rat] :
% 8.83/5.03 ? [Z2: rat] :
% 8.83/5.03 ! [X3: rat] :
% 8.83/5.03 ( ( ord_less_rat @ Z2 @ X3 )
% 8.83/5.03 => ( X3 != T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2839_pinf_I4_J,axiom,
% 8.83/5.03 ! [T: num] :
% 8.83/5.03 ? [Z2: num] :
% 8.83/5.03 ! [X3: num] :
% 8.83/5.03 ( ( ord_less_num @ Z2 @ X3 )
% 8.83/5.03 => ( X3 != T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2840_pinf_I4_J,axiom,
% 8.83/5.03 ! [T: nat] :
% 8.83/5.03 ? [Z2: nat] :
% 8.83/5.03 ! [X3: nat] :
% 8.83/5.03 ( ( ord_less_nat @ Z2 @ X3 )
% 8.83/5.03 => ( X3 != T ) ) ).
% 8.83/5.03
% 8.83/5.03 thf(fact_2841_pinf_I4_J,axiom,
% 8.83/5.03 ! [T: int] :
% 8.83/5.03 ? [Z2: int] :
% 8.83/5.03 ! [X3: int] :
% 8.83/5.03 ( ( ord_less_int @ Z2 @ X3 )
% 8.83/5.04 => ( X3 != T ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2842_pinf_I3_J,axiom,
% 8.83/5.04 ! [T: real] :
% 8.83/5.04 ? [Z2: real] :
% 8.83/5.04 ! [X3: real] :
% 8.83/5.04 ( ( ord_less_real @ Z2 @ X3 )
% 8.83/5.04 => ( X3 != T ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2843_pinf_I3_J,axiom,
% 8.83/5.04 ! [T: rat] :
% 8.83/5.04 ? [Z2: rat] :
% 8.83/5.04 ! [X3: rat] :
% 8.83/5.04 ( ( ord_less_rat @ Z2 @ X3 )
% 8.83/5.04 => ( X3 != T ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2844_pinf_I3_J,axiom,
% 8.83/5.04 ! [T: num] :
% 8.83/5.04 ? [Z2: num] :
% 8.83/5.04 ! [X3: num] :
% 8.83/5.04 ( ( ord_less_num @ Z2 @ X3 )
% 8.83/5.04 => ( X3 != T ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2845_pinf_I3_J,axiom,
% 8.83/5.04 ! [T: nat] :
% 8.83/5.04 ? [Z2: nat] :
% 8.83/5.04 ! [X3: nat] :
% 8.83/5.04 ( ( ord_less_nat @ Z2 @ X3 )
% 8.83/5.04 => ( X3 != T ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2846_pinf_I3_J,axiom,
% 8.83/5.04 ! [T: int] :
% 8.83/5.04 ? [Z2: int] :
% 8.83/5.04 ! [X3: int] :
% 8.83/5.04 ( ( ord_less_int @ Z2 @ X3 )
% 8.83/5.04 => ( X3 != T ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2847_pinf_I2_J,axiom,
% 8.83/5.04 ! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
% 8.83/5.04 ( ? [Z4: real] :
% 8.83/5.04 ! [X5: real] :
% 8.83/5.04 ( ( ord_less_real @ Z4 @ X5 )
% 8.83/5.04 => ( ( P @ X5 )
% 8.83/5.04 = ( P5 @ X5 ) ) )
% 8.83/5.04 => ( ? [Z4: real] :
% 8.83/5.04 ! [X5: real] :
% 8.83/5.04 ( ( ord_less_real @ Z4 @ X5 )
% 8.83/5.04 => ( ( Q @ X5 )
% 8.83/5.04 = ( Q6 @ X5 ) ) )
% 8.83/5.04 => ? [Z2: real] :
% 8.83/5.04 ! [X3: real] :
% 8.83/5.04 ( ( ord_less_real @ Z2 @ X3 )
% 8.83/5.04 => ( ( ( P @ X3 )
% 8.83/5.04 | ( Q @ X3 ) )
% 8.83/5.04 = ( ( P5 @ X3 )
% 8.83/5.04 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2848_pinf_I2_J,axiom,
% 8.83/5.04 ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 8.83/5.04 ( ? [Z4: rat] :
% 8.83/5.04 ! [X5: rat] :
% 8.83/5.04 ( ( ord_less_rat @ Z4 @ X5 )
% 8.83/5.04 => ( ( P @ X5 )
% 8.83/5.04 = ( P5 @ X5 ) ) )
% 8.83/5.04 => ( ? [Z4: rat] :
% 8.83/5.04 ! [X5: rat] :
% 8.83/5.04 ( ( ord_less_rat @ Z4 @ X5 )
% 8.83/5.04 => ( ( Q @ X5 )
% 8.83/5.04 = ( Q6 @ X5 ) ) )
% 8.83/5.04 => ? [Z2: rat] :
% 8.83/5.04 ! [X3: rat] :
% 8.83/5.04 ( ( ord_less_rat @ Z2 @ X3 )
% 8.83/5.04 => ( ( ( P @ X3 )
% 8.83/5.04 | ( Q @ X3 ) )
% 8.83/5.04 = ( ( P5 @ X3 )
% 8.83/5.04 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2849_pinf_I2_J,axiom,
% 8.83/5.04 ! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
% 8.83/5.04 ( ? [Z4: num] :
% 8.83/5.04 ! [X5: num] :
% 8.83/5.04 ( ( ord_less_num @ Z4 @ X5 )
% 8.83/5.04 => ( ( P @ X5 )
% 8.83/5.04 = ( P5 @ X5 ) ) )
% 8.83/5.04 => ( ? [Z4: num] :
% 8.83/5.04 ! [X5: num] :
% 8.83/5.04 ( ( ord_less_num @ Z4 @ X5 )
% 8.83/5.04 => ( ( Q @ X5 )
% 8.83/5.04 = ( Q6 @ X5 ) ) )
% 8.83/5.04 => ? [Z2: num] :
% 8.83/5.04 ! [X3: num] :
% 8.83/5.04 ( ( ord_less_num @ Z2 @ X3 )
% 8.83/5.04 => ( ( ( P @ X3 )
% 8.83/5.04 | ( Q @ X3 ) )
% 8.83/5.04 = ( ( P5 @ X3 )
% 8.83/5.04 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2850_pinf_I2_J,axiom,
% 8.83/5.04 ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 8.83/5.04 ( ? [Z4: nat] :
% 8.83/5.04 ! [X5: nat] :
% 8.83/5.04 ( ( ord_less_nat @ Z4 @ X5 )
% 8.83/5.04 => ( ( P @ X5 )
% 8.83/5.04 = ( P5 @ X5 ) ) )
% 8.83/5.04 => ( ? [Z4: nat] :
% 8.83/5.04 ! [X5: nat] :
% 8.83/5.04 ( ( ord_less_nat @ Z4 @ X5 )
% 8.83/5.04 => ( ( Q @ X5 )
% 8.83/5.04 = ( Q6 @ X5 ) ) )
% 8.83/5.04 => ? [Z2: nat] :
% 8.83/5.04 ! [X3: nat] :
% 8.83/5.04 ( ( ord_less_nat @ Z2 @ X3 )
% 8.83/5.04 => ( ( ( P @ X3 )
% 8.83/5.04 | ( Q @ X3 ) )
% 8.83/5.04 = ( ( P5 @ X3 )
% 8.83/5.04 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2851_pinf_I2_J,axiom,
% 8.83/5.04 ! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
% 8.83/5.04 ( ? [Z4: int] :
% 8.83/5.04 ! [X5: int] :
% 8.83/5.04 ( ( ord_less_int @ Z4 @ X5 )
% 8.83/5.04 => ( ( P @ X5 )
% 8.83/5.04 = ( P5 @ X5 ) ) )
% 8.83/5.04 => ( ? [Z4: int] :
% 8.83/5.04 ! [X5: int] :
% 8.83/5.04 ( ( ord_less_int @ Z4 @ X5 )
% 8.83/5.04 => ( ( Q @ X5 )
% 8.83/5.04 = ( Q6 @ X5 ) ) )
% 8.83/5.04 => ? [Z2: int] :
% 8.83/5.04 ! [X3: int] :
% 8.83/5.04 ( ( ord_less_int @ Z2 @ X3 )
% 8.83/5.04 => ( ( ( P @ X3 )
% 8.83/5.04 | ( Q @ X3 ) )
% 8.83/5.04 = ( ( P5 @ X3 )
% 8.83/5.04 | ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2852_pinf_I1_J,axiom,
% 8.83/5.04 ! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
% 8.83/5.04 ( ? [Z4: real] :
% 8.83/5.04 ! [X5: real] :
% 8.83/5.04 ( ( ord_less_real @ Z4 @ X5 )
% 8.83/5.04 => ( ( P @ X5 )
% 8.83/5.04 = ( P5 @ X5 ) ) )
% 8.83/5.04 => ( ? [Z4: real] :
% 8.83/5.04 ! [X5: real] :
% 8.83/5.04 ( ( ord_less_real @ Z4 @ X5 )
% 8.83/5.04 => ( ( Q @ X5 )
% 8.83/5.04 = ( Q6 @ X5 ) ) )
% 8.83/5.04 => ? [Z2: real] :
% 8.83/5.04 ! [X3: real] :
% 8.83/5.04 ( ( ord_less_real @ Z2 @ X3 )
% 8.83/5.04 => ( ( ( P @ X3 )
% 8.83/5.04 & ( Q @ X3 ) )
% 8.83/5.04 = ( ( P5 @ X3 )
% 8.83/5.04 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2853_pinf_I1_J,axiom,
% 8.83/5.04 ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 8.83/5.04 ( ? [Z4: rat] :
% 8.83/5.04 ! [X5: rat] :
% 8.83/5.04 ( ( ord_less_rat @ Z4 @ X5 )
% 8.83/5.04 => ( ( P @ X5 )
% 8.83/5.04 = ( P5 @ X5 ) ) )
% 8.83/5.04 => ( ? [Z4: rat] :
% 8.83/5.04 ! [X5: rat] :
% 8.83/5.04 ( ( ord_less_rat @ Z4 @ X5 )
% 8.83/5.04 => ( ( Q @ X5 )
% 8.83/5.04 = ( Q6 @ X5 ) ) )
% 8.83/5.04 => ? [Z2: rat] :
% 8.83/5.04 ! [X3: rat] :
% 8.83/5.04 ( ( ord_less_rat @ Z2 @ X3 )
% 8.83/5.04 => ( ( ( P @ X3 )
% 8.83/5.04 & ( Q @ X3 ) )
% 8.83/5.04 = ( ( P5 @ X3 )
% 8.83/5.04 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2854_pinf_I1_J,axiom,
% 8.83/5.04 ! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
% 8.83/5.04 ( ? [Z4: num] :
% 8.83/5.04 ! [X5: num] :
% 8.83/5.04 ( ( ord_less_num @ Z4 @ X5 )
% 8.83/5.04 => ( ( P @ X5 )
% 8.83/5.04 = ( P5 @ X5 ) ) )
% 8.83/5.04 => ( ? [Z4: num] :
% 8.83/5.04 ! [X5: num] :
% 8.83/5.04 ( ( ord_less_num @ Z4 @ X5 )
% 8.83/5.04 => ( ( Q @ X5 )
% 8.83/5.04 = ( Q6 @ X5 ) ) )
% 8.83/5.04 => ? [Z2: num] :
% 8.83/5.04 ! [X3: num] :
% 8.83/5.04 ( ( ord_less_num @ Z2 @ X3 )
% 8.83/5.04 => ( ( ( P @ X3 )
% 8.83/5.04 & ( Q @ X3 ) )
% 8.83/5.04 = ( ( P5 @ X3 )
% 8.83/5.04 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2855_pinf_I1_J,axiom,
% 8.83/5.04 ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 8.83/5.04 ( ? [Z4: nat] :
% 8.83/5.04 ! [X5: nat] :
% 8.83/5.04 ( ( ord_less_nat @ Z4 @ X5 )
% 8.83/5.04 => ( ( P @ X5 )
% 8.83/5.04 = ( P5 @ X5 ) ) )
% 8.83/5.04 => ( ? [Z4: nat] :
% 8.83/5.04 ! [X5: nat] :
% 8.83/5.04 ( ( ord_less_nat @ Z4 @ X5 )
% 8.83/5.04 => ( ( Q @ X5 )
% 8.83/5.04 = ( Q6 @ X5 ) ) )
% 8.83/5.04 => ? [Z2: nat] :
% 8.83/5.04 ! [X3: nat] :
% 8.83/5.04 ( ( ord_less_nat @ Z2 @ X3 )
% 8.83/5.04 => ( ( ( P @ X3 )
% 8.83/5.04 & ( Q @ X3 ) )
% 8.83/5.04 = ( ( P5 @ X3 )
% 8.83/5.04 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2856_pinf_I1_J,axiom,
% 8.83/5.04 ! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
% 8.83/5.04 ( ? [Z4: int] :
% 8.83/5.04 ! [X5: int] :
% 8.83/5.04 ( ( ord_less_int @ Z4 @ X5 )
% 8.83/5.04 => ( ( P @ X5 )
% 8.83/5.04 = ( P5 @ X5 ) ) )
% 8.83/5.04 => ( ? [Z4: int] :
% 8.83/5.04 ! [X5: int] :
% 8.83/5.04 ( ( ord_less_int @ Z4 @ X5 )
% 8.83/5.04 => ( ( Q @ X5 )
% 8.83/5.04 = ( Q6 @ X5 ) ) )
% 8.83/5.04 => ? [Z2: int] :
% 8.83/5.04 ! [X3: int] :
% 8.83/5.04 ( ( ord_less_int @ Z2 @ X3 )
% 8.83/5.04 => ( ( ( P @ X3 )
% 8.83/5.04 & ( Q @ X3 ) )
% 8.83/5.04 = ( ( P5 @ X3 )
% 8.83/5.04 & ( Q6 @ X3 ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2857_bounded__Max__nat,axiom,
% 8.83/5.04 ! [P: nat > $o,X2: nat,M7: nat] :
% 8.83/5.04 ( ( P @ X2 )
% 8.83/5.04 => ( ! [X5: nat] :
% 8.83/5.04 ( ( P @ X5 )
% 8.83/5.04 => ( ord_less_eq_nat @ X5 @ M7 ) )
% 8.83/5.04 => ~ ! [M4: nat] :
% 8.83/5.04 ( ( P @ M4 )
% 8.83/5.04 => ~ ! [X3: nat] :
% 8.83/5.04 ( ( P @ X3 )
% 8.83/5.04 => ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2858_fold__atLeastAtMost__nat_Ocases,axiom,
% 8.83/5.04 ! [X2: produc3368934014287244435at_num] :
% 8.83/5.04 ~ ! [F2: nat > num > num,A4: nat,B4: nat,Acc: num] :
% 8.83/5.04 ( X2
% 8.83/5.04 != ( produc851828971589881931at_num @ F2 @ ( produc1195630363706982562at_num @ A4 @ ( product_Pair_nat_num @ B4 @ Acc ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2859_fold__atLeastAtMost__nat_Ocases,axiom,
% 8.83/5.04 ! [X2: produc4471711990508489141at_nat] :
% 8.83/5.04 ~ ! [F2: nat > nat > nat,A4: nat,B4: nat,Acc: nat] :
% 8.83/5.04 ( X2
% 8.83/5.04 != ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A4 @ ( product_Pair_nat_nat @ B4 @ Acc ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2860_periodic__finite__ex,axiom,
% 8.83/5.04 ! [D: int,P: int > $o] :
% 8.83/5.04 ( ( ord_less_int @ zero_zero_int @ D )
% 8.83/5.04 => ( ! [X5: int,K3: int] :
% 8.83/5.04 ( ( P @ X5 )
% 8.83/5.04 = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) )
% 8.83/5.04 => ( ( ? [X4: int] : ( P @ X4 ) )
% 8.83/5.04 = ( ? [X: int] :
% 8.83/5.04 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 8.83/5.04 & ( P @ X ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2861_bset_I3_J,axiom,
% 8.83/5.04 ! [D4: int,T: int,B3: set_int] :
% 8.83/5.04 ( ( ord_less_int @ zero_zero_int @ D4 )
% 8.83/5.04 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B3 )
% 8.83/5.04 => ! [X3: int] :
% 8.83/5.04 ( ! [Xa3: int] :
% 8.83/5.04 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.04 => ! [Xb3: int] :
% 8.83/5.04 ( ( member_int @ Xb3 @ B3 )
% 8.83/5.04 => ( X3
% 8.83/5.04 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.83/5.04 => ( ( X3 = T )
% 8.83/5.04 => ( ( minus_minus_int @ X3 @ D4 )
% 8.83/5.04 = T ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2862_bset_I4_J,axiom,
% 8.83/5.04 ! [D4: int,T: int,B3: set_int] :
% 8.83/5.04 ( ( ord_less_int @ zero_zero_int @ D4 )
% 8.83/5.04 => ( ( member_int @ T @ B3 )
% 8.83/5.04 => ! [X3: int] :
% 8.83/5.04 ( ! [Xa3: int] :
% 8.83/5.04 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.04 => ! [Xb3: int] :
% 8.83/5.04 ( ( member_int @ Xb3 @ B3 )
% 8.83/5.04 => ( X3
% 8.83/5.04 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.83/5.04 => ( ( X3 != T )
% 8.83/5.04 => ( ( minus_minus_int @ X3 @ D4 )
% 8.83/5.04 != T ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2863_bset_I5_J,axiom,
% 8.83/5.04 ! [D4: int,B3: set_int,T: int] :
% 8.83/5.04 ( ( ord_less_int @ zero_zero_int @ D4 )
% 8.83/5.04 => ! [X3: int] :
% 8.83/5.04 ( ! [Xa3: int] :
% 8.83/5.04 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.04 => ! [Xb3: int] :
% 8.83/5.04 ( ( member_int @ Xb3 @ B3 )
% 8.83/5.04 => ( X3
% 8.83/5.04 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.83/5.04 => ( ( ord_less_int @ X3 @ T )
% 8.83/5.04 => ( ord_less_int @ ( minus_minus_int @ X3 @ D4 ) @ T ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2864_bset_I7_J,axiom,
% 8.83/5.04 ! [D4: int,T: int,B3: set_int] :
% 8.83/5.04 ( ( ord_less_int @ zero_zero_int @ D4 )
% 8.83/5.04 => ( ( member_int @ T @ B3 )
% 8.83/5.04 => ! [X3: int] :
% 8.83/5.04 ( ! [Xa3: int] :
% 8.83/5.04 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.04 => ! [Xb3: int] :
% 8.83/5.04 ( ( member_int @ Xb3 @ B3 )
% 8.83/5.04 => ( X3
% 8.83/5.04 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.83/5.04 => ( ( ord_less_int @ T @ X3 )
% 8.83/5.04 => ( ord_less_int @ T @ ( minus_minus_int @ X3 @ D4 ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2865_aset_I3_J,axiom,
% 8.83/5.04 ! [D4: int,T: int,A2: set_int] :
% 8.83/5.04 ( ( ord_less_int @ zero_zero_int @ D4 )
% 8.83/5.04 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 8.83/5.04 => ! [X3: int] :
% 8.83/5.04 ( ! [Xa3: int] :
% 8.83/5.04 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.04 => ! [Xb3: int] :
% 8.83/5.04 ( ( member_int @ Xb3 @ A2 )
% 8.83/5.04 => ( X3
% 8.83/5.04 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.83/5.04 => ( ( X3 = T )
% 8.83/5.04 => ( ( plus_plus_int @ X3 @ D4 )
% 8.83/5.04 = T ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2866_aset_I4_J,axiom,
% 8.83/5.04 ! [D4: int,T: int,A2: set_int] :
% 8.83/5.04 ( ( ord_less_int @ zero_zero_int @ D4 )
% 8.83/5.04 => ( ( member_int @ T @ A2 )
% 8.83/5.04 => ! [X3: int] :
% 8.83/5.04 ( ! [Xa3: int] :
% 8.83/5.04 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.04 => ! [Xb3: int] :
% 8.83/5.04 ( ( member_int @ Xb3 @ A2 )
% 8.83/5.04 => ( X3
% 8.83/5.04 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.83/5.04 => ( ( X3 != T )
% 8.83/5.04 => ( ( plus_plus_int @ X3 @ D4 )
% 8.83/5.04 != T ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2867_aset_I5_J,axiom,
% 8.83/5.04 ! [D4: int,T: int,A2: set_int] :
% 8.83/5.04 ( ( ord_less_int @ zero_zero_int @ D4 )
% 8.83/5.04 => ( ( member_int @ T @ A2 )
% 8.83/5.04 => ! [X3: int] :
% 8.83/5.04 ( ! [Xa3: int] :
% 8.83/5.04 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.04 => ! [Xb3: int] :
% 8.83/5.04 ( ( member_int @ Xb3 @ A2 )
% 8.83/5.04 => ( X3
% 8.83/5.04 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.83/5.04 => ( ( ord_less_int @ X3 @ T )
% 8.83/5.04 => ( ord_less_int @ ( plus_plus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2868_aset_I7_J,axiom,
% 8.83/5.04 ! [D4: int,A2: set_int,T: int] :
% 8.83/5.04 ( ( ord_less_int @ zero_zero_int @ D4 )
% 8.83/5.04 => ! [X3: int] :
% 8.83/5.04 ( ! [Xa3: int] :
% 8.83/5.04 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.04 => ! [Xb3: int] :
% 8.83/5.04 ( ( member_int @ Xb3 @ A2 )
% 8.83/5.04 => ( X3
% 8.83/5.04 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.83/5.04 => ( ( ord_less_int @ T @ X3 )
% 8.83/5.04 => ( ord_less_int @ T @ ( plus_plus_int @ X3 @ D4 ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2869_bset_I6_J,axiom,
% 8.83/5.04 ! [D4: int,B3: set_int,T: int] :
% 8.83/5.04 ( ( ord_less_int @ zero_zero_int @ D4 )
% 8.83/5.04 => ! [X3: int] :
% 8.83/5.04 ( ! [Xa3: int] :
% 8.83/5.04 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.04 => ! [Xb3: int] :
% 8.83/5.04 ( ( member_int @ Xb3 @ B3 )
% 8.83/5.04 => ( X3
% 8.83/5.04 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.83/5.04 => ( ( ord_less_eq_int @ X3 @ T )
% 8.83/5.04 => ( ord_less_eq_int @ ( minus_minus_int @ X3 @ D4 ) @ T ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2870_bset_I8_J,axiom,
% 8.83/5.04 ! [D4: int,T: int,B3: set_int] :
% 8.83/5.04 ( ( ord_less_int @ zero_zero_int @ D4 )
% 8.83/5.04 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B3 )
% 8.83/5.04 => ! [X3: int] :
% 8.83/5.04 ( ! [Xa3: int] :
% 8.83/5.04 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.04 => ! [Xb3: int] :
% 8.83/5.04 ( ( member_int @ Xb3 @ B3 )
% 8.83/5.04 => ( X3
% 8.83/5.04 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.83/5.04 => ( ( ord_less_eq_int @ T @ X3 )
% 8.83/5.04 => ( ord_less_eq_int @ T @ ( minus_minus_int @ X3 @ D4 ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2871_aset_I6_J,axiom,
% 8.83/5.04 ! [D4: int,T: int,A2: set_int] :
% 8.83/5.04 ( ( ord_less_int @ zero_zero_int @ D4 )
% 8.83/5.04 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 8.83/5.04 => ! [X3: int] :
% 8.83/5.04 ( ! [Xa3: int] :
% 8.83/5.04 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.04 => ! [Xb3: int] :
% 8.83/5.04 ( ( member_int @ Xb3 @ A2 )
% 8.83/5.04 => ( X3
% 8.83/5.04 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.83/5.04 => ( ( ord_less_eq_int @ X3 @ T )
% 8.83/5.04 => ( ord_less_eq_int @ ( plus_plus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2872_aset_I8_J,axiom,
% 8.83/5.04 ! [D4: int,A2: set_int,T: int] :
% 8.83/5.04 ( ( ord_less_int @ zero_zero_int @ D4 )
% 8.83/5.04 => ! [X3: int] :
% 8.83/5.04 ( ! [Xa3: int] :
% 8.83/5.04 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.04 => ! [Xb3: int] :
% 8.83/5.04 ( ( member_int @ Xb3 @ A2 )
% 8.83/5.04 => ( X3
% 8.83/5.04 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.83/5.04 => ( ( ord_less_eq_int @ T @ X3 )
% 8.83/5.04 => ( ord_less_eq_int @ T @ ( plus_plus_int @ X3 @ D4 ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2873_cppi,axiom,
% 8.83/5.04 ! [D4: int,P: int > $o,P5: int > $o,A2: set_int] :
% 8.83/5.04 ( ( ord_less_int @ zero_zero_int @ D4 )
% 8.83/5.04 => ( ? [Z4: int] :
% 8.83/5.04 ! [X5: int] :
% 8.83/5.04 ( ( ord_less_int @ Z4 @ X5 )
% 8.83/5.04 => ( ( P @ X5 )
% 8.83/5.04 = ( P5 @ X5 ) ) )
% 8.83/5.04 => ( ! [X5: int] :
% 8.83/5.04 ( ! [Xa: int] :
% 8.83/5.04 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.04 => ! [Xb2: int] :
% 8.83/5.04 ( ( member_int @ Xb2 @ A2 )
% 8.83/5.04 => ( X5
% 8.83/5.04 != ( minus_minus_int @ Xb2 @ Xa ) ) ) )
% 8.83/5.04 => ( ( P @ X5 )
% 8.83/5.04 => ( P @ ( plus_plus_int @ X5 @ D4 ) ) ) )
% 8.83/5.04 => ( ! [X5: int,K3: int] :
% 8.83/5.04 ( ( P5 @ X5 )
% 8.83/5.04 = ( P5 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 8.83/5.04 => ( ( ? [X4: int] : ( P @ X4 ) )
% 8.83/5.04 = ( ? [X: int] :
% 8.83/5.04 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.04 & ( P5 @ X ) )
% 8.83/5.04 | ? [X: int] :
% 8.83/5.04 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.04 & ? [Y: int] :
% 8.83/5.04 ( ( member_int @ Y @ A2 )
% 8.83/5.04 & ( P @ ( minus_minus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2874_cpmi,axiom,
% 8.83/5.04 ! [D4: int,P: int > $o,P5: int > $o,B3: set_int] :
% 8.83/5.04 ( ( ord_less_int @ zero_zero_int @ D4 )
% 8.83/5.04 => ( ? [Z4: int] :
% 8.83/5.04 ! [X5: int] :
% 8.83/5.04 ( ( ord_less_int @ X5 @ Z4 )
% 8.83/5.04 => ( ( P @ X5 )
% 8.83/5.04 = ( P5 @ X5 ) ) )
% 8.83/5.04 => ( ! [X5: int] :
% 8.83/5.04 ( ! [Xa: int] :
% 8.83/5.04 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.04 => ! [Xb2: int] :
% 8.83/5.04 ( ( member_int @ Xb2 @ B3 )
% 8.83/5.04 => ( X5
% 8.83/5.04 != ( plus_plus_int @ Xb2 @ Xa ) ) ) )
% 8.83/5.04 => ( ( P @ X5 )
% 8.83/5.04 => ( P @ ( minus_minus_int @ X5 @ D4 ) ) ) )
% 8.83/5.04 => ( ! [X5: int,K3: int] :
% 8.83/5.04 ( ( P5 @ X5 )
% 8.83/5.04 = ( P5 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 8.83/5.04 => ( ( ? [X4: int] : ( P @ X4 ) )
% 8.83/5.04 = ( ? [X: int] :
% 8.83/5.04 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.04 & ( P5 @ X ) )
% 8.83/5.04 | ? [X: int] :
% 8.83/5.04 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.83/5.04 & ? [Y: int] :
% 8.83/5.04 ( ( member_int @ Y @ B3 )
% 8.83/5.04 & ( P @ ( plus_plus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2875_pinf_I6_J,axiom,
% 8.83/5.04 ! [T: real] :
% 8.83/5.04 ? [Z2: real] :
% 8.83/5.04 ! [X3: real] :
% 8.83/5.04 ( ( ord_less_real @ Z2 @ X3 )
% 8.83/5.04 => ~ ( ord_less_eq_real @ X3 @ T ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2876_pinf_I6_J,axiom,
% 8.83/5.04 ! [T: rat] :
% 8.83/5.04 ? [Z2: rat] :
% 8.83/5.04 ! [X3: rat] :
% 8.83/5.04 ( ( ord_less_rat @ Z2 @ X3 )
% 8.83/5.04 => ~ ( ord_less_eq_rat @ X3 @ T ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2877_pinf_I6_J,axiom,
% 8.83/5.04 ! [T: num] :
% 8.83/5.04 ? [Z2: num] :
% 8.83/5.04 ! [X3: num] :
% 8.83/5.04 ( ( ord_less_num @ Z2 @ X3 )
% 8.83/5.04 => ~ ( ord_less_eq_num @ X3 @ T ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2878_pinf_I6_J,axiom,
% 8.83/5.04 ! [T: nat] :
% 8.83/5.04 ? [Z2: nat] :
% 8.83/5.04 ! [X3: nat] :
% 8.83/5.04 ( ( ord_less_nat @ Z2 @ X3 )
% 8.83/5.04 => ~ ( ord_less_eq_nat @ X3 @ T ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2879_pinf_I6_J,axiom,
% 8.83/5.04 ! [T: int] :
% 8.83/5.04 ? [Z2: int] :
% 8.83/5.04 ! [X3: int] :
% 8.83/5.04 ( ( ord_less_int @ Z2 @ X3 )
% 8.83/5.04 => ~ ( ord_less_eq_int @ X3 @ T ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2880_pinf_I8_J,axiom,
% 8.83/5.04 ! [T: real] :
% 8.83/5.04 ? [Z2: real] :
% 8.83/5.04 ! [X3: real] :
% 8.83/5.04 ( ( ord_less_real @ Z2 @ X3 )
% 8.83/5.04 => ( ord_less_eq_real @ T @ X3 ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2881_pinf_I8_J,axiom,
% 8.83/5.04 ! [T: rat] :
% 8.83/5.04 ? [Z2: rat] :
% 8.83/5.04 ! [X3: rat] :
% 8.83/5.04 ( ( ord_less_rat @ Z2 @ X3 )
% 8.83/5.04 => ( ord_less_eq_rat @ T @ X3 ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2882_pinf_I8_J,axiom,
% 8.83/5.04 ! [T: num] :
% 8.83/5.04 ? [Z2: num] :
% 8.83/5.04 ! [X3: num] :
% 8.83/5.04 ( ( ord_less_num @ Z2 @ X3 )
% 8.83/5.04 => ( ord_less_eq_num @ T @ X3 ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2883_pinf_I8_J,axiom,
% 8.83/5.04 ! [T: nat] :
% 8.83/5.04 ? [Z2: nat] :
% 8.83/5.04 ! [X3: nat] :
% 8.83/5.04 ( ( ord_less_nat @ Z2 @ X3 )
% 8.83/5.04 => ( ord_less_eq_nat @ T @ X3 ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2884_pinf_I8_J,axiom,
% 8.83/5.04 ! [T: int] :
% 8.83/5.04 ? [Z2: int] :
% 8.83/5.04 ! [X3: int] :
% 8.83/5.04 ( ( ord_less_int @ Z2 @ X3 )
% 8.83/5.04 => ( ord_less_eq_int @ T @ X3 ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2885_minf_I6_J,axiom,
% 8.83/5.04 ! [T: real] :
% 8.83/5.04 ? [Z2: real] :
% 8.83/5.04 ! [X3: real] :
% 8.83/5.04 ( ( ord_less_real @ X3 @ Z2 )
% 8.83/5.04 => ( ord_less_eq_real @ X3 @ T ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2886_minf_I6_J,axiom,
% 8.83/5.04 ! [T: rat] :
% 8.83/5.04 ? [Z2: rat] :
% 8.83/5.04 ! [X3: rat] :
% 8.83/5.04 ( ( ord_less_rat @ X3 @ Z2 )
% 8.83/5.04 => ( ord_less_eq_rat @ X3 @ T ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2887_minf_I6_J,axiom,
% 8.83/5.04 ! [T: num] :
% 8.83/5.04 ? [Z2: num] :
% 8.83/5.04 ! [X3: num] :
% 8.83/5.04 ( ( ord_less_num @ X3 @ Z2 )
% 8.83/5.04 => ( ord_less_eq_num @ X3 @ T ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2888_minf_I6_J,axiom,
% 8.83/5.04 ! [T: nat] :
% 8.83/5.04 ? [Z2: nat] :
% 8.83/5.04 ! [X3: nat] :
% 8.83/5.04 ( ( ord_less_nat @ X3 @ Z2 )
% 8.83/5.04 => ( ord_less_eq_nat @ X3 @ T ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2889_minf_I6_J,axiom,
% 8.83/5.04 ! [T: int] :
% 8.83/5.04 ? [Z2: int] :
% 8.83/5.04 ! [X3: int] :
% 8.83/5.04 ( ( ord_less_int @ X3 @ Z2 )
% 8.83/5.04 => ( ord_less_eq_int @ X3 @ T ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2890_minf_I8_J,axiom,
% 8.83/5.04 ! [T: real] :
% 8.83/5.04 ? [Z2: real] :
% 8.83/5.04 ! [X3: real] :
% 8.83/5.04 ( ( ord_less_real @ X3 @ Z2 )
% 8.83/5.04 => ~ ( ord_less_eq_real @ T @ X3 ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2891_minf_I8_J,axiom,
% 8.83/5.04 ! [T: rat] :
% 8.83/5.04 ? [Z2: rat] :
% 8.83/5.04 ! [X3: rat] :
% 8.83/5.04 ( ( ord_less_rat @ X3 @ Z2 )
% 8.83/5.04 => ~ ( ord_less_eq_rat @ T @ X3 ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2892_minf_I8_J,axiom,
% 8.83/5.04 ! [T: num] :
% 8.83/5.04 ? [Z2: num] :
% 8.83/5.04 ! [X3: num] :
% 8.83/5.04 ( ( ord_less_num @ X3 @ Z2 )
% 8.83/5.04 => ~ ( ord_less_eq_num @ T @ X3 ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2893_minf_I8_J,axiom,
% 8.83/5.04 ! [T: nat] :
% 8.83/5.04 ? [Z2: nat] :
% 8.83/5.04 ! [X3: nat] :
% 8.83/5.04 ( ( ord_less_nat @ X3 @ Z2 )
% 8.83/5.04 => ~ ( ord_less_eq_nat @ T @ X3 ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2894_minf_I8_J,axiom,
% 8.83/5.04 ! [T: int] :
% 8.83/5.04 ? [Z2: int] :
% 8.83/5.04 ! [X3: int] :
% 8.83/5.04 ( ( ord_less_int @ X3 @ Z2 )
% 8.83/5.04 => ~ ( ord_less_eq_int @ T @ X3 ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2895_inf__period_I2_J,axiom,
% 8.83/5.04 ! [P: real > $o,D4: real,Q: real > $o] :
% 8.83/5.04 ( ! [X5: real,K3: real] :
% 8.83/5.04 ( ( P @ X5 )
% 8.83/5.04 = ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 8.83/5.04 => ( ! [X5: real,K3: real] :
% 8.83/5.04 ( ( Q @ X5 )
% 8.83/5.04 = ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 8.83/5.04 => ! [X3: real,K4: real] :
% 8.83/5.04 ( ( ( P @ X3 )
% 8.83/5.04 | ( Q @ X3 ) )
% 8.83/5.04 = ( ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) )
% 8.83/5.04 | ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2896_inf__period_I2_J,axiom,
% 8.83/5.04 ! [P: rat > $o,D4: rat,Q: rat > $o] :
% 8.83/5.04 ( ! [X5: rat,K3: rat] :
% 8.83/5.04 ( ( P @ X5 )
% 8.83/5.04 = ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 8.83/5.04 => ( ! [X5: rat,K3: rat] :
% 8.83/5.04 ( ( Q @ X5 )
% 8.83/5.04 = ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 8.83/5.04 => ! [X3: rat,K4: rat] :
% 8.83/5.04 ( ( ( P @ X3 )
% 8.83/5.04 | ( Q @ X3 ) )
% 8.83/5.04 = ( ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) )
% 8.83/5.04 | ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2897_inf__period_I2_J,axiom,
% 8.83/5.04 ! [P: int > $o,D4: int,Q: int > $o] :
% 8.83/5.04 ( ! [X5: int,K3: int] :
% 8.83/5.04 ( ( P @ X5 )
% 8.83/5.04 = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 8.83/5.04 => ( ! [X5: int,K3: int] :
% 8.83/5.04 ( ( Q @ X5 )
% 8.83/5.04 = ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 8.83/5.04 => ! [X3: int,K4: int] :
% 8.83/5.04 ( ( ( P @ X3 )
% 8.83/5.04 | ( Q @ X3 ) )
% 8.83/5.04 = ( ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) )
% 8.83/5.04 | ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2898_inf__period_I1_J,axiom,
% 8.83/5.04 ! [P: real > $o,D4: real,Q: real > $o] :
% 8.83/5.04 ( ! [X5: real,K3: real] :
% 8.83/5.04 ( ( P @ X5 )
% 8.83/5.04 = ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 8.83/5.04 => ( ! [X5: real,K3: real] :
% 8.83/5.04 ( ( Q @ X5 )
% 8.83/5.04 = ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 8.83/5.04 => ! [X3: real,K4: real] :
% 8.83/5.04 ( ( ( P @ X3 )
% 8.83/5.04 & ( Q @ X3 ) )
% 8.83/5.04 = ( ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) )
% 8.83/5.04 & ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2899_inf__period_I1_J,axiom,
% 8.83/5.04 ! [P: rat > $o,D4: rat,Q: rat > $o] :
% 8.83/5.04 ( ! [X5: rat,K3: rat] :
% 8.83/5.04 ( ( P @ X5 )
% 8.83/5.04 = ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 8.83/5.04 => ( ! [X5: rat,K3: rat] :
% 8.83/5.04 ( ( Q @ X5 )
% 8.83/5.04 = ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 8.83/5.04 => ! [X3: rat,K4: rat] :
% 8.83/5.04 ( ( ( P @ X3 )
% 8.83/5.04 & ( Q @ X3 ) )
% 8.83/5.04 = ( ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) )
% 8.83/5.04 & ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2900_inf__period_I1_J,axiom,
% 8.83/5.04 ! [P: int > $o,D4: int,Q: int > $o] :
% 8.83/5.04 ( ! [X5: int,K3: int] :
% 8.83/5.04 ( ( P @ X5 )
% 8.83/5.04 = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 8.83/5.04 => ( ! [X5: int,K3: int] :
% 8.83/5.04 ( ( Q @ X5 )
% 8.83/5.04 = ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 8.83/5.04 => ! [X3: int,K4: int] :
% 8.83/5.04 ( ( ( P @ X3 )
% 8.83/5.04 & ( Q @ X3 ) )
% 8.83/5.04 = ( ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) )
% 8.83/5.04 & ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2901_conj__le__cong,axiom,
% 8.83/5.04 ! [X2: int,X7: int,P: $o,P5: $o] :
% 8.83/5.04 ( ( X2 = X7 )
% 8.83/5.04 => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 8.83/5.04 => ( P = P5 ) )
% 8.83/5.04 => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.83/5.04 & P )
% 8.83/5.04 = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 8.83/5.04 & P5 ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2902_imp__le__cong,axiom,
% 8.83/5.04 ! [X2: int,X7: int,P: $o,P5: $o] :
% 8.83/5.04 ( ( X2 = X7 )
% 8.83/5.04 => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 8.83/5.04 => ( P = P5 ) )
% 8.83/5.04 => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.83/5.04 => P )
% 8.83/5.04 = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 8.83/5.04 => P5 ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2903_atLeastatMost__psubset__iff,axiom,
% 8.83/5.04 ! [A: set_int,B: set_int,C: set_int,D: set_int] :
% 8.83/5.04 ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 8.83/5.04 = ( ( ~ ( ord_less_eq_set_int @ A @ B )
% 8.83/5.04 | ( ( ord_less_eq_set_int @ C @ A )
% 8.83/5.04 & ( ord_less_eq_set_int @ B @ D )
% 8.83/5.04 & ( ( ord_less_set_int @ C @ A )
% 8.83/5.04 | ( ord_less_set_int @ B @ D ) ) ) )
% 8.83/5.04 & ( ord_less_eq_set_int @ C @ D ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2904_atLeastatMost__psubset__iff,axiom,
% 8.83/5.04 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.83/5.04 ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 8.83/5.04 = ( ( ~ ( ord_less_eq_rat @ A @ B )
% 8.83/5.04 | ( ( ord_less_eq_rat @ C @ A )
% 8.83/5.04 & ( ord_less_eq_rat @ B @ D )
% 8.83/5.04 & ( ( ord_less_rat @ C @ A )
% 8.83/5.04 | ( ord_less_rat @ B @ D ) ) ) )
% 8.83/5.04 & ( ord_less_eq_rat @ C @ D ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2905_atLeastatMost__psubset__iff,axiom,
% 8.83/5.04 ! [A: num,B: num,C: num,D: num] :
% 8.83/5.04 ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 8.83/5.04 = ( ( ~ ( ord_less_eq_num @ A @ B )
% 8.83/5.04 | ( ( ord_less_eq_num @ C @ A )
% 8.83/5.04 & ( ord_less_eq_num @ B @ D )
% 8.83/5.04 & ( ( ord_less_num @ C @ A )
% 8.83/5.04 | ( ord_less_num @ B @ D ) ) ) )
% 8.83/5.04 & ( ord_less_eq_num @ C @ D ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2906_atLeastatMost__psubset__iff,axiom,
% 8.83/5.04 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.83/5.04 ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 8.83/5.04 = ( ( ~ ( ord_less_eq_nat @ A @ B )
% 8.83/5.04 | ( ( ord_less_eq_nat @ C @ A )
% 8.83/5.04 & ( ord_less_eq_nat @ B @ D )
% 8.83/5.04 & ( ( ord_less_nat @ C @ A )
% 8.83/5.04 | ( ord_less_nat @ B @ D ) ) ) )
% 8.83/5.04 & ( ord_less_eq_nat @ C @ D ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2907_atLeastatMost__psubset__iff,axiom,
% 8.83/5.04 ! [A: int,B: int,C: int,D: int] :
% 8.83/5.04 ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 8.83/5.04 = ( ( ~ ( ord_less_eq_int @ A @ B )
% 8.83/5.04 | ( ( ord_less_eq_int @ C @ A )
% 8.83/5.04 & ( ord_less_eq_int @ B @ D )
% 8.83/5.04 & ( ( ord_less_int @ C @ A )
% 8.83/5.04 | ( ord_less_int @ B @ D ) ) ) )
% 8.83/5.04 & ( ord_less_eq_int @ C @ D ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2908_atLeastatMost__psubset__iff,axiom,
% 8.83/5.04 ! [A: real,B: real,C: real,D: real] :
% 8.83/5.04 ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 8.83/5.04 = ( ( ~ ( ord_less_eq_real @ A @ B )
% 8.83/5.04 | ( ( ord_less_eq_real @ C @ A )
% 8.83/5.04 & ( ord_less_eq_real @ B @ D )
% 8.83/5.04 & ( ( ord_less_real @ C @ A )
% 8.83/5.04 | ( ord_less_real @ B @ D ) ) ) )
% 8.83/5.04 & ( ord_less_eq_real @ C @ D ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2909_minusinfinity,axiom,
% 8.83/5.04 ! [D: int,P1: int > $o,P: int > $o] :
% 8.83/5.04 ( ( ord_less_int @ zero_zero_int @ D )
% 8.83/5.04 => ( ! [X5: int,K3: int] :
% 8.83/5.04 ( ( P1 @ X5 )
% 8.83/5.04 = ( P1 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) )
% 8.83/5.04 => ( ? [Z4: int] :
% 8.83/5.04 ! [X5: int] :
% 8.83/5.04 ( ( ord_less_int @ X5 @ Z4 )
% 8.83/5.04 => ( ( P @ X5 )
% 8.83/5.04 = ( P1 @ X5 ) ) )
% 8.83/5.04 => ( ? [X_1: int] : ( P1 @ X_1 )
% 8.83/5.04 => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2910_plusinfinity,axiom,
% 8.83/5.04 ! [D: int,P5: int > $o,P: int > $o] :
% 8.83/5.04 ( ( ord_less_int @ zero_zero_int @ D )
% 8.83/5.04 => ( ! [X5: int,K3: int] :
% 8.83/5.04 ( ( P5 @ X5 )
% 8.83/5.04 = ( P5 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) )
% 8.83/5.04 => ( ? [Z4: int] :
% 8.83/5.04 ! [X5: int] :
% 8.83/5.04 ( ( ord_less_int @ Z4 @ X5 )
% 8.83/5.04 => ( ( P @ X5 )
% 8.83/5.04 = ( P5 @ X5 ) ) )
% 8.83/5.04 => ( ? [X_1: int] : ( P5 @ X_1 )
% 8.83/5.04 => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2911_Diff__eq__empty__iff,axiom,
% 8.83/5.04 ! [A2: set_real,B3: set_real] :
% 8.83/5.04 ( ( ( minus_minus_set_real @ A2 @ B3 )
% 8.83/5.04 = bot_bot_set_real )
% 8.83/5.04 = ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2912_Diff__eq__empty__iff,axiom,
% 8.83/5.04 ! [A2: set_nat,B3: set_nat] :
% 8.83/5.04 ( ( ( minus_minus_set_nat @ A2 @ B3 )
% 8.83/5.04 = bot_bot_set_nat )
% 8.83/5.04 = ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2913_Diff__eq__empty__iff,axiom,
% 8.83/5.04 ! [A2: set_int,B3: set_int] :
% 8.83/5.04 ( ( ( minus_minus_set_int @ A2 @ B3 )
% 8.83/5.04 = bot_bot_set_int )
% 8.83/5.04 = ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2914_empty__subsetI,axiom,
% 8.83/5.04 ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2915_empty__subsetI,axiom,
% 8.83/5.04 ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2916_empty__subsetI,axiom,
% 8.83/5.04 ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2917_subset__empty,axiom,
% 8.83/5.04 ! [A2: set_nat] :
% 8.83/5.04 ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 8.83/5.04 = ( A2 = bot_bot_set_nat ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2918_subset__empty,axiom,
% 8.83/5.04 ! [A2: set_real] :
% 8.83/5.04 ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
% 8.83/5.04 = ( A2 = bot_bot_set_real ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2919_subset__empty,axiom,
% 8.83/5.04 ! [A2: set_int] :
% 8.83/5.04 ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 8.83/5.04 = ( A2 = bot_bot_set_int ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2920_Bolzano,axiom,
% 8.83/5.04 ! [A: real,B: real,P: real > real > $o] :
% 8.83/5.04 ( ( ord_less_eq_real @ A @ B )
% 8.83/5.04 => ( ! [A4: real,B4: real,C3: real] :
% 8.83/5.04 ( ( P @ A4 @ B4 )
% 8.83/5.04 => ( ( P @ B4 @ C3 )
% 8.83/5.04 => ( ( ord_less_eq_real @ A4 @ B4 )
% 8.83/5.04 => ( ( ord_less_eq_real @ B4 @ C3 )
% 8.83/5.04 => ( P @ A4 @ C3 ) ) ) ) )
% 8.83/5.04 => ( ! [X5: real] :
% 8.83/5.04 ( ( ord_less_eq_real @ A @ X5 )
% 8.83/5.04 => ( ( ord_less_eq_real @ X5 @ B )
% 8.83/5.04 => ? [D5: real] :
% 8.83/5.04 ( ( ord_less_real @ zero_zero_real @ D5 )
% 8.83/5.04 & ! [A4: real,B4: real] :
% 8.83/5.04 ( ( ( ord_less_eq_real @ A4 @ X5 )
% 8.83/5.04 & ( ord_less_eq_real @ X5 @ B4 )
% 8.83/5.04 & ( ord_less_real @ ( minus_minus_real @ B4 @ A4 ) @ D5 ) )
% 8.83/5.04 => ( P @ A4 @ B4 ) ) ) ) )
% 8.83/5.04 => ( P @ A @ B ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2921_divides__aux__eq,axiom,
% 8.83/5.04 ! [Q3: nat,R2: nat] :
% 8.83/5.04 ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
% 8.83/5.04 = ( R2 = zero_zero_nat ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2922_divides__aux__eq,axiom,
% 8.83/5.04 ! [Q3: int,R2: int] :
% 8.83/5.04 ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 8.83/5.04 = ( R2 = zero_zero_int ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2923_neg__eucl__rel__int__mult__2,axiom,
% 8.83/5.04 ! [B: int,A: int,Q3: int,R2: int] :
% 8.83/5.04 ( ( ord_less_eq_int @ B @ zero_zero_int )
% 8.83/5.04 => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 8.83/5.04 => ( 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 @ Q3 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2924_low__def,axiom,
% 8.83/5.04 ( vEBT_VEBT_low
% 8.83/5.04 = ( ^ [X: nat,N3: nat] : ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2925_subsetI,axiom,
% 8.83/5.04 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 8.83/5.04 ( ! [X5: product_prod_nat_nat] :
% 8.83/5.04 ( ( member8440522571783428010at_nat @ X5 @ A2 )
% 8.83/5.04 => ( member8440522571783428010at_nat @ X5 @ B3 ) )
% 8.83/5.04 => ( ord_le3146513528884898305at_nat @ A2 @ B3 ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2926_subsetI,axiom,
% 8.83/5.04 ! [A2: set_complex,B3: set_complex] :
% 8.83/5.04 ( ! [X5: complex] :
% 8.83/5.04 ( ( member_complex @ X5 @ A2 )
% 8.83/5.04 => ( member_complex @ X5 @ B3 ) )
% 8.83/5.04 => ( ord_le211207098394363844omplex @ A2 @ B3 ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2927_subsetI,axiom,
% 8.83/5.04 ! [A2: set_real,B3: set_real] :
% 8.83/5.04 ( ! [X5: real] :
% 8.83/5.04 ( ( member_real @ X5 @ A2 )
% 8.83/5.04 => ( member_real @ X5 @ B3 ) )
% 8.83/5.04 => ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2928_subsetI,axiom,
% 8.83/5.04 ! [A2: set_nat,B3: set_nat] :
% 8.83/5.04 ( ! [X5: nat] :
% 8.83/5.04 ( ( member_nat @ X5 @ A2 )
% 8.83/5.04 => ( member_nat @ X5 @ B3 ) )
% 8.83/5.04 => ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2929_subsetI,axiom,
% 8.83/5.04 ! [A2: set_int,B3: set_int] :
% 8.83/5.04 ( ! [X5: int] :
% 8.83/5.04 ( ( member_int @ X5 @ A2 )
% 8.83/5.04 => ( member_int @ X5 @ B3 ) )
% 8.83/5.04 => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2930_subset__antisym,axiom,
% 8.83/5.04 ! [A2: set_int,B3: set_int] :
% 8.83/5.04 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 8.83/5.04 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 8.83/5.04 => ( A2 = B3 ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2931_mod__mod__trivial,axiom,
% 8.83/5.04 ! [A: nat,B: nat] :
% 8.83/5.04 ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 8.83/5.04 = ( modulo_modulo_nat @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2932_mod__mod__trivial,axiom,
% 8.83/5.04 ! [A: int,B: int] :
% 8.83/5.04 ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 8.83/5.04 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2933_mod__mod__trivial,axiom,
% 8.83/5.04 ! [A: code_integer,B: code_integer] :
% 8.83/5.04 ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 8.83/5.04 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2934_Diff__idemp,axiom,
% 8.83/5.04 ! [A2: set_nat,B3: set_nat] :
% 8.83/5.04 ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ B3 )
% 8.83/5.04 = ( minus_minus_set_nat @ A2 @ B3 ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2935_Diff__iff,axiom,
% 8.83/5.04 ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 8.83/5.04 ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B3 ) )
% 8.83/5.04 = ( ( member8440522571783428010at_nat @ C @ A2 )
% 8.83/5.04 & ~ ( member8440522571783428010at_nat @ C @ B3 ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2936_Diff__iff,axiom,
% 8.83/5.04 ! [C: complex,A2: set_complex,B3: set_complex] :
% 8.83/5.04 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 8.83/5.04 = ( ( member_complex @ C @ A2 )
% 8.83/5.04 & ~ ( member_complex @ C @ B3 ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2937_Diff__iff,axiom,
% 8.83/5.04 ! [C: real,A2: set_real,B3: set_real] :
% 8.83/5.04 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 8.83/5.04 = ( ( member_real @ C @ A2 )
% 8.83/5.04 & ~ ( member_real @ C @ B3 ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2938_Diff__iff,axiom,
% 8.83/5.04 ! [C: int,A2: set_int,B3: set_int] :
% 8.83/5.04 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
% 8.83/5.04 = ( ( member_int @ C @ A2 )
% 8.83/5.04 & ~ ( member_int @ C @ B3 ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2939_Diff__iff,axiom,
% 8.83/5.04 ! [C: nat,A2: set_nat,B3: set_nat] :
% 8.83/5.04 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 8.83/5.04 = ( ( member_nat @ C @ A2 )
% 8.83/5.04 & ~ ( member_nat @ C @ B3 ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2940_DiffI,axiom,
% 8.83/5.04 ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 8.83/5.04 ( ( member8440522571783428010at_nat @ C @ A2 )
% 8.83/5.04 => ( ~ ( member8440522571783428010at_nat @ C @ B3 )
% 8.83/5.04 => ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B3 ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2941_DiffI,axiom,
% 8.83/5.04 ! [C: complex,A2: set_complex,B3: set_complex] :
% 8.83/5.04 ( ( member_complex @ C @ A2 )
% 8.83/5.04 => ( ~ ( member_complex @ C @ B3 )
% 8.83/5.04 => ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2942_DiffI,axiom,
% 8.83/5.04 ! [C: real,A2: set_real,B3: set_real] :
% 8.83/5.04 ( ( member_real @ C @ A2 )
% 8.83/5.04 => ( ~ ( member_real @ C @ B3 )
% 8.83/5.04 => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2943_DiffI,axiom,
% 8.83/5.04 ! [C: int,A2: set_int,B3: set_int] :
% 8.83/5.04 ( ( member_int @ C @ A2 )
% 8.83/5.04 => ( ~ ( member_int @ C @ B3 )
% 8.83/5.04 => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2944_DiffI,axiom,
% 8.83/5.04 ! [C: nat,A2: set_nat,B3: set_nat] :
% 8.83/5.04 ( ( member_nat @ C @ A2 )
% 8.83/5.04 => ( ~ ( member_nat @ C @ B3 )
% 8.83/5.04 => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2945_mod__0,axiom,
% 8.83/5.04 ! [A: nat] :
% 8.83/5.04 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 8.83/5.04 = zero_zero_nat ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2946_mod__0,axiom,
% 8.83/5.04 ! [A: int] :
% 8.83/5.04 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 8.83/5.04 = zero_zero_int ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2947_mod__0,axiom,
% 8.83/5.04 ! [A: code_integer] :
% 8.83/5.04 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 8.83/5.04 = zero_z3403309356797280102nteger ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2948_mod__by__0,axiom,
% 8.83/5.04 ! [A: nat] :
% 8.83/5.04 ( ( modulo_modulo_nat @ A @ zero_zero_nat )
% 8.83/5.04 = A ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2949_mod__by__0,axiom,
% 8.83/5.04 ! [A: int] :
% 8.83/5.04 ( ( modulo_modulo_int @ A @ zero_zero_int )
% 8.83/5.04 = A ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2950_mod__by__0,axiom,
% 8.83/5.04 ! [A: code_integer] :
% 8.83/5.04 ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
% 8.83/5.04 = A ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2951_mod__self,axiom,
% 8.83/5.04 ! [A: nat] :
% 8.83/5.04 ( ( modulo_modulo_nat @ A @ A )
% 8.83/5.04 = zero_zero_nat ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2952_mod__self,axiom,
% 8.83/5.04 ! [A: int] :
% 8.83/5.04 ( ( modulo_modulo_int @ A @ A )
% 8.83/5.04 = zero_zero_int ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2953_mod__self,axiom,
% 8.83/5.04 ! [A: code_integer] :
% 8.83/5.04 ( ( modulo364778990260209775nteger @ A @ A )
% 8.83/5.04 = zero_z3403309356797280102nteger ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2954_bits__mod__0,axiom,
% 8.83/5.04 ! [A: nat] :
% 8.83/5.04 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 8.83/5.04 = zero_zero_nat ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2955_bits__mod__0,axiom,
% 8.83/5.04 ! [A: int] :
% 8.83/5.04 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 8.83/5.04 = zero_zero_int ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2956_bits__mod__0,axiom,
% 8.83/5.04 ! [A: code_integer] :
% 8.83/5.04 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 8.83/5.04 = zero_z3403309356797280102nteger ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2957_mod__add__self1,axiom,
% 8.83/5.04 ! [B: nat,A: nat] :
% 8.83/5.04 ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 8.83/5.04 = ( modulo_modulo_nat @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2958_mod__add__self1,axiom,
% 8.83/5.04 ! [B: int,A: int] :
% 8.83/5.04 ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
% 8.83/5.04 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2959_mod__add__self1,axiom,
% 8.83/5.04 ! [B: code_integer,A: code_integer] :
% 8.83/5.04 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
% 8.83/5.04 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2960_mod__add__self2,axiom,
% 8.83/5.04 ! [A: nat,B: nat] :
% 8.83/5.04 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 8.83/5.04 = ( modulo_modulo_nat @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2961_mod__add__self2,axiom,
% 8.83/5.04 ! [A: int,B: int] :
% 8.83/5.04 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
% 8.83/5.04 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2962_mod__add__self2,axiom,
% 8.83/5.04 ! [A: code_integer,B: code_integer] :
% 8.83/5.04 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
% 8.83/5.04 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2963_minus__mod__self2,axiom,
% 8.83/5.04 ! [A: int,B: int] :
% 8.83/5.04 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
% 8.83/5.04 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2964_minus__mod__self2,axiom,
% 8.83/5.04 ! [A: code_integer,B: code_integer] :
% 8.83/5.04 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
% 8.83/5.04 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2965_Diff__empty,axiom,
% 8.83/5.04 ! [A2: set_int] :
% 8.83/5.04 ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
% 8.83/5.04 = A2 ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2966_Diff__empty,axiom,
% 8.83/5.04 ! [A2: set_real] :
% 8.83/5.04 ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
% 8.83/5.04 = A2 ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2967_Diff__empty,axiom,
% 8.83/5.04 ! [A2: set_nat] :
% 8.83/5.04 ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
% 8.83/5.04 = A2 ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2968_empty__Diff,axiom,
% 8.83/5.04 ! [A2: set_int] :
% 8.83/5.04 ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
% 8.83/5.04 = bot_bot_set_int ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2969_empty__Diff,axiom,
% 8.83/5.04 ! [A2: set_real] :
% 8.83/5.04 ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
% 8.83/5.04 = bot_bot_set_real ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2970_empty__Diff,axiom,
% 8.83/5.04 ! [A2: set_nat] :
% 8.83/5.04 ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
% 8.83/5.04 = bot_bot_set_nat ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2971_Diff__cancel,axiom,
% 8.83/5.04 ! [A2: set_int] :
% 8.83/5.04 ( ( minus_minus_set_int @ A2 @ A2 )
% 8.83/5.04 = bot_bot_set_int ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2972_Diff__cancel,axiom,
% 8.83/5.04 ! [A2: set_real] :
% 8.83/5.04 ( ( minus_minus_set_real @ A2 @ A2 )
% 8.83/5.04 = bot_bot_set_real ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2973_Diff__cancel,axiom,
% 8.83/5.04 ! [A2: set_nat] :
% 8.83/5.04 ( ( minus_minus_set_nat @ A2 @ A2 )
% 8.83/5.04 = bot_bot_set_nat ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2974_psubsetI,axiom,
% 8.83/5.04 ! [A2: set_int,B3: set_int] :
% 8.83/5.04 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 8.83/5.04 => ( ( A2 != B3 )
% 8.83/5.04 => ( ord_less_set_int @ A2 @ B3 ) ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2975_mod__less,axiom,
% 8.83/5.04 ! [M: nat,N: nat] :
% 8.83/5.04 ( ( ord_less_nat @ M @ N )
% 8.83/5.04 => ( ( modulo_modulo_nat @ M @ N )
% 8.83/5.04 = M ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2976_mod__mult__self1__is__0,axiom,
% 8.83/5.04 ! [B: nat,A: nat] :
% 8.83/5.04 ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
% 8.83/5.04 = zero_zero_nat ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2977_mod__mult__self1__is__0,axiom,
% 8.83/5.04 ! [B: int,A: int] :
% 8.83/5.04 ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
% 8.83/5.04 = zero_zero_int ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2978_mod__mult__self1__is__0,axiom,
% 8.83/5.04 ! [B: code_integer,A: code_integer] :
% 8.83/5.04 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
% 8.83/5.04 = zero_z3403309356797280102nteger ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2979_mod__mult__self2__is__0,axiom,
% 8.83/5.04 ! [A: nat,B: nat] :
% 8.83/5.04 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
% 8.83/5.04 = zero_zero_nat ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2980_mod__mult__self2__is__0,axiom,
% 8.83/5.04 ! [A: int,B: int] :
% 8.83/5.04 ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
% 8.83/5.04 = zero_zero_int ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2981_mod__mult__self2__is__0,axiom,
% 8.83/5.04 ! [A: code_integer,B: code_integer] :
% 8.83/5.04 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 8.83/5.04 = zero_z3403309356797280102nteger ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2982_bits__mod__by__1,axiom,
% 8.83/5.04 ! [A: nat] :
% 8.83/5.04 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 8.83/5.04 = zero_zero_nat ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2983_bits__mod__by__1,axiom,
% 8.83/5.04 ! [A: int] :
% 8.83/5.04 ( ( modulo_modulo_int @ A @ one_one_int )
% 8.83/5.04 = zero_zero_int ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2984_bits__mod__by__1,axiom,
% 8.83/5.04 ! [A: code_integer] :
% 8.83/5.04 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 8.83/5.04 = zero_z3403309356797280102nteger ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2985_mod__by__1,axiom,
% 8.83/5.04 ! [A: nat] :
% 8.83/5.04 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 8.83/5.04 = zero_zero_nat ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2986_mod__by__1,axiom,
% 8.83/5.04 ! [A: int] :
% 8.83/5.04 ( ( modulo_modulo_int @ A @ one_one_int )
% 8.83/5.04 = zero_zero_int ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2987_mod__by__1,axiom,
% 8.83/5.04 ! [A: code_integer] :
% 8.83/5.04 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 8.83/5.04 = zero_z3403309356797280102nteger ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2988_bits__mod__div__trivial,axiom,
% 8.83/5.04 ! [A: nat,B: nat] :
% 8.83/5.04 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 8.83/5.04 = zero_zero_nat ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2989_bits__mod__div__trivial,axiom,
% 8.83/5.04 ! [A: int,B: int] :
% 8.83/5.04 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 8.83/5.04 = zero_zero_int ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2990_bits__mod__div__trivial,axiom,
% 8.83/5.04 ! [A: code_integer,B: code_integer] :
% 8.83/5.04 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 8.83/5.04 = zero_z3403309356797280102nteger ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2991_mod__div__trivial,axiom,
% 8.83/5.04 ! [A: nat,B: nat] :
% 8.83/5.04 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 8.83/5.04 = zero_zero_nat ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2992_mod__div__trivial,axiom,
% 8.83/5.04 ! [A: int,B: int] :
% 8.83/5.04 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 8.83/5.04 = zero_zero_int ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2993_mod__div__trivial,axiom,
% 8.83/5.04 ! [A: code_integer,B: code_integer] :
% 8.83/5.04 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 8.83/5.04 = zero_z3403309356797280102nteger ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2994_mod__mult__self1,axiom,
% 8.83/5.04 ! [A: nat,C: nat,B: nat] :
% 8.83/5.04 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 8.83/5.04 = ( modulo_modulo_nat @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2995_mod__mult__self1,axiom,
% 8.83/5.04 ! [A: int,C: int,B: int] :
% 8.83/5.04 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 8.83/5.04 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2996_mod__mult__self1,axiom,
% 8.83/5.04 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.83/5.04 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
% 8.83/5.04 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2997_mod__mult__self2,axiom,
% 8.83/5.04 ! [A: nat,B: nat,C: nat] :
% 8.83/5.04 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 8.83/5.04 = ( modulo_modulo_nat @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2998_mod__mult__self2,axiom,
% 8.83/5.04 ! [A: int,B: int,C: int] :
% 8.83/5.04 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 8.83/5.04 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_2999_mod__mult__self2,axiom,
% 8.83/5.04 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.83/5.04 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
% 8.83/5.04 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_3000_mod__mult__self3,axiom,
% 8.83/5.04 ! [C: nat,B: nat,A: nat] :
% 8.83/5.04 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 8.83/5.04 = ( modulo_modulo_nat @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_3001_mod__mult__self3,axiom,
% 8.83/5.04 ! [C: int,B: int,A: int] :
% 8.83/5.04 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 8.83/5.04 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_3002_mod__mult__self3,axiom,
% 8.83/5.04 ! [C: code_integer,B: code_integer,A: code_integer] :
% 8.83/5.04 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
% 8.83/5.04 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_3003_mod__mult__self4,axiom,
% 8.83/5.04 ! [B: nat,C: nat,A: nat] :
% 8.83/5.04 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 8.83/5.04 = ( modulo_modulo_nat @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_3004_mod__mult__self4,axiom,
% 8.83/5.04 ! [B: int,C: int,A: int] :
% 8.83/5.04 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 8.83/5.04 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_3005_mod__mult__self4,axiom,
% 8.83/5.04 ! [B: code_integer,C: code_integer,A: code_integer] :
% 8.83/5.04 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
% 8.83/5.04 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_3006_mod__by__Suc__0,axiom,
% 8.83/5.04 ! [M: nat] :
% 8.83/5.04 ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 8.83/5.04 = zero_zero_nat ) ).
% 8.83/5.04
% 8.83/5.04 thf(fact_3007_Suc__mod__mult__self1,axiom,
% 8.90/5.04 ! [M: nat,K: nat,N: nat] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
% 8.90/5.04 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3008_Suc__mod__mult__self2,axiom,
% 8.90/5.04 ! [M: nat,N: nat,K: nat] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
% 8.90/5.04 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3009_Suc__mod__mult__self3,axiom,
% 8.90/5.04 ! [K: nat,N: nat,M: nat] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
% 8.90/5.04 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3010_Suc__mod__mult__self4,axiom,
% 8.90/5.04 ! [N: nat,K: nat,M: nat] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
% 8.90/5.04 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3011_bits__one__mod__two__eq__one,axiom,
% 8.90/5.04 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.04 = one_one_nat ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3012_bits__one__mod__two__eq__one,axiom,
% 8.90/5.04 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.04 = one_one_int ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3013_bits__one__mod__two__eq__one,axiom,
% 8.90/5.04 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.04 = one_one_Code_integer ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3014_one__mod__two__eq__one,axiom,
% 8.90/5.04 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.04 = one_one_nat ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3015_one__mod__two__eq__one,axiom,
% 8.90/5.04 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.04 = one_one_int ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3016_one__mod__two__eq__one,axiom,
% 8.90/5.04 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.04 = one_one_Code_integer ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3017_mod2__Suc__Suc,axiom,
% 8.90/5.04 ! [M: nat] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.04 = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3018_Suc__times__numeral__mod__eq,axiom,
% 8.90/5.04 ! [K: num,N: nat] :
% 8.90/5.04 ( ( ( numeral_numeral_nat @ K )
% 8.90/5.04 != one_one_nat )
% 8.90/5.04 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
% 8.90/5.04 = one_one_nat ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3019_not__mod__2__eq__1__eq__0,axiom,
% 8.90/5.04 ! [A: nat] :
% 8.90/5.04 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.04 != one_one_nat )
% 8.90/5.04 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.04 = zero_zero_nat ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3020_not__mod__2__eq__1__eq__0,axiom,
% 8.90/5.04 ! [A: int] :
% 8.90/5.04 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.04 != one_one_int )
% 8.90/5.04 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.04 = zero_zero_int ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3021_not__mod__2__eq__1__eq__0,axiom,
% 8.90/5.04 ! [A: code_integer] :
% 8.90/5.04 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.04 != one_one_Code_integer )
% 8.90/5.04 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.04 = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3022_not__mod__2__eq__0__eq__1,axiom,
% 8.90/5.04 ! [A: nat] :
% 8.90/5.04 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.04 != zero_zero_nat )
% 8.90/5.04 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.04 = one_one_nat ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3023_not__mod__2__eq__0__eq__1,axiom,
% 8.90/5.04 ! [A: int] :
% 8.90/5.04 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.04 != zero_zero_int )
% 8.90/5.04 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.04 = one_one_int ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3024_not__mod__2__eq__0__eq__1,axiom,
% 8.90/5.04 ! [A: code_integer] :
% 8.90/5.04 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.04 != zero_z3403309356797280102nteger )
% 8.90/5.04 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.04 = one_one_Code_integer ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3025_not__mod2__eq__Suc__0__eq__0,axiom,
% 8.90/5.04 ! [N: nat] :
% 8.90/5.04 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.04 != ( suc @ zero_zero_nat ) )
% 8.90/5.04 = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.04 = zero_zero_nat ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3026_add__self__mod__2,axiom,
% 8.90/5.04 ! [M: nat] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.04 = zero_zero_nat ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3027_mod2__gr__0,axiom,
% 8.90/5.04 ! [M: nat] :
% 8.90/5.04 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.04 = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.04 = one_one_nat ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3028_minus__set__def,axiom,
% 8.90/5.04 ( minus_1356011639430497352at_nat
% 8.90/5.04 = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 8.90/5.04 ( collec3392354462482085612at_nat
% 8.90/5.04 @ ( minus_2270307095948843157_nat_o
% 8.90/5.04 @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A6 )
% 8.90/5.04 @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3029_minus__set__def,axiom,
% 8.90/5.04 ( minus_811609699411566653omplex
% 8.90/5.04 = ( ^ [A6: set_complex,B6: set_complex] :
% 8.90/5.04 ( collect_complex
% 8.90/5.04 @ ( minus_8727706125548526216plex_o
% 8.90/5.04 @ ^ [X: complex] : ( member_complex @ X @ A6 )
% 8.90/5.04 @ ^ [X: complex] : ( member_complex @ X @ B6 ) ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3030_minus__set__def,axiom,
% 8.90/5.04 ( minus_minus_set_real
% 8.90/5.04 = ( ^ [A6: set_real,B6: set_real] :
% 8.90/5.04 ( collect_real
% 8.90/5.04 @ ( minus_minus_real_o
% 8.90/5.04 @ ^ [X: real] : ( member_real @ X @ A6 )
% 8.90/5.04 @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3031_minus__set__def,axiom,
% 8.90/5.04 ( minus_7954133019191499631st_nat
% 8.90/5.04 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 8.90/5.04 ( collect_list_nat
% 8.90/5.04 @ ( minus_1139252259498527702_nat_o
% 8.90/5.04 @ ^ [X: list_nat] : ( member_list_nat @ X @ A6 )
% 8.90/5.04 @ ^ [X: list_nat] : ( member_list_nat @ X @ B6 ) ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3032_minus__set__def,axiom,
% 8.90/5.04 ( minus_minus_set_int
% 8.90/5.04 = ( ^ [A6: set_int,B6: set_int] :
% 8.90/5.04 ( collect_int
% 8.90/5.04 @ ( minus_minus_int_o
% 8.90/5.04 @ ^ [X: int] : ( member_int @ X @ A6 )
% 8.90/5.04 @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3033_minus__set__def,axiom,
% 8.90/5.04 ( minus_minus_set_nat
% 8.90/5.04 = ( ^ [A6: set_nat,B6: set_nat] :
% 8.90/5.04 ( collect_nat
% 8.90/5.04 @ ( minus_minus_nat_o
% 8.90/5.04 @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 8.90/5.04 @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3034_set__diff__eq,axiom,
% 8.90/5.04 ( minus_1356011639430497352at_nat
% 8.90/5.04 = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 8.90/5.04 ( collec3392354462482085612at_nat
% 8.90/5.04 @ ^ [X: product_prod_nat_nat] :
% 8.90/5.04 ( ( member8440522571783428010at_nat @ X @ A6 )
% 8.90/5.04 & ~ ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3035_set__diff__eq,axiom,
% 8.90/5.04 ( minus_811609699411566653omplex
% 8.90/5.04 = ( ^ [A6: set_complex,B6: set_complex] :
% 8.90/5.04 ( collect_complex
% 8.90/5.04 @ ^ [X: complex] :
% 8.90/5.04 ( ( member_complex @ X @ A6 )
% 8.90/5.04 & ~ ( member_complex @ X @ B6 ) ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3036_set__diff__eq,axiom,
% 8.90/5.04 ( minus_minus_set_real
% 8.90/5.04 = ( ^ [A6: set_real,B6: set_real] :
% 8.90/5.04 ( collect_real
% 8.90/5.04 @ ^ [X: real] :
% 8.90/5.04 ( ( member_real @ X @ A6 )
% 8.90/5.04 & ~ ( member_real @ X @ B6 ) ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3037_set__diff__eq,axiom,
% 8.90/5.04 ( minus_7954133019191499631st_nat
% 8.90/5.04 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 8.90/5.04 ( collect_list_nat
% 8.90/5.04 @ ^ [X: list_nat] :
% 8.90/5.04 ( ( member_list_nat @ X @ A6 )
% 8.90/5.04 & ~ ( member_list_nat @ X @ B6 ) ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3038_set__diff__eq,axiom,
% 8.90/5.04 ( minus_minus_set_int
% 8.90/5.04 = ( ^ [A6: set_int,B6: set_int] :
% 8.90/5.04 ( collect_int
% 8.90/5.04 @ ^ [X: int] :
% 8.90/5.04 ( ( member_int @ X @ A6 )
% 8.90/5.04 & ~ ( member_int @ X @ B6 ) ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3039_set__diff__eq,axiom,
% 8.90/5.04 ( minus_minus_set_nat
% 8.90/5.04 = ( ^ [A6: set_nat,B6: set_nat] :
% 8.90/5.04 ( collect_nat
% 8.90/5.04 @ ^ [X: nat] :
% 8.90/5.04 ( ( member_nat @ X @ A6 )
% 8.90/5.04 & ~ ( member_nat @ X @ B6 ) ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3040_DiffD2,axiom,
% 8.90/5.04 ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 8.90/5.04 ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B3 ) )
% 8.90/5.04 => ~ ( member8440522571783428010at_nat @ C @ B3 ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3041_DiffD2,axiom,
% 8.90/5.04 ! [C: complex,A2: set_complex,B3: set_complex] :
% 8.90/5.04 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 8.90/5.04 => ~ ( member_complex @ C @ B3 ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3042_DiffD2,axiom,
% 8.90/5.04 ! [C: real,A2: set_real,B3: set_real] :
% 8.90/5.04 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 8.90/5.04 => ~ ( member_real @ C @ B3 ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3043_DiffD2,axiom,
% 8.90/5.04 ! [C: int,A2: set_int,B3: set_int] :
% 8.90/5.04 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
% 8.90/5.04 => ~ ( member_int @ C @ B3 ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3044_DiffD2,axiom,
% 8.90/5.04 ! [C: nat,A2: set_nat,B3: set_nat] :
% 8.90/5.04 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 8.90/5.04 => ~ ( member_nat @ C @ B3 ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3045_DiffD1,axiom,
% 8.90/5.04 ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 8.90/5.04 ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B3 ) )
% 8.90/5.04 => ( member8440522571783428010at_nat @ C @ A2 ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3046_DiffD1,axiom,
% 8.90/5.04 ! [C: complex,A2: set_complex,B3: set_complex] :
% 8.90/5.04 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 8.90/5.04 => ( member_complex @ C @ A2 ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3047_DiffD1,axiom,
% 8.90/5.04 ! [C: real,A2: set_real,B3: set_real] :
% 8.90/5.04 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 8.90/5.04 => ( member_real @ C @ A2 ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3048_DiffD1,axiom,
% 8.90/5.04 ! [C: int,A2: set_int,B3: set_int] :
% 8.90/5.04 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
% 8.90/5.04 => ( member_int @ C @ A2 ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3049_DiffD1,axiom,
% 8.90/5.04 ! [C: nat,A2: set_nat,B3: set_nat] :
% 8.90/5.04 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 8.90/5.04 => ( member_nat @ C @ A2 ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3050_DiffE,axiom,
% 8.90/5.04 ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 8.90/5.04 ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B3 ) )
% 8.90/5.04 => ~ ( ( member8440522571783428010at_nat @ C @ A2 )
% 8.90/5.04 => ( member8440522571783428010at_nat @ C @ B3 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3051_DiffE,axiom,
% 8.90/5.04 ! [C: complex,A2: set_complex,B3: set_complex] :
% 8.90/5.04 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 8.90/5.04 => ~ ( ( member_complex @ C @ A2 )
% 8.90/5.04 => ( member_complex @ C @ B3 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3052_DiffE,axiom,
% 8.90/5.04 ! [C: real,A2: set_real,B3: set_real] :
% 8.90/5.04 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 8.90/5.04 => ~ ( ( member_real @ C @ A2 )
% 8.90/5.04 => ( member_real @ C @ B3 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3053_DiffE,axiom,
% 8.90/5.04 ! [C: int,A2: set_int,B3: set_int] :
% 8.90/5.04 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
% 8.90/5.04 => ~ ( ( member_int @ C @ A2 )
% 8.90/5.04 => ( member_int @ C @ B3 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3054_DiffE,axiom,
% 8.90/5.04 ! [C: nat,A2: set_nat,B3: set_nat] :
% 8.90/5.04 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 8.90/5.04 => ~ ( ( member_nat @ C @ A2 )
% 8.90/5.04 => ( member_nat @ C @ B3 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3055_psubset__imp__ex__mem,axiom,
% 8.90/5.04 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 8.90/5.04 ( ( ord_le7866589430770878221at_nat @ A2 @ B3 )
% 8.90/5.04 => ? [B4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B4 @ ( minus_1356011639430497352at_nat @ B3 @ A2 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3056_psubset__imp__ex__mem,axiom,
% 8.90/5.04 ! [A2: set_complex,B3: set_complex] :
% 8.90/5.04 ( ( ord_less_set_complex @ A2 @ B3 )
% 8.90/5.04 => ? [B4: complex] : ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B3 @ A2 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3057_psubset__imp__ex__mem,axiom,
% 8.90/5.04 ! [A2: set_real,B3: set_real] :
% 8.90/5.04 ( ( ord_less_set_real @ A2 @ B3 )
% 8.90/5.04 => ? [B4: real] : ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A2 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3058_psubset__imp__ex__mem,axiom,
% 8.90/5.04 ! [A2: set_int,B3: set_int] :
% 8.90/5.04 ( ( ord_less_set_int @ A2 @ B3 )
% 8.90/5.04 => ? [B4: int] : ( member_int @ B4 @ ( minus_minus_set_int @ B3 @ A2 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3059_psubset__imp__ex__mem,axiom,
% 8.90/5.04 ! [A2: set_nat,B3: set_nat] :
% 8.90/5.04 ( ( ord_less_set_nat @ A2 @ B3 )
% 8.90/5.04 => ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B3 @ A2 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3060_mod__mult__eq,axiom,
% 8.90/5.04 ! [A: nat,C: nat,B: nat] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 8.90/5.04 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3061_mod__mult__eq,axiom,
% 8.90/5.04 ! [A: int,C: int,B: int] :
% 8.90/5.04 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 8.90/5.04 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3062_mod__mult__eq,axiom,
% 8.90/5.04 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.90/5.04 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 8.90/5.04 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3063_mod__mult__cong,axiom,
% 8.90/5.04 ! [A: nat,C: nat,A5: nat,B: nat,B5: nat] :
% 8.90/5.04 ( ( ( modulo_modulo_nat @ A @ C )
% 8.90/5.04 = ( modulo_modulo_nat @ A5 @ C ) )
% 8.90/5.04 => ( ( ( modulo_modulo_nat @ B @ C )
% 8.90/5.04 = ( modulo_modulo_nat @ B5 @ C ) )
% 8.90/5.04 => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
% 8.90/5.04 = ( modulo_modulo_nat @ ( times_times_nat @ A5 @ B5 ) @ C ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3064_mod__mult__cong,axiom,
% 8.90/5.04 ! [A: int,C: int,A5: int,B: int,B5: int] :
% 8.90/5.04 ( ( ( modulo_modulo_int @ A @ C )
% 8.90/5.04 = ( modulo_modulo_int @ A5 @ C ) )
% 8.90/5.04 => ( ( ( modulo_modulo_int @ B @ C )
% 8.90/5.04 = ( modulo_modulo_int @ B5 @ C ) )
% 8.90/5.04 => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
% 8.90/5.04 = ( modulo_modulo_int @ ( times_times_int @ A5 @ B5 ) @ C ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3065_mod__mult__cong,axiom,
% 8.90/5.04 ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B5: code_integer] :
% 8.90/5.04 ( ( ( modulo364778990260209775nteger @ A @ C )
% 8.90/5.04 = ( modulo364778990260209775nteger @ A5 @ C ) )
% 8.90/5.04 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 8.90/5.04 = ( modulo364778990260209775nteger @ B5 @ C ) )
% 8.90/5.04 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 8.90/5.04 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A5 @ B5 ) @ C ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3066_mod__mult__mult2,axiom,
% 8.90/5.04 ! [A: nat,C: nat,B: nat] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 8.90/5.04 = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3067_mod__mult__mult2,axiom,
% 8.90/5.04 ! [A: int,C: int,B: int] :
% 8.90/5.04 ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 8.90/5.04 = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3068_mod__mult__mult2,axiom,
% 8.90/5.04 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.90/5.04 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 8.90/5.04 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3069_mult__mod__right,axiom,
% 8.90/5.04 ! [C: nat,A: nat,B: nat] :
% 8.90/5.04 ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 8.90/5.04 = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3070_mult__mod__right,axiom,
% 8.90/5.04 ! [C: int,A: int,B: int] :
% 8.90/5.04 ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 8.90/5.04 = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3071_mult__mod__right,axiom,
% 8.90/5.04 ! [C: code_integer,A: code_integer,B: code_integer] :
% 8.90/5.04 ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 8.90/5.04 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3072_mod__mult__left__eq,axiom,
% 8.90/5.04 ! [A: nat,C: nat,B: nat] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 8.90/5.04 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3073_mod__mult__left__eq,axiom,
% 8.90/5.04 ! [A: int,C: int,B: int] :
% 8.90/5.04 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 8.90/5.04 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3074_mod__mult__left__eq,axiom,
% 8.90/5.04 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.90/5.04 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 8.90/5.04 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3075_mod__mult__right__eq,axiom,
% 8.90/5.04 ! [A: nat,B: nat,C: nat] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 8.90/5.04 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3076_mod__mult__right__eq,axiom,
% 8.90/5.04 ! [A: int,B: int,C: int] :
% 8.90/5.04 ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 8.90/5.04 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3077_mod__mult__right__eq,axiom,
% 8.90/5.04 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.04 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 8.90/5.04 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3078_mod__add__eq,axiom,
% 8.90/5.04 ! [A: nat,C: nat,B: nat] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 8.90/5.04 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3079_mod__add__eq,axiom,
% 8.90/5.04 ! [A: int,C: int,B: int] :
% 8.90/5.04 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 8.90/5.04 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3080_mod__add__eq,axiom,
% 8.90/5.04 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.90/5.04 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 8.90/5.04 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3081_mod__add__cong,axiom,
% 8.90/5.04 ! [A: nat,C: nat,A5: nat,B: nat,B5: nat] :
% 8.90/5.04 ( ( ( modulo_modulo_nat @ A @ C )
% 8.90/5.04 = ( modulo_modulo_nat @ A5 @ C ) )
% 8.90/5.04 => ( ( ( modulo_modulo_nat @ B @ C )
% 8.90/5.04 = ( modulo_modulo_nat @ B5 @ C ) )
% 8.90/5.04 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 8.90/5.04 = ( modulo_modulo_nat @ ( plus_plus_nat @ A5 @ B5 ) @ C ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3082_mod__add__cong,axiom,
% 8.90/5.04 ! [A: int,C: int,A5: int,B: int,B5: int] :
% 8.90/5.04 ( ( ( modulo_modulo_int @ A @ C )
% 8.90/5.04 = ( modulo_modulo_int @ A5 @ C ) )
% 8.90/5.04 => ( ( ( modulo_modulo_int @ B @ C )
% 8.90/5.04 = ( modulo_modulo_int @ B5 @ C ) )
% 8.90/5.04 => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.90/5.04 = ( modulo_modulo_int @ ( plus_plus_int @ A5 @ B5 ) @ C ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3083_mod__add__cong,axiom,
% 8.90/5.04 ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B5: code_integer] :
% 8.90/5.04 ( ( ( modulo364778990260209775nteger @ A @ C )
% 8.90/5.04 = ( modulo364778990260209775nteger @ A5 @ C ) )
% 8.90/5.04 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 8.90/5.04 = ( modulo364778990260209775nteger @ B5 @ C ) )
% 8.90/5.04 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 8.90/5.04 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A5 @ B5 ) @ C ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3084_mod__add__left__eq,axiom,
% 8.90/5.04 ! [A: nat,C: nat,B: nat] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 8.90/5.04 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3085_mod__add__left__eq,axiom,
% 8.90/5.04 ! [A: int,C: int,B: int] :
% 8.90/5.04 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 8.90/5.04 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3086_mod__add__left__eq,axiom,
% 8.90/5.04 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.90/5.04 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 8.90/5.04 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3087_mod__add__right__eq,axiom,
% 8.90/5.04 ! [A: nat,B: nat,C: nat] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 8.90/5.04 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3088_mod__add__right__eq,axiom,
% 8.90/5.04 ! [A: int,B: int,C: int] :
% 8.90/5.04 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 8.90/5.04 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3089_mod__add__right__eq,axiom,
% 8.90/5.04 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.04 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 8.90/5.04 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3090_mod__diff__eq,axiom,
% 8.90/5.04 ! [A: int,C: int,B: int] :
% 8.90/5.04 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 8.90/5.04 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3091_mod__diff__eq,axiom,
% 8.90/5.04 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.90/5.04 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 8.90/5.04 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3092_mod__diff__cong,axiom,
% 8.90/5.04 ! [A: int,C: int,A5: int,B: int,B5: int] :
% 8.90/5.04 ( ( ( modulo_modulo_int @ A @ C )
% 8.90/5.04 = ( modulo_modulo_int @ A5 @ C ) )
% 8.90/5.04 => ( ( ( modulo_modulo_int @ B @ C )
% 8.90/5.04 = ( modulo_modulo_int @ B5 @ C ) )
% 8.90/5.04 => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
% 8.90/5.04 = ( modulo_modulo_int @ ( minus_minus_int @ A5 @ B5 ) @ C ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3093_mod__diff__cong,axiom,
% 8.90/5.04 ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B5: code_integer] :
% 8.90/5.04 ( ( ( modulo364778990260209775nteger @ A @ C )
% 8.90/5.04 = ( modulo364778990260209775nteger @ A5 @ C ) )
% 8.90/5.04 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 8.90/5.04 = ( modulo364778990260209775nteger @ B5 @ C ) )
% 8.90/5.04 => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 8.90/5.04 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A5 @ B5 ) @ C ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3094_mod__diff__left__eq,axiom,
% 8.90/5.04 ! [A: int,C: int,B: int] :
% 8.90/5.04 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 8.90/5.04 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3095_mod__diff__left__eq,axiom,
% 8.90/5.04 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.90/5.04 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 8.90/5.04 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3096_mod__diff__right__eq,axiom,
% 8.90/5.04 ! [A: int,B: int,C: int] :
% 8.90/5.04 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 8.90/5.04 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3097_mod__diff__right__eq,axiom,
% 8.90/5.04 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.04 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 8.90/5.04 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3098_power__mod,axiom,
% 8.90/5.04 ! [A: nat,B: nat,N: nat] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N ) @ B )
% 8.90/5.04 = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3099_power__mod,axiom,
% 8.90/5.04 ! [A: int,B: int,N: nat] :
% 8.90/5.04 ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N ) @ B )
% 8.90/5.04 = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3100_power__mod,axiom,
% 8.90/5.04 ! [A: code_integer,B: code_integer,N: nat] :
% 8.90/5.04 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N ) @ B )
% 8.90/5.04 = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3101_mod__Suc__Suc__eq,axiom,
% 8.90/5.04 ! [M: nat,N: nat] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
% 8.90/5.04 = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3102_mod__Suc__eq,axiom,
% 8.90/5.04 ! [M: nat,N: nat] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
% 8.90/5.04 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3103_mod__less__eq__dividend,axiom,
% 8.90/5.04 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3104_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 8.90/5.04 ! [A: code_integer,B: code_integer] :
% 8.90/5.04 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 8.90/5.04 => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3105_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 8.90/5.04 ! [A: nat,B: nat] :
% 8.90/5.04 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.90/5.04 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3106_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 8.90/5.04 ! [A: int,B: int] :
% 8.90/5.04 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.90/5.04 => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3107_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 8.90/5.04 ! [B: nat,A: nat] :
% 8.90/5.04 ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.90/5.04 => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3108_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 8.90/5.04 ! [B: int,A: int] :
% 8.90/5.04 ( ( ord_less_int @ zero_zero_int @ B )
% 8.90/5.04 => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3109_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 8.90/5.04 ! [B: code_integer,A: code_integer] :
% 8.90/5.04 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 8.90/5.04 => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3110_mod__eq__self__iff__div__eq__0,axiom,
% 8.90/5.04 ! [A: nat,B: nat] :
% 8.90/5.04 ( ( ( modulo_modulo_nat @ A @ B )
% 8.90/5.04 = A )
% 8.90/5.04 = ( ( divide_divide_nat @ A @ B )
% 8.90/5.04 = zero_zero_nat ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3111_mod__eq__self__iff__div__eq__0,axiom,
% 8.90/5.04 ! [A: int,B: int] :
% 8.90/5.04 ( ( ( modulo_modulo_int @ A @ B )
% 8.90/5.04 = A )
% 8.90/5.04 = ( ( divide_divide_int @ A @ B )
% 8.90/5.04 = zero_zero_int ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3112_mod__eq__self__iff__div__eq__0,axiom,
% 8.90/5.04 ! [A: code_integer,B: code_integer] :
% 8.90/5.04 ( ( ( modulo364778990260209775nteger @ A @ B )
% 8.90/5.04 = A )
% 8.90/5.04 = ( ( divide6298287555418463151nteger @ A @ B )
% 8.90/5.04 = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3113_cong__exp__iff__simps_I9_J,axiom,
% 8.90/5.04 ! [M: num,Q3: num,N: num] :
% 8.90/5.04 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 8.90/5.04 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 8.90/5.04 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
% 8.90/5.04 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3114_cong__exp__iff__simps_I9_J,axiom,
% 8.90/5.04 ! [M: num,Q3: num,N: num] :
% 8.90/5.04 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 8.90/5.04 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 8.90/5.04 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
% 8.90/5.04 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3115_cong__exp__iff__simps_I9_J,axiom,
% 8.90/5.04 ! [M: num,Q3: num,N: num] :
% 8.90/5.04 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 8.90/5.04 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 8.90/5.04 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 8.90/5.04 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3116_cong__exp__iff__simps_I4_J,axiom,
% 8.90/5.04 ! [M: num,N: num] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
% 8.90/5.04 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3117_cong__exp__iff__simps_I4_J,axiom,
% 8.90/5.04 ! [M: num,N: num] :
% 8.90/5.04 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
% 8.90/5.04 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3118_cong__exp__iff__simps_I4_J,axiom,
% 8.90/5.04 ! [M: num,N: num] :
% 8.90/5.04 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
% 8.90/5.04 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3119_mod__eqE,axiom,
% 8.90/5.04 ! [A: int,C: int,B: int] :
% 8.90/5.04 ( ( ( modulo_modulo_int @ A @ C )
% 8.90/5.04 = ( modulo_modulo_int @ B @ C ) )
% 8.90/5.04 => ~ ! [D3: int] :
% 8.90/5.04 ( B
% 8.90/5.04 != ( plus_plus_int @ A @ ( times_times_int @ C @ D3 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3120_mod__eqE,axiom,
% 8.90/5.04 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.90/5.04 ( ( ( modulo364778990260209775nteger @ A @ C )
% 8.90/5.04 = ( modulo364778990260209775nteger @ B @ C ) )
% 8.90/5.04 => ~ ! [D3: code_integer] :
% 8.90/5.04 ( B
% 8.90/5.04 != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D3 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3121_div__add1__eq,axiom,
% 8.90/5.04 ! [A: nat,B: nat,C: nat] :
% 8.90/5.04 ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 8.90/5.04 = ( 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 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3122_div__add1__eq,axiom,
% 8.90/5.04 ! [A: int,B: int,C: int] :
% 8.90/5.04 ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.90/5.04 = ( 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 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3123_div__add1__eq,axiom,
% 8.90/5.04 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.04 ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 8.90/5.04 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3124_mod__Suc,axiom,
% 8.90/5.04 ! [M: nat,N: nat] :
% 8.90/5.04 ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 8.90/5.04 = N )
% 8.90/5.04 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 8.90/5.04 = zero_zero_nat ) )
% 8.90/5.04 & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 8.90/5.04 != N )
% 8.90/5.04 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 8.90/5.04 = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3125_mod__induct,axiom,
% 8.90/5.04 ! [P: nat > $o,N: nat,P6: nat,M: nat] :
% 8.90/5.04 ( ( P @ N )
% 8.90/5.04 => ( ( ord_less_nat @ N @ P6 )
% 8.90/5.04 => ( ( ord_less_nat @ M @ P6 )
% 8.90/5.04 => ( ! [N2: nat] :
% 8.90/5.04 ( ( ord_less_nat @ N2 @ P6 )
% 8.90/5.04 => ( ( P @ N2 )
% 8.90/5.04 => ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P6 ) ) ) )
% 8.90/5.04 => ( P @ M ) ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3126_mod__less__divisor,axiom,
% 8.90/5.04 ! [N: nat,M: nat] :
% 8.90/5.04 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.04 => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3127_mod__Suc__le__divisor,axiom,
% 8.90/5.04 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3128_mod__eq__0D,axiom,
% 8.90/5.04 ! [M: nat,D: nat] :
% 8.90/5.04 ( ( ( modulo_modulo_nat @ M @ D )
% 8.90/5.04 = zero_zero_nat )
% 8.90/5.04 => ? [Q2: nat] :
% 8.90/5.04 ( M
% 8.90/5.04 = ( times_times_nat @ D @ Q2 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3129_mod__geq,axiom,
% 8.90/5.04 ! [M: nat,N: nat] :
% 8.90/5.04 ( ~ ( ord_less_nat @ M @ N )
% 8.90/5.04 => ( ( modulo_modulo_nat @ M @ N )
% 8.90/5.04 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3130_mod__if,axiom,
% 8.90/5.04 ( modulo_modulo_nat
% 8.90/5.04 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N3 ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N3 ) @ N3 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3131_le__mod__geq,axiom,
% 8.90/5.04 ! [N: nat,M: nat] :
% 8.90/5.04 ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.04 => ( ( modulo_modulo_nat @ M @ N )
% 8.90/5.04 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3132_nat__mod__eq__iff,axiom,
% 8.90/5.04 ! [X2: nat,N: nat,Y3: nat] :
% 8.90/5.04 ( ( ( modulo_modulo_nat @ X2 @ N )
% 8.90/5.04 = ( modulo_modulo_nat @ Y3 @ N ) )
% 8.90/5.04 = ( ? [Q1: nat,Q22: nat] :
% 8.90/5.04 ( ( plus_plus_nat @ X2 @ ( times_times_nat @ N @ Q1 ) )
% 8.90/5.04 = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3133_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 8.90/5.04 ! [B: code_integer,A: code_integer] :
% 8.90/5.04 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 8.90/5.04 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3134_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 8.90/5.04 ! [B: nat,A: nat] :
% 8.90/5.04 ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.90/5.04 => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3135_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 8.90/5.04 ! [B: int,A: int] :
% 8.90/5.04 ( ( ord_less_int @ zero_zero_int @ B )
% 8.90/5.04 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3136_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 8.90/5.04 ! [A: code_integer,B: code_integer] :
% 8.90/5.04 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 8.90/5.04 => ( ( ord_le6747313008572928689nteger @ A @ B )
% 8.90/5.04 => ( ( modulo364778990260209775nteger @ A @ B )
% 8.90/5.04 = A ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3137_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 8.90/5.04 ! [A: nat,B: nat] :
% 8.90/5.04 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.90/5.04 => ( ( ord_less_nat @ A @ B )
% 8.90/5.04 => ( ( modulo_modulo_nat @ A @ B )
% 8.90/5.04 = A ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3138_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 8.90/5.04 ! [A: int,B: int] :
% 8.90/5.04 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.90/5.04 => ( ( ord_less_int @ A @ B )
% 8.90/5.04 => ( ( modulo_modulo_int @ A @ B )
% 8.90/5.04 = A ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3139_cong__exp__iff__simps_I2_J,axiom,
% 8.90/5.04 ! [N: num,Q3: num] :
% 8.90/5.04 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 8.90/5.04 = zero_zero_nat )
% 8.90/5.04 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
% 8.90/5.04 = zero_zero_nat ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3140_cong__exp__iff__simps_I2_J,axiom,
% 8.90/5.04 ! [N: num,Q3: num] :
% 8.90/5.04 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 8.90/5.04 = zero_zero_int )
% 8.90/5.04 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
% 8.90/5.04 = zero_zero_int ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3141_cong__exp__iff__simps_I2_J,axiom,
% 8.90/5.04 ! [N: num,Q3: num] :
% 8.90/5.04 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 8.90/5.04 = zero_z3403309356797280102nteger )
% 8.90/5.04 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 8.90/5.04 = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3142_cong__exp__iff__simps_I1_J,axiom,
% 8.90/5.04 ! [N: num] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
% 8.90/5.04 = zero_zero_nat ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3143_cong__exp__iff__simps_I1_J,axiom,
% 8.90/5.04 ! [N: num] :
% 8.90/5.04 ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
% 8.90/5.04 = zero_zero_int ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3144_cong__exp__iff__simps_I1_J,axiom,
% 8.90/5.04 ! [N: num] :
% 8.90/5.04 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
% 8.90/5.04 = zero_z3403309356797280102nteger ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3145_cong__exp__iff__simps_I6_J,axiom,
% 8.90/5.04 ! [Q3: num,N: num] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 8.90/5.04 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3146_cong__exp__iff__simps_I6_J,axiom,
% 8.90/5.04 ! [Q3: num,N: num] :
% 8.90/5.04 ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 8.90/5.04 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3147_cong__exp__iff__simps_I6_J,axiom,
% 8.90/5.04 ! [Q3: num,N: num] :
% 8.90/5.04 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 8.90/5.04 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3148_cong__exp__iff__simps_I8_J,axiom,
% 8.90/5.04 ! [M: num,Q3: num] :
% 8.90/5.04 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 8.90/5.04 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3149_cong__exp__iff__simps_I8_J,axiom,
% 8.90/5.04 ! [M: num,Q3: num] :
% 8.90/5.04 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 8.90/5.04 != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3150_cong__exp__iff__simps_I8_J,axiom,
% 8.90/5.04 ! [M: num,Q3: num] :
% 8.90/5.04 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 8.90/5.04 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3151_div__mult1__eq,axiom,
% 8.90/5.04 ! [A: nat,B: nat,C: nat] :
% 8.90/5.04 ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
% 8.90/5.04 = ( 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 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3152_div__mult1__eq,axiom,
% 8.90/5.04 ! [A: int,B: int,C: int] :
% 8.90/5.04 ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
% 8.90/5.04 = ( 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 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3153_div__mult1__eq,axiom,
% 8.90/5.04 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.04 ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 8.90/5.04 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3154_mult__div__mod__eq,axiom,
% 8.90/5.04 ! [B: nat,A: nat] :
% 8.90/5.04 ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
% 8.90/5.04 = A ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3155_mult__div__mod__eq,axiom,
% 8.90/5.04 ! [B: int,A: int] :
% 8.90/5.04 ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
% 8.90/5.04 = A ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3156_mult__div__mod__eq,axiom,
% 8.90/5.04 ! [B: code_integer,A: code_integer] :
% 8.90/5.04 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 8.90/5.04 = A ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3157_mod__mult__div__eq,axiom,
% 8.90/5.04 ! [A: nat,B: nat] :
% 8.90/5.04 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 8.90/5.04 = A ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3158_mod__mult__div__eq,axiom,
% 8.90/5.04 ! [A: int,B: int] :
% 8.90/5.04 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 8.90/5.04 = A ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3159_mod__mult__div__eq,axiom,
% 8.90/5.04 ! [A: code_integer,B: code_integer] :
% 8.90/5.04 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 8.90/5.04 = A ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3160_mod__div__mult__eq,axiom,
% 8.90/5.04 ! [A: nat,B: nat] :
% 8.90/5.04 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 8.90/5.04 = A ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3161_mod__div__mult__eq,axiom,
% 8.90/5.04 ! [A: int,B: int] :
% 8.90/5.04 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 8.90/5.04 = A ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3162_mod__div__mult__eq,axiom,
% 8.90/5.04 ! [A: code_integer,B: code_integer] :
% 8.90/5.04 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 8.90/5.04 = A ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3163_div__mult__mod__eq,axiom,
% 8.90/5.04 ! [A: nat,B: nat] :
% 8.90/5.04 ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
% 8.90/5.04 = A ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3164_div__mult__mod__eq,axiom,
% 8.90/5.04 ! [A: int,B: int] :
% 8.90/5.04 ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
% 8.90/5.04 = A ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3165_div__mult__mod__eq,axiom,
% 8.90/5.04 ! [A: code_integer,B: code_integer] :
% 8.90/5.04 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 8.90/5.04 = A ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3166_mod__div__decomp,axiom,
% 8.90/5.04 ! [A: nat,B: nat] :
% 8.90/5.04 ( A
% 8.90/5.04 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3167_mod__div__decomp,axiom,
% 8.90/5.04 ! [A: int,B: int] :
% 8.90/5.04 ( A
% 8.90/5.04 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3168_mod__div__decomp,axiom,
% 8.90/5.04 ! [A: code_integer,B: code_integer] :
% 8.90/5.04 ( A
% 8.90/5.04 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3169_cancel__div__mod__rules_I1_J,axiom,
% 8.90/5.04 ! [A: nat,B: nat,C: nat] :
% 8.90/5.04 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 8.90/5.04 = ( plus_plus_nat @ A @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3170_cancel__div__mod__rules_I1_J,axiom,
% 8.90/5.04 ! [A: int,B: int,C: int] :
% 8.90/5.04 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 8.90/5.04 = ( plus_plus_int @ A @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3171_cancel__div__mod__rules_I1_J,axiom,
% 8.90/5.04 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.04 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 8.90/5.04 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3172_cancel__div__mod__rules_I2_J,axiom,
% 8.90/5.04 ! [B: nat,A: nat,C: nat] :
% 8.90/5.04 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 8.90/5.04 = ( plus_plus_nat @ A @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3173_cancel__div__mod__rules_I2_J,axiom,
% 8.90/5.04 ! [B: int,A: int,C: int] :
% 8.90/5.04 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 8.90/5.04 = ( plus_plus_int @ A @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3174_cancel__div__mod__rules_I2_J,axiom,
% 8.90/5.04 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.90/5.04 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 8.90/5.04 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3175_minus__mult__div__eq__mod,axiom,
% 8.90/5.04 ! [A: nat,B: nat] :
% 8.90/5.04 ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 8.90/5.04 = ( modulo_modulo_nat @ A @ B ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3176_minus__mult__div__eq__mod,axiom,
% 8.90/5.04 ! [A: int,B: int] :
% 8.90/5.04 ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 8.90/5.04 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3177_minus__mult__div__eq__mod,axiom,
% 8.90/5.04 ! [A: code_integer,B: code_integer] :
% 8.90/5.04 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 8.90/5.04 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3178_minus__mod__eq__mult__div,axiom,
% 8.90/5.04 ! [A: nat,B: nat] :
% 8.90/5.04 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 8.90/5.04 = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3179_minus__mod__eq__mult__div,axiom,
% 8.90/5.04 ! [A: int,B: int] :
% 8.90/5.04 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 8.90/5.04 = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3180_minus__mod__eq__mult__div,axiom,
% 8.90/5.04 ! [A: code_integer,B: code_integer] :
% 8.90/5.04 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 8.90/5.04 = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3181_minus__mod__eq__div__mult,axiom,
% 8.90/5.04 ! [A: nat,B: nat] :
% 8.90/5.04 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 8.90/5.04 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3182_minus__mod__eq__div__mult,axiom,
% 8.90/5.04 ! [A: int,B: int] :
% 8.90/5.04 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 8.90/5.04 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3183_minus__mod__eq__div__mult,axiom,
% 8.90/5.04 ! [A: code_integer,B: code_integer] :
% 8.90/5.04 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 8.90/5.04 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3184_minus__div__mult__eq__mod,axiom,
% 8.90/5.04 ! [A: nat,B: nat] :
% 8.90/5.04 ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 8.90/5.04 = ( modulo_modulo_nat @ A @ B ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3185_minus__div__mult__eq__mod,axiom,
% 8.90/5.04 ! [A: int,B: int] :
% 8.90/5.04 ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 8.90/5.04 = ( modulo_modulo_int @ A @ B ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3186_minus__div__mult__eq__mod,axiom,
% 8.90/5.04 ! [A: code_integer,B: code_integer] :
% 8.90/5.04 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 8.90/5.04 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3187_mod__le__divisor,axiom,
% 8.90/5.04 ! [N: nat,M: nat] :
% 8.90/5.04 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.04 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3188_mod__eq__nat1E,axiom,
% 8.90/5.04 ! [M: nat,Q3: nat,N: nat] :
% 8.90/5.04 ( ( ( modulo_modulo_nat @ M @ Q3 )
% 8.90/5.04 = ( modulo_modulo_nat @ N @ Q3 ) )
% 8.90/5.04 => ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.04 => ~ ! [S2: nat] :
% 8.90/5.04 ( M
% 8.90/5.04 != ( plus_plus_nat @ N @ ( times_times_nat @ Q3 @ S2 ) ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3189_mod__eq__nat2E,axiom,
% 8.90/5.04 ! [M: nat,Q3: nat,N: nat] :
% 8.90/5.04 ( ( ( modulo_modulo_nat @ M @ Q3 )
% 8.90/5.04 = ( modulo_modulo_nat @ N @ Q3 ) )
% 8.90/5.04 => ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.04 => ~ ! [S2: nat] :
% 8.90/5.04 ( N
% 8.90/5.04 != ( plus_plus_nat @ M @ ( times_times_nat @ Q3 @ S2 ) ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3190_nat__mod__eq__lemma,axiom,
% 8.90/5.04 ! [X2: nat,N: nat,Y3: nat] :
% 8.90/5.04 ( ( ( modulo_modulo_nat @ X2 @ N )
% 8.90/5.04 = ( modulo_modulo_nat @ Y3 @ N ) )
% 8.90/5.04 => ( ( ord_less_eq_nat @ Y3 @ X2 )
% 8.90/5.04 => ? [Q2: nat] :
% 8.90/5.04 ( X2
% 8.90/5.04 = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q2 ) ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3191_eucl__rel__int__dividesI,axiom,
% 8.90/5.04 ! [L: int,K: int,Q3: int] :
% 8.90/5.04 ( ( L != zero_zero_int )
% 8.90/5.04 => ( ( K
% 8.90/5.04 = ( times_times_int @ Q3 @ L ) )
% 8.90/5.04 => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ zero_zero_int ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3192_mod__mult2__eq,axiom,
% 8.90/5.04 ! [M: nat,N: nat,Q3: nat] :
% 8.90/5.04 ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q3 ) )
% 8.90/5.04 = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q3 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3193_modulo__nat__def,axiom,
% 8.90/5.04 ( modulo_modulo_nat
% 8.90/5.04 = ( ^ [M3: nat,N3: nat] : ( minus_minus_nat @ M3 @ ( times_times_nat @ ( divide_divide_nat @ M3 @ N3 ) @ N3 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3194_in__mono,axiom,
% 8.90/5.04 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 8.90/5.04 ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
% 8.90/5.04 => ( ( member8440522571783428010at_nat @ X2 @ A2 )
% 8.90/5.04 => ( member8440522571783428010at_nat @ X2 @ B3 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3195_in__mono,axiom,
% 8.90/5.04 ! [A2: set_complex,B3: set_complex,X2: complex] :
% 8.90/5.04 ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 8.90/5.04 => ( ( member_complex @ X2 @ A2 )
% 8.90/5.04 => ( member_complex @ X2 @ B3 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3196_in__mono,axiom,
% 8.90/5.04 ! [A2: set_real,B3: set_real,X2: real] :
% 8.90/5.04 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 8.90/5.04 => ( ( member_real @ X2 @ A2 )
% 8.90/5.04 => ( member_real @ X2 @ B3 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3197_in__mono,axiom,
% 8.90/5.04 ! [A2: set_nat,B3: set_nat,X2: nat] :
% 8.90/5.04 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 8.90/5.04 => ( ( member_nat @ X2 @ A2 )
% 8.90/5.04 => ( member_nat @ X2 @ B3 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3198_in__mono,axiom,
% 8.90/5.04 ! [A2: set_int,B3: set_int,X2: int] :
% 8.90/5.04 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 8.90/5.04 => ( ( member_int @ X2 @ A2 )
% 8.90/5.04 => ( member_int @ X2 @ B3 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3199_subsetD,axiom,
% 8.90/5.04 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
% 8.90/5.04 ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
% 8.90/5.04 => ( ( member8440522571783428010at_nat @ C @ A2 )
% 8.90/5.04 => ( member8440522571783428010at_nat @ C @ B3 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3200_subsetD,axiom,
% 8.90/5.04 ! [A2: set_complex,B3: set_complex,C: complex] :
% 8.90/5.04 ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 8.90/5.04 => ( ( member_complex @ C @ A2 )
% 8.90/5.04 => ( member_complex @ C @ B3 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3201_subsetD,axiom,
% 8.90/5.04 ! [A2: set_real,B3: set_real,C: real] :
% 8.90/5.04 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 8.90/5.04 => ( ( member_real @ C @ A2 )
% 8.90/5.04 => ( member_real @ C @ B3 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3202_subsetD,axiom,
% 8.90/5.04 ! [A2: set_nat,B3: set_nat,C: nat] :
% 8.90/5.04 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 8.90/5.04 => ( ( member_nat @ C @ A2 )
% 8.90/5.04 => ( member_nat @ C @ B3 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3203_subsetD,axiom,
% 8.90/5.04 ! [A2: set_int,B3: set_int,C: int] :
% 8.90/5.04 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 8.90/5.04 => ( ( member_int @ C @ A2 )
% 8.90/5.04 => ( member_int @ C @ B3 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3204_equalityE,axiom,
% 8.90/5.04 ! [A2: set_int,B3: set_int] :
% 8.90/5.04 ( ( A2 = B3 )
% 8.90/5.04 => ~ ( ( ord_less_eq_set_int @ A2 @ B3 )
% 8.90/5.04 => ~ ( ord_less_eq_set_int @ B3 @ A2 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3205_subset__eq,axiom,
% 8.90/5.04 ( ord_le3146513528884898305at_nat
% 8.90/5.04 = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 8.90/5.04 ! [X: product_prod_nat_nat] :
% 8.90/5.04 ( ( member8440522571783428010at_nat @ X @ A6 )
% 8.90/5.04 => ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3206_subset__eq,axiom,
% 8.90/5.04 ( ord_le211207098394363844omplex
% 8.90/5.04 = ( ^ [A6: set_complex,B6: set_complex] :
% 8.90/5.04 ! [X: complex] :
% 8.90/5.04 ( ( member_complex @ X @ A6 )
% 8.90/5.04 => ( member_complex @ X @ B6 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3207_subset__eq,axiom,
% 8.90/5.04 ( ord_less_eq_set_real
% 8.90/5.04 = ( ^ [A6: set_real,B6: set_real] :
% 8.90/5.04 ! [X: real] :
% 8.90/5.04 ( ( member_real @ X @ A6 )
% 8.90/5.04 => ( member_real @ X @ B6 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3208_subset__eq,axiom,
% 8.90/5.04 ( ord_less_eq_set_nat
% 8.90/5.04 = ( ^ [A6: set_nat,B6: set_nat] :
% 8.90/5.04 ! [X: nat] :
% 8.90/5.04 ( ( member_nat @ X @ A6 )
% 8.90/5.04 => ( member_nat @ X @ B6 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3209_subset__eq,axiom,
% 8.90/5.04 ( ord_less_eq_set_int
% 8.90/5.04 = ( ^ [A6: set_int,B6: set_int] :
% 8.90/5.04 ! [X: int] :
% 8.90/5.04 ( ( member_int @ X @ A6 )
% 8.90/5.04 => ( member_int @ X @ B6 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3210_equalityD1,axiom,
% 8.90/5.04 ! [A2: set_int,B3: set_int] :
% 8.90/5.04 ( ( A2 = B3 )
% 8.90/5.04 => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3211_equalityD2,axiom,
% 8.90/5.04 ! [A2: set_int,B3: set_int] :
% 8.90/5.04 ( ( A2 = B3 )
% 8.90/5.04 => ( ord_less_eq_set_int @ B3 @ A2 ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3212_subset__iff,axiom,
% 8.90/5.04 ( ord_le3146513528884898305at_nat
% 8.90/5.04 = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 8.90/5.04 ! [T2: product_prod_nat_nat] :
% 8.90/5.04 ( ( member8440522571783428010at_nat @ T2 @ A6 )
% 8.90/5.04 => ( member8440522571783428010at_nat @ T2 @ B6 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3213_subset__iff,axiom,
% 8.90/5.04 ( ord_le211207098394363844omplex
% 8.90/5.04 = ( ^ [A6: set_complex,B6: set_complex] :
% 8.90/5.04 ! [T2: complex] :
% 8.90/5.04 ( ( member_complex @ T2 @ A6 )
% 8.90/5.04 => ( member_complex @ T2 @ B6 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3214_subset__iff,axiom,
% 8.90/5.04 ( ord_less_eq_set_real
% 8.90/5.04 = ( ^ [A6: set_real,B6: set_real] :
% 8.90/5.04 ! [T2: real] :
% 8.90/5.04 ( ( member_real @ T2 @ A6 )
% 8.90/5.04 => ( member_real @ T2 @ B6 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3215_subset__iff,axiom,
% 8.90/5.04 ( ord_less_eq_set_nat
% 8.90/5.04 = ( ^ [A6: set_nat,B6: set_nat] :
% 8.90/5.04 ! [T2: nat] :
% 8.90/5.04 ( ( member_nat @ T2 @ A6 )
% 8.90/5.04 => ( member_nat @ T2 @ B6 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3216_subset__iff,axiom,
% 8.90/5.04 ( ord_less_eq_set_int
% 8.90/5.04 = ( ^ [A6: set_int,B6: set_int] :
% 8.90/5.04 ! [T2: int] :
% 8.90/5.04 ( ( member_int @ T2 @ A6 )
% 8.90/5.04 => ( member_int @ T2 @ B6 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3217_subset__refl,axiom,
% 8.90/5.04 ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3218_Collect__mono,axiom,
% 8.90/5.04 ! [P: complex > $o,Q: complex > $o] :
% 8.90/5.04 ( ! [X5: complex] :
% 8.90/5.04 ( ( P @ X5 )
% 8.90/5.04 => ( Q @ X5 ) )
% 8.90/5.04 => ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3219_Collect__mono,axiom,
% 8.90/5.04 ! [P: real > $o,Q: real > $o] :
% 8.90/5.04 ( ! [X5: real] :
% 8.90/5.04 ( ( P @ X5 )
% 8.90/5.04 => ( Q @ X5 ) )
% 8.90/5.04 => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3220_Collect__mono,axiom,
% 8.90/5.04 ! [P: list_nat > $o,Q: list_nat > $o] :
% 8.90/5.04 ( ! [X5: list_nat] :
% 8.90/5.04 ( ( P @ X5 )
% 8.90/5.04 => ( Q @ X5 ) )
% 8.90/5.04 => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3221_Collect__mono,axiom,
% 8.90/5.04 ! [P: nat > $o,Q: nat > $o] :
% 8.90/5.04 ( ! [X5: nat] :
% 8.90/5.04 ( ( P @ X5 )
% 8.90/5.04 => ( Q @ X5 ) )
% 8.90/5.04 => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3222_Collect__mono,axiom,
% 8.90/5.04 ! [P: int > $o,Q: int > $o] :
% 8.90/5.04 ( ! [X5: int] :
% 8.90/5.04 ( ( P @ X5 )
% 8.90/5.04 => ( Q @ X5 ) )
% 8.90/5.04 => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3223_subset__trans,axiom,
% 8.90/5.04 ! [A2: set_int,B3: set_int,C4: set_int] :
% 8.90/5.04 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 8.90/5.04 => ( ( ord_less_eq_set_int @ B3 @ C4 )
% 8.90/5.04 => ( ord_less_eq_set_int @ A2 @ C4 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3224_set__eq__subset,axiom,
% 8.90/5.04 ( ( ^ [Y6: set_int,Z5: set_int] : ( Y6 = Z5 ) )
% 8.90/5.04 = ( ^ [A6: set_int,B6: set_int] :
% 8.90/5.04 ( ( ord_less_eq_set_int @ A6 @ B6 )
% 8.90/5.04 & ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3225_Collect__mono__iff,axiom,
% 8.90/5.04 ! [P: complex > $o,Q: complex > $o] :
% 8.90/5.04 ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
% 8.90/5.04 = ( ! [X: complex] :
% 8.90/5.04 ( ( P @ X )
% 8.90/5.04 => ( Q @ X ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3226_Collect__mono__iff,axiom,
% 8.90/5.04 ! [P: real > $o,Q: real > $o] :
% 8.90/5.04 ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
% 8.90/5.04 = ( ! [X: real] :
% 8.90/5.04 ( ( P @ X )
% 8.90/5.04 => ( Q @ X ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3227_Collect__mono__iff,axiom,
% 8.90/5.04 ! [P: list_nat > $o,Q: list_nat > $o] :
% 8.90/5.04 ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
% 8.90/5.04 = ( ! [X: list_nat] :
% 8.90/5.04 ( ( P @ X )
% 8.90/5.04 => ( Q @ X ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3228_Collect__mono__iff,axiom,
% 8.90/5.04 ! [P: nat > $o,Q: nat > $o] :
% 8.90/5.04 ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 8.90/5.04 = ( ! [X: nat] :
% 8.90/5.04 ( ( P @ X )
% 8.90/5.04 => ( Q @ X ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3229_Collect__mono__iff,axiom,
% 8.90/5.04 ! [P: int > $o,Q: int > $o] :
% 8.90/5.04 ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
% 8.90/5.04 = ( ! [X: int] :
% 8.90/5.04 ( ( P @ X )
% 8.90/5.04 => ( Q @ X ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3230_subset__iff__psubset__eq,axiom,
% 8.90/5.04 ( ord_less_eq_set_int
% 8.90/5.04 = ( ^ [A6: set_int,B6: set_int] :
% 8.90/5.04 ( ( ord_less_set_int @ A6 @ B6 )
% 8.90/5.04 | ( A6 = B6 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3231_subset__psubset__trans,axiom,
% 8.90/5.04 ! [A2: set_int,B3: set_int,C4: set_int] :
% 8.90/5.04 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 8.90/5.04 => ( ( ord_less_set_int @ B3 @ C4 )
% 8.90/5.04 => ( ord_less_set_int @ A2 @ C4 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3232_subset__not__subset__eq,axiom,
% 8.90/5.04 ( ord_less_set_int
% 8.90/5.04 = ( ^ [A6: set_int,B6: set_int] :
% 8.90/5.04 ( ( ord_less_eq_set_int @ A6 @ B6 )
% 8.90/5.04 & ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3233_psubset__subset__trans,axiom,
% 8.90/5.04 ! [A2: set_int,B3: set_int,C4: set_int] :
% 8.90/5.04 ( ( ord_less_set_int @ A2 @ B3 )
% 8.90/5.04 => ( ( ord_less_eq_set_int @ B3 @ C4 )
% 8.90/5.04 => ( ord_less_set_int @ A2 @ C4 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3234_psubset__imp__subset,axiom,
% 8.90/5.04 ! [A2: set_int,B3: set_int] :
% 8.90/5.04 ( ( ord_less_set_int @ A2 @ B3 )
% 8.90/5.04 => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3235_psubset__eq,axiom,
% 8.90/5.04 ( ord_less_set_int
% 8.90/5.04 = ( ^ [A6: set_int,B6: set_int] :
% 8.90/5.04 ( ( ord_less_eq_set_int @ A6 @ B6 )
% 8.90/5.04 & ( A6 != B6 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3236_psubsetE,axiom,
% 8.90/5.04 ! [A2: set_int,B3: set_int] :
% 8.90/5.04 ( ( ord_less_set_int @ A2 @ B3 )
% 8.90/5.04 => ~ ( ( ord_less_eq_set_int @ A2 @ B3 )
% 8.90/5.04 => ( ord_less_eq_set_int @ B3 @ A2 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3237_Diff__mono,axiom,
% 8.90/5.04 ! [A2: set_nat,C4: set_nat,D4: set_nat,B3: set_nat] :
% 8.90/5.04 ( ( ord_less_eq_set_nat @ A2 @ C4 )
% 8.90/5.04 => ( ( ord_less_eq_set_nat @ D4 @ B3 )
% 8.90/5.04 => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ ( minus_minus_set_nat @ C4 @ D4 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3238_Diff__mono,axiom,
% 8.90/5.04 ! [A2: set_int,C4: set_int,D4: set_int,B3: set_int] :
% 8.90/5.04 ( ( ord_less_eq_set_int @ A2 @ C4 )
% 8.90/5.04 => ( ( ord_less_eq_set_int @ D4 @ B3 )
% 8.90/5.04 => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B3 ) @ ( minus_minus_set_int @ C4 @ D4 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3239_Diff__subset,axiom,
% 8.90/5.04 ! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ A2 ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3240_Diff__subset,axiom,
% 8.90/5.04 ! [A2: set_int,B3: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B3 ) @ A2 ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3241_double__diff,axiom,
% 8.90/5.04 ! [A2: set_nat,B3: set_nat,C4: set_nat] :
% 8.90/5.04 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 8.90/5.04 => ( ( ord_less_eq_set_nat @ B3 @ C4 )
% 8.90/5.04 => ( ( minus_minus_set_nat @ B3 @ ( minus_minus_set_nat @ C4 @ A2 ) )
% 8.90/5.04 = A2 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3242_double__diff,axiom,
% 8.90/5.04 ! [A2: set_int,B3: set_int,C4: set_int] :
% 8.90/5.04 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 8.90/5.04 => ( ( ord_less_eq_set_int @ B3 @ C4 )
% 8.90/5.04 => ( ( minus_minus_set_int @ B3 @ ( minus_minus_set_int @ C4 @ A2 ) )
% 8.90/5.04 = A2 ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3243_split__mod,axiom,
% 8.90/5.04 ! [P: nat > $o,M: nat,N: nat] :
% 8.90/5.04 ( ( P @ ( modulo_modulo_nat @ M @ N ) )
% 8.90/5.04 = ( ( ( N = zero_zero_nat )
% 8.90/5.04 => ( P @ M ) )
% 8.90/5.04 & ( ( N != zero_zero_nat )
% 8.90/5.04 => ! [I4: nat,J3: nat] :
% 8.90/5.04 ( ( ord_less_nat @ J3 @ N )
% 8.90/5.04 => ( ( M
% 8.90/5.04 = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 8.90/5.04 => ( P @ J3 ) ) ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3244_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 8.90/5.04 ! [C: code_integer,A: code_integer,B: code_integer] :
% 8.90/5.04 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 8.90/5.04 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 8.90/5.04 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3245_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 8.90/5.04 ! [C: nat,A: nat,B: nat] :
% 8.90/5.04 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 8.90/5.04 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
% 8.90/5.04 = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3246_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 8.90/5.04 ! [C: int,A: int,B: int] :
% 8.90/5.04 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.90/5.04 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 8.90/5.04 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3247_less__eq__set__def,axiom,
% 8.90/5.04 ( ord_le3146513528884898305at_nat
% 8.90/5.04 = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 8.90/5.04 ( ord_le704812498762024988_nat_o
% 8.90/5.04 @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A6 )
% 8.90/5.04 @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3248_less__eq__set__def,axiom,
% 8.90/5.04 ( ord_le211207098394363844omplex
% 8.90/5.04 = ( ^ [A6: set_complex,B6: set_complex] :
% 8.90/5.04 ( ord_le4573692005234683329plex_o
% 8.90/5.04 @ ^ [X: complex] : ( member_complex @ X @ A6 )
% 8.90/5.04 @ ^ [X: complex] : ( member_complex @ X @ B6 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3249_less__eq__set__def,axiom,
% 8.90/5.04 ( ord_less_eq_set_real
% 8.90/5.04 = ( ^ [A6: set_real,B6: set_real] :
% 8.90/5.04 ( ord_less_eq_real_o
% 8.90/5.04 @ ^ [X: real] : ( member_real @ X @ A6 )
% 8.90/5.04 @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3250_less__eq__set__def,axiom,
% 8.90/5.04 ( ord_less_eq_set_nat
% 8.90/5.04 = ( ^ [A6: set_nat,B6: set_nat] :
% 8.90/5.04 ( ord_less_eq_nat_o
% 8.90/5.04 @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 8.90/5.04 @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3251_less__eq__set__def,axiom,
% 8.90/5.04 ( ord_less_eq_set_int
% 8.90/5.04 = ( ^ [A6: set_int,B6: set_int] :
% 8.90/5.04 ( ord_less_eq_int_o
% 8.90/5.04 @ ^ [X: int] : ( member_int @ X @ A6 )
% 8.90/5.04 @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3252_Collect__subset,axiom,
% 8.90/5.04 ! [A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
% 8.90/5.04 ( ord_le3146513528884898305at_nat
% 8.90/5.04 @ ( collec3392354462482085612at_nat
% 8.90/5.04 @ ^ [X: product_prod_nat_nat] :
% 8.90/5.04 ( ( member8440522571783428010at_nat @ X @ A2 )
% 8.90/5.04 & ( P @ X ) ) )
% 8.90/5.04 @ A2 ) ).
% 8.90/5.04
% 8.90/5.04 thf(fact_3253_Collect__subset,axiom,
% 8.90/5.04 ! [A2: set_complex,P: complex > $o] :
% 8.90/5.04 ( ord_le211207098394363844omplex
% 8.90/5.04 @ ( collect_complex
% 8.90/5.05 @ ^ [X: complex] :
% 8.90/5.05 ( ( member_complex @ X @ A2 )
% 8.90/5.05 & ( P @ X ) ) )
% 8.90/5.05 @ A2 ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3254_Collect__subset,axiom,
% 8.90/5.05 ! [A2: set_real,P: real > $o] :
% 8.90/5.05 ( ord_less_eq_set_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [X: real] :
% 8.90/5.05 ( ( member_real @ X @ A2 )
% 8.90/5.05 & ( P @ X ) ) )
% 8.90/5.05 @ A2 ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3255_Collect__subset,axiom,
% 8.90/5.05 ! [A2: set_list_nat,P: list_nat > $o] :
% 8.90/5.05 ( ord_le6045566169113846134st_nat
% 8.90/5.05 @ ( collect_list_nat
% 8.90/5.05 @ ^ [X: list_nat] :
% 8.90/5.05 ( ( member_list_nat @ X @ A2 )
% 8.90/5.05 & ( P @ X ) ) )
% 8.90/5.05 @ A2 ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3256_Collect__subset,axiom,
% 8.90/5.05 ! [A2: set_nat,P: nat > $o] :
% 8.90/5.05 ( ord_less_eq_set_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [X: nat] :
% 8.90/5.05 ( ( member_nat @ X @ A2 )
% 8.90/5.05 & ( P @ X ) ) )
% 8.90/5.05 @ A2 ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3257_Collect__subset,axiom,
% 8.90/5.05 ! [A2: set_int,P: int > $o] :
% 8.90/5.05 ( ord_less_eq_set_int
% 8.90/5.05 @ ( collect_int
% 8.90/5.05 @ ^ [X: int] :
% 8.90/5.05 ( ( member_int @ X @ A2 )
% 8.90/5.05 & ( P @ X ) ) )
% 8.90/5.05 @ A2 ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3258_Suc__times__mod__eq,axiom,
% 8.90/5.05 ! [M: nat,N: nat] :
% 8.90/5.05 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 8.90/5.05 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
% 8.90/5.05 = one_one_nat ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3259_divmod__digit__0_I2_J,axiom,
% 8.90/5.05 ! [B: nat,A: nat] :
% 8.90/5.05 ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.90/5.05 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 8.90/5.05 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
% 8.90/5.05 = ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3260_divmod__digit__0_I2_J,axiom,
% 8.90/5.05 ! [B: int,A: int] :
% 8.90/5.05 ( ( ord_less_int @ zero_zero_int @ B )
% 8.90/5.05 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 8.90/5.05 => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
% 8.90/5.05 = ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3261_divmod__digit__0_I2_J,axiom,
% 8.90/5.05 ! [B: code_integer,A: code_integer] :
% 8.90/5.05 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 8.90/5.05 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 8.90/5.05 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
% 8.90/5.05 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3262_bits__stable__imp__add__self,axiom,
% 8.90/5.05 ! [A: nat] :
% 8.90/5.05 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.05 = A )
% 8.90/5.05 => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.05 = zero_zero_nat ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3263_bits__stable__imp__add__self,axiom,
% 8.90/5.05 ! [A: int] :
% 8.90/5.05 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.05 = A )
% 8.90/5.05 => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 8.90/5.05 = zero_zero_int ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3264_bits__stable__imp__add__self,axiom,
% 8.90/5.05 ! [A: code_integer] :
% 8.90/5.05 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.05 = A )
% 8.90/5.05 => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 8.90/5.05 = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3265_div__exp__mod__exp__eq,axiom,
% 8.90/5.05 ! [A: nat,N: nat,M: nat] :
% 8.90/5.05 ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.90/5.05 = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3266_div__exp__mod__exp__eq,axiom,
% 8.90/5.05 ! [A: int,N: nat,M: nat] :
% 8.90/5.05 ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 8.90/5.05 = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3267_div__exp__mod__exp__eq,axiom,
% 8.90/5.05 ! [A: code_integer,N: nat,M: nat] :
% 8.90/5.05 ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 8.90/5.05 = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3268_divmod__digit__0_I1_J,axiom,
% 8.90/5.05 ! [B: nat,A: nat] :
% 8.90/5.05 ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.90/5.05 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 8.90/5.05 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 8.90/5.05 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3269_divmod__digit__0_I1_J,axiom,
% 8.90/5.05 ! [B: int,A: int] :
% 8.90/5.05 ( ( ord_less_int @ zero_zero_int @ B )
% 8.90/5.05 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 8.90/5.05 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 8.90/5.05 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3270_divmod__digit__0_I1_J,axiom,
% 8.90/5.05 ! [B: code_integer,A: code_integer] :
% 8.90/5.05 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 8.90/5.05 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 8.90/5.05 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 8.90/5.05 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3271_mult__exp__mod__exp__eq,axiom,
% 8.90/5.05 ! [M: nat,N: nat,A: nat] :
% 8.90/5.05 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.05 => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.05 = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3272_mult__exp__mod__exp__eq,axiom,
% 8.90/5.05 ! [M: nat,N: nat,A: int] :
% 8.90/5.05 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.05 => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.05 = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3273_mult__exp__mod__exp__eq,axiom,
% 8.90/5.05 ! [M: nat,N: nat,A: code_integer] :
% 8.90/5.05 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.05 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.05 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3274_eucl__rel__int__iff,axiom,
% 8.90/5.05 ! [K: int,L: int,Q3: int,R2: int] :
% 8.90/5.05 ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 8.90/5.05 = ( ( K
% 8.90/5.05 = ( plus_plus_int @ ( times_times_int @ L @ Q3 ) @ R2 ) )
% 8.90/5.05 & ( ( ord_less_int @ zero_zero_int @ L )
% 8.90/5.05 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 8.90/5.05 & ( ord_less_int @ R2 @ L ) ) )
% 8.90/5.05 & ( ~ ( ord_less_int @ zero_zero_int @ L )
% 8.90/5.05 => ( ( ( ord_less_int @ L @ zero_zero_int )
% 8.90/5.05 => ( ( ord_less_int @ L @ R2 )
% 8.90/5.05 & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
% 8.90/5.05 & ( ~ ( ord_less_int @ L @ zero_zero_int )
% 8.90/5.05 => ( Q3 = zero_zero_int ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3275_mod__double__modulus,axiom,
% 8.90/5.05 ! [M: code_integer,X2: code_integer] :
% 8.90/5.05 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 8.90/5.05 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X2 )
% 8.90/5.05 => ( ( ( modulo364778990260209775nteger @ X2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 8.90/5.05 = ( modulo364778990260209775nteger @ X2 @ M ) )
% 8.90/5.05 | ( ( modulo364778990260209775nteger @ X2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 8.90/5.05 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X2 @ M ) @ M ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3276_mod__double__modulus,axiom,
% 8.90/5.05 ! [M: nat,X2: nat] :
% 8.90/5.05 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.90/5.05 => ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
% 8.90/5.05 => ( ( ( modulo_modulo_nat @ X2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.90/5.05 = ( modulo_modulo_nat @ X2 @ M ) )
% 8.90/5.05 | ( ( modulo_modulo_nat @ X2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 8.90/5.05 = ( plus_plus_nat @ ( modulo_modulo_nat @ X2 @ M ) @ M ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3277_mod__double__modulus,axiom,
% 8.90/5.05 ! [M: int,X2: int] :
% 8.90/5.05 ( ( ord_less_int @ zero_zero_int @ M )
% 8.90/5.05 => ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.90/5.05 => ( ( ( modulo_modulo_int @ X2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 8.90/5.05 = ( modulo_modulo_int @ X2 @ M ) )
% 8.90/5.05 | ( ( modulo_modulo_int @ X2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 8.90/5.05 = ( plus_plus_int @ ( modulo_modulo_int @ X2 @ M ) @ M ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3278_divmod__digit__1_I2_J,axiom,
% 8.90/5.05 ! [A: code_integer,B: code_integer] :
% 8.90/5.05 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 8.90/5.05 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 8.90/5.05 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 8.90/5.05 => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 8.90/5.05 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3279_divmod__digit__1_I2_J,axiom,
% 8.90/5.05 ! [A: nat,B: nat] :
% 8.90/5.05 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.90/5.05 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.90/5.05 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 8.90/5.05 => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 8.90/5.05 = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3280_divmod__digit__1_I2_J,axiom,
% 8.90/5.05 ! [A: int,B: int] :
% 8.90/5.05 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.90/5.05 => ( ( ord_less_int @ zero_zero_int @ B )
% 8.90/5.05 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 8.90/5.05 => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 8.90/5.05 = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3281_unset__bit__Suc,axiom,
% 8.90/5.05 ! [N: nat,A: code_integer] :
% 8.90/5.05 ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
% 8.90/5.05 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3282_unset__bit__Suc,axiom,
% 8.90/5.05 ! [N: nat,A: int] :
% 8.90/5.05 ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
% 8.90/5.05 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3283_unset__bit__Suc,axiom,
% 8.90/5.05 ! [N: nat,A: nat] :
% 8.90/5.05 ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
% 8.90/5.05 = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3284_set__bit__Suc,axiom,
% 8.90/5.05 ! [N: nat,A: code_integer] :
% 8.90/5.05 ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
% 8.90/5.05 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3285_set__bit__Suc,axiom,
% 8.90/5.05 ! [N: nat,A: int] :
% 8.90/5.05 ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
% 8.90/5.05 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3286_set__bit__Suc,axiom,
% 8.90/5.05 ! [N: nat,A: nat] :
% 8.90/5.05 ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
% 8.90/5.05 = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3287_divmod__digit__1_I1_J,axiom,
% 8.90/5.05 ! [A: code_integer,B: code_integer] :
% 8.90/5.05 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 8.90/5.05 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 8.90/5.05 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 8.90/5.05 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
% 8.90/5.05 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3288_divmod__digit__1_I1_J,axiom,
% 8.90/5.05 ! [A: nat,B: nat] :
% 8.90/5.05 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 8.90/5.05 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 8.90/5.05 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 8.90/5.05 => ( ( 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 )
% 8.90/5.05 = ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3289_divmod__digit__1_I1_J,axiom,
% 8.90/5.05 ! [A: int,B: int] :
% 8.90/5.05 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.90/5.05 => ( ( ord_less_int @ zero_zero_int @ B )
% 8.90/5.05 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 8.90/5.05 => ( ( 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 )
% 8.90/5.05 = ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3290_pos__eucl__rel__int__mult__2,axiom,
% 8.90/5.05 ! [B: int,A: int,Q3: int,R2: int] :
% 8.90/5.05 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.90/5.05 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 8.90/5.05 => ( 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 @ Q3 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3291_verit__le__mono__div,axiom,
% 8.90/5.05 ! [A2: nat,B3: nat,N: nat] :
% 8.90/5.05 ( ( ord_less_nat @ A2 @ B3 )
% 8.90/5.05 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.05 => ( ord_less_eq_nat
% 8.90/5.05 @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N )
% 8.90/5.05 @ ( if_nat
% 8.90/5.05 @ ( ( modulo_modulo_nat @ B3 @ N )
% 8.90/5.05 = zero_zero_nat )
% 8.90/5.05 @ one_one_nat
% 8.90/5.05 @ zero_zero_nat ) )
% 8.90/5.05 @ ( divide_divide_nat @ B3 @ N ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3292_div__mod__decomp,axiom,
% 8.90/5.05 ! [A2: nat,N: nat] :
% 8.90/5.05 ( A2
% 8.90/5.05 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N ) @ N ) @ ( modulo_modulo_nat @ A2 @ N ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3293_flip__bit__Suc,axiom,
% 8.90/5.05 ! [N: nat,A: code_integer] :
% 8.90/5.05 ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
% 8.90/5.05 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3294_flip__bit__Suc,axiom,
% 8.90/5.05 ! [N: nat,A: int] :
% 8.90/5.05 ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
% 8.90/5.05 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3295_flip__bit__Suc,axiom,
% 8.90/5.05 ! [N: nat,A: nat] :
% 8.90/5.05 ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
% 8.90/5.05 = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3296_div__less__mono,axiom,
% 8.90/5.05 ! [A2: nat,B3: nat,N: nat] :
% 8.90/5.05 ( ( ord_less_nat @ A2 @ B3 )
% 8.90/5.05 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.05 => ( ( ( modulo_modulo_nat @ A2 @ N )
% 8.90/5.05 = zero_zero_nat )
% 8.90/5.05 => ( ( ( modulo_modulo_nat @ B3 @ N )
% 8.90/5.05 = zero_zero_nat )
% 8.90/5.05 => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B3 @ N ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3297_product__nth,axiom,
% 8.90/5.05 ! [N: nat,Xs2: list_num,Ys: list_num] :
% 8.90/5.05 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_num @ Xs2 ) @ ( size_size_list_num @ Ys ) ) )
% 8.90/5.05 => ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs2 @ Ys ) @ N )
% 8.90/5.05 = ( product_Pair_num_num @ ( nth_num @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys ) ) ) @ ( nth_num @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3298_product__nth,axiom,
% 8.90/5.05 ! [N: nat,Xs2: list_Code_integer,Ys: list_o] :
% 8.90/5.05 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s3445333598471063425nteger @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
% 8.90/5.05 => ( ( nth_Pr8522763379788166057eger_o @ ( produc3607205314601156340eger_o @ Xs2 @ Ys ) @ N )
% 8.90/5.05 = ( produc6677183202524767010eger_o @ ( nth_Code_integer @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3299_product__nth,axiom,
% 8.90/5.05 ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 8.90/5.05 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 8.90/5.05 => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) @ N )
% 8.90/5.05 = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3300_product__nth,axiom,
% 8.90/5.05 ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_o] :
% 8.90/5.05 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
% 8.90/5.05 => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) @ N )
% 8.90/5.05 = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3301_product__nth,axiom,
% 8.90/5.05 ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_nat] :
% 8.90/5.05 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
% 8.90/5.05 => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) @ N )
% 8.90/5.05 = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3302_product__nth,axiom,
% 8.90/5.05 ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_int] :
% 8.90/5.05 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
% 8.90/5.05 => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys ) @ N )
% 8.90/5.05 = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3303_product__nth,axiom,
% 8.90/5.05 ! [N: nat,Xs2: list_o,Ys: list_VEBT_VEBT] :
% 8.90/5.05 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 8.90/5.05 => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys ) @ N )
% 8.90/5.05 = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3304_product__nth,axiom,
% 8.90/5.05 ! [N: nat,Xs2: list_o,Ys: list_o] :
% 8.90/5.05 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
% 8.90/5.05 => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs2 @ Ys ) @ N )
% 8.90/5.05 = ( product_Pair_o_o @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3305_product__nth,axiom,
% 8.90/5.05 ! [N: nat,Xs2: list_o,Ys: list_nat] :
% 8.90/5.05 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
% 8.90/5.05 => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs2 @ Ys ) @ N )
% 8.90/5.05 = ( product_Pair_o_nat @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3306_product__nth,axiom,
% 8.90/5.05 ! [N: nat,Xs2: list_o,Ys: list_int] :
% 8.90/5.05 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
% 8.90/5.05 => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs2 @ Ys ) @ N )
% 8.90/5.05 = ( product_Pair_o_int @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3307_obtain__set__succ,axiom,
% 8.90/5.05 ! [X2: nat,Z: nat,A2: set_nat,B3: set_nat] :
% 8.90/5.05 ( ( ord_less_nat @ X2 @ Z )
% 8.90/5.05 => ( ( vEBT_VEBT_max_in_set @ A2 @ Z )
% 8.90/5.05 => ( ( finite_finite_nat @ B3 )
% 8.90/5.05 => ( ( A2 = B3 )
% 8.90/5.05 => ? [X_12: nat] : ( vEBT_is_succ_in_set @ A2 @ X2 @ X_12 ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3308_obtain__set__pred,axiom,
% 8.90/5.05 ! [Z: nat,X2: nat,A2: set_nat] :
% 8.90/5.05 ( ( ord_less_nat @ Z @ X2 )
% 8.90/5.05 => ( ( vEBT_VEBT_min_in_set @ A2 @ Z )
% 8.90/5.05 => ( ( finite_finite_nat @ A2 )
% 8.90/5.05 => ? [X_12: nat] : ( vEBT_is_pred_in_set @ A2 @ X2 @ X_12 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3309_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
% 8.90/5.05 ! [A: $o,B: $o,X2: nat] :
% 8.90/5.05 ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Leaf @ A @ B ) @ X2 )
% 8.90/5.05 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( X2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3310_set__vebt__finite,axiom,
% 8.90/5.05 ! [T: vEBT_VEBT,N: nat] :
% 8.90/5.05 ( ( vEBT_invar_vebt @ T @ N )
% 8.90/5.05 => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3311_pred__none__empty,axiom,
% 8.90/5.05 ! [Xs2: set_nat,A: nat] :
% 8.90/5.05 ( ~ ? [X_12: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A @ X_12 )
% 8.90/5.05 => ( ( finite_finite_nat @ Xs2 )
% 8.90/5.05 => ~ ? [X3: nat] :
% 8.90/5.05 ( ( member_nat @ X3 @ Xs2 )
% 8.90/5.05 & ( ord_less_nat @ X3 @ A ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3312_succ__none__empty,axiom,
% 8.90/5.05 ! [Xs2: set_nat,A: nat] :
% 8.90/5.05 ( ~ ? [X_12: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A @ X_12 )
% 8.90/5.05 => ( ( finite_finite_nat @ Xs2 )
% 8.90/5.05 => ~ ? [X3: nat] :
% 8.90/5.05 ( ( member_nat @ X3 @ Xs2 )
% 8.90/5.05 & ( ord_less_nat @ A @ X3 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3313_verit__eq__simplify_I8_J,axiom,
% 8.90/5.05 ! [X22: num,Y2: num] :
% 8.90/5.05 ( ( ( bit0 @ X22 )
% 8.90/5.05 = ( bit0 @ Y2 ) )
% 8.90/5.05 = ( X22 = Y2 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3314_List_Ofinite__set,axiom,
% 8.90/5.05 ! [Xs2: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3315_List_Ofinite__set,axiom,
% 8.90/5.05 ! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3316_List_Ofinite__set,axiom,
% 8.90/5.05 ! [Xs2: list_int] : ( finite_finite_int @ ( set_int2 @ Xs2 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3317_List_Ofinite__set,axiom,
% 8.90/5.05 ! [Xs2: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs2 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3318_flip__bit__nonnegative__int__iff,axiom,
% 8.90/5.05 ! [N: nat,K: int] :
% 8.90/5.05 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
% 8.90/5.05 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3319_flip__bit__negative__int__iff,axiom,
% 8.90/5.05 ! [N: nat,K: int] :
% 8.90/5.05 ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
% 8.90/5.05 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3320_infinite__Icc__iff,axiom,
% 8.90/5.05 ! [A: rat,B: rat] :
% 8.90/5.05 ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
% 8.90/5.05 = ( ord_less_rat @ A @ B ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3321_infinite__Icc__iff,axiom,
% 8.90/5.05 ! [A: real,B: real] :
% 8.90/5.05 ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
% 8.90/5.05 = ( ord_less_real @ A @ B ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3322_mod__neg__neg__trivial,axiom,
% 8.90/5.05 ! [K: int,L: int] :
% 8.90/5.05 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 8.90/5.05 => ( ( ord_less_int @ L @ K )
% 8.90/5.05 => ( ( modulo_modulo_int @ K @ L )
% 8.90/5.05 = K ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3323_mod__pos__pos__trivial,axiom,
% 8.90/5.05 ! [K: int,L: int] :
% 8.90/5.05 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.90/5.05 => ( ( ord_less_int @ K @ L )
% 8.90/5.05 => ( ( modulo_modulo_int @ K @ L )
% 8.90/5.05 = K ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3324_length__product,axiom,
% 8.90/5.05 ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 8.90/5.05 ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) )
% 8.90/5.05 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3325_length__product,axiom,
% 8.90/5.05 ! [Xs2: list_VEBT_VEBT,Ys: list_o] :
% 8.90/5.05 ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) )
% 8.90/5.05 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3326_length__product,axiom,
% 8.90/5.05 ! [Xs2: list_VEBT_VEBT,Ys: list_nat] :
% 8.90/5.05 ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) )
% 8.90/5.05 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3327_length__product,axiom,
% 8.90/5.05 ! [Xs2: list_VEBT_VEBT,Ys: list_int] :
% 8.90/5.05 ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys ) )
% 8.90/5.05 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3328_length__product,axiom,
% 8.90/5.05 ! [Xs2: list_o,Ys: list_VEBT_VEBT] :
% 8.90/5.05 ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys ) )
% 8.90/5.05 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3329_length__product,axiom,
% 8.90/5.05 ! [Xs2: list_o,Ys: list_o] :
% 8.90/5.05 ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs2 @ Ys ) )
% 8.90/5.05 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3330_length__product,axiom,
% 8.90/5.05 ! [Xs2: list_o,Ys: list_nat] :
% 8.90/5.05 ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs2 @ Ys ) )
% 8.90/5.05 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3331_length__product,axiom,
% 8.90/5.05 ! [Xs2: list_o,Ys: list_int] :
% 8.90/5.05 ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs2 @ Ys ) )
% 8.90/5.05 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3332_length__product,axiom,
% 8.90/5.05 ! [Xs2: list_nat,Ys: list_VEBT_VEBT] :
% 8.90/5.05 ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys ) )
% 8.90/5.05 = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3333_length__product,axiom,
% 8.90/5.05 ! [Xs2: list_nat,Ys: list_o] :
% 8.90/5.05 ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs2 @ Ys ) )
% 8.90/5.05 = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3334_zmod__numeral__Bit0,axiom,
% 8.90/5.05 ! [V: num,W: num] :
% 8.90/5.05 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 8.90/5.05 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3335_bounded__nat__set__is__finite,axiom,
% 8.90/5.05 ! [N5: set_nat,N: nat] :
% 8.90/5.05 ( ! [X5: nat] :
% 8.90/5.05 ( ( member_nat @ X5 @ N5 )
% 8.90/5.05 => ( ord_less_nat @ X5 @ N ) )
% 8.90/5.05 => ( finite_finite_nat @ N5 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3336_finite__nat__set__iff__bounded,axiom,
% 8.90/5.05 ( finite_finite_nat
% 8.90/5.05 = ( ^ [N6: set_nat] :
% 8.90/5.05 ? [M3: nat] :
% 8.90/5.05 ! [X: nat] :
% 8.90/5.05 ( ( member_nat @ X @ N6 )
% 8.90/5.05 => ( ord_less_nat @ X @ M3 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3337_finite__nat__set__iff__bounded__le,axiom,
% 8.90/5.05 ( finite_finite_nat
% 8.90/5.05 = ( ^ [N6: set_nat] :
% 8.90/5.05 ? [M3: nat] :
% 8.90/5.05 ! [X: nat] :
% 8.90/5.05 ( ( member_nat @ X @ N6 )
% 8.90/5.05 => ( ord_less_eq_nat @ X @ M3 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3338_finite__list,axiom,
% 8.90/5.05 ! [A2: set_VEBT_VEBT] :
% 8.90/5.05 ( ( finite5795047828879050333T_VEBT @ A2 )
% 8.90/5.05 => ? [Xs3: list_VEBT_VEBT] :
% 8.90/5.05 ( ( set_VEBT_VEBT2 @ Xs3 )
% 8.90/5.05 = A2 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3339_finite__list,axiom,
% 8.90/5.05 ! [A2: set_nat] :
% 8.90/5.05 ( ( finite_finite_nat @ A2 )
% 8.90/5.05 => ? [Xs3: list_nat] :
% 8.90/5.05 ( ( set_nat2 @ Xs3 )
% 8.90/5.05 = A2 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3340_finite__list,axiom,
% 8.90/5.05 ! [A2: set_int] :
% 8.90/5.05 ( ( finite_finite_int @ A2 )
% 8.90/5.05 => ? [Xs3: list_int] :
% 8.90/5.05 ( ( set_int2 @ Xs3 )
% 8.90/5.05 = A2 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3341_finite__list,axiom,
% 8.90/5.05 ! [A2: set_complex] :
% 8.90/5.05 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.05 => ? [Xs3: list_complex] :
% 8.90/5.05 ( ( set_complex2 @ Xs3 )
% 8.90/5.05 = A2 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3342_finite__M__bounded__by__nat,axiom,
% 8.90/5.05 ! [P: nat > $o,I2: nat] :
% 8.90/5.05 ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [K2: nat] :
% 8.90/5.05 ( ( P @ K2 )
% 8.90/5.05 & ( ord_less_nat @ K2 @ I2 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3343_finite__less__ub,axiom,
% 8.90/5.05 ! [F: nat > nat,U: nat] :
% 8.90/5.05 ( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
% 8.90/5.05 => ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3344_finite__lists__length__eq,axiom,
% 8.90/5.05 ! [A2: set_complex,N: nat] :
% 8.90/5.05 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.05 => ( finite8712137658972009173omplex
% 8.90/5.05 @ ( collect_list_complex
% 8.90/5.05 @ ^ [Xs: list_complex] :
% 8.90/5.05 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
% 8.90/5.05 & ( ( size_s3451745648224563538omplex @ Xs )
% 8.90/5.05 = N ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3345_finite__lists__length__eq,axiom,
% 8.90/5.05 ! [A2: set_VEBT_VEBT,N: nat] :
% 8.90/5.05 ( ( finite5795047828879050333T_VEBT @ A2 )
% 8.90/5.05 => ( finite3004134309566078307T_VEBT
% 8.90/5.05 @ ( collec5608196760682091941T_VEBT
% 8.90/5.05 @ ^ [Xs: list_VEBT_VEBT] :
% 8.90/5.05 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
% 8.90/5.05 & ( ( size_s6755466524823107622T_VEBT @ Xs )
% 8.90/5.05 = N ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3346_finite__lists__length__eq,axiom,
% 8.90/5.05 ! [A2: set_o,N: nat] :
% 8.90/5.05 ( ( finite_finite_o @ A2 )
% 8.90/5.05 => ( finite_finite_list_o
% 8.90/5.05 @ ( collect_list_o
% 8.90/5.05 @ ^ [Xs: list_o] :
% 8.90/5.05 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
% 8.90/5.05 & ( ( size_size_list_o @ Xs )
% 8.90/5.05 = N ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3347_finite__lists__length__eq,axiom,
% 8.90/5.05 ! [A2: set_nat,N: nat] :
% 8.90/5.05 ( ( finite_finite_nat @ A2 )
% 8.90/5.05 => ( finite8100373058378681591st_nat
% 8.90/5.05 @ ( collect_list_nat
% 8.90/5.05 @ ^ [Xs: list_nat] :
% 8.90/5.05 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
% 8.90/5.05 & ( ( size_size_list_nat @ Xs )
% 8.90/5.05 = N ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3348_finite__lists__length__eq,axiom,
% 8.90/5.05 ! [A2: set_int,N: nat] :
% 8.90/5.05 ( ( finite_finite_int @ A2 )
% 8.90/5.05 => ( finite3922522038869484883st_int
% 8.90/5.05 @ ( collect_list_int
% 8.90/5.05 @ ^ [Xs: list_int] :
% 8.90/5.05 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
% 8.90/5.05 & ( ( size_size_list_int @ Xs )
% 8.90/5.05 = N ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3349_zmod__le__nonneg__dividend,axiom,
% 8.90/5.05 ! [M: int,K: int] :
% 8.90/5.05 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 8.90/5.05 => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3350_Euclidean__Division_Opos__mod__bound,axiom,
% 8.90/5.05 ! [L: int,K: int] :
% 8.90/5.05 ( ( ord_less_int @ zero_zero_int @ L )
% 8.90/5.05 => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3351_neg__mod__bound,axiom,
% 8.90/5.05 ! [L: int,K: int] :
% 8.90/5.05 ( ( ord_less_int @ L @ zero_zero_int )
% 8.90/5.05 => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3352_zmod__eq__0__iff,axiom,
% 8.90/5.05 ! [M: int,D: int] :
% 8.90/5.05 ( ( ( modulo_modulo_int @ M @ D )
% 8.90/5.05 = zero_zero_int )
% 8.90/5.05 = ( ? [Q4: int] :
% 8.90/5.05 ( M
% 8.90/5.05 = ( times_times_int @ D @ Q4 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3353_zmod__eq__0D,axiom,
% 8.90/5.05 ! [M: int,D: int] :
% 8.90/5.05 ( ( ( modulo_modulo_int @ M @ D )
% 8.90/5.05 = zero_zero_int )
% 8.90/5.05 => ? [Q2: int] :
% 8.90/5.05 ( M
% 8.90/5.05 = ( times_times_int @ D @ Q2 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3354_infinite__Icc,axiom,
% 8.90/5.05 ! [A: rat,B: rat] :
% 8.90/5.05 ( ( ord_less_rat @ A @ B )
% 8.90/5.05 => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3355_infinite__Icc,axiom,
% 8.90/5.05 ! [A: real,B: real] :
% 8.90/5.05 ( ( ord_less_real @ A @ B )
% 8.90/5.05 => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3356_finite__lists__length__le,axiom,
% 8.90/5.05 ! [A2: set_complex,N: nat] :
% 8.90/5.05 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.05 => ( finite8712137658972009173omplex
% 8.90/5.05 @ ( collect_list_complex
% 8.90/5.05 @ ^ [Xs: list_complex] :
% 8.90/5.05 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
% 8.90/5.05 & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3357_finite__lists__length__le,axiom,
% 8.90/5.05 ! [A2: set_VEBT_VEBT,N: nat] :
% 8.90/5.05 ( ( finite5795047828879050333T_VEBT @ A2 )
% 8.90/5.05 => ( finite3004134309566078307T_VEBT
% 8.90/5.05 @ ( collec5608196760682091941T_VEBT
% 8.90/5.05 @ ^ [Xs: list_VEBT_VEBT] :
% 8.90/5.05 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
% 8.90/5.05 & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3358_finite__lists__length__le,axiom,
% 8.90/5.05 ! [A2: set_o,N: nat] :
% 8.90/5.05 ( ( finite_finite_o @ A2 )
% 8.90/5.05 => ( finite_finite_list_o
% 8.90/5.05 @ ( collect_list_o
% 8.90/5.05 @ ^ [Xs: list_o] :
% 8.90/5.05 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
% 8.90/5.05 & ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3359_finite__lists__length__le,axiom,
% 8.90/5.05 ! [A2: set_nat,N: nat] :
% 8.90/5.05 ( ( finite_finite_nat @ A2 )
% 8.90/5.05 => ( finite8100373058378681591st_nat
% 8.90/5.05 @ ( collect_list_nat
% 8.90/5.05 @ ^ [Xs: list_nat] :
% 8.90/5.05 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
% 8.90/5.05 & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3360_finite__lists__length__le,axiom,
% 8.90/5.05 ! [A2: set_int,N: nat] :
% 8.90/5.05 ( ( finite_finite_int @ A2 )
% 8.90/5.05 => ( finite3922522038869484883st_int
% 8.90/5.05 @ ( collect_list_int
% 8.90/5.05 @ ^ [Xs: list_int] :
% 8.90/5.05 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
% 8.90/5.05 & ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3361_div__mod__decomp__int,axiom,
% 8.90/5.05 ! [A2: int,N: int] :
% 8.90/5.05 ( A2
% 8.90/5.05 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N ) @ N ) @ ( modulo_modulo_int @ A2 @ N ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3362_neg__mod__conj,axiom,
% 8.90/5.05 ! [B: int,A: int] :
% 8.90/5.05 ( ( ord_less_int @ B @ zero_zero_int )
% 8.90/5.05 => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
% 8.90/5.05 & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3363_pos__mod__conj,axiom,
% 8.90/5.05 ! [B: int,A: int] :
% 8.90/5.05 ( ( ord_less_int @ zero_zero_int @ B )
% 8.90/5.05 => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
% 8.90/5.05 & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3364_zmod__trivial__iff,axiom,
% 8.90/5.05 ! [I2: int,K: int] :
% 8.90/5.05 ( ( ( modulo_modulo_int @ I2 @ K )
% 8.90/5.05 = I2 )
% 8.90/5.05 = ( ( K = zero_zero_int )
% 8.90/5.05 | ( ( ord_less_eq_int @ zero_zero_int @ I2 )
% 8.90/5.05 & ( ord_less_int @ I2 @ K ) )
% 8.90/5.05 | ( ( ord_less_eq_int @ I2 @ zero_zero_int )
% 8.90/5.05 & ( ord_less_int @ K @ I2 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3365_neg__mod__sign,axiom,
% 8.90/5.05 ! [L: int,K: int] :
% 8.90/5.05 ( ( ord_less_int @ L @ zero_zero_int )
% 8.90/5.05 => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3366_Euclidean__Division_Opos__mod__sign,axiom,
% 8.90/5.05 ! [L: int,K: int] :
% 8.90/5.05 ( ( ord_less_int @ zero_zero_int @ L )
% 8.90/5.05 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3367_verit__la__disequality,axiom,
% 8.90/5.05 ! [A: rat,B: rat] :
% 8.90/5.05 ( ( A = B )
% 8.90/5.05 | ~ ( ord_less_eq_rat @ A @ B )
% 8.90/5.05 | ~ ( ord_less_eq_rat @ B @ A ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3368_verit__la__disequality,axiom,
% 8.90/5.05 ! [A: num,B: num] :
% 8.90/5.05 ( ( A = B )
% 8.90/5.05 | ~ ( ord_less_eq_num @ A @ B )
% 8.90/5.05 | ~ ( ord_less_eq_num @ B @ A ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3369_verit__la__disequality,axiom,
% 8.90/5.05 ! [A: nat,B: nat] :
% 8.90/5.05 ( ( A = B )
% 8.90/5.05 | ~ ( ord_less_eq_nat @ A @ B )
% 8.90/5.05 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3370_verit__la__disequality,axiom,
% 8.90/5.05 ! [A: int,B: int] :
% 8.90/5.05 ( ( A = B )
% 8.90/5.05 | ~ ( ord_less_eq_int @ A @ B )
% 8.90/5.05 | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3371_verit__comp__simplify1_I2_J,axiom,
% 8.90/5.05 ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3372_verit__comp__simplify1_I2_J,axiom,
% 8.90/5.05 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3373_verit__comp__simplify1_I2_J,axiom,
% 8.90/5.05 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3374_verit__comp__simplify1_I2_J,axiom,
% 8.90/5.05 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3375_verit__comp__simplify1_I2_J,axiom,
% 8.90/5.05 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3376_verit__comp__simplify1_I1_J,axiom,
% 8.90/5.05 ! [A: real] :
% 8.90/5.05 ~ ( ord_less_real @ A @ A ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3377_verit__comp__simplify1_I1_J,axiom,
% 8.90/5.05 ! [A: rat] :
% 8.90/5.05 ~ ( ord_less_rat @ A @ A ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3378_verit__comp__simplify1_I1_J,axiom,
% 8.90/5.05 ! [A: num] :
% 8.90/5.05 ~ ( ord_less_num @ A @ A ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3379_verit__comp__simplify1_I1_J,axiom,
% 8.90/5.05 ! [A: nat] :
% 8.90/5.05 ~ ( ord_less_nat @ A @ A ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3380_verit__comp__simplify1_I1_J,axiom,
% 8.90/5.05 ! [A: int] :
% 8.90/5.05 ~ ( ord_less_int @ A @ A ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3381_verit__la__generic,axiom,
% 8.90/5.05 ! [A: int,X2: int] :
% 8.90/5.05 ( ( ord_less_eq_int @ A @ X2 )
% 8.90/5.05 | ( A = X2 )
% 8.90/5.05 | ( ord_less_eq_int @ X2 @ A ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3382_subset__eq__atLeast0__atMost__finite,axiom,
% 8.90/5.05 ! [N5: set_nat,N: nat] :
% 8.90/5.05 ( ( ord_less_eq_set_nat @ N5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 8.90/5.05 => ( finite_finite_nat @ N5 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3383_mod__pos__neg__trivial,axiom,
% 8.90/5.05 ! [K: int,L: int] :
% 8.90/5.05 ( ( ord_less_int @ zero_zero_int @ K )
% 8.90/5.05 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 8.90/5.05 => ( ( modulo_modulo_int @ K @ L )
% 8.90/5.05 = ( plus_plus_int @ K @ L ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3384_mod__pos__geq,axiom,
% 8.90/5.05 ! [L: int,K: int] :
% 8.90/5.05 ( ( ord_less_int @ zero_zero_int @ L )
% 8.90/5.05 => ( ( ord_less_eq_int @ L @ K )
% 8.90/5.05 => ( ( modulo_modulo_int @ K @ L )
% 8.90/5.05 = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3385_verit__le__mono__div__int,axiom,
% 8.90/5.05 ! [A2: int,B3: int,N: int] :
% 8.90/5.05 ( ( ord_less_int @ A2 @ B3 )
% 8.90/5.05 => ( ( ord_less_int @ zero_zero_int @ N )
% 8.90/5.05 => ( ord_less_eq_int
% 8.90/5.05 @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N )
% 8.90/5.05 @ ( if_int
% 8.90/5.05 @ ( ( modulo_modulo_int @ B3 @ N )
% 8.90/5.05 = zero_zero_int )
% 8.90/5.05 @ one_one_int
% 8.90/5.05 @ zero_zero_int ) )
% 8.90/5.05 @ ( divide_divide_int @ B3 @ N ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3386_int__mod__pos__eq,axiom,
% 8.90/5.05 ! [A: int,B: int,Q3: int,R2: int] :
% 8.90/5.05 ( ( A
% 8.90/5.05 = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 8.90/5.05 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 8.90/5.05 => ( ( ord_less_int @ R2 @ B )
% 8.90/5.05 => ( ( modulo_modulo_int @ A @ B )
% 8.90/5.05 = R2 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3387_int__mod__neg__eq,axiom,
% 8.90/5.05 ! [A: int,B: int,Q3: int,R2: int] :
% 8.90/5.05 ( ( A
% 8.90/5.05 = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
% 8.90/5.05 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 8.90/5.05 => ( ( ord_less_int @ B @ R2 )
% 8.90/5.05 => ( ( modulo_modulo_int @ A @ B )
% 8.90/5.05 = R2 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3388_split__zmod,axiom,
% 8.90/5.05 ! [P: int > $o,N: int,K: int] :
% 8.90/5.05 ( ( P @ ( modulo_modulo_int @ N @ K ) )
% 8.90/5.05 = ( ( ( K = zero_zero_int )
% 8.90/5.05 => ( P @ N ) )
% 8.90/5.05 & ( ( ord_less_int @ zero_zero_int @ K )
% 8.90/5.05 => ! [I4: int,J3: int] :
% 8.90/5.05 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 8.90/5.05 & ( ord_less_int @ J3 @ K )
% 8.90/5.05 & ( N
% 8.90/5.05 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 8.90/5.05 => ( P @ J3 ) ) )
% 8.90/5.05 & ( ( ord_less_int @ K @ zero_zero_int )
% 8.90/5.05 => ! [I4: int,J3: int] :
% 8.90/5.05 ( ( ( ord_less_int @ K @ J3 )
% 8.90/5.05 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 8.90/5.05 & ( N
% 8.90/5.05 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 8.90/5.05 => ( P @ J3 ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3389_zmod__zmult2__eq,axiom,
% 8.90/5.05 ! [C: int,A: int,B: int] :
% 8.90/5.05 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 8.90/5.05 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 8.90/5.05 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3390_split__pos__lemma,axiom,
% 8.90/5.05 ! [K: int,P: int > int > $o,N: int] :
% 8.90/5.05 ( ( ord_less_int @ zero_zero_int @ K )
% 8.90/5.05 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 8.90/5.05 = ( ! [I4: int,J3: int] :
% 8.90/5.05 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 8.90/5.05 & ( ord_less_int @ J3 @ K )
% 8.90/5.05 & ( N
% 8.90/5.05 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 8.90/5.05 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3391_split__neg__lemma,axiom,
% 8.90/5.05 ! [K: int,P: int > int > $o,N: int] :
% 8.90/5.05 ( ( ord_less_int @ K @ zero_zero_int )
% 8.90/5.05 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 8.90/5.05 = ( ! [I4: int,J3: int] :
% 8.90/5.05 ( ( ( ord_less_int @ K @ J3 )
% 8.90/5.05 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 8.90/5.05 & ( N
% 8.90/5.05 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 8.90/5.05 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3392_verit__comp__simplify1_I3_J,axiom,
% 8.90/5.05 ! [B5: real,A5: real] :
% 8.90/5.05 ( ( ~ ( ord_less_eq_real @ B5 @ A5 ) )
% 8.90/5.05 = ( ord_less_real @ A5 @ B5 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3393_verit__comp__simplify1_I3_J,axiom,
% 8.90/5.05 ! [B5: rat,A5: rat] :
% 8.90/5.05 ( ( ~ ( ord_less_eq_rat @ B5 @ A5 ) )
% 8.90/5.05 = ( ord_less_rat @ A5 @ B5 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3394_verit__comp__simplify1_I3_J,axiom,
% 8.90/5.05 ! [B5: num,A5: num] :
% 8.90/5.05 ( ( ~ ( ord_less_eq_num @ B5 @ A5 ) )
% 8.90/5.05 = ( ord_less_num @ A5 @ B5 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3395_verit__comp__simplify1_I3_J,axiom,
% 8.90/5.05 ! [B5: nat,A5: nat] :
% 8.90/5.05 ( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
% 8.90/5.05 = ( ord_less_nat @ A5 @ B5 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3396_verit__comp__simplify1_I3_J,axiom,
% 8.90/5.05 ! [B5: int,A5: int] :
% 8.90/5.05 ( ( ~ ( ord_less_eq_int @ B5 @ A5 ) )
% 8.90/5.05 = ( ord_less_int @ A5 @ B5 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3397_verit__sum__simplify,axiom,
% 8.90/5.05 ! [A: complex] :
% 8.90/5.05 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 8.90/5.05 = A ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3398_verit__sum__simplify,axiom,
% 8.90/5.05 ! [A: real] :
% 8.90/5.05 ( ( plus_plus_real @ A @ zero_zero_real )
% 8.90/5.05 = A ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3399_verit__sum__simplify,axiom,
% 8.90/5.05 ! [A: rat] :
% 8.90/5.05 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 8.90/5.05 = A ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3400_verit__sum__simplify,axiom,
% 8.90/5.05 ! [A: nat] :
% 8.90/5.05 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 8.90/5.05 = A ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3401_verit__sum__simplify,axiom,
% 8.90/5.05 ! [A: int] :
% 8.90/5.05 ( ( plus_plus_int @ A @ zero_zero_int )
% 8.90/5.05 = A ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3402_verit__eq__simplify_I10_J,axiom,
% 8.90/5.05 ! [X22: num] :
% 8.90/5.05 ( one
% 8.90/5.05 != ( bit0 @ X22 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3403_pos__zmod__mult__2,axiom,
% 8.90/5.05 ! [A: int,B: int] :
% 8.90/5.05 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.90/5.05 => ( ( 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 ) )
% 8.90/5.05 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3404_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
% 8.90/5.05 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
% 8.90/5.05 ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X2 )
% 8.90/5.05 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3405_neg__zmod__mult__2,axiom,
% 8.90/5.05 ! [A: int,B: int] :
% 8.90/5.05 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.90/5.05 => ( ( 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 ) )
% 8.90/5.05 = ( 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 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3406_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
% 8.90/5.05 ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
% 8.90/5.05 ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X2 )
% 8.90/5.05 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3407_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
% 8.90/5.05 ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
% 8.90/5.05 ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X2 )
% 8.90/5.05 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3408_finite__Collect__le__nat,axiom,
% 8.90/5.05 ! [K: nat] :
% 8.90/5.05 ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3409_finite__Collect__less__nat,axiom,
% 8.90/5.05 ! [K: nat] :
% 8.90/5.05 ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3410_finite__Collect__subsets,axiom,
% 8.90/5.05 ! [A2: set_nat] :
% 8.90/5.05 ( ( finite_finite_nat @ A2 )
% 8.90/5.05 => ( finite1152437895449049373et_nat
% 8.90/5.05 @ ( collect_set_nat
% 8.90/5.05 @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A2 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3411_finite__Collect__subsets,axiom,
% 8.90/5.05 ! [A2: set_complex] :
% 8.90/5.05 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.05 => ( finite6551019134538273531omplex
% 8.90/5.05 @ ( collect_set_complex
% 8.90/5.05 @ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A2 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3412_finite__Collect__subsets,axiom,
% 8.90/5.05 ! [A2: set_int] :
% 8.90/5.05 ( ( finite_finite_int @ A2 )
% 8.90/5.05 => ( finite6197958912794628473et_int
% 8.90/5.05 @ ( collect_set_int
% 8.90/5.05 @ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A2 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3413_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
% 8.90/5.05 ! [X2: vEBT_VEBT,Xa2: nat,Y3: nat] :
% 8.90/5.05 ( ( ( vEBT_T_m_e_m_b_e_r @ X2 @ Xa2 )
% 8.90/5.05 = Y3 )
% 8.90/5.05 => ( ( ? [A4: $o,B4: $o] :
% 8.90/5.05 ( X2
% 8.90/5.05 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.90/5.05 => ( Y3
% 8.90/5.05 != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
% 8.90/5.05 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.90/5.05 ( X2
% 8.90/5.05 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 8.90/5.05 => ( Y3
% 8.90/5.05 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.05 => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 8.90/5.05 ( X2
% 8.90/5.05 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 8.90/5.05 => ( Y3
% 8.90/5.05 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.05 => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.90/5.05 ( X2
% 8.90/5.05 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 8.90/5.05 => ( Y3
% 8.90/5.05 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.05 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
% 8.90/5.05 ( ? [Summary2: vEBT_VEBT] :
% 8.90/5.05 ( X2
% 8.90/5.05 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.90/5.05 => ( Y3
% 8.90/5.05 != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa2 = Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa2 = Ma2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa2 ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3414_finite__roots__unity,axiom,
% 8.90/5.05 ! [N: nat] :
% 8.90/5.05 ( ( ord_less_eq_nat @ one_one_nat @ N )
% 8.90/5.05 => ( finite_finite_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [Z3: real] :
% 8.90/5.05 ( ( power_power_real @ Z3 @ N )
% 8.90/5.05 = one_one_real ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3415_finite__roots__unity,axiom,
% 8.90/5.05 ! [N: nat] :
% 8.90/5.05 ( ( ord_less_eq_nat @ one_one_nat @ N )
% 8.90/5.05 => ( finite3207457112153483333omplex
% 8.90/5.05 @ ( collect_complex
% 8.90/5.05 @ ^ [Z3: complex] :
% 8.90/5.05 ( ( power_power_complex @ Z3 @ N )
% 8.90/5.05 = one_one_complex ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3416_finite__Diff,axiom,
% 8.90/5.05 ! [A2: set_int,B3: set_int] :
% 8.90/5.05 ( ( finite_finite_int @ A2 )
% 8.90/5.05 => ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3417_finite__Diff,axiom,
% 8.90/5.05 ! [A2: set_complex,B3: set_complex] :
% 8.90/5.05 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.05 => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3418_finite__Diff,axiom,
% 8.90/5.05 ! [A2: set_nat,B3: set_nat] :
% 8.90/5.05 ( ( finite_finite_nat @ A2 )
% 8.90/5.05 => ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3419_finite__Diff2,axiom,
% 8.90/5.05 ! [B3: set_int,A2: set_int] :
% 8.90/5.05 ( ( finite_finite_int @ B3 )
% 8.90/5.05 => ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B3 ) )
% 8.90/5.05 = ( finite_finite_int @ A2 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3420_finite__Diff2,axiom,
% 8.90/5.05 ! [B3: set_complex,A2: set_complex] :
% 8.90/5.05 ( ( finite3207457112153483333omplex @ B3 )
% 8.90/5.05 => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 8.90/5.05 = ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3421_finite__Diff2,axiom,
% 8.90/5.05 ! [B3: set_nat,A2: set_nat] :
% 8.90/5.05 ( ( finite_finite_nat @ B3 )
% 8.90/5.05 => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 8.90/5.05 = ( finite_finite_nat @ A2 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3422_semiring__norm_I90_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( ( bit1 @ M )
% 8.90/5.05 = ( bit1 @ N ) )
% 8.90/5.05 = ( M = N ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3423_finite__interval__int1,axiom,
% 8.90/5.05 ! [A: int,B: int] :
% 8.90/5.05 ( finite_finite_int
% 8.90/5.05 @ ( collect_int
% 8.90/5.05 @ ^ [I4: int] :
% 8.90/5.05 ( ( ord_less_eq_int @ A @ I4 )
% 8.90/5.05 & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3424_semiring__norm_I88_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( bit0 @ M )
% 8.90/5.05 != ( bit1 @ N ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3425_semiring__norm_I89_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( bit1 @ M )
% 8.90/5.05 != ( bit0 @ N ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3426_semiring__norm_I84_J,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( one
% 8.90/5.05 != ( bit1 @ N ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3427_semiring__norm_I86_J,axiom,
% 8.90/5.05 ! [M: num] :
% 8.90/5.05 ( ( bit1 @ M )
% 8.90/5.05 != one ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3428_semiring__norm_I80_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( ord_less_num @ M @ N ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3429_semiring__norm_I73_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( ord_less_eq_num @ M @ N ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3430_finite__interval__int3,axiom,
% 8.90/5.05 ! [A: int,B: int] :
% 8.90/5.05 ( finite_finite_int
% 8.90/5.05 @ ( collect_int
% 8.90/5.05 @ ^ [I4: int] :
% 8.90/5.05 ( ( ord_less_int @ A @ I4 )
% 8.90/5.05 & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3431_finite__interval__int2,axiom,
% 8.90/5.05 ! [A: int,B: int] :
% 8.90/5.05 ( finite_finite_int
% 8.90/5.05 @ ( collect_int
% 8.90/5.05 @ ^ [I4: int] :
% 8.90/5.05 ( ( ord_less_eq_int @ A @ I4 )
% 8.90/5.05 & ( ord_less_int @ I4 @ B ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3432_semiring__norm_I7_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3433_semiring__norm_I9_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.90/5.05 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3434_semiring__norm_I15_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.90/5.05 = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3435_semiring__norm_I14_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3436_semiring__norm_I81_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.90/5.05 = ( ord_less_num @ M @ N ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3437_semiring__norm_I72_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( ord_less_eq_num @ M @ N ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3438_semiring__norm_I77_J,axiom,
% 8.90/5.05 ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3439_semiring__norm_I70_J,axiom,
% 8.90/5.05 ! [M: num] :
% 8.90/5.05 ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3440_zdiv__numeral__Bit1,axiom,
% 8.90/5.05 ! [V: num,W: num] :
% 8.90/5.05 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 8.90/5.05 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3441_semiring__norm_I3_J,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( plus_plus_num @ one @ ( bit0 @ N ) )
% 8.90/5.05 = ( bit1 @ N ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3442_semiring__norm_I4_J,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( plus_plus_num @ one @ ( bit1 @ N ) )
% 8.90/5.05 = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3443_semiring__norm_I5_J,axiom,
% 8.90/5.05 ! [M: num] :
% 8.90/5.05 ( ( plus_plus_num @ ( bit0 @ M ) @ one )
% 8.90/5.05 = ( bit1 @ M ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3444_semiring__norm_I8_J,axiom,
% 8.90/5.05 ! [M: num] :
% 8.90/5.05 ( ( plus_plus_num @ ( bit1 @ M ) @ one )
% 8.90/5.05 = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3445_semiring__norm_I10_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3446_semiring__norm_I16_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3447_semiring__norm_I79_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( ord_less_eq_num @ M @ N ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3448_semiring__norm_I74_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.90/5.05 = ( ord_less_num @ M @ N ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3449_Suc__div__eq__add3__div__numeral,axiom,
% 8.90/5.05 ! [M: nat,V: num] :
% 8.90/5.05 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 8.90/5.05 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3450_div__Suc__eq__div__add3,axiom,
% 8.90/5.05 ! [M: nat,N: nat] :
% 8.90/5.05 ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 8.90/5.05 = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3451_Suc__mod__eq__add3__mod__numeral,axiom,
% 8.90/5.05 ! [M: nat,V: num] :
% 8.90/5.05 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 8.90/5.05 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3452_mod__Suc__eq__mod__add3,axiom,
% 8.90/5.05 ! [M: nat,N: nat] :
% 8.90/5.05 ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 8.90/5.05 = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3453_zmod__numeral__Bit1,axiom,
% 8.90/5.05 ! [V: num,W: num] :
% 8.90/5.05 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 8.90/5.05 = ( 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 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3454_verit__eq__simplify_I14_J,axiom,
% 8.90/5.05 ! [X22: num,X32: num] :
% 8.90/5.05 ( ( bit0 @ X22 )
% 8.90/5.05 != ( bit1 @ X32 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3455_verit__eq__simplify_I12_J,axiom,
% 8.90/5.05 ! [X32: num] :
% 8.90/5.05 ( one
% 8.90/5.05 != ( bit1 @ X32 ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3456_xor__num_Ocases,axiom,
% 8.90/5.05 ! [X2: product_prod_num_num] :
% 8.90/5.05 ( ( X2
% 8.90/5.05 != ( product_Pair_num_num @ one @ one ) )
% 8.90/5.05 => ( ! [N2: num] :
% 8.90/5.05 ( X2
% 8.90/5.05 != ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) )
% 8.90/5.05 => ( ! [N2: num] :
% 8.90/5.05 ( X2
% 8.90/5.05 != ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) )
% 8.90/5.05 => ( ! [M4: num] :
% 8.90/5.05 ( X2
% 8.90/5.05 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) )
% 8.90/5.05 => ( ! [M4: num,N2: num] :
% 8.90/5.05 ( X2
% 8.90/5.05 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) )
% 8.90/5.05 => ( ! [M4: num,N2: num] :
% 8.90/5.05 ( X2
% 8.90/5.05 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) )
% 8.90/5.05 => ( ! [M4: num] :
% 8.90/5.05 ( X2
% 8.90/5.05 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) )
% 8.90/5.05 => ( ! [M4: num,N2: num] :
% 8.90/5.05 ( X2
% 8.90/5.05 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) )
% 8.90/5.05 => ~ ! [M4: num,N2: num] :
% 8.90/5.05 ( X2
% 8.90/5.05 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3457_num_Oexhaust,axiom,
% 8.90/5.05 ! [Y3: num] :
% 8.90/5.05 ( ( Y3 != one )
% 8.90/5.05 => ( ! [X23: num] :
% 8.90/5.05 ( Y3
% 8.90/5.05 != ( bit0 @ X23 ) )
% 8.90/5.05 => ~ ! [X33: num] :
% 8.90/5.05 ( Y3
% 8.90/5.05 != ( bit1 @ X33 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3458_finite__maxlen,axiom,
% 8.90/5.05 ! [M7: set_list_VEBT_VEBT] :
% 8.90/5.05 ( ( finite3004134309566078307T_VEBT @ M7 )
% 8.90/5.05 => ? [N2: nat] :
% 8.90/5.05 ! [X3: list_VEBT_VEBT] :
% 8.90/5.05 ( ( member2936631157270082147T_VEBT @ X3 @ M7 )
% 8.90/5.05 => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X3 ) @ N2 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3459_finite__maxlen,axiom,
% 8.90/5.05 ! [M7: set_list_o] :
% 8.90/5.05 ( ( finite_finite_list_o @ M7 )
% 8.90/5.05 => ? [N2: nat] :
% 8.90/5.05 ! [X3: list_o] :
% 8.90/5.05 ( ( member_list_o @ X3 @ M7 )
% 8.90/5.05 => ( ord_less_nat @ ( size_size_list_o @ X3 ) @ N2 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3460_finite__maxlen,axiom,
% 8.90/5.05 ! [M7: set_list_nat] :
% 8.90/5.05 ( ( finite8100373058378681591st_nat @ M7 )
% 8.90/5.05 => ? [N2: nat] :
% 8.90/5.05 ! [X3: list_nat] :
% 8.90/5.05 ( ( member_list_nat @ X3 @ M7 )
% 8.90/5.05 => ( ord_less_nat @ ( size_size_list_nat @ X3 ) @ N2 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3461_finite__maxlen,axiom,
% 8.90/5.05 ! [M7: set_list_int] :
% 8.90/5.05 ( ( finite3922522038869484883st_int @ M7 )
% 8.90/5.05 => ? [N2: nat] :
% 8.90/5.05 ! [X3: list_int] :
% 8.90/5.05 ( ( member_list_int @ X3 @ M7 )
% 8.90/5.05 => ( ord_less_nat @ ( size_size_list_int @ X3 ) @ N2 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3462_numeral__Bit1,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 8.90/5.05 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3463_numeral__Bit1,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 8.90/5.05 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3464_numeral__Bit1,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 8.90/5.05 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3465_numeral__Bit1,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 8.90/5.05 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3466_numeral__Bit1,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 8.90/5.05 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3467_eval__nat__numeral_I3_J,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 8.90/5.05 = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3468_cong__exp__iff__simps_I10_J,axiom,
% 8.90/5.05 ! [M: num,Q3: num,N: num] :
% 8.90/5.05 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 8.90/5.05 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3469_cong__exp__iff__simps_I10_J,axiom,
% 8.90/5.05 ! [M: num,Q3: num,N: num] :
% 8.90/5.05 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 8.90/5.05 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3470_cong__exp__iff__simps_I10_J,axiom,
% 8.90/5.05 ! [M: num,Q3: num,N: num] :
% 8.90/5.05 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 8.90/5.05 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3471_cong__exp__iff__simps_I12_J,axiom,
% 8.90/5.05 ! [M: num,Q3: num,N: num] :
% 8.90/5.05 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 8.90/5.05 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3472_cong__exp__iff__simps_I12_J,axiom,
% 8.90/5.05 ! [M: num,Q3: num,N: num] :
% 8.90/5.05 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 8.90/5.05 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3473_cong__exp__iff__simps_I12_J,axiom,
% 8.90/5.05 ! [M: num,Q3: num,N: num] :
% 8.90/5.05 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 8.90/5.05 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3474_cong__exp__iff__simps_I13_J,axiom,
% 8.90/5.05 ! [M: num,Q3: num,N: num] :
% 8.90/5.05 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 8.90/5.05 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 8.90/5.05 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
% 8.90/5.05 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3475_cong__exp__iff__simps_I13_J,axiom,
% 8.90/5.05 ! [M: num,Q3: num,N: num] :
% 8.90/5.05 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 8.90/5.05 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 8.90/5.05 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
% 8.90/5.05 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3476_cong__exp__iff__simps_I13_J,axiom,
% 8.90/5.05 ! [M: num,Q3: num,N: num] :
% 8.90/5.05 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 8.90/5.05 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 8.90/5.05 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 8.90/5.05 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3477_numeral__code_I3_J,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 8.90/5.05 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3478_numeral__code_I3_J,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 8.90/5.05 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3479_numeral__code_I3_J,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 8.90/5.05 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3480_numeral__code_I3_J,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 8.90/5.05 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3481_numeral__code_I3_J,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 8.90/5.05 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3482_power__numeral__odd,axiom,
% 8.90/5.05 ! [Z: complex,W: num] :
% 8.90/5.05 ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 8.90/5.05 = ( times_times_complex @ ( times_times_complex @ Z @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3483_power__numeral__odd,axiom,
% 8.90/5.05 ! [Z: real,W: num] :
% 8.90/5.05 ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 8.90/5.05 = ( times_times_real @ ( times_times_real @ Z @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3484_power__numeral__odd,axiom,
% 8.90/5.05 ! [Z: rat,W: num] :
% 8.90/5.05 ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 8.90/5.05 = ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3485_power__numeral__odd,axiom,
% 8.90/5.05 ! [Z: nat,W: num] :
% 8.90/5.05 ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 8.90/5.05 = ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3486_power__numeral__odd,axiom,
% 8.90/5.05 ! [Z: int,W: num] :
% 8.90/5.05 ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 8.90/5.05 = ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3487_numeral__Bit1__div__2,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.05 = ( numeral_numeral_nat @ N ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3488_numeral__Bit1__div__2,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.05 = ( numeral_numeral_int @ N ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3489_cong__exp__iff__simps_I3_J,axiom,
% 8.90/5.05 ! [N: num,Q3: num] :
% 8.90/5.05 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 8.90/5.05 != zero_zero_nat ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3490_cong__exp__iff__simps_I3_J,axiom,
% 8.90/5.05 ! [N: num,Q3: num] :
% 8.90/5.05 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 8.90/5.05 != zero_zero_int ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3491_cong__exp__iff__simps_I3_J,axiom,
% 8.90/5.05 ! [N: num,Q3: num] :
% 8.90/5.05 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 8.90/5.05 != zero_z3403309356797280102nteger ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3492_power3__eq__cube,axiom,
% 8.90/5.05 ! [A: complex] :
% 8.90/5.05 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 8.90/5.05 = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3493_power3__eq__cube,axiom,
% 8.90/5.05 ! [A: real] :
% 8.90/5.05 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 8.90/5.05 = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3494_power3__eq__cube,axiom,
% 8.90/5.05 ! [A: rat] :
% 8.90/5.05 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 8.90/5.05 = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3495_power3__eq__cube,axiom,
% 8.90/5.05 ! [A: nat] :
% 8.90/5.05 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 8.90/5.05 = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3496_power3__eq__cube,axiom,
% 8.90/5.05 ! [A: int] :
% 8.90/5.05 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 8.90/5.05 = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3497_numeral__3__eq__3,axiom,
% 8.90/5.05 ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 8.90/5.05 = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3498_Suc3__eq__add__3,axiom,
% 8.90/5.05 ! [N: nat] :
% 8.90/5.05 ( ( suc @ ( suc @ ( suc @ N ) ) )
% 8.90/5.05 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3499_maxt__bound,axiom,
% 8.90/5.05 ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_a_x_t @ T ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3500_mint__bound,axiom,
% 8.90/5.05 ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_i_n_t @ T ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3501_num_Osize_I6_J,axiom,
% 8.90/5.05 ! [X32: num] :
% 8.90/5.05 ( ( size_size_num @ ( bit1 @ X32 ) )
% 8.90/5.05 = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3502_cong__exp__iff__simps_I7_J,axiom,
% 8.90/5.05 ! [Q3: num,N: num] :
% 8.90/5.05 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 8.90/5.05 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 8.90/5.05 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
% 8.90/5.05 = zero_zero_nat ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3503_cong__exp__iff__simps_I7_J,axiom,
% 8.90/5.05 ! [Q3: num,N: num] :
% 8.90/5.05 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 8.90/5.05 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 8.90/5.05 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
% 8.90/5.05 = zero_zero_int ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3504_cong__exp__iff__simps_I7_J,axiom,
% 8.90/5.05 ! [Q3: num,N: num] :
% 8.90/5.05 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 8.90/5.05 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 8.90/5.05 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 8.90/5.05 = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3505_cong__exp__iff__simps_I11_J,axiom,
% 8.90/5.05 ! [M: num,Q3: num] :
% 8.90/5.05 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 8.90/5.05 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 8.90/5.05 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
% 8.90/5.05 = zero_zero_nat ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3506_cong__exp__iff__simps_I11_J,axiom,
% 8.90/5.05 ! [M: num,Q3: num] :
% 8.90/5.05 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 8.90/5.05 = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 8.90/5.05 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
% 8.90/5.05 = zero_zero_int ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3507_cong__exp__iff__simps_I11_J,axiom,
% 8.90/5.05 ! [M: num,Q3: num] :
% 8.90/5.05 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 8.90/5.05 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 8.90/5.05 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 8.90/5.05 = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3508_Suc__div__eq__add3__div,axiom,
% 8.90/5.05 ! [M: nat,N: nat] :
% 8.90/5.05 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 8.90/5.05 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3509_Suc__mod__eq__add3__mod,axiom,
% 8.90/5.05 ! [M: nat,N: nat] :
% 8.90/5.05 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 8.90/5.05 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3510_finite__has__maximal2,axiom,
% 8.90/5.05 ! [A2: set_real,A: real] :
% 8.90/5.05 ( ( finite_finite_real @ A2 )
% 8.90/5.05 => ( ( member_real @ A @ A2 )
% 8.90/5.05 => ? [X5: real] :
% 8.90/5.05 ( ( member_real @ X5 @ A2 )
% 8.90/5.05 & ( ord_less_eq_real @ A @ X5 )
% 8.90/5.05 & ! [Xa: real] :
% 8.90/5.05 ( ( member_real @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_real @ X5 @ Xa )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3511_finite__has__maximal2,axiom,
% 8.90/5.05 ! [A2: set_set_int,A: set_int] :
% 8.90/5.05 ( ( finite6197958912794628473et_int @ A2 )
% 8.90/5.05 => ( ( member_set_int @ A @ A2 )
% 8.90/5.05 => ? [X5: set_int] :
% 8.90/5.05 ( ( member_set_int @ X5 @ A2 )
% 8.90/5.05 & ( ord_less_eq_set_int @ A @ X5 )
% 8.90/5.05 & ! [Xa: set_int] :
% 8.90/5.05 ( ( member_set_int @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_set_int @ X5 @ Xa )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3512_finite__has__maximal2,axiom,
% 8.90/5.05 ! [A2: set_rat,A: rat] :
% 8.90/5.05 ( ( finite_finite_rat @ A2 )
% 8.90/5.05 => ( ( member_rat @ A @ A2 )
% 8.90/5.05 => ? [X5: rat] :
% 8.90/5.05 ( ( member_rat @ X5 @ A2 )
% 8.90/5.05 & ( ord_less_eq_rat @ A @ X5 )
% 8.90/5.05 & ! [Xa: rat] :
% 8.90/5.05 ( ( member_rat @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_rat @ X5 @ Xa )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3513_finite__has__maximal2,axiom,
% 8.90/5.05 ! [A2: set_num,A: num] :
% 8.90/5.05 ( ( finite_finite_num @ A2 )
% 8.90/5.05 => ( ( member_num @ A @ A2 )
% 8.90/5.05 => ? [X5: num] :
% 8.90/5.05 ( ( member_num @ X5 @ A2 )
% 8.90/5.05 & ( ord_less_eq_num @ A @ X5 )
% 8.90/5.05 & ! [Xa: num] :
% 8.90/5.05 ( ( member_num @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_num @ X5 @ Xa )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3514_finite__has__maximal2,axiom,
% 8.90/5.05 ! [A2: set_nat,A: nat] :
% 8.90/5.05 ( ( finite_finite_nat @ A2 )
% 8.90/5.05 => ( ( member_nat @ A @ A2 )
% 8.90/5.05 => ? [X5: nat] :
% 8.90/5.05 ( ( member_nat @ X5 @ A2 )
% 8.90/5.05 & ( ord_less_eq_nat @ A @ X5 )
% 8.90/5.05 & ! [Xa: nat] :
% 8.90/5.05 ( ( member_nat @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_nat @ X5 @ Xa )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3515_finite__has__maximal2,axiom,
% 8.90/5.05 ! [A2: set_int,A: int] :
% 8.90/5.05 ( ( finite_finite_int @ A2 )
% 8.90/5.05 => ( ( member_int @ A @ A2 )
% 8.90/5.05 => ? [X5: int] :
% 8.90/5.05 ( ( member_int @ X5 @ A2 )
% 8.90/5.05 & ( ord_less_eq_int @ A @ X5 )
% 8.90/5.05 & ! [Xa: int] :
% 8.90/5.05 ( ( member_int @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_int @ X5 @ Xa )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3516_finite__has__minimal2,axiom,
% 8.90/5.05 ! [A2: set_real,A: real] :
% 8.90/5.05 ( ( finite_finite_real @ A2 )
% 8.90/5.05 => ( ( member_real @ A @ A2 )
% 8.90/5.05 => ? [X5: real] :
% 8.90/5.05 ( ( member_real @ X5 @ A2 )
% 8.90/5.05 & ( ord_less_eq_real @ X5 @ A )
% 8.90/5.05 & ! [Xa: real] :
% 8.90/5.05 ( ( member_real @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_real @ Xa @ X5 )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3517_finite__has__minimal2,axiom,
% 8.90/5.05 ! [A2: set_set_int,A: set_int] :
% 8.90/5.05 ( ( finite6197958912794628473et_int @ A2 )
% 8.90/5.05 => ( ( member_set_int @ A @ A2 )
% 8.90/5.05 => ? [X5: set_int] :
% 8.90/5.05 ( ( member_set_int @ X5 @ A2 )
% 8.90/5.05 & ( ord_less_eq_set_int @ X5 @ A )
% 8.90/5.05 & ! [Xa: set_int] :
% 8.90/5.05 ( ( member_set_int @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_set_int @ Xa @ X5 )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3518_finite__has__minimal2,axiom,
% 8.90/5.05 ! [A2: set_rat,A: rat] :
% 8.90/5.05 ( ( finite_finite_rat @ A2 )
% 8.90/5.05 => ( ( member_rat @ A @ A2 )
% 8.90/5.05 => ? [X5: rat] :
% 8.90/5.05 ( ( member_rat @ X5 @ A2 )
% 8.90/5.05 & ( ord_less_eq_rat @ X5 @ A )
% 8.90/5.05 & ! [Xa: rat] :
% 8.90/5.05 ( ( member_rat @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_rat @ Xa @ X5 )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3519_finite__has__minimal2,axiom,
% 8.90/5.05 ! [A2: set_num,A: num] :
% 8.90/5.05 ( ( finite_finite_num @ A2 )
% 8.90/5.05 => ( ( member_num @ A @ A2 )
% 8.90/5.05 => ? [X5: num] :
% 8.90/5.05 ( ( member_num @ X5 @ A2 )
% 8.90/5.05 & ( ord_less_eq_num @ X5 @ A )
% 8.90/5.05 & ! [Xa: num] :
% 8.90/5.05 ( ( member_num @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_num @ Xa @ X5 )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3520_finite__has__minimal2,axiom,
% 8.90/5.05 ! [A2: set_nat,A: nat] :
% 8.90/5.05 ( ( finite_finite_nat @ A2 )
% 8.90/5.05 => ( ( member_nat @ A @ A2 )
% 8.90/5.05 => ? [X5: nat] :
% 8.90/5.05 ( ( member_nat @ X5 @ A2 )
% 8.90/5.05 & ( ord_less_eq_nat @ X5 @ A )
% 8.90/5.05 & ! [Xa: nat] :
% 8.90/5.05 ( ( member_nat @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_nat @ Xa @ X5 )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3521_finite__has__minimal2,axiom,
% 8.90/5.05 ! [A2: set_int,A: int] :
% 8.90/5.05 ( ( finite_finite_int @ A2 )
% 8.90/5.05 => ( ( member_int @ A @ A2 )
% 8.90/5.05 => ? [X5: int] :
% 8.90/5.05 ( ( member_int @ X5 @ A2 )
% 8.90/5.05 & ( ord_less_eq_int @ X5 @ A )
% 8.90/5.05 & ! [Xa: int] :
% 8.90/5.05 ( ( member_int @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_int @ Xa @ X5 )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3522_finite__subset,axiom,
% 8.90/5.05 ! [A2: set_nat,B3: set_nat] :
% 8.90/5.05 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 8.90/5.05 => ( ( finite_finite_nat @ B3 )
% 8.90/5.05 => ( finite_finite_nat @ A2 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3523_finite__subset,axiom,
% 8.90/5.05 ! [A2: set_complex,B3: set_complex] :
% 8.90/5.05 ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 8.90/5.05 => ( ( finite3207457112153483333omplex @ B3 )
% 8.90/5.05 => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3524_finite__subset,axiom,
% 8.90/5.05 ! [A2: set_int,B3: set_int] :
% 8.90/5.05 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 8.90/5.05 => ( ( finite_finite_int @ B3 )
% 8.90/5.05 => ( finite_finite_int @ A2 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3525_infinite__super,axiom,
% 8.90/5.05 ! [S3: set_nat,T3: set_nat] :
% 8.90/5.05 ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 8.90/5.05 => ( ~ ( finite_finite_nat @ S3 )
% 8.90/5.05 => ~ ( finite_finite_nat @ T3 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3526_infinite__super,axiom,
% 8.90/5.05 ! [S3: set_complex,T3: set_complex] :
% 8.90/5.05 ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 8.90/5.05 => ( ~ ( finite3207457112153483333omplex @ S3 )
% 8.90/5.05 => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3527_infinite__super,axiom,
% 8.90/5.05 ! [S3: set_int,T3: set_int] :
% 8.90/5.05 ( ( ord_less_eq_set_int @ S3 @ T3 )
% 8.90/5.05 => ( ~ ( finite_finite_int @ S3 )
% 8.90/5.05 => ~ ( finite_finite_int @ T3 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3528_rev__finite__subset,axiom,
% 8.90/5.05 ! [B3: set_nat,A2: set_nat] :
% 8.90/5.05 ( ( finite_finite_nat @ B3 )
% 8.90/5.05 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 8.90/5.05 => ( finite_finite_nat @ A2 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3529_rev__finite__subset,axiom,
% 8.90/5.05 ! [B3: set_complex,A2: set_complex] :
% 8.90/5.05 ( ( finite3207457112153483333omplex @ B3 )
% 8.90/5.05 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 8.90/5.05 => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3530_rev__finite__subset,axiom,
% 8.90/5.05 ! [B3: set_int,A2: set_int] :
% 8.90/5.05 ( ( finite_finite_int @ B3 )
% 8.90/5.05 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 8.90/5.05 => ( finite_finite_int @ A2 ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3531_Diff__infinite__finite,axiom,
% 8.90/5.05 ! [T3: set_int,S3: set_int] :
% 8.90/5.05 ( ( finite_finite_int @ T3 )
% 8.90/5.05 => ( ~ ( finite_finite_int @ S3 )
% 8.90/5.05 => ~ ( finite_finite_int @ ( minus_minus_set_int @ S3 @ T3 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3532_Diff__infinite__finite,axiom,
% 8.90/5.05 ! [T3: set_complex,S3: set_complex] :
% 8.90/5.05 ( ( finite3207457112153483333omplex @ T3 )
% 8.90/5.05 => ( ~ ( finite3207457112153483333omplex @ S3 )
% 8.90/5.05 => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S3 @ T3 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3533_Diff__infinite__finite,axiom,
% 8.90/5.05 ! [T3: set_nat,S3: set_nat] :
% 8.90/5.05 ( ( finite_finite_nat @ T3 )
% 8.90/5.05 => ( ~ ( finite_finite_nat @ S3 )
% 8.90/5.05 => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ T3 ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3534_member__bound__height,axiom,
% 8.90/5.05 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 8.90/5.05 ( ( vEBT_invar_vebt @ T @ N )
% 8.90/5.05 => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r @ T @ X2 ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3535_finite__has__maximal,axiom,
% 8.90/5.05 ! [A2: set_real] :
% 8.90/5.05 ( ( finite_finite_real @ A2 )
% 8.90/5.05 => ( ( A2 != bot_bot_set_real )
% 8.90/5.05 => ? [X5: real] :
% 8.90/5.05 ( ( member_real @ X5 @ A2 )
% 8.90/5.05 & ! [Xa: real] :
% 8.90/5.05 ( ( member_real @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_real @ X5 @ Xa )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3536_finite__has__maximal,axiom,
% 8.90/5.05 ! [A2: set_set_int] :
% 8.90/5.05 ( ( finite6197958912794628473et_int @ A2 )
% 8.90/5.05 => ( ( A2 != bot_bot_set_set_int )
% 8.90/5.05 => ? [X5: set_int] :
% 8.90/5.05 ( ( member_set_int @ X5 @ A2 )
% 8.90/5.05 & ! [Xa: set_int] :
% 8.90/5.05 ( ( member_set_int @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_set_int @ X5 @ Xa )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3537_finite__has__maximal,axiom,
% 8.90/5.05 ! [A2: set_rat] :
% 8.90/5.05 ( ( finite_finite_rat @ A2 )
% 8.90/5.05 => ( ( A2 != bot_bot_set_rat )
% 8.90/5.05 => ? [X5: rat] :
% 8.90/5.05 ( ( member_rat @ X5 @ A2 )
% 8.90/5.05 & ! [Xa: rat] :
% 8.90/5.05 ( ( member_rat @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_rat @ X5 @ Xa )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3538_finite__has__maximal,axiom,
% 8.90/5.05 ! [A2: set_num] :
% 8.90/5.05 ( ( finite_finite_num @ A2 )
% 8.90/5.05 => ( ( A2 != bot_bot_set_num )
% 8.90/5.05 => ? [X5: num] :
% 8.90/5.05 ( ( member_num @ X5 @ A2 )
% 8.90/5.05 & ! [Xa: num] :
% 8.90/5.05 ( ( member_num @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_num @ X5 @ Xa )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3539_finite__has__maximal,axiom,
% 8.90/5.05 ! [A2: set_nat] :
% 8.90/5.05 ( ( finite_finite_nat @ A2 )
% 8.90/5.05 => ( ( A2 != bot_bot_set_nat )
% 8.90/5.05 => ? [X5: nat] :
% 8.90/5.05 ( ( member_nat @ X5 @ A2 )
% 8.90/5.05 & ! [Xa: nat] :
% 8.90/5.05 ( ( member_nat @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_nat @ X5 @ Xa )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3540_finite__has__maximal,axiom,
% 8.90/5.05 ! [A2: set_int] :
% 8.90/5.05 ( ( finite_finite_int @ A2 )
% 8.90/5.05 => ( ( A2 != bot_bot_set_int )
% 8.90/5.05 => ? [X5: int] :
% 8.90/5.05 ( ( member_int @ X5 @ A2 )
% 8.90/5.05 & ! [Xa: int] :
% 8.90/5.05 ( ( member_int @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_int @ X5 @ Xa )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3541_finite__has__minimal,axiom,
% 8.90/5.05 ! [A2: set_real] :
% 8.90/5.05 ( ( finite_finite_real @ A2 )
% 8.90/5.05 => ( ( A2 != bot_bot_set_real )
% 8.90/5.05 => ? [X5: real] :
% 8.90/5.05 ( ( member_real @ X5 @ A2 )
% 8.90/5.05 & ! [Xa: real] :
% 8.90/5.05 ( ( member_real @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_real @ Xa @ X5 )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3542_finite__has__minimal,axiom,
% 8.90/5.05 ! [A2: set_set_int] :
% 8.90/5.05 ( ( finite6197958912794628473et_int @ A2 )
% 8.90/5.05 => ( ( A2 != bot_bot_set_set_int )
% 8.90/5.05 => ? [X5: set_int] :
% 8.90/5.05 ( ( member_set_int @ X5 @ A2 )
% 8.90/5.05 & ! [Xa: set_int] :
% 8.90/5.05 ( ( member_set_int @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_set_int @ Xa @ X5 )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3543_finite__has__minimal,axiom,
% 8.90/5.05 ! [A2: set_rat] :
% 8.90/5.05 ( ( finite_finite_rat @ A2 )
% 8.90/5.05 => ( ( A2 != bot_bot_set_rat )
% 8.90/5.05 => ? [X5: rat] :
% 8.90/5.05 ( ( member_rat @ X5 @ A2 )
% 8.90/5.05 & ! [Xa: rat] :
% 8.90/5.05 ( ( member_rat @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_rat @ Xa @ X5 )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3544_finite__has__minimal,axiom,
% 8.90/5.05 ! [A2: set_num] :
% 8.90/5.05 ( ( finite_finite_num @ A2 )
% 8.90/5.05 => ( ( A2 != bot_bot_set_num )
% 8.90/5.05 => ? [X5: num] :
% 8.90/5.05 ( ( member_num @ X5 @ A2 )
% 8.90/5.05 & ! [Xa: num] :
% 8.90/5.05 ( ( member_num @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_num @ Xa @ X5 )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3545_finite__has__minimal,axiom,
% 8.90/5.05 ! [A2: set_nat] :
% 8.90/5.05 ( ( finite_finite_nat @ A2 )
% 8.90/5.05 => ( ( A2 != bot_bot_set_nat )
% 8.90/5.05 => ? [X5: nat] :
% 8.90/5.05 ( ( member_nat @ X5 @ A2 )
% 8.90/5.05 & ! [Xa: nat] :
% 8.90/5.05 ( ( member_nat @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_nat @ Xa @ X5 )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3546_finite__has__minimal,axiom,
% 8.90/5.05 ! [A2: set_int] :
% 8.90/5.05 ( ( finite_finite_int @ A2 )
% 8.90/5.05 => ( ( A2 != bot_bot_set_int )
% 8.90/5.05 => ? [X5: int] :
% 8.90/5.05 ( ( member_int @ X5 @ A2 )
% 8.90/5.05 & ! [Xa: int] :
% 8.90/5.05 ( ( member_int @ Xa @ A2 )
% 8.90/5.05 => ( ( ord_less_eq_int @ Xa @ X5 )
% 8.90/5.05 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3547_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
% 8.90/5.05 ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 8.90/5.05 ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 8.90/5.05 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( X2 = Mi ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( X2 = Ma ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma @ X2 ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3548_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
% 8.90/5.05 ! [X2: vEBT_VEBT,Xa2: nat,Y3: nat] :
% 8.90/5.05 ( ( ( vEBT_T_m_e_m_b_e_r @ X2 @ Xa2 )
% 8.90/5.05 = Y3 )
% 8.90/5.05 => ( ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 8.90/5.05 => ( ! [A4: $o,B4: $o] :
% 8.90/5.05 ( ( X2
% 8.90/5.05 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.90/5.05 => ( ( Y3
% 8.90/5.05 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
% 8.90/5.05 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
% 8.90/5.05 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.90/5.05 ( ( X2
% 8.90/5.05 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 8.90/5.05 => ( ( Y3
% 8.90/5.05 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.05 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 8.90/5.05 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 8.90/5.05 ( ( X2
% 8.90/5.05 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 8.90/5.05 => ( ( Y3
% 8.90/5.05 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.05 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 8.90/5.05 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.90/5.05 ( ( X2
% 8.90/5.05 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 8.90/5.05 => ( ( Y3
% 8.90/5.05 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.05 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
% 8.90/5.05 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.90/5.05 ( ( X2
% 8.90/5.05 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.90/5.05 => ( ( Y3
% 8.90/5.05 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa2 = Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa2 = Ma2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa2 ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) )
% 8.90/5.05 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3549_mod__exhaust__less__4,axiom,
% 8.90/5.05 ! [M: nat] :
% 8.90/5.05 ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.90/5.05 = zero_zero_nat )
% 8.90/5.05 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.90/5.05 = one_one_nat )
% 8.90/5.05 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.90/5.05 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.05 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.90/5.05 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3550_arcosh__1,axiom,
% 8.90/5.05 ( ( arcosh_real @ one_one_real )
% 8.90/5.05 = zero_zero_real ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3551_finite__nth__roots,axiom,
% 8.90/5.05 ! [N: nat,C: complex] :
% 8.90/5.05 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.05 => ( finite3207457112153483333omplex
% 8.90/5.05 @ ( collect_complex
% 8.90/5.05 @ ^ [Z3: complex] :
% 8.90/5.05 ( ( power_power_complex @ Z3 @ N )
% 8.90/5.05 = C ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3552_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
% 8.90/5.05 ! [X2: vEBT_VEBT,Xa2: nat,Y3: nat] :
% 8.90/5.05 ( ( ( vEBT_T_i_n_s_e_r_t @ X2 @ Xa2 )
% 8.90/5.05 = Y3 )
% 8.90/5.05 => ( ( ? [A4: $o,B4: $o] :
% 8.90/5.05 ( X2
% 8.90/5.05 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.90/5.05 => ( Y3
% 8.90/5.05 != ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
% 8.90/5.05 => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 8.90/5.05 ( X2
% 8.90/5.05 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 8.90/5.05 => ( Y3 != one_one_nat ) )
% 8.90/5.05 => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 8.90/5.05 ( X2
% 8.90/5.05 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 8.90/5.05 => ( Y3 != one_one_nat ) )
% 8.90/5.05 => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.90/5.05 ( X2
% 8.90/5.05 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
% 8.90/5.05 => ( Y3
% 8.90/5.05 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.05 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.90/5.05 ( ( X2
% 8.90/5.05 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.90/5.05 => ( Y3
% 8.90/5.05 != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 8.90/5.05 @ ( if_nat
% 8.90/5.05 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.90/5.05 & ~ ( ( Xa2 = Mi2 )
% 8.90/5.05 | ( Xa2 = Ma2 ) ) )
% 8.90/5.05 @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
% 8.90/5.05 @ one_one_nat ) ) ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3553_insert__bound__height,axiom,
% 8.90/5.05 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 8.90/5.05 ( ( vEBT_invar_vebt @ T @ N )
% 8.90/5.05 => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ X2 ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3554_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
% 8.90/5.05 ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 8.90/5.05 ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 8.90/5.05 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 8.90/5.05 @ ( if_nat
% 8.90/5.05 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 8.90/5.05 & ~ ( ( X2 = Mi )
% 8.90/5.05 | ( X2 = Ma ) ) )
% 8.90/5.05 @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
% 8.90/5.05 @ one_one_nat ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3555_divmod__algorithm__code_I8_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( ( ord_less_num @ M @ N )
% 8.90/5.05 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
% 8.90/5.05 & ( ~ ( ord_less_num @ M @ N )
% 8.90/5.05 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3556_divmod__algorithm__code_I8_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( ( ord_less_num @ M @ N )
% 8.90/5.05 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
% 8.90/5.05 & ( ~ ( ord_less_num @ M @ N )
% 8.90/5.05 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3557_divmod__algorithm__code_I8_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( ( ord_less_num @ M @ N )
% 8.90/5.05 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
% 8.90/5.05 & ( ~ ( ord_less_num @ M @ N )
% 8.90/5.05 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3558_divmod__algorithm__code_I2_J,axiom,
% 8.90/5.05 ! [M: num] :
% 8.90/5.05 ( ( unique5052692396658037445od_int @ M @ one )
% 8.90/5.05 = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3559_divmod__algorithm__code_I2_J,axiom,
% 8.90/5.05 ! [M: num] :
% 8.90/5.05 ( ( unique5055182867167087721od_nat @ M @ one )
% 8.90/5.05 = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3560_divmod__algorithm__code_I2_J,axiom,
% 8.90/5.05 ! [M: num] :
% 8.90/5.05 ( ( unique3479559517661332726nteger @ M @ one )
% 8.90/5.05 = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3561_divmod__algorithm__code_I3_J,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
% 8.90/5.05 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3562_divmod__algorithm__code_I3_J,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
% 8.90/5.05 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3563_divmod__algorithm__code_I3_J,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
% 8.90/5.05 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3564_divmod__algorithm__code_I4_J,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
% 8.90/5.05 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3565_divmod__algorithm__code_I4_J,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
% 8.90/5.05 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3566_divmod__algorithm__code_I4_J,axiom,
% 8.90/5.05 ! [N: num] :
% 8.90/5.05 ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
% 8.90/5.05 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3567_divmod__algorithm__code_I7_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( ( ord_less_eq_num @ M @ N )
% 8.90/5.05 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
% 8.90/5.05 & ( ~ ( ord_less_eq_num @ M @ N )
% 8.90/5.05 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3568_divmod__algorithm__code_I7_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( ( ord_less_eq_num @ M @ N )
% 8.90/5.05 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
% 8.90/5.05 & ( ~ ( ord_less_eq_num @ M @ N )
% 8.90/5.05 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3569_divmod__algorithm__code_I7_J,axiom,
% 8.90/5.05 ! [M: num,N: num] :
% 8.90/5.05 ( ( ( ord_less_eq_num @ M @ N )
% 8.90/5.05 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
% 8.90/5.05 & ( ~ ( ord_less_eq_num @ M @ N )
% 8.90/5.05 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.90/5.05 = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3570_minNull__bound,axiom,
% 8.90/5.05 ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_i_n_N_u_l_l @ T ) @ one_one_nat ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3571_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I1_J,axiom,
% 8.90/5.05 ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $false @ $false ) )
% 8.90/5.05 = one_one_nat ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3572_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I2_J,axiom,
% 8.90/5.05 ! [Uv: $o] :
% 8.90/5.05 ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $true @ Uv ) )
% 8.90/5.05 = one_one_nat ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3573_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
% 8.90/5.05 ! [Uu: $o] :
% 8.90/5.05 ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ Uu @ $true ) )
% 8.90/5.05 = one_one_nat ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3574_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
% 8.90/5.05 ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
% 8.90/5.05 ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S ) @ X2 )
% 8.90/5.05 = one_one_nat ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3575_divmod__int__def,axiom,
% 8.90/5.05 ( unique5052692396658037445od_int
% 8.90/5.05 = ( ^ [M3: num,N3: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3576_divmod__def,axiom,
% 8.90/5.05 ( unique5052692396658037445od_int
% 8.90/5.05 = ( ^ [M3: num,N3: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3577_divmod__def,axiom,
% 8.90/5.05 ( unique5055182867167087721od_nat
% 8.90/5.05 = ( ^ [M3: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3578_divmod__def,axiom,
% 8.90/5.05 ( unique3479559517661332726nteger
% 8.90/5.05 = ( ^ [M3: num,N3: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N3 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N3 ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3579_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
% 8.90/5.05 ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 8.90/5.05 ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) )
% 8.90/5.05 = one_one_nat ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3580_divmod_H__nat__def,axiom,
% 8.90/5.05 ( unique5055182867167087721od_nat
% 8.90/5.05 = ( ^ [M3: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3581_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
% 8.90/5.05 ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
% 8.90/5.05 ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
% 8.90/5.05 = one_one_nat ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3582_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
% 8.90/5.05 ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
% 8.90/5.05 ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ X2 )
% 8.90/5.05 = one_one_nat ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3583_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
% 8.90/5.05 ! [A: $o,B: $o,X2: nat] :
% 8.90/5.05 ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Leaf @ A @ B ) @ X2 )
% 8.90/5.05 = ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( X2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3584_eq__diff__eq_H,axiom,
% 8.90/5.05 ! [X2: real,Y3: real,Z: real] :
% 8.90/5.05 ( ( X2
% 8.90/5.05 = ( minus_minus_real @ Y3 @ Z ) )
% 8.90/5.05 = ( Y3
% 8.90/5.05 = ( plus_plus_real @ X2 @ Z ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3585_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
% 8.90/5.05 ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 8.90/5.05 ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X2 )
% 8.90/5.05 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3586_divmod__divmod__step,axiom,
% 8.90/5.05 ( unique5055182867167087721od_nat
% 8.90/5.05 = ( ^ [M3: num,N3: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M3 @ N3 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M3 ) ) @ ( unique5026877609467782581ep_nat @ N3 @ ( unique5055182867167087721od_nat @ M3 @ ( bit0 @ N3 ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3587_divmod__divmod__step,axiom,
% 8.90/5.05 ( unique5052692396658037445od_int
% 8.90/5.05 = ( ^ [M3: num,N3: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M3 @ N3 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M3 ) ) @ ( unique5024387138958732305ep_int @ N3 @ ( unique5052692396658037445od_int @ M3 @ ( bit0 @ N3 ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3588_divmod__divmod__step,axiom,
% 8.90/5.05 ( unique3479559517661332726nteger
% 8.90/5.05 = ( ^ [M3: num,N3: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M3 @ N3 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M3 ) ) @ ( unique4921790084139445826nteger @ N3 @ ( unique3479559517661332726nteger @ M3 @ ( bit0 @ N3 ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3589_insersimp,axiom,
% 8.90/5.05 ! [T: vEBT_VEBT,N: nat,Y3: nat] :
% 8.90/5.05 ( ( vEBT_invar_vebt @ T @ N )
% 8.90/5.05 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 )
% 8.90/5.05 => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ Y3 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3590_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
% 8.90/5.05 ! [X2: vEBT_VEBT,Y3: nat] :
% 8.90/5.05 ( ( ( vEBT_T_m_i_n_N_u_l_l @ X2 )
% 8.90/5.05 = Y3 )
% 8.90/5.05 => ( ( ( X2
% 8.90/5.05 = ( vEBT_Leaf @ $false @ $false ) )
% 8.90/5.05 => ( Y3 != one_one_nat ) )
% 8.90/5.05 => ( ( ? [Uv2: $o] :
% 8.90/5.05 ( X2
% 8.90/5.05 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 8.90/5.05 => ( Y3 != one_one_nat ) )
% 8.90/5.05 => ( ( ? [Uu2: $o] :
% 8.90/5.05 ( X2
% 8.90/5.05 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 8.90/5.05 => ( Y3 != one_one_nat ) )
% 8.90/5.05 => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 8.90/5.05 ( X2
% 8.90/5.05 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 8.90/5.05 => ( Y3 != one_one_nat ) )
% 8.90/5.05 => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.90/5.05 ( X2
% 8.90/5.05 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
% 8.90/5.05 => ( Y3 != one_one_nat ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3591_insertsimp,axiom,
% 8.90/5.05 ! [T: vEBT_VEBT,N: nat,L: nat] :
% 8.90/5.05 ( ( vEBT_invar_vebt @ T @ N )
% 8.90/5.05 => ( ( vEBT_VEBT_minNull @ T )
% 8.90/5.05 => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ L ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3592_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
% 8.90/5.05 ! [X2: vEBT_VEBT,Xa2: nat,Y3: nat] :
% 8.90/5.05 ( ( ( vEBT_T_i_n_s_e_r_t @ X2 @ Xa2 )
% 8.90/5.05 = Y3 )
% 8.90/5.05 => ( ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 8.90/5.05 => ( ! [A4: $o,B4: $o] :
% 8.90/5.05 ( ( X2
% 8.90/5.05 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.90/5.05 => ( ( Y3
% 8.90/5.05 = ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
% 8.90/5.05 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
% 8.90/5.05 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 8.90/5.05 ( ( X2
% 8.90/5.05 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 8.90/5.05 => ( ( Y3 = one_one_nat )
% 8.90/5.05 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) @ Xa2 ) ) ) )
% 8.90/5.05 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 8.90/5.05 ( ( X2
% 8.90/5.05 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 8.90/5.05 => ( ( Y3 = one_one_nat )
% 8.90/5.05 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) @ Xa2 ) ) ) )
% 8.90/5.05 => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.90/5.05 ( ( X2
% 8.90/5.05 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
% 8.90/5.05 => ( ( Y3
% 8.90/5.05 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.05 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
% 8.90/5.05 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.90/5.05 ( ( X2
% 8.90/5.05 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.90/5.05 => ( ( Y3
% 8.90/5.05 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 8.90/5.05 @ ( if_nat
% 8.90/5.05 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.90/5.05 & ~ ( ( Xa2 = Mi2 )
% 8.90/5.05 | ( Xa2 = Ma2 ) ) )
% 8.90/5.05 @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
% 8.90/5.05 @ one_one_nat ) ) )
% 8.90/5.05 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3593_prod_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_real,X2: real > complex,Y3: real > complex] :
% 8.90/5.05 ( ( finite_finite_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [I4: real] :
% 8.90/5.05 ( ( member_real @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != one_one_complex ) ) ) )
% 8.90/5.05 => ( ( finite_finite_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [I4: real] :
% 8.90/5.05 ( ( member_real @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != one_one_complex ) ) ) )
% 8.90/5.05 => ( finite_finite_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [I4: real] :
% 8.90/5.05 ( ( member_real @ I4 @ I5 )
% 8.90/5.05 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != one_one_complex ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3594_prod_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_nat,X2: nat > complex,Y3: nat > complex] :
% 8.90/5.05 ( ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [I4: nat] :
% 8.90/5.05 ( ( member_nat @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != one_one_complex ) ) ) )
% 8.90/5.05 => ( ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [I4: nat] :
% 8.90/5.05 ( ( member_nat @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != one_one_complex ) ) ) )
% 8.90/5.05 => ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [I4: nat] :
% 8.90/5.05 ( ( member_nat @ I4 @ I5 )
% 8.90/5.05 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != one_one_complex ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3595_prod_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_int,X2: int > complex,Y3: int > complex] :
% 8.90/5.05 ( ( finite_finite_int
% 8.90/5.05 @ ( collect_int
% 8.90/5.05 @ ^ [I4: int] :
% 8.90/5.05 ( ( member_int @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != one_one_complex ) ) ) )
% 8.90/5.05 => ( ( finite_finite_int
% 8.90/5.05 @ ( collect_int
% 8.90/5.05 @ ^ [I4: int] :
% 8.90/5.05 ( ( member_int @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != one_one_complex ) ) ) )
% 8.90/5.05 => ( finite_finite_int
% 8.90/5.05 @ ( collect_int
% 8.90/5.05 @ ^ [I4: int] :
% 8.90/5.05 ( ( member_int @ I4 @ I5 )
% 8.90/5.05 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != one_one_complex ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3596_prod_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_complex,X2: complex > complex,Y3: complex > complex] :
% 8.90/5.05 ( ( finite3207457112153483333omplex
% 8.90/5.05 @ ( collect_complex
% 8.90/5.05 @ ^ [I4: complex] :
% 8.90/5.05 ( ( member_complex @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != one_one_complex ) ) ) )
% 8.90/5.05 => ( ( finite3207457112153483333omplex
% 8.90/5.05 @ ( collect_complex
% 8.90/5.05 @ ^ [I4: complex] :
% 8.90/5.05 ( ( member_complex @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != one_one_complex ) ) ) )
% 8.90/5.05 => ( finite3207457112153483333omplex
% 8.90/5.05 @ ( collect_complex
% 8.90/5.05 @ ^ [I4: complex] :
% 8.90/5.05 ( ( member_complex @ I4 @ I5 )
% 8.90/5.05 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != one_one_complex ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3597_prod_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_real,X2: real > real,Y3: real > real] :
% 8.90/5.05 ( ( finite_finite_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [I4: real] :
% 8.90/5.05 ( ( member_real @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != one_one_real ) ) ) )
% 8.90/5.05 => ( ( finite_finite_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [I4: real] :
% 8.90/5.05 ( ( member_real @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != one_one_real ) ) ) )
% 8.90/5.05 => ( finite_finite_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [I4: real] :
% 8.90/5.05 ( ( member_real @ I4 @ I5 )
% 8.90/5.05 & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != one_one_real ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3598_prod_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_nat,X2: nat > real,Y3: nat > real] :
% 8.90/5.05 ( ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [I4: nat] :
% 8.90/5.05 ( ( member_nat @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != one_one_real ) ) ) )
% 8.90/5.05 => ( ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [I4: nat] :
% 8.90/5.05 ( ( member_nat @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != one_one_real ) ) ) )
% 8.90/5.05 => ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [I4: nat] :
% 8.90/5.05 ( ( member_nat @ I4 @ I5 )
% 8.90/5.05 & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != one_one_real ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3599_prod_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_int,X2: int > real,Y3: int > real] :
% 8.90/5.05 ( ( finite_finite_int
% 8.90/5.05 @ ( collect_int
% 8.90/5.05 @ ^ [I4: int] :
% 8.90/5.05 ( ( member_int @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != one_one_real ) ) ) )
% 8.90/5.05 => ( ( finite_finite_int
% 8.90/5.05 @ ( collect_int
% 8.90/5.05 @ ^ [I4: int] :
% 8.90/5.05 ( ( member_int @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != one_one_real ) ) ) )
% 8.90/5.05 => ( finite_finite_int
% 8.90/5.05 @ ( collect_int
% 8.90/5.05 @ ^ [I4: int] :
% 8.90/5.05 ( ( member_int @ I4 @ I5 )
% 8.90/5.05 & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != one_one_real ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3600_prod_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_complex,X2: complex > real,Y3: complex > real] :
% 8.90/5.05 ( ( finite3207457112153483333omplex
% 8.90/5.05 @ ( collect_complex
% 8.90/5.05 @ ^ [I4: complex] :
% 8.90/5.05 ( ( member_complex @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != one_one_real ) ) ) )
% 8.90/5.05 => ( ( finite3207457112153483333omplex
% 8.90/5.05 @ ( collect_complex
% 8.90/5.05 @ ^ [I4: complex] :
% 8.90/5.05 ( ( member_complex @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != one_one_real ) ) ) )
% 8.90/5.05 => ( finite3207457112153483333omplex
% 8.90/5.05 @ ( collect_complex
% 8.90/5.05 @ ^ [I4: complex] :
% 8.90/5.05 ( ( member_complex @ I4 @ I5 )
% 8.90/5.05 & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != one_one_real ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3601_prod_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_real,X2: real > rat,Y3: real > rat] :
% 8.90/5.05 ( ( finite_finite_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [I4: real] :
% 8.90/5.05 ( ( member_real @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != one_one_rat ) ) ) )
% 8.90/5.05 => ( ( finite_finite_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [I4: real] :
% 8.90/5.05 ( ( member_real @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != one_one_rat ) ) ) )
% 8.90/5.05 => ( finite_finite_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [I4: real] :
% 8.90/5.05 ( ( member_real @ I4 @ I5 )
% 8.90/5.05 & ( ( times_times_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != one_one_rat ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3602_prod_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_nat,X2: nat > rat,Y3: nat > rat] :
% 8.90/5.05 ( ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [I4: nat] :
% 8.90/5.05 ( ( member_nat @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != one_one_rat ) ) ) )
% 8.90/5.05 => ( ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [I4: nat] :
% 8.90/5.05 ( ( member_nat @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != one_one_rat ) ) ) )
% 8.90/5.05 => ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [I4: nat] :
% 8.90/5.05 ( ( member_nat @ I4 @ I5 )
% 8.90/5.05 & ( ( times_times_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != one_one_rat ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3603_sum_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_real,X2: real > complex,Y3: real > complex] :
% 8.90/5.05 ( ( finite_finite_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [I4: real] :
% 8.90/5.05 ( ( member_real @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != zero_zero_complex ) ) ) )
% 8.90/5.05 => ( ( finite_finite_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [I4: real] :
% 8.90/5.05 ( ( member_real @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != zero_zero_complex ) ) ) )
% 8.90/5.05 => ( finite_finite_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [I4: real] :
% 8.90/5.05 ( ( member_real @ I4 @ I5 )
% 8.90/5.05 & ( ( plus_plus_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != zero_zero_complex ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3604_sum_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_nat,X2: nat > complex,Y3: nat > complex] :
% 8.90/5.05 ( ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [I4: nat] :
% 8.90/5.05 ( ( member_nat @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != zero_zero_complex ) ) ) )
% 8.90/5.05 => ( ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [I4: nat] :
% 8.90/5.05 ( ( member_nat @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != zero_zero_complex ) ) ) )
% 8.90/5.05 => ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [I4: nat] :
% 8.90/5.05 ( ( member_nat @ I4 @ I5 )
% 8.90/5.05 & ( ( plus_plus_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != zero_zero_complex ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3605_sum_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_int,X2: int > complex,Y3: int > complex] :
% 8.90/5.05 ( ( finite_finite_int
% 8.90/5.05 @ ( collect_int
% 8.90/5.05 @ ^ [I4: int] :
% 8.90/5.05 ( ( member_int @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != zero_zero_complex ) ) ) )
% 8.90/5.05 => ( ( finite_finite_int
% 8.90/5.05 @ ( collect_int
% 8.90/5.05 @ ^ [I4: int] :
% 8.90/5.05 ( ( member_int @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != zero_zero_complex ) ) ) )
% 8.90/5.05 => ( finite_finite_int
% 8.90/5.05 @ ( collect_int
% 8.90/5.05 @ ^ [I4: int] :
% 8.90/5.05 ( ( member_int @ I4 @ I5 )
% 8.90/5.05 & ( ( plus_plus_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != zero_zero_complex ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3606_sum_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_complex,X2: complex > complex,Y3: complex > complex] :
% 8.90/5.05 ( ( finite3207457112153483333omplex
% 8.90/5.05 @ ( collect_complex
% 8.90/5.05 @ ^ [I4: complex] :
% 8.90/5.05 ( ( member_complex @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != zero_zero_complex ) ) ) )
% 8.90/5.05 => ( ( finite3207457112153483333omplex
% 8.90/5.05 @ ( collect_complex
% 8.90/5.05 @ ^ [I4: complex] :
% 8.90/5.05 ( ( member_complex @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != zero_zero_complex ) ) ) )
% 8.90/5.05 => ( finite3207457112153483333omplex
% 8.90/5.05 @ ( collect_complex
% 8.90/5.05 @ ^ [I4: complex] :
% 8.90/5.05 ( ( member_complex @ I4 @ I5 )
% 8.90/5.05 & ( ( plus_plus_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != zero_zero_complex ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3607_sum_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_real,X2: real > real,Y3: real > real] :
% 8.90/5.05 ( ( finite_finite_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [I4: real] :
% 8.90/5.05 ( ( member_real @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != zero_zero_real ) ) ) )
% 8.90/5.05 => ( ( finite_finite_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [I4: real] :
% 8.90/5.05 ( ( member_real @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != zero_zero_real ) ) ) )
% 8.90/5.05 => ( finite_finite_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [I4: real] :
% 8.90/5.05 ( ( member_real @ I4 @ I5 )
% 8.90/5.05 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != zero_zero_real ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3608_sum_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_nat,X2: nat > real,Y3: nat > real] :
% 8.90/5.05 ( ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [I4: nat] :
% 8.90/5.05 ( ( member_nat @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != zero_zero_real ) ) ) )
% 8.90/5.05 => ( ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [I4: nat] :
% 8.90/5.05 ( ( member_nat @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != zero_zero_real ) ) ) )
% 8.90/5.05 => ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [I4: nat] :
% 8.90/5.05 ( ( member_nat @ I4 @ I5 )
% 8.90/5.05 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != zero_zero_real ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3609_sum_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_int,X2: int > real,Y3: int > real] :
% 8.90/5.05 ( ( finite_finite_int
% 8.90/5.05 @ ( collect_int
% 8.90/5.05 @ ^ [I4: int] :
% 8.90/5.05 ( ( member_int @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != zero_zero_real ) ) ) )
% 8.90/5.05 => ( ( finite_finite_int
% 8.90/5.05 @ ( collect_int
% 8.90/5.05 @ ^ [I4: int] :
% 8.90/5.05 ( ( member_int @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != zero_zero_real ) ) ) )
% 8.90/5.05 => ( finite_finite_int
% 8.90/5.05 @ ( collect_int
% 8.90/5.05 @ ^ [I4: int] :
% 8.90/5.05 ( ( member_int @ I4 @ I5 )
% 8.90/5.05 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != zero_zero_real ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3610_sum_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_complex,X2: complex > real,Y3: complex > real] :
% 8.90/5.05 ( ( finite3207457112153483333omplex
% 8.90/5.05 @ ( collect_complex
% 8.90/5.05 @ ^ [I4: complex] :
% 8.90/5.05 ( ( member_complex @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != zero_zero_real ) ) ) )
% 8.90/5.05 => ( ( finite3207457112153483333omplex
% 8.90/5.05 @ ( collect_complex
% 8.90/5.05 @ ^ [I4: complex] :
% 8.90/5.05 ( ( member_complex @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != zero_zero_real ) ) ) )
% 8.90/5.05 => ( finite3207457112153483333omplex
% 8.90/5.05 @ ( collect_complex
% 8.90/5.05 @ ^ [I4: complex] :
% 8.90/5.05 ( ( member_complex @ I4 @ I5 )
% 8.90/5.05 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != zero_zero_real ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3611_sum_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_real,X2: real > rat,Y3: real > rat] :
% 8.90/5.05 ( ( finite_finite_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [I4: real] :
% 8.90/5.05 ( ( member_real @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != zero_zero_rat ) ) ) )
% 8.90/5.05 => ( ( finite_finite_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [I4: real] :
% 8.90/5.05 ( ( member_real @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != zero_zero_rat ) ) ) )
% 8.90/5.05 => ( finite_finite_real
% 8.90/5.05 @ ( collect_real
% 8.90/5.05 @ ^ [I4: real] :
% 8.90/5.05 ( ( member_real @ I4 @ I5 )
% 8.90/5.05 & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != zero_zero_rat ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3612_sum_Ofinite__Collect__op,axiom,
% 8.90/5.05 ! [I5: set_nat,X2: nat > rat,Y3: nat > rat] :
% 8.90/5.05 ( ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [I4: nat] :
% 8.90/5.05 ( ( member_nat @ I4 @ I5 )
% 8.90/5.05 & ( ( X2 @ I4 )
% 8.90/5.05 != zero_zero_rat ) ) ) )
% 8.90/5.05 => ( ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [I4: nat] :
% 8.90/5.05 ( ( member_nat @ I4 @ I5 )
% 8.90/5.05 & ( ( Y3 @ I4 )
% 8.90/5.05 != zero_zero_rat ) ) ) )
% 8.90/5.05 => ( finite_finite_nat
% 8.90/5.05 @ ( collect_nat
% 8.90/5.05 @ ^ [I4: nat] :
% 8.90/5.05 ( ( member_nat @ I4 @ I5 )
% 8.90/5.05 & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 8.90/5.05 != zero_zero_rat ) ) ) ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3613_dbl__inc__simps_I3_J,axiom,
% 8.90/5.05 ( ( neg_nu8557863876264182079omplex @ one_one_complex )
% 8.90/5.05 = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3614_dbl__inc__simps_I3_J,axiom,
% 8.90/5.05 ( ( neg_nu8295874005876285629c_real @ one_one_real )
% 8.90/5.05 = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3615_dbl__inc__simps_I3_J,axiom,
% 8.90/5.05 ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
% 8.90/5.05 = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3616_dbl__inc__simps_I3_J,axiom,
% 8.90/5.05 ( ( neg_nu5851722552734809277nc_int @ one_one_int )
% 8.90/5.05 = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% 8.90/5.05
% 8.90/5.05 thf(fact_3617_prod_Oinject,axiom,
% 8.90/5.05 ! [X1: code_integer,X22: $o,Y1: code_integer,Y2: $o] :
% 8.90/5.05 ( ( ( produc6677183202524767010eger_o @ X1 @ X22 )
% 8.90/5.05 = ( produc6677183202524767010eger_o @ Y1 @ Y2 ) )
% 8.90/5.05 = ( ( X1 = Y1 )
% 8.90/5.05 & ( X22 = Y2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3618_prod_Oinject,axiom,
% 8.90/5.06 ! [X1: num,X22: num,Y1: num,Y2: num] :
% 8.90/5.06 ( ( ( product_Pair_num_num @ X1 @ X22 )
% 8.90/5.06 = ( product_Pair_num_num @ Y1 @ Y2 ) )
% 8.90/5.06 = ( ( X1 = Y1 )
% 8.90/5.06 & ( X22 = Y2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3619_prod_Oinject,axiom,
% 8.90/5.06 ! [X1: nat,X22: num,Y1: nat,Y2: num] :
% 8.90/5.06 ( ( ( product_Pair_nat_num @ X1 @ X22 )
% 8.90/5.06 = ( product_Pair_nat_num @ Y1 @ Y2 ) )
% 8.90/5.06 = ( ( X1 = Y1 )
% 8.90/5.06 & ( X22 = Y2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3620_prod_Oinject,axiom,
% 8.90/5.06 ! [X1: nat,X22: nat,Y1: nat,Y2: nat] :
% 8.90/5.06 ( ( ( product_Pair_nat_nat @ X1 @ X22 )
% 8.90/5.06 = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
% 8.90/5.06 = ( ( X1 = Y1 )
% 8.90/5.06 & ( X22 = Y2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3621_prod_Oinject,axiom,
% 8.90/5.06 ! [X1: int,X22: int,Y1: int,Y2: int] :
% 8.90/5.06 ( ( ( product_Pair_int_int @ X1 @ X22 )
% 8.90/5.06 = ( product_Pair_int_int @ Y1 @ Y2 ) )
% 8.90/5.06 = ( ( X1 = Y1 )
% 8.90/5.06 & ( X22 = Y2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3622_old_Oprod_Oinject,axiom,
% 8.90/5.06 ! [A: code_integer,B: $o,A5: code_integer,B5: $o] :
% 8.90/5.06 ( ( ( produc6677183202524767010eger_o @ A @ B )
% 8.90/5.06 = ( produc6677183202524767010eger_o @ A5 @ B5 ) )
% 8.90/5.06 = ( ( A = A5 )
% 8.90/5.06 & ( B = B5 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3623_old_Oprod_Oinject,axiom,
% 8.90/5.06 ! [A: num,B: num,A5: num,B5: num] :
% 8.90/5.06 ( ( ( product_Pair_num_num @ A @ B )
% 8.90/5.06 = ( product_Pair_num_num @ A5 @ B5 ) )
% 8.90/5.06 = ( ( A = A5 )
% 8.90/5.06 & ( B = B5 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3624_old_Oprod_Oinject,axiom,
% 8.90/5.06 ! [A: nat,B: num,A5: nat,B5: num] :
% 8.90/5.06 ( ( ( product_Pair_nat_num @ A @ B )
% 8.90/5.06 = ( product_Pair_nat_num @ A5 @ B5 ) )
% 8.90/5.06 = ( ( A = A5 )
% 8.90/5.06 & ( B = B5 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3625_old_Oprod_Oinject,axiom,
% 8.90/5.06 ! [A: nat,B: nat,A5: nat,B5: nat] :
% 8.90/5.06 ( ( ( product_Pair_nat_nat @ A @ B )
% 8.90/5.06 = ( product_Pair_nat_nat @ A5 @ B5 ) )
% 8.90/5.06 = ( ( A = A5 )
% 8.90/5.06 & ( B = B5 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3626_old_Oprod_Oinject,axiom,
% 8.90/5.06 ! [A: int,B: int,A5: int,B5: int] :
% 8.90/5.06 ( ( ( product_Pair_int_int @ A @ B )
% 8.90/5.06 = ( product_Pair_int_int @ A5 @ B5 ) )
% 8.90/5.06 = ( ( A = A5 )
% 8.90/5.06 & ( B = B5 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3627_dbl__inc__simps_I2_J,axiom,
% 8.90/5.06 ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
% 8.90/5.06 = one_one_complex ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3628_dbl__inc__simps_I2_J,axiom,
% 8.90/5.06 ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
% 8.90/5.06 = one_one_real ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3629_dbl__inc__simps_I2_J,axiom,
% 8.90/5.06 ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
% 8.90/5.06 = one_one_rat ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3630_dbl__inc__simps_I2_J,axiom,
% 8.90/5.06 ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 8.90/5.06 = one_one_int ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3631_dbl__inc__simps_I5_J,axiom,
% 8.90/5.06 ! [K: num] :
% 8.90/5.06 ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
% 8.90/5.06 = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3632_dbl__inc__simps_I5_J,axiom,
% 8.90/5.06 ! [K: num] :
% 8.90/5.06 ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
% 8.90/5.06 = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3633_dbl__inc__simps_I5_J,axiom,
% 8.90/5.06 ! [K: num] :
% 8.90/5.06 ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
% 8.90/5.06 = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3634_dbl__inc__simps_I5_J,axiom,
% 8.90/5.06 ! [K: num] :
% 8.90/5.06 ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
% 8.90/5.06 = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3635_old_Oprod_Oexhaust,axiom,
% 8.90/5.06 ! [Y3: produc6271795597528267376eger_o] :
% 8.90/5.06 ~ ! [A4: code_integer,B4: $o] :
% 8.90/5.06 ( Y3
% 8.90/5.06 != ( produc6677183202524767010eger_o @ A4 @ B4 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3636_old_Oprod_Oexhaust,axiom,
% 8.90/5.06 ! [Y3: product_prod_num_num] :
% 8.90/5.06 ~ ! [A4: num,B4: num] :
% 8.90/5.06 ( Y3
% 8.90/5.06 != ( product_Pair_num_num @ A4 @ B4 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3637_old_Oprod_Oexhaust,axiom,
% 8.90/5.06 ! [Y3: product_prod_nat_num] :
% 8.90/5.06 ~ ! [A4: nat,B4: num] :
% 8.90/5.06 ( Y3
% 8.90/5.06 != ( product_Pair_nat_num @ A4 @ B4 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3638_old_Oprod_Oexhaust,axiom,
% 8.90/5.06 ! [Y3: product_prod_nat_nat] :
% 8.90/5.06 ~ ! [A4: nat,B4: nat] :
% 8.90/5.06 ( Y3
% 8.90/5.06 != ( product_Pair_nat_nat @ A4 @ B4 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3639_old_Oprod_Oexhaust,axiom,
% 8.90/5.06 ! [Y3: product_prod_int_int] :
% 8.90/5.06 ~ ! [A4: int,B4: int] :
% 8.90/5.06 ( Y3
% 8.90/5.06 != ( product_Pair_int_int @ A4 @ B4 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3640_surj__pair,axiom,
% 8.90/5.06 ! [P6: produc6271795597528267376eger_o] :
% 8.90/5.06 ? [X5: code_integer,Y4: $o] :
% 8.90/5.06 ( P6
% 8.90/5.06 = ( produc6677183202524767010eger_o @ X5 @ Y4 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3641_surj__pair,axiom,
% 8.90/5.06 ! [P6: product_prod_num_num] :
% 8.90/5.06 ? [X5: num,Y4: num] :
% 8.90/5.06 ( P6
% 8.90/5.06 = ( product_Pair_num_num @ X5 @ Y4 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3642_surj__pair,axiom,
% 8.90/5.06 ! [P6: product_prod_nat_num] :
% 8.90/5.06 ? [X5: nat,Y4: num] :
% 8.90/5.06 ( P6
% 8.90/5.06 = ( product_Pair_nat_num @ X5 @ Y4 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3643_surj__pair,axiom,
% 8.90/5.06 ! [P6: product_prod_nat_nat] :
% 8.90/5.06 ? [X5: nat,Y4: nat] :
% 8.90/5.06 ( P6
% 8.90/5.06 = ( product_Pair_nat_nat @ X5 @ Y4 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3644_surj__pair,axiom,
% 8.90/5.06 ! [P6: product_prod_int_int] :
% 8.90/5.06 ? [X5: int,Y4: int] :
% 8.90/5.06 ( P6
% 8.90/5.06 = ( product_Pair_int_int @ X5 @ Y4 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3645_prod__cases,axiom,
% 8.90/5.06 ! [P: produc6271795597528267376eger_o > $o,P6: produc6271795597528267376eger_o] :
% 8.90/5.06 ( ! [A4: code_integer,B4: $o] : ( P @ ( produc6677183202524767010eger_o @ A4 @ B4 ) )
% 8.90/5.06 => ( P @ P6 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3646_prod__cases,axiom,
% 8.90/5.06 ! [P: product_prod_num_num > $o,P6: product_prod_num_num] :
% 8.90/5.06 ( ! [A4: num,B4: num] : ( P @ ( product_Pair_num_num @ A4 @ B4 ) )
% 8.90/5.06 => ( P @ P6 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3647_prod__cases,axiom,
% 8.90/5.06 ! [P: product_prod_nat_num > $o,P6: product_prod_nat_num] :
% 8.90/5.06 ( ! [A4: nat,B4: num] : ( P @ ( product_Pair_nat_num @ A4 @ B4 ) )
% 8.90/5.06 => ( P @ P6 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3648_prod__cases,axiom,
% 8.90/5.06 ! [P: product_prod_nat_nat > $o,P6: product_prod_nat_nat] :
% 8.90/5.06 ( ! [A4: nat,B4: nat] : ( P @ ( product_Pair_nat_nat @ A4 @ B4 ) )
% 8.90/5.06 => ( P @ P6 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3649_prod__cases,axiom,
% 8.90/5.06 ! [P: product_prod_int_int > $o,P6: product_prod_int_int] :
% 8.90/5.06 ( ! [A4: int,B4: int] : ( P @ ( product_Pair_int_int @ A4 @ B4 ) )
% 8.90/5.06 => ( P @ P6 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3650_Pair__inject,axiom,
% 8.90/5.06 ! [A: code_integer,B: $o,A5: code_integer,B5: $o] :
% 8.90/5.06 ( ( ( produc6677183202524767010eger_o @ A @ B )
% 8.90/5.06 = ( produc6677183202524767010eger_o @ A5 @ B5 ) )
% 8.90/5.06 => ~ ( ( A = A5 )
% 8.90/5.06 => ( B = ~ B5 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3651_Pair__inject,axiom,
% 8.90/5.06 ! [A: num,B: num,A5: num,B5: num] :
% 8.90/5.06 ( ( ( product_Pair_num_num @ A @ B )
% 8.90/5.06 = ( product_Pair_num_num @ A5 @ B5 ) )
% 8.90/5.06 => ~ ( ( A = A5 )
% 8.90/5.06 => ( B != B5 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3652_Pair__inject,axiom,
% 8.90/5.06 ! [A: nat,B: num,A5: nat,B5: num] :
% 8.90/5.06 ( ( ( product_Pair_nat_num @ A @ B )
% 8.90/5.06 = ( product_Pair_nat_num @ A5 @ B5 ) )
% 8.90/5.06 => ~ ( ( A = A5 )
% 8.90/5.06 => ( B != B5 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3653_Pair__inject,axiom,
% 8.90/5.06 ! [A: nat,B: nat,A5: nat,B5: nat] :
% 8.90/5.06 ( ( ( product_Pair_nat_nat @ A @ B )
% 8.90/5.06 = ( product_Pair_nat_nat @ A5 @ B5 ) )
% 8.90/5.06 => ~ ( ( A = A5 )
% 8.90/5.06 => ( B != B5 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3654_Pair__inject,axiom,
% 8.90/5.06 ! [A: int,B: int,A5: int,B5: int] :
% 8.90/5.06 ( ( ( product_Pair_int_int @ A @ B )
% 8.90/5.06 = ( product_Pair_int_int @ A5 @ B5 ) )
% 8.90/5.06 => ~ ( ( A = A5 )
% 8.90/5.06 => ( B != B5 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3655_dbl__inc__def,axiom,
% 8.90/5.06 ( neg_nu8557863876264182079omplex
% 8.90/5.06 = ( ^ [X: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3656_dbl__inc__def,axiom,
% 8.90/5.06 ( neg_nu8295874005876285629c_real
% 8.90/5.06 = ( ^ [X: real] : ( plus_plus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3657_dbl__inc__def,axiom,
% 8.90/5.06 ( neg_nu5219082963157363817nc_rat
% 8.90/5.06 = ( ^ [X: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3658_dbl__inc__def,axiom,
% 8.90/5.06 ( neg_nu5851722552734809277nc_int
% 8.90/5.06 = ( ^ [X: int] : ( plus_plus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3659_VEBT__internal_Oheight_Osimps_I1_J,axiom,
% 8.90/5.06 ! [A: $o,B: $o] :
% 8.90/5.06 ( ( vEBT_VEBT_height @ ( vEBT_Leaf @ A @ B ) )
% 8.90/5.06 = zero_zero_nat ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3660_divmod__BitM__2__eq,axiom,
% 8.90/5.06 ! [M: num] :
% 8.90/5.06 ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 8.90/5.06 = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3661_insert__simp__excp,axiom,
% 8.90/5.06 ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X2: nat,Ma: nat,Summary: vEBT_VEBT] :
% 8.90/5.06 ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 8.90/5.06 => ( ( ord_less_nat @ X2 @ Mi )
% 8.90/5.06 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 8.90/5.06 => ( ( X2 != Ma )
% 8.90/5.06 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 8.90/5.06 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X2 @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( 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 ) ) ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3662_insert__simp__norm,axiom,
% 8.90/5.06 ! [X2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 8.90/5.06 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 8.90/5.06 => ( ( ord_less_nat @ Mi @ X2 )
% 8.90/5.06 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 8.90/5.06 => ( ( X2 != Ma )
% 8.90/5.06 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X2 )
% 8.90/5.06 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X2 @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3663_dual__order_Orefl,axiom,
% 8.90/5.06 ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3664_dual__order_Orefl,axiom,
% 8.90/5.06 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3665_dual__order_Orefl,axiom,
% 8.90/5.06 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3666_dual__order_Orefl,axiom,
% 8.90/5.06 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3667_dual__order_Orefl,axiom,
% 8.90/5.06 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3668_order__refl,axiom,
% 8.90/5.06 ! [X2: set_int] : ( ord_less_eq_set_int @ X2 @ X2 ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3669_order__refl,axiom,
% 8.90/5.06 ! [X2: rat] : ( ord_less_eq_rat @ X2 @ X2 ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3670_order__refl,axiom,
% 8.90/5.06 ! [X2: num] : ( ord_less_eq_num @ X2 @ X2 ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3671_order__refl,axiom,
% 8.90/5.06 ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3672_order__refl,axiom,
% 8.90/5.06 ! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3673_max_Oabsorb1,axiom,
% 8.90/5.06 ! [B: extended_enat,A: extended_enat] :
% 8.90/5.06 ( ( ord_le2932123472753598470d_enat @ B @ A )
% 8.90/5.06 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 8.90/5.06 = A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3674_max_Oabsorb1,axiom,
% 8.90/5.06 ! [B: rat,A: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ B @ A )
% 8.90/5.06 => ( ( ord_max_rat @ A @ B )
% 8.90/5.06 = A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3675_max_Oabsorb1,axiom,
% 8.90/5.06 ! [B: num,A: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ B @ A )
% 8.90/5.06 => ( ( ord_max_num @ A @ B )
% 8.90/5.06 = A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3676_max_Oabsorb1,axiom,
% 8.90/5.06 ! [B: nat,A: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ B @ A )
% 8.90/5.06 => ( ( ord_max_nat @ A @ B )
% 8.90/5.06 = A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3677_max_Oabsorb1,axiom,
% 8.90/5.06 ! [B: int,A: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ B @ A )
% 8.90/5.06 => ( ( ord_max_int @ A @ B )
% 8.90/5.06 = A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3678_max_Oabsorb2,axiom,
% 8.90/5.06 ! [A: extended_enat,B: extended_enat] :
% 8.90/5.06 ( ( ord_le2932123472753598470d_enat @ A @ B )
% 8.90/5.06 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 8.90/5.06 = B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3679_max_Oabsorb2,axiom,
% 8.90/5.06 ! [A: rat,B: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.06 => ( ( ord_max_rat @ A @ B )
% 8.90/5.06 = B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3680_max_Oabsorb2,axiom,
% 8.90/5.06 ! [A: num,B: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ A @ B )
% 8.90/5.06 => ( ( ord_max_num @ A @ B )
% 8.90/5.06 = B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3681_max_Oabsorb2,axiom,
% 8.90/5.06 ! [A: nat,B: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ A @ B )
% 8.90/5.06 => ( ( ord_max_nat @ A @ B )
% 8.90/5.06 = B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3682_max_Oabsorb2,axiom,
% 8.90/5.06 ! [A: int,B: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ A @ B )
% 8.90/5.06 => ( ( ord_max_int @ A @ B )
% 8.90/5.06 = B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3683_max_Obounded__iff,axiom,
% 8.90/5.06 ! [B: extended_enat,C: extended_enat,A: extended_enat] :
% 8.90/5.06 ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
% 8.90/5.06 = ( ( ord_le2932123472753598470d_enat @ B @ A )
% 8.90/5.06 & ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3684_max_Obounded__iff,axiom,
% 8.90/5.06 ! [B: rat,C: rat,A: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 8.90/5.06 = ( ( ord_less_eq_rat @ B @ A )
% 8.90/5.06 & ( ord_less_eq_rat @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3685_max_Obounded__iff,axiom,
% 8.90/5.06 ! [B: num,C: num,A: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 8.90/5.06 = ( ( ord_less_eq_num @ B @ A )
% 8.90/5.06 & ( ord_less_eq_num @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3686_max_Obounded__iff,axiom,
% 8.90/5.06 ! [B: nat,C: nat,A: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 8.90/5.06 = ( ( ord_less_eq_nat @ B @ A )
% 8.90/5.06 & ( ord_less_eq_nat @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3687_max_Obounded__iff,axiom,
% 8.90/5.06 ! [B: int,C: int,A: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 8.90/5.06 = ( ( ord_less_eq_int @ B @ A )
% 8.90/5.06 & ( ord_less_eq_int @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3688_max__less__iff__conj,axiom,
% 8.90/5.06 ! [X2: extended_enat,Y3: extended_enat,Z: extended_enat] :
% 8.90/5.06 ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ X2 @ Y3 ) @ Z )
% 8.90/5.06 = ( ( ord_le72135733267957522d_enat @ X2 @ Z )
% 8.90/5.06 & ( ord_le72135733267957522d_enat @ Y3 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3689_max__less__iff__conj,axiom,
% 8.90/5.06 ! [X2: real,Y3: real,Z: real] :
% 8.90/5.06 ( ( ord_less_real @ ( ord_max_real @ X2 @ Y3 ) @ Z )
% 8.90/5.06 = ( ( ord_less_real @ X2 @ Z )
% 8.90/5.06 & ( ord_less_real @ Y3 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3690_max__less__iff__conj,axiom,
% 8.90/5.06 ! [X2: rat,Y3: rat,Z: rat] :
% 8.90/5.06 ( ( ord_less_rat @ ( ord_max_rat @ X2 @ Y3 ) @ Z )
% 8.90/5.06 = ( ( ord_less_rat @ X2 @ Z )
% 8.90/5.06 & ( ord_less_rat @ Y3 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3691_max__less__iff__conj,axiom,
% 8.90/5.06 ! [X2: num,Y3: num,Z: num] :
% 8.90/5.06 ( ( ord_less_num @ ( ord_max_num @ X2 @ Y3 ) @ Z )
% 8.90/5.06 = ( ( ord_less_num @ X2 @ Z )
% 8.90/5.06 & ( ord_less_num @ Y3 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3692_max__less__iff__conj,axiom,
% 8.90/5.06 ! [X2: nat,Y3: nat,Z: nat] :
% 8.90/5.06 ( ( ord_less_nat @ ( ord_max_nat @ X2 @ Y3 ) @ Z )
% 8.90/5.06 = ( ( ord_less_nat @ X2 @ Z )
% 8.90/5.06 & ( ord_less_nat @ Y3 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3693_max__less__iff__conj,axiom,
% 8.90/5.06 ! [X2: int,Y3: int,Z: int] :
% 8.90/5.06 ( ( ord_less_int @ ( ord_max_int @ X2 @ Y3 ) @ Z )
% 8.90/5.06 = ( ( ord_less_int @ X2 @ Z )
% 8.90/5.06 & ( ord_less_int @ Y3 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3694_max_Oabsorb4,axiom,
% 8.90/5.06 ! [A: extended_enat,B: extended_enat] :
% 8.90/5.06 ( ( ord_le72135733267957522d_enat @ A @ B )
% 8.90/5.06 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 8.90/5.06 = B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3695_max_Oabsorb4,axiom,
% 8.90/5.06 ! [A: real,B: real] :
% 8.90/5.06 ( ( ord_less_real @ A @ B )
% 8.90/5.06 => ( ( ord_max_real @ A @ B )
% 8.90/5.06 = B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3696_max_Oabsorb4,axiom,
% 8.90/5.06 ! [A: rat,B: rat] :
% 8.90/5.06 ( ( ord_less_rat @ A @ B )
% 8.90/5.06 => ( ( ord_max_rat @ A @ B )
% 8.90/5.06 = B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3697_max_Oabsorb4,axiom,
% 8.90/5.06 ! [A: num,B: num] :
% 8.90/5.06 ( ( ord_less_num @ A @ B )
% 8.90/5.06 => ( ( ord_max_num @ A @ B )
% 8.90/5.06 = B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3698_max_Oabsorb4,axiom,
% 8.90/5.06 ! [A: nat,B: nat] :
% 8.90/5.06 ( ( ord_less_nat @ A @ B )
% 8.90/5.06 => ( ( ord_max_nat @ A @ B )
% 8.90/5.06 = B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3699_max_Oabsorb4,axiom,
% 8.90/5.06 ! [A: int,B: int] :
% 8.90/5.06 ( ( ord_less_int @ A @ B )
% 8.90/5.06 => ( ( ord_max_int @ A @ B )
% 8.90/5.06 = B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3700_max_Oabsorb3,axiom,
% 8.90/5.06 ! [B: extended_enat,A: extended_enat] :
% 8.90/5.06 ( ( ord_le72135733267957522d_enat @ B @ A )
% 8.90/5.06 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 8.90/5.06 = A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3701_max_Oabsorb3,axiom,
% 8.90/5.06 ! [B: real,A: real] :
% 8.90/5.06 ( ( ord_less_real @ B @ A )
% 8.90/5.06 => ( ( ord_max_real @ A @ B )
% 8.90/5.06 = A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3702_max_Oabsorb3,axiom,
% 8.90/5.06 ! [B: rat,A: rat] :
% 8.90/5.06 ( ( ord_less_rat @ B @ A )
% 8.90/5.06 => ( ( ord_max_rat @ A @ B )
% 8.90/5.06 = A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3703_max_Oabsorb3,axiom,
% 8.90/5.06 ! [B: num,A: num] :
% 8.90/5.06 ( ( ord_less_num @ B @ A )
% 8.90/5.06 => ( ( ord_max_num @ A @ B )
% 8.90/5.06 = A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3704_max_Oabsorb3,axiom,
% 8.90/5.06 ! [B: nat,A: nat] :
% 8.90/5.06 ( ( ord_less_nat @ B @ A )
% 8.90/5.06 => ( ( ord_max_nat @ A @ B )
% 8.90/5.06 = A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3705_max_Oabsorb3,axiom,
% 8.90/5.06 ! [B: int,A: int] :
% 8.90/5.06 ( ( ord_less_int @ B @ A )
% 8.90/5.06 => ( ( ord_max_int @ A @ B )
% 8.90/5.06 = A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3706_length__list__update,axiom,
% 8.90/5.06 ! [Xs2: list_VEBT_VEBT,I2: nat,X2: vEBT_VEBT] :
% 8.90/5.06 ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X2 ) )
% 8.90/5.06 = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3707_length__list__update,axiom,
% 8.90/5.06 ! [Xs2: list_o,I2: nat,X2: $o] :
% 8.90/5.06 ( ( size_size_list_o @ ( list_update_o @ Xs2 @ I2 @ X2 ) )
% 8.90/5.06 = ( size_size_list_o @ Xs2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3708_length__list__update,axiom,
% 8.90/5.06 ! [Xs2: list_nat,I2: nat,X2: nat] :
% 8.90/5.06 ( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I2 @ X2 ) )
% 8.90/5.06 = ( size_size_list_nat @ Xs2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3709_length__list__update,axiom,
% 8.90/5.06 ! [Xs2: list_int,I2: nat,X2: int] :
% 8.90/5.06 ( ( size_size_list_int @ ( list_update_int @ Xs2 @ I2 @ X2 ) )
% 8.90/5.06 = ( size_size_list_int @ Xs2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3710_max__Suc__Suc,axiom,
% 8.90/5.06 ! [M: nat,N: nat] :
% 8.90/5.06 ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
% 8.90/5.06 = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3711_max__0R,axiom,
% 8.90/5.06 ! [N: nat] :
% 8.90/5.06 ( ( ord_max_nat @ N @ zero_zero_nat )
% 8.90/5.06 = N ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3712_max__0L,axiom,
% 8.90/5.06 ! [N: nat] :
% 8.90/5.06 ( ( ord_max_nat @ zero_zero_nat @ N )
% 8.90/5.06 = N ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3713_max__nat_Oright__neutral,axiom,
% 8.90/5.06 ! [A: nat] :
% 8.90/5.06 ( ( ord_max_nat @ A @ zero_zero_nat )
% 8.90/5.06 = A ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3714_max__nat_Oneutr__eq__iff,axiom,
% 8.90/5.06 ! [A: nat,B: nat] :
% 8.90/5.06 ( ( zero_zero_nat
% 8.90/5.06 = ( ord_max_nat @ A @ B ) )
% 8.90/5.06 = ( ( A = zero_zero_nat )
% 8.90/5.06 & ( B = zero_zero_nat ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3715_max__nat_Oleft__neutral,axiom,
% 8.90/5.06 ! [A: nat] :
% 8.90/5.06 ( ( ord_max_nat @ zero_zero_nat @ A )
% 8.90/5.06 = A ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3716_max__nat_Oeq__neutr__iff,axiom,
% 8.90/5.06 ! [A: nat,B: nat] :
% 8.90/5.06 ( ( ( ord_max_nat @ A @ B )
% 8.90/5.06 = zero_zero_nat )
% 8.90/5.06 = ( ( A = zero_zero_nat )
% 8.90/5.06 & ( B = zero_zero_nat ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3717_list__update__id,axiom,
% 8.90/5.06 ! [Xs2: list_int,I2: nat] :
% 8.90/5.06 ( ( list_update_int @ Xs2 @ I2 @ ( nth_int @ Xs2 @ I2 ) )
% 8.90/5.06 = Xs2 ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3718_list__update__id,axiom,
% 8.90/5.06 ! [Xs2: list_nat,I2: nat] :
% 8.90/5.06 ( ( list_update_nat @ Xs2 @ I2 @ ( nth_nat @ Xs2 @ I2 ) )
% 8.90/5.06 = Xs2 ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3719_list__update__id,axiom,
% 8.90/5.06 ! [Xs2: list_VEBT_VEBT,I2: nat] :
% 8.90/5.06 ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) )
% 8.90/5.06 = Xs2 ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3720_nth__list__update__neq,axiom,
% 8.90/5.06 ! [I2: nat,J: nat,Xs2: list_int,X2: int] :
% 8.90/5.06 ( ( I2 != J )
% 8.90/5.06 => ( ( nth_int @ ( list_update_int @ Xs2 @ I2 @ X2 ) @ J )
% 8.90/5.06 = ( nth_int @ Xs2 @ J ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3721_nth__list__update__neq,axiom,
% 8.90/5.06 ! [I2: nat,J: nat,Xs2: list_nat,X2: nat] :
% 8.90/5.06 ( ( I2 != J )
% 8.90/5.06 => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I2 @ X2 ) @ J )
% 8.90/5.06 = ( nth_nat @ Xs2 @ J ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3722_nth__list__update__neq,axiom,
% 8.90/5.06 ! [I2: nat,J: nat,Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 8.90/5.06 ( ( I2 != J )
% 8.90/5.06 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X2 ) @ J )
% 8.90/5.06 = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3723_max__number__of_I1_J,axiom,
% 8.90/5.06 ! [U: num,V: num] :
% 8.90/5.06 ( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 8.90/5.06 => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 8.90/5.06 = ( numera1916890842035813515d_enat @ V ) ) )
% 8.90/5.06 & ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 8.90/5.06 => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 8.90/5.06 = ( numera1916890842035813515d_enat @ U ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3724_max__number__of_I1_J,axiom,
% 8.90/5.06 ! [U: num,V: num] :
% 8.90/5.06 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 8.90/5.06 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 8.90/5.06 = ( numeral_numeral_real @ V ) ) )
% 8.90/5.06 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 8.90/5.06 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 8.90/5.06 = ( numeral_numeral_real @ U ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3725_max__number__of_I1_J,axiom,
% 8.90/5.06 ! [U: num,V: num] :
% 8.90/5.06 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 8.90/5.06 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 8.90/5.06 = ( numeral_numeral_rat @ V ) ) )
% 8.90/5.06 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 8.90/5.06 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 8.90/5.06 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3726_max__number__of_I1_J,axiom,
% 8.90/5.06 ! [U: num,V: num] :
% 8.90/5.06 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 8.90/5.06 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 8.90/5.06 = ( numeral_numeral_nat @ V ) ) )
% 8.90/5.06 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 8.90/5.06 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 8.90/5.06 = ( numeral_numeral_nat @ U ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3727_max__number__of_I1_J,axiom,
% 8.90/5.06 ! [U: num,V: num] :
% 8.90/5.06 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 8.90/5.06 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 8.90/5.06 = ( numeral_numeral_int @ V ) ) )
% 8.90/5.06 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 8.90/5.06 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 8.90/5.06 = ( numeral_numeral_int @ U ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3728_max__0__1_I3_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X2 ) )
% 8.90/5.06 = ( numera1916890842035813515d_enat @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3729_max__0__1_I3_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X2 ) )
% 8.90/5.06 = ( numeral_numeral_real @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3730_max__0__1_I3_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X2 ) )
% 8.90/5.06 = ( numeral_numeral_rat @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3731_max__0__1_I3_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X2 ) )
% 8.90/5.06 = ( numeral_numeral_nat @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3732_max__0__1_I3_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X2 ) )
% 8.90/5.06 = ( numeral_numeral_int @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3733_max__0__1_I4_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X2 ) @ zero_z5237406670263579293d_enat )
% 8.90/5.06 = ( numera1916890842035813515d_enat @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3734_max__0__1_I4_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_max_real @ ( numeral_numeral_real @ X2 ) @ zero_zero_real )
% 8.90/5.06 = ( numeral_numeral_real @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3735_max__0__1_I4_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_max_rat @ ( numeral_numeral_rat @ X2 ) @ zero_zero_rat )
% 8.90/5.06 = ( numeral_numeral_rat @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3736_max__0__1_I4_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_max_nat @ ( numeral_numeral_nat @ X2 ) @ zero_zero_nat )
% 8.90/5.06 = ( numeral_numeral_nat @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3737_max__0__1_I4_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_max_int @ ( numeral_numeral_int @ X2 ) @ zero_zero_int )
% 8.90/5.06 = ( numeral_numeral_int @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3738_max__0__1_I2_J,axiom,
% 8.90/5.06 ( ( ord_max_real @ one_one_real @ zero_zero_real )
% 8.90/5.06 = one_one_real ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3739_max__0__1_I2_J,axiom,
% 8.90/5.06 ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
% 8.90/5.06 = one_one_rat ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3740_max__0__1_I2_J,axiom,
% 8.90/5.06 ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
% 8.90/5.06 = one_one_nat ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3741_max__0__1_I2_J,axiom,
% 8.90/5.06 ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
% 8.90/5.06 = one_on7984719198319812577d_enat ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3742_max__0__1_I2_J,axiom,
% 8.90/5.06 ( ( ord_max_int @ one_one_int @ zero_zero_int )
% 8.90/5.06 = one_one_int ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3743_max__0__1_I1_J,axiom,
% 8.90/5.06 ( ( ord_max_real @ zero_zero_real @ one_one_real )
% 8.90/5.06 = one_one_real ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3744_max__0__1_I1_J,axiom,
% 8.90/5.06 ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
% 8.90/5.06 = one_one_rat ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3745_max__0__1_I1_J,axiom,
% 8.90/5.06 ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
% 8.90/5.06 = one_one_nat ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3746_max__0__1_I1_J,axiom,
% 8.90/5.06 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
% 8.90/5.06 = one_on7984719198319812577d_enat ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3747_max__0__1_I1_J,axiom,
% 8.90/5.06 ( ( ord_max_int @ zero_zero_int @ one_one_int )
% 8.90/5.06 = one_one_int ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3748_max__0__1_I5_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X2 ) )
% 8.90/5.06 = ( numera1916890842035813515d_enat @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3749_max__0__1_I5_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X2 ) )
% 8.90/5.06 = ( numeral_numeral_real @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3750_max__0__1_I5_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X2 ) )
% 8.90/5.06 = ( numeral_numeral_rat @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3751_max__0__1_I5_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X2 ) )
% 8.90/5.06 = ( numeral_numeral_nat @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3752_max__0__1_I5_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X2 ) )
% 8.90/5.06 = ( numeral_numeral_int @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3753_max__0__1_I6_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X2 ) @ one_on7984719198319812577d_enat )
% 8.90/5.06 = ( numera1916890842035813515d_enat @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3754_max__0__1_I6_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_max_real @ ( numeral_numeral_real @ X2 ) @ one_one_real )
% 8.90/5.06 = ( numeral_numeral_real @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3755_max__0__1_I6_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_max_rat @ ( numeral_numeral_rat @ X2 ) @ one_one_rat )
% 8.90/5.06 = ( numeral_numeral_rat @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3756_max__0__1_I6_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_max_nat @ ( numeral_numeral_nat @ X2 ) @ one_one_nat )
% 8.90/5.06 = ( numeral_numeral_nat @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3757_max__0__1_I6_J,axiom,
% 8.90/5.06 ! [X2: num] :
% 8.90/5.06 ( ( ord_max_int @ ( numeral_numeral_int @ X2 ) @ one_one_int )
% 8.90/5.06 = ( numeral_numeral_int @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3758_list__update__beyond,axiom,
% 8.90/5.06 ! [Xs2: list_VEBT_VEBT,I2: nat,X2: vEBT_VEBT] :
% 8.90/5.06 ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ I2 )
% 8.90/5.06 => ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X2 )
% 8.90/5.06 = Xs2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3759_list__update__beyond,axiom,
% 8.90/5.06 ! [Xs2: list_o,I2: nat,X2: $o] :
% 8.90/5.06 ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I2 )
% 8.90/5.06 => ( ( list_update_o @ Xs2 @ I2 @ X2 )
% 8.90/5.06 = Xs2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3760_list__update__beyond,axiom,
% 8.90/5.06 ! [Xs2: list_nat,I2: nat,X2: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I2 )
% 8.90/5.06 => ( ( list_update_nat @ Xs2 @ I2 @ X2 )
% 8.90/5.06 = Xs2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3761_list__update__beyond,axiom,
% 8.90/5.06 ! [Xs2: list_int,I2: nat,X2: int] :
% 8.90/5.06 ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ I2 )
% 8.90/5.06 => ( ( list_update_int @ Xs2 @ I2 @ X2 )
% 8.90/5.06 = Xs2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3762_nth__list__update__eq,axiom,
% 8.90/5.06 ! [I2: nat,Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 8.90/5.06 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 8.90/5.06 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X2 ) @ I2 )
% 8.90/5.06 = X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3763_nth__list__update__eq,axiom,
% 8.90/5.06 ! [I2: nat,Xs2: list_o,X2: $o] :
% 8.90/5.06 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
% 8.90/5.06 => ( ( nth_o @ ( list_update_o @ Xs2 @ I2 @ X2 ) @ I2 )
% 8.90/5.06 = X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3764_nth__list__update__eq,axiom,
% 8.90/5.06 ! [I2: nat,Xs2: list_nat,X2: nat] :
% 8.90/5.06 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 8.90/5.06 => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I2 @ X2 ) @ I2 )
% 8.90/5.06 = X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3765_nth__list__update__eq,axiom,
% 8.90/5.06 ! [I2: nat,Xs2: list_int,X2: int] :
% 8.90/5.06 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
% 8.90/5.06 => ( ( nth_int @ ( list_update_int @ Xs2 @ I2 @ X2 ) @ I2 )
% 8.90/5.06 = X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3766_set__swap,axiom,
% 8.90/5.06 ! [I2: nat,Xs2: list_VEBT_VEBT,J: nat] :
% 8.90/5.06 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 8.90/5.06 => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 8.90/5.06 => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) )
% 8.90/5.06 = ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3767_set__swap,axiom,
% 8.90/5.06 ! [I2: nat,Xs2: list_o,J: nat] :
% 8.90/5.06 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
% 8.90/5.06 => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
% 8.90/5.06 => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs2 @ I2 @ ( nth_o @ Xs2 @ J ) ) @ J @ ( nth_o @ Xs2 @ I2 ) ) )
% 8.90/5.06 = ( set_o2 @ Xs2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3768_set__swap,axiom,
% 8.90/5.06 ! [I2: nat,Xs2: list_nat,J: nat] :
% 8.90/5.06 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 8.90/5.06 => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
% 8.90/5.06 => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I2 @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I2 ) ) )
% 8.90/5.06 = ( set_nat2 @ Xs2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3769_set__swap,axiom,
% 8.90/5.06 ! [I2: nat,Xs2: list_int,J: nat] :
% 8.90/5.06 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
% 8.90/5.06 => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
% 8.90/5.06 => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs2 @ I2 @ ( nth_int @ Xs2 @ J ) ) @ J @ ( nth_int @ Xs2 @ I2 ) ) )
% 8.90/5.06 = ( set_int2 @ Xs2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3770_max__def,axiom,
% 8.90/5.06 ( ord_ma741700101516333627d_enat
% 8.90/5.06 = ( ^ [A3: extended_enat,B2: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3771_max__def,axiom,
% 8.90/5.06 ( ord_max_set_int
% 8.90/5.06 = ( ^ [A3: set_int,B2: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3772_max__def,axiom,
% 8.90/5.06 ( ord_max_rat
% 8.90/5.06 = ( ^ [A3: rat,B2: rat] : ( if_rat @ ( ord_less_eq_rat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3773_max__def,axiom,
% 8.90/5.06 ( ord_max_num
% 8.90/5.06 = ( ^ [A3: num,B2: num] : ( if_num @ ( ord_less_eq_num @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3774_max__def,axiom,
% 8.90/5.06 ( ord_max_nat
% 8.90/5.06 = ( ^ [A3: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3775_max__def,axiom,
% 8.90/5.06 ( ord_max_int
% 8.90/5.06 = ( ^ [A3: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3776_max__absorb1,axiom,
% 8.90/5.06 ! [Y3: extended_enat,X2: extended_enat] :
% 8.90/5.06 ( ( ord_le2932123472753598470d_enat @ Y3 @ X2 )
% 8.90/5.06 => ( ( ord_ma741700101516333627d_enat @ X2 @ Y3 )
% 8.90/5.06 = X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3777_max__absorb1,axiom,
% 8.90/5.06 ! [Y3: set_int,X2: set_int] :
% 8.90/5.06 ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 8.90/5.06 => ( ( ord_max_set_int @ X2 @ Y3 )
% 8.90/5.06 = X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3778_max__absorb1,axiom,
% 8.90/5.06 ! [Y3: rat,X2: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 8.90/5.06 => ( ( ord_max_rat @ X2 @ Y3 )
% 8.90/5.06 = X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3779_max__absorb1,axiom,
% 8.90/5.06 ! [Y3: num,X2: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ Y3 @ X2 )
% 8.90/5.06 => ( ( ord_max_num @ X2 @ Y3 )
% 8.90/5.06 = X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3780_max__absorb1,axiom,
% 8.90/5.06 ! [Y3: nat,X2: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 8.90/5.06 => ( ( ord_max_nat @ X2 @ Y3 )
% 8.90/5.06 = X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3781_max__absorb1,axiom,
% 8.90/5.06 ! [Y3: int,X2: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ Y3 @ X2 )
% 8.90/5.06 => ( ( ord_max_int @ X2 @ Y3 )
% 8.90/5.06 = X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3782_max__absorb2,axiom,
% 8.90/5.06 ! [X2: extended_enat,Y3: extended_enat] :
% 8.90/5.06 ( ( ord_le2932123472753598470d_enat @ X2 @ Y3 )
% 8.90/5.06 => ( ( ord_ma741700101516333627d_enat @ X2 @ Y3 )
% 8.90/5.06 = Y3 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3783_max__absorb2,axiom,
% 8.90/5.06 ! [X2: set_int,Y3: set_int] :
% 8.90/5.06 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 8.90/5.06 => ( ( ord_max_set_int @ X2 @ Y3 )
% 8.90/5.06 = Y3 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3784_max__absorb2,axiom,
% 8.90/5.06 ! [X2: rat,Y3: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 8.90/5.06 => ( ( ord_max_rat @ X2 @ Y3 )
% 8.90/5.06 = Y3 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3785_max__absorb2,axiom,
% 8.90/5.06 ! [X2: num,Y3: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X2 @ Y3 )
% 8.90/5.06 => ( ( ord_max_num @ X2 @ Y3 )
% 8.90/5.06 = Y3 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3786_max__absorb2,axiom,
% 8.90/5.06 ! [X2: nat,Y3: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 8.90/5.06 => ( ( ord_max_nat @ X2 @ Y3 )
% 8.90/5.06 = Y3 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3787_max__absorb2,axiom,
% 8.90/5.06 ! [X2: int,Y3: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ X2 @ Y3 )
% 8.90/5.06 => ( ( ord_max_int @ X2 @ Y3 )
% 8.90/5.06 = Y3 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3788_max_Omono,axiom,
% 8.90/5.06 ! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
% 8.90/5.06 ( ( ord_le2932123472753598470d_enat @ C @ A )
% 8.90/5.06 => ( ( ord_le2932123472753598470d_enat @ D @ B )
% 8.90/5.06 => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ C @ D ) @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3789_max_Omono,axiom,
% 8.90/5.06 ! [C: rat,A: rat,D: rat,B: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ C @ A )
% 8.90/5.06 => ( ( ord_less_eq_rat @ D @ B )
% 8.90/5.06 => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3790_max_Omono,axiom,
% 8.90/5.06 ! [C: num,A: num,D: num,B: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ C @ A )
% 8.90/5.06 => ( ( ord_less_eq_num @ D @ B )
% 8.90/5.06 => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3791_max_Omono,axiom,
% 8.90/5.06 ! [C: nat,A: nat,D: nat,B: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ C @ A )
% 8.90/5.06 => ( ( ord_less_eq_nat @ D @ B )
% 8.90/5.06 => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3792_max_Omono,axiom,
% 8.90/5.06 ! [C: int,A: int,D: int,B: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ C @ A )
% 8.90/5.06 => ( ( ord_less_eq_int @ D @ B )
% 8.90/5.06 => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3793_max_OorderE,axiom,
% 8.90/5.06 ! [B: extended_enat,A: extended_enat] :
% 8.90/5.06 ( ( ord_le2932123472753598470d_enat @ B @ A )
% 8.90/5.06 => ( A
% 8.90/5.06 = ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3794_max_OorderE,axiom,
% 8.90/5.06 ! [B: rat,A: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ B @ A )
% 8.90/5.06 => ( A
% 8.90/5.06 = ( ord_max_rat @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3795_max_OorderE,axiom,
% 8.90/5.06 ! [B: num,A: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ B @ A )
% 8.90/5.06 => ( A
% 8.90/5.06 = ( ord_max_num @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3796_max_OorderE,axiom,
% 8.90/5.06 ! [B: nat,A: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ B @ A )
% 8.90/5.06 => ( A
% 8.90/5.06 = ( ord_max_nat @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3797_max_OorderE,axiom,
% 8.90/5.06 ! [B: int,A: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ B @ A )
% 8.90/5.06 => ( A
% 8.90/5.06 = ( ord_max_int @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3798_max_OorderI,axiom,
% 8.90/5.06 ! [A: extended_enat,B: extended_enat] :
% 8.90/5.06 ( ( A
% 8.90/5.06 = ( ord_ma741700101516333627d_enat @ A @ B ) )
% 8.90/5.06 => ( ord_le2932123472753598470d_enat @ B @ A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3799_max_OorderI,axiom,
% 8.90/5.06 ! [A: rat,B: rat] :
% 8.90/5.06 ( ( A
% 8.90/5.06 = ( ord_max_rat @ A @ B ) )
% 8.90/5.06 => ( ord_less_eq_rat @ B @ A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3800_max_OorderI,axiom,
% 8.90/5.06 ! [A: num,B: num] :
% 8.90/5.06 ( ( A
% 8.90/5.06 = ( ord_max_num @ A @ B ) )
% 8.90/5.06 => ( ord_less_eq_num @ B @ A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3801_max_OorderI,axiom,
% 8.90/5.06 ! [A: nat,B: nat] :
% 8.90/5.06 ( ( A
% 8.90/5.06 = ( ord_max_nat @ A @ B ) )
% 8.90/5.06 => ( ord_less_eq_nat @ B @ A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3802_max_OorderI,axiom,
% 8.90/5.06 ! [A: int,B: int] :
% 8.90/5.06 ( ( A
% 8.90/5.06 = ( ord_max_int @ A @ B ) )
% 8.90/5.06 => ( ord_less_eq_int @ B @ A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3803_max_OboundedE,axiom,
% 8.90/5.06 ! [B: extended_enat,C: extended_enat,A: extended_enat] :
% 8.90/5.06 ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
% 8.90/5.06 => ~ ( ( ord_le2932123472753598470d_enat @ B @ A )
% 8.90/5.06 => ~ ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3804_max_OboundedE,axiom,
% 8.90/5.06 ! [B: rat,C: rat,A: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 8.90/5.06 => ~ ( ( ord_less_eq_rat @ B @ A )
% 8.90/5.06 => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3805_max_OboundedE,axiom,
% 8.90/5.06 ! [B: num,C: num,A: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 8.90/5.06 => ~ ( ( ord_less_eq_num @ B @ A )
% 8.90/5.06 => ~ ( ord_less_eq_num @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3806_max_OboundedE,axiom,
% 8.90/5.06 ! [B: nat,C: nat,A: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 8.90/5.06 => ~ ( ( ord_less_eq_nat @ B @ A )
% 8.90/5.06 => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3807_max_OboundedE,axiom,
% 8.90/5.06 ! [B: int,C: int,A: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 8.90/5.06 => ~ ( ( ord_less_eq_int @ B @ A )
% 8.90/5.06 => ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3808_max_OboundedI,axiom,
% 8.90/5.06 ! [B: extended_enat,A: extended_enat,C: extended_enat] :
% 8.90/5.06 ( ( ord_le2932123472753598470d_enat @ B @ A )
% 8.90/5.06 => ( ( ord_le2932123472753598470d_enat @ C @ A )
% 8.90/5.06 => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3809_max_OboundedI,axiom,
% 8.90/5.06 ! [B: rat,A: rat,C: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ B @ A )
% 8.90/5.06 => ( ( ord_less_eq_rat @ C @ A )
% 8.90/5.06 => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3810_max_OboundedI,axiom,
% 8.90/5.06 ! [B: num,A: num,C: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ B @ A )
% 8.90/5.06 => ( ( ord_less_eq_num @ C @ A )
% 8.90/5.06 => ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3811_max_OboundedI,axiom,
% 8.90/5.06 ! [B: nat,A: nat,C: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ B @ A )
% 8.90/5.06 => ( ( ord_less_eq_nat @ C @ A )
% 8.90/5.06 => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3812_max_OboundedI,axiom,
% 8.90/5.06 ! [B: int,A: int,C: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ B @ A )
% 8.90/5.06 => ( ( ord_less_eq_int @ C @ A )
% 8.90/5.06 => ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3813_max_Oorder__iff,axiom,
% 8.90/5.06 ( ord_le2932123472753598470d_enat
% 8.90/5.06 = ( ^ [B2: extended_enat,A3: extended_enat] :
% 8.90/5.06 ( A3
% 8.90/5.06 = ( ord_ma741700101516333627d_enat @ A3 @ B2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3814_max_Oorder__iff,axiom,
% 8.90/5.06 ( ord_less_eq_rat
% 8.90/5.06 = ( ^ [B2: rat,A3: rat] :
% 8.90/5.06 ( A3
% 8.90/5.06 = ( ord_max_rat @ A3 @ B2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3815_max_Oorder__iff,axiom,
% 8.90/5.06 ( ord_less_eq_num
% 8.90/5.06 = ( ^ [B2: num,A3: num] :
% 8.90/5.06 ( A3
% 8.90/5.06 = ( ord_max_num @ A3 @ B2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3816_max_Oorder__iff,axiom,
% 8.90/5.06 ( ord_less_eq_nat
% 8.90/5.06 = ( ^ [B2: nat,A3: nat] :
% 8.90/5.06 ( A3
% 8.90/5.06 = ( ord_max_nat @ A3 @ B2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3817_max_Oorder__iff,axiom,
% 8.90/5.06 ( ord_less_eq_int
% 8.90/5.06 = ( ^ [B2: int,A3: int] :
% 8.90/5.06 ( A3
% 8.90/5.06 = ( ord_max_int @ A3 @ B2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3818_max_Ocobounded1,axiom,
% 8.90/5.06 ! [A: extended_enat,B: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3819_max_Ocobounded1,axiom,
% 8.90/5.06 ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3820_max_Ocobounded1,axiom,
% 8.90/5.06 ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3821_max_Ocobounded1,axiom,
% 8.90/5.06 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3822_max_Ocobounded1,axiom,
% 8.90/5.06 ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3823_max_Ocobounded2,axiom,
% 8.90/5.06 ! [B: extended_enat,A: extended_enat] : ( ord_le2932123472753598470d_enat @ B @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3824_max_Ocobounded2,axiom,
% 8.90/5.06 ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3825_max_Ocobounded2,axiom,
% 8.90/5.06 ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3826_max_Ocobounded2,axiom,
% 8.90/5.06 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3827_max_Ocobounded2,axiom,
% 8.90/5.06 ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3828_le__max__iff__disj,axiom,
% 8.90/5.06 ! [Z: extended_enat,X2: extended_enat,Y3: extended_enat] :
% 8.90/5.06 ( ( ord_le2932123472753598470d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X2 @ Y3 ) )
% 8.90/5.06 = ( ( ord_le2932123472753598470d_enat @ Z @ X2 )
% 8.90/5.06 | ( ord_le2932123472753598470d_enat @ Z @ Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3829_le__max__iff__disj,axiom,
% 8.90/5.06 ! [Z: rat,X2: rat,Y3: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X2 @ Y3 ) )
% 8.90/5.06 = ( ( ord_less_eq_rat @ Z @ X2 )
% 8.90/5.06 | ( ord_less_eq_rat @ Z @ Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3830_le__max__iff__disj,axiom,
% 8.90/5.06 ! [Z: num,X2: num,Y3: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X2 @ Y3 ) )
% 8.90/5.06 = ( ( ord_less_eq_num @ Z @ X2 )
% 8.90/5.06 | ( ord_less_eq_num @ Z @ Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3831_le__max__iff__disj,axiom,
% 8.90/5.06 ! [Z: nat,X2: nat,Y3: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X2 @ Y3 ) )
% 8.90/5.06 = ( ( ord_less_eq_nat @ Z @ X2 )
% 8.90/5.06 | ( ord_less_eq_nat @ Z @ Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3832_le__max__iff__disj,axiom,
% 8.90/5.06 ! [Z: int,X2: int,Y3: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X2 @ Y3 ) )
% 8.90/5.06 = ( ( ord_less_eq_int @ Z @ X2 )
% 8.90/5.06 | ( ord_less_eq_int @ Z @ Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3833_max_Oabsorb__iff1,axiom,
% 8.90/5.06 ( ord_le2932123472753598470d_enat
% 8.90/5.06 = ( ^ [B2: extended_enat,A3: extended_enat] :
% 8.90/5.06 ( ( ord_ma741700101516333627d_enat @ A3 @ B2 )
% 8.90/5.06 = A3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3834_max_Oabsorb__iff1,axiom,
% 8.90/5.06 ( ord_less_eq_rat
% 8.90/5.06 = ( ^ [B2: rat,A3: rat] :
% 8.90/5.06 ( ( ord_max_rat @ A3 @ B2 )
% 8.90/5.06 = A3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3835_max_Oabsorb__iff1,axiom,
% 8.90/5.06 ( ord_less_eq_num
% 8.90/5.06 = ( ^ [B2: num,A3: num] :
% 8.90/5.06 ( ( ord_max_num @ A3 @ B2 )
% 8.90/5.06 = A3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3836_max_Oabsorb__iff1,axiom,
% 8.90/5.06 ( ord_less_eq_nat
% 8.90/5.06 = ( ^ [B2: nat,A3: nat] :
% 8.90/5.06 ( ( ord_max_nat @ A3 @ B2 )
% 8.90/5.06 = A3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3837_max_Oabsorb__iff1,axiom,
% 8.90/5.06 ( ord_less_eq_int
% 8.90/5.06 = ( ^ [B2: int,A3: int] :
% 8.90/5.06 ( ( ord_max_int @ A3 @ B2 )
% 8.90/5.06 = A3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3838_max_Oabsorb__iff2,axiom,
% 8.90/5.06 ( ord_le2932123472753598470d_enat
% 8.90/5.06 = ( ^ [A3: extended_enat,B2: extended_enat] :
% 8.90/5.06 ( ( ord_ma741700101516333627d_enat @ A3 @ B2 )
% 8.90/5.06 = B2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3839_max_Oabsorb__iff2,axiom,
% 8.90/5.06 ( ord_less_eq_rat
% 8.90/5.06 = ( ^ [A3: rat,B2: rat] :
% 8.90/5.06 ( ( ord_max_rat @ A3 @ B2 )
% 8.90/5.06 = B2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3840_max_Oabsorb__iff2,axiom,
% 8.90/5.06 ( ord_less_eq_num
% 8.90/5.06 = ( ^ [A3: num,B2: num] :
% 8.90/5.06 ( ( ord_max_num @ A3 @ B2 )
% 8.90/5.06 = B2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3841_max_Oabsorb__iff2,axiom,
% 8.90/5.06 ( ord_less_eq_nat
% 8.90/5.06 = ( ^ [A3: nat,B2: nat] :
% 8.90/5.06 ( ( ord_max_nat @ A3 @ B2 )
% 8.90/5.06 = B2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3842_max_Oabsorb__iff2,axiom,
% 8.90/5.06 ( ord_less_eq_int
% 8.90/5.06 = ( ^ [A3: int,B2: int] :
% 8.90/5.06 ( ( ord_max_int @ A3 @ B2 )
% 8.90/5.06 = B2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3843_max_OcoboundedI1,axiom,
% 8.90/5.06 ! [C: extended_enat,A: extended_enat,B: extended_enat] :
% 8.90/5.06 ( ( ord_le2932123472753598470d_enat @ C @ A )
% 8.90/5.06 => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3844_max_OcoboundedI1,axiom,
% 8.90/5.06 ! [C: rat,A: rat,B: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ C @ A )
% 8.90/5.06 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3845_max_OcoboundedI1,axiom,
% 8.90/5.06 ! [C: num,A: num,B: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ C @ A )
% 8.90/5.06 => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3846_max_OcoboundedI1,axiom,
% 8.90/5.06 ! [C: nat,A: nat,B: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ C @ A )
% 8.90/5.06 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3847_max_OcoboundedI1,axiom,
% 8.90/5.06 ! [C: int,A: int,B: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ C @ A )
% 8.90/5.06 => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3848_max_OcoboundedI2,axiom,
% 8.90/5.06 ! [C: extended_enat,B: extended_enat,A: extended_enat] :
% 8.90/5.06 ( ( ord_le2932123472753598470d_enat @ C @ B )
% 8.90/5.06 => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3849_max_OcoboundedI2,axiom,
% 8.90/5.06 ! [C: rat,B: rat,A: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ C @ B )
% 8.90/5.06 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3850_max_OcoboundedI2,axiom,
% 8.90/5.06 ! [C: num,B: num,A: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ C @ B )
% 8.90/5.06 => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3851_max_OcoboundedI2,axiom,
% 8.90/5.06 ! [C: nat,B: nat,A: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ C @ B )
% 8.90/5.06 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3852_max_OcoboundedI2,axiom,
% 8.90/5.06 ! [C: int,B: int,A: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ C @ B )
% 8.90/5.06 => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3853_max_Ostrict__coboundedI2,axiom,
% 8.90/5.06 ! [C: extended_enat,B: extended_enat,A: extended_enat] :
% 8.90/5.06 ( ( ord_le72135733267957522d_enat @ C @ B )
% 8.90/5.06 => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3854_max_Ostrict__coboundedI2,axiom,
% 8.90/5.06 ! [C: real,B: real,A: real] :
% 8.90/5.06 ( ( ord_less_real @ C @ B )
% 8.90/5.06 => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3855_max_Ostrict__coboundedI2,axiom,
% 8.90/5.06 ! [C: rat,B: rat,A: rat] :
% 8.90/5.06 ( ( ord_less_rat @ C @ B )
% 8.90/5.06 => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3856_max_Ostrict__coboundedI2,axiom,
% 8.90/5.06 ! [C: num,B: num,A: num] :
% 8.90/5.06 ( ( ord_less_num @ C @ B )
% 8.90/5.06 => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3857_max_Ostrict__coboundedI2,axiom,
% 8.90/5.06 ! [C: nat,B: nat,A: nat] :
% 8.90/5.06 ( ( ord_less_nat @ C @ B )
% 8.90/5.06 => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3858_max_Ostrict__coboundedI2,axiom,
% 8.90/5.06 ! [C: int,B: int,A: int] :
% 8.90/5.06 ( ( ord_less_int @ C @ B )
% 8.90/5.06 => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3859_max_Ostrict__coboundedI1,axiom,
% 8.90/5.06 ! [C: extended_enat,A: extended_enat,B: extended_enat] :
% 8.90/5.06 ( ( ord_le72135733267957522d_enat @ C @ A )
% 8.90/5.06 => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3860_max_Ostrict__coboundedI1,axiom,
% 8.90/5.06 ! [C: real,A: real,B: real] :
% 8.90/5.06 ( ( ord_less_real @ C @ A )
% 8.90/5.06 => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3861_max_Ostrict__coboundedI1,axiom,
% 8.90/5.06 ! [C: rat,A: rat,B: rat] :
% 8.90/5.06 ( ( ord_less_rat @ C @ A )
% 8.90/5.06 => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3862_max_Ostrict__coboundedI1,axiom,
% 8.90/5.06 ! [C: num,A: num,B: num] :
% 8.90/5.06 ( ( ord_less_num @ C @ A )
% 8.90/5.06 => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3863_max_Ostrict__coboundedI1,axiom,
% 8.90/5.06 ! [C: nat,A: nat,B: nat] :
% 8.90/5.06 ( ( ord_less_nat @ C @ A )
% 8.90/5.06 => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3864_max_Ostrict__coboundedI1,axiom,
% 8.90/5.06 ! [C: int,A: int,B: int] :
% 8.90/5.06 ( ( ord_less_int @ C @ A )
% 8.90/5.06 => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3865_max_Ostrict__order__iff,axiom,
% 8.90/5.06 ( ord_le72135733267957522d_enat
% 8.90/5.06 = ( ^ [B2: extended_enat,A3: extended_enat] :
% 8.90/5.06 ( ( A3
% 8.90/5.06 = ( ord_ma741700101516333627d_enat @ A3 @ B2 ) )
% 8.90/5.06 & ( A3 != B2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3866_max_Ostrict__order__iff,axiom,
% 8.90/5.06 ( ord_less_real
% 8.90/5.06 = ( ^ [B2: real,A3: real] :
% 8.90/5.06 ( ( A3
% 8.90/5.06 = ( ord_max_real @ A3 @ B2 ) )
% 8.90/5.06 & ( A3 != B2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3867_max_Ostrict__order__iff,axiom,
% 8.90/5.06 ( ord_less_rat
% 8.90/5.06 = ( ^ [B2: rat,A3: rat] :
% 8.90/5.06 ( ( A3
% 8.90/5.06 = ( ord_max_rat @ A3 @ B2 ) )
% 8.90/5.06 & ( A3 != B2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3868_max_Ostrict__order__iff,axiom,
% 8.90/5.06 ( ord_less_num
% 8.90/5.06 = ( ^ [B2: num,A3: num] :
% 8.90/5.06 ( ( A3
% 8.90/5.06 = ( ord_max_num @ A3 @ B2 ) )
% 8.90/5.06 & ( A3 != B2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3869_max_Ostrict__order__iff,axiom,
% 8.90/5.06 ( ord_less_nat
% 8.90/5.06 = ( ^ [B2: nat,A3: nat] :
% 8.90/5.06 ( ( A3
% 8.90/5.06 = ( ord_max_nat @ A3 @ B2 ) )
% 8.90/5.06 & ( A3 != B2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3870_max_Ostrict__order__iff,axiom,
% 8.90/5.06 ( ord_less_int
% 8.90/5.06 = ( ^ [B2: int,A3: int] :
% 8.90/5.06 ( ( A3
% 8.90/5.06 = ( ord_max_int @ A3 @ B2 ) )
% 8.90/5.06 & ( A3 != B2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3871_max_Ostrict__boundedE,axiom,
% 8.90/5.06 ! [B: extended_enat,C: extended_enat,A: extended_enat] :
% 8.90/5.06 ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
% 8.90/5.06 => ~ ( ( ord_le72135733267957522d_enat @ B @ A )
% 8.90/5.06 => ~ ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3872_max_Ostrict__boundedE,axiom,
% 8.90/5.06 ! [B: real,C: real,A: real] :
% 8.90/5.06 ( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
% 8.90/5.06 => ~ ( ( ord_less_real @ B @ A )
% 8.90/5.06 => ~ ( ord_less_real @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3873_max_Ostrict__boundedE,axiom,
% 8.90/5.06 ! [B: rat,C: rat,A: rat] :
% 8.90/5.06 ( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
% 8.90/5.06 => ~ ( ( ord_less_rat @ B @ A )
% 8.90/5.06 => ~ ( ord_less_rat @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3874_max_Ostrict__boundedE,axiom,
% 8.90/5.06 ! [B: num,C: num,A: num] :
% 8.90/5.06 ( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
% 8.90/5.06 => ~ ( ( ord_less_num @ B @ A )
% 8.90/5.06 => ~ ( ord_less_num @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3875_max_Ostrict__boundedE,axiom,
% 8.90/5.06 ! [B: nat,C: nat,A: nat] :
% 8.90/5.06 ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
% 8.90/5.06 => ~ ( ( ord_less_nat @ B @ A )
% 8.90/5.06 => ~ ( ord_less_nat @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3876_max_Ostrict__boundedE,axiom,
% 8.90/5.06 ! [B: int,C: int,A: int] :
% 8.90/5.06 ( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
% 8.90/5.06 => ~ ( ( ord_less_int @ B @ A )
% 8.90/5.06 => ~ ( ord_less_int @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3877_less__max__iff__disj,axiom,
% 8.90/5.06 ! [Z: extended_enat,X2: extended_enat,Y3: extended_enat] :
% 8.90/5.06 ( ( ord_le72135733267957522d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X2 @ Y3 ) )
% 8.90/5.06 = ( ( ord_le72135733267957522d_enat @ Z @ X2 )
% 8.90/5.06 | ( ord_le72135733267957522d_enat @ Z @ Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3878_less__max__iff__disj,axiom,
% 8.90/5.06 ! [Z: real,X2: real,Y3: real] :
% 8.90/5.06 ( ( ord_less_real @ Z @ ( ord_max_real @ X2 @ Y3 ) )
% 8.90/5.06 = ( ( ord_less_real @ Z @ X2 )
% 8.90/5.06 | ( ord_less_real @ Z @ Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3879_less__max__iff__disj,axiom,
% 8.90/5.06 ! [Z: rat,X2: rat,Y3: rat] :
% 8.90/5.06 ( ( ord_less_rat @ Z @ ( ord_max_rat @ X2 @ Y3 ) )
% 8.90/5.06 = ( ( ord_less_rat @ Z @ X2 )
% 8.90/5.06 | ( ord_less_rat @ Z @ Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3880_less__max__iff__disj,axiom,
% 8.90/5.06 ! [Z: num,X2: num,Y3: num] :
% 8.90/5.06 ( ( ord_less_num @ Z @ ( ord_max_num @ X2 @ Y3 ) )
% 8.90/5.06 = ( ( ord_less_num @ Z @ X2 )
% 8.90/5.06 | ( ord_less_num @ Z @ Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3881_less__max__iff__disj,axiom,
% 8.90/5.06 ! [Z: nat,X2: nat,Y3: nat] :
% 8.90/5.06 ( ( ord_less_nat @ Z @ ( ord_max_nat @ X2 @ Y3 ) )
% 8.90/5.06 = ( ( ord_less_nat @ Z @ X2 )
% 8.90/5.06 | ( ord_less_nat @ Z @ Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3882_less__max__iff__disj,axiom,
% 8.90/5.06 ! [Z: int,X2: int,Y3: int] :
% 8.90/5.06 ( ( ord_less_int @ Z @ ( ord_max_int @ X2 @ Y3 ) )
% 8.90/5.06 = ( ( ord_less_int @ Z @ X2 )
% 8.90/5.06 | ( ord_less_int @ Z @ Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3883_max__add__distrib__right,axiom,
% 8.90/5.06 ! [X2: real,Y3: real,Z: real] :
% 8.90/5.06 ( ( plus_plus_real @ X2 @ ( ord_max_real @ Y3 @ Z ) )
% 8.90/5.06 = ( ord_max_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( plus_plus_real @ X2 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3884_max__add__distrib__right,axiom,
% 8.90/5.06 ! [X2: rat,Y3: rat,Z: rat] :
% 8.90/5.06 ( ( plus_plus_rat @ X2 @ ( ord_max_rat @ Y3 @ Z ) )
% 8.90/5.06 = ( ord_max_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ ( plus_plus_rat @ X2 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3885_max__add__distrib__right,axiom,
% 8.90/5.06 ! [X2: nat,Y3: nat,Z: nat] :
% 8.90/5.06 ( ( plus_plus_nat @ X2 @ ( ord_max_nat @ Y3 @ Z ) )
% 8.90/5.06 = ( ord_max_nat @ ( plus_plus_nat @ X2 @ Y3 ) @ ( plus_plus_nat @ X2 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3886_max__add__distrib__right,axiom,
% 8.90/5.06 ! [X2: int,Y3: int,Z: int] :
% 8.90/5.06 ( ( plus_plus_int @ X2 @ ( ord_max_int @ Y3 @ Z ) )
% 8.90/5.06 = ( ord_max_int @ ( plus_plus_int @ X2 @ Y3 ) @ ( plus_plus_int @ X2 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3887_max__add__distrib__left,axiom,
% 8.90/5.06 ! [X2: real,Y3: real,Z: real] :
% 8.90/5.06 ( ( plus_plus_real @ ( ord_max_real @ X2 @ Y3 ) @ Z )
% 8.90/5.06 = ( ord_max_real @ ( plus_plus_real @ X2 @ Z ) @ ( plus_plus_real @ Y3 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3888_max__add__distrib__left,axiom,
% 8.90/5.06 ! [X2: rat,Y3: rat,Z: rat] :
% 8.90/5.06 ( ( plus_plus_rat @ ( ord_max_rat @ X2 @ Y3 ) @ Z )
% 8.90/5.06 = ( ord_max_rat @ ( plus_plus_rat @ X2 @ Z ) @ ( plus_plus_rat @ Y3 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3889_max__add__distrib__left,axiom,
% 8.90/5.06 ! [X2: nat,Y3: nat,Z: nat] :
% 8.90/5.06 ( ( plus_plus_nat @ ( ord_max_nat @ X2 @ Y3 ) @ Z )
% 8.90/5.06 = ( ord_max_nat @ ( plus_plus_nat @ X2 @ Z ) @ ( plus_plus_nat @ Y3 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3890_max__add__distrib__left,axiom,
% 8.90/5.06 ! [X2: int,Y3: int,Z: int] :
% 8.90/5.06 ( ( plus_plus_int @ ( ord_max_int @ X2 @ Y3 ) @ Z )
% 8.90/5.06 = ( ord_max_int @ ( plus_plus_int @ X2 @ Z ) @ ( plus_plus_int @ Y3 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3891_max__diff__distrib__left,axiom,
% 8.90/5.06 ! [X2: real,Y3: real,Z: real] :
% 8.90/5.06 ( ( minus_minus_real @ ( ord_max_real @ X2 @ Y3 ) @ Z )
% 8.90/5.06 = ( ord_max_real @ ( minus_minus_real @ X2 @ Z ) @ ( minus_minus_real @ Y3 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3892_max__diff__distrib__left,axiom,
% 8.90/5.06 ! [X2: rat,Y3: rat,Z: rat] :
% 8.90/5.06 ( ( minus_minus_rat @ ( ord_max_rat @ X2 @ Y3 ) @ Z )
% 8.90/5.06 = ( ord_max_rat @ ( minus_minus_rat @ X2 @ Z ) @ ( minus_minus_rat @ Y3 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3893_max__diff__distrib__left,axiom,
% 8.90/5.06 ! [X2: int,Y3: int,Z: int] :
% 8.90/5.06 ( ( minus_minus_int @ ( ord_max_int @ X2 @ Y3 ) @ Z )
% 8.90/5.06 = ( ord_max_int @ ( minus_minus_int @ X2 @ Z ) @ ( minus_minus_int @ Y3 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3894_nat__add__max__right,axiom,
% 8.90/5.06 ! [M: nat,N: nat,Q3: nat] :
% 8.90/5.06 ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
% 8.90/5.06 = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3895_nat__add__max__left,axiom,
% 8.90/5.06 ! [M: nat,N: nat,Q3: nat] :
% 8.90/5.06 ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
% 8.90/5.06 = ( ord_max_nat @ ( plus_plus_nat @ M @ Q3 ) @ ( plus_plus_nat @ N @ Q3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3896_nat__mult__max__right,axiom,
% 8.90/5.06 ! [M: nat,N: nat,Q3: nat] :
% 8.90/5.06 ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
% 8.90/5.06 = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3897_nat__mult__max__left,axiom,
% 8.90/5.06 ! [M: nat,N: nat,Q3: nat] :
% 8.90/5.06 ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
% 8.90/5.06 = ( ord_max_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3898_semiring__norm_I26_J,axiom,
% 8.90/5.06 ( ( bitM @ one )
% 8.90/5.06 = one ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3899_max__def__raw,axiom,
% 8.90/5.06 ( ord_ma741700101516333627d_enat
% 8.90/5.06 = ( ^ [A3: extended_enat,B2: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3900_max__def__raw,axiom,
% 8.90/5.06 ( ord_max_set_int
% 8.90/5.06 = ( ^ [A3: set_int,B2: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3901_max__def__raw,axiom,
% 8.90/5.06 ( ord_max_rat
% 8.90/5.06 = ( ^ [A3: rat,B2: rat] : ( if_rat @ ( ord_less_eq_rat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3902_max__def__raw,axiom,
% 8.90/5.06 ( ord_max_num
% 8.90/5.06 = ( ^ [A3: num,B2: num] : ( if_num @ ( ord_less_eq_num @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3903_max__def__raw,axiom,
% 8.90/5.06 ( ord_max_nat
% 8.90/5.06 = ( ^ [A3: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3904_max__def__raw,axiom,
% 8.90/5.06 ( ord_max_int
% 8.90/5.06 = ( ^ [A3: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3905_nat__minus__add__max,axiom,
% 8.90/5.06 ! [N: nat,M: nat] :
% 8.90/5.06 ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 8.90/5.06 = ( ord_max_nat @ N @ M ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3906_set__update__subsetI,axiom,
% 8.90/5.06 ! [Xs2: list_P6011104703257516679at_nat,A2: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,I2: nat] :
% 8.90/5.06 ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 )
% 8.90/5.06 => ( ( member8440522571783428010at_nat @ X2 @ A2 )
% 8.90/5.06 => ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs2 @ I2 @ X2 ) ) @ A2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3907_set__update__subsetI,axiom,
% 8.90/5.06 ! [Xs2: list_complex,A2: set_complex,X2: complex,I2: nat] :
% 8.90/5.06 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
% 8.90/5.06 => ( ( member_complex @ X2 @ A2 )
% 8.90/5.06 => ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs2 @ I2 @ X2 ) ) @ A2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3908_set__update__subsetI,axiom,
% 8.90/5.06 ! [Xs2: list_real,A2: set_real,X2: real,I2: nat] :
% 8.90/5.06 ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A2 )
% 8.90/5.06 => ( ( member_real @ X2 @ A2 )
% 8.90/5.06 => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I2 @ X2 ) ) @ A2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3909_set__update__subsetI,axiom,
% 8.90/5.06 ! [Xs2: list_nat,A2: set_nat,X2: nat,I2: nat] :
% 8.90/5.06 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
% 8.90/5.06 => ( ( member_nat @ X2 @ A2 )
% 8.90/5.06 => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I2 @ X2 ) ) @ A2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3910_set__update__subsetI,axiom,
% 8.90/5.06 ! [Xs2: list_VEBT_VEBT,A2: set_VEBT_VEBT,X2: vEBT_VEBT,I2: nat] :
% 8.90/5.06 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
% 8.90/5.06 => ( ( member_VEBT_VEBT @ X2 @ A2 )
% 8.90/5.06 => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X2 ) ) @ A2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3911_set__update__subsetI,axiom,
% 8.90/5.06 ! [Xs2: list_int,A2: set_int,X2: int,I2: nat] :
% 8.90/5.06 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
% 8.90/5.06 => ( ( member_int @ X2 @ A2 )
% 8.90/5.06 => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I2 @ X2 ) ) @ A2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3912_semiring__norm_I28_J,axiom,
% 8.90/5.06 ! [N: num] :
% 8.90/5.06 ( ( bitM @ ( bit1 @ N ) )
% 8.90/5.06 = ( bit1 @ ( bit0 @ N ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3913_semiring__norm_I27_J,axiom,
% 8.90/5.06 ! [N: num] :
% 8.90/5.06 ( ( bitM @ ( bit0 @ N ) )
% 8.90/5.06 = ( bit1 @ ( bitM @ N ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3914_set__update__memI,axiom,
% 8.90/5.06 ! [N: nat,Xs2: list_P6011104703257516679at_nat,X2: product_prod_nat_nat] :
% 8.90/5.06 ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
% 8.90/5.06 => ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs2 @ N @ X2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3915_set__update__memI,axiom,
% 8.90/5.06 ! [N: nat,Xs2: list_complex,X2: complex] :
% 8.90/5.06 ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 8.90/5.06 => ( member_complex @ X2 @ ( set_complex2 @ ( list_update_complex @ Xs2 @ N @ X2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3916_set__update__memI,axiom,
% 8.90/5.06 ! [N: nat,Xs2: list_real,X2: real] :
% 8.90/5.06 ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 8.90/5.06 => ( member_real @ X2 @ ( set_real2 @ ( list_update_real @ Xs2 @ N @ X2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3917_set__update__memI,axiom,
% 8.90/5.06 ! [N: nat,Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 8.90/5.06 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 8.90/5.06 => ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N @ X2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3918_set__update__memI,axiom,
% 8.90/5.06 ! [N: nat,Xs2: list_o,X2: $o] :
% 8.90/5.06 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 8.90/5.06 => ( member_o @ X2 @ ( set_o2 @ ( list_update_o @ Xs2 @ N @ X2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3919_set__update__memI,axiom,
% 8.90/5.06 ! [N: nat,Xs2: list_nat,X2: nat] :
% 8.90/5.06 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 8.90/5.06 => ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3920_set__update__memI,axiom,
% 8.90/5.06 ! [N: nat,Xs2: list_int,X2: int] :
% 8.90/5.06 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 8.90/5.06 => ( member_int @ X2 @ ( set_int2 @ ( list_update_int @ Xs2 @ N @ X2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3921_order__antisym__conv,axiom,
% 8.90/5.06 ! [Y3: set_int,X2: set_int] :
% 8.90/5.06 ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 8.90/5.06 => ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 8.90/5.06 = ( X2 = Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3922_order__antisym__conv,axiom,
% 8.90/5.06 ! [Y3: rat,X2: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 8.90/5.06 => ( ( ord_less_eq_rat @ X2 @ Y3 )
% 8.90/5.06 = ( X2 = Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3923_order__antisym__conv,axiom,
% 8.90/5.06 ! [Y3: num,X2: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ Y3 @ X2 )
% 8.90/5.06 => ( ( ord_less_eq_num @ X2 @ Y3 )
% 8.90/5.06 = ( X2 = Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3924_order__antisym__conv,axiom,
% 8.90/5.06 ! [Y3: nat,X2: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 8.90/5.06 => ( ( ord_less_eq_nat @ X2 @ Y3 )
% 8.90/5.06 = ( X2 = Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3925_order__antisym__conv,axiom,
% 8.90/5.06 ! [Y3: int,X2: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ Y3 @ X2 )
% 8.90/5.06 => ( ( ord_less_eq_int @ X2 @ Y3 )
% 8.90/5.06 = ( X2 = Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3926_linorder__le__cases,axiom,
% 8.90/5.06 ! [X2: rat,Y3: rat] :
% 8.90/5.06 ( ~ ( ord_less_eq_rat @ X2 @ Y3 )
% 8.90/5.06 => ( ord_less_eq_rat @ Y3 @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3927_linorder__le__cases,axiom,
% 8.90/5.06 ! [X2: num,Y3: num] :
% 8.90/5.06 ( ~ ( ord_less_eq_num @ X2 @ Y3 )
% 8.90/5.06 => ( ord_less_eq_num @ Y3 @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3928_linorder__le__cases,axiom,
% 8.90/5.06 ! [X2: nat,Y3: nat] :
% 8.90/5.06 ( ~ ( ord_less_eq_nat @ X2 @ Y3 )
% 8.90/5.06 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3929_linorder__le__cases,axiom,
% 8.90/5.06 ! [X2: int,Y3: int] :
% 8.90/5.06 ( ~ ( ord_less_eq_int @ X2 @ Y3 )
% 8.90/5.06 => ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3930_ord__le__eq__subst,axiom,
% 8.90/5.06 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.06 => ( ( ( F @ B )
% 8.90/5.06 = C )
% 8.90/5.06 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3931_ord__le__eq__subst,axiom,
% 8.90/5.06 ! [A: rat,B: rat,F: rat > num,C: num] :
% 8.90/5.06 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.06 => ( ( ( F @ B )
% 8.90/5.06 = C )
% 8.90/5.06 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3932_ord__le__eq__subst,axiom,
% 8.90/5.06 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.06 => ( ( ( F @ B )
% 8.90/5.06 = C )
% 8.90/5.06 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3933_ord__le__eq__subst,axiom,
% 8.90/5.06 ! [A: rat,B: rat,F: rat > int,C: int] :
% 8.90/5.06 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.06 => ( ( ( F @ B )
% 8.90/5.06 = C )
% 8.90/5.06 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3934_ord__le__eq__subst,axiom,
% 8.90/5.06 ! [A: num,B: num,F: num > rat,C: rat] :
% 8.90/5.06 ( ( ord_less_eq_num @ A @ B )
% 8.90/5.06 => ( ( ( F @ B )
% 8.90/5.06 = C )
% 8.90/5.06 => ( ! [X5: num,Y4: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3935_ord__le__eq__subst,axiom,
% 8.90/5.06 ! [A: num,B: num,F: num > num,C: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ A @ B )
% 8.90/5.06 => ( ( ( F @ B )
% 8.90/5.06 = C )
% 8.90/5.06 => ( ! [X5: num,Y4: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3936_ord__le__eq__subst,axiom,
% 8.90/5.06 ! [A: num,B: num,F: num > nat,C: nat] :
% 8.90/5.06 ( ( ord_less_eq_num @ A @ B )
% 8.90/5.06 => ( ( ( F @ B )
% 8.90/5.06 = C )
% 8.90/5.06 => ( ! [X5: num,Y4: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3937_ord__le__eq__subst,axiom,
% 8.90/5.06 ! [A: num,B: num,F: num > int,C: int] :
% 8.90/5.06 ( ( ord_less_eq_num @ A @ B )
% 8.90/5.06 => ( ( ( F @ B )
% 8.90/5.06 = C )
% 8.90/5.06 => ( ! [X5: num,Y4: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3938_ord__le__eq__subst,axiom,
% 8.90/5.06 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ A @ B )
% 8.90/5.06 => ( ( ( F @ B )
% 8.90/5.06 = C )
% 8.90/5.06 => ( ! [X5: nat,Y4: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3939_ord__le__eq__subst,axiom,
% 8.90/5.06 ! [A: nat,B: nat,F: nat > num,C: num] :
% 8.90/5.06 ( ( ord_less_eq_nat @ A @ B )
% 8.90/5.06 => ( ( ( F @ B )
% 8.90/5.06 = C )
% 8.90/5.06 => ( ! [X5: nat,Y4: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3940_ord__eq__le__subst,axiom,
% 8.90/5.06 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 8.90/5.06 ( ( A
% 8.90/5.06 = ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_rat @ B @ C )
% 8.90/5.06 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3941_ord__eq__le__subst,axiom,
% 8.90/5.06 ! [A: num,F: rat > num,B: rat,C: rat] :
% 8.90/5.06 ( ( A
% 8.90/5.06 = ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_rat @ B @ C )
% 8.90/5.06 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3942_ord__eq__le__subst,axiom,
% 8.90/5.06 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 8.90/5.06 ( ( A
% 8.90/5.06 = ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_rat @ B @ C )
% 8.90/5.06 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3943_ord__eq__le__subst,axiom,
% 8.90/5.06 ! [A: int,F: rat > int,B: rat,C: rat] :
% 8.90/5.06 ( ( A
% 8.90/5.06 = ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_rat @ B @ C )
% 8.90/5.06 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3944_ord__eq__le__subst,axiom,
% 8.90/5.06 ! [A: rat,F: num > rat,B: num,C: num] :
% 8.90/5.06 ( ( A
% 8.90/5.06 = ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_num @ B @ C )
% 8.90/5.06 => ( ! [X5: num,Y4: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3945_ord__eq__le__subst,axiom,
% 8.90/5.06 ! [A: num,F: num > num,B: num,C: num] :
% 8.90/5.06 ( ( A
% 8.90/5.06 = ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_num @ B @ C )
% 8.90/5.06 => ( ! [X5: num,Y4: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3946_ord__eq__le__subst,axiom,
% 8.90/5.06 ! [A: nat,F: num > nat,B: num,C: num] :
% 8.90/5.06 ( ( A
% 8.90/5.06 = ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_num @ B @ C )
% 8.90/5.06 => ( ! [X5: num,Y4: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3947_ord__eq__le__subst,axiom,
% 8.90/5.06 ! [A: int,F: num > int,B: num,C: num] :
% 8.90/5.06 ( ( A
% 8.90/5.06 = ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_num @ B @ C )
% 8.90/5.06 => ( ! [X5: num,Y4: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3948_ord__eq__le__subst,axiom,
% 8.90/5.06 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 8.90/5.06 ( ( A
% 8.90/5.06 = ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_nat @ B @ C )
% 8.90/5.06 => ( ! [X5: nat,Y4: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3949_ord__eq__le__subst,axiom,
% 8.90/5.06 ! [A: num,F: nat > num,B: nat,C: nat] :
% 8.90/5.06 ( ( A
% 8.90/5.06 = ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_nat @ B @ C )
% 8.90/5.06 => ( ! [X5: nat,Y4: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3950_linorder__linear,axiom,
% 8.90/5.06 ! [X2: rat,Y3: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 8.90/5.06 | ( ord_less_eq_rat @ Y3 @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3951_linorder__linear,axiom,
% 8.90/5.06 ! [X2: num,Y3: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X2 @ Y3 )
% 8.90/5.06 | ( ord_less_eq_num @ Y3 @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3952_linorder__linear,axiom,
% 8.90/5.06 ! [X2: nat,Y3: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 8.90/5.06 | ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3953_linorder__linear,axiom,
% 8.90/5.06 ! [X2: int,Y3: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ X2 @ Y3 )
% 8.90/5.06 | ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3954_order__eq__refl,axiom,
% 8.90/5.06 ! [X2: set_int,Y3: set_int] :
% 8.90/5.06 ( ( X2 = Y3 )
% 8.90/5.06 => ( ord_less_eq_set_int @ X2 @ Y3 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3955_order__eq__refl,axiom,
% 8.90/5.06 ! [X2: rat,Y3: rat] :
% 8.90/5.06 ( ( X2 = Y3 )
% 8.90/5.06 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3956_order__eq__refl,axiom,
% 8.90/5.06 ! [X2: num,Y3: num] :
% 8.90/5.06 ( ( X2 = Y3 )
% 8.90/5.06 => ( ord_less_eq_num @ X2 @ Y3 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3957_order__eq__refl,axiom,
% 8.90/5.06 ! [X2: nat,Y3: nat] :
% 8.90/5.06 ( ( X2 = Y3 )
% 8.90/5.06 => ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3958_order__eq__refl,axiom,
% 8.90/5.06 ! [X2: int,Y3: int] :
% 8.90/5.06 ( ( X2 = Y3 )
% 8.90/5.06 => ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3959_order__subst2,axiom,
% 8.90/5.06 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 8.90/5.06 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3960_order__subst2,axiom,
% 8.90/5.06 ! [A: rat,B: rat,F: rat > num,C: num] :
% 8.90/5.06 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 8.90/5.06 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3961_order__subst2,axiom,
% 8.90/5.06 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 8.90/5.06 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3962_order__subst2,axiom,
% 8.90/5.06 ! [A: rat,B: rat,F: rat > int,C: int] :
% 8.90/5.06 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 8.90/5.06 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3963_order__subst2,axiom,
% 8.90/5.06 ! [A: num,B: num,F: num > rat,C: rat] :
% 8.90/5.06 ( ( ord_less_eq_num @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 8.90/5.06 => ( ! [X5: num,Y4: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3964_order__subst2,axiom,
% 8.90/5.06 ! [A: num,B: num,F: num > num,C: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 8.90/5.06 => ( ! [X5: num,Y4: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3965_order__subst2,axiom,
% 8.90/5.06 ! [A: num,B: num,F: num > nat,C: nat] :
% 8.90/5.06 ( ( ord_less_eq_num @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 8.90/5.06 => ( ! [X5: num,Y4: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3966_order__subst2,axiom,
% 8.90/5.06 ! [A: num,B: num,F: num > int,C: int] :
% 8.90/5.06 ( ( ord_less_eq_num @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 8.90/5.06 => ( ! [X5: num,Y4: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3967_order__subst2,axiom,
% 8.90/5.06 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 8.90/5.06 => ( ! [X5: nat,Y4: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3968_order__subst2,axiom,
% 8.90/5.06 ! [A: nat,B: nat,F: nat > num,C: num] :
% 8.90/5.06 ( ( ord_less_eq_nat @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 8.90/5.06 => ( ! [X5: nat,Y4: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3969_order__subst1,axiom,
% 8.90/5.06 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_rat @ B @ C )
% 8.90/5.06 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3970_order__subst1,axiom,
% 8.90/5.06 ! [A: rat,F: num > rat,B: num,C: num] :
% 8.90/5.06 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_num @ B @ C )
% 8.90/5.06 => ( ! [X5: num,Y4: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3971_order__subst1,axiom,
% 8.90/5.06 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_nat @ B @ C )
% 8.90/5.06 => ( ! [X5: nat,Y4: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3972_order__subst1,axiom,
% 8.90/5.06 ! [A: rat,F: int > rat,B: int,C: int] :
% 8.90/5.06 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_int @ B @ C )
% 8.90/5.06 => ( ! [X5: int,Y4: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3973_order__subst1,axiom,
% 8.90/5.06 ! [A: num,F: rat > num,B: rat,C: rat] :
% 8.90/5.06 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_rat @ B @ C )
% 8.90/5.06 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3974_order__subst1,axiom,
% 8.90/5.06 ! [A: num,F: num > num,B: num,C: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_num @ B @ C )
% 8.90/5.06 => ( ! [X5: num,Y4: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3975_order__subst1,axiom,
% 8.90/5.06 ! [A: num,F: nat > num,B: nat,C: nat] :
% 8.90/5.06 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_nat @ B @ C )
% 8.90/5.06 => ( ! [X5: nat,Y4: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3976_order__subst1,axiom,
% 8.90/5.06 ! [A: num,F: int > num,B: int,C: int] :
% 8.90/5.06 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_int @ B @ C )
% 8.90/5.06 => ( ! [X5: int,Y4: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3977_order__subst1,axiom,
% 8.90/5.06 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_rat @ B @ C )
% 8.90/5.06 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3978_order__subst1,axiom,
% 8.90/5.06 ! [A: nat,F: num > nat,B: num,C: num] :
% 8.90/5.06 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 8.90/5.06 => ( ( ord_less_eq_num @ B @ C )
% 8.90/5.06 => ( ! [X5: num,Y4: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.06 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.06 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3979_Orderings_Oorder__eq__iff,axiom,
% 8.90/5.06 ( ( ^ [Y6: set_int,Z5: set_int] : ( Y6 = Z5 ) )
% 8.90/5.06 = ( ^ [A3: set_int,B2: set_int] :
% 8.90/5.06 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 8.90/5.06 & ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3980_Orderings_Oorder__eq__iff,axiom,
% 8.90/5.06 ( ( ^ [Y6: rat,Z5: rat] : ( Y6 = Z5 ) )
% 8.90/5.06 = ( ^ [A3: rat,B2: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ A3 @ B2 )
% 8.90/5.06 & ( ord_less_eq_rat @ B2 @ A3 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3981_Orderings_Oorder__eq__iff,axiom,
% 8.90/5.06 ( ( ^ [Y6: num,Z5: num] : ( Y6 = Z5 ) )
% 8.90/5.06 = ( ^ [A3: num,B2: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ A3 @ B2 )
% 8.90/5.06 & ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3982_Orderings_Oorder__eq__iff,axiom,
% 8.90/5.06 ( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 8.90/5.06 = ( ^ [A3: nat,B2: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ A3 @ B2 )
% 8.90/5.06 & ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3983_Orderings_Oorder__eq__iff,axiom,
% 8.90/5.06 ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 8.90/5.06 = ( ^ [A3: int,B2: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ A3 @ B2 )
% 8.90/5.06 & ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3984_antisym,axiom,
% 8.90/5.06 ! [A: set_int,B: set_int] :
% 8.90/5.06 ( ( ord_less_eq_set_int @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_set_int @ B @ A )
% 8.90/5.06 => ( A = B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3985_antisym,axiom,
% 8.90/5.06 ! [A: rat,B: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_rat @ B @ A )
% 8.90/5.06 => ( A = B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3986_antisym,axiom,
% 8.90/5.06 ! [A: num,B: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_num @ B @ A )
% 8.90/5.06 => ( A = B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3987_antisym,axiom,
% 8.90/5.06 ! [A: nat,B: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_nat @ B @ A )
% 8.90/5.06 => ( A = B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3988_antisym,axiom,
% 8.90/5.06 ! [A: int,B: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_int @ B @ A )
% 8.90/5.06 => ( A = B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3989_dual__order_Otrans,axiom,
% 8.90/5.06 ! [B: set_int,A: set_int,C: set_int] :
% 8.90/5.06 ( ( ord_less_eq_set_int @ B @ A )
% 8.90/5.06 => ( ( ord_less_eq_set_int @ C @ B )
% 8.90/5.06 => ( ord_less_eq_set_int @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3990_dual__order_Otrans,axiom,
% 8.90/5.06 ! [B: rat,A: rat,C: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ B @ A )
% 8.90/5.06 => ( ( ord_less_eq_rat @ C @ B )
% 8.90/5.06 => ( ord_less_eq_rat @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3991_dual__order_Otrans,axiom,
% 8.90/5.06 ! [B: num,A: num,C: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ B @ A )
% 8.90/5.06 => ( ( ord_less_eq_num @ C @ B )
% 8.90/5.06 => ( ord_less_eq_num @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3992_dual__order_Otrans,axiom,
% 8.90/5.06 ! [B: nat,A: nat,C: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ B @ A )
% 8.90/5.06 => ( ( ord_less_eq_nat @ C @ B )
% 8.90/5.06 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3993_dual__order_Otrans,axiom,
% 8.90/5.06 ! [B: int,A: int,C: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ B @ A )
% 8.90/5.06 => ( ( ord_less_eq_int @ C @ B )
% 8.90/5.06 => ( ord_less_eq_int @ C @ A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3994_dual__order_Oantisym,axiom,
% 8.90/5.06 ! [B: set_int,A: set_int] :
% 8.90/5.06 ( ( ord_less_eq_set_int @ B @ A )
% 8.90/5.06 => ( ( ord_less_eq_set_int @ A @ B )
% 8.90/5.06 => ( A = B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3995_dual__order_Oantisym,axiom,
% 8.90/5.06 ! [B: rat,A: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ B @ A )
% 8.90/5.06 => ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.06 => ( A = B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3996_dual__order_Oantisym,axiom,
% 8.90/5.06 ! [B: num,A: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ B @ A )
% 8.90/5.06 => ( ( ord_less_eq_num @ A @ B )
% 8.90/5.06 => ( A = B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3997_dual__order_Oantisym,axiom,
% 8.90/5.06 ! [B: nat,A: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ B @ A )
% 8.90/5.06 => ( ( ord_less_eq_nat @ A @ B )
% 8.90/5.06 => ( A = B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3998_dual__order_Oantisym,axiom,
% 8.90/5.06 ! [B: int,A: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ B @ A )
% 8.90/5.06 => ( ( ord_less_eq_int @ A @ B )
% 8.90/5.06 => ( A = B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_3999_dual__order_Oeq__iff,axiom,
% 8.90/5.06 ( ( ^ [Y6: set_int,Z5: set_int] : ( Y6 = Z5 ) )
% 8.90/5.06 = ( ^ [A3: set_int,B2: set_int] :
% 8.90/5.06 ( ( ord_less_eq_set_int @ B2 @ A3 )
% 8.90/5.06 & ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4000_dual__order_Oeq__iff,axiom,
% 8.90/5.06 ( ( ^ [Y6: rat,Z5: rat] : ( Y6 = Z5 ) )
% 8.90/5.06 = ( ^ [A3: rat,B2: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ B2 @ A3 )
% 8.90/5.06 & ( ord_less_eq_rat @ A3 @ B2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4001_dual__order_Oeq__iff,axiom,
% 8.90/5.06 ( ( ^ [Y6: num,Z5: num] : ( Y6 = Z5 ) )
% 8.90/5.06 = ( ^ [A3: num,B2: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ B2 @ A3 )
% 8.90/5.06 & ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4002_dual__order_Oeq__iff,axiom,
% 8.90/5.06 ( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 8.90/5.06 = ( ^ [A3: nat,B2: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ B2 @ A3 )
% 8.90/5.06 & ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4003_dual__order_Oeq__iff,axiom,
% 8.90/5.06 ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 8.90/5.06 = ( ^ [A3: int,B2: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ B2 @ A3 )
% 8.90/5.06 & ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4004_linorder__wlog,axiom,
% 8.90/5.06 ! [P: rat > rat > $o,A: rat,B: rat] :
% 8.90/5.06 ( ! [A4: rat,B4: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ A4 @ B4 )
% 8.90/5.06 => ( P @ A4 @ B4 ) )
% 8.90/5.06 => ( ! [A4: rat,B4: rat] :
% 8.90/5.06 ( ( P @ B4 @ A4 )
% 8.90/5.06 => ( P @ A4 @ B4 ) )
% 8.90/5.06 => ( P @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4005_linorder__wlog,axiom,
% 8.90/5.06 ! [P: num > num > $o,A: num,B: num] :
% 8.90/5.06 ( ! [A4: num,B4: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ A4 @ B4 )
% 8.90/5.06 => ( P @ A4 @ B4 ) )
% 8.90/5.06 => ( ! [A4: num,B4: num] :
% 8.90/5.06 ( ( P @ B4 @ A4 )
% 8.90/5.06 => ( P @ A4 @ B4 ) )
% 8.90/5.06 => ( P @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4006_linorder__wlog,axiom,
% 8.90/5.06 ! [P: nat > nat > $o,A: nat,B: nat] :
% 8.90/5.06 ( ! [A4: nat,B4: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ A4 @ B4 )
% 8.90/5.06 => ( P @ A4 @ B4 ) )
% 8.90/5.06 => ( ! [A4: nat,B4: nat] :
% 8.90/5.06 ( ( P @ B4 @ A4 )
% 8.90/5.06 => ( P @ A4 @ B4 ) )
% 8.90/5.06 => ( P @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4007_linorder__wlog,axiom,
% 8.90/5.06 ! [P: int > int > $o,A: int,B: int] :
% 8.90/5.06 ( ! [A4: int,B4: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ A4 @ B4 )
% 8.90/5.06 => ( P @ A4 @ B4 ) )
% 8.90/5.06 => ( ! [A4: int,B4: int] :
% 8.90/5.06 ( ( P @ B4 @ A4 )
% 8.90/5.06 => ( P @ A4 @ B4 ) )
% 8.90/5.06 => ( P @ A @ B ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4008_order__trans,axiom,
% 8.90/5.06 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 8.90/5.06 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 8.90/5.06 => ( ( ord_less_eq_set_int @ Y3 @ Z )
% 8.90/5.06 => ( ord_less_eq_set_int @ X2 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4009_order__trans,axiom,
% 8.90/5.06 ! [X2: rat,Y3: rat,Z: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 8.90/5.06 => ( ( ord_less_eq_rat @ Y3 @ Z )
% 8.90/5.06 => ( ord_less_eq_rat @ X2 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4010_order__trans,axiom,
% 8.90/5.06 ! [X2: num,Y3: num,Z: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X2 @ Y3 )
% 8.90/5.06 => ( ( ord_less_eq_num @ Y3 @ Z )
% 8.90/5.06 => ( ord_less_eq_num @ X2 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4011_order__trans,axiom,
% 8.90/5.06 ! [X2: nat,Y3: nat,Z: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 8.90/5.06 => ( ( ord_less_eq_nat @ Y3 @ Z )
% 8.90/5.06 => ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4012_order__trans,axiom,
% 8.90/5.06 ! [X2: int,Y3: int,Z: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ X2 @ Y3 )
% 8.90/5.06 => ( ( ord_less_eq_int @ Y3 @ Z )
% 8.90/5.06 => ( ord_less_eq_int @ X2 @ Z ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4013_order_Otrans,axiom,
% 8.90/5.06 ! [A: set_int,B: set_int,C: set_int] :
% 8.90/5.06 ( ( ord_less_eq_set_int @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_set_int @ B @ C )
% 8.90/5.06 => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4014_order_Otrans,axiom,
% 8.90/5.06 ! [A: rat,B: rat,C: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_rat @ B @ C )
% 8.90/5.06 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4015_order_Otrans,axiom,
% 8.90/5.06 ! [A: num,B: num,C: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_num @ B @ C )
% 8.90/5.06 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4016_order_Otrans,axiom,
% 8.90/5.06 ! [A: nat,B: nat,C: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_nat @ B @ C )
% 8.90/5.06 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4017_order_Otrans,axiom,
% 8.90/5.06 ! [A: int,B: int,C: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ A @ B )
% 8.90/5.06 => ( ( ord_less_eq_int @ B @ C )
% 8.90/5.06 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4018_order__antisym,axiom,
% 8.90/5.06 ! [X2: set_int,Y3: set_int] :
% 8.90/5.06 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 8.90/5.06 => ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 8.90/5.06 => ( X2 = Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4019_order__antisym,axiom,
% 8.90/5.06 ! [X2: rat,Y3: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 8.90/5.06 => ( ( ord_less_eq_rat @ Y3 @ X2 )
% 8.90/5.06 => ( X2 = Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4020_order__antisym,axiom,
% 8.90/5.06 ! [X2: num,Y3: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X2 @ Y3 )
% 8.90/5.06 => ( ( ord_less_eq_num @ Y3 @ X2 )
% 8.90/5.06 => ( X2 = Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4021_order__antisym,axiom,
% 8.90/5.06 ! [X2: nat,Y3: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 8.90/5.06 => ( ( ord_less_eq_nat @ Y3 @ X2 )
% 8.90/5.06 => ( X2 = Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4022_order__antisym,axiom,
% 8.90/5.06 ! [X2: int,Y3: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ X2 @ Y3 )
% 8.90/5.06 => ( ( ord_less_eq_int @ Y3 @ X2 )
% 8.90/5.06 => ( X2 = Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4023_ord__le__eq__trans,axiom,
% 8.90/5.06 ! [A: set_int,B: set_int,C: set_int] :
% 8.90/5.06 ( ( ord_less_eq_set_int @ A @ B )
% 8.90/5.06 => ( ( B = C )
% 8.90/5.06 => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4024_ord__le__eq__trans,axiom,
% 8.90/5.06 ! [A: rat,B: rat,C: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.06 => ( ( B = C )
% 8.90/5.06 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4025_ord__le__eq__trans,axiom,
% 8.90/5.06 ! [A: num,B: num,C: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ A @ B )
% 8.90/5.06 => ( ( B = C )
% 8.90/5.06 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4026_ord__le__eq__trans,axiom,
% 8.90/5.06 ! [A: nat,B: nat,C: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ A @ B )
% 8.90/5.06 => ( ( B = C )
% 8.90/5.06 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4027_ord__le__eq__trans,axiom,
% 8.90/5.06 ! [A: int,B: int,C: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ A @ B )
% 8.90/5.06 => ( ( B = C )
% 8.90/5.06 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4028_ord__eq__le__trans,axiom,
% 8.90/5.06 ! [A: set_int,B: set_int,C: set_int] :
% 8.90/5.06 ( ( A = B )
% 8.90/5.06 => ( ( ord_less_eq_set_int @ B @ C )
% 8.90/5.06 => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4029_ord__eq__le__trans,axiom,
% 8.90/5.06 ! [A: rat,B: rat,C: rat] :
% 8.90/5.06 ( ( A = B )
% 8.90/5.06 => ( ( ord_less_eq_rat @ B @ C )
% 8.90/5.06 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4030_ord__eq__le__trans,axiom,
% 8.90/5.06 ! [A: num,B: num,C: num] :
% 8.90/5.06 ( ( A = B )
% 8.90/5.06 => ( ( ord_less_eq_num @ B @ C )
% 8.90/5.06 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4031_ord__eq__le__trans,axiom,
% 8.90/5.06 ! [A: nat,B: nat,C: nat] :
% 8.90/5.06 ( ( A = B )
% 8.90/5.06 => ( ( ord_less_eq_nat @ B @ C )
% 8.90/5.06 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4032_ord__eq__le__trans,axiom,
% 8.90/5.06 ! [A: int,B: int,C: int] :
% 8.90/5.06 ( ( A = B )
% 8.90/5.06 => ( ( ord_less_eq_int @ B @ C )
% 8.90/5.06 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4033_order__class_Oorder__eq__iff,axiom,
% 8.90/5.06 ( ( ^ [Y6: set_int,Z5: set_int] : ( Y6 = Z5 ) )
% 8.90/5.06 = ( ^ [X: set_int,Y: set_int] :
% 8.90/5.06 ( ( ord_less_eq_set_int @ X @ Y )
% 8.90/5.06 & ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4034_order__class_Oorder__eq__iff,axiom,
% 8.90/5.06 ( ( ^ [Y6: rat,Z5: rat] : ( Y6 = Z5 ) )
% 8.90/5.06 = ( ^ [X: rat,Y: rat] :
% 8.90/5.06 ( ( ord_less_eq_rat @ X @ Y )
% 8.90/5.06 & ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4035_order__class_Oorder__eq__iff,axiom,
% 8.90/5.06 ( ( ^ [Y6: num,Z5: num] : ( Y6 = Z5 ) )
% 8.90/5.06 = ( ^ [X: num,Y: num] :
% 8.90/5.06 ( ( ord_less_eq_num @ X @ Y )
% 8.90/5.06 & ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4036_order__class_Oorder__eq__iff,axiom,
% 8.90/5.06 ( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 8.90/5.06 = ( ^ [X: nat,Y: nat] :
% 8.90/5.06 ( ( ord_less_eq_nat @ X @ Y )
% 8.90/5.06 & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4037_order__class_Oorder__eq__iff,axiom,
% 8.90/5.06 ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 8.90/5.06 = ( ^ [X: int,Y: int] :
% 8.90/5.06 ( ( ord_less_eq_int @ X @ Y )
% 8.90/5.06 & ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4038_le__cases3,axiom,
% 8.90/5.06 ! [X2: rat,Y3: rat,Z: rat] :
% 8.90/5.06 ( ( ( ord_less_eq_rat @ X2 @ Y3 )
% 8.90/5.06 => ~ ( ord_less_eq_rat @ Y3 @ Z ) )
% 8.90/5.06 => ( ( ( ord_less_eq_rat @ Y3 @ X2 )
% 8.90/5.06 => ~ ( ord_less_eq_rat @ X2 @ Z ) )
% 8.90/5.06 => ( ( ( ord_less_eq_rat @ X2 @ Z )
% 8.90/5.06 => ~ ( ord_less_eq_rat @ Z @ Y3 ) )
% 8.90/5.06 => ( ( ( ord_less_eq_rat @ Z @ Y3 )
% 8.90/5.06 => ~ ( ord_less_eq_rat @ Y3 @ X2 ) )
% 8.90/5.06 => ( ( ( ord_less_eq_rat @ Y3 @ Z )
% 8.90/5.06 => ~ ( ord_less_eq_rat @ Z @ X2 ) )
% 8.90/5.06 => ~ ( ( ord_less_eq_rat @ Z @ X2 )
% 8.90/5.06 => ~ ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4039_le__cases3,axiom,
% 8.90/5.06 ! [X2: num,Y3: num,Z: num] :
% 8.90/5.06 ( ( ( ord_less_eq_num @ X2 @ Y3 )
% 8.90/5.06 => ~ ( ord_less_eq_num @ Y3 @ Z ) )
% 8.90/5.06 => ( ( ( ord_less_eq_num @ Y3 @ X2 )
% 8.90/5.06 => ~ ( ord_less_eq_num @ X2 @ Z ) )
% 8.90/5.06 => ( ( ( ord_less_eq_num @ X2 @ Z )
% 8.90/5.06 => ~ ( ord_less_eq_num @ Z @ Y3 ) )
% 8.90/5.06 => ( ( ( ord_less_eq_num @ Z @ Y3 )
% 8.90/5.06 => ~ ( ord_less_eq_num @ Y3 @ X2 ) )
% 8.90/5.06 => ( ( ( ord_less_eq_num @ Y3 @ Z )
% 8.90/5.06 => ~ ( ord_less_eq_num @ Z @ X2 ) )
% 8.90/5.06 => ~ ( ( ord_less_eq_num @ Z @ X2 )
% 8.90/5.06 => ~ ( ord_less_eq_num @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4040_le__cases3,axiom,
% 8.90/5.06 ! [X2: nat,Y3: nat,Z: nat] :
% 8.90/5.06 ( ( ( ord_less_eq_nat @ X2 @ Y3 )
% 8.90/5.06 => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
% 8.90/5.06 => ( ( ( ord_less_eq_nat @ Y3 @ X2 )
% 8.90/5.06 => ~ ( ord_less_eq_nat @ X2 @ Z ) )
% 8.90/5.06 => ( ( ( ord_less_eq_nat @ X2 @ Z )
% 8.90/5.06 => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
% 8.90/5.06 => ( ( ( ord_less_eq_nat @ Z @ Y3 )
% 8.90/5.06 => ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
% 8.90/5.06 => ( ( ( ord_less_eq_nat @ Y3 @ Z )
% 8.90/5.06 => ~ ( ord_less_eq_nat @ Z @ X2 ) )
% 8.90/5.06 => ~ ( ( ord_less_eq_nat @ Z @ X2 )
% 8.90/5.06 => ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4041_le__cases3,axiom,
% 8.90/5.06 ! [X2: int,Y3: int,Z: int] :
% 8.90/5.06 ( ( ( ord_less_eq_int @ X2 @ Y3 )
% 8.90/5.06 => ~ ( ord_less_eq_int @ Y3 @ Z ) )
% 8.90/5.06 => ( ( ( ord_less_eq_int @ Y3 @ X2 )
% 8.90/5.06 => ~ ( ord_less_eq_int @ X2 @ Z ) )
% 8.90/5.06 => ( ( ( ord_less_eq_int @ X2 @ Z )
% 8.90/5.06 => ~ ( ord_less_eq_int @ Z @ Y3 ) )
% 8.90/5.06 => ( ( ( ord_less_eq_int @ Z @ Y3 )
% 8.90/5.06 => ~ ( ord_less_eq_int @ Y3 @ X2 ) )
% 8.90/5.06 => ( ( ( ord_less_eq_int @ Y3 @ Z )
% 8.90/5.06 => ~ ( ord_less_eq_int @ Z @ X2 ) )
% 8.90/5.06 => ~ ( ( ord_less_eq_int @ Z @ X2 )
% 8.90/5.06 => ~ ( ord_less_eq_int @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4042_nle__le,axiom,
% 8.90/5.06 ! [A: rat,B: rat] :
% 8.90/5.06 ( ( ~ ( ord_less_eq_rat @ A @ B ) )
% 8.90/5.06 = ( ( ord_less_eq_rat @ B @ A )
% 8.90/5.06 & ( B != A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4043_nle__le,axiom,
% 8.90/5.06 ! [A: num,B: num] :
% 8.90/5.06 ( ( ~ ( ord_less_eq_num @ A @ B ) )
% 8.90/5.06 = ( ( ord_less_eq_num @ B @ A )
% 8.90/5.06 & ( B != A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4044_nle__le,axiom,
% 8.90/5.06 ! [A: nat,B: nat] :
% 8.90/5.06 ( ( ~ ( ord_less_eq_nat @ A @ B ) )
% 8.90/5.06 = ( ( ord_less_eq_nat @ B @ A )
% 8.90/5.06 & ( B != A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4045_nle__le,axiom,
% 8.90/5.06 ! [A: int,B: int] :
% 8.90/5.06 ( ( ~ ( ord_less_eq_int @ A @ B ) )
% 8.90/5.06 = ( ( ord_less_eq_int @ B @ A )
% 8.90/5.06 & ( B != A ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4046_nth__list__update,axiom,
% 8.90/5.06 ! [I2: nat,Xs2: list_VEBT_VEBT,J: nat,X2: vEBT_VEBT] :
% 8.90/5.06 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 8.90/5.06 => ( ( ( I2 = J )
% 8.90/5.06 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X2 ) @ J )
% 8.90/5.06 = X2 ) )
% 8.90/5.06 & ( ( I2 != J )
% 8.90/5.06 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X2 ) @ J )
% 8.90/5.06 = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4047_nth__list__update,axiom,
% 8.90/5.06 ! [I2: nat,Xs2: list_o,X2: $o,J: nat] :
% 8.90/5.06 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
% 8.90/5.06 => ( ( nth_o @ ( list_update_o @ Xs2 @ I2 @ X2 ) @ J )
% 8.90/5.06 = ( ( ( I2 = J )
% 8.90/5.06 => X2 )
% 8.90/5.06 & ( ( I2 != J )
% 8.90/5.06 => ( nth_o @ Xs2 @ J ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4048_nth__list__update,axiom,
% 8.90/5.06 ! [I2: nat,Xs2: list_nat,J: nat,X2: nat] :
% 8.90/5.06 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 8.90/5.06 => ( ( ( I2 = J )
% 8.90/5.06 => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I2 @ X2 ) @ J )
% 8.90/5.06 = X2 ) )
% 8.90/5.06 & ( ( I2 != J )
% 8.90/5.06 => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I2 @ X2 ) @ J )
% 8.90/5.06 = ( nth_nat @ Xs2 @ J ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4049_nth__list__update,axiom,
% 8.90/5.06 ! [I2: nat,Xs2: list_int,J: nat,X2: int] :
% 8.90/5.06 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
% 8.90/5.06 => ( ( ( I2 = J )
% 8.90/5.06 => ( ( nth_int @ ( list_update_int @ Xs2 @ I2 @ X2 ) @ J )
% 8.90/5.06 = X2 ) )
% 8.90/5.06 & ( ( I2 != J )
% 8.90/5.06 => ( ( nth_int @ ( list_update_int @ Xs2 @ I2 @ X2 ) @ J )
% 8.90/5.06 = ( nth_int @ Xs2 @ J ) ) ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4050_list__update__same__conv,axiom,
% 8.90/5.06 ! [I2: nat,Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 8.90/5.06 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 8.90/5.06 => ( ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X2 )
% 8.90/5.06 = Xs2 )
% 8.90/5.06 = ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
% 8.90/5.06 = X2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4051_list__update__same__conv,axiom,
% 8.90/5.06 ! [I2: nat,Xs2: list_o,X2: $o] :
% 8.90/5.06 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
% 8.90/5.06 => ( ( ( list_update_o @ Xs2 @ I2 @ X2 )
% 8.90/5.06 = Xs2 )
% 8.90/5.06 = ( ( nth_o @ Xs2 @ I2 )
% 8.90/5.06 = X2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4052_list__update__same__conv,axiom,
% 8.90/5.06 ! [I2: nat,Xs2: list_nat,X2: nat] :
% 8.90/5.06 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 8.90/5.06 => ( ( ( list_update_nat @ Xs2 @ I2 @ X2 )
% 8.90/5.06 = Xs2 )
% 8.90/5.06 = ( ( nth_nat @ Xs2 @ I2 )
% 8.90/5.06 = X2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4053_list__update__same__conv,axiom,
% 8.90/5.06 ! [I2: nat,Xs2: list_int,X2: int] :
% 8.90/5.06 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
% 8.90/5.06 => ( ( ( list_update_int @ Xs2 @ I2 @ X2 )
% 8.90/5.06 = Xs2 )
% 8.90/5.06 = ( ( nth_int @ Xs2 @ I2 )
% 8.90/5.06 = X2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4054_lt__ex,axiom,
% 8.90/5.06 ! [X2: real] :
% 8.90/5.06 ? [Y4: real] : ( ord_less_real @ Y4 @ X2 ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4055_lt__ex,axiom,
% 8.90/5.06 ! [X2: rat] :
% 8.90/5.06 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X2 ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4056_lt__ex,axiom,
% 8.90/5.06 ! [X2: int] :
% 8.90/5.06 ? [Y4: int] : ( ord_less_int @ Y4 @ X2 ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4057_gt__ex,axiom,
% 8.90/5.06 ! [X2: real] :
% 8.90/5.06 ? [X_12: real] : ( ord_less_real @ X2 @ X_12 ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4058_gt__ex,axiom,
% 8.90/5.06 ! [X2: rat] :
% 8.90/5.06 ? [X_12: rat] : ( ord_less_rat @ X2 @ X_12 ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4059_gt__ex,axiom,
% 8.90/5.06 ! [X2: nat] :
% 8.90/5.06 ? [X_12: nat] : ( ord_less_nat @ X2 @ X_12 ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4060_gt__ex,axiom,
% 8.90/5.06 ! [X2: int] :
% 8.90/5.06 ? [X_12: int] : ( ord_less_int @ X2 @ X_12 ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4061_dense,axiom,
% 8.90/5.06 ! [X2: real,Y3: real] :
% 8.90/5.06 ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.06 => ? [Z2: real] :
% 8.90/5.06 ( ( ord_less_real @ X2 @ Z2 )
% 8.90/5.06 & ( ord_less_real @ Z2 @ Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4062_dense,axiom,
% 8.90/5.06 ! [X2: rat,Y3: rat] :
% 8.90/5.06 ( ( ord_less_rat @ X2 @ Y3 )
% 8.90/5.06 => ? [Z2: rat] :
% 8.90/5.06 ( ( ord_less_rat @ X2 @ Z2 )
% 8.90/5.06 & ( ord_less_rat @ Z2 @ Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4063_less__imp__neq,axiom,
% 8.90/5.06 ! [X2: real,Y3: real] :
% 8.90/5.06 ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.06 => ( X2 != Y3 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4064_less__imp__neq,axiom,
% 8.90/5.06 ! [X2: rat,Y3: rat] :
% 8.90/5.06 ( ( ord_less_rat @ X2 @ Y3 )
% 8.90/5.06 => ( X2 != Y3 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4065_less__imp__neq,axiom,
% 8.90/5.06 ! [X2: num,Y3: num] :
% 8.90/5.06 ( ( ord_less_num @ X2 @ Y3 )
% 8.90/5.06 => ( X2 != Y3 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4066_less__imp__neq,axiom,
% 8.90/5.06 ! [X2: nat,Y3: nat] :
% 8.90/5.06 ( ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.06 => ( X2 != Y3 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4067_less__imp__neq,axiom,
% 8.90/5.06 ! [X2: int,Y3: int] :
% 8.90/5.06 ( ( ord_less_int @ X2 @ Y3 )
% 8.90/5.06 => ( X2 != Y3 ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4068_order_Oasym,axiom,
% 8.90/5.06 ! [A: real,B: real] :
% 8.90/5.06 ( ( ord_less_real @ A @ B )
% 8.90/5.06 => ~ ( ord_less_real @ B @ A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4069_order_Oasym,axiom,
% 8.90/5.06 ! [A: rat,B: rat] :
% 8.90/5.06 ( ( ord_less_rat @ A @ B )
% 8.90/5.06 => ~ ( ord_less_rat @ B @ A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4070_order_Oasym,axiom,
% 8.90/5.06 ! [A: num,B: num] :
% 8.90/5.06 ( ( ord_less_num @ A @ B )
% 8.90/5.06 => ~ ( ord_less_num @ B @ A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4071_order_Oasym,axiom,
% 8.90/5.06 ! [A: nat,B: nat] :
% 8.90/5.06 ( ( ord_less_nat @ A @ B )
% 8.90/5.06 => ~ ( ord_less_nat @ B @ A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4072_order_Oasym,axiom,
% 8.90/5.06 ! [A: int,B: int] :
% 8.90/5.06 ( ( ord_less_int @ A @ B )
% 8.90/5.06 => ~ ( ord_less_int @ B @ A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4073_ord__eq__less__trans,axiom,
% 8.90/5.06 ! [A: real,B: real,C: real] :
% 8.90/5.06 ( ( A = B )
% 8.90/5.06 => ( ( ord_less_real @ B @ C )
% 8.90/5.06 => ( ord_less_real @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4074_ord__eq__less__trans,axiom,
% 8.90/5.06 ! [A: rat,B: rat,C: rat] :
% 8.90/5.06 ( ( A = B )
% 8.90/5.06 => ( ( ord_less_rat @ B @ C )
% 8.90/5.06 => ( ord_less_rat @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4075_ord__eq__less__trans,axiom,
% 8.90/5.06 ! [A: num,B: num,C: num] :
% 8.90/5.06 ( ( A = B )
% 8.90/5.06 => ( ( ord_less_num @ B @ C )
% 8.90/5.06 => ( ord_less_num @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4076_ord__eq__less__trans,axiom,
% 8.90/5.06 ! [A: nat,B: nat,C: nat] :
% 8.90/5.06 ( ( A = B )
% 8.90/5.06 => ( ( ord_less_nat @ B @ C )
% 8.90/5.06 => ( ord_less_nat @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4077_ord__eq__less__trans,axiom,
% 8.90/5.06 ! [A: int,B: int,C: int] :
% 8.90/5.06 ( ( A = B )
% 8.90/5.06 => ( ( ord_less_int @ B @ C )
% 8.90/5.06 => ( ord_less_int @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4078_ord__less__eq__trans,axiom,
% 8.90/5.06 ! [A: real,B: real,C: real] :
% 8.90/5.06 ( ( ord_less_real @ A @ B )
% 8.90/5.06 => ( ( B = C )
% 8.90/5.06 => ( ord_less_real @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4079_ord__less__eq__trans,axiom,
% 8.90/5.06 ! [A: rat,B: rat,C: rat] :
% 8.90/5.06 ( ( ord_less_rat @ A @ B )
% 8.90/5.06 => ( ( B = C )
% 8.90/5.06 => ( ord_less_rat @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4080_ord__less__eq__trans,axiom,
% 8.90/5.06 ! [A: num,B: num,C: num] :
% 8.90/5.06 ( ( ord_less_num @ A @ B )
% 8.90/5.06 => ( ( B = C )
% 8.90/5.06 => ( ord_less_num @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4081_ord__less__eq__trans,axiom,
% 8.90/5.06 ! [A: nat,B: nat,C: nat] :
% 8.90/5.06 ( ( ord_less_nat @ A @ B )
% 8.90/5.06 => ( ( B = C )
% 8.90/5.06 => ( ord_less_nat @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4082_ord__less__eq__trans,axiom,
% 8.90/5.06 ! [A: int,B: int,C: int] :
% 8.90/5.06 ( ( ord_less_int @ A @ B )
% 8.90/5.06 => ( ( B = C )
% 8.90/5.06 => ( ord_less_int @ A @ C ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4083_less__induct,axiom,
% 8.90/5.06 ! [P: nat > $o,A: nat] :
% 8.90/5.06 ( ! [X5: nat] :
% 8.90/5.06 ( ! [Y5: nat] :
% 8.90/5.06 ( ( ord_less_nat @ Y5 @ X5 )
% 8.90/5.06 => ( P @ Y5 ) )
% 8.90/5.06 => ( P @ X5 ) )
% 8.90/5.06 => ( P @ A ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4084_antisym__conv3,axiom,
% 8.90/5.06 ! [Y3: real,X2: real] :
% 8.90/5.06 ( ~ ( ord_less_real @ Y3 @ X2 )
% 8.90/5.06 => ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
% 8.90/5.06 = ( X2 = Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4085_antisym__conv3,axiom,
% 8.90/5.06 ! [Y3: rat,X2: rat] :
% 8.90/5.06 ( ~ ( ord_less_rat @ Y3 @ X2 )
% 8.90/5.06 => ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
% 8.90/5.06 = ( X2 = Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4086_antisym__conv3,axiom,
% 8.90/5.06 ! [Y3: num,X2: num] :
% 8.90/5.06 ( ~ ( ord_less_num @ Y3 @ X2 )
% 8.90/5.06 => ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
% 8.90/5.06 = ( X2 = Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4087_antisym__conv3,axiom,
% 8.90/5.06 ! [Y3: nat,X2: nat] :
% 8.90/5.06 ( ~ ( ord_less_nat @ Y3 @ X2 )
% 8.90/5.06 => ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
% 8.90/5.06 = ( X2 = Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4088_antisym__conv3,axiom,
% 8.90/5.06 ! [Y3: int,X2: int] :
% 8.90/5.06 ( ~ ( ord_less_int @ Y3 @ X2 )
% 8.90/5.06 => ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
% 8.90/5.06 = ( X2 = Y3 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4089_linorder__cases,axiom,
% 8.90/5.06 ! [X2: real,Y3: real] :
% 8.90/5.06 ( ~ ( ord_less_real @ X2 @ Y3 )
% 8.90/5.06 => ( ( X2 != Y3 )
% 8.90/5.06 => ( ord_less_real @ Y3 @ X2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4090_linorder__cases,axiom,
% 8.90/5.06 ! [X2: rat,Y3: rat] :
% 8.90/5.06 ( ~ ( ord_less_rat @ X2 @ Y3 )
% 8.90/5.06 => ( ( X2 != Y3 )
% 8.90/5.06 => ( ord_less_rat @ Y3 @ X2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4091_linorder__cases,axiom,
% 8.90/5.06 ! [X2: num,Y3: num] :
% 8.90/5.06 ( ~ ( ord_less_num @ X2 @ Y3 )
% 8.90/5.06 => ( ( X2 != Y3 )
% 8.90/5.06 => ( ord_less_num @ Y3 @ X2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4092_linorder__cases,axiom,
% 8.90/5.06 ! [X2: nat,Y3: nat] :
% 8.90/5.06 ( ~ ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.06 => ( ( X2 != Y3 )
% 8.90/5.06 => ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4093_linorder__cases,axiom,
% 8.90/5.06 ! [X2: int,Y3: int] :
% 8.90/5.06 ( ~ ( ord_less_int @ X2 @ Y3 )
% 8.90/5.06 => ( ( X2 != Y3 )
% 8.90/5.06 => ( ord_less_int @ Y3 @ X2 ) ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4094_dual__order_Oasym,axiom,
% 8.90/5.06 ! [B: real,A: real] :
% 8.90/5.06 ( ( ord_less_real @ B @ A )
% 8.90/5.06 => ~ ( ord_less_real @ A @ B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4095_dual__order_Oasym,axiom,
% 8.90/5.06 ! [B: rat,A: rat] :
% 8.90/5.06 ( ( ord_less_rat @ B @ A )
% 8.90/5.06 => ~ ( ord_less_rat @ A @ B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4096_dual__order_Oasym,axiom,
% 8.90/5.06 ! [B: num,A: num] :
% 8.90/5.06 ( ( ord_less_num @ B @ A )
% 8.90/5.06 => ~ ( ord_less_num @ A @ B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4097_dual__order_Oasym,axiom,
% 8.90/5.06 ! [B: nat,A: nat] :
% 8.90/5.06 ( ( ord_less_nat @ B @ A )
% 8.90/5.06 => ~ ( ord_less_nat @ A @ B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4098_dual__order_Oasym,axiom,
% 8.90/5.06 ! [B: int,A: int] :
% 8.90/5.06 ( ( ord_less_int @ B @ A )
% 8.90/5.06 => ~ ( ord_less_int @ A @ B ) ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4099_dual__order_Oirrefl,axiom,
% 8.90/5.06 ! [A: real] :
% 8.90/5.06 ~ ( ord_less_real @ A @ A ) ).
% 8.90/5.06
% 8.90/5.06 thf(fact_4100_dual__order_Oirrefl,axiom,
% 8.90/5.06 ! [A: rat] :
% 8.90/5.06 ~ ( ord_less_rat @ A @ A ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4101_dual__order_Oirrefl,axiom,
% 8.90/5.07 ! [A: num] :
% 8.90/5.07 ~ ( ord_less_num @ A @ A ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4102_dual__order_Oirrefl,axiom,
% 8.90/5.07 ! [A: nat] :
% 8.90/5.07 ~ ( ord_less_nat @ A @ A ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4103_dual__order_Oirrefl,axiom,
% 8.90/5.07 ! [A: int] :
% 8.90/5.07 ~ ( ord_less_int @ A @ A ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4104_exists__least__iff,axiom,
% 8.90/5.07 ( ( ^ [P2: nat > $o] :
% 8.90/5.07 ? [X6: nat] : ( P2 @ X6 ) )
% 8.90/5.07 = ( ^ [P3: nat > $o] :
% 8.90/5.07 ? [N3: nat] :
% 8.90/5.07 ( ( P3 @ N3 )
% 8.90/5.07 & ! [M3: nat] :
% 8.90/5.07 ( ( ord_less_nat @ M3 @ N3 )
% 8.90/5.07 => ~ ( P3 @ M3 ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4105_linorder__less__wlog,axiom,
% 8.90/5.07 ! [P: real > real > $o,A: real,B: real] :
% 8.90/5.07 ( ! [A4: real,B4: real] :
% 8.90/5.07 ( ( ord_less_real @ A4 @ B4 )
% 8.90/5.07 => ( P @ A4 @ B4 ) )
% 8.90/5.07 => ( ! [A4: real] : ( P @ A4 @ A4 )
% 8.90/5.07 => ( ! [A4: real,B4: real] :
% 8.90/5.07 ( ( P @ B4 @ A4 )
% 8.90/5.07 => ( P @ A4 @ B4 ) )
% 8.90/5.07 => ( P @ A @ B ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4106_linorder__less__wlog,axiom,
% 8.90/5.07 ! [P: rat > rat > $o,A: rat,B: rat] :
% 8.90/5.07 ( ! [A4: rat,B4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ A4 @ B4 )
% 8.90/5.07 => ( P @ A4 @ B4 ) )
% 8.90/5.07 => ( ! [A4: rat] : ( P @ A4 @ A4 )
% 8.90/5.07 => ( ! [A4: rat,B4: rat] :
% 8.90/5.07 ( ( P @ B4 @ A4 )
% 8.90/5.07 => ( P @ A4 @ B4 ) )
% 8.90/5.07 => ( P @ A @ B ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4107_linorder__less__wlog,axiom,
% 8.90/5.07 ! [P: num > num > $o,A: num,B: num] :
% 8.90/5.07 ( ! [A4: num,B4: num] :
% 8.90/5.07 ( ( ord_less_num @ A4 @ B4 )
% 8.90/5.07 => ( P @ A4 @ B4 ) )
% 8.90/5.07 => ( ! [A4: num] : ( P @ A4 @ A4 )
% 8.90/5.07 => ( ! [A4: num,B4: num] :
% 8.90/5.07 ( ( P @ B4 @ A4 )
% 8.90/5.07 => ( P @ A4 @ B4 ) )
% 8.90/5.07 => ( P @ A @ B ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4108_linorder__less__wlog,axiom,
% 8.90/5.07 ! [P: nat > nat > $o,A: nat,B: nat] :
% 8.90/5.07 ( ! [A4: nat,B4: nat] :
% 8.90/5.07 ( ( ord_less_nat @ A4 @ B4 )
% 8.90/5.07 => ( P @ A4 @ B4 ) )
% 8.90/5.07 => ( ! [A4: nat] : ( P @ A4 @ A4 )
% 8.90/5.07 => ( ! [A4: nat,B4: nat] :
% 8.90/5.07 ( ( P @ B4 @ A4 )
% 8.90/5.07 => ( P @ A4 @ B4 ) )
% 8.90/5.07 => ( P @ A @ B ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4109_linorder__less__wlog,axiom,
% 8.90/5.07 ! [P: int > int > $o,A: int,B: int] :
% 8.90/5.07 ( ! [A4: int,B4: int] :
% 8.90/5.07 ( ( ord_less_int @ A4 @ B4 )
% 8.90/5.07 => ( P @ A4 @ B4 ) )
% 8.90/5.07 => ( ! [A4: int] : ( P @ A4 @ A4 )
% 8.90/5.07 => ( ! [A4: int,B4: int] :
% 8.90/5.07 ( ( P @ B4 @ A4 )
% 8.90/5.07 => ( P @ A4 @ B4 ) )
% 8.90/5.07 => ( P @ A @ B ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4110_order_Ostrict__trans,axiom,
% 8.90/5.07 ! [A: real,B: real,C: real] :
% 8.90/5.07 ( ( ord_less_real @ A @ B )
% 8.90/5.07 => ( ( ord_less_real @ B @ C )
% 8.90/5.07 => ( ord_less_real @ A @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4111_order_Ostrict__trans,axiom,
% 8.90/5.07 ! [A: rat,B: rat,C: rat] :
% 8.90/5.07 ( ( ord_less_rat @ A @ B )
% 8.90/5.07 => ( ( ord_less_rat @ B @ C )
% 8.90/5.07 => ( ord_less_rat @ A @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4112_order_Ostrict__trans,axiom,
% 8.90/5.07 ! [A: num,B: num,C: num] :
% 8.90/5.07 ( ( ord_less_num @ A @ B )
% 8.90/5.07 => ( ( ord_less_num @ B @ C )
% 8.90/5.07 => ( ord_less_num @ A @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4113_order_Ostrict__trans,axiom,
% 8.90/5.07 ! [A: nat,B: nat,C: nat] :
% 8.90/5.07 ( ( ord_less_nat @ A @ B )
% 8.90/5.07 => ( ( ord_less_nat @ B @ C )
% 8.90/5.07 => ( ord_less_nat @ A @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4114_order_Ostrict__trans,axiom,
% 8.90/5.07 ! [A: int,B: int,C: int] :
% 8.90/5.07 ( ( ord_less_int @ A @ B )
% 8.90/5.07 => ( ( ord_less_int @ B @ C )
% 8.90/5.07 => ( ord_less_int @ A @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4115_not__less__iff__gr__or__eq,axiom,
% 8.90/5.07 ! [X2: real,Y3: real] :
% 8.90/5.07 ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
% 8.90/5.07 = ( ( ord_less_real @ Y3 @ X2 )
% 8.90/5.07 | ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4116_not__less__iff__gr__or__eq,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat] :
% 8.90/5.07 ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
% 8.90/5.07 = ( ( ord_less_rat @ Y3 @ X2 )
% 8.90/5.07 | ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4117_not__less__iff__gr__or__eq,axiom,
% 8.90/5.07 ! [X2: num,Y3: num] :
% 8.90/5.07 ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
% 8.90/5.07 = ( ( ord_less_num @ Y3 @ X2 )
% 8.90/5.07 | ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4118_not__less__iff__gr__or__eq,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat] :
% 8.90/5.07 ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
% 8.90/5.07 = ( ( ord_less_nat @ Y3 @ X2 )
% 8.90/5.07 | ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4119_not__less__iff__gr__or__eq,axiom,
% 8.90/5.07 ! [X2: int,Y3: int] :
% 8.90/5.07 ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
% 8.90/5.07 = ( ( ord_less_int @ Y3 @ X2 )
% 8.90/5.07 | ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4120_dual__order_Ostrict__trans,axiom,
% 8.90/5.07 ! [B: real,A: real,C: real] :
% 8.90/5.07 ( ( ord_less_real @ B @ A )
% 8.90/5.07 => ( ( ord_less_real @ C @ B )
% 8.90/5.07 => ( ord_less_real @ C @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4121_dual__order_Ostrict__trans,axiom,
% 8.90/5.07 ! [B: rat,A: rat,C: rat] :
% 8.90/5.07 ( ( ord_less_rat @ B @ A )
% 8.90/5.07 => ( ( ord_less_rat @ C @ B )
% 8.90/5.07 => ( ord_less_rat @ C @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4122_dual__order_Ostrict__trans,axiom,
% 8.90/5.07 ! [B: num,A: num,C: num] :
% 8.90/5.07 ( ( ord_less_num @ B @ A )
% 8.90/5.07 => ( ( ord_less_num @ C @ B )
% 8.90/5.07 => ( ord_less_num @ C @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4123_dual__order_Ostrict__trans,axiom,
% 8.90/5.07 ! [B: nat,A: nat,C: nat] :
% 8.90/5.07 ( ( ord_less_nat @ B @ A )
% 8.90/5.07 => ( ( ord_less_nat @ C @ B )
% 8.90/5.07 => ( ord_less_nat @ C @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4124_dual__order_Ostrict__trans,axiom,
% 8.90/5.07 ! [B: int,A: int,C: int] :
% 8.90/5.07 ( ( ord_less_int @ B @ A )
% 8.90/5.07 => ( ( ord_less_int @ C @ B )
% 8.90/5.07 => ( ord_less_int @ C @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4125_order_Ostrict__implies__not__eq,axiom,
% 8.90/5.07 ! [A: real,B: real] :
% 8.90/5.07 ( ( ord_less_real @ A @ B )
% 8.90/5.07 => ( A != B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4126_order_Ostrict__implies__not__eq,axiom,
% 8.90/5.07 ! [A: rat,B: rat] :
% 8.90/5.07 ( ( ord_less_rat @ A @ B )
% 8.90/5.07 => ( A != B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4127_order_Ostrict__implies__not__eq,axiom,
% 8.90/5.07 ! [A: num,B: num] :
% 8.90/5.07 ( ( ord_less_num @ A @ B )
% 8.90/5.07 => ( A != B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4128_order_Ostrict__implies__not__eq,axiom,
% 8.90/5.07 ! [A: nat,B: nat] :
% 8.90/5.07 ( ( ord_less_nat @ A @ B )
% 8.90/5.07 => ( A != B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4129_order_Ostrict__implies__not__eq,axiom,
% 8.90/5.07 ! [A: int,B: int] :
% 8.90/5.07 ( ( ord_less_int @ A @ B )
% 8.90/5.07 => ( A != B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4130_dual__order_Ostrict__implies__not__eq,axiom,
% 8.90/5.07 ! [B: real,A: real] :
% 8.90/5.07 ( ( ord_less_real @ B @ A )
% 8.90/5.07 => ( A != B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4131_dual__order_Ostrict__implies__not__eq,axiom,
% 8.90/5.07 ! [B: rat,A: rat] :
% 8.90/5.07 ( ( ord_less_rat @ B @ A )
% 8.90/5.07 => ( A != B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4132_dual__order_Ostrict__implies__not__eq,axiom,
% 8.90/5.07 ! [B: num,A: num] :
% 8.90/5.07 ( ( ord_less_num @ B @ A )
% 8.90/5.07 => ( A != B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4133_dual__order_Ostrict__implies__not__eq,axiom,
% 8.90/5.07 ! [B: nat,A: nat] :
% 8.90/5.07 ( ( ord_less_nat @ B @ A )
% 8.90/5.07 => ( A != B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4134_dual__order_Ostrict__implies__not__eq,axiom,
% 8.90/5.07 ! [B: int,A: int] :
% 8.90/5.07 ( ( ord_less_int @ B @ A )
% 8.90/5.07 => ( A != B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4135_linorder__neqE,axiom,
% 8.90/5.07 ! [X2: real,Y3: real] :
% 8.90/5.07 ( ( X2 != Y3 )
% 8.90/5.07 => ( ~ ( ord_less_real @ X2 @ Y3 )
% 8.90/5.07 => ( ord_less_real @ Y3 @ X2 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4136_linorder__neqE,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat] :
% 8.90/5.07 ( ( X2 != Y3 )
% 8.90/5.07 => ( ~ ( ord_less_rat @ X2 @ Y3 )
% 8.90/5.07 => ( ord_less_rat @ Y3 @ X2 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4137_linorder__neqE,axiom,
% 8.90/5.07 ! [X2: num,Y3: num] :
% 8.90/5.07 ( ( X2 != Y3 )
% 8.90/5.07 => ( ~ ( ord_less_num @ X2 @ Y3 )
% 8.90/5.07 => ( ord_less_num @ Y3 @ X2 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4138_linorder__neqE,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat] :
% 8.90/5.07 ( ( X2 != Y3 )
% 8.90/5.07 => ( ~ ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.07 => ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4139_linorder__neqE,axiom,
% 8.90/5.07 ! [X2: int,Y3: int] :
% 8.90/5.07 ( ( X2 != Y3 )
% 8.90/5.07 => ( ~ ( ord_less_int @ X2 @ Y3 )
% 8.90/5.07 => ( ord_less_int @ Y3 @ X2 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4140_order__less__asym,axiom,
% 8.90/5.07 ! [X2: real,Y3: real] :
% 8.90/5.07 ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.07 => ~ ( ord_less_real @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4141_order__less__asym,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X2 @ Y3 )
% 8.90/5.07 => ~ ( ord_less_rat @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4142_order__less__asym,axiom,
% 8.90/5.07 ! [X2: num,Y3: num] :
% 8.90/5.07 ( ( ord_less_num @ X2 @ Y3 )
% 8.90/5.07 => ~ ( ord_less_num @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4143_order__less__asym,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat] :
% 8.90/5.07 ( ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.07 => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4144_order__less__asym,axiom,
% 8.90/5.07 ! [X2: int,Y3: int] :
% 8.90/5.07 ( ( ord_less_int @ X2 @ Y3 )
% 8.90/5.07 => ~ ( ord_less_int @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4145_linorder__neq__iff,axiom,
% 8.90/5.07 ! [X2: real,Y3: real] :
% 8.90/5.07 ( ( X2 != Y3 )
% 8.90/5.07 = ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.07 | ( ord_less_real @ Y3 @ X2 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4146_linorder__neq__iff,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat] :
% 8.90/5.07 ( ( X2 != Y3 )
% 8.90/5.07 = ( ( ord_less_rat @ X2 @ Y3 )
% 8.90/5.07 | ( ord_less_rat @ Y3 @ X2 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4147_linorder__neq__iff,axiom,
% 8.90/5.07 ! [X2: num,Y3: num] :
% 8.90/5.07 ( ( X2 != Y3 )
% 8.90/5.07 = ( ( ord_less_num @ X2 @ Y3 )
% 8.90/5.07 | ( ord_less_num @ Y3 @ X2 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4148_linorder__neq__iff,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat] :
% 8.90/5.07 ( ( X2 != Y3 )
% 8.90/5.07 = ( ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.07 | ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4149_linorder__neq__iff,axiom,
% 8.90/5.07 ! [X2: int,Y3: int] :
% 8.90/5.07 ( ( X2 != Y3 )
% 8.90/5.07 = ( ( ord_less_int @ X2 @ Y3 )
% 8.90/5.07 | ( ord_less_int @ Y3 @ X2 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4150_order__less__asym_H,axiom,
% 8.90/5.07 ! [A: real,B: real] :
% 8.90/5.07 ( ( ord_less_real @ A @ B )
% 8.90/5.07 => ~ ( ord_less_real @ B @ A ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4151_order__less__asym_H,axiom,
% 8.90/5.07 ! [A: rat,B: rat] :
% 8.90/5.07 ( ( ord_less_rat @ A @ B )
% 8.90/5.07 => ~ ( ord_less_rat @ B @ A ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4152_order__less__asym_H,axiom,
% 8.90/5.07 ! [A: num,B: num] :
% 8.90/5.07 ( ( ord_less_num @ A @ B )
% 8.90/5.07 => ~ ( ord_less_num @ B @ A ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4153_order__less__asym_H,axiom,
% 8.90/5.07 ! [A: nat,B: nat] :
% 8.90/5.07 ( ( ord_less_nat @ A @ B )
% 8.90/5.07 => ~ ( ord_less_nat @ B @ A ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4154_order__less__asym_H,axiom,
% 8.90/5.07 ! [A: int,B: int] :
% 8.90/5.07 ( ( ord_less_int @ A @ B )
% 8.90/5.07 => ~ ( ord_less_int @ B @ A ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4155_order__less__trans,axiom,
% 8.90/5.07 ! [X2: real,Y3: real,Z: real] :
% 8.90/5.07 ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_real @ Y3 @ Z )
% 8.90/5.07 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4156_order__less__trans,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat,Z: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_rat @ Y3 @ Z )
% 8.90/5.07 => ( ord_less_rat @ X2 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4157_order__less__trans,axiom,
% 8.90/5.07 ! [X2: num,Y3: num,Z: num] :
% 8.90/5.07 ( ( ord_less_num @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_num @ Y3 @ Z )
% 8.90/5.07 => ( ord_less_num @ X2 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4158_order__less__trans,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat,Z: nat] :
% 8.90/5.07 ( ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_nat @ Y3 @ Z )
% 8.90/5.07 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4159_order__less__trans,axiom,
% 8.90/5.07 ! [X2: int,Y3: int,Z: int] :
% 8.90/5.07 ( ( ord_less_int @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_int @ Y3 @ Z )
% 8.90/5.07 => ( ord_less_int @ X2 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4160_ord__eq__less__subst,axiom,
% 8.90/5.07 ! [A: real,F: real > real,B: real,C: real] :
% 8.90/5.07 ( ( A
% 8.90/5.07 = ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_real @ B @ C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4161_ord__eq__less__subst,axiom,
% 8.90/5.07 ! [A: rat,F: real > rat,B: real,C: real] :
% 8.90/5.07 ( ( A
% 8.90/5.07 = ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_real @ B @ C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4162_ord__eq__less__subst,axiom,
% 8.90/5.07 ! [A: num,F: real > num,B: real,C: real] :
% 8.90/5.07 ( ( A
% 8.90/5.07 = ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_real @ B @ C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4163_ord__eq__less__subst,axiom,
% 8.90/5.07 ! [A: nat,F: real > nat,B: real,C: real] :
% 8.90/5.07 ( ( A
% 8.90/5.07 = ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_real @ B @ C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4164_ord__eq__less__subst,axiom,
% 8.90/5.07 ! [A: int,F: real > int,B: real,C: real] :
% 8.90/5.07 ( ( A
% 8.90/5.07 = ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_real @ B @ C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4165_ord__eq__less__subst,axiom,
% 8.90/5.07 ! [A: real,F: rat > real,B: rat,C: rat] :
% 8.90/5.07 ( ( A
% 8.90/5.07 = ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_rat @ B @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4166_ord__eq__less__subst,axiom,
% 8.90/5.07 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 8.90/5.07 ( ( A
% 8.90/5.07 = ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_rat @ B @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4167_ord__eq__less__subst,axiom,
% 8.90/5.07 ! [A: num,F: rat > num,B: rat,C: rat] :
% 8.90/5.07 ( ( A
% 8.90/5.07 = ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_rat @ B @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4168_ord__eq__less__subst,axiom,
% 8.90/5.07 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 8.90/5.07 ( ( A
% 8.90/5.07 = ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_rat @ B @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4169_ord__eq__less__subst,axiom,
% 8.90/5.07 ! [A: int,F: rat > int,B: rat,C: rat] :
% 8.90/5.07 ( ( A
% 8.90/5.07 = ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_rat @ B @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4170_ord__less__eq__subst,axiom,
% 8.90/5.07 ! [A: real,B: real,F: real > real,C: real] :
% 8.90/5.07 ( ( ord_less_real @ A @ B )
% 8.90/5.07 => ( ( ( F @ B )
% 8.90/5.07 = C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4171_ord__less__eq__subst,axiom,
% 8.90/5.07 ! [A: real,B: real,F: real > rat,C: rat] :
% 8.90/5.07 ( ( ord_less_real @ A @ B )
% 8.90/5.07 => ( ( ( F @ B )
% 8.90/5.07 = C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4172_ord__less__eq__subst,axiom,
% 8.90/5.07 ! [A: real,B: real,F: real > num,C: num] :
% 8.90/5.07 ( ( ord_less_real @ A @ B )
% 8.90/5.07 => ( ( ( F @ B )
% 8.90/5.07 = C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4173_ord__less__eq__subst,axiom,
% 8.90/5.07 ! [A: real,B: real,F: real > nat,C: nat] :
% 8.90/5.07 ( ( ord_less_real @ A @ B )
% 8.90/5.07 => ( ( ( F @ B )
% 8.90/5.07 = C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4174_ord__less__eq__subst,axiom,
% 8.90/5.07 ! [A: real,B: real,F: real > int,C: int] :
% 8.90/5.07 ( ( ord_less_real @ A @ B )
% 8.90/5.07 => ( ( ( F @ B )
% 8.90/5.07 = C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4175_ord__less__eq__subst,axiom,
% 8.90/5.07 ! [A: rat,B: rat,F: rat > real,C: real] :
% 8.90/5.07 ( ( ord_less_rat @ A @ B )
% 8.90/5.07 => ( ( ( F @ B )
% 8.90/5.07 = C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4176_ord__less__eq__subst,axiom,
% 8.90/5.07 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 8.90/5.07 ( ( ord_less_rat @ A @ B )
% 8.90/5.07 => ( ( ( F @ B )
% 8.90/5.07 = C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4177_ord__less__eq__subst,axiom,
% 8.90/5.07 ! [A: rat,B: rat,F: rat > num,C: num] :
% 8.90/5.07 ( ( ord_less_rat @ A @ B )
% 8.90/5.07 => ( ( ( F @ B )
% 8.90/5.07 = C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4178_ord__less__eq__subst,axiom,
% 8.90/5.07 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 8.90/5.07 ( ( ord_less_rat @ A @ B )
% 8.90/5.07 => ( ( ( F @ B )
% 8.90/5.07 = C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4179_ord__less__eq__subst,axiom,
% 8.90/5.07 ! [A: rat,B: rat,F: rat > int,C: int] :
% 8.90/5.07 ( ( ord_less_rat @ A @ B )
% 8.90/5.07 => ( ( ( F @ B )
% 8.90/5.07 = C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4180_order__less__irrefl,axiom,
% 8.90/5.07 ! [X2: real] :
% 8.90/5.07 ~ ( ord_less_real @ X2 @ X2 ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4181_order__less__irrefl,axiom,
% 8.90/5.07 ! [X2: rat] :
% 8.90/5.07 ~ ( ord_less_rat @ X2 @ X2 ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4182_order__less__irrefl,axiom,
% 8.90/5.07 ! [X2: num] :
% 8.90/5.07 ~ ( ord_less_num @ X2 @ X2 ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4183_order__less__irrefl,axiom,
% 8.90/5.07 ! [X2: nat] :
% 8.90/5.07 ~ ( ord_less_nat @ X2 @ X2 ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4184_order__less__irrefl,axiom,
% 8.90/5.07 ! [X2: int] :
% 8.90/5.07 ~ ( ord_less_int @ X2 @ X2 ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4185_order__less__subst1,axiom,
% 8.90/5.07 ! [A: real,F: real > real,B: real,C: real] :
% 8.90/5.07 ( ( ord_less_real @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_real @ B @ C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4186_order__less__subst1,axiom,
% 8.90/5.07 ! [A: real,F: rat > real,B: rat,C: rat] :
% 8.90/5.07 ( ( ord_less_real @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_rat @ B @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4187_order__less__subst1,axiom,
% 8.90/5.07 ! [A: real,F: num > real,B: num,C: num] :
% 8.90/5.07 ( ( ord_less_real @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_num @ B @ C )
% 8.90/5.07 => ( ! [X5: num,Y4: num] :
% 8.90/5.07 ( ( ord_less_num @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4188_order__less__subst1,axiom,
% 8.90/5.07 ! [A: real,F: nat > real,B: nat,C: nat] :
% 8.90/5.07 ( ( ord_less_real @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_nat @ B @ C )
% 8.90/5.07 => ( ! [X5: nat,Y4: nat] :
% 8.90/5.07 ( ( ord_less_nat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4189_order__less__subst1,axiom,
% 8.90/5.07 ! [A: real,F: int > real,B: int,C: int] :
% 8.90/5.07 ( ( ord_less_real @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_int @ B @ C )
% 8.90/5.07 => ( ! [X5: int,Y4: int] :
% 8.90/5.07 ( ( ord_less_int @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4190_order__less__subst1,axiom,
% 8.90/5.07 ! [A: rat,F: real > rat,B: real,C: real] :
% 8.90/5.07 ( ( ord_less_rat @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_real @ B @ C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4191_order__less__subst1,axiom,
% 8.90/5.07 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 8.90/5.07 ( ( ord_less_rat @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_rat @ B @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4192_order__less__subst1,axiom,
% 8.90/5.07 ! [A: rat,F: num > rat,B: num,C: num] :
% 8.90/5.07 ( ( ord_less_rat @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_num @ B @ C )
% 8.90/5.07 => ( ! [X5: num,Y4: num] :
% 8.90/5.07 ( ( ord_less_num @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4193_order__less__subst1,axiom,
% 8.90/5.07 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 8.90/5.07 ( ( ord_less_rat @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_nat @ B @ C )
% 8.90/5.07 => ( ! [X5: nat,Y4: nat] :
% 8.90/5.07 ( ( ord_less_nat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4194_order__less__subst1,axiom,
% 8.90/5.07 ! [A: rat,F: int > rat,B: int,C: int] :
% 8.90/5.07 ( ( ord_less_rat @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_int @ B @ C )
% 8.90/5.07 => ( ! [X5: int,Y4: int] :
% 8.90/5.07 ( ( ord_less_int @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4195_order__less__subst2,axiom,
% 8.90/5.07 ! [A: real,B: real,F: real > real,C: real] :
% 8.90/5.07 ( ( ord_less_real @ A @ B )
% 8.90/5.07 => ( ( ord_less_real @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4196_order__less__subst2,axiom,
% 8.90/5.07 ! [A: real,B: real,F: real > rat,C: rat] :
% 8.90/5.07 ( ( ord_less_real @ A @ B )
% 8.90/5.07 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4197_order__less__subst2,axiom,
% 8.90/5.07 ! [A: real,B: real,F: real > num,C: num] :
% 8.90/5.07 ( ( ord_less_real @ A @ B )
% 8.90/5.07 => ( ( ord_less_num @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4198_order__less__subst2,axiom,
% 8.90/5.07 ! [A: real,B: real,F: real > nat,C: nat] :
% 8.90/5.07 ( ( ord_less_real @ A @ B )
% 8.90/5.07 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4199_order__less__subst2,axiom,
% 8.90/5.07 ! [A: real,B: real,F: real > int,C: int] :
% 8.90/5.07 ( ( ord_less_real @ A @ B )
% 8.90/5.07 => ( ( ord_less_int @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4200_order__less__subst2,axiom,
% 8.90/5.07 ! [A: rat,B: rat,F: rat > real,C: real] :
% 8.90/5.07 ( ( ord_less_rat @ A @ B )
% 8.90/5.07 => ( ( ord_less_real @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4201_order__less__subst2,axiom,
% 8.90/5.07 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 8.90/5.07 ( ( ord_less_rat @ A @ B )
% 8.90/5.07 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4202_order__less__subst2,axiom,
% 8.90/5.07 ! [A: rat,B: rat,F: rat > num,C: num] :
% 8.90/5.07 ( ( ord_less_rat @ A @ B )
% 8.90/5.07 => ( ( ord_less_num @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4203_order__less__subst2,axiom,
% 8.90/5.07 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 8.90/5.07 ( ( ord_less_rat @ A @ B )
% 8.90/5.07 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4204_order__less__subst2,axiom,
% 8.90/5.07 ! [A: rat,B: rat,F: rat > int,C: int] :
% 8.90/5.07 ( ( ord_less_rat @ A @ B )
% 8.90/5.07 => ( ( ord_less_int @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4205_order__less__not__sym,axiom,
% 8.90/5.07 ! [X2: real,Y3: real] :
% 8.90/5.07 ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.07 => ~ ( ord_less_real @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4206_order__less__not__sym,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X2 @ Y3 )
% 8.90/5.07 => ~ ( ord_less_rat @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4207_order__less__not__sym,axiom,
% 8.90/5.07 ! [X2: num,Y3: num] :
% 8.90/5.07 ( ( ord_less_num @ X2 @ Y3 )
% 8.90/5.07 => ~ ( ord_less_num @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4208_order__less__not__sym,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat] :
% 8.90/5.07 ( ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.07 => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4209_order__less__not__sym,axiom,
% 8.90/5.07 ! [X2: int,Y3: int] :
% 8.90/5.07 ( ( ord_less_int @ X2 @ Y3 )
% 8.90/5.07 => ~ ( ord_less_int @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4210_order__less__imp__triv,axiom,
% 8.90/5.07 ! [X2: real,Y3: real,P: $o] :
% 8.90/5.07 ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_real @ Y3 @ X2 )
% 8.90/5.07 => P ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4211_order__less__imp__triv,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat,P: $o] :
% 8.90/5.07 ( ( ord_less_rat @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_rat @ Y3 @ X2 )
% 8.90/5.07 => P ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4212_order__less__imp__triv,axiom,
% 8.90/5.07 ! [X2: num,Y3: num,P: $o] :
% 8.90/5.07 ( ( ord_less_num @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_num @ Y3 @ X2 )
% 8.90/5.07 => P ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4213_order__less__imp__triv,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat,P: $o] :
% 8.90/5.07 ( ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_nat @ Y3 @ X2 )
% 8.90/5.07 => P ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4214_order__less__imp__triv,axiom,
% 8.90/5.07 ! [X2: int,Y3: int,P: $o] :
% 8.90/5.07 ( ( ord_less_int @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_int @ Y3 @ X2 )
% 8.90/5.07 => P ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4215_linorder__less__linear,axiom,
% 8.90/5.07 ! [X2: real,Y3: real] :
% 8.90/5.07 ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.07 | ( X2 = Y3 )
% 8.90/5.07 | ( ord_less_real @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4216_linorder__less__linear,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X2 @ Y3 )
% 8.90/5.07 | ( X2 = Y3 )
% 8.90/5.07 | ( ord_less_rat @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4217_linorder__less__linear,axiom,
% 8.90/5.07 ! [X2: num,Y3: num] :
% 8.90/5.07 ( ( ord_less_num @ X2 @ Y3 )
% 8.90/5.07 | ( X2 = Y3 )
% 8.90/5.07 | ( ord_less_num @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4218_linorder__less__linear,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat] :
% 8.90/5.07 ( ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.07 | ( X2 = Y3 )
% 8.90/5.07 | ( ord_less_nat @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4219_linorder__less__linear,axiom,
% 8.90/5.07 ! [X2: int,Y3: int] :
% 8.90/5.07 ( ( ord_less_int @ X2 @ Y3 )
% 8.90/5.07 | ( X2 = Y3 )
% 8.90/5.07 | ( ord_less_int @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4220_order__less__imp__not__eq,axiom,
% 8.90/5.07 ! [X2: real,Y3: real] :
% 8.90/5.07 ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.07 => ( X2 != Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4221_order__less__imp__not__eq,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X2 @ Y3 )
% 8.90/5.07 => ( X2 != Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4222_order__less__imp__not__eq,axiom,
% 8.90/5.07 ! [X2: num,Y3: num] :
% 8.90/5.07 ( ( ord_less_num @ X2 @ Y3 )
% 8.90/5.07 => ( X2 != Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4223_order__less__imp__not__eq,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat] :
% 8.90/5.07 ( ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.07 => ( X2 != Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4224_order__less__imp__not__eq,axiom,
% 8.90/5.07 ! [X2: int,Y3: int] :
% 8.90/5.07 ( ( ord_less_int @ X2 @ Y3 )
% 8.90/5.07 => ( X2 != Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4225_order__less__imp__not__eq2,axiom,
% 8.90/5.07 ! [X2: real,Y3: real] :
% 8.90/5.07 ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.07 => ( Y3 != X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4226_order__less__imp__not__eq2,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X2 @ Y3 )
% 8.90/5.07 => ( Y3 != X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4227_order__less__imp__not__eq2,axiom,
% 8.90/5.07 ! [X2: num,Y3: num] :
% 8.90/5.07 ( ( ord_less_num @ X2 @ Y3 )
% 8.90/5.07 => ( Y3 != X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4228_order__less__imp__not__eq2,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat] :
% 8.90/5.07 ( ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.07 => ( Y3 != X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4229_order__less__imp__not__eq2,axiom,
% 8.90/5.07 ! [X2: int,Y3: int] :
% 8.90/5.07 ( ( ord_less_int @ X2 @ Y3 )
% 8.90/5.07 => ( Y3 != X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4230_order__less__imp__not__less,axiom,
% 8.90/5.07 ! [X2: real,Y3: real] :
% 8.90/5.07 ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.07 => ~ ( ord_less_real @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4231_order__less__imp__not__less,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X2 @ Y3 )
% 8.90/5.07 => ~ ( ord_less_rat @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4232_order__less__imp__not__less,axiom,
% 8.90/5.07 ! [X2: num,Y3: num] :
% 8.90/5.07 ( ( ord_less_num @ X2 @ Y3 )
% 8.90/5.07 => ~ ( ord_less_num @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4233_order__less__imp__not__less,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat] :
% 8.90/5.07 ( ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.07 => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4234_order__less__imp__not__less,axiom,
% 8.90/5.07 ! [X2: int,Y3: int] :
% 8.90/5.07 ( ( ord_less_int @ X2 @ Y3 )
% 8.90/5.07 => ~ ( ord_less_int @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4235_eval__nat__numeral_I2_J,axiom,
% 8.90/5.07 ! [N: num] :
% 8.90/5.07 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 8.90/5.07 = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4236_BitM__plus__one,axiom,
% 8.90/5.07 ! [N: num] :
% 8.90/5.07 ( ( plus_plus_num @ ( bitM @ N ) @ one )
% 8.90/5.07 = ( bit0 @ N ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4237_one__plus__BitM,axiom,
% 8.90/5.07 ! [N: num] :
% 8.90/5.07 ( ( plus_plus_num @ one @ ( bitM @ N ) )
% 8.90/5.07 = ( bit0 @ N ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4238_numeral__BitM,axiom,
% 8.90/5.07 ! [N: num] :
% 8.90/5.07 ( ( numera6690914467698888265omplex @ ( bitM @ N ) )
% 8.90/5.07 = ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N ) ) @ one_one_complex ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4239_numeral__BitM,axiom,
% 8.90/5.07 ! [N: num] :
% 8.90/5.07 ( ( numeral_numeral_real @ ( bitM @ N ) )
% 8.90/5.07 = ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N ) ) @ one_one_real ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4240_numeral__BitM,axiom,
% 8.90/5.07 ! [N: num] :
% 8.90/5.07 ( ( numeral_numeral_rat @ ( bitM @ N ) )
% 8.90/5.07 = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N ) ) @ one_one_rat ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4241_numeral__BitM,axiom,
% 8.90/5.07 ! [N: num] :
% 8.90/5.07 ( ( numeral_numeral_int @ ( bitM @ N ) )
% 8.90/5.07 = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4242_order__le__imp__less__or__eq,axiom,
% 8.90/5.07 ! [X2: real,Y3: real] :
% 8.90/5.07 ( ( ord_less_eq_real @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.07 | ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4243_order__le__imp__less__or__eq,axiom,
% 8.90/5.07 ! [X2: set_int,Y3: set_int] :
% 8.90/5.07 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_set_int @ X2 @ Y3 )
% 8.90/5.07 | ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4244_order__le__imp__less__or__eq,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_rat @ X2 @ Y3 )
% 8.90/5.07 | ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4245_order__le__imp__less__or__eq,axiom,
% 8.90/5.07 ! [X2: num,Y3: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_num @ X2 @ Y3 )
% 8.90/5.07 | ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4246_order__le__imp__less__or__eq,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat] :
% 8.90/5.07 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.07 | ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4247_order__le__imp__less__or__eq,axiom,
% 8.90/5.07 ! [X2: int,Y3: int] :
% 8.90/5.07 ( ( ord_less_eq_int @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_int @ X2 @ Y3 )
% 8.90/5.07 | ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4248_linorder__le__less__linear,axiom,
% 8.90/5.07 ! [X2: real,Y3: real] :
% 8.90/5.07 ( ( ord_less_eq_real @ X2 @ Y3 )
% 8.90/5.07 | ( ord_less_real @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4249_linorder__le__less__linear,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 8.90/5.07 | ( ord_less_rat @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4250_linorder__le__less__linear,axiom,
% 8.90/5.07 ! [X2: num,Y3: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ X2 @ Y3 )
% 8.90/5.07 | ( ord_less_num @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4251_linorder__le__less__linear,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat] :
% 8.90/5.07 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 8.90/5.07 | ( ord_less_nat @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4252_linorder__le__less__linear,axiom,
% 8.90/5.07 ! [X2: int,Y3: int] :
% 8.90/5.07 ( ( ord_less_eq_int @ X2 @ Y3 )
% 8.90/5.07 | ( ord_less_int @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4253_order__less__le__subst2,axiom,
% 8.90/5.07 ! [A: real,B: real,F: real > real,C: real] :
% 8.90/5.07 ( ( ord_less_real @ A @ B )
% 8.90/5.07 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4254_order__less__le__subst2,axiom,
% 8.90/5.07 ! [A: rat,B: rat,F: rat > real,C: real] :
% 8.90/5.07 ( ( ord_less_rat @ A @ B )
% 8.90/5.07 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4255_order__less__le__subst2,axiom,
% 8.90/5.07 ! [A: num,B: num,F: num > real,C: real] :
% 8.90/5.07 ( ( ord_less_num @ A @ B )
% 8.90/5.07 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: num,Y4: num] :
% 8.90/5.07 ( ( ord_less_num @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4256_order__less__le__subst2,axiom,
% 8.90/5.07 ! [A: nat,B: nat,F: nat > real,C: real] :
% 8.90/5.07 ( ( ord_less_nat @ A @ B )
% 8.90/5.07 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: nat,Y4: nat] :
% 8.90/5.07 ( ( ord_less_nat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4257_order__less__le__subst2,axiom,
% 8.90/5.07 ! [A: int,B: int,F: int > real,C: real] :
% 8.90/5.07 ( ( ord_less_int @ A @ B )
% 8.90/5.07 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: int,Y4: int] :
% 8.90/5.07 ( ( ord_less_int @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4258_order__less__le__subst2,axiom,
% 8.90/5.07 ! [A: real,B: real,F: real > rat,C: rat] :
% 8.90/5.07 ( ( ord_less_real @ A @ B )
% 8.90/5.07 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4259_order__less__le__subst2,axiom,
% 8.90/5.07 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 8.90/5.07 ( ( ord_less_rat @ A @ B )
% 8.90/5.07 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4260_order__less__le__subst2,axiom,
% 8.90/5.07 ! [A: num,B: num,F: num > rat,C: rat] :
% 8.90/5.07 ( ( ord_less_num @ A @ B )
% 8.90/5.07 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: num,Y4: num] :
% 8.90/5.07 ( ( ord_less_num @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4261_order__less__le__subst2,axiom,
% 8.90/5.07 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 8.90/5.07 ( ( ord_less_nat @ A @ B )
% 8.90/5.07 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: nat,Y4: nat] :
% 8.90/5.07 ( ( ord_less_nat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4262_order__less__le__subst2,axiom,
% 8.90/5.07 ! [A: int,B: int,F: int > rat,C: rat] :
% 8.90/5.07 ( ( ord_less_int @ A @ B )
% 8.90/5.07 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: int,Y4: int] :
% 8.90/5.07 ( ( ord_less_int @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4263_order__less__le__subst1,axiom,
% 8.90/5.07 ! [A: real,F: rat > real,B: rat,C: rat] :
% 8.90/5.07 ( ( ord_less_real @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_eq_rat @ B @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4264_order__less__le__subst1,axiom,
% 8.90/5.07 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 8.90/5.07 ( ( ord_less_rat @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_eq_rat @ B @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4265_order__less__le__subst1,axiom,
% 8.90/5.07 ! [A: num,F: rat > num,B: rat,C: rat] :
% 8.90/5.07 ( ( ord_less_num @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_eq_rat @ B @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4266_order__less__le__subst1,axiom,
% 8.90/5.07 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 8.90/5.07 ( ( ord_less_nat @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_eq_rat @ B @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4267_order__less__le__subst1,axiom,
% 8.90/5.07 ! [A: int,F: rat > int,B: rat,C: rat] :
% 8.90/5.07 ( ( ord_less_int @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_eq_rat @ B @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4268_order__less__le__subst1,axiom,
% 8.90/5.07 ! [A: real,F: num > real,B: num,C: num] :
% 8.90/5.07 ( ( ord_less_real @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_eq_num @ B @ C )
% 8.90/5.07 => ( ! [X5: num,Y4: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4269_order__less__le__subst1,axiom,
% 8.90/5.07 ! [A: rat,F: num > rat,B: num,C: num] :
% 8.90/5.07 ( ( ord_less_rat @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_eq_num @ B @ C )
% 8.90/5.07 => ( ! [X5: num,Y4: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4270_order__less__le__subst1,axiom,
% 8.90/5.07 ! [A: num,F: num > num,B: num,C: num] :
% 8.90/5.07 ( ( ord_less_num @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_eq_num @ B @ C )
% 8.90/5.07 => ( ! [X5: num,Y4: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4271_order__less__le__subst1,axiom,
% 8.90/5.07 ! [A: nat,F: num > nat,B: num,C: num] :
% 8.90/5.07 ( ( ord_less_nat @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_eq_num @ B @ C )
% 8.90/5.07 => ( ! [X5: num,Y4: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4272_order__less__le__subst1,axiom,
% 8.90/5.07 ! [A: int,F: num > int,B: num,C: num] :
% 8.90/5.07 ( ( ord_less_int @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_eq_num @ B @ C )
% 8.90/5.07 => ( ! [X5: num,Y4: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4273_order__le__less__subst2,axiom,
% 8.90/5.07 ! [A: rat,B: rat,F: rat > real,C: real] :
% 8.90/5.07 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.07 => ( ( ord_less_real @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4274_order__le__less__subst2,axiom,
% 8.90/5.07 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.07 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4275_order__le__less__subst2,axiom,
% 8.90/5.07 ! [A: rat,B: rat,F: rat > num,C: num] :
% 8.90/5.07 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.07 => ( ( ord_less_num @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4276_order__le__less__subst2,axiom,
% 8.90/5.07 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.07 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4277_order__le__less__subst2,axiom,
% 8.90/5.07 ! [A: rat,B: rat,F: rat > int,C: int] :
% 8.90/5.07 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.07 => ( ( ord_less_int @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4278_order__le__less__subst2,axiom,
% 8.90/5.07 ! [A: num,B: num,F: num > real,C: real] :
% 8.90/5.07 ( ( ord_less_eq_num @ A @ B )
% 8.90/5.07 => ( ( ord_less_real @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: num,Y4: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4279_order__le__less__subst2,axiom,
% 8.90/5.07 ! [A: num,B: num,F: num > rat,C: rat] :
% 8.90/5.07 ( ( ord_less_eq_num @ A @ B )
% 8.90/5.07 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: num,Y4: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4280_order__le__less__subst2,axiom,
% 8.90/5.07 ! [A: num,B: num,F: num > num,C: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ A @ B )
% 8.90/5.07 => ( ( ord_less_num @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: num,Y4: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4281_order__le__less__subst2,axiom,
% 8.90/5.07 ! [A: num,B: num,F: num > nat,C: nat] :
% 8.90/5.07 ( ( ord_less_eq_num @ A @ B )
% 8.90/5.07 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: num,Y4: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4282_order__le__less__subst2,axiom,
% 8.90/5.07 ! [A: num,B: num,F: num > int,C: int] :
% 8.90/5.07 ( ( ord_less_eq_num @ A @ B )
% 8.90/5.07 => ( ( ord_less_int @ ( F @ B ) @ C )
% 8.90/5.07 => ( ! [X5: num,Y4: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4283_order__le__less__subst1,axiom,
% 8.90/5.07 ! [A: real,F: real > real,B: real,C: real] :
% 8.90/5.07 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_real @ B @ C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4284_order__le__less__subst1,axiom,
% 8.90/5.07 ! [A: real,F: rat > real,B: rat,C: rat] :
% 8.90/5.07 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_rat @ B @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4285_order__le__less__subst1,axiom,
% 8.90/5.07 ! [A: real,F: num > real,B: num,C: num] :
% 8.90/5.07 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_num @ B @ C )
% 8.90/5.07 => ( ! [X5: num,Y4: num] :
% 8.90/5.07 ( ( ord_less_num @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4286_order__le__less__subst1,axiom,
% 8.90/5.07 ! [A: real,F: nat > real,B: nat,C: nat] :
% 8.90/5.07 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_nat @ B @ C )
% 8.90/5.07 => ( ! [X5: nat,Y4: nat] :
% 8.90/5.07 ( ( ord_less_nat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4287_order__le__less__subst1,axiom,
% 8.90/5.07 ! [A: real,F: int > real,B: int,C: int] :
% 8.90/5.07 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_int @ B @ C )
% 8.90/5.07 => ( ! [X5: int,Y4: int] :
% 8.90/5.07 ( ( ord_less_int @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4288_order__le__less__subst1,axiom,
% 8.90/5.07 ! [A: rat,F: real > rat,B: real,C: real] :
% 8.90/5.07 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_real @ B @ C )
% 8.90/5.07 => ( ! [X5: real,Y4: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4289_order__le__less__subst1,axiom,
% 8.90/5.07 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_rat @ B @ C )
% 8.90/5.07 => ( ! [X5: rat,Y4: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4290_order__le__less__subst1,axiom,
% 8.90/5.07 ! [A: rat,F: num > rat,B: num,C: num] :
% 8.90/5.07 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_num @ B @ C )
% 8.90/5.07 => ( ! [X5: num,Y4: num] :
% 8.90/5.07 ( ( ord_less_num @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4291_order__le__less__subst1,axiom,
% 8.90/5.07 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_nat @ B @ C )
% 8.90/5.07 => ( ! [X5: nat,Y4: nat] :
% 8.90/5.07 ( ( ord_less_nat @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4292_order__le__less__subst1,axiom,
% 8.90/5.07 ! [A: rat,F: int > rat,B: int,C: int] :
% 8.90/5.07 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 8.90/5.07 => ( ( ord_less_int @ B @ C )
% 8.90/5.07 => ( ! [X5: int,Y4: int] :
% 8.90/5.07 ( ( ord_less_int @ X5 @ Y4 )
% 8.90/5.07 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.90/5.07 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4293_order__less__le__trans,axiom,
% 8.90/5.07 ! [X2: real,Y3: real,Z: real] :
% 8.90/5.07 ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_eq_real @ Y3 @ Z )
% 8.90/5.07 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4294_order__less__le__trans,axiom,
% 8.90/5.07 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 8.90/5.07 ( ( ord_less_set_int @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_eq_set_int @ Y3 @ Z )
% 8.90/5.07 => ( ord_less_set_int @ X2 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4295_order__less__le__trans,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat,Z: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_eq_rat @ Y3 @ Z )
% 8.90/5.07 => ( ord_less_rat @ X2 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4296_order__less__le__trans,axiom,
% 8.90/5.07 ! [X2: num,Y3: num,Z: num] :
% 8.90/5.07 ( ( ord_less_num @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_eq_num @ Y3 @ Z )
% 8.90/5.07 => ( ord_less_num @ X2 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4297_order__less__le__trans,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat,Z: nat] :
% 8.90/5.07 ( ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_eq_nat @ Y3 @ Z )
% 8.90/5.07 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4298_order__less__le__trans,axiom,
% 8.90/5.07 ! [X2: int,Y3: int,Z: int] :
% 8.90/5.07 ( ( ord_less_int @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_eq_int @ Y3 @ Z )
% 8.90/5.07 => ( ord_less_int @ X2 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4299_order__le__less__trans,axiom,
% 8.90/5.07 ! [X2: real,Y3: real,Z: real] :
% 8.90/5.07 ( ( ord_less_eq_real @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_real @ Y3 @ Z )
% 8.90/5.07 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4300_order__le__less__trans,axiom,
% 8.90/5.07 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 8.90/5.07 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_set_int @ Y3 @ Z )
% 8.90/5.07 => ( ord_less_set_int @ X2 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4301_order__le__less__trans,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat,Z: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_rat @ Y3 @ Z )
% 8.90/5.07 => ( ord_less_rat @ X2 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4302_order__le__less__trans,axiom,
% 8.90/5.07 ! [X2: num,Y3: num,Z: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_num @ Y3 @ Z )
% 8.90/5.07 => ( ord_less_num @ X2 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4303_order__le__less__trans,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat,Z: nat] :
% 8.90/5.07 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_nat @ Y3 @ Z )
% 8.90/5.07 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4304_order__le__less__trans,axiom,
% 8.90/5.07 ! [X2: int,Y3: int,Z: int] :
% 8.90/5.07 ( ( ord_less_eq_int @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_int @ Y3 @ Z )
% 8.90/5.07 => ( ord_less_int @ X2 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4305_order__neq__le__trans,axiom,
% 8.90/5.07 ! [A: real,B: real] :
% 8.90/5.07 ( ( A != B )
% 8.90/5.07 => ( ( ord_less_eq_real @ A @ B )
% 8.90/5.07 => ( ord_less_real @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4306_order__neq__le__trans,axiom,
% 8.90/5.07 ! [A: set_int,B: set_int] :
% 8.90/5.07 ( ( A != B )
% 8.90/5.07 => ( ( ord_less_eq_set_int @ A @ B )
% 8.90/5.07 => ( ord_less_set_int @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4307_order__neq__le__trans,axiom,
% 8.90/5.07 ! [A: rat,B: rat] :
% 8.90/5.07 ( ( A != B )
% 8.90/5.07 => ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.07 => ( ord_less_rat @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4308_order__neq__le__trans,axiom,
% 8.90/5.07 ! [A: num,B: num] :
% 8.90/5.07 ( ( A != B )
% 8.90/5.07 => ( ( ord_less_eq_num @ A @ B )
% 8.90/5.07 => ( ord_less_num @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4309_order__neq__le__trans,axiom,
% 8.90/5.07 ! [A: nat,B: nat] :
% 8.90/5.07 ( ( A != B )
% 8.90/5.07 => ( ( ord_less_eq_nat @ A @ B )
% 8.90/5.07 => ( ord_less_nat @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4310_order__neq__le__trans,axiom,
% 8.90/5.07 ! [A: int,B: int] :
% 8.90/5.07 ( ( A != B )
% 8.90/5.07 => ( ( ord_less_eq_int @ A @ B )
% 8.90/5.07 => ( ord_less_int @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4311_order__le__neq__trans,axiom,
% 8.90/5.07 ! [A: real,B: real] :
% 8.90/5.07 ( ( ord_less_eq_real @ A @ B )
% 8.90/5.07 => ( ( A != B )
% 8.90/5.07 => ( ord_less_real @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4312_order__le__neq__trans,axiom,
% 8.90/5.07 ! [A: set_int,B: set_int] :
% 8.90/5.07 ( ( ord_less_eq_set_int @ A @ B )
% 8.90/5.07 => ( ( A != B )
% 8.90/5.07 => ( ord_less_set_int @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4313_order__le__neq__trans,axiom,
% 8.90/5.07 ! [A: rat,B: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.07 => ( ( A != B )
% 8.90/5.07 => ( ord_less_rat @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4314_order__le__neq__trans,axiom,
% 8.90/5.07 ! [A: num,B: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ A @ B )
% 8.90/5.07 => ( ( A != B )
% 8.90/5.07 => ( ord_less_num @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4315_order__le__neq__trans,axiom,
% 8.90/5.07 ! [A: nat,B: nat] :
% 8.90/5.07 ( ( ord_less_eq_nat @ A @ B )
% 8.90/5.07 => ( ( A != B )
% 8.90/5.07 => ( ord_less_nat @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4316_order__le__neq__trans,axiom,
% 8.90/5.07 ! [A: int,B: int] :
% 8.90/5.07 ( ( ord_less_eq_int @ A @ B )
% 8.90/5.07 => ( ( A != B )
% 8.90/5.07 => ( ord_less_int @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4317_order__less__imp__le,axiom,
% 8.90/5.07 ! [X2: real,Y3: real] :
% 8.90/5.07 ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.07 => ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4318_order__less__imp__le,axiom,
% 8.90/5.07 ! [X2: set_int,Y3: set_int] :
% 8.90/5.07 ( ( ord_less_set_int @ X2 @ Y3 )
% 8.90/5.07 => ( ord_less_eq_set_int @ X2 @ Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4319_order__less__imp__le,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X2 @ Y3 )
% 8.90/5.07 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4320_order__less__imp__le,axiom,
% 8.90/5.07 ! [X2: num,Y3: num] :
% 8.90/5.07 ( ( ord_less_num @ X2 @ Y3 )
% 8.90/5.07 => ( ord_less_eq_num @ X2 @ Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4321_order__less__imp__le,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat] :
% 8.90/5.07 ( ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.07 => ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4322_order__less__imp__le,axiom,
% 8.90/5.07 ! [X2: int,Y3: int] :
% 8.90/5.07 ( ( ord_less_int @ X2 @ Y3 )
% 8.90/5.07 => ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4323_linorder__not__less,axiom,
% 8.90/5.07 ! [X2: real,Y3: real] :
% 8.90/5.07 ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
% 8.90/5.07 = ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4324_linorder__not__less,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat] :
% 8.90/5.07 ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
% 8.90/5.07 = ( ord_less_eq_rat @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4325_linorder__not__less,axiom,
% 8.90/5.07 ! [X2: num,Y3: num] :
% 8.90/5.07 ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
% 8.90/5.07 = ( ord_less_eq_num @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4326_linorder__not__less,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat] :
% 8.90/5.07 ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
% 8.90/5.07 = ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4327_linorder__not__less,axiom,
% 8.90/5.07 ! [X2: int,Y3: int] :
% 8.90/5.07 ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
% 8.90/5.07 = ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4328_linorder__not__le,axiom,
% 8.90/5.07 ! [X2: real,Y3: real] :
% 8.90/5.07 ( ( ~ ( ord_less_eq_real @ X2 @ Y3 ) )
% 8.90/5.07 = ( ord_less_real @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4329_linorder__not__le,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat] :
% 8.90/5.07 ( ( ~ ( ord_less_eq_rat @ X2 @ Y3 ) )
% 8.90/5.07 = ( ord_less_rat @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4330_linorder__not__le,axiom,
% 8.90/5.07 ! [X2: num,Y3: num] :
% 8.90/5.07 ( ( ~ ( ord_less_eq_num @ X2 @ Y3 ) )
% 8.90/5.07 = ( ord_less_num @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4331_linorder__not__le,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat] :
% 8.90/5.07 ( ( ~ ( ord_less_eq_nat @ X2 @ Y3 ) )
% 8.90/5.07 = ( ord_less_nat @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4332_linorder__not__le,axiom,
% 8.90/5.07 ! [X2: int,Y3: int] :
% 8.90/5.07 ( ( ~ ( ord_less_eq_int @ X2 @ Y3 ) )
% 8.90/5.07 = ( ord_less_int @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4333_order__less__le,axiom,
% 8.90/5.07 ( ord_less_real
% 8.90/5.07 = ( ^ [X: real,Y: real] :
% 8.90/5.07 ( ( ord_less_eq_real @ X @ Y )
% 8.90/5.07 & ( X != Y ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4334_order__less__le,axiom,
% 8.90/5.07 ( ord_less_set_int
% 8.90/5.07 = ( ^ [X: set_int,Y: set_int] :
% 8.90/5.07 ( ( ord_less_eq_set_int @ X @ Y )
% 8.90/5.07 & ( X != Y ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4335_order__less__le,axiom,
% 8.90/5.07 ( ord_less_rat
% 8.90/5.07 = ( ^ [X: rat,Y: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ X @ Y )
% 8.90/5.07 & ( X != Y ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4336_order__less__le,axiom,
% 8.90/5.07 ( ord_less_num
% 8.90/5.07 = ( ^ [X: num,Y: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ X @ Y )
% 8.90/5.07 & ( X != Y ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4337_order__less__le,axiom,
% 8.90/5.07 ( ord_less_nat
% 8.90/5.07 = ( ^ [X: nat,Y: nat] :
% 8.90/5.07 ( ( ord_less_eq_nat @ X @ Y )
% 8.90/5.07 & ( X != Y ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4338_order__less__le,axiom,
% 8.90/5.07 ( ord_less_int
% 8.90/5.07 = ( ^ [X: int,Y: int] :
% 8.90/5.07 ( ( ord_less_eq_int @ X @ Y )
% 8.90/5.07 & ( X != Y ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4339_order__le__less,axiom,
% 8.90/5.07 ( ord_less_eq_real
% 8.90/5.07 = ( ^ [X: real,Y: real] :
% 8.90/5.07 ( ( ord_less_real @ X @ Y )
% 8.90/5.07 | ( X = Y ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4340_order__le__less,axiom,
% 8.90/5.07 ( ord_less_eq_set_int
% 8.90/5.07 = ( ^ [X: set_int,Y: set_int] :
% 8.90/5.07 ( ( ord_less_set_int @ X @ Y )
% 8.90/5.07 | ( X = Y ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4341_order__le__less,axiom,
% 8.90/5.07 ( ord_less_eq_rat
% 8.90/5.07 = ( ^ [X: rat,Y: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X @ Y )
% 8.90/5.07 | ( X = Y ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4342_order__le__less,axiom,
% 8.90/5.07 ( ord_less_eq_num
% 8.90/5.07 = ( ^ [X: num,Y: num] :
% 8.90/5.07 ( ( ord_less_num @ X @ Y )
% 8.90/5.07 | ( X = Y ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4343_order__le__less,axiom,
% 8.90/5.07 ( ord_less_eq_nat
% 8.90/5.07 = ( ^ [X: nat,Y: nat] :
% 8.90/5.07 ( ( ord_less_nat @ X @ Y )
% 8.90/5.07 | ( X = Y ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4344_order__le__less,axiom,
% 8.90/5.07 ( ord_less_eq_int
% 8.90/5.07 = ( ^ [X: int,Y: int] :
% 8.90/5.07 ( ( ord_less_int @ X @ Y )
% 8.90/5.07 | ( X = Y ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4345_dual__order_Ostrict__implies__order,axiom,
% 8.90/5.07 ! [B: real,A: real] :
% 8.90/5.07 ( ( ord_less_real @ B @ A )
% 8.90/5.07 => ( ord_less_eq_real @ B @ A ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4346_dual__order_Ostrict__implies__order,axiom,
% 8.90/5.07 ! [B: set_int,A: set_int] :
% 8.90/5.07 ( ( ord_less_set_int @ B @ A )
% 8.90/5.07 => ( ord_less_eq_set_int @ B @ A ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4347_dual__order_Ostrict__implies__order,axiom,
% 8.90/5.07 ! [B: rat,A: rat] :
% 8.90/5.07 ( ( ord_less_rat @ B @ A )
% 8.90/5.07 => ( ord_less_eq_rat @ B @ A ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4348_dual__order_Ostrict__implies__order,axiom,
% 8.90/5.07 ! [B: num,A: num] :
% 8.90/5.07 ( ( ord_less_num @ B @ A )
% 8.90/5.07 => ( ord_less_eq_num @ B @ A ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4349_dual__order_Ostrict__implies__order,axiom,
% 8.90/5.07 ! [B: nat,A: nat] :
% 8.90/5.07 ( ( ord_less_nat @ B @ A )
% 8.90/5.07 => ( ord_less_eq_nat @ B @ A ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4350_dual__order_Ostrict__implies__order,axiom,
% 8.90/5.07 ! [B: int,A: int] :
% 8.90/5.07 ( ( ord_less_int @ B @ A )
% 8.90/5.07 => ( ord_less_eq_int @ B @ A ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4351_order_Ostrict__implies__order,axiom,
% 8.90/5.07 ! [A: real,B: real] :
% 8.90/5.07 ( ( ord_less_real @ A @ B )
% 8.90/5.07 => ( ord_less_eq_real @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4352_order_Ostrict__implies__order,axiom,
% 8.90/5.07 ! [A: set_int,B: set_int] :
% 8.90/5.07 ( ( ord_less_set_int @ A @ B )
% 8.90/5.07 => ( ord_less_eq_set_int @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4353_order_Ostrict__implies__order,axiom,
% 8.90/5.07 ! [A: rat,B: rat] :
% 8.90/5.07 ( ( ord_less_rat @ A @ B )
% 8.90/5.07 => ( ord_less_eq_rat @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4354_order_Ostrict__implies__order,axiom,
% 8.90/5.07 ! [A: num,B: num] :
% 8.90/5.07 ( ( ord_less_num @ A @ B )
% 8.90/5.07 => ( ord_less_eq_num @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4355_order_Ostrict__implies__order,axiom,
% 8.90/5.07 ! [A: nat,B: nat] :
% 8.90/5.07 ( ( ord_less_nat @ A @ B )
% 8.90/5.07 => ( ord_less_eq_nat @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4356_order_Ostrict__implies__order,axiom,
% 8.90/5.07 ! [A: int,B: int] :
% 8.90/5.07 ( ( ord_less_int @ A @ B )
% 8.90/5.07 => ( ord_less_eq_int @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4357_dual__order_Ostrict__iff__not,axiom,
% 8.90/5.07 ( ord_less_real
% 8.90/5.07 = ( ^ [B2: real,A3: real] :
% 8.90/5.07 ( ( ord_less_eq_real @ B2 @ A3 )
% 8.90/5.07 & ~ ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4358_dual__order_Ostrict__iff__not,axiom,
% 8.90/5.07 ( ord_less_set_int
% 8.90/5.07 = ( ^ [B2: set_int,A3: set_int] :
% 8.90/5.07 ( ( ord_less_eq_set_int @ B2 @ A3 )
% 8.90/5.07 & ~ ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4359_dual__order_Ostrict__iff__not,axiom,
% 8.90/5.07 ( ord_less_rat
% 8.90/5.07 = ( ^ [B2: rat,A3: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ B2 @ A3 )
% 8.90/5.07 & ~ ( ord_less_eq_rat @ A3 @ B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4360_dual__order_Ostrict__iff__not,axiom,
% 8.90/5.07 ( ord_less_num
% 8.90/5.07 = ( ^ [B2: num,A3: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ B2 @ A3 )
% 8.90/5.07 & ~ ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4361_dual__order_Ostrict__iff__not,axiom,
% 8.90/5.07 ( ord_less_nat
% 8.90/5.07 = ( ^ [B2: nat,A3: nat] :
% 8.90/5.07 ( ( ord_less_eq_nat @ B2 @ A3 )
% 8.90/5.07 & ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4362_dual__order_Ostrict__iff__not,axiom,
% 8.90/5.07 ( ord_less_int
% 8.90/5.07 = ( ^ [B2: int,A3: int] :
% 8.90/5.07 ( ( ord_less_eq_int @ B2 @ A3 )
% 8.90/5.07 & ~ ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4363_dual__order_Ostrict__trans2,axiom,
% 8.90/5.07 ! [B: real,A: real,C: real] :
% 8.90/5.07 ( ( ord_less_real @ B @ A )
% 8.90/5.07 => ( ( ord_less_eq_real @ C @ B )
% 8.90/5.07 => ( ord_less_real @ C @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4364_dual__order_Ostrict__trans2,axiom,
% 8.90/5.07 ! [B: set_int,A: set_int,C: set_int] :
% 8.90/5.07 ( ( ord_less_set_int @ B @ A )
% 8.90/5.07 => ( ( ord_less_eq_set_int @ C @ B )
% 8.90/5.07 => ( ord_less_set_int @ C @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4365_dual__order_Ostrict__trans2,axiom,
% 8.90/5.07 ! [B: rat,A: rat,C: rat] :
% 8.90/5.07 ( ( ord_less_rat @ B @ A )
% 8.90/5.07 => ( ( ord_less_eq_rat @ C @ B )
% 8.90/5.07 => ( ord_less_rat @ C @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4366_dual__order_Ostrict__trans2,axiom,
% 8.90/5.07 ! [B: num,A: num,C: num] :
% 8.90/5.07 ( ( ord_less_num @ B @ A )
% 8.90/5.07 => ( ( ord_less_eq_num @ C @ B )
% 8.90/5.07 => ( ord_less_num @ C @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4367_dual__order_Ostrict__trans2,axiom,
% 8.90/5.07 ! [B: nat,A: nat,C: nat] :
% 8.90/5.07 ( ( ord_less_nat @ B @ A )
% 8.90/5.07 => ( ( ord_less_eq_nat @ C @ B )
% 8.90/5.07 => ( ord_less_nat @ C @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4368_dual__order_Ostrict__trans2,axiom,
% 8.90/5.07 ! [B: int,A: int,C: int] :
% 8.90/5.07 ( ( ord_less_int @ B @ A )
% 8.90/5.07 => ( ( ord_less_eq_int @ C @ B )
% 8.90/5.07 => ( ord_less_int @ C @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4369_dual__order_Ostrict__trans1,axiom,
% 8.90/5.07 ! [B: real,A: real,C: real] :
% 8.90/5.07 ( ( ord_less_eq_real @ B @ A )
% 8.90/5.07 => ( ( ord_less_real @ C @ B )
% 8.90/5.07 => ( ord_less_real @ C @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4370_dual__order_Ostrict__trans1,axiom,
% 8.90/5.07 ! [B: set_int,A: set_int,C: set_int] :
% 8.90/5.07 ( ( ord_less_eq_set_int @ B @ A )
% 8.90/5.07 => ( ( ord_less_set_int @ C @ B )
% 8.90/5.07 => ( ord_less_set_int @ C @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4371_dual__order_Ostrict__trans1,axiom,
% 8.90/5.07 ! [B: rat,A: rat,C: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ B @ A )
% 8.90/5.07 => ( ( ord_less_rat @ C @ B )
% 8.90/5.07 => ( ord_less_rat @ C @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4372_dual__order_Ostrict__trans1,axiom,
% 8.90/5.07 ! [B: num,A: num,C: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ B @ A )
% 8.90/5.07 => ( ( ord_less_num @ C @ B )
% 8.90/5.07 => ( ord_less_num @ C @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4373_dual__order_Ostrict__trans1,axiom,
% 8.90/5.07 ! [B: nat,A: nat,C: nat] :
% 8.90/5.07 ( ( ord_less_eq_nat @ B @ A )
% 8.90/5.07 => ( ( ord_less_nat @ C @ B )
% 8.90/5.07 => ( ord_less_nat @ C @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4374_dual__order_Ostrict__trans1,axiom,
% 8.90/5.07 ! [B: int,A: int,C: int] :
% 8.90/5.07 ( ( ord_less_eq_int @ B @ A )
% 8.90/5.07 => ( ( ord_less_int @ C @ B )
% 8.90/5.07 => ( ord_less_int @ C @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4375_dual__order_Ostrict__iff__order,axiom,
% 8.90/5.07 ( ord_less_real
% 8.90/5.07 = ( ^ [B2: real,A3: real] :
% 8.90/5.07 ( ( ord_less_eq_real @ B2 @ A3 )
% 8.90/5.07 & ( A3 != B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4376_dual__order_Ostrict__iff__order,axiom,
% 8.90/5.07 ( ord_less_set_int
% 8.90/5.07 = ( ^ [B2: set_int,A3: set_int] :
% 8.90/5.07 ( ( ord_less_eq_set_int @ B2 @ A3 )
% 8.90/5.07 & ( A3 != B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4377_dual__order_Ostrict__iff__order,axiom,
% 8.90/5.07 ( ord_less_rat
% 8.90/5.07 = ( ^ [B2: rat,A3: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ B2 @ A3 )
% 8.90/5.07 & ( A3 != B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4378_dual__order_Ostrict__iff__order,axiom,
% 8.90/5.07 ( ord_less_num
% 8.90/5.07 = ( ^ [B2: num,A3: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ B2 @ A3 )
% 8.90/5.07 & ( A3 != B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4379_dual__order_Ostrict__iff__order,axiom,
% 8.90/5.07 ( ord_less_nat
% 8.90/5.07 = ( ^ [B2: nat,A3: nat] :
% 8.90/5.07 ( ( ord_less_eq_nat @ B2 @ A3 )
% 8.90/5.07 & ( A3 != B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4380_dual__order_Ostrict__iff__order,axiom,
% 8.90/5.07 ( ord_less_int
% 8.90/5.07 = ( ^ [B2: int,A3: int] :
% 8.90/5.07 ( ( ord_less_eq_int @ B2 @ A3 )
% 8.90/5.07 & ( A3 != B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4381_dual__order_Oorder__iff__strict,axiom,
% 8.90/5.07 ( ord_less_eq_real
% 8.90/5.07 = ( ^ [B2: real,A3: real] :
% 8.90/5.07 ( ( ord_less_real @ B2 @ A3 )
% 8.90/5.07 | ( A3 = B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4382_dual__order_Oorder__iff__strict,axiom,
% 8.90/5.07 ( ord_less_eq_set_int
% 8.90/5.07 = ( ^ [B2: set_int,A3: set_int] :
% 8.90/5.07 ( ( ord_less_set_int @ B2 @ A3 )
% 8.90/5.07 | ( A3 = B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4383_dual__order_Oorder__iff__strict,axiom,
% 8.90/5.07 ( ord_less_eq_rat
% 8.90/5.07 = ( ^ [B2: rat,A3: rat] :
% 8.90/5.07 ( ( ord_less_rat @ B2 @ A3 )
% 8.90/5.07 | ( A3 = B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4384_dual__order_Oorder__iff__strict,axiom,
% 8.90/5.07 ( ord_less_eq_num
% 8.90/5.07 = ( ^ [B2: num,A3: num] :
% 8.90/5.07 ( ( ord_less_num @ B2 @ A3 )
% 8.90/5.07 | ( A3 = B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4385_dual__order_Oorder__iff__strict,axiom,
% 8.90/5.07 ( ord_less_eq_nat
% 8.90/5.07 = ( ^ [B2: nat,A3: nat] :
% 8.90/5.07 ( ( ord_less_nat @ B2 @ A3 )
% 8.90/5.07 | ( A3 = B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4386_dual__order_Oorder__iff__strict,axiom,
% 8.90/5.07 ( ord_less_eq_int
% 8.90/5.07 = ( ^ [B2: int,A3: int] :
% 8.90/5.07 ( ( ord_less_int @ B2 @ A3 )
% 8.90/5.07 | ( A3 = B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4387_dense__le__bounded,axiom,
% 8.90/5.07 ! [X2: real,Y3: real,Z: real] :
% 8.90/5.07 ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.07 => ( ! [W2: real] :
% 8.90/5.07 ( ( ord_less_real @ X2 @ W2 )
% 8.90/5.07 => ( ( ord_less_real @ W2 @ Y3 )
% 8.90/5.07 => ( ord_less_eq_real @ W2 @ Z ) ) )
% 8.90/5.07 => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4388_dense__le__bounded,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat,Z: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X2 @ Y3 )
% 8.90/5.07 => ( ! [W2: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X2 @ W2 )
% 8.90/5.07 => ( ( ord_less_rat @ W2 @ Y3 )
% 8.90/5.07 => ( ord_less_eq_rat @ W2 @ Z ) ) )
% 8.90/5.07 => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4389_dense__ge__bounded,axiom,
% 8.90/5.07 ! [Z: real,X2: real,Y3: real] :
% 8.90/5.07 ( ( ord_less_real @ Z @ X2 )
% 8.90/5.07 => ( ! [W2: real] :
% 8.90/5.07 ( ( ord_less_real @ Z @ W2 )
% 8.90/5.07 => ( ( ord_less_real @ W2 @ X2 )
% 8.90/5.07 => ( ord_less_eq_real @ Y3 @ W2 ) ) )
% 8.90/5.07 => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4390_dense__ge__bounded,axiom,
% 8.90/5.07 ! [Z: rat,X2: rat,Y3: rat] :
% 8.90/5.07 ( ( ord_less_rat @ Z @ X2 )
% 8.90/5.07 => ( ! [W2: rat] :
% 8.90/5.07 ( ( ord_less_rat @ Z @ W2 )
% 8.90/5.07 => ( ( ord_less_rat @ W2 @ X2 )
% 8.90/5.07 => ( ord_less_eq_rat @ Y3 @ W2 ) ) )
% 8.90/5.07 => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4391_order_Ostrict__iff__not,axiom,
% 8.90/5.07 ( ord_less_real
% 8.90/5.07 = ( ^ [A3: real,B2: real] :
% 8.90/5.07 ( ( ord_less_eq_real @ A3 @ B2 )
% 8.90/5.07 & ~ ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4392_order_Ostrict__iff__not,axiom,
% 8.90/5.07 ( ord_less_set_int
% 8.90/5.07 = ( ^ [A3: set_int,B2: set_int] :
% 8.90/5.07 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 8.90/5.07 & ~ ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4393_order_Ostrict__iff__not,axiom,
% 8.90/5.07 ( ord_less_rat
% 8.90/5.07 = ( ^ [A3: rat,B2: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ A3 @ B2 )
% 8.90/5.07 & ~ ( ord_less_eq_rat @ B2 @ A3 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4394_order_Ostrict__iff__not,axiom,
% 8.90/5.07 ( ord_less_num
% 8.90/5.07 = ( ^ [A3: num,B2: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ A3 @ B2 )
% 8.90/5.07 & ~ ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4395_order_Ostrict__iff__not,axiom,
% 8.90/5.07 ( ord_less_nat
% 8.90/5.07 = ( ^ [A3: nat,B2: nat] :
% 8.90/5.07 ( ( ord_less_eq_nat @ A3 @ B2 )
% 8.90/5.07 & ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4396_order_Ostrict__iff__not,axiom,
% 8.90/5.07 ( ord_less_int
% 8.90/5.07 = ( ^ [A3: int,B2: int] :
% 8.90/5.07 ( ( ord_less_eq_int @ A3 @ B2 )
% 8.90/5.07 & ~ ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4397_order_Ostrict__trans2,axiom,
% 8.90/5.07 ! [A: real,B: real,C: real] :
% 8.90/5.07 ( ( ord_less_real @ A @ B )
% 8.90/5.07 => ( ( ord_less_eq_real @ B @ C )
% 8.90/5.07 => ( ord_less_real @ A @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4398_order_Ostrict__trans2,axiom,
% 8.90/5.07 ! [A: set_int,B: set_int,C: set_int] :
% 8.90/5.07 ( ( ord_less_set_int @ A @ B )
% 8.90/5.07 => ( ( ord_less_eq_set_int @ B @ C )
% 8.90/5.07 => ( ord_less_set_int @ A @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4399_order_Ostrict__trans2,axiom,
% 8.90/5.07 ! [A: rat,B: rat,C: rat] :
% 8.90/5.07 ( ( ord_less_rat @ A @ B )
% 8.90/5.07 => ( ( ord_less_eq_rat @ B @ C )
% 8.90/5.07 => ( ord_less_rat @ A @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4400_order_Ostrict__trans2,axiom,
% 8.90/5.07 ! [A: num,B: num,C: num] :
% 8.90/5.07 ( ( ord_less_num @ A @ B )
% 8.90/5.07 => ( ( ord_less_eq_num @ B @ C )
% 8.90/5.07 => ( ord_less_num @ A @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4401_order_Ostrict__trans2,axiom,
% 8.90/5.07 ! [A: nat,B: nat,C: nat] :
% 8.90/5.07 ( ( ord_less_nat @ A @ B )
% 8.90/5.07 => ( ( ord_less_eq_nat @ B @ C )
% 8.90/5.07 => ( ord_less_nat @ A @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4402_order_Ostrict__trans2,axiom,
% 8.90/5.07 ! [A: int,B: int,C: int] :
% 8.90/5.07 ( ( ord_less_int @ A @ B )
% 8.90/5.07 => ( ( ord_less_eq_int @ B @ C )
% 8.90/5.07 => ( ord_less_int @ A @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4403_order_Ostrict__trans1,axiom,
% 8.90/5.07 ! [A: real,B: real,C: real] :
% 8.90/5.07 ( ( ord_less_eq_real @ A @ B )
% 8.90/5.07 => ( ( ord_less_real @ B @ C )
% 8.90/5.07 => ( ord_less_real @ A @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4404_order_Ostrict__trans1,axiom,
% 8.90/5.07 ! [A: set_int,B: set_int,C: set_int] :
% 8.90/5.07 ( ( ord_less_eq_set_int @ A @ B )
% 8.90/5.07 => ( ( ord_less_set_int @ B @ C )
% 8.90/5.07 => ( ord_less_set_int @ A @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4405_order_Ostrict__trans1,axiom,
% 8.90/5.07 ! [A: rat,B: rat,C: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.07 => ( ( ord_less_rat @ B @ C )
% 8.90/5.07 => ( ord_less_rat @ A @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4406_order_Ostrict__trans1,axiom,
% 8.90/5.07 ! [A: num,B: num,C: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ A @ B )
% 8.90/5.07 => ( ( ord_less_num @ B @ C )
% 8.90/5.07 => ( ord_less_num @ A @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4407_order_Ostrict__trans1,axiom,
% 8.90/5.07 ! [A: nat,B: nat,C: nat] :
% 8.90/5.07 ( ( ord_less_eq_nat @ A @ B )
% 8.90/5.07 => ( ( ord_less_nat @ B @ C )
% 8.90/5.07 => ( ord_less_nat @ A @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4408_order_Ostrict__trans1,axiom,
% 8.90/5.07 ! [A: int,B: int,C: int] :
% 8.90/5.07 ( ( ord_less_eq_int @ A @ B )
% 8.90/5.07 => ( ( ord_less_int @ B @ C )
% 8.90/5.07 => ( ord_less_int @ A @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4409_order_Ostrict__iff__order,axiom,
% 8.90/5.07 ( ord_less_real
% 8.90/5.07 = ( ^ [A3: real,B2: real] :
% 8.90/5.07 ( ( ord_less_eq_real @ A3 @ B2 )
% 8.90/5.07 & ( A3 != B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4410_order_Ostrict__iff__order,axiom,
% 8.90/5.07 ( ord_less_set_int
% 8.90/5.07 = ( ^ [A3: set_int,B2: set_int] :
% 8.90/5.07 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 8.90/5.07 & ( A3 != B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4411_order_Ostrict__iff__order,axiom,
% 8.90/5.07 ( ord_less_rat
% 8.90/5.07 = ( ^ [A3: rat,B2: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ A3 @ B2 )
% 8.90/5.07 & ( A3 != B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4412_order_Ostrict__iff__order,axiom,
% 8.90/5.07 ( ord_less_num
% 8.90/5.07 = ( ^ [A3: num,B2: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ A3 @ B2 )
% 8.90/5.07 & ( A3 != B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4413_order_Ostrict__iff__order,axiom,
% 8.90/5.07 ( ord_less_nat
% 8.90/5.07 = ( ^ [A3: nat,B2: nat] :
% 8.90/5.07 ( ( ord_less_eq_nat @ A3 @ B2 )
% 8.90/5.07 & ( A3 != B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4414_order_Ostrict__iff__order,axiom,
% 8.90/5.07 ( ord_less_int
% 8.90/5.07 = ( ^ [A3: int,B2: int] :
% 8.90/5.07 ( ( ord_less_eq_int @ A3 @ B2 )
% 8.90/5.07 & ( A3 != B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4415_order_Oorder__iff__strict,axiom,
% 8.90/5.07 ( ord_less_eq_real
% 8.90/5.07 = ( ^ [A3: real,B2: real] :
% 8.90/5.07 ( ( ord_less_real @ A3 @ B2 )
% 8.90/5.07 | ( A3 = B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4416_order_Oorder__iff__strict,axiom,
% 8.90/5.07 ( ord_less_eq_set_int
% 8.90/5.07 = ( ^ [A3: set_int,B2: set_int] :
% 8.90/5.07 ( ( ord_less_set_int @ A3 @ B2 )
% 8.90/5.07 | ( A3 = B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4417_order_Oorder__iff__strict,axiom,
% 8.90/5.07 ( ord_less_eq_rat
% 8.90/5.07 = ( ^ [A3: rat,B2: rat] :
% 8.90/5.07 ( ( ord_less_rat @ A3 @ B2 )
% 8.90/5.07 | ( A3 = B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4418_order_Oorder__iff__strict,axiom,
% 8.90/5.07 ( ord_less_eq_num
% 8.90/5.07 = ( ^ [A3: num,B2: num] :
% 8.90/5.07 ( ( ord_less_num @ A3 @ B2 )
% 8.90/5.07 | ( A3 = B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4419_order_Oorder__iff__strict,axiom,
% 8.90/5.07 ( ord_less_eq_nat
% 8.90/5.07 = ( ^ [A3: nat,B2: nat] :
% 8.90/5.07 ( ( ord_less_nat @ A3 @ B2 )
% 8.90/5.07 | ( A3 = B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4420_order_Oorder__iff__strict,axiom,
% 8.90/5.07 ( ord_less_eq_int
% 8.90/5.07 = ( ^ [A3: int,B2: int] :
% 8.90/5.07 ( ( ord_less_int @ A3 @ B2 )
% 8.90/5.07 | ( A3 = B2 ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4421_not__le__imp__less,axiom,
% 8.90/5.07 ! [Y3: real,X2: real] :
% 8.90/5.07 ( ~ ( ord_less_eq_real @ Y3 @ X2 )
% 8.90/5.07 => ( ord_less_real @ X2 @ Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4422_not__le__imp__less,axiom,
% 8.90/5.07 ! [Y3: rat,X2: rat] :
% 8.90/5.07 ( ~ ( ord_less_eq_rat @ Y3 @ X2 )
% 8.90/5.07 => ( ord_less_rat @ X2 @ Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4423_not__le__imp__less,axiom,
% 8.90/5.07 ! [Y3: num,X2: num] :
% 8.90/5.07 ( ~ ( ord_less_eq_num @ Y3 @ X2 )
% 8.90/5.07 => ( ord_less_num @ X2 @ Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4424_not__le__imp__less,axiom,
% 8.90/5.07 ! [Y3: nat,X2: nat] :
% 8.90/5.07 ( ~ ( ord_less_eq_nat @ Y3 @ X2 )
% 8.90/5.07 => ( ord_less_nat @ X2 @ Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4425_not__le__imp__less,axiom,
% 8.90/5.07 ! [Y3: int,X2: int] :
% 8.90/5.07 ( ~ ( ord_less_eq_int @ Y3 @ X2 )
% 8.90/5.07 => ( ord_less_int @ X2 @ Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4426_less__le__not__le,axiom,
% 8.90/5.07 ( ord_less_real
% 8.90/5.07 = ( ^ [X: real,Y: real] :
% 8.90/5.07 ( ( ord_less_eq_real @ X @ Y )
% 8.90/5.07 & ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4427_less__le__not__le,axiom,
% 8.90/5.07 ( ord_less_set_int
% 8.90/5.07 = ( ^ [X: set_int,Y: set_int] :
% 8.90/5.07 ( ( ord_less_eq_set_int @ X @ Y )
% 8.90/5.07 & ~ ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4428_less__le__not__le,axiom,
% 8.90/5.07 ( ord_less_rat
% 8.90/5.07 = ( ^ [X: rat,Y: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ X @ Y )
% 8.90/5.07 & ~ ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4429_less__le__not__le,axiom,
% 8.90/5.07 ( ord_less_num
% 8.90/5.07 = ( ^ [X: num,Y: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ X @ Y )
% 8.90/5.07 & ~ ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4430_less__le__not__le,axiom,
% 8.90/5.07 ( ord_less_nat
% 8.90/5.07 = ( ^ [X: nat,Y: nat] :
% 8.90/5.07 ( ( ord_less_eq_nat @ X @ Y )
% 8.90/5.07 & ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4431_less__le__not__le,axiom,
% 8.90/5.07 ( ord_less_int
% 8.90/5.07 = ( ^ [X: int,Y: int] :
% 8.90/5.07 ( ( ord_less_eq_int @ X @ Y )
% 8.90/5.07 & ~ ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4432_dense__le,axiom,
% 8.90/5.07 ! [Y3: real,Z: real] :
% 8.90/5.07 ( ! [X5: real] :
% 8.90/5.07 ( ( ord_less_real @ X5 @ Y3 )
% 8.90/5.07 => ( ord_less_eq_real @ X5 @ Z ) )
% 8.90/5.07 => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4433_dense__le,axiom,
% 8.90/5.07 ! [Y3: rat,Z: rat] :
% 8.90/5.07 ( ! [X5: rat] :
% 8.90/5.07 ( ( ord_less_rat @ X5 @ Y3 )
% 8.90/5.07 => ( ord_less_eq_rat @ X5 @ Z ) )
% 8.90/5.07 => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4434_dense__ge,axiom,
% 8.90/5.07 ! [Z: real,Y3: real] :
% 8.90/5.07 ( ! [X5: real] :
% 8.90/5.07 ( ( ord_less_real @ Z @ X5 )
% 8.90/5.07 => ( ord_less_eq_real @ Y3 @ X5 ) )
% 8.90/5.07 => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4435_dense__ge,axiom,
% 8.90/5.07 ! [Z: rat,Y3: rat] :
% 8.90/5.07 ( ! [X5: rat] :
% 8.90/5.07 ( ( ord_less_rat @ Z @ X5 )
% 8.90/5.07 => ( ord_less_eq_rat @ Y3 @ X5 ) )
% 8.90/5.07 => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4436_antisym__conv2,axiom,
% 8.90/5.07 ! [X2: real,Y3: real] :
% 8.90/5.07 ( ( ord_less_eq_real @ X2 @ Y3 )
% 8.90/5.07 => ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
% 8.90/5.07 = ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4437_antisym__conv2,axiom,
% 8.90/5.07 ! [X2: set_int,Y3: set_int] :
% 8.90/5.07 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 8.90/5.07 => ( ( ~ ( ord_less_set_int @ X2 @ Y3 ) )
% 8.90/5.07 = ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4438_antisym__conv2,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 8.90/5.07 => ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
% 8.90/5.07 = ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4439_antisym__conv2,axiom,
% 8.90/5.07 ! [X2: num,Y3: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ X2 @ Y3 )
% 8.90/5.07 => ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
% 8.90/5.07 = ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4440_antisym__conv2,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat] :
% 8.90/5.07 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 8.90/5.07 => ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
% 8.90/5.07 = ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4441_antisym__conv2,axiom,
% 8.90/5.07 ! [X2: int,Y3: int] :
% 8.90/5.07 ( ( ord_less_eq_int @ X2 @ Y3 )
% 8.90/5.07 => ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
% 8.90/5.07 = ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4442_antisym__conv1,axiom,
% 8.90/5.07 ! [X2: real,Y3: real] :
% 8.90/5.07 ( ~ ( ord_less_real @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 8.90/5.07 = ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4443_antisym__conv1,axiom,
% 8.90/5.07 ! [X2: set_int,Y3: set_int] :
% 8.90/5.07 ( ~ ( ord_less_set_int @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 8.90/5.07 = ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4444_antisym__conv1,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat] :
% 8.90/5.07 ( ~ ( ord_less_rat @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_eq_rat @ X2 @ Y3 )
% 8.90/5.07 = ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4445_antisym__conv1,axiom,
% 8.90/5.07 ! [X2: num,Y3: num] :
% 8.90/5.07 ( ~ ( ord_less_num @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_eq_num @ X2 @ Y3 )
% 8.90/5.07 = ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4446_antisym__conv1,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat] :
% 8.90/5.07 ( ~ ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_eq_nat @ X2 @ Y3 )
% 8.90/5.07 = ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4447_antisym__conv1,axiom,
% 8.90/5.07 ! [X2: int,Y3: int] :
% 8.90/5.07 ( ~ ( ord_less_int @ X2 @ Y3 )
% 8.90/5.07 => ( ( ord_less_eq_int @ X2 @ Y3 )
% 8.90/5.07 = ( X2 = Y3 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4448_nless__le,axiom,
% 8.90/5.07 ! [A: real,B: real] :
% 8.90/5.07 ( ( ~ ( ord_less_real @ A @ B ) )
% 8.90/5.07 = ( ~ ( ord_less_eq_real @ A @ B )
% 8.90/5.07 | ( A = B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4449_nless__le,axiom,
% 8.90/5.07 ! [A: set_int,B: set_int] :
% 8.90/5.07 ( ( ~ ( ord_less_set_int @ A @ B ) )
% 8.90/5.07 = ( ~ ( ord_less_eq_set_int @ A @ B )
% 8.90/5.07 | ( A = B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4450_nless__le,axiom,
% 8.90/5.07 ! [A: rat,B: rat] :
% 8.90/5.07 ( ( ~ ( ord_less_rat @ A @ B ) )
% 8.90/5.07 = ( ~ ( ord_less_eq_rat @ A @ B )
% 8.90/5.07 | ( A = B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4451_nless__le,axiom,
% 8.90/5.07 ! [A: num,B: num] :
% 8.90/5.07 ( ( ~ ( ord_less_num @ A @ B ) )
% 8.90/5.07 = ( ~ ( ord_less_eq_num @ A @ B )
% 8.90/5.07 | ( A = B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4452_nless__le,axiom,
% 8.90/5.07 ! [A: nat,B: nat] :
% 8.90/5.07 ( ( ~ ( ord_less_nat @ A @ B ) )
% 8.90/5.07 = ( ~ ( ord_less_eq_nat @ A @ B )
% 8.90/5.07 | ( A = B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4453_nless__le,axiom,
% 8.90/5.07 ! [A: int,B: int] :
% 8.90/5.07 ( ( ~ ( ord_less_int @ A @ B ) )
% 8.90/5.07 = ( ~ ( ord_less_eq_int @ A @ B )
% 8.90/5.07 | ( A = B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4454_leI,axiom,
% 8.90/5.07 ! [X2: real,Y3: real] :
% 8.90/5.07 ( ~ ( ord_less_real @ X2 @ Y3 )
% 8.90/5.07 => ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4455_leI,axiom,
% 8.90/5.07 ! [X2: rat,Y3: rat] :
% 8.90/5.07 ( ~ ( ord_less_rat @ X2 @ Y3 )
% 8.90/5.07 => ( ord_less_eq_rat @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4456_leI,axiom,
% 8.90/5.07 ! [X2: num,Y3: num] :
% 8.90/5.07 ( ~ ( ord_less_num @ X2 @ Y3 )
% 8.90/5.07 => ( ord_less_eq_num @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4457_leI,axiom,
% 8.90/5.07 ! [X2: nat,Y3: nat] :
% 8.90/5.07 ( ~ ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.07 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4458_leI,axiom,
% 8.90/5.07 ! [X2: int,Y3: int] :
% 8.90/5.07 ( ~ ( ord_less_int @ X2 @ Y3 )
% 8.90/5.07 => ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4459_leD,axiom,
% 8.90/5.07 ! [Y3: real,X2: real] :
% 8.90/5.07 ( ( ord_less_eq_real @ Y3 @ X2 )
% 8.90/5.07 => ~ ( ord_less_real @ X2 @ Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4460_leD,axiom,
% 8.90/5.07 ! [Y3: set_int,X2: set_int] :
% 8.90/5.07 ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 8.90/5.07 => ~ ( ord_less_set_int @ X2 @ Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4461_leD,axiom,
% 8.90/5.07 ! [Y3: rat,X2: rat] :
% 8.90/5.07 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 8.90/5.07 => ~ ( ord_less_rat @ X2 @ Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4462_leD,axiom,
% 8.90/5.07 ! [Y3: num,X2: num] :
% 8.90/5.07 ( ( ord_less_eq_num @ Y3 @ X2 )
% 8.90/5.07 => ~ ( ord_less_num @ X2 @ Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4463_leD,axiom,
% 8.90/5.07 ! [Y3: nat,X2: nat] :
% 8.90/5.07 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 8.90/5.07 => ~ ( ord_less_nat @ X2 @ Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4464_leD,axiom,
% 8.90/5.07 ! [Y3: int,X2: int] :
% 8.90/5.07 ( ( ord_less_eq_int @ Y3 @ X2 )
% 8.90/5.07 => ~ ( ord_less_int @ X2 @ Y3 ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4465_bot_Oextremum__uniqueI,axiom,
% 8.90/5.07 ! [A: set_nat] :
% 8.90/5.07 ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 8.90/5.07 => ( A = bot_bot_set_nat ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4466_bot_Oextremum__uniqueI,axiom,
% 8.90/5.07 ! [A: extended_enat] :
% 8.90/5.07 ( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
% 8.90/5.07 => ( A = bot_bo4199563552545308370d_enat ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4467_bot_Oextremum__uniqueI,axiom,
% 8.90/5.07 ! [A: set_real] :
% 8.90/5.07 ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 8.90/5.07 => ( A = bot_bot_set_real ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4468_bot_Oextremum__uniqueI,axiom,
% 8.90/5.07 ! [A: set_int] :
% 8.90/5.07 ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 8.90/5.07 => ( A = bot_bot_set_int ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4469_bot_Oextremum__uniqueI,axiom,
% 8.90/5.07 ! [A: nat] :
% 8.90/5.07 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 8.90/5.07 => ( A = bot_bot_nat ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4470_bot_Oextremum__unique,axiom,
% 8.90/5.07 ! [A: set_nat] :
% 8.90/5.07 ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 8.90/5.07 = ( A = bot_bot_set_nat ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4471_bot_Oextremum__unique,axiom,
% 8.90/5.07 ! [A: extended_enat] :
% 8.90/5.07 ( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
% 8.90/5.07 = ( A = bot_bo4199563552545308370d_enat ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4472_bot_Oextremum__unique,axiom,
% 8.90/5.07 ! [A: set_real] :
% 8.90/5.07 ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 8.90/5.07 = ( A = bot_bot_set_real ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4473_bot_Oextremum__unique,axiom,
% 8.90/5.07 ! [A: set_int] :
% 8.90/5.07 ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 8.90/5.07 = ( A = bot_bot_set_int ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4474_bot_Oextremum__unique,axiom,
% 8.90/5.07 ! [A: nat] :
% 8.90/5.07 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 8.90/5.07 = ( A = bot_bot_nat ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4475_bot_Oextremum,axiom,
% 8.90/5.07 ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4476_bot_Oextremum,axiom,
% 8.90/5.07 ! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ bot_bo4199563552545308370d_enat @ A ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4477_bot_Oextremum,axiom,
% 8.90/5.07 ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4478_bot_Oextremum,axiom,
% 8.90/5.07 ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4479_bot_Oextremum,axiom,
% 8.90/5.07 ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4480_bot_Oextremum__strict,axiom,
% 8.90/5.07 ! [A: set_nat] :
% 8.90/5.07 ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4481_bot_Oextremum__strict,axiom,
% 8.90/5.07 ! [A: extended_enat] :
% 8.90/5.07 ~ ( ord_le72135733267957522d_enat @ A @ bot_bo4199563552545308370d_enat ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4482_bot_Oextremum__strict,axiom,
% 8.90/5.07 ! [A: set_int] :
% 8.90/5.07 ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4483_bot_Oextremum__strict,axiom,
% 8.90/5.07 ! [A: set_real] :
% 8.90/5.07 ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4484_bot_Oextremum__strict,axiom,
% 8.90/5.07 ! [A: nat] :
% 8.90/5.07 ~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4485_bot_Onot__eq__extremum,axiom,
% 8.90/5.07 ! [A: set_nat] :
% 8.90/5.07 ( ( A != bot_bot_set_nat )
% 8.90/5.07 = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4486_bot_Onot__eq__extremum,axiom,
% 8.90/5.07 ! [A: extended_enat] :
% 8.90/5.07 ( ( A != bot_bo4199563552545308370d_enat )
% 8.90/5.07 = ( ord_le72135733267957522d_enat @ bot_bo4199563552545308370d_enat @ A ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4487_bot_Onot__eq__extremum,axiom,
% 8.90/5.07 ! [A: set_int] :
% 8.90/5.07 ( ( A != bot_bot_set_int )
% 8.90/5.07 = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4488_bot_Onot__eq__extremum,axiom,
% 8.90/5.07 ! [A: set_real] :
% 8.90/5.07 ( ( A != bot_bot_set_real )
% 8.90/5.07 = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4489_bot_Onot__eq__extremum,axiom,
% 8.90/5.07 ! [A: nat] :
% 8.90/5.07 ( ( A != bot_bot_nat )
% 8.90/5.07 = ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4490_VEBT__internal_Oheight_Ocases,axiom,
% 8.90/5.07 ! [X2: vEBT_VEBT] :
% 8.90/5.07 ( ! [A4: $o,B4: $o] :
% 8.90/5.07 ( X2
% 8.90/5.07 != ( vEBT_Leaf @ A4 @ B4 ) )
% 8.90/5.07 => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.90/5.07 ( X2
% 8.90/5.07 != ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4491_vebt__insert_Osimps_I5_J,axiom,
% 8.90/5.07 ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 8.90/5.07 ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X2 )
% 8.90/5.07 = ( if_VEBT_VEBT
% 8.90/5.07 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 8.90/5.07 & ~ ( ( X2 = Mi )
% 8.90/5.07 | ( X2 = Ma ) ) )
% 8.90/5.07 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ X2 @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
% 8.90/5.07 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4492_vebt__insert_Oelims,axiom,
% 8.90/5.07 ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 8.90/5.07 ( ( ( vEBT_vebt_insert @ X2 @ Xa2 )
% 8.90/5.07 = Y3 )
% 8.90/5.07 => ( ! [A4: $o,B4: $o] :
% 8.90/5.07 ( ( X2
% 8.90/5.07 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.90/5.07 => ~ ( ( ( Xa2 = zero_zero_nat )
% 8.90/5.07 => ( Y3
% 8.90/5.07 = ( vEBT_Leaf @ $true @ B4 ) ) )
% 8.90/5.07 & ( ( Xa2 != zero_zero_nat )
% 8.90/5.07 => ( ( ( Xa2 = one_one_nat )
% 8.90/5.07 => ( Y3
% 8.90/5.07 = ( vEBT_Leaf @ A4 @ $true ) ) )
% 8.90/5.07 & ( ( Xa2 != one_one_nat )
% 8.90/5.07 => ( Y3
% 8.90/5.07 = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ) ) )
% 8.90/5.07 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 8.90/5.07 ( ( X2
% 8.90/5.07 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 8.90/5.07 => ( Y3
% 8.90/5.07 != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) ) )
% 8.90/5.07 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 8.90/5.07 ( ( X2
% 8.90/5.07 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 8.90/5.07 => ( Y3
% 8.90/5.07 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) ) )
% 8.90/5.07 => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.90/5.07 ( ( X2
% 8.90/5.07 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
% 8.90/5.07 => ( Y3
% 8.90/5.07 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) ) )
% 8.90/5.07 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.90/5.07 ( ( X2
% 8.90/5.07 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.90/5.07 => ( Y3
% 8.90/5.07 != ( if_VEBT_VEBT
% 8.90/5.07 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.90/5.07 & ~ ( ( Xa2 = Mi2 )
% 8.90/5.07 | ( Xa2 = Ma2 ) ) )
% 8.90/5.07 @ ( 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 @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 8.90/5.07 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4493_vebt__insert_Opelims,axiom,
% 8.90/5.07 ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 8.90/5.07 ( ( ( vEBT_vebt_insert @ X2 @ Xa2 )
% 8.90/5.07 = Y3 )
% 8.90/5.07 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 8.90/5.07 => ( ! [A4: $o,B4: $o] :
% 8.90/5.07 ( ( X2
% 8.90/5.07 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.90/5.07 => ( ( ( ( Xa2 = zero_zero_nat )
% 8.90/5.07 => ( Y3
% 8.90/5.07 = ( vEBT_Leaf @ $true @ B4 ) ) )
% 8.90/5.07 & ( ( Xa2 != zero_zero_nat )
% 8.90/5.07 => ( ( ( Xa2 = one_one_nat )
% 8.90/5.07 => ( Y3
% 8.90/5.07 = ( vEBT_Leaf @ A4 @ $true ) ) )
% 8.90/5.07 & ( ( Xa2 != one_one_nat )
% 8.90/5.07 => ( Y3
% 8.90/5.07 = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ) )
% 8.90/5.07 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
% 8.90/5.07 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 8.90/5.07 ( ( X2
% 8.90/5.07 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 8.90/5.07 => ( ( Y3
% 8.90/5.07 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 8.90/5.07 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) @ Xa2 ) ) ) )
% 8.90/5.07 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 8.90/5.07 ( ( X2
% 8.90/5.07 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 8.90/5.07 => ( ( Y3
% 8.90/5.07 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 8.90/5.07 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) @ Xa2 ) ) ) )
% 8.90/5.07 => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.90/5.07 ( ( X2
% 8.90/5.07 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
% 8.90/5.07 => ( ( Y3
% 8.90/5.07 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
% 8.90/5.07 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
% 8.90/5.07 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.90/5.07 ( ( X2
% 8.90/5.07 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
% 8.90/5.07 => ( ( Y3
% 8.90/5.07 = ( if_VEBT_VEBT
% 8.90/5.07 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 8.90/5.07 & ~ ( ( Xa2 = Mi2 )
% 8.90/5.07 | ( Xa2 = Ma2 ) ) )
% 8.90/5.07 @ ( 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 @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 8.90/5.07 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
% 8.90/5.07 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4494_gcd__nat__induct,axiom,
% 8.90/5.07 ! [P: nat > nat > $o,M: nat,N: nat] :
% 8.90/5.07 ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
% 8.90/5.07 => ( ! [M4: nat,N2: nat] :
% 8.90/5.07 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 8.90/5.07 => ( ( P @ N2 @ ( modulo_modulo_nat @ M4 @ N2 ) )
% 8.90/5.07 => ( P @ M4 @ N2 ) ) )
% 8.90/5.07 => ( P @ M @ N ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4495_concat__bit__Suc,axiom,
% 8.90/5.07 ! [N: nat,K: int,L: int] :
% 8.90/5.07 ( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
% 8.90/5.07 = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4496_dbl__simps_I3_J,axiom,
% 8.90/5.07 ( ( neg_nu7009210354673126013omplex @ one_one_complex )
% 8.90/5.07 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4497_dbl__simps_I3_J,axiom,
% 8.90/5.07 ( ( neg_numeral_dbl_real @ one_one_real )
% 8.90/5.07 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4498_dbl__simps_I3_J,axiom,
% 8.90/5.07 ( ( neg_numeral_dbl_rat @ one_one_rat )
% 8.90/5.07 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4499_dbl__simps_I3_J,axiom,
% 8.90/5.07 ( ( neg_numeral_dbl_int @ one_one_int )
% 8.90/5.07 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4500_even__succ__mod__exp,axiom,
% 8.90/5.07 ! [A: nat,N: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.07 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.07 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.07 = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4501_even__succ__mod__exp,axiom,
% 8.90/5.07 ! [A: int,N: nat] :
% 8.90/5.07 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.07 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.07 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.07 = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4502_even__succ__mod__exp,axiom,
% 8.90/5.07 ! [A: code_integer,N: nat] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.07 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.07 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.07 = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4503_nat__dvd__1__iff__1,axiom,
% 8.90/5.07 ! [M: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ M @ one_one_nat )
% 8.90/5.07 = ( M = one_one_nat ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4504_max__enat__simps_I3_J,axiom,
% 8.90/5.07 ! [Q3: extended_enat] :
% 8.90/5.07 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q3 )
% 8.90/5.07 = Q3 ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4505_max__enat__simps_I2_J,axiom,
% 8.90/5.07 ! [Q3: extended_enat] :
% 8.90/5.07 ( ( ord_ma741700101516333627d_enat @ Q3 @ zero_z5237406670263579293d_enat )
% 8.90/5.07 = Q3 ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4506_dvd__0__right,axiom,
% 8.90/5.07 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4507_dvd__0__right,axiom,
% 8.90/5.07 ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4508_dvd__0__right,axiom,
% 8.90/5.07 ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4509_dvd__0__right,axiom,
% 8.90/5.07 ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4510_dvd__0__right,axiom,
% 8.90/5.07 ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4511_dvd__0__right,axiom,
% 8.90/5.07 ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4512_dvd__0__left__iff,axiom,
% 8.90/5.07 ! [A: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 8.90/5.07 = ( A = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4513_dvd__0__left__iff,axiom,
% 8.90/5.07 ! [A: complex] :
% 8.90/5.07 ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 8.90/5.07 = ( A = zero_zero_complex ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4514_dvd__0__left__iff,axiom,
% 8.90/5.07 ! [A: real] :
% 8.90/5.07 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 8.90/5.07 = ( A = zero_zero_real ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4515_dvd__0__left__iff,axiom,
% 8.90/5.07 ! [A: rat] :
% 8.90/5.07 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 8.90/5.07 = ( A = zero_zero_rat ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4516_dvd__0__left__iff,axiom,
% 8.90/5.07 ! [A: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 8.90/5.07 = ( A = zero_zero_nat ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4517_dvd__0__left__iff,axiom,
% 8.90/5.07 ! [A: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 8.90/5.07 = ( A = zero_zero_int ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4518_dvd__1__left,axiom,
% 8.90/5.07 ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4519_dvd__1__iff__1,axiom,
% 8.90/5.07 ! [M: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 8.90/5.07 = ( M
% 8.90/5.07 = ( suc @ zero_zero_nat ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4520_dvd__add__triv__left__iff,axiom,
% 8.90/5.07 ! [A: code_integer,B: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 8.90/5.07 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4521_dvd__add__triv__left__iff,axiom,
% 8.90/5.07 ! [A: real,B: real] :
% 8.90/5.07 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
% 8.90/5.07 = ( dvd_dvd_real @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4522_dvd__add__triv__left__iff,axiom,
% 8.90/5.07 ! [A: rat,B: rat] :
% 8.90/5.07 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 8.90/5.07 = ( dvd_dvd_rat @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4523_dvd__add__triv__left__iff,axiom,
% 8.90/5.07 ! [A: nat,B: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 8.90/5.07 = ( dvd_dvd_nat @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4524_dvd__add__triv__left__iff,axiom,
% 8.90/5.07 ! [A: int,B: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
% 8.90/5.07 = ( dvd_dvd_int @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4525_dvd__add__triv__right__iff,axiom,
% 8.90/5.07 ! [A: code_integer,B: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
% 8.90/5.07 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4526_dvd__add__triv__right__iff,axiom,
% 8.90/5.07 ! [A: real,B: real] :
% 8.90/5.07 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
% 8.90/5.07 = ( dvd_dvd_real @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4527_dvd__add__triv__right__iff,axiom,
% 8.90/5.07 ! [A: rat,B: rat] :
% 8.90/5.07 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 8.90/5.07 = ( dvd_dvd_rat @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4528_dvd__add__triv__right__iff,axiom,
% 8.90/5.07 ! [A: nat,B: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 8.90/5.07 = ( dvd_dvd_nat @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4529_dvd__add__triv__right__iff,axiom,
% 8.90/5.07 ! [A: int,B: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
% 8.90/5.07 = ( dvd_dvd_int @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4530_div__dvd__div,axiom,
% 8.90/5.07 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.90/5.07 => ( ( dvd_dvd_Code_integer @ A @ C )
% 8.90/5.07 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
% 8.90/5.07 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4531_div__dvd__div,axiom,
% 8.90/5.07 ! [A: nat,B: nat,C: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ A @ B )
% 8.90/5.07 => ( ( dvd_dvd_nat @ A @ C )
% 8.90/5.07 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
% 8.90/5.07 = ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4532_div__dvd__div,axiom,
% 8.90/5.07 ! [A: int,B: int,C: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ A @ B )
% 8.90/5.07 => ( ( dvd_dvd_int @ A @ C )
% 8.90/5.07 => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
% 8.90/5.07 = ( dvd_dvd_int @ B @ C ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4533_nat__mult__dvd__cancel__disj,axiom,
% 8.90/5.07 ! [K: nat,M: nat,N: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.90/5.07 = ( ( K = zero_zero_nat )
% 8.90/5.07 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4534_concat__bit__0,axiom,
% 8.90/5.07 ! [K: int,L: int] :
% 8.90/5.07 ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
% 8.90/5.07 = L ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4535_dbl__simps_I2_J,axiom,
% 8.90/5.07 ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
% 8.90/5.07 = zero_zero_complex ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4536_dbl__simps_I2_J,axiom,
% 8.90/5.07 ( ( neg_numeral_dbl_real @ zero_zero_real )
% 8.90/5.07 = zero_zero_real ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4537_dbl__simps_I2_J,axiom,
% 8.90/5.07 ( ( neg_numeral_dbl_rat @ zero_zero_rat )
% 8.90/5.07 = zero_zero_rat ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4538_dbl__simps_I2_J,axiom,
% 8.90/5.07 ( ( neg_numeral_dbl_int @ zero_zero_int )
% 8.90/5.07 = zero_zero_int ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4539_dvd__mult__cancel__left,axiom,
% 8.90/5.07 ! [C: code_integer,A: code_integer,B: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 8.90/5.07 = ( ( C = zero_z3403309356797280102nteger )
% 8.90/5.07 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4540_dvd__mult__cancel__left,axiom,
% 8.90/5.07 ! [C: complex,A: complex,B: complex] :
% 8.90/5.07 ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 8.90/5.07 = ( ( C = zero_zero_complex )
% 8.90/5.07 | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4541_dvd__mult__cancel__left,axiom,
% 8.90/5.07 ! [C: real,A: real,B: real] :
% 8.90/5.07 ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 8.90/5.07 = ( ( C = zero_zero_real )
% 8.90/5.07 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4542_dvd__mult__cancel__left,axiom,
% 8.90/5.07 ! [C: rat,A: rat,B: rat] :
% 8.90/5.07 ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 8.90/5.07 = ( ( C = zero_zero_rat )
% 8.90/5.07 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4543_dvd__mult__cancel__left,axiom,
% 8.90/5.07 ! [C: int,A: int,B: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.90/5.07 = ( ( C = zero_zero_int )
% 8.90/5.07 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4544_dvd__mult__cancel__right,axiom,
% 8.90/5.07 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 8.90/5.07 = ( ( C = zero_z3403309356797280102nteger )
% 8.90/5.07 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4545_dvd__mult__cancel__right,axiom,
% 8.90/5.07 ! [A: complex,C: complex,B: complex] :
% 8.90/5.07 ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 8.90/5.07 = ( ( C = zero_zero_complex )
% 8.90/5.07 | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4546_dvd__mult__cancel__right,axiom,
% 8.90/5.07 ! [A: real,C: real,B: real] :
% 8.90/5.07 ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 8.90/5.07 = ( ( C = zero_zero_real )
% 8.90/5.07 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4547_dvd__mult__cancel__right,axiom,
% 8.90/5.07 ! [A: rat,C: rat,B: rat] :
% 8.90/5.07 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 8.90/5.07 = ( ( C = zero_zero_rat )
% 8.90/5.07 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4548_dvd__mult__cancel__right,axiom,
% 8.90/5.07 ! [A: int,C: int,B: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 8.90/5.07 = ( ( C = zero_zero_int )
% 8.90/5.07 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4549_dvd__times__left__cancel__iff,axiom,
% 8.90/5.07 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.07 ( ( A != zero_z3403309356797280102nteger )
% 8.90/5.07 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 8.90/5.07 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4550_dvd__times__left__cancel__iff,axiom,
% 8.90/5.07 ! [A: nat,B: nat,C: nat] :
% 8.90/5.07 ( ( A != zero_zero_nat )
% 8.90/5.07 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
% 8.90/5.07 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4551_dvd__times__left__cancel__iff,axiom,
% 8.90/5.07 ! [A: int,B: int,C: int] :
% 8.90/5.07 ( ( A != zero_zero_int )
% 8.90/5.07 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
% 8.90/5.07 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4552_dvd__times__right__cancel__iff,axiom,
% 8.90/5.07 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.07 ( ( A != zero_z3403309356797280102nteger )
% 8.90/5.07 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 8.90/5.07 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4553_dvd__times__right__cancel__iff,axiom,
% 8.90/5.07 ! [A: nat,B: nat,C: nat] :
% 8.90/5.07 ( ( A != zero_zero_nat )
% 8.90/5.07 => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
% 8.90/5.07 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4554_dvd__times__right__cancel__iff,axiom,
% 8.90/5.07 ! [A: int,B: int,C: int] :
% 8.90/5.07 ( ( A != zero_zero_int )
% 8.90/5.07 => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
% 8.90/5.07 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4555_unit__prod,axiom,
% 8.90/5.07 ! [A: code_integer,B: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.90/5.07 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.90/5.07 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4556_unit__prod,axiom,
% 8.90/5.07 ! [A: nat,B: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.90/5.07 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.90/5.07 => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4557_unit__prod,axiom,
% 8.90/5.07 ! [A: int,B: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.90/5.07 => ( ( dvd_dvd_int @ B @ one_one_int )
% 8.90/5.07 => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4558_dvd__add__times__triv__right__iff,axiom,
% 8.90/5.07 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
% 8.90/5.07 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4559_dvd__add__times__triv__right__iff,axiom,
% 8.90/5.07 ! [A: real,B: real,C: real] :
% 8.90/5.07 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
% 8.90/5.07 = ( dvd_dvd_real @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4560_dvd__add__times__triv__right__iff,axiom,
% 8.90/5.07 ! [A: rat,B: rat,C: rat] :
% 8.90/5.07 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
% 8.90/5.07 = ( dvd_dvd_rat @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4561_dvd__add__times__triv__right__iff,axiom,
% 8.90/5.07 ! [A: nat,B: nat,C: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
% 8.90/5.07 = ( dvd_dvd_nat @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4562_dvd__add__times__triv__right__iff,axiom,
% 8.90/5.07 ! [A: int,B: int,C: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
% 8.90/5.07 = ( dvd_dvd_int @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4563_dvd__add__times__triv__left__iff,axiom,
% 8.90/5.07 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
% 8.90/5.07 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4564_dvd__add__times__triv__left__iff,axiom,
% 8.90/5.07 ! [A: real,C: real,B: real] :
% 8.90/5.07 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
% 8.90/5.07 = ( dvd_dvd_real @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4565_dvd__add__times__triv__left__iff,axiom,
% 8.90/5.07 ! [A: rat,C: rat,B: rat] :
% 8.90/5.07 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
% 8.90/5.07 = ( dvd_dvd_rat @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4566_dvd__add__times__triv__left__iff,axiom,
% 8.90/5.07 ! [A: nat,C: nat,B: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
% 8.90/5.07 = ( dvd_dvd_nat @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4567_dvd__add__times__triv__left__iff,axiom,
% 8.90/5.07 ! [A: int,C: int,B: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
% 8.90/5.07 = ( dvd_dvd_int @ A @ B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4568_dvd__mult__div__cancel,axiom,
% 8.90/5.07 ! [A: code_integer,B: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.90/5.07 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
% 8.90/5.07 = B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4569_dvd__mult__div__cancel,axiom,
% 8.90/5.07 ! [A: nat,B: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ A @ B )
% 8.90/5.07 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
% 8.90/5.07 = B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4570_dvd__mult__div__cancel,axiom,
% 8.90/5.07 ! [A: int,B: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ A @ B )
% 8.90/5.07 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
% 8.90/5.07 = B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4571_dvd__div__mult__self,axiom,
% 8.90/5.07 ! [A: code_integer,B: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.90/5.07 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 8.90/5.07 = B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4572_dvd__div__mult__self,axiom,
% 8.90/5.07 ! [A: nat,B: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ A @ B )
% 8.90/5.07 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 8.90/5.07 = B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4573_dvd__div__mult__self,axiom,
% 8.90/5.07 ! [A: int,B: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ A @ B )
% 8.90/5.07 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 8.90/5.07 = B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4574_unit__div,axiom,
% 8.90/5.07 ! [A: code_integer,B: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.90/5.07 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.90/5.07 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4575_unit__div,axiom,
% 8.90/5.07 ! [A: nat,B: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.90/5.07 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.90/5.07 => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4576_unit__div,axiom,
% 8.90/5.07 ! [A: int,B: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.90/5.07 => ( ( dvd_dvd_int @ B @ one_one_int )
% 8.90/5.07 => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4577_unit__div__1__unit,axiom,
% 8.90/5.07 ! [A: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.90/5.07 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4578_unit__div__1__unit,axiom,
% 8.90/5.07 ! [A: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.90/5.07 => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4579_unit__div__1__unit,axiom,
% 8.90/5.07 ! [A: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.90/5.07 => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4580_unit__div__1__div__1,axiom,
% 8.90/5.07 ! [A: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.90/5.07 => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 8.90/5.07 = A ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4581_unit__div__1__div__1,axiom,
% 8.90/5.07 ! [A: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.90/5.07 => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
% 8.90/5.07 = A ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4582_unit__div__1__div__1,axiom,
% 8.90/5.07 ! [A: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.90/5.07 => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
% 8.90/5.07 = A ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4583_div__add,axiom,
% 8.90/5.07 ! [C: code_integer,A: code_integer,B: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ C @ A )
% 8.90/5.07 => ( ( dvd_dvd_Code_integer @ C @ B )
% 8.90/5.07 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 8.90/5.07 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4584_div__add,axiom,
% 8.90/5.07 ! [C: nat,A: nat,B: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ C @ A )
% 8.90/5.07 => ( ( dvd_dvd_nat @ C @ B )
% 8.90/5.07 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 8.90/5.07 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4585_div__add,axiom,
% 8.90/5.07 ! [C: int,A: int,B: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ C @ A )
% 8.90/5.07 => ( ( dvd_dvd_int @ C @ B )
% 8.90/5.07 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.90/5.07 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4586_div__diff,axiom,
% 8.90/5.07 ! [C: code_integer,A: code_integer,B: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ C @ A )
% 8.90/5.07 => ( ( dvd_dvd_Code_integer @ C @ B )
% 8.90/5.07 => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 8.90/5.07 = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4587_div__diff,axiom,
% 8.90/5.07 ! [C: int,A: int,B: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ C @ A )
% 8.90/5.07 => ( ( dvd_dvd_int @ C @ B )
% 8.90/5.07 => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
% 8.90/5.07 = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4588_dvd__imp__mod__0,axiom,
% 8.90/5.07 ! [A: nat,B: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ A @ B )
% 8.90/5.07 => ( ( modulo_modulo_nat @ B @ A )
% 8.90/5.07 = zero_zero_nat ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4589_dvd__imp__mod__0,axiom,
% 8.90/5.07 ! [A: int,B: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ A @ B )
% 8.90/5.07 => ( ( modulo_modulo_int @ B @ A )
% 8.90/5.07 = zero_zero_int ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4590_dvd__imp__mod__0,axiom,
% 8.90/5.07 ! [A: code_integer,B: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.90/5.07 => ( ( modulo364778990260209775nteger @ B @ A )
% 8.90/5.07 = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4591_concat__bit__nonnegative__iff,axiom,
% 8.90/5.07 ! [N: nat,K: int,L: int] :
% 8.90/5.07 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
% 8.90/5.07 = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4592_concat__bit__negative__iff,axiom,
% 8.90/5.07 ! [N: nat,K: int,L: int] :
% 8.90/5.07 ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
% 8.90/5.07 = ( ord_less_int @ L @ zero_zero_int ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4593_dbl__simps_I5_J,axiom,
% 8.90/5.07 ! [K: num] :
% 8.90/5.07 ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
% 8.90/5.07 = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4594_dbl__simps_I5_J,axiom,
% 8.90/5.07 ! [K: num] :
% 8.90/5.07 ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
% 8.90/5.07 = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4595_dbl__simps_I5_J,axiom,
% 8.90/5.07 ! [K: num] :
% 8.90/5.07 ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
% 8.90/5.07 = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4596_dbl__simps_I5_J,axiom,
% 8.90/5.07 ! [K: num] :
% 8.90/5.07 ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
% 8.90/5.07 = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4597_even__Suc__Suc__iff,axiom,
% 8.90/5.07 ! [N: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
% 8.90/5.07 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4598_even__Suc,axiom,
% 8.90/5.07 ! [N: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
% 8.90/5.07 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4599_unit__div__mult__self,axiom,
% 8.90/5.07 ! [A: code_integer,B: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.90/5.07 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 8.90/5.07 = B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4600_unit__div__mult__self,axiom,
% 8.90/5.07 ! [A: nat,B: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.90/5.07 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 8.90/5.07 = B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4601_unit__div__mult__self,axiom,
% 8.90/5.07 ! [A: int,B: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.90/5.07 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 8.90/5.07 = B ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4602_unit__mult__div__div,axiom,
% 8.90/5.07 ! [A: code_integer,B: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.90/5.07 => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 8.90/5.07 = ( divide6298287555418463151nteger @ B @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4603_unit__mult__div__div,axiom,
% 8.90/5.07 ! [A: nat,B: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.90/5.07 => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
% 8.90/5.07 = ( divide_divide_nat @ B @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4604_unit__mult__div__div,axiom,
% 8.90/5.07 ! [A: int,B: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.90/5.07 => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
% 8.90/5.07 = ( divide_divide_int @ B @ A ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4605_pow__divides__pow__iff,axiom,
% 8.90/5.07 ! [N: nat,A: nat,B: nat] :
% 8.90/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.07 => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 8.90/5.07 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4606_pow__divides__pow__iff,axiom,
% 8.90/5.07 ! [N: nat,A: int,B: int] :
% 8.90/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.07 => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 8.90/5.07 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4607_even__mult__iff,axiom,
% 8.90/5.07 ! [A: code_integer,B: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
% 8.90/5.07 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.07 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4608_even__mult__iff,axiom,
% 8.90/5.07 ! [A: nat,B: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
% 8.90/5.07 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.07 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4609_even__mult__iff,axiom,
% 8.90/5.07 ! [A: int,B: int] :
% 8.90/5.07 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
% 8.90/5.07 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.07 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4610_even__add,axiom,
% 8.90/5.07 ! [A: code_integer,B: code_integer] :
% 8.90/5.07 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 8.90/5.07 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.07 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.90/5.07
% 8.90/5.07 thf(fact_4611_even__add,axiom,
% 8.90/5.07 ! [A: nat,B: nat] :
% 8.90/5.07 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
% 8.90/5.07 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4612_even__add,axiom,
% 8.90/5.08 ! [A: int,B: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
% 8.90/5.08 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4613_odd__add,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer] :
% 8.90/5.08 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
% 8.90/5.08 = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 8.90/5.08 != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4614_odd__add,axiom,
% 8.90/5.08 ! [A: nat,B: nat] :
% 8.90/5.08 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
% 8.90/5.08 = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 8.90/5.08 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4615_odd__add,axiom,
% 8.90/5.08 ! [A: int,B: int] :
% 8.90/5.08 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
% 8.90/5.08 = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 8.90/5.08 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4616_even__mod__2__iff,axiom,
% 8.90/5.08 ! [A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.08 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4617_even__mod__2__iff,axiom,
% 8.90/5.08 ! [A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 8.90/5.08 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4618_even__mod__2__iff,axiom,
% 8.90/5.08 ! [A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4619_even__Suc__div__two,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.08 = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4620_odd__Suc__div__two,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.08 = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4621_dvd__numeral__simp,axiom,
% 8.90/5.08 ! [M: num,N: num] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 8.90/5.08 = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N @ M ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4622_dvd__numeral__simp,axiom,
% 8.90/5.08 ! [M: num,N: num] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 8.90/5.08 = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N @ M ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4623_dvd__numeral__simp,axiom,
% 8.90/5.08 ! [M: num,N: num] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
% 8.90/5.08 = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N @ M ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4624_zero__le__power__eq__numeral,axiom,
% 8.90/5.08 ! [A: real,W: num] :
% 8.90/5.08 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 8.90/5.08 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4625_zero__le__power__eq__numeral,axiom,
% 8.90/5.08 ! [A: rat,W: num] :
% 8.90/5.08 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 8.90/5.08 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4626_zero__le__power__eq__numeral,axiom,
% 8.90/5.08 ! [A: int,W: num] :
% 8.90/5.08 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 8.90/5.08 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4627_power__less__zero__eq__numeral,axiom,
% 8.90/5.08 ! [A: real,W: num] :
% 8.90/5.08 ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 8.90/5.08 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4628_power__less__zero__eq__numeral,axiom,
% 8.90/5.08 ! [A: rat,W: num] :
% 8.90/5.08 ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 8.90/5.08 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4629_power__less__zero__eq__numeral,axiom,
% 8.90/5.08 ! [A: int,W: num] :
% 8.90/5.08 ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 8.90/5.08 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4630_power__less__zero__eq,axiom,
% 8.90/5.08 ! [A: real,N: nat] :
% 8.90/5.08 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 8.90/5.08 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4631_power__less__zero__eq,axiom,
% 8.90/5.08 ! [A: rat,N: nat] :
% 8.90/5.08 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 8.90/5.08 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4632_power__less__zero__eq,axiom,
% 8.90/5.08 ! [A: int,N: nat] :
% 8.90/5.08 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 8.90/5.08 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4633_even__plus__one__iff,axiom,
% 8.90/5.08 ! [A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
% 8.90/5.08 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4634_even__plus__one__iff,axiom,
% 8.90/5.08 ! [A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
% 8.90/5.08 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4635_even__plus__one__iff,axiom,
% 8.90/5.08 ! [A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
% 8.90/5.08 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4636_even__diff,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4637_even__diff,axiom,
% 8.90/5.08 ! [A: int,B: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
% 8.90/5.08 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4638_odd__Suc__minus__one,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 8.90/5.08 = N ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4639_even__diff__nat,axiom,
% 8.90/5.08 ! [M: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
% 8.90/5.08 = ( ( ord_less_nat @ M @ N )
% 8.90/5.08 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4640_zero__less__power__eq__numeral,axiom,
% 8.90/5.08 ! [A: real,W: num] :
% 8.90/5.08 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 8.90/5.08 = ( ( ( numeral_numeral_nat @ W )
% 8.90/5.08 = zero_zero_nat )
% 8.90/5.08 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( A != zero_zero_real ) )
% 8.90/5.08 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4641_zero__less__power__eq__numeral,axiom,
% 8.90/5.08 ! [A: rat,W: num] :
% 8.90/5.08 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 8.90/5.08 = ( ( ( numeral_numeral_nat @ W )
% 8.90/5.08 = zero_zero_nat )
% 8.90/5.08 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( A != zero_zero_rat ) )
% 8.90/5.08 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4642_zero__less__power__eq__numeral,axiom,
% 8.90/5.08 ! [A: int,W: num] :
% 8.90/5.08 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 8.90/5.08 = ( ( ( numeral_numeral_nat @ W )
% 8.90/5.08 = zero_zero_nat )
% 8.90/5.08 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( A != zero_zero_int ) )
% 8.90/5.08 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4643_odd__succ__div__two,axiom,
% 8.90/5.08 ! [A: code_integer] :
% 8.90/5.08 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.08 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4644_odd__succ__div__two,axiom,
% 8.90/5.08 ! [A: nat] :
% 8.90/5.08 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.08 = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4645_odd__succ__div__two,axiom,
% 8.90/5.08 ! [A: int] :
% 8.90/5.08 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.08 = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4646_even__succ__div__two,axiom,
% 8.90/5.08 ! [A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.08 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4647_even__succ__div__two,axiom,
% 8.90/5.08 ! [A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.08 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4648_even__succ__div__two,axiom,
% 8.90/5.08 ! [A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.08 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4649_even__succ__div__2,axiom,
% 8.90/5.08 ! [A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.08 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4650_even__succ__div__2,axiom,
% 8.90/5.08 ! [A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.08 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4651_even__succ__div__2,axiom,
% 8.90/5.08 ! [A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.08 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4652_even__power,axiom,
% 8.90/5.08 ! [A: code_integer,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 8.90/5.08 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4653_even__power,axiom,
% 8.90/5.08 ! [A: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
% 8.90/5.08 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4654_even__power,axiom,
% 8.90/5.08 ! [A: int,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
% 8.90/5.08 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4655_odd__two__times__div__two__nat,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.08 = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4656_odd__two__times__div__two__succ,axiom,
% 8.90/5.08 ! [A: code_integer] :
% 8.90/5.08 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
% 8.90/5.08 = A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4657_odd__two__times__div__two__succ,axiom,
% 8.90/5.08 ! [A: nat] :
% 8.90/5.08 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
% 8.90/5.08 = A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4658_odd__two__times__div__two__succ,axiom,
% 8.90/5.08 ! [A: int] :
% 8.90/5.08 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
% 8.90/5.08 = A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4659_power__le__zero__eq__numeral,axiom,
% 8.90/5.08 ! [A: real,W: num] :
% 8.90/5.08 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 8.90/5.08 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 8.90/5.08 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( A = zero_zero_real ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4660_power__le__zero__eq__numeral,axiom,
% 8.90/5.08 ! [A: rat,W: num] :
% 8.90/5.08 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 8.90/5.08 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 8.90/5.08 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( A = zero_zero_rat ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4661_power__le__zero__eq__numeral,axiom,
% 8.90/5.08 ! [A: int,W: num] :
% 8.90/5.08 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 8.90/5.08 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 8.90/5.08 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.08 & ( A = zero_zero_int ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4662_semiring__parity__class_Oeven__mask__iff,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
% 8.90/5.08 = ( N = zero_zero_nat ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4663_semiring__parity__class_Oeven__mask__iff,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
% 8.90/5.08 = ( N = zero_zero_nat ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4664_semiring__parity__class_Oeven__mask__iff,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 8.90/5.08 = ( N = zero_zero_nat ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4665_even__succ__div__exp,axiom,
% 8.90/5.08 ! [A: code_integer,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.08 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.08 = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4666_even__succ__div__exp,axiom,
% 8.90/5.08 ! [A: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.08 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.08 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4667_even__succ__div__exp,axiom,
% 8.90/5.08 ! [A: int,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.08 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.08 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4668_dvd__refl,axiom,
% 8.90/5.08 ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4669_dvd__refl,axiom,
% 8.90/5.08 ! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4670_dvd__refl,axiom,
% 8.90/5.08 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4671_dvd__trans,axiom,
% 8.90/5.08 ! [A: nat,B: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_nat @ B @ C )
% 8.90/5.08 => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4672_dvd__trans,axiom,
% 8.90/5.08 ! [A: int,B: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_int @ B @ C )
% 8.90/5.08 => ( dvd_dvd_int @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4673_dvd__trans,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ B @ C )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4674_division__decomp,axiom,
% 8.90/5.08 ! [A: nat,B: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 8.90/5.08 => ? [B7: nat,C5: nat] :
% 8.90/5.08 ( ( A
% 8.90/5.08 = ( times_times_nat @ B7 @ C5 ) )
% 8.90/5.08 & ( dvd_dvd_nat @ B7 @ B )
% 8.90/5.08 & ( dvd_dvd_nat @ C5 @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4675_division__decomp,axiom,
% 8.90/5.08 ! [A: int,B: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 8.90/5.08 => ? [B7: int,C5: int] :
% 8.90/5.08 ( ( A
% 8.90/5.08 = ( times_times_int @ B7 @ C5 ) )
% 8.90/5.08 & ( dvd_dvd_int @ B7 @ B )
% 8.90/5.08 & ( dvd_dvd_int @ C5 @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4676_dvd__productE,axiom,
% 8.90/5.08 ! [P6: nat,A: nat,B: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ P6 @ ( times_times_nat @ A @ B ) )
% 8.90/5.08 => ~ ! [X5: nat,Y4: nat] :
% 8.90/5.08 ( ( P6
% 8.90/5.08 = ( times_times_nat @ X5 @ Y4 ) )
% 8.90/5.08 => ( ( dvd_dvd_nat @ X5 @ A )
% 8.90/5.08 => ~ ( dvd_dvd_nat @ Y4 @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4677_dvd__productE,axiom,
% 8.90/5.08 ! [P6: int,A: int,B: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ P6 @ ( times_times_int @ A @ B ) )
% 8.90/5.08 => ~ ! [X5: int,Y4: int] :
% 8.90/5.08 ( ( P6
% 8.90/5.08 = ( times_times_int @ X5 @ Y4 ) )
% 8.90/5.08 => ( ( dvd_dvd_int @ X5 @ A )
% 8.90/5.08 => ~ ( dvd_dvd_int @ Y4 @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4678_bot__nat__def,axiom,
% 8.90/5.08 bot_bot_nat = zero_zero_nat ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4679_dvd__0__left,axiom,
% 8.90/5.08 ! [A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 8.90/5.08 => ( A = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4680_dvd__0__left,axiom,
% 8.90/5.08 ! [A: complex] :
% 8.90/5.08 ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 8.90/5.08 => ( A = zero_zero_complex ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4681_dvd__0__left,axiom,
% 8.90/5.08 ! [A: real] :
% 8.90/5.08 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 8.90/5.08 => ( A = zero_zero_real ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4682_dvd__0__left,axiom,
% 8.90/5.08 ! [A: rat] :
% 8.90/5.08 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 8.90/5.08 => ( A = zero_zero_rat ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4683_dvd__0__left,axiom,
% 8.90/5.08 ! [A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 8.90/5.08 => ( A = zero_zero_nat ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4684_dvd__0__left,axiom,
% 8.90/5.08 ! [A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 8.90/5.08 => ( A = zero_zero_int ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4685_dvd__triv__right,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4686_dvd__triv__right,axiom,
% 8.90/5.08 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4687_dvd__triv__right,axiom,
% 8.90/5.08 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4688_dvd__triv__right,axiom,
% 8.90/5.08 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4689_dvd__triv__right,axiom,
% 8.90/5.08 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4690_dvd__mult__right,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ B @ C ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4691_dvd__mult__right,axiom,
% 8.90/5.08 ! [A: real,B: real,C: real] :
% 8.90/5.08 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 8.90/5.08 => ( dvd_dvd_real @ B @ C ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4692_dvd__mult__right,axiom,
% 8.90/5.08 ! [A: rat,B: rat,C: rat] :
% 8.90/5.08 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 8.90/5.08 => ( dvd_dvd_rat @ B @ C ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4693_dvd__mult__right,axiom,
% 8.90/5.08 ! [A: nat,B: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 8.90/5.08 => ( dvd_dvd_nat @ B @ C ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4694_dvd__mult__right,axiom,
% 8.90/5.08 ! [A: int,B: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 8.90/5.08 => ( dvd_dvd_int @ B @ C ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4695_mult__dvd__mono,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ C @ D )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4696_mult__dvd__mono,axiom,
% 8.90/5.08 ! [A: real,B: real,C: real,D: real] :
% 8.90/5.08 ( ( dvd_dvd_real @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_real @ C @ D )
% 8.90/5.08 => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4697_mult__dvd__mono,axiom,
% 8.90/5.08 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.90/5.08 ( ( dvd_dvd_rat @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_rat @ C @ D )
% 8.90/5.08 => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4698_mult__dvd__mono,axiom,
% 8.90/5.08 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_nat @ C @ D )
% 8.90/5.08 => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4699_mult__dvd__mono,axiom,
% 8.90/5.08 ! [A: int,B: int,C: int,D: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_int @ C @ D )
% 8.90/5.08 => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4700_dvd__triv__left,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4701_dvd__triv__left,axiom,
% 8.90/5.08 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4702_dvd__triv__left,axiom,
% 8.90/5.08 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4703_dvd__triv__left,axiom,
% 8.90/5.08 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4704_dvd__triv__left,axiom,
% 8.90/5.08 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4705_dvd__mult__left,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ A @ C ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4706_dvd__mult__left,axiom,
% 8.90/5.08 ! [A: real,B: real,C: real] :
% 8.90/5.08 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 8.90/5.08 => ( dvd_dvd_real @ A @ C ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4707_dvd__mult__left,axiom,
% 8.90/5.08 ! [A: rat,B: rat,C: rat] :
% 8.90/5.08 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 8.90/5.08 => ( dvd_dvd_rat @ A @ C ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4708_dvd__mult__left,axiom,
% 8.90/5.08 ! [A: nat,B: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 8.90/5.08 => ( dvd_dvd_nat @ A @ C ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4709_dvd__mult__left,axiom,
% 8.90/5.08 ! [A: int,B: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 8.90/5.08 => ( dvd_dvd_int @ A @ C ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4710_dvd__mult2,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4711_dvd__mult2,axiom,
% 8.90/5.08 ! [A: real,B: real,C: real] :
% 8.90/5.08 ( ( dvd_dvd_real @ A @ B )
% 8.90/5.08 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4712_dvd__mult2,axiom,
% 8.90/5.08 ! [A: rat,B: rat,C: rat] :
% 8.90/5.08 ( ( dvd_dvd_rat @ A @ B )
% 8.90/5.08 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4713_dvd__mult2,axiom,
% 8.90/5.08 ! [A: nat,B: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ A @ B )
% 8.90/5.08 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4714_dvd__mult2,axiom,
% 8.90/5.08 ! [A: int,B: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ A @ B )
% 8.90/5.08 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4715_dvd__mult,axiom,
% 8.90/5.08 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ A @ C )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4716_dvd__mult,axiom,
% 8.90/5.08 ! [A: real,C: real,B: real] :
% 8.90/5.08 ( ( dvd_dvd_real @ A @ C )
% 8.90/5.08 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4717_dvd__mult,axiom,
% 8.90/5.08 ! [A: rat,C: rat,B: rat] :
% 8.90/5.08 ( ( dvd_dvd_rat @ A @ C )
% 8.90/5.08 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4718_dvd__mult,axiom,
% 8.90/5.08 ! [A: nat,C: nat,B: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ A @ C )
% 8.90/5.08 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4719_dvd__mult,axiom,
% 8.90/5.08 ! [A: int,C: int,B: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ A @ C )
% 8.90/5.08 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4720_dvd__def,axiom,
% 8.90/5.08 ( dvd_dvd_Code_integer
% 8.90/5.08 = ( ^ [B2: code_integer,A3: code_integer] :
% 8.90/5.08 ? [K2: code_integer] :
% 8.90/5.08 ( A3
% 8.90/5.08 = ( times_3573771949741848930nteger @ B2 @ K2 ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4721_dvd__def,axiom,
% 8.90/5.08 ( dvd_dvd_real
% 8.90/5.08 = ( ^ [B2: real,A3: real] :
% 8.90/5.08 ? [K2: real] :
% 8.90/5.08 ( A3
% 8.90/5.08 = ( times_times_real @ B2 @ K2 ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4722_dvd__def,axiom,
% 8.90/5.08 ( dvd_dvd_rat
% 8.90/5.08 = ( ^ [B2: rat,A3: rat] :
% 8.90/5.08 ? [K2: rat] :
% 8.90/5.08 ( A3
% 8.90/5.08 = ( times_times_rat @ B2 @ K2 ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4723_dvd__def,axiom,
% 8.90/5.08 ( dvd_dvd_nat
% 8.90/5.08 = ( ^ [B2: nat,A3: nat] :
% 8.90/5.08 ? [K2: nat] :
% 8.90/5.08 ( A3
% 8.90/5.08 = ( times_times_nat @ B2 @ K2 ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4724_dvd__def,axiom,
% 8.90/5.08 ( dvd_dvd_int
% 8.90/5.08 = ( ^ [B2: int,A3: int] :
% 8.90/5.08 ? [K2: int] :
% 8.90/5.08 ( A3
% 8.90/5.08 = ( times_times_int @ B2 @ K2 ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4725_dvdI,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer,K: code_integer] :
% 8.90/5.08 ( ( A
% 8.90/5.08 = ( times_3573771949741848930nteger @ B @ K ) )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4726_dvdI,axiom,
% 8.90/5.08 ! [A: real,B: real,K: real] :
% 8.90/5.08 ( ( A
% 8.90/5.08 = ( times_times_real @ B @ K ) )
% 8.90/5.08 => ( dvd_dvd_real @ B @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4727_dvdI,axiom,
% 8.90/5.08 ! [A: rat,B: rat,K: rat] :
% 8.90/5.08 ( ( A
% 8.90/5.08 = ( times_times_rat @ B @ K ) )
% 8.90/5.08 => ( dvd_dvd_rat @ B @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4728_dvdI,axiom,
% 8.90/5.08 ! [A: nat,B: nat,K: nat] :
% 8.90/5.08 ( ( A
% 8.90/5.08 = ( times_times_nat @ B @ K ) )
% 8.90/5.08 => ( dvd_dvd_nat @ B @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4729_dvdI,axiom,
% 8.90/5.08 ! [A: int,B: int,K: int] :
% 8.90/5.08 ( ( A
% 8.90/5.08 = ( times_times_int @ B @ K ) )
% 8.90/5.08 => ( dvd_dvd_int @ B @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4730_dvdE,axiom,
% 8.90/5.08 ! [B: code_integer,A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ B @ A )
% 8.90/5.08 => ~ ! [K3: code_integer] :
% 8.90/5.08 ( A
% 8.90/5.08 != ( times_3573771949741848930nteger @ B @ K3 ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4731_dvdE,axiom,
% 8.90/5.08 ! [B: real,A: real] :
% 8.90/5.08 ( ( dvd_dvd_real @ B @ A )
% 8.90/5.08 => ~ ! [K3: real] :
% 8.90/5.08 ( A
% 8.90/5.08 != ( times_times_real @ B @ K3 ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4732_dvdE,axiom,
% 8.90/5.08 ! [B: rat,A: rat] :
% 8.90/5.08 ( ( dvd_dvd_rat @ B @ A )
% 8.90/5.08 => ~ ! [K3: rat] :
% 8.90/5.08 ( A
% 8.90/5.08 != ( times_times_rat @ B @ K3 ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4733_dvdE,axiom,
% 8.90/5.08 ! [B: nat,A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ B @ A )
% 8.90/5.08 => ~ ! [K3: nat] :
% 8.90/5.08 ( A
% 8.90/5.08 != ( times_times_nat @ B @ K3 ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4734_dvdE,axiom,
% 8.90/5.08 ! [B: int,A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ B @ A )
% 8.90/5.08 => ~ ! [K3: int] :
% 8.90/5.08 ( A
% 8.90/5.08 != ( times_times_int @ B @ K3 ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4735_one__dvd,axiom,
% 8.90/5.08 ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4736_one__dvd,axiom,
% 8.90/5.08 ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4737_one__dvd,axiom,
% 8.90/5.08 ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4738_one__dvd,axiom,
% 8.90/5.08 ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4739_one__dvd,axiom,
% 8.90/5.08 ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4740_one__dvd,axiom,
% 8.90/5.08 ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4741_unit__imp__dvd,axiom,
% 8.90/5.08 ! [B: code_integer,A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4742_unit__imp__dvd,axiom,
% 8.90/5.08 ! [B: nat,A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.90/5.08 => ( dvd_dvd_nat @ B @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4743_unit__imp__dvd,axiom,
% 8.90/5.08 ! [B: int,A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.90/5.08 => ( dvd_dvd_int @ B @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4744_dvd__unit__imp__unit,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4745_dvd__unit__imp__unit,axiom,
% 8.90/5.08 ! [A: nat,B: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.90/5.08 => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4746_dvd__unit__imp__unit,axiom,
% 8.90/5.08 ! [A: int,B: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_int @ B @ one_one_int )
% 8.90/5.08 => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4747_dvd__add,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ A @ C )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4748_dvd__add,axiom,
% 8.90/5.08 ! [A: real,B: real,C: real] :
% 8.90/5.08 ( ( dvd_dvd_real @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_real @ A @ C )
% 8.90/5.08 => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4749_dvd__add,axiom,
% 8.90/5.08 ! [A: rat,B: rat,C: rat] :
% 8.90/5.08 ( ( dvd_dvd_rat @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_rat @ A @ C )
% 8.90/5.08 => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4750_dvd__add,axiom,
% 8.90/5.08 ! [A: nat,B: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_nat @ A @ C )
% 8.90/5.08 => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4751_dvd__add,axiom,
% 8.90/5.08 ! [A: int,B: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_int @ A @ C )
% 8.90/5.08 => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4752_dvd__add__left__iff,axiom,
% 8.90/5.08 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ A @ C )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4753_dvd__add__left__iff,axiom,
% 8.90/5.08 ! [A: real,C: real,B: real] :
% 8.90/5.08 ( ( dvd_dvd_real @ A @ C )
% 8.90/5.08 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 8.90/5.08 = ( dvd_dvd_real @ A @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4754_dvd__add__left__iff,axiom,
% 8.90/5.08 ! [A: rat,C: rat,B: rat] :
% 8.90/5.08 ( ( dvd_dvd_rat @ A @ C )
% 8.90/5.08 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 8.90/5.08 = ( dvd_dvd_rat @ A @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4755_dvd__add__left__iff,axiom,
% 8.90/5.08 ! [A: nat,C: nat,B: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ A @ C )
% 8.90/5.08 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 8.90/5.08 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4756_dvd__add__left__iff,axiom,
% 8.90/5.08 ! [A: int,C: int,B: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ A @ C )
% 8.90/5.08 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 8.90/5.08 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4757_dvd__add__right__iff,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4758_dvd__add__right__iff,axiom,
% 8.90/5.08 ! [A: real,B: real,C: real] :
% 8.90/5.08 ( ( dvd_dvd_real @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 8.90/5.08 = ( dvd_dvd_real @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4759_dvd__add__right__iff,axiom,
% 8.90/5.08 ! [A: rat,B: rat,C: rat] :
% 8.90/5.08 ( ( dvd_dvd_rat @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 8.90/5.08 = ( dvd_dvd_rat @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4760_dvd__add__right__iff,axiom,
% 8.90/5.08 ! [A: nat,B: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 8.90/5.08 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4761_dvd__add__right__iff,axiom,
% 8.90/5.08 ! [A: int,B: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 8.90/5.08 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4762_dvd__diff__commute,axiom,
% 8.90/5.08 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4763_dvd__diff__commute,axiom,
% 8.90/5.08 ! [A: int,C: int,B: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
% 8.90/5.08 = ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4764_dvd__diff,axiom,
% 8.90/5.08 ! [X2: code_integer,Y3: code_integer,Z: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ X2 @ Y3 )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ X2 @ Z )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ X2 @ ( minus_8373710615458151222nteger @ Y3 @ Z ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4765_dvd__diff,axiom,
% 8.90/5.08 ! [X2: real,Y3: real,Z: real] :
% 8.90/5.08 ( ( dvd_dvd_real @ X2 @ Y3 )
% 8.90/5.08 => ( ( dvd_dvd_real @ X2 @ Z )
% 8.90/5.08 => ( dvd_dvd_real @ X2 @ ( minus_minus_real @ Y3 @ Z ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4766_dvd__diff,axiom,
% 8.90/5.08 ! [X2: rat,Y3: rat,Z: rat] :
% 8.90/5.08 ( ( dvd_dvd_rat @ X2 @ Y3 )
% 8.90/5.08 => ( ( dvd_dvd_rat @ X2 @ Z )
% 8.90/5.08 => ( dvd_dvd_rat @ X2 @ ( minus_minus_rat @ Y3 @ Z ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4767_dvd__diff,axiom,
% 8.90/5.08 ! [X2: int,Y3: int,Z: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ X2 @ Y3 )
% 8.90/5.08 => ( ( dvd_dvd_int @ X2 @ Z )
% 8.90/5.08 => ( dvd_dvd_int @ X2 @ ( minus_minus_int @ Y3 @ Z ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4768_div__div__div__same,axiom,
% 8.90/5.08 ! [D: code_integer,B: code_integer,A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ D @ B )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ B @ A )
% 8.90/5.08 => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
% 8.90/5.08 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4769_div__div__div__same,axiom,
% 8.90/5.08 ! [D: nat,B: nat,A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ D @ B )
% 8.90/5.08 => ( ( dvd_dvd_nat @ B @ A )
% 8.90/5.08 => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
% 8.90/5.08 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4770_div__div__div__same,axiom,
% 8.90/5.08 ! [D: int,B: int,A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ D @ B )
% 8.90/5.08 => ( ( dvd_dvd_int @ B @ A )
% 8.90/5.08 => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
% 8.90/5.08 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4771_dvd__div__eq__cancel,axiom,
% 8.90/5.08 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.90/5.08 ( ( ( divide6298287555418463151nteger @ A @ C )
% 8.90/5.08 = ( divide6298287555418463151nteger @ B @ C ) )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ C @ A )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ C @ B )
% 8.90/5.08 => ( A = B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4772_dvd__div__eq__cancel,axiom,
% 8.90/5.08 ! [A: complex,C: complex,B: complex] :
% 8.90/5.08 ( ( ( divide1717551699836669952omplex @ A @ C )
% 8.90/5.08 = ( divide1717551699836669952omplex @ B @ C ) )
% 8.90/5.08 => ( ( dvd_dvd_complex @ C @ A )
% 8.90/5.08 => ( ( dvd_dvd_complex @ C @ B )
% 8.90/5.08 => ( A = B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4773_dvd__div__eq__cancel,axiom,
% 8.90/5.08 ! [A: real,C: real,B: real] :
% 8.90/5.08 ( ( ( divide_divide_real @ A @ C )
% 8.90/5.08 = ( divide_divide_real @ B @ C ) )
% 8.90/5.08 => ( ( dvd_dvd_real @ C @ A )
% 8.90/5.08 => ( ( dvd_dvd_real @ C @ B )
% 8.90/5.08 => ( A = B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4774_dvd__div__eq__cancel,axiom,
% 8.90/5.08 ! [A: rat,C: rat,B: rat] :
% 8.90/5.08 ( ( ( divide_divide_rat @ A @ C )
% 8.90/5.08 = ( divide_divide_rat @ B @ C ) )
% 8.90/5.08 => ( ( dvd_dvd_rat @ C @ A )
% 8.90/5.08 => ( ( dvd_dvd_rat @ C @ B )
% 8.90/5.08 => ( A = B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4775_dvd__div__eq__cancel,axiom,
% 8.90/5.08 ! [A: nat,C: nat,B: nat] :
% 8.90/5.08 ( ( ( divide_divide_nat @ A @ C )
% 8.90/5.08 = ( divide_divide_nat @ B @ C ) )
% 8.90/5.08 => ( ( dvd_dvd_nat @ C @ A )
% 8.90/5.08 => ( ( dvd_dvd_nat @ C @ B )
% 8.90/5.08 => ( A = B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4776_dvd__div__eq__cancel,axiom,
% 8.90/5.08 ! [A: int,C: int,B: int] :
% 8.90/5.08 ( ( ( divide_divide_int @ A @ C )
% 8.90/5.08 = ( divide_divide_int @ B @ C ) )
% 8.90/5.08 => ( ( dvd_dvd_int @ C @ A )
% 8.90/5.08 => ( ( dvd_dvd_int @ C @ B )
% 8.90/5.08 => ( A = B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4777_dvd__div__eq__iff,axiom,
% 8.90/5.08 ! [C: code_integer,A: code_integer,B: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ C @ A )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ C @ B )
% 8.90/5.08 => ( ( ( divide6298287555418463151nteger @ A @ C )
% 8.90/5.08 = ( divide6298287555418463151nteger @ B @ C ) )
% 8.90/5.08 = ( A = B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4778_dvd__div__eq__iff,axiom,
% 8.90/5.08 ! [C: complex,A: complex,B: complex] :
% 8.90/5.08 ( ( dvd_dvd_complex @ C @ A )
% 8.90/5.08 => ( ( dvd_dvd_complex @ C @ B )
% 8.90/5.08 => ( ( ( divide1717551699836669952omplex @ A @ C )
% 8.90/5.08 = ( divide1717551699836669952omplex @ B @ C ) )
% 8.90/5.08 = ( A = B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4779_dvd__div__eq__iff,axiom,
% 8.90/5.08 ! [C: real,A: real,B: real] :
% 8.90/5.08 ( ( dvd_dvd_real @ C @ A )
% 8.90/5.08 => ( ( dvd_dvd_real @ C @ B )
% 8.90/5.08 => ( ( ( divide_divide_real @ A @ C )
% 8.90/5.08 = ( divide_divide_real @ B @ C ) )
% 8.90/5.08 = ( A = B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4780_dvd__div__eq__iff,axiom,
% 8.90/5.08 ! [C: rat,A: rat,B: rat] :
% 8.90/5.08 ( ( dvd_dvd_rat @ C @ A )
% 8.90/5.08 => ( ( dvd_dvd_rat @ C @ B )
% 8.90/5.08 => ( ( ( divide_divide_rat @ A @ C )
% 8.90/5.08 = ( divide_divide_rat @ B @ C ) )
% 8.90/5.08 = ( A = B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4781_dvd__div__eq__iff,axiom,
% 8.90/5.08 ! [C: nat,A: nat,B: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ C @ A )
% 8.90/5.08 => ( ( dvd_dvd_nat @ C @ B )
% 8.90/5.08 => ( ( ( divide_divide_nat @ A @ C )
% 8.90/5.08 = ( divide_divide_nat @ B @ C ) )
% 8.90/5.08 = ( A = B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4782_dvd__div__eq__iff,axiom,
% 8.90/5.08 ! [C: int,A: int,B: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ C @ A )
% 8.90/5.08 => ( ( dvd_dvd_int @ C @ B )
% 8.90/5.08 => ( ( ( divide_divide_int @ A @ C )
% 8.90/5.08 = ( divide_divide_int @ B @ C ) )
% 8.90/5.08 = ( A = B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4783_dvd__power__same,axiom,
% 8.90/5.08 ! [X2: code_integer,Y3: code_integer,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ X2 @ Y3 )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X2 @ N ) @ ( power_8256067586552552935nteger @ Y3 @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4784_dvd__power__same,axiom,
% 8.90/5.08 ! [X2: nat,Y3: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ X2 @ Y3 )
% 8.90/5.08 => ( dvd_dvd_nat @ ( power_power_nat @ X2 @ N ) @ ( power_power_nat @ Y3 @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4785_dvd__power__same,axiom,
% 8.90/5.08 ! [X2: real,Y3: real,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_real @ X2 @ Y3 )
% 8.90/5.08 => ( dvd_dvd_real @ ( power_power_real @ X2 @ N ) @ ( power_power_real @ Y3 @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4786_dvd__power__same,axiom,
% 8.90/5.08 ! [X2: int,Y3: int,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_int @ X2 @ Y3 )
% 8.90/5.08 => ( dvd_dvd_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y3 @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4787_dvd__power__same,axiom,
% 8.90/5.08 ! [X2: complex,Y3: complex,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_complex @ X2 @ Y3 )
% 8.90/5.08 => ( dvd_dvd_complex @ ( power_power_complex @ X2 @ N ) @ ( power_power_complex @ Y3 @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4788_dvd__mod__iff,axiom,
% 8.90/5.08 ! [C: nat,B: nat,A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ C @ B )
% 8.90/5.08 => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 8.90/5.08 = ( dvd_dvd_nat @ C @ A ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4789_dvd__mod__iff,axiom,
% 8.90/5.08 ! [C: int,B: int,A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ C @ B )
% 8.90/5.08 => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 8.90/5.08 = ( dvd_dvd_int @ C @ A ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4790_dvd__mod__iff,axiom,
% 8.90/5.08 ! [C: code_integer,B: code_integer,A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ C @ B )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4791_dvd__mod__imp__dvd,axiom,
% 8.90/5.08 ! [C: nat,A: nat,B: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 8.90/5.08 => ( ( dvd_dvd_nat @ C @ B )
% 8.90/5.08 => ( dvd_dvd_nat @ C @ A ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4792_dvd__mod__imp__dvd,axiom,
% 8.90/5.08 ! [C: int,A: int,B: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 8.90/5.08 => ( ( dvd_dvd_int @ C @ B )
% 8.90/5.08 => ( dvd_dvd_int @ C @ A ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4793_dvd__mod__imp__dvd,axiom,
% 8.90/5.08 ! [C: code_integer,A: code_integer,B: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ C @ B )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4794_dvd__mod,axiom,
% 8.90/5.08 ! [K: nat,M: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ K @ M )
% 8.90/5.08 => ( ( dvd_dvd_nat @ K @ N )
% 8.90/5.08 => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4795_dvd__mod,axiom,
% 8.90/5.08 ! [K: int,M: int,N: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ K @ M )
% 8.90/5.08 => ( ( dvd_dvd_int @ K @ N )
% 8.90/5.08 => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4796_dvd__mod,axiom,
% 8.90/5.08 ! [K: code_integer,M: code_integer,N: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ K @ M )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ K @ N )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4797_mod__mod__cancel,axiom,
% 8.90/5.08 ! [C: nat,B: nat,A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ C @ B )
% 8.90/5.08 => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ C )
% 8.90/5.08 = ( modulo_modulo_nat @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4798_mod__mod__cancel,axiom,
% 8.90/5.08 ! [C: int,B: int,A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ C @ B )
% 8.90/5.08 => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ C )
% 8.90/5.08 = ( modulo_modulo_int @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4799_mod__mod__cancel,axiom,
% 8.90/5.08 ! [C: code_integer,B: code_integer,A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ C @ B )
% 8.90/5.08 => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C )
% 8.90/5.08 = ( modulo364778990260209775nteger @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4800_dvd__diff__nat,axiom,
% 8.90/5.08 ! [K: nat,M: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ K @ M )
% 8.90/5.08 => ( ( dvd_dvd_nat @ K @ N )
% 8.90/5.08 => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4801_zdvd__zdiffD,axiom,
% 8.90/5.08 ! [K: int,M: int,N: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
% 8.90/5.08 => ( ( dvd_dvd_int @ K @ N )
% 8.90/5.08 => ( dvd_dvd_int @ K @ M ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4802_bot__enat__def,axiom,
% 8.90/5.08 bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4803_dvd__pos__nat,axiom,
% 8.90/5.08 ! [N: nat,M: nat] :
% 8.90/5.08 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.08 => ( ( dvd_dvd_nat @ M @ N )
% 8.90/5.08 => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4804_bezout__lemma__nat,axiom,
% 8.90/5.08 ! [D: nat,A: nat,B: nat,X2: nat,Y3: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ D @ A )
% 8.90/5.08 => ( ( dvd_dvd_nat @ D @ B )
% 8.90/5.08 => ( ( ( ( times_times_nat @ A @ X2 )
% 8.90/5.08 = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D ) )
% 8.90/5.08 | ( ( times_times_nat @ B @ X2 )
% 8.90/5.08 = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) )
% 8.90/5.08 => ? [X5: nat,Y4: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ D @ A )
% 8.90/5.08 & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
% 8.90/5.08 & ( ( ( times_times_nat @ A @ X5 )
% 8.90/5.08 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y4 ) @ D ) )
% 8.90/5.08 | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X5 )
% 8.90/5.08 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4805_bezout__add__nat,axiom,
% 8.90/5.08 ! [A: nat,B: nat] :
% 8.90/5.08 ? [D3: nat,X5: nat,Y4: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ D3 @ A )
% 8.90/5.08 & ( dvd_dvd_nat @ D3 @ B )
% 8.90/5.08 & ( ( ( times_times_nat @ A @ X5 )
% 8.90/5.08 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D3 ) )
% 8.90/5.08 | ( ( times_times_nat @ B @ X5 )
% 8.90/5.08 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D3 ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4806_bezout1__nat,axiom,
% 8.90/5.08 ! [A: nat,B: nat] :
% 8.90/5.08 ? [D3: nat,X5: nat,Y4: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ D3 @ A )
% 8.90/5.08 & ( dvd_dvd_nat @ D3 @ B )
% 8.90/5.08 & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X5 ) @ ( times_times_nat @ B @ Y4 ) )
% 8.90/5.08 = D3 )
% 8.90/5.08 | ( ( minus_minus_nat @ ( times_times_nat @ B @ X5 ) @ ( times_times_nat @ A @ Y4 ) )
% 8.90/5.08 = D3 ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4807_subset__divisors__dvd,axiom,
% 8.90/5.08 ! [A: complex,B: complex] :
% 8.90/5.08 ( ( ord_le211207098394363844omplex
% 8.90/5.08 @ ( collect_complex
% 8.90/5.08 @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ A ) )
% 8.90/5.08 @ ( collect_complex
% 8.90/5.08 @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ B ) ) )
% 8.90/5.08 = ( dvd_dvd_complex @ A @ B ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4808_subset__divisors__dvd,axiom,
% 8.90/5.08 ! [A: real,B: real] :
% 8.90/5.08 ( ( ord_less_eq_set_real
% 8.90/5.08 @ ( collect_real
% 8.90/5.08 @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ A ) )
% 8.90/5.08 @ ( collect_real
% 8.90/5.08 @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ B ) ) )
% 8.90/5.08 = ( dvd_dvd_real @ A @ B ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4809_subset__divisors__dvd,axiom,
% 8.90/5.08 ! [A: nat,B: nat] :
% 8.90/5.08 ( ( ord_less_eq_set_nat
% 8.90/5.08 @ ( collect_nat
% 8.90/5.08 @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
% 8.90/5.08 @ ( collect_nat
% 8.90/5.08 @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B ) ) )
% 8.90/5.08 = ( dvd_dvd_nat @ A @ B ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4810_subset__divisors__dvd,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer] :
% 8.90/5.08 ( ( ord_le7084787975880047091nteger
% 8.90/5.08 @ ( collect_Code_integer
% 8.90/5.08 @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ A ) )
% 8.90/5.08 @ ( collect_Code_integer
% 8.90/5.08 @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ B ) ) )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4811_subset__divisors__dvd,axiom,
% 8.90/5.08 ! [A: int,B: int] :
% 8.90/5.08 ( ( ord_less_eq_set_int
% 8.90/5.08 @ ( collect_int
% 8.90/5.08 @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
% 8.90/5.08 @ ( collect_int
% 8.90/5.08 @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B ) ) )
% 8.90/5.08 = ( dvd_dvd_int @ A @ B ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4812_concat__bit__assoc,axiom,
% 8.90/5.08 ! [N: nat,K: int,M: nat,L: int,R2: int] :
% 8.90/5.08 ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L @ R2 ) )
% 8.90/5.08 = ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R2 ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4813_strict__subset__divisors__dvd,axiom,
% 8.90/5.08 ! [A: complex,B: complex] :
% 8.90/5.08 ( ( ord_less_set_complex
% 8.90/5.08 @ ( collect_complex
% 8.90/5.08 @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ A ) )
% 8.90/5.08 @ ( collect_complex
% 8.90/5.08 @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ B ) ) )
% 8.90/5.08 = ( ( dvd_dvd_complex @ A @ B )
% 8.90/5.08 & ~ ( dvd_dvd_complex @ B @ A ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4814_strict__subset__divisors__dvd,axiom,
% 8.90/5.08 ! [A: real,B: real] :
% 8.90/5.08 ( ( ord_less_set_real
% 8.90/5.08 @ ( collect_real
% 8.90/5.08 @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ A ) )
% 8.90/5.08 @ ( collect_real
% 8.90/5.08 @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ B ) ) )
% 8.90/5.08 = ( ( dvd_dvd_real @ A @ B )
% 8.90/5.08 & ~ ( dvd_dvd_real @ B @ A ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4815_strict__subset__divisors__dvd,axiom,
% 8.90/5.08 ! [A: nat,B: nat] :
% 8.90/5.08 ( ( ord_less_set_nat
% 8.90/5.08 @ ( collect_nat
% 8.90/5.08 @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
% 8.90/5.08 @ ( collect_nat
% 8.90/5.08 @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B ) ) )
% 8.90/5.08 = ( ( dvd_dvd_nat @ A @ B )
% 8.90/5.08 & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4816_strict__subset__divisors__dvd,axiom,
% 8.90/5.08 ! [A: int,B: int] :
% 8.90/5.08 ( ( ord_less_set_int
% 8.90/5.08 @ ( collect_int
% 8.90/5.08 @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
% 8.90/5.08 @ ( collect_int
% 8.90/5.08 @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B ) ) )
% 8.90/5.08 = ( ( dvd_dvd_int @ A @ B )
% 8.90/5.08 & ~ ( dvd_dvd_int @ B @ A ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4817_strict__subset__divisors__dvd,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer] :
% 8.90/5.08 ( ( ord_le1307284697595431911nteger
% 8.90/5.08 @ ( collect_Code_integer
% 8.90/5.08 @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ A ) )
% 8.90/5.08 @ ( collect_Code_integer
% 8.90/5.08 @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ B ) ) )
% 8.90/5.08 = ( ( dvd_dvd_Code_integer @ A @ B )
% 8.90/5.08 & ~ ( dvd_dvd_Code_integer @ B @ A ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4818_not__is__unit__0,axiom,
% 8.90/5.08 ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4819_not__is__unit__0,axiom,
% 8.90/5.08 ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4820_not__is__unit__0,axiom,
% 8.90/5.08 ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4821_minf_I10_J,axiom,
% 8.90/5.08 ! [D: code_integer,S: code_integer] :
% 8.90/5.08 ? [Z2: code_integer] :
% 8.90/5.08 ! [X3: code_integer] :
% 8.90/5.08 ( ( ord_le6747313008572928689nteger @ X3 @ Z2 )
% 8.90/5.08 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) ) )
% 8.90/5.08 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4822_minf_I10_J,axiom,
% 8.90/5.08 ! [D: real,S: real] :
% 8.90/5.08 ? [Z2: real] :
% 8.90/5.08 ! [X3: real] :
% 8.90/5.08 ( ( ord_less_real @ X3 @ Z2 )
% 8.90/5.08 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) )
% 8.90/5.08 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4823_minf_I10_J,axiom,
% 8.90/5.08 ! [D: rat,S: rat] :
% 8.90/5.08 ? [Z2: rat] :
% 8.90/5.08 ! [X3: rat] :
% 8.90/5.08 ( ( ord_less_rat @ X3 @ Z2 )
% 8.90/5.08 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) )
% 8.90/5.08 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4824_minf_I10_J,axiom,
% 8.90/5.08 ! [D: nat,S: nat] :
% 8.90/5.08 ? [Z2: nat] :
% 8.90/5.08 ! [X3: nat] :
% 8.90/5.08 ( ( ord_less_nat @ X3 @ Z2 )
% 8.90/5.08 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) )
% 8.90/5.08 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4825_minf_I10_J,axiom,
% 8.90/5.08 ! [D: int,S: int] :
% 8.90/5.08 ? [Z2: int] :
% 8.90/5.08 ! [X3: int] :
% 8.90/5.08 ( ( ord_less_int @ X3 @ Z2 )
% 8.90/5.08 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) )
% 8.90/5.08 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4826_minf_I9_J,axiom,
% 8.90/5.08 ! [D: code_integer,S: code_integer] :
% 8.90/5.08 ? [Z2: code_integer] :
% 8.90/5.08 ! [X3: code_integer] :
% 8.90/5.08 ( ( ord_le6747313008572928689nteger @ X3 @ Z2 )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4827_minf_I9_J,axiom,
% 8.90/5.08 ! [D: real,S: real] :
% 8.90/5.08 ? [Z2: real] :
% 8.90/5.08 ! [X3: real] :
% 8.90/5.08 ( ( ord_less_real @ X3 @ Z2 )
% 8.90/5.08 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) )
% 8.90/5.08 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4828_minf_I9_J,axiom,
% 8.90/5.08 ! [D: rat,S: rat] :
% 8.90/5.08 ? [Z2: rat] :
% 8.90/5.08 ! [X3: rat] :
% 8.90/5.08 ( ( ord_less_rat @ X3 @ Z2 )
% 8.90/5.08 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) )
% 8.90/5.08 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4829_minf_I9_J,axiom,
% 8.90/5.08 ! [D: nat,S: nat] :
% 8.90/5.08 ? [Z2: nat] :
% 8.90/5.08 ! [X3: nat] :
% 8.90/5.08 ( ( ord_less_nat @ X3 @ Z2 )
% 8.90/5.08 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) )
% 8.90/5.08 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4830_minf_I9_J,axiom,
% 8.90/5.08 ! [D: int,S: int] :
% 8.90/5.08 ? [Z2: int] :
% 8.90/5.08 ! [X3: int] :
% 8.90/5.08 ( ( ord_less_int @ X3 @ Z2 )
% 8.90/5.08 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) )
% 8.90/5.08 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4831_pinf_I10_J,axiom,
% 8.90/5.08 ! [D: code_integer,S: code_integer] :
% 8.90/5.08 ? [Z2: code_integer] :
% 8.90/5.08 ! [X3: code_integer] :
% 8.90/5.08 ( ( ord_le6747313008572928689nteger @ Z2 @ X3 )
% 8.90/5.08 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) ) )
% 8.90/5.08 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4832_pinf_I10_J,axiom,
% 8.90/5.08 ! [D: real,S: real] :
% 8.90/5.08 ? [Z2: real] :
% 8.90/5.08 ! [X3: real] :
% 8.90/5.08 ( ( ord_less_real @ Z2 @ X3 )
% 8.90/5.08 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) )
% 8.90/5.08 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4833_pinf_I10_J,axiom,
% 8.90/5.08 ! [D: rat,S: rat] :
% 8.90/5.08 ? [Z2: rat] :
% 8.90/5.08 ! [X3: rat] :
% 8.90/5.08 ( ( ord_less_rat @ Z2 @ X3 )
% 8.90/5.08 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) )
% 8.90/5.08 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4834_pinf_I10_J,axiom,
% 8.90/5.08 ! [D: nat,S: nat] :
% 8.90/5.08 ? [Z2: nat] :
% 8.90/5.08 ! [X3: nat] :
% 8.90/5.08 ( ( ord_less_nat @ Z2 @ X3 )
% 8.90/5.08 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) )
% 8.90/5.08 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4835_pinf_I10_J,axiom,
% 8.90/5.08 ! [D: int,S: int] :
% 8.90/5.08 ? [Z2: int] :
% 8.90/5.08 ! [X3: int] :
% 8.90/5.08 ( ( ord_less_int @ Z2 @ X3 )
% 8.90/5.08 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) )
% 8.90/5.08 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4836_pinf_I9_J,axiom,
% 8.90/5.08 ! [D: code_integer,S: code_integer] :
% 8.90/5.08 ? [Z2: code_integer] :
% 8.90/5.08 ! [X3: code_integer] :
% 8.90/5.08 ( ( ord_le6747313008572928689nteger @ Z2 @ X3 )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4837_pinf_I9_J,axiom,
% 8.90/5.08 ! [D: real,S: real] :
% 8.90/5.08 ? [Z2: real] :
% 8.90/5.08 ! [X3: real] :
% 8.90/5.08 ( ( ord_less_real @ Z2 @ X3 )
% 8.90/5.08 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) )
% 8.90/5.08 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4838_pinf_I9_J,axiom,
% 8.90/5.08 ! [D: rat,S: rat] :
% 8.90/5.08 ? [Z2: rat] :
% 8.90/5.08 ! [X3: rat] :
% 8.90/5.08 ( ( ord_less_rat @ Z2 @ X3 )
% 8.90/5.08 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) )
% 8.90/5.08 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4839_pinf_I9_J,axiom,
% 8.90/5.08 ! [D: nat,S: nat] :
% 8.90/5.08 ? [Z2: nat] :
% 8.90/5.08 ! [X3: nat] :
% 8.90/5.08 ( ( ord_less_nat @ Z2 @ X3 )
% 8.90/5.08 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) )
% 8.90/5.08 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4840_pinf_I9_J,axiom,
% 8.90/5.08 ! [D: int,S: int] :
% 8.90/5.08 ? [Z2: int] :
% 8.90/5.08 ! [X3: int] :
% 8.90/5.08 ( ( ord_less_int @ Z2 @ X3 )
% 8.90/5.08 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) )
% 8.90/5.08 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4841_dvd__div__eq__0__iff,axiom,
% 8.90/5.08 ! [B: code_integer,A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ B @ A )
% 8.90/5.08 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 8.90/5.08 = zero_z3403309356797280102nteger )
% 8.90/5.08 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4842_dvd__div__eq__0__iff,axiom,
% 8.90/5.08 ! [B: complex,A: complex] :
% 8.90/5.08 ( ( dvd_dvd_complex @ B @ A )
% 8.90/5.08 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 8.90/5.08 = zero_zero_complex )
% 8.90/5.08 = ( A = zero_zero_complex ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4843_dvd__div__eq__0__iff,axiom,
% 8.90/5.08 ! [B: real,A: real] :
% 8.90/5.08 ( ( dvd_dvd_real @ B @ A )
% 8.90/5.08 => ( ( ( divide_divide_real @ A @ B )
% 8.90/5.08 = zero_zero_real )
% 8.90/5.08 = ( A = zero_zero_real ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4844_dvd__div__eq__0__iff,axiom,
% 8.90/5.08 ! [B: rat,A: rat] :
% 8.90/5.08 ( ( dvd_dvd_rat @ B @ A )
% 8.90/5.08 => ( ( ( divide_divide_rat @ A @ B )
% 8.90/5.08 = zero_zero_rat )
% 8.90/5.08 = ( A = zero_zero_rat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4845_dvd__div__eq__0__iff,axiom,
% 8.90/5.08 ! [B: nat,A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ B @ A )
% 8.90/5.08 => ( ( ( divide_divide_nat @ A @ B )
% 8.90/5.08 = zero_zero_nat )
% 8.90/5.08 = ( A = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4846_dvd__div__eq__0__iff,axiom,
% 8.90/5.08 ! [B: int,A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ B @ A )
% 8.90/5.08 => ( ( ( divide_divide_int @ A @ B )
% 8.90/5.08 = zero_zero_int )
% 8.90/5.08 = ( A = zero_zero_int ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4847_is__unit__mult__iff,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
% 8.90/5.08 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.90/5.08 & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4848_is__unit__mult__iff,axiom,
% 8.90/5.08 ! [A: nat,B: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
% 8.90/5.08 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.90/5.08 & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4849_is__unit__mult__iff,axiom,
% 8.90/5.08 ! [A: int,B: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
% 8.90/5.08 = ( ( dvd_dvd_int @ A @ one_one_int )
% 8.90/5.08 & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4850_dvd__mult__unit__iff,axiom,
% 8.90/5.08 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4851_dvd__mult__unit__iff,axiom,
% 8.90/5.08 ! [B: nat,A: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.90/5.08 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
% 8.90/5.08 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4852_dvd__mult__unit__iff,axiom,
% 8.90/5.08 ! [B: int,A: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.90/5.08 => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
% 8.90/5.08 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4853_mult__unit__dvd__iff,axiom,
% 8.90/5.08 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4854_mult__unit__dvd__iff,axiom,
% 8.90/5.08 ! [B: nat,A: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.90/5.08 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 8.90/5.08 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4855_mult__unit__dvd__iff,axiom,
% 8.90/5.08 ! [B: int,A: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.90/5.08 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 8.90/5.08 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4856_dvd__mult__unit__iff_H,axiom,
% 8.90/5.08 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4857_dvd__mult__unit__iff_H,axiom,
% 8.90/5.08 ! [B: nat,A: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.90/5.08 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 8.90/5.08 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4858_dvd__mult__unit__iff_H,axiom,
% 8.90/5.08 ! [B: int,A: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.90/5.08 => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 8.90/5.08 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4859_mult__unit__dvd__iff_H,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4860_mult__unit__dvd__iff_H,axiom,
% 8.90/5.08 ! [A: nat,B: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.90/5.08 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 8.90/5.08 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4861_mult__unit__dvd__iff_H,axiom,
% 8.90/5.08 ! [A: int,B: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.90/5.08 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 8.90/5.08 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4862_unit__mult__left__cancel,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.90/5.08 => ( ( ( times_3573771949741848930nteger @ A @ B )
% 8.90/5.08 = ( times_3573771949741848930nteger @ A @ C ) )
% 8.90/5.08 = ( B = C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4863_unit__mult__left__cancel,axiom,
% 8.90/5.08 ! [A: nat,B: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.90/5.08 => ( ( ( times_times_nat @ A @ B )
% 8.90/5.08 = ( times_times_nat @ A @ C ) )
% 8.90/5.08 = ( B = C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4864_unit__mult__left__cancel,axiom,
% 8.90/5.08 ! [A: int,B: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.90/5.08 => ( ( ( times_times_int @ A @ B )
% 8.90/5.08 = ( times_times_int @ A @ C ) )
% 8.90/5.08 = ( B = C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4865_unit__mult__right__cancel,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.90/5.08 => ( ( ( times_3573771949741848930nteger @ B @ A )
% 8.90/5.08 = ( times_3573771949741848930nteger @ C @ A ) )
% 8.90/5.08 = ( B = C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4866_unit__mult__right__cancel,axiom,
% 8.90/5.08 ! [A: nat,B: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.90/5.08 => ( ( ( times_times_nat @ B @ A )
% 8.90/5.08 = ( times_times_nat @ C @ A ) )
% 8.90/5.08 = ( B = C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4867_unit__mult__right__cancel,axiom,
% 8.90/5.08 ! [A: int,B: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.90/5.08 => ( ( ( times_times_int @ B @ A )
% 8.90/5.08 = ( times_times_int @ C @ A ) )
% 8.90/5.08 = ( B = C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4868_div__mult__div__if__dvd,axiom,
% 8.90/5.08 ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ B @ A )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ D @ C )
% 8.90/5.08 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
% 8.90/5.08 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4869_div__mult__div__if__dvd,axiom,
% 8.90/5.08 ! [B: nat,A: nat,D: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ B @ A )
% 8.90/5.08 => ( ( dvd_dvd_nat @ D @ C )
% 8.90/5.08 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
% 8.90/5.08 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4870_div__mult__div__if__dvd,axiom,
% 8.90/5.08 ! [B: int,A: int,D: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ B @ A )
% 8.90/5.08 => ( ( dvd_dvd_int @ D @ C )
% 8.90/5.08 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
% 8.90/5.08 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4871_dvd__mult__imp__div,axiom,
% 8.90/5.08 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4872_dvd__mult__imp__div,axiom,
% 8.90/5.08 ! [A: nat,C: nat,B: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
% 8.90/5.08 => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4873_dvd__mult__imp__div,axiom,
% 8.90/5.08 ! [A: int,C: int,B: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
% 8.90/5.08 => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4874_dvd__div__mult2__eq,axiom,
% 8.90/5.08 ! [B: code_integer,C: code_integer,A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
% 8.90/5.08 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 8.90/5.08 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4875_dvd__div__mult2__eq,axiom,
% 8.90/5.08 ! [B: nat,C: nat,A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
% 8.90/5.08 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 8.90/5.08 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4876_dvd__div__mult2__eq,axiom,
% 8.90/5.08 ! [B: int,C: int,A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
% 8.90/5.08 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 8.90/5.08 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4877_div__div__eq__right,axiom,
% 8.90/5.08 ! [C: code_integer,B: code_integer,A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ C @ B )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ B @ A )
% 8.90/5.08 => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 8.90/5.08 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4878_div__div__eq__right,axiom,
% 8.90/5.08 ! [C: nat,B: nat,A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ C @ B )
% 8.90/5.08 => ( ( dvd_dvd_nat @ B @ A )
% 8.90/5.08 => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 8.90/5.08 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4879_div__div__eq__right,axiom,
% 8.90/5.08 ! [C: int,B: int,A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ C @ B )
% 8.90/5.08 => ( ( dvd_dvd_int @ B @ A )
% 8.90/5.08 => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
% 8.90/5.08 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4880_div__mult__swap,axiom,
% 8.90/5.08 ! [C: code_integer,B: code_integer,A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ C @ B )
% 8.90/5.08 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 8.90/5.08 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4881_div__mult__swap,axiom,
% 8.90/5.08 ! [C: nat,B: nat,A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ C @ B )
% 8.90/5.08 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 8.90/5.08 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4882_div__mult__swap,axiom,
% 8.90/5.08 ! [C: int,B: int,A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ C @ B )
% 8.90/5.08 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 8.90/5.08 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4883_dvd__div__mult,axiom,
% 8.90/5.08 ! [C: code_integer,B: code_integer,A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ C @ B )
% 8.90/5.08 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
% 8.90/5.08 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4884_dvd__div__mult,axiom,
% 8.90/5.08 ! [C: nat,B: nat,A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ C @ B )
% 8.90/5.08 => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
% 8.90/5.08 = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4885_dvd__div__mult,axiom,
% 8.90/5.08 ! [C: int,B: int,A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ C @ B )
% 8.90/5.08 => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
% 8.90/5.08 = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4886_unit__div__cancel,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.90/5.08 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 8.90/5.08 = ( divide6298287555418463151nteger @ C @ A ) )
% 8.90/5.08 = ( B = C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4887_unit__div__cancel,axiom,
% 8.90/5.08 ! [A: nat,B: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.90/5.08 => ( ( ( divide_divide_nat @ B @ A )
% 8.90/5.08 = ( divide_divide_nat @ C @ A ) )
% 8.90/5.08 = ( B = C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4888_unit__div__cancel,axiom,
% 8.90/5.08 ! [A: int,B: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.90/5.08 => ( ( ( divide_divide_int @ B @ A )
% 8.90/5.08 = ( divide_divide_int @ C @ A ) )
% 8.90/5.08 = ( B = C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4889_div__unit__dvd__iff,axiom,
% 8.90/5.08 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4890_div__unit__dvd__iff,axiom,
% 8.90/5.08 ! [B: nat,A: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.90/5.08 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 8.90/5.08 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4891_div__unit__dvd__iff,axiom,
% 8.90/5.08 ! [B: int,A: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.90/5.08 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 8.90/5.08 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4892_dvd__div__unit__iff,axiom,
% 8.90/5.08 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4893_dvd__div__unit__iff,axiom,
% 8.90/5.08 ! [B: nat,A: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.90/5.08 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
% 8.90/5.08 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4894_dvd__div__unit__iff,axiom,
% 8.90/5.08 ! [B: int,A: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.90/5.08 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
% 8.90/5.08 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4895_div__plus__div__distrib__dvd__left,axiom,
% 8.90/5.08 ! [C: code_integer,A: code_integer,B: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ C @ A )
% 8.90/5.08 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 8.90/5.08 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4896_div__plus__div__distrib__dvd__left,axiom,
% 8.90/5.08 ! [C: nat,A: nat,B: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ C @ A )
% 8.90/5.08 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 8.90/5.08 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4897_div__plus__div__distrib__dvd__left,axiom,
% 8.90/5.08 ! [C: int,A: int,B: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ C @ A )
% 8.90/5.08 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.90/5.08 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4898_div__plus__div__distrib__dvd__right,axiom,
% 8.90/5.08 ! [C: code_integer,B: code_integer,A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ C @ B )
% 8.90/5.08 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 8.90/5.08 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4899_div__plus__div__distrib__dvd__right,axiom,
% 8.90/5.08 ! [C: nat,B: nat,A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ C @ B )
% 8.90/5.08 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 8.90/5.08 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4900_div__plus__div__distrib__dvd__right,axiom,
% 8.90/5.08 ! [C: int,B: int,A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ C @ B )
% 8.90/5.08 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 8.90/5.08 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4901_div__power,axiom,
% 8.90/5.08 ! [B: code_integer,A: code_integer,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ B @ A )
% 8.90/5.08 => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N )
% 8.90/5.08 = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4902_div__power,axiom,
% 8.90/5.08 ! [B: nat,A: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ B @ A )
% 8.90/5.08 => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N )
% 8.90/5.08 = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4903_div__power,axiom,
% 8.90/5.08 ! [B: int,A: int,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_int @ B @ A )
% 8.90/5.08 => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N )
% 8.90/5.08 = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4904_mod__eq__0__iff__dvd,axiom,
% 8.90/5.08 ! [A: nat,B: nat] :
% 8.90/5.08 ( ( ( modulo_modulo_nat @ A @ B )
% 8.90/5.08 = zero_zero_nat )
% 8.90/5.08 = ( dvd_dvd_nat @ B @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4905_mod__eq__0__iff__dvd,axiom,
% 8.90/5.08 ! [A: int,B: int] :
% 8.90/5.08 ( ( ( modulo_modulo_int @ A @ B )
% 8.90/5.08 = zero_zero_int )
% 8.90/5.08 = ( dvd_dvd_int @ B @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4906_mod__eq__0__iff__dvd,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer] :
% 8.90/5.08 ( ( ( modulo364778990260209775nteger @ A @ B )
% 8.90/5.08 = zero_z3403309356797280102nteger )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4907_dvd__eq__mod__eq__0,axiom,
% 8.90/5.08 ( dvd_dvd_nat
% 8.90/5.08 = ( ^ [A3: nat,B2: nat] :
% 8.90/5.08 ( ( modulo_modulo_nat @ B2 @ A3 )
% 8.90/5.08 = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4908_dvd__eq__mod__eq__0,axiom,
% 8.90/5.08 ( dvd_dvd_int
% 8.90/5.08 = ( ^ [A3: int,B2: int] :
% 8.90/5.08 ( ( modulo_modulo_int @ B2 @ A3 )
% 8.90/5.08 = zero_zero_int ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4909_dvd__eq__mod__eq__0,axiom,
% 8.90/5.08 ( dvd_dvd_Code_integer
% 8.90/5.08 = ( ^ [A3: code_integer,B2: code_integer] :
% 8.90/5.08 ( ( modulo364778990260209775nteger @ B2 @ A3 )
% 8.90/5.08 = zero_z3403309356797280102nteger ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4910_mod__0__imp__dvd,axiom,
% 8.90/5.08 ! [A: nat,B: nat] :
% 8.90/5.08 ( ( ( modulo_modulo_nat @ A @ B )
% 8.90/5.08 = zero_zero_nat )
% 8.90/5.08 => ( dvd_dvd_nat @ B @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4911_mod__0__imp__dvd,axiom,
% 8.90/5.08 ! [A: int,B: int] :
% 8.90/5.08 ( ( ( modulo_modulo_int @ A @ B )
% 8.90/5.08 = zero_zero_int )
% 8.90/5.08 => ( dvd_dvd_int @ B @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4912_mod__0__imp__dvd,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer] :
% 8.90/5.08 ( ( ( modulo364778990260209775nteger @ A @ B )
% 8.90/5.08 = zero_z3403309356797280102nteger )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4913_dvd__power__le,axiom,
% 8.90/5.08 ! [X2: code_integer,Y3: code_integer,N: nat,M: nat] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ X2 @ Y3 )
% 8.90/5.08 => ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X2 @ N ) @ ( power_8256067586552552935nteger @ Y3 @ M ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4914_dvd__power__le,axiom,
% 8.90/5.08 ! [X2: nat,Y3: nat,N: nat,M: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ X2 @ Y3 )
% 8.90/5.08 => ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.08 => ( dvd_dvd_nat @ ( power_power_nat @ X2 @ N ) @ ( power_power_nat @ Y3 @ M ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4915_dvd__power__le,axiom,
% 8.90/5.08 ! [X2: real,Y3: real,N: nat,M: nat] :
% 8.90/5.08 ( ( dvd_dvd_real @ X2 @ Y3 )
% 8.90/5.08 => ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.08 => ( dvd_dvd_real @ ( power_power_real @ X2 @ N ) @ ( power_power_real @ Y3 @ M ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4916_dvd__power__le,axiom,
% 8.90/5.08 ! [X2: int,Y3: int,N: nat,M: nat] :
% 8.90/5.08 ( ( dvd_dvd_int @ X2 @ Y3 )
% 8.90/5.08 => ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.08 => ( dvd_dvd_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y3 @ M ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4917_dvd__power__le,axiom,
% 8.90/5.08 ! [X2: complex,Y3: complex,N: nat,M: nat] :
% 8.90/5.08 ( ( dvd_dvd_complex @ X2 @ Y3 )
% 8.90/5.08 => ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.08 => ( dvd_dvd_complex @ ( power_power_complex @ X2 @ N ) @ ( power_power_complex @ Y3 @ M ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4918_power__le__dvd,axiom,
% 8.90/5.08 ! [A: code_integer,N: nat,B: code_integer,M: nat] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
% 8.90/5.08 => ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4919_power__le__dvd,axiom,
% 8.90/5.08 ! [A: nat,N: nat,B: nat,M: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
% 8.90/5.08 => ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.08 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4920_power__le__dvd,axiom,
% 8.90/5.08 ! [A: real,N: nat,B: real,M: nat] :
% 8.90/5.08 ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
% 8.90/5.08 => ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.08 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4921_power__le__dvd,axiom,
% 8.90/5.08 ! [A: int,N: nat,B: int,M: nat] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
% 8.90/5.08 => ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.08 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4922_power__le__dvd,axiom,
% 8.90/5.08 ! [A: complex,N: nat,B: complex,M: nat] :
% 8.90/5.08 ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
% 8.90/5.08 => ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.08 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4923_le__imp__power__dvd,axiom,
% 8.90/5.08 ! [M: nat,N: nat,A: code_integer] :
% 8.90/5.08 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4924_le__imp__power__dvd,axiom,
% 8.90/5.08 ! [M: nat,N: nat,A: nat] :
% 8.90/5.08 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.08 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4925_le__imp__power__dvd,axiom,
% 8.90/5.08 ! [M: nat,N: nat,A: real] :
% 8.90/5.08 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.08 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4926_le__imp__power__dvd,axiom,
% 8.90/5.08 ! [M: nat,N: nat,A: int] :
% 8.90/5.08 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.08 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4927_le__imp__power__dvd,axiom,
% 8.90/5.08 ! [M: nat,N: nat,A: complex] :
% 8.90/5.08 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.08 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4928_dvd__minus__mod,axiom,
% 8.90/5.08 ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4929_dvd__minus__mod,axiom,
% 8.90/5.08 ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4930_dvd__minus__mod,axiom,
% 8.90/5.08 ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4931_mod__eq__dvd__iff,axiom,
% 8.90/5.08 ! [A: int,C: int,B: int] :
% 8.90/5.08 ( ( ( modulo_modulo_int @ A @ C )
% 8.90/5.08 = ( modulo_modulo_int @ B @ C ) )
% 8.90/5.08 = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4932_mod__eq__dvd__iff,axiom,
% 8.90/5.08 ! [A: code_integer,C: code_integer,B: code_integer] :
% 8.90/5.08 ( ( ( modulo364778990260209775nteger @ A @ C )
% 8.90/5.08 = ( modulo364778990260209775nteger @ B @ C ) )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4933_nat__dvd__not__less,axiom,
% 8.90/5.08 ! [M: nat,N: nat] :
% 8.90/5.08 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.90/5.08 => ( ( ord_less_nat @ M @ N )
% 8.90/5.08 => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4934_bezout__add__strong__nat,axiom,
% 8.90/5.08 ! [A: nat,B: nat] :
% 8.90/5.08 ( ( A != zero_zero_nat )
% 8.90/5.08 => ? [D3: nat,X5: nat,Y4: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ D3 @ A )
% 8.90/5.08 & ( dvd_dvd_nat @ D3 @ B )
% 8.90/5.08 & ( ( times_times_nat @ A @ X5 )
% 8.90/5.08 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D3 ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4935_dvd__minus__self,axiom,
% 8.90/5.08 ! [M: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
% 8.90/5.08 = ( ( ord_less_nat @ N @ M )
% 8.90/5.08 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4936_zdvd__antisym__nonneg,axiom,
% 8.90/5.08 ! [M: int,N: int] :
% 8.90/5.08 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 8.90/5.08 => ( ( ord_less_eq_int @ zero_zero_int @ N )
% 8.90/5.08 => ( ( dvd_dvd_int @ M @ N )
% 8.90/5.08 => ( ( dvd_dvd_int @ N @ M )
% 8.90/5.08 => ( M = N ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4937_less__eq__dvd__minus,axiom,
% 8.90/5.08 ! [M: nat,N: nat] :
% 8.90/5.08 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.08 => ( ( dvd_dvd_nat @ M @ N )
% 8.90/5.08 = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4938_dvd__diffD1,axiom,
% 8.90/5.08 ! [K: nat,M: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 8.90/5.08 => ( ( dvd_dvd_nat @ K @ M )
% 8.90/5.08 => ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.08 => ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4939_dvd__diffD,axiom,
% 8.90/5.08 ! [K: nat,M: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 8.90/5.08 => ( ( dvd_dvd_nat @ K @ N )
% 8.90/5.08 => ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.08 => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4940_zdvd__mult__cancel,axiom,
% 8.90/5.08 ! [K: int,M: int,N: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
% 8.90/5.08 => ( ( K != zero_zero_int )
% 8.90/5.08 => ( dvd_dvd_int @ M @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4941_zdvd__mono,axiom,
% 8.90/5.08 ! [K: int,M: int,T: int] :
% 8.90/5.08 ( ( K != zero_zero_int )
% 8.90/5.08 => ( ( dvd_dvd_int @ M @ T )
% 8.90/5.08 = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4942_dbl__def,axiom,
% 8.90/5.08 ( neg_numeral_dbl_real
% 8.90/5.08 = ( ^ [X: real] : ( plus_plus_real @ X @ X ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4943_dbl__def,axiom,
% 8.90/5.08 ( neg_numeral_dbl_rat
% 8.90/5.08 = ( ^ [X: rat] : ( plus_plus_rat @ X @ X ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4944_dbl__def,axiom,
% 8.90/5.08 ( neg_numeral_dbl_int
% 8.90/5.08 = ( ^ [X: int] : ( plus_plus_int @ X @ X ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4945_zdvd__reduce,axiom,
% 8.90/5.08 ! [K: int,N: int,M: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
% 8.90/5.08 = ( dvd_dvd_int @ K @ N ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4946_zdvd__period,axiom,
% 8.90/5.08 ! [A: int,D: int,X2: int,T: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ A @ D )
% 8.90/5.08 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X2 @ T ) )
% 8.90/5.08 = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X2 @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4947_finite__divisors__nat,axiom,
% 8.90/5.08 ! [M: nat] :
% 8.90/5.08 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.90/5.08 => ( finite_finite_nat
% 8.90/5.08 @ ( collect_nat
% 8.90/5.08 @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4948_div2__even__ext__nat,axiom,
% 8.90/5.08 ! [X2: nat,Y3: nat] :
% 8.90/5.08 ( ( ( divide_divide_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.08 = ( divide_divide_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.08 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 )
% 8.90/5.08 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y3 ) )
% 8.90/5.08 => ( X2 = Y3 ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4949_unit__dvdE,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.90/5.08 => ~ ( ( A != zero_z3403309356797280102nteger )
% 8.90/5.08 => ! [C3: code_integer] :
% 8.90/5.08 ( B
% 8.90/5.08 != ( times_3573771949741848930nteger @ A @ C3 ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4950_unit__dvdE,axiom,
% 8.90/5.08 ! [A: nat,B: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.90/5.08 => ~ ( ( A != zero_zero_nat )
% 8.90/5.08 => ! [C3: nat] :
% 8.90/5.08 ( B
% 8.90/5.08 != ( times_times_nat @ A @ C3 ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4951_unit__dvdE,axiom,
% 8.90/5.08 ! [A: int,B: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.90/5.08 => ~ ( ( A != zero_zero_int )
% 8.90/5.08 => ! [C3: int] :
% 8.90/5.08 ( B
% 8.90/5.08 != ( times_times_int @ A @ C3 ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4952_unity__coeff__ex,axiom,
% 8.90/5.08 ! [P: code_integer > $o,L: code_integer] :
% 8.90/5.08 ( ( ? [X: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X ) ) )
% 8.90/5.08 = ( ? [X: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X @ zero_z3403309356797280102nteger ) )
% 8.90/5.08 & ( P @ X ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4953_unity__coeff__ex,axiom,
% 8.90/5.08 ! [P: complex > $o,L: complex] :
% 8.90/5.08 ( ( ? [X: complex] : ( P @ ( times_times_complex @ L @ X ) ) )
% 8.90/5.08 = ( ? [X: complex] :
% 8.90/5.08 ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X @ zero_zero_complex ) )
% 8.90/5.08 & ( P @ X ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4954_unity__coeff__ex,axiom,
% 8.90/5.08 ! [P: real > $o,L: real] :
% 8.90/5.08 ( ( ? [X: real] : ( P @ ( times_times_real @ L @ X ) ) )
% 8.90/5.08 = ( ? [X: real] :
% 8.90/5.08 ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X @ zero_zero_real ) )
% 8.90/5.08 & ( P @ X ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4955_unity__coeff__ex,axiom,
% 8.90/5.08 ! [P: rat > $o,L: rat] :
% 8.90/5.08 ( ( ? [X: rat] : ( P @ ( times_times_rat @ L @ X ) ) )
% 8.90/5.08 = ( ? [X: rat] :
% 8.90/5.08 ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X @ zero_zero_rat ) )
% 8.90/5.08 & ( P @ X ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4956_unity__coeff__ex,axiom,
% 8.90/5.08 ! [P: nat > $o,L: nat] :
% 8.90/5.08 ( ( ? [X: nat] : ( P @ ( times_times_nat @ L @ X ) ) )
% 8.90/5.08 = ( ? [X: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X @ zero_zero_nat ) )
% 8.90/5.08 & ( P @ X ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4957_unity__coeff__ex,axiom,
% 8.90/5.08 ! [P: int > $o,L: int] :
% 8.90/5.08 ( ( ? [X: int] : ( P @ ( times_times_int @ L @ X ) ) )
% 8.90/5.08 = ( ? [X: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X @ zero_zero_int ) )
% 8.90/5.08 & ( P @ X ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4958_dvd__div__div__eq__mult,axiom,
% 8.90/5.08 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 8.90/5.08 ( ( A != zero_z3403309356797280102nteger )
% 8.90/5.08 => ( ( C != zero_z3403309356797280102nteger )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ C @ D )
% 8.90/5.08 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 8.90/5.08 = ( divide6298287555418463151nteger @ D @ C ) )
% 8.90/5.08 = ( ( times_3573771949741848930nteger @ B @ C )
% 8.90/5.08 = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4959_dvd__div__div__eq__mult,axiom,
% 8.90/5.08 ! [A: nat,C: nat,B: nat,D: nat] :
% 8.90/5.08 ( ( A != zero_zero_nat )
% 8.90/5.08 => ( ( C != zero_zero_nat )
% 8.90/5.08 => ( ( dvd_dvd_nat @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_nat @ C @ D )
% 8.90/5.08 => ( ( ( divide_divide_nat @ B @ A )
% 8.90/5.08 = ( divide_divide_nat @ D @ C ) )
% 8.90/5.08 = ( ( times_times_nat @ B @ C )
% 8.90/5.08 = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4960_dvd__div__div__eq__mult,axiom,
% 8.90/5.08 ! [A: int,C: int,B: int,D: int] :
% 8.90/5.08 ( ( A != zero_zero_int )
% 8.90/5.08 => ( ( C != zero_zero_int )
% 8.90/5.08 => ( ( dvd_dvd_int @ A @ B )
% 8.90/5.08 => ( ( dvd_dvd_int @ C @ D )
% 8.90/5.08 => ( ( ( divide_divide_int @ B @ A )
% 8.90/5.08 = ( divide_divide_int @ D @ C ) )
% 8.90/5.08 = ( ( times_times_int @ B @ C )
% 8.90/5.08 = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4961_dvd__div__iff__mult,axiom,
% 8.90/5.08 ! [C: code_integer,B: code_integer,A: code_integer] :
% 8.90/5.08 ( ( C != zero_z3403309356797280102nteger )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ C @ B )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4962_dvd__div__iff__mult,axiom,
% 8.90/5.08 ! [C: nat,B: nat,A: nat] :
% 8.90/5.08 ( ( C != zero_zero_nat )
% 8.90/5.08 => ( ( dvd_dvd_nat @ C @ B )
% 8.90/5.08 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 8.90/5.08 = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4963_dvd__div__iff__mult,axiom,
% 8.90/5.08 ! [C: int,B: int,A: int] :
% 8.90/5.08 ( ( C != zero_zero_int )
% 8.90/5.08 => ( ( dvd_dvd_int @ C @ B )
% 8.90/5.08 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
% 8.90/5.08 = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4964_div__dvd__iff__mult,axiom,
% 8.90/5.08 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.90/5.08 ( ( B != zero_z3403309356797280102nteger )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ B @ A )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4965_div__dvd__iff__mult,axiom,
% 8.90/5.08 ! [B: nat,A: nat,C: nat] :
% 8.90/5.08 ( ( B != zero_zero_nat )
% 8.90/5.08 => ( ( dvd_dvd_nat @ B @ A )
% 8.90/5.08 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 8.90/5.08 = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4966_div__dvd__iff__mult,axiom,
% 8.90/5.08 ! [B: int,A: int,C: int] :
% 8.90/5.08 ( ( B != zero_zero_int )
% 8.90/5.08 => ( ( dvd_dvd_int @ B @ A )
% 8.90/5.08 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 8.90/5.08 = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4967_dvd__div__eq__mult,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer,C: code_integer] :
% 8.90/5.08 ( ( A != zero_z3403309356797280102nteger )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ A @ B )
% 8.90/5.08 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 8.90/5.08 = C )
% 8.90/5.08 = ( B
% 8.90/5.08 = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4968_dvd__div__eq__mult,axiom,
% 8.90/5.08 ! [A: nat,B: nat,C: nat] :
% 8.90/5.08 ( ( A != zero_zero_nat )
% 8.90/5.08 => ( ( dvd_dvd_nat @ A @ B )
% 8.90/5.08 => ( ( ( divide_divide_nat @ B @ A )
% 8.90/5.08 = C )
% 8.90/5.08 = ( B
% 8.90/5.08 = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4969_dvd__div__eq__mult,axiom,
% 8.90/5.08 ! [A: int,B: int,C: int] :
% 8.90/5.08 ( ( A != zero_zero_int )
% 8.90/5.08 => ( ( dvd_dvd_int @ A @ B )
% 8.90/5.08 => ( ( ( divide_divide_int @ B @ A )
% 8.90/5.08 = C )
% 8.90/5.08 = ( B
% 8.90/5.08 = ( times_times_int @ C @ A ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4970_unit__div__eq__0__iff,axiom,
% 8.90/5.08 ! [B: code_integer,A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.90/5.08 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 8.90/5.08 = zero_z3403309356797280102nteger )
% 8.90/5.08 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4971_unit__div__eq__0__iff,axiom,
% 8.90/5.08 ! [B: nat,A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.90/5.08 => ( ( ( divide_divide_nat @ A @ B )
% 8.90/5.08 = zero_zero_nat )
% 8.90/5.08 = ( A = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4972_unit__div__eq__0__iff,axiom,
% 8.90/5.08 ! [B: int,A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.90/5.08 => ( ( ( divide_divide_int @ A @ B )
% 8.90/5.08 = zero_zero_int )
% 8.90/5.08 = ( A = zero_zero_int ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4973_even__numeral,axiom,
% 8.90/5.08 ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4974_even__numeral,axiom,
% 8.90/5.08 ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4975_even__numeral,axiom,
% 8.90/5.08 ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4976_inf__period_I3_J,axiom,
% 8.90/5.08 ! [D: code_integer,D4: code_integer,T: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ D @ D4 )
% 8.90/5.08 => ! [X3: code_integer,K4: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ T ) )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4977_inf__period_I3_J,axiom,
% 8.90/5.08 ! [D: real,D4: real,T: real] :
% 8.90/5.08 ( ( dvd_dvd_real @ D @ D4 )
% 8.90/5.08 => ! [X3: real,K4: real] :
% 8.90/5.08 ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ T ) )
% 8.90/5.08 = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4978_inf__period_I3_J,axiom,
% 8.90/5.08 ! [D: rat,D4: rat,T: rat] :
% 8.90/5.08 ( ( dvd_dvd_rat @ D @ D4 )
% 8.90/5.08 => ! [X3: rat,K4: rat] :
% 8.90/5.08 ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ T ) )
% 8.90/5.08 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4979_inf__period_I3_J,axiom,
% 8.90/5.08 ! [D: int,D4: int,T: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ D @ D4 )
% 8.90/5.08 => ! [X3: int,K4: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) )
% 8.90/5.08 = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4980_inf__period_I4_J,axiom,
% 8.90/5.08 ! [D: code_integer,D4: code_integer,T: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ D @ D4 )
% 8.90/5.08 => ! [X3: code_integer,K4: code_integer] :
% 8.90/5.08 ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ T ) ) )
% 8.90/5.08 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4981_inf__period_I4_J,axiom,
% 8.90/5.08 ! [D: real,D4: real,T: real] :
% 8.90/5.08 ( ( dvd_dvd_real @ D @ D4 )
% 8.90/5.08 => ! [X3: real,K4: real] :
% 8.90/5.08 ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ T ) ) )
% 8.90/5.08 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4982_inf__period_I4_J,axiom,
% 8.90/5.08 ! [D: rat,D4: rat,T: rat] :
% 8.90/5.08 ( ( dvd_dvd_rat @ D @ D4 )
% 8.90/5.08 => ! [X3: rat,K4: rat] :
% 8.90/5.08 ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ T ) ) )
% 8.90/5.08 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4983_inf__period_I4_J,axiom,
% 8.90/5.08 ! [D: int,D4: int,T: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ D @ D4 )
% 8.90/5.08 => ! [X3: int,K4: int] :
% 8.90/5.08 ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) ) )
% 8.90/5.08 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4984_is__unit__div__mult2__eq,axiom,
% 8.90/5.08 ! [B: code_integer,C: code_integer,A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 8.90/5.08 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 8.90/5.08 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4985_is__unit__div__mult2__eq,axiom,
% 8.90/5.08 ! [B: nat,C: nat,A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.90/5.08 => ( ( dvd_dvd_nat @ C @ one_one_nat )
% 8.90/5.08 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 8.90/5.08 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4986_is__unit__div__mult2__eq,axiom,
% 8.90/5.08 ! [B: int,C: int,A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.90/5.08 => ( ( dvd_dvd_int @ C @ one_one_int )
% 8.90/5.08 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 8.90/5.08 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4987_unit__div__mult__swap,axiom,
% 8.90/5.08 ! [C: code_integer,A: code_integer,B: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 8.90/5.08 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 8.90/5.08 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4988_unit__div__mult__swap,axiom,
% 8.90/5.08 ! [C: nat,A: nat,B: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 8.90/5.08 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 8.90/5.08 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4989_unit__div__mult__swap,axiom,
% 8.90/5.08 ! [C: int,A: int,B: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ C @ one_one_int )
% 8.90/5.08 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 8.90/5.08 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4990_unit__div__commute,axiom,
% 8.90/5.08 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.90/5.08 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 8.90/5.08 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4991_unit__div__commute,axiom,
% 8.90/5.08 ! [B: nat,A: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.90/5.08 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 8.90/5.08 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4992_unit__div__commute,axiom,
% 8.90/5.08 ! [B: int,A: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.90/5.08 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
% 8.90/5.08 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4993_div__mult__unit2,axiom,
% 8.90/5.08 ! [C: code_integer,B: code_integer,A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ B @ A )
% 8.90/5.08 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 8.90/5.08 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4994_div__mult__unit2,axiom,
% 8.90/5.08 ! [C: nat,B: nat,A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 8.90/5.08 => ( ( dvd_dvd_nat @ B @ A )
% 8.90/5.08 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 8.90/5.08 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4995_div__mult__unit2,axiom,
% 8.90/5.08 ! [C: int,B: int,A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ C @ one_one_int )
% 8.90/5.08 => ( ( dvd_dvd_int @ B @ A )
% 8.90/5.08 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 8.90/5.08 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4996_unit__eq__div2,axiom,
% 8.90/5.08 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.90/5.08 => ( ( A
% 8.90/5.08 = ( divide6298287555418463151nteger @ C @ B ) )
% 8.90/5.08 = ( ( times_3573771949741848930nteger @ A @ B )
% 8.90/5.08 = C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4997_unit__eq__div2,axiom,
% 8.90/5.08 ! [B: nat,A: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.90/5.08 => ( ( A
% 8.90/5.08 = ( divide_divide_nat @ C @ B ) )
% 8.90/5.08 = ( ( times_times_nat @ A @ B )
% 8.90/5.08 = C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4998_unit__eq__div2,axiom,
% 8.90/5.08 ! [B: int,A: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.90/5.08 => ( ( A
% 8.90/5.08 = ( divide_divide_int @ C @ B ) )
% 8.90/5.08 = ( ( times_times_int @ A @ B )
% 8.90/5.08 = C ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_4999_unit__eq__div1,axiom,
% 8.90/5.08 ! [B: code_integer,A: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.90/5.08 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 8.90/5.08 = C )
% 8.90/5.08 = ( A
% 8.90/5.08 = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5000_unit__eq__div1,axiom,
% 8.90/5.08 ! [B: nat,A: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.90/5.08 => ( ( ( divide_divide_nat @ A @ B )
% 8.90/5.08 = C )
% 8.90/5.08 = ( A
% 8.90/5.08 = ( times_times_nat @ C @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5001_unit__eq__div1,axiom,
% 8.90/5.08 ! [B: int,A: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.90/5.08 => ( ( ( divide_divide_int @ A @ B )
% 8.90/5.08 = C )
% 8.90/5.08 = ( A
% 8.90/5.08 = ( times_times_int @ C @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5002_is__unit__power__iff,axiom,
% 8.90/5.08 ! [A: code_integer,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
% 8.90/5.08 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.90/5.08 | ( N = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5003_is__unit__power__iff,axiom,
% 8.90/5.08 ! [A: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
% 8.90/5.08 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.90/5.08 | ( N = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5004_is__unit__power__iff,axiom,
% 8.90/5.08 ! [A: int,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
% 8.90/5.08 = ( ( dvd_dvd_int @ A @ one_one_int )
% 8.90/5.08 | ( N = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5005_unit__imp__mod__eq__0,axiom,
% 8.90/5.08 ! [B: nat,A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.90/5.08 => ( ( modulo_modulo_nat @ A @ B )
% 8.90/5.08 = zero_zero_nat ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5006_unit__imp__mod__eq__0,axiom,
% 8.90/5.08 ! [B: int,A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ B @ one_one_int )
% 8.90/5.08 => ( ( modulo_modulo_int @ A @ B )
% 8.90/5.08 = zero_zero_int ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5007_unit__imp__mod__eq__0,axiom,
% 8.90/5.08 ! [B: code_integer,A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.90/5.08 => ( ( modulo364778990260209775nteger @ A @ B )
% 8.90/5.08 = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5008_dvd__imp__le,axiom,
% 8.90/5.08 ! [K: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ K @ N )
% 8.90/5.08 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.08 => ( ord_less_eq_nat @ K @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5009_nat__mult__dvd__cancel1,axiom,
% 8.90/5.08 ! [K: nat,M: nat,N: nat] :
% 8.90/5.08 ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.90/5.08 => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.90/5.08 = ( dvd_dvd_nat @ M @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5010_dvd__mult__cancel,axiom,
% 8.90/5.08 ! [K: nat,M: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 8.90/5.08 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.90/5.08 => ( dvd_dvd_nat @ M @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5011_zdvd__imp__le,axiom,
% 8.90/5.08 ! [Z: int,N: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ Z @ N )
% 8.90/5.08 => ( ( ord_less_int @ zero_zero_int @ N )
% 8.90/5.08 => ( ord_less_eq_int @ Z @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5012_mod__greater__zero__iff__not__dvd,axiom,
% 8.90/5.08 ! [M: nat,N: nat] :
% 8.90/5.08 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
% 8.90/5.08 = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5013_mod__eq__dvd__iff__nat,axiom,
% 8.90/5.08 ! [N: nat,M: nat,Q3: nat] :
% 8.90/5.08 ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.08 => ( ( ( modulo_modulo_nat @ M @ Q3 )
% 8.90/5.08 = ( modulo_modulo_nat @ N @ Q3 ) )
% 8.90/5.08 = ( dvd_dvd_nat @ Q3 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5014_prod__decode__aux_Ocases,axiom,
% 8.90/5.08 ! [X2: product_prod_nat_nat] :
% 8.90/5.08 ~ ! [K3: nat,M4: nat] :
% 8.90/5.08 ( X2
% 8.90/5.08 != ( product_Pair_nat_nat @ K3 @ M4 ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5015_even__zero,axiom,
% 8.90/5.08 dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5016_even__zero,axiom,
% 8.90/5.08 dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5017_even__zero,axiom,
% 8.90/5.08 dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5018_is__unitE,axiom,
% 8.90/5.08 ! [A: code_integer,C: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 8.90/5.08 => ~ ( ( A != zero_z3403309356797280102nteger )
% 8.90/5.08 => ! [B4: code_integer] :
% 8.90/5.08 ( ( B4 != zero_z3403309356797280102nteger )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ B4 @ one_one_Code_integer )
% 8.90/5.08 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 8.90/5.08 = B4 )
% 8.90/5.08 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B4 )
% 8.90/5.08 = A )
% 8.90/5.08 => ( ( ( times_3573771949741848930nteger @ A @ B4 )
% 8.90/5.08 = one_one_Code_integer )
% 8.90/5.08 => ( ( divide6298287555418463151nteger @ C @ A )
% 8.90/5.08 != ( times_3573771949741848930nteger @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5019_is__unitE,axiom,
% 8.90/5.08 ! [A: nat,C: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 8.90/5.08 => ~ ( ( A != zero_zero_nat )
% 8.90/5.08 => ! [B4: nat] :
% 8.90/5.08 ( ( B4 != zero_zero_nat )
% 8.90/5.08 => ( ( dvd_dvd_nat @ B4 @ one_one_nat )
% 8.90/5.08 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 8.90/5.08 = B4 )
% 8.90/5.08 => ( ( ( divide_divide_nat @ one_one_nat @ B4 )
% 8.90/5.08 = A )
% 8.90/5.08 => ( ( ( times_times_nat @ A @ B4 )
% 8.90/5.08 = one_one_nat )
% 8.90/5.08 => ( ( divide_divide_nat @ C @ A )
% 8.90/5.08 != ( times_times_nat @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5020_is__unitE,axiom,
% 8.90/5.08 ! [A: int,C: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ A @ one_one_int )
% 8.90/5.08 => ~ ( ( A != zero_zero_int )
% 8.90/5.08 => ! [B4: int] :
% 8.90/5.08 ( ( B4 != zero_zero_int )
% 8.90/5.08 => ( ( dvd_dvd_int @ B4 @ one_one_int )
% 8.90/5.08 => ( ( ( divide_divide_int @ one_one_int @ A )
% 8.90/5.08 = B4 )
% 8.90/5.08 => ( ( ( divide_divide_int @ one_one_int @ B4 )
% 8.90/5.08 = A )
% 8.90/5.08 => ( ( ( times_times_int @ A @ B4 )
% 8.90/5.08 = one_one_int )
% 8.90/5.08 => ( ( divide_divide_int @ C @ A )
% 8.90/5.08 != ( times_times_int @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5021_is__unit__div__mult__cancel__left,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer] :
% 8.90/5.08 ( ( A != zero_z3403309356797280102nteger )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.90/5.08 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
% 8.90/5.08 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5022_is__unit__div__mult__cancel__left,axiom,
% 8.90/5.08 ! [A: nat,B: nat] :
% 8.90/5.08 ( ( A != zero_zero_nat )
% 8.90/5.08 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.90/5.08 => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
% 8.90/5.08 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5023_is__unit__div__mult__cancel__left,axiom,
% 8.90/5.08 ! [A: int,B: int] :
% 8.90/5.08 ( ( A != zero_zero_int )
% 8.90/5.08 => ( ( dvd_dvd_int @ B @ one_one_int )
% 8.90/5.08 => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
% 8.90/5.08 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5024_is__unit__div__mult__cancel__right,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer] :
% 8.90/5.08 ( ( A != zero_z3403309356797280102nteger )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 8.90/5.08 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
% 8.90/5.08 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5025_is__unit__div__mult__cancel__right,axiom,
% 8.90/5.08 ! [A: nat,B: nat] :
% 8.90/5.08 ( ( A != zero_zero_nat )
% 8.90/5.08 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 8.90/5.08 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
% 8.90/5.08 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5026_is__unit__div__mult__cancel__right,axiom,
% 8.90/5.08 ! [A: int,B: int] :
% 8.90/5.08 ( ( A != zero_zero_int )
% 8.90/5.08 => ( ( dvd_dvd_int @ B @ one_one_int )
% 8.90/5.08 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
% 8.90/5.08 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5027_evenE,axiom,
% 8.90/5.08 ! [A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ~ ! [B4: code_integer] :
% 8.90/5.08 ( A
% 8.90/5.08 != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5028_evenE,axiom,
% 8.90/5.08 ! [A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ~ ! [B4: nat] :
% 8.90/5.08 ( A
% 8.90/5.08 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5029_evenE,axiom,
% 8.90/5.08 ! [A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ~ ! [B4: int] :
% 8.90/5.08 ( A
% 8.90/5.08 != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5030_odd__one,axiom,
% 8.90/5.08 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5031_odd__one,axiom,
% 8.90/5.08 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5032_odd__one,axiom,
% 8.90/5.08 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5033_odd__even__add,axiom,
% 8.90/5.08 ! [A: code_integer,B: code_integer] :
% 8.90/5.08 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5034_odd__even__add,axiom,
% 8.90/5.08 ! [A: nat,B: nat] :
% 8.90/5.08 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 8.90/5.08 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5035_odd__even__add,axiom,
% 8.90/5.08 ! [A: int,B: int] :
% 8.90/5.08 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
% 8.90/5.08 => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5036_bit__eq__rec,axiom,
% 8.90/5.08 ( ( ^ [Y6: code_integer,Z5: code_integer] : ( Y6 = Z5 ) )
% 8.90/5.08 = ( ^ [A3: code_integer,B2: code_integer] :
% 8.90/5.08 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
% 8.90/5.08 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) )
% 8.90/5.08 & ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.08 = ( divide6298287555418463151nteger @ B2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5037_bit__eq__rec,axiom,
% 8.90/5.08 ( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 8.90/5.08 = ( ^ [A3: nat,B2: nat] :
% 8.90/5.08 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
% 8.90/5.08 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
% 8.90/5.08 & ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.08 = ( divide_divide_nat @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5038_bit__eq__rec,axiom,
% 8.90/5.08 ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 8.90/5.08 = ( ^ [A3: int,B2: int] :
% 8.90/5.08 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
% 8.90/5.08 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
% 8.90/5.08 & ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.08 = ( divide_divide_int @ B2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5039_dvd__power__iff,axiom,
% 8.90/5.08 ! [X2: code_integer,M: nat,N: nat] :
% 8.90/5.08 ( ( X2 != zero_z3403309356797280102nteger )
% 8.90/5.08 => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X2 @ M ) @ ( power_8256067586552552935nteger @ X2 @ N ) )
% 8.90/5.08 = ( ( dvd_dvd_Code_integer @ X2 @ one_one_Code_integer )
% 8.90/5.08 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5040_dvd__power__iff,axiom,
% 8.90/5.08 ! [X2: nat,M: nat,N: nat] :
% 8.90/5.08 ( ( X2 != zero_zero_nat )
% 8.90/5.08 => ( ( dvd_dvd_nat @ ( power_power_nat @ X2 @ M ) @ ( power_power_nat @ X2 @ N ) )
% 8.90/5.08 = ( ( dvd_dvd_nat @ X2 @ one_one_nat )
% 8.90/5.08 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5041_dvd__power__iff,axiom,
% 8.90/5.08 ! [X2: int,M: nat,N: nat] :
% 8.90/5.08 ( ( X2 != zero_zero_int )
% 8.90/5.08 => ( ( dvd_dvd_int @ ( power_power_int @ X2 @ M ) @ ( power_power_int @ X2 @ N ) )
% 8.90/5.08 = ( ( dvd_dvd_int @ X2 @ one_one_int )
% 8.90/5.08 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5042_odd__numeral,axiom,
% 8.90/5.08 ! [N: num] :
% 8.90/5.08 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5043_odd__numeral,axiom,
% 8.90/5.08 ! [N: num] :
% 8.90/5.08 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5044_odd__numeral,axiom,
% 8.90/5.08 ! [N: num] :
% 8.90/5.08 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5045_dvd__power,axiom,
% 8.90/5.08 ! [N: nat,X2: code_integer] :
% 8.90/5.08 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.08 | ( X2 = one_one_Code_integer ) )
% 8.90/5.08 => ( dvd_dvd_Code_integer @ X2 @ ( power_8256067586552552935nteger @ X2 @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5046_dvd__power,axiom,
% 8.90/5.08 ! [N: nat,X2: rat] :
% 8.90/5.08 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.08 | ( X2 = one_one_rat ) )
% 8.90/5.08 => ( dvd_dvd_rat @ X2 @ ( power_power_rat @ X2 @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5047_dvd__power,axiom,
% 8.90/5.08 ! [N: nat,X2: nat] :
% 8.90/5.08 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.08 | ( X2 = one_one_nat ) )
% 8.90/5.08 => ( dvd_dvd_nat @ X2 @ ( power_power_nat @ X2 @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5048_dvd__power,axiom,
% 8.90/5.08 ! [N: nat,X2: real] :
% 8.90/5.08 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.08 | ( X2 = one_one_real ) )
% 8.90/5.08 => ( dvd_dvd_real @ X2 @ ( power_power_real @ X2 @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5049_dvd__power,axiom,
% 8.90/5.08 ! [N: nat,X2: int] :
% 8.90/5.08 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.08 | ( X2 = one_one_int ) )
% 8.90/5.08 => ( dvd_dvd_int @ X2 @ ( power_power_int @ X2 @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5050_dvd__power,axiom,
% 8.90/5.08 ! [N: nat,X2: complex] :
% 8.90/5.08 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.08 | ( X2 = one_one_complex ) )
% 8.90/5.08 => ( dvd_dvd_complex @ X2 @ ( power_power_complex @ X2 @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5051_even__even__mod__4__iff,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5052_dvd__mult__cancel2,axiom,
% 8.90/5.08 ! [M: nat,N: nat] :
% 8.90/5.08 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.90/5.08 => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
% 8.90/5.08 = ( N = one_one_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5053_dvd__mult__cancel1,axiom,
% 8.90/5.08 ! [M: nat,N: nat] :
% 8.90/5.08 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.90/5.08 => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
% 8.90/5.08 = ( N = one_one_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5054_odd__numeral__BitM,axiom,
% 8.90/5.08 ! [W: num] :
% 8.90/5.08 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5055_odd__numeral__BitM,axiom,
% 8.90/5.08 ! [W: num] :
% 8.90/5.08 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5056_odd__numeral__BitM,axiom,
% 8.90/5.08 ! [W: num] :
% 8.90/5.08 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5057_dvd__minus__add,axiom,
% 8.90/5.08 ! [Q3: nat,N: nat,R2: nat,M: nat] :
% 8.90/5.08 ( ( ord_less_eq_nat @ Q3 @ N )
% 8.90/5.08 => ( ( ord_less_eq_nat @ Q3 @ ( times_times_nat @ R2 @ M ) )
% 8.90/5.08 => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q3 ) )
% 8.90/5.08 = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q3 ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5058_power__dvd__imp__le,axiom,
% 8.90/5.08 ! [I2: nat,M: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N ) )
% 8.90/5.08 => ( ( ord_less_nat @ one_one_nat @ I2 )
% 8.90/5.08 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5059_mod__nat__eqI,axiom,
% 8.90/5.08 ! [R2: nat,N: nat,M: nat] :
% 8.90/5.08 ( ( ord_less_nat @ R2 @ N )
% 8.90/5.08 => ( ( ord_less_eq_nat @ R2 @ M )
% 8.90/5.08 => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R2 ) )
% 8.90/5.08 => ( ( modulo_modulo_nat @ M @ N )
% 8.90/5.08 = R2 ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5060_mod__int__pos__iff,axiom,
% 8.90/5.08 ! [K: int,L: int] :
% 8.90/5.08 ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
% 8.90/5.08 = ( ( dvd_dvd_int @ L @ K )
% 8.90/5.08 | ( ( L = zero_zero_int )
% 8.90/5.08 & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 8.90/5.08 | ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5061_bset_I9_J,axiom,
% 8.90/5.08 ! [D: int,D4: int,B3: set_int,T: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ D @ D4 )
% 8.90/5.08 => ! [X3: int] :
% 8.90/5.08 ( ! [Xa3: int] :
% 8.90/5.08 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.90/5.08 => ! [Xb3: int] :
% 8.90/5.08 ( ( member_int @ Xb3 @ B3 )
% 8.90/5.08 => ( X3
% 8.90/5.08 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.90/5.08 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) )
% 8.90/5.08 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5062_bset_I10_J,axiom,
% 8.90/5.08 ! [D: int,D4: int,B3: set_int,T: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ D @ D4 )
% 8.90/5.08 => ! [X3: int] :
% 8.90/5.08 ( ! [Xa3: int] :
% 8.90/5.08 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.90/5.08 => ! [Xb3: int] :
% 8.90/5.08 ( ( member_int @ Xb3 @ B3 )
% 8.90/5.08 => ( X3
% 8.90/5.08 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 8.90/5.08 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) )
% 8.90/5.08 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5063_aset_I9_J,axiom,
% 8.90/5.08 ! [D: int,D4: int,A2: set_int,T: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ D @ D4 )
% 8.90/5.08 => ! [X3: int] :
% 8.90/5.08 ( ! [Xa3: int] :
% 8.90/5.08 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.90/5.08 => ! [Xb3: int] :
% 8.90/5.08 ( ( member_int @ Xb3 @ A2 )
% 8.90/5.08 => ( X3
% 8.90/5.08 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.90/5.08 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) )
% 8.90/5.08 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5064_aset_I10_J,axiom,
% 8.90/5.08 ! [D: int,D4: int,A2: set_int,T: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ D @ D4 )
% 8.90/5.08 => ! [X3: int] :
% 8.90/5.08 ( ! [Xa3: int] :
% 8.90/5.08 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 8.90/5.08 => ! [Xb3: int] :
% 8.90/5.08 ( ( member_int @ Xb3 @ A2 )
% 8.90/5.08 => ( X3
% 8.90/5.08 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 8.90/5.08 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) )
% 8.90/5.08 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5065_even__two__times__div__two,axiom,
% 8.90/5.08 ! [A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 8.90/5.08 = A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5066_even__two__times__div__two,axiom,
% 8.90/5.08 ! [A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.08 = A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5067_even__two__times__div__two,axiom,
% 8.90/5.08 ! [A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 8.90/5.08 = A ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5068_even__iff__mod__2__eq__zero,axiom,
% 8.90/5.08 ! [A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.08 = zero_zero_nat ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5069_even__iff__mod__2__eq__zero,axiom,
% 8.90/5.08 ! [A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.08 = zero_zero_int ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5070_even__iff__mod__2__eq__zero,axiom,
% 8.90/5.08 ! [A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.08 = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5071_odd__iff__mod__2__eq__one,axiom,
% 8.90/5.08 ! [A: nat] :
% 8.90/5.08 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 8.90/5.08 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.08 = one_one_nat ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5072_odd__iff__mod__2__eq__one,axiom,
% 8.90/5.08 ! [A: int] :
% 8.90/5.08 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 8.90/5.08 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.08 = one_one_int ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5073_odd__iff__mod__2__eq__one,axiom,
% 8.90/5.08 ! [A: code_integer] :
% 8.90/5.08 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 8.90/5.08 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.08 = one_one_Code_integer ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5074_power__mono__odd,axiom,
% 8.90/5.08 ! [N: nat,A: real,B: real] :
% 8.90/5.08 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 => ( ( ord_less_eq_real @ A @ B )
% 8.90/5.08 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5075_power__mono__odd,axiom,
% 8.90/5.08 ! [N: nat,A: rat,B: rat] :
% 8.90/5.08 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 => ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.08 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5076_power__mono__odd,axiom,
% 8.90/5.08 ! [N: nat,A: int,B: int] :
% 8.90/5.08 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 => ( ( ord_less_eq_int @ A @ B )
% 8.90/5.08 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5077_odd__pos,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5078_dvd__power__iff__le,axiom,
% 8.90/5.08 ! [K: nat,M: nat,N: nat] :
% 8.90/5.08 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 8.90/5.08 => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
% 8.90/5.08 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5079_even__unset__bit__iff,axiom,
% 8.90/5.08 ! [M: nat,A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 8.90/5.08 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 | ( M = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5080_even__unset__bit__iff,axiom,
% 8.90/5.08 ! [M: nat,A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 8.90/5.08 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 | ( M = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5081_even__unset__bit__iff,axiom,
% 8.90/5.08 ! [M: nat,A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 8.90/5.08 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 | ( M = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5082_even__set__bit__iff,axiom,
% 8.90/5.08 ! [M: nat,A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 8.90/5.08 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 & ( M != zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5083_even__set__bit__iff,axiom,
% 8.90/5.08 ! [M: nat,A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 8.90/5.08 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 & ( M != zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5084_even__set__bit__iff,axiom,
% 8.90/5.08 ! [M: nat,A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 8.90/5.08 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 & ( M != zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5085_even__flip__bit__iff,axiom,
% 8.90/5.08 ! [M: nat,A: code_integer] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 8.90/5.08 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 != ( M = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5086_even__flip__bit__iff,axiom,
% 8.90/5.08 ! [M: nat,A: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 8.90/5.08 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 != ( M = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5087_even__flip__bit__iff,axiom,
% 8.90/5.08 ! [M: nat,A: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 8.90/5.08 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 != ( M = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5088_even__diff__iff,axiom,
% 8.90/5.08 ! [K: int,L: int] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
% 8.90/5.08 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5089_oddE,axiom,
% 8.90/5.08 ! [A: code_integer] :
% 8.90/5.08 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ~ ! [B4: code_integer] :
% 8.90/5.08 ( A
% 8.90/5.08 != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) @ one_one_Code_integer ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5090_oddE,axiom,
% 8.90/5.08 ! [A: nat] :
% 8.90/5.08 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ~ ! [B4: nat] :
% 8.90/5.08 ( A
% 8.90/5.08 != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) @ one_one_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5091_oddE,axiom,
% 8.90/5.08 ! [A: int] :
% 8.90/5.08 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ~ ! [B4: int] :
% 8.90/5.08 ( A
% 8.90/5.08 != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) @ one_one_int ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5092_parity__cases,axiom,
% 8.90/5.08 ! [A: nat] :
% 8.90/5.08 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.08 != zero_zero_nat ) )
% 8.90/5.08 => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.08 != one_one_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5093_parity__cases,axiom,
% 8.90/5.08 ! [A: int] :
% 8.90/5.08 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.08 != zero_zero_int ) )
% 8.90/5.08 => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.08 != one_one_int ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5094_parity__cases,axiom,
% 8.90/5.08 ! [A: code_integer] :
% 8.90/5.08 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.08 != zero_z3403309356797280102nteger ) )
% 8.90/5.08 => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.08 != one_one_Code_integer ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5095_mod2__eq__if,axiom,
% 8.90/5.08 ! [A: nat] :
% 8.90/5.08 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.08 = zero_zero_nat ) )
% 8.90/5.08 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.08 = one_one_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5096_mod2__eq__if,axiom,
% 8.90/5.08 ! [A: int] :
% 8.90/5.08 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.08 = zero_zero_int ) )
% 8.90/5.08 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.08 = one_one_int ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5097_mod2__eq__if,axiom,
% 8.90/5.08 ! [A: code_integer] :
% 8.90/5.08 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.08 = zero_z3403309356797280102nteger ) )
% 8.90/5.08 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 8.90/5.08 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.08 = one_one_Code_integer ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5098_zero__le__even__power,axiom,
% 8.90/5.08 ! [N: nat,A: real] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5099_zero__le__even__power,axiom,
% 8.90/5.08 ! [N: nat,A: rat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5100_zero__le__even__power,axiom,
% 8.90/5.08 ! [N: nat,A: int] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5101_zero__le__odd__power,axiom,
% 8.90/5.08 ! [N: nat,A: real] :
% 8.90/5.08 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 8.90/5.08 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5102_zero__le__odd__power,axiom,
% 8.90/5.08 ! [N: nat,A: rat] :
% 8.90/5.08 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 8.90/5.08 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5103_zero__le__odd__power,axiom,
% 8.90/5.08 ! [N: nat,A: int] :
% 8.90/5.08 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 8.90/5.08 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5104_zero__le__power__eq,axiom,
% 8.90/5.08 ! [A: real,N: nat] :
% 8.90/5.08 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 8.90/5.08 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5105_zero__le__power__eq,axiom,
% 8.90/5.08 ! [A: rat,N: nat] :
% 8.90/5.08 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 8.90/5.08 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5106_zero__le__power__eq,axiom,
% 8.90/5.08 ! [A: int,N: nat] :
% 8.90/5.08 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 8.90/5.08 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5107_list__decode_Ocases,axiom,
% 8.90/5.08 ! [X2: nat] :
% 8.90/5.08 ( ( X2 != zero_zero_nat )
% 8.90/5.08 => ~ ! [N2: nat] :
% 8.90/5.08 ( X2
% 8.90/5.08 != ( suc @ N2 ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5108_zero__less__power__eq,axiom,
% 8.90/5.08 ! [A: real,N: nat] :
% 8.90/5.08 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 8.90/5.08 = ( ( N = zero_zero_nat )
% 8.90/5.08 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 & ( A != zero_zero_real ) )
% 8.90/5.08 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5109_zero__less__power__eq,axiom,
% 8.90/5.08 ! [A: rat,N: nat] :
% 8.90/5.08 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 8.90/5.08 = ( ( N = zero_zero_nat )
% 8.90/5.08 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 & ( A != zero_zero_rat ) )
% 8.90/5.08 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5110_zero__less__power__eq,axiom,
% 8.90/5.08 ! [A: int,N: nat] :
% 8.90/5.08 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 8.90/5.08 = ( ( N = zero_zero_nat )
% 8.90/5.08 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 & ( A != zero_zero_int ) )
% 8.90/5.08 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5111_Euclid__induct,axiom,
% 8.90/5.08 ! [P: nat > nat > $o,A: nat,B: nat] :
% 8.90/5.08 ( ! [A4: nat,B4: nat] :
% 8.90/5.08 ( ( P @ A4 @ B4 )
% 8.90/5.08 = ( P @ B4 @ A4 ) )
% 8.90/5.08 => ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
% 8.90/5.08 => ( ! [A4: nat,B4: nat] :
% 8.90/5.08 ( ( P @ A4 @ B4 )
% 8.90/5.08 => ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
% 8.90/5.08 => ( P @ A @ B ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5112_even__mask__div__iff_H,axiom,
% 8.90/5.08 ! [M: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 8.90/5.08 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5113_even__mask__div__iff_H,axiom,
% 8.90/5.08 ! [M: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.90/5.08 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5114_even__mask__div__iff_H,axiom,
% 8.90/5.08 ! [M: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 8.90/5.08 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5115_power__le__zero__eq,axiom,
% 8.90/5.08 ! [A: real,N: nat] :
% 8.90/5.08 ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 8.90/5.08 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.08 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 8.90/5.08 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 & ( A = zero_zero_real ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5116_power__le__zero__eq,axiom,
% 8.90/5.08 ! [A: rat,N: nat] :
% 8.90/5.08 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 8.90/5.08 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.08 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 8.90/5.08 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 & ( A = zero_zero_rat ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5117_power__le__zero__eq,axiom,
% 8.90/5.08 ! [A: int,N: nat] :
% 8.90/5.08 ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 8.90/5.08 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.08 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 8.90/5.08 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 & ( A = zero_zero_int ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5118_even__mod__4__div__2,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.90/5.08 = ( suc @ zero_zero_nat ) )
% 8.90/5.08 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5119_even__mask__div__iff,axiom,
% 8.90/5.08 ! [M: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 8.90/5.08 = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 = zero_z3403309356797280102nteger )
% 8.90/5.08 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5120_even__mask__div__iff,axiom,
% 8.90/5.08 ! [M: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.90/5.08 = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 = zero_zero_nat )
% 8.90/5.08 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5121_even__mask__div__iff,axiom,
% 8.90/5.08 ! [M: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 8.90/5.08 = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 = zero_zero_int )
% 8.90/5.08 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5122_odd__mod__4__div__2,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.90/5.08 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 8.90/5.08 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5123_even__mult__exp__div__exp__iff,axiom,
% 8.90/5.08 ! [A: code_integer,M: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 8.90/5.08 = ( ( ord_less_nat @ N @ M )
% 8.90/5.08 | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 = zero_z3403309356797280102nteger )
% 8.90/5.08 | ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.08 & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5124_even__mult__exp__div__exp__iff,axiom,
% 8.90/5.08 ! [A: nat,M: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.90/5.08 = ( ( ord_less_nat @ N @ M )
% 8.90/5.08 | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 = zero_zero_nat )
% 8.90/5.08 | ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.08 & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5125_even__mult__exp__div__exp__iff,axiom,
% 8.90/5.08 ! [A: int,M: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 8.90/5.08 = ( ( ord_less_nat @ N @ M )
% 8.90/5.08 | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 8.90/5.08 = zero_zero_int )
% 8.90/5.08 | ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.08 & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5126_infinite__growing,axiom,
% 8.90/5.08 ! [X8: set_real] :
% 8.90/5.08 ( ( X8 != bot_bot_set_real )
% 8.90/5.08 => ( ! [X5: real] :
% 8.90/5.08 ( ( member_real @ X5 @ X8 )
% 8.90/5.08 => ? [Xa: real] :
% 8.90/5.08 ( ( member_real @ Xa @ X8 )
% 8.90/5.08 & ( ord_less_real @ X5 @ Xa ) ) )
% 8.90/5.08 => ~ ( finite_finite_real @ X8 ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5127_infinite__growing,axiom,
% 8.90/5.08 ! [X8: set_rat] :
% 8.90/5.08 ( ( X8 != bot_bot_set_rat )
% 8.90/5.08 => ( ! [X5: rat] :
% 8.90/5.08 ( ( member_rat @ X5 @ X8 )
% 8.90/5.08 => ? [Xa: rat] :
% 8.90/5.08 ( ( member_rat @ Xa @ X8 )
% 8.90/5.08 & ( ord_less_rat @ X5 @ Xa ) ) )
% 8.90/5.08 => ~ ( finite_finite_rat @ X8 ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5128_infinite__growing,axiom,
% 8.90/5.08 ! [X8: set_num] :
% 8.90/5.08 ( ( X8 != bot_bot_set_num )
% 8.90/5.08 => ( ! [X5: num] :
% 8.90/5.08 ( ( member_num @ X5 @ X8 )
% 8.90/5.08 => ? [Xa: num] :
% 8.90/5.08 ( ( member_num @ Xa @ X8 )
% 8.90/5.08 & ( ord_less_num @ X5 @ Xa ) ) )
% 8.90/5.08 => ~ ( finite_finite_num @ X8 ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5129_infinite__growing,axiom,
% 8.90/5.08 ! [X8: set_nat] :
% 8.90/5.08 ( ( X8 != bot_bot_set_nat )
% 8.90/5.08 => ( ! [X5: nat] :
% 8.90/5.08 ( ( member_nat @ X5 @ X8 )
% 8.90/5.08 => ? [Xa: nat] :
% 8.90/5.08 ( ( member_nat @ Xa @ X8 )
% 8.90/5.08 & ( ord_less_nat @ X5 @ Xa ) ) )
% 8.90/5.08 => ~ ( finite_finite_nat @ X8 ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5130_infinite__growing,axiom,
% 8.90/5.08 ! [X8: set_int] :
% 8.90/5.08 ( ( X8 != bot_bot_set_int )
% 8.90/5.08 => ( ! [X5: int] :
% 8.90/5.08 ( ( member_int @ X5 @ X8 )
% 8.90/5.08 => ? [Xa: int] :
% 8.90/5.08 ( ( member_int @ Xa @ X8 )
% 8.90/5.08 & ( ord_less_int @ X5 @ Xa ) ) )
% 8.90/5.08 => ~ ( finite_finite_int @ X8 ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5131_ex__min__if__finite,axiom,
% 8.90/5.08 ! [S3: set_real] :
% 8.90/5.08 ( ( finite_finite_real @ S3 )
% 8.90/5.08 => ( ( S3 != bot_bot_set_real )
% 8.90/5.08 => ? [X5: real] :
% 8.90/5.08 ( ( member_real @ X5 @ S3 )
% 8.90/5.08 & ~ ? [Xa: real] :
% 8.90/5.08 ( ( member_real @ Xa @ S3 )
% 8.90/5.08 & ( ord_less_real @ Xa @ X5 ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5132_ex__min__if__finite,axiom,
% 8.90/5.08 ! [S3: set_rat] :
% 8.90/5.08 ( ( finite_finite_rat @ S3 )
% 8.90/5.08 => ( ( S3 != bot_bot_set_rat )
% 8.90/5.08 => ? [X5: rat] :
% 8.90/5.08 ( ( member_rat @ X5 @ S3 )
% 8.90/5.08 & ~ ? [Xa: rat] :
% 8.90/5.08 ( ( member_rat @ Xa @ S3 )
% 8.90/5.08 & ( ord_less_rat @ Xa @ X5 ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5133_ex__min__if__finite,axiom,
% 8.90/5.08 ! [S3: set_num] :
% 8.90/5.08 ( ( finite_finite_num @ S3 )
% 8.90/5.08 => ( ( S3 != bot_bot_set_num )
% 8.90/5.08 => ? [X5: num] :
% 8.90/5.08 ( ( member_num @ X5 @ S3 )
% 8.90/5.08 & ~ ? [Xa: num] :
% 8.90/5.08 ( ( member_num @ Xa @ S3 )
% 8.90/5.08 & ( ord_less_num @ Xa @ X5 ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5134_ex__min__if__finite,axiom,
% 8.90/5.08 ! [S3: set_nat] :
% 8.90/5.08 ( ( finite_finite_nat @ S3 )
% 8.90/5.08 => ( ( S3 != bot_bot_set_nat )
% 8.90/5.08 => ? [X5: nat] :
% 8.90/5.08 ( ( member_nat @ X5 @ S3 )
% 8.90/5.08 & ~ ? [Xa: nat] :
% 8.90/5.08 ( ( member_nat @ Xa @ S3 )
% 8.90/5.08 & ( ord_less_nat @ Xa @ X5 ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5135_ex__min__if__finite,axiom,
% 8.90/5.08 ! [S3: set_int] :
% 8.90/5.08 ( ( finite_finite_int @ S3 )
% 8.90/5.08 => ( ( S3 != bot_bot_set_int )
% 8.90/5.08 => ? [X5: int] :
% 8.90/5.08 ( ( member_int @ X5 @ S3 )
% 8.90/5.08 & ~ ? [Xa: int] :
% 8.90/5.08 ( ( member_int @ Xa @ S3 )
% 8.90/5.08 & ( ord_less_int @ Xa @ X5 ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5136_triangle__def,axiom,
% 8.90/5.08 ( nat_triangle
% 8.90/5.08 = ( ^ [N3: nat] : ( divide_divide_nat @ ( times_times_nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5137_vebt__buildup_Oelims,axiom,
% 8.90/5.08 ! [X2: nat,Y3: vEBT_VEBT] :
% 8.90/5.08 ( ( ( vEBT_vebt_buildup @ X2 )
% 8.90/5.08 = Y3 )
% 8.90/5.08 => ( ( ( X2 = zero_zero_nat )
% 8.90/5.08 => ( Y3
% 8.90/5.08 != ( vEBT_Leaf @ $false @ $false ) ) )
% 8.90/5.08 => ( ( ( X2
% 8.90/5.08 = ( suc @ zero_zero_nat ) )
% 8.90/5.08 => ( Y3
% 8.90/5.08 != ( vEBT_Leaf @ $false @ $false ) ) )
% 8.90/5.08 => ~ ! [Va3: nat] :
% 8.90/5.08 ( ( X2
% 8.90/5.08 = ( suc @ ( suc @ Va3 ) ) )
% 8.90/5.08 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 8.90/5.08 => ( Y3
% 8.90/5.08 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 8.90/5.08 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 8.90/5.08 => ( Y3
% 8.90/5.08 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5138_divmod__algorithm__code_I6_J,axiom,
% 8.90/5.08 ! [M: num,N: num] :
% 8.90/5.08 ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.90/5.08 = ( produc4245557441103728435nt_int
% 8.90/5.08 @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
% 8.90/5.08 @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5139_divmod__algorithm__code_I6_J,axiom,
% 8.90/5.08 ! [M: num,N: num] :
% 8.90/5.08 ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.90/5.08 = ( produc2626176000494625587at_nat
% 8.90/5.08 @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
% 8.90/5.08 @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5140_divmod__algorithm__code_I6_J,axiom,
% 8.90/5.08 ! [M: num,N: num] :
% 8.90/5.08 ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.90/5.08 = ( produc6916734918728496179nteger
% 8.90/5.08 @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
% 8.90/5.08 @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5141_flip__bit__0,axiom,
% 8.90/5.08 ! [A: code_integer] :
% 8.90/5.08 ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 8.90/5.08 = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5142_flip__bit__0,axiom,
% 8.90/5.08 ! [A: int] :
% 8.90/5.08 ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 8.90/5.08 = ( 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 ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5143_flip__bit__0,axiom,
% 8.90/5.08 ! [A: nat] :
% 8.90/5.08 ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 8.90/5.08 = ( 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 ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5144_option_Osize__gen_I2_J,axiom,
% 8.90/5.08 ! [X2: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
% 8.90/5.08 ( ( size_o8335143837870341156at_nat @ X2 @ ( some_P7363390416028606310at_nat @ X22 ) )
% 8.90/5.08 = ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5145_option_Osize__gen_I2_J,axiom,
% 8.90/5.08 ! [X2: nat > nat,X22: nat] :
% 8.90/5.08 ( ( size_option_nat @ X2 @ ( some_nat @ X22 ) )
% 8.90/5.08 = ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5146_option_Osize__gen_I2_J,axiom,
% 8.90/5.08 ! [X2: num > nat,X22: num] :
% 8.90/5.08 ( ( size_option_num @ X2 @ ( some_num @ X22 ) )
% 8.90/5.08 = ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5147_signed__take__bit__Suc,axiom,
% 8.90/5.08 ! [N: nat,A: code_integer] :
% 8.90/5.08 ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
% 8.90/5.08 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5148_signed__take__bit__Suc,axiom,
% 8.90/5.08 ! [N: nat,A: int] :
% 8.90/5.08 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
% 8.90/5.08 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5149_set__decode__Suc,axiom,
% 8.90/5.08 ! [N: nat,X2: nat] :
% 8.90/5.08 ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X2 ) )
% 8.90/5.08 = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5150_intind,axiom,
% 8.90/5.08 ! [I2: nat,N: nat,P: int > $o,X2: int] :
% 8.90/5.08 ( ( ord_less_nat @ I2 @ N )
% 8.90/5.08 => ( ( P @ X2 )
% 8.90/5.08 => ( P @ ( nth_int @ ( replicate_int @ N @ X2 ) @ I2 ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5151_intind,axiom,
% 8.90/5.08 ! [I2: nat,N: nat,P: nat > $o,X2: nat] :
% 8.90/5.08 ( ( ord_less_nat @ I2 @ N )
% 8.90/5.08 => ( ( P @ X2 )
% 8.90/5.08 => ( P @ ( nth_nat @ ( replicate_nat @ N @ X2 ) @ I2 ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5152_intind,axiom,
% 8.90/5.08 ! [I2: nat,N: nat,P: vEBT_VEBT > $o,X2: vEBT_VEBT] :
% 8.90/5.08 ( ( ord_less_nat @ I2 @ N )
% 8.90/5.08 => ( ( P @ X2 )
% 8.90/5.08 => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X2 ) @ I2 ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5153_of__bool__less__eq__iff,axiom,
% 8.90/5.08 ! [P: $o,Q: $o] :
% 8.90/5.08 ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 8.90/5.08 = ( P
% 8.90/5.08 => Q ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5154_of__bool__less__eq__iff,axiom,
% 8.90/5.08 ! [P: $o,Q: $o] :
% 8.90/5.08 ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 8.90/5.08 = ( P
% 8.90/5.08 => Q ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5155_of__bool__less__eq__iff,axiom,
% 8.90/5.08 ! [P: $o,Q: $o] :
% 8.90/5.08 ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 8.90/5.08 = ( P
% 8.90/5.08 => Q ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5156_of__bool__less__eq__iff,axiom,
% 8.90/5.08 ! [P: $o,Q: $o] :
% 8.90/5.08 ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 8.90/5.08 = ( P
% 8.90/5.08 => Q ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5157_of__bool__eq__0__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ( zero_n1201886186963655149omplex @ P )
% 8.90/5.08 = zero_zero_complex )
% 8.90/5.08 = ~ P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5158_of__bool__eq__0__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ( zero_n3304061248610475627l_real @ P )
% 8.90/5.08 = zero_zero_real )
% 8.90/5.08 = ~ P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5159_of__bool__eq__0__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 8.90/5.08 = zero_zero_rat )
% 8.90/5.08 = ~ P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5160_of__bool__eq__0__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 8.90/5.08 = zero_zero_nat )
% 8.90/5.08 = ~ P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5161_of__bool__eq__0__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ( zero_n2684676970156552555ol_int @ P )
% 8.90/5.08 = zero_zero_int )
% 8.90/5.08 = ~ P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5162_of__bool__eq__0__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ( zero_n356916108424825756nteger @ P )
% 8.90/5.08 = zero_z3403309356797280102nteger )
% 8.90/5.08 = ~ P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5163_of__bool__eq_I1_J,axiom,
% 8.90/5.08 ( ( zero_n1201886186963655149omplex @ $false )
% 8.90/5.08 = zero_zero_complex ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5164_of__bool__eq_I1_J,axiom,
% 8.90/5.08 ( ( zero_n3304061248610475627l_real @ $false )
% 8.90/5.08 = zero_zero_real ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5165_of__bool__eq_I1_J,axiom,
% 8.90/5.08 ( ( zero_n2052037380579107095ol_rat @ $false )
% 8.90/5.08 = zero_zero_rat ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5166_of__bool__eq_I1_J,axiom,
% 8.90/5.08 ( ( zero_n2687167440665602831ol_nat @ $false )
% 8.90/5.08 = zero_zero_nat ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5167_of__bool__eq_I1_J,axiom,
% 8.90/5.08 ( ( zero_n2684676970156552555ol_int @ $false )
% 8.90/5.08 = zero_zero_int ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5168_of__bool__eq_I1_J,axiom,
% 8.90/5.08 ( ( zero_n356916108424825756nteger @ $false )
% 8.90/5.08 = zero_z3403309356797280102nteger ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5169_of__bool__less__iff,axiom,
% 8.90/5.08 ! [P: $o,Q: $o] :
% 8.90/5.08 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 8.90/5.08 = ( ~ P
% 8.90/5.08 & Q ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5170_of__bool__less__iff,axiom,
% 8.90/5.08 ! [P: $o,Q: $o] :
% 8.90/5.08 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 8.90/5.08 = ( ~ P
% 8.90/5.08 & Q ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5171_of__bool__less__iff,axiom,
% 8.90/5.08 ! [P: $o,Q: $o] :
% 8.90/5.08 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 8.90/5.08 = ( ~ P
% 8.90/5.08 & Q ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5172_of__bool__less__iff,axiom,
% 8.90/5.08 ! [P: $o,Q: $o] :
% 8.90/5.08 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 8.90/5.08 = ( ~ P
% 8.90/5.08 & Q ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5173_of__bool__less__iff,axiom,
% 8.90/5.08 ! [P: $o,Q: $o] :
% 8.90/5.08 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 8.90/5.08 = ( ~ P
% 8.90/5.08 & Q ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5174_of__bool__eq__1__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ( zero_n1201886186963655149omplex @ P )
% 8.90/5.08 = one_one_complex )
% 8.90/5.08 = P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5175_of__bool__eq__1__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ( zero_n3304061248610475627l_real @ P )
% 8.90/5.08 = one_one_real )
% 8.90/5.08 = P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5176_of__bool__eq__1__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 8.90/5.08 = one_one_rat )
% 8.90/5.08 = P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5177_of__bool__eq__1__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 8.90/5.08 = one_one_nat )
% 8.90/5.08 = P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5178_of__bool__eq__1__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ( zero_n2684676970156552555ol_int @ P )
% 8.90/5.08 = one_one_int )
% 8.90/5.08 = P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5179_of__bool__eq__1__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ( zero_n356916108424825756nteger @ P )
% 8.90/5.08 = one_one_Code_integer )
% 8.90/5.08 = P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5180_of__bool__eq_I2_J,axiom,
% 8.90/5.08 ( ( zero_n1201886186963655149omplex @ $true )
% 8.90/5.08 = one_one_complex ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5181_of__bool__eq_I2_J,axiom,
% 8.90/5.08 ( ( zero_n3304061248610475627l_real @ $true )
% 8.90/5.08 = one_one_real ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5182_of__bool__eq_I2_J,axiom,
% 8.90/5.08 ( ( zero_n2052037380579107095ol_rat @ $true )
% 8.90/5.08 = one_one_rat ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5183_of__bool__eq_I2_J,axiom,
% 8.90/5.08 ( ( zero_n2687167440665602831ol_nat @ $true )
% 8.90/5.08 = one_one_nat ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5184_of__bool__eq_I2_J,axiom,
% 8.90/5.08 ( ( zero_n2684676970156552555ol_int @ $true )
% 8.90/5.08 = one_one_int ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5185_of__bool__eq_I2_J,axiom,
% 8.90/5.08 ( ( zero_n356916108424825756nteger @ $true )
% 8.90/5.08 = one_one_Code_integer ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5186_signed__take__bit__of__0,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
% 8.90/5.08 = zero_zero_int ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5187_length__replicate,axiom,
% 8.90/5.08 ! [N: nat,X2: vEBT_VEBT] :
% 8.90/5.08 ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N @ X2 ) )
% 8.90/5.08 = N ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5188_length__replicate,axiom,
% 8.90/5.08 ! [N: nat,X2: $o] :
% 8.90/5.08 ( ( size_size_list_o @ ( replicate_o @ N @ X2 ) )
% 8.90/5.08 = N ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5189_length__replicate,axiom,
% 8.90/5.08 ! [N: nat,X2: nat] :
% 8.90/5.08 ( ( size_size_list_nat @ ( replicate_nat @ N @ X2 ) )
% 8.90/5.08 = N ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5190_length__replicate,axiom,
% 8.90/5.08 ! [N: nat,X2: int] :
% 8.90/5.08 ( ( size_size_list_int @ ( replicate_int @ N @ X2 ) )
% 8.90/5.08 = N ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5191_of__bool__or__iff,axiom,
% 8.90/5.08 ! [P: $o,Q: $o] :
% 8.90/5.08 ( ( zero_n2687167440665602831ol_nat
% 8.90/5.08 @ ( P
% 8.90/5.08 | Q ) )
% 8.90/5.08 = ( ord_max_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5192_of__bool__or__iff,axiom,
% 8.90/5.08 ! [P: $o,Q: $o] :
% 8.90/5.08 ( ( zero_n2684676970156552555ol_int
% 8.90/5.08 @ ( P
% 8.90/5.08 | Q ) )
% 8.90/5.08 = ( ord_max_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5193_of__bool__or__iff,axiom,
% 8.90/5.08 ! [P: $o,Q: $o] :
% 8.90/5.08 ( ( zero_n356916108424825756nteger
% 8.90/5.08 @ ( P
% 8.90/5.08 | Q ) )
% 8.90/5.08 = ( ord_max_Code_integer @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5194_case__prod__conv,axiom,
% 8.90/5.08 ! [F: nat > nat > nat,A: nat,B: nat] :
% 8.90/5.08 ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 8.90/5.08 = ( F @ A @ B ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5195_case__prod__conv,axiom,
% 8.90/5.08 ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
% 8.90/5.08 ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 8.90/5.08 = ( F @ A @ B ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5196_case__prod__conv,axiom,
% 8.90/5.08 ! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
% 8.90/5.08 ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 8.90/5.08 = ( F @ A @ B ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5197_case__prod__conv,axiom,
% 8.90/5.08 ! [F: int > int > product_prod_int_int,A: int,B: int] :
% 8.90/5.08 ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
% 8.90/5.08 = ( F @ A @ B ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5198_case__prod__conv,axiom,
% 8.90/5.08 ! [F: int > int > $o,A: int,B: int] :
% 8.90/5.08 ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
% 8.90/5.08 = ( F @ A @ B ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5199_zero__less__of__bool__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 8.90/5.08 = P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5200_zero__less__of__bool__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 8.90/5.08 = P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5201_zero__less__of__bool__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 8.90/5.08 = P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5202_zero__less__of__bool__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 8.90/5.08 = P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5203_zero__less__of__bool__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
% 8.90/5.08 = P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5204_of__bool__less__one__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 8.90/5.08 = ~ P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5205_of__bool__less__one__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 8.90/5.08 = ~ P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5206_of__bool__less__one__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 8.90/5.08 = ~ P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5207_of__bool__less__one__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 8.90/5.08 = ~ P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5208_of__bool__less__one__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
% 8.90/5.08 = ~ P ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5209_of__bool__not__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( zero_n1201886186963655149omplex @ ~ P )
% 8.90/5.08 = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5210_of__bool__not__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( zero_n3304061248610475627l_real @ ~ P )
% 8.90/5.08 = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5211_of__bool__not__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( zero_n2052037380579107095ol_rat @ ~ P )
% 8.90/5.08 = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5212_of__bool__not__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( zero_n2684676970156552555ol_int @ ~ P )
% 8.90/5.08 = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5213_of__bool__not__iff,axiom,
% 8.90/5.08 ! [P: $o] :
% 8.90/5.08 ( ( zero_n356916108424825756nteger @ ~ P )
% 8.90/5.08 = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5214_Suc__0__mod__eq,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.90/5.08 = ( zero_n2687167440665602831ol_nat
% 8.90/5.08 @ ( N
% 8.90/5.08 != ( suc @ zero_zero_nat ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5215_signed__take__bit__Suc__1,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
% 8.90/5.08 = one_one_int ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5216_signed__take__bit__numeral__of__1,axiom,
% 8.90/5.08 ! [K: num] :
% 8.90/5.08 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
% 8.90/5.08 = one_one_int ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5217_in__set__replicate,axiom,
% 8.90/5.08 ! [X2: product_prod_nat_nat,N: nat,Y3: product_prod_nat_nat] :
% 8.90/5.08 ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ Y3 ) ) )
% 8.90/5.08 = ( ( X2 = Y3 )
% 8.90/5.08 & ( N != zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5218_in__set__replicate,axiom,
% 8.90/5.08 ! [X2: complex,N: nat,Y3: complex] :
% 8.90/5.08 ( ( member_complex @ X2 @ ( set_complex2 @ ( replicate_complex @ N @ Y3 ) ) )
% 8.90/5.08 = ( ( X2 = Y3 )
% 8.90/5.08 & ( N != zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5219_in__set__replicate,axiom,
% 8.90/5.08 ! [X2: real,N: nat,Y3: real] :
% 8.90/5.08 ( ( member_real @ X2 @ ( set_real2 @ ( replicate_real @ N @ Y3 ) ) )
% 8.90/5.08 = ( ( X2 = Y3 )
% 8.90/5.08 & ( N != zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5220_in__set__replicate,axiom,
% 8.90/5.08 ! [X2: int,N: nat,Y3: int] :
% 8.90/5.08 ( ( member_int @ X2 @ ( set_int2 @ ( replicate_int @ N @ Y3 ) ) )
% 8.90/5.08 = ( ( X2 = Y3 )
% 8.90/5.08 & ( N != zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5221_in__set__replicate,axiom,
% 8.90/5.08 ! [X2: nat,N: nat,Y3: nat] :
% 8.90/5.08 ( ( member_nat @ X2 @ ( set_nat2 @ ( replicate_nat @ N @ Y3 ) ) )
% 8.90/5.08 = ( ( X2 = Y3 )
% 8.90/5.08 & ( N != zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5222_in__set__replicate,axiom,
% 8.90/5.08 ! [X2: vEBT_VEBT,N: nat,Y3: vEBT_VEBT] :
% 8.90/5.08 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y3 ) ) )
% 8.90/5.08 = ( ( X2 = Y3 )
% 8.90/5.08 & ( N != zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5223_Bex__set__replicate,axiom,
% 8.90/5.08 ! [N: nat,A: int,P: int > $o] :
% 8.90/5.08 ( ( ? [X: int] :
% 8.90/5.08 ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
% 8.90/5.08 & ( P @ X ) ) )
% 8.90/5.08 = ( ( P @ A )
% 8.90/5.08 & ( N != zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5224_Bex__set__replicate,axiom,
% 8.90/5.08 ! [N: nat,A: nat,P: nat > $o] :
% 8.90/5.08 ( ( ? [X: nat] :
% 8.90/5.08 ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
% 8.90/5.08 & ( P @ X ) ) )
% 8.90/5.08 = ( ( P @ A )
% 8.90/5.08 & ( N != zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5225_Bex__set__replicate,axiom,
% 8.90/5.08 ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 8.90/5.08 ( ( ? [X: vEBT_VEBT] :
% 8.90/5.08 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
% 8.90/5.08 & ( P @ X ) ) )
% 8.90/5.08 = ( ( P @ A )
% 8.90/5.08 & ( N != zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5226_Ball__set__replicate,axiom,
% 8.90/5.08 ! [N: nat,A: int,P: int > $o] :
% 8.90/5.08 ( ( ! [X: int] :
% 8.90/5.08 ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
% 8.90/5.08 => ( P @ X ) ) )
% 8.90/5.08 = ( ( P @ A )
% 8.90/5.08 | ( N = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5227_Ball__set__replicate,axiom,
% 8.90/5.08 ! [N: nat,A: nat,P: nat > $o] :
% 8.90/5.08 ( ( ! [X: nat] :
% 8.90/5.08 ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
% 8.90/5.08 => ( P @ X ) ) )
% 8.90/5.08 = ( ( P @ A )
% 8.90/5.08 | ( N = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5228_Ball__set__replicate,axiom,
% 8.90/5.08 ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 8.90/5.08 ( ( ! [X: vEBT_VEBT] :
% 8.90/5.08 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
% 8.90/5.08 => ( P @ X ) ) )
% 8.90/5.08 = ( ( P @ A )
% 8.90/5.08 | ( N = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5229_nth__replicate,axiom,
% 8.90/5.08 ! [I2: nat,N: nat,X2: int] :
% 8.90/5.08 ( ( ord_less_nat @ I2 @ N )
% 8.90/5.08 => ( ( nth_int @ ( replicate_int @ N @ X2 ) @ I2 )
% 8.90/5.08 = X2 ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5230_nth__replicate,axiom,
% 8.90/5.08 ! [I2: nat,N: nat,X2: nat] :
% 8.90/5.08 ( ( ord_less_nat @ I2 @ N )
% 8.90/5.08 => ( ( nth_nat @ ( replicate_nat @ N @ X2 ) @ I2 )
% 8.90/5.08 = X2 ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5231_nth__replicate,axiom,
% 8.90/5.08 ! [I2: nat,N: nat,X2: vEBT_VEBT] :
% 8.90/5.08 ( ( ord_less_nat @ I2 @ N )
% 8.90/5.08 => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X2 ) @ I2 )
% 8.90/5.08 = X2 ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5232_triangle__Suc,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( nat_triangle @ ( suc @ N ) )
% 8.90/5.08 = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5233_signed__take__bit__Suc__bit0,axiom,
% 8.90/5.08 ! [N: nat,K: num] :
% 8.90/5.08 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 8.90/5.08 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5234_odd__of__bool__self,axiom,
% 8.90/5.08 ! [P6: $o] :
% 8.90/5.08 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P6 ) ) )
% 8.90/5.08 = P6 ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5235_odd__of__bool__self,axiom,
% 8.90/5.08 ! [P6: $o] :
% 8.90/5.08 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P6 ) ) )
% 8.90/5.08 = P6 ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5236_odd__of__bool__self,axiom,
% 8.90/5.08 ! [P6: $o] :
% 8.90/5.08 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P6 ) ) )
% 8.90/5.08 = P6 ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5237_of__bool__half__eq__0,axiom,
% 8.90/5.08 ! [B: $o] :
% 8.90/5.08 ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.08 = zero_zero_nat ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5238_of__bool__half__eq__0,axiom,
% 8.90/5.08 ! [B: $o] :
% 8.90/5.08 ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.08 = zero_zero_int ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5239_of__bool__half__eq__0,axiom,
% 8.90/5.08 ! [B: $o] :
% 8.90/5.08 ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.08 = zero_z3403309356797280102nteger ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5240_set__decode__0,axiom,
% 8.90/5.08 ! [X2: nat] :
% 8.90/5.08 ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X2 ) )
% 8.90/5.08 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5241_divmod__algorithm__code_I5_J,axiom,
% 8.90/5.08 ! [M: num,N: num] :
% 8.90/5.08 ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.90/5.08 = ( produc4245557441103728435nt_int
% 8.90/5.08 @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
% 8.90/5.08 @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5242_divmod__algorithm__code_I5_J,axiom,
% 8.90/5.08 ! [M: num,N: num] :
% 8.90/5.08 ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.90/5.08 = ( produc2626176000494625587at_nat
% 8.90/5.08 @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
% 8.90/5.08 @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5243_divmod__algorithm__code_I5_J,axiom,
% 8.90/5.08 ! [M: num,N: num] :
% 8.90/5.08 ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.90/5.08 = ( produc6916734918728496179nteger
% 8.90/5.08 @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
% 8.90/5.08 @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5244_one__div__2__pow__eq,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.08 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5245_one__div__2__pow__eq,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.08 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5246_one__div__2__pow__eq,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.08 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5247_bits__1__div__exp,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.08 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5248_bits__1__div__exp,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.08 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5249_bits__1__div__exp,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.08 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5250_signed__take__bit__Suc__bit1,axiom,
% 8.90/5.08 ! [N: nat,K: num] :
% 8.90/5.08 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 8.90/5.08 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5251_one__mod__2__pow__eq,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.08 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5252_one__mod__2__pow__eq,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.08 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5253_one__mod__2__pow__eq,axiom,
% 8.90/5.08 ! [N: nat] :
% 8.90/5.08 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.08 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5254_dvd__antisym,axiom,
% 8.90/5.08 ! [M: nat,N: nat] :
% 8.90/5.08 ( ( dvd_dvd_nat @ M @ N )
% 8.90/5.08 => ( ( dvd_dvd_nat @ N @ M )
% 8.90/5.08 => ( M = N ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5255_of__bool__eq__iff,axiom,
% 8.90/5.08 ! [P6: $o,Q3: $o] :
% 8.90/5.08 ( ( ( zero_n2687167440665602831ol_nat @ P6 )
% 8.90/5.08 = ( zero_n2687167440665602831ol_nat @ Q3 ) )
% 8.90/5.08 = ( P6 = Q3 ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5256_of__bool__eq__iff,axiom,
% 8.90/5.08 ! [P6: $o,Q3: $o] :
% 8.90/5.08 ( ( ( zero_n2684676970156552555ol_int @ P6 )
% 8.90/5.08 = ( zero_n2684676970156552555ol_int @ Q3 ) )
% 8.90/5.08 = ( P6 = Q3 ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5257_of__bool__eq__iff,axiom,
% 8.90/5.08 ! [P6: $o,Q3: $o] :
% 8.90/5.08 ( ( ( zero_n356916108424825756nteger @ P6 )
% 8.90/5.08 = ( zero_n356916108424825756nteger @ Q3 ) )
% 8.90/5.08 = ( P6 = Q3 ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5258_of__bool__conj,axiom,
% 8.90/5.08 ! [P: $o,Q: $o] :
% 8.90/5.08 ( ( zero_n3304061248610475627l_real
% 8.90/5.08 @ ( P
% 8.90/5.08 & Q ) )
% 8.90/5.08 = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5259_of__bool__conj,axiom,
% 8.90/5.08 ! [P: $o,Q: $o] :
% 8.90/5.08 ( ( zero_n2052037380579107095ol_rat
% 8.90/5.08 @ ( P
% 8.90/5.08 & Q ) )
% 8.90/5.08 = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5260_of__bool__conj,axiom,
% 8.90/5.08 ! [P: $o,Q: $o] :
% 8.90/5.08 ( ( zero_n2687167440665602831ol_nat
% 8.90/5.08 @ ( P
% 8.90/5.08 & Q ) )
% 8.90/5.08 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5261_of__bool__conj,axiom,
% 8.90/5.08 ! [P: $o,Q: $o] :
% 8.90/5.08 ( ( zero_n2684676970156552555ol_int
% 8.90/5.08 @ ( P
% 8.90/5.08 & Q ) )
% 8.90/5.08 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5262_of__bool__conj,axiom,
% 8.90/5.08 ! [P: $o,Q: $o] :
% 8.90/5.08 ( ( zero_n356916108424825756nteger
% 8.90/5.08 @ ( P
% 8.90/5.08 & Q ) )
% 8.90/5.08 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5263_old_Oprod_Ocase,axiom,
% 8.90/5.08 ! [F: nat > nat > nat,X1: nat,X22: nat] :
% 8.90/5.08 ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 8.90/5.08 = ( F @ X1 @ X22 ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5264_old_Oprod_Ocase,axiom,
% 8.90/5.08 ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X22: nat] :
% 8.90/5.08 ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 8.90/5.08 = ( F @ X1 @ X22 ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5265_old_Oprod_Ocase,axiom,
% 8.90/5.08 ! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X22: nat] :
% 8.90/5.08 ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 8.90/5.08 = ( F @ X1 @ X22 ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5266_old_Oprod_Ocase,axiom,
% 8.90/5.08 ! [F: int > int > product_prod_int_int,X1: int,X22: int] :
% 8.90/5.08 ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
% 8.90/5.08 = ( F @ X1 @ X22 ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5267_old_Oprod_Ocase,axiom,
% 8.90/5.08 ! [F: int > int > $o,X1: int,X22: int] :
% 8.90/5.08 ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
% 8.90/5.08 = ( F @ X1 @ X22 ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5268_signed__take__bit__mult,axiom,
% 8.90/5.08 ! [N: nat,K: int,L: int] :
% 8.90/5.08 ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 8.90/5.08 = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5269_signed__take__bit__add,axiom,
% 8.90/5.08 ! [N: nat,K: int,L: int] :
% 8.90/5.08 ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 8.90/5.08 = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5270_signed__take__bit__diff,axiom,
% 8.90/5.08 ! [N: nat,K: int,L: int] :
% 8.90/5.08 ( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 8.90/5.08 = ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5271_case__prodE2,axiom,
% 8.90/5.08 ! [Q: nat > $o,P: nat > nat > nat,Z: product_prod_nat_nat] :
% 8.90/5.08 ( ( Q @ ( produc6842872674320459806at_nat @ P @ Z ) )
% 8.90/5.08 => ~ ! [X5: nat,Y4: nat] :
% 8.90/5.08 ( ( Z
% 8.90/5.08 = ( product_Pair_nat_nat @ X5 @ Y4 ) )
% 8.90/5.08 => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5272_case__prodE2,axiom,
% 8.90/5.08 ! [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] :
% 8.90/5.08 ( ( Q @ ( produc27273713700761075at_nat @ P @ Z ) )
% 8.90/5.08 => ~ ! [X5: nat,Y4: nat] :
% 8.90/5.08 ( ( Z
% 8.90/5.08 = ( product_Pair_nat_nat @ X5 @ Y4 ) )
% 8.90/5.08 => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5273_case__prodE2,axiom,
% 8.90/5.08 ! [Q: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat] :
% 8.90/5.08 ( ( Q @ ( produc8739625826339149834_nat_o @ P @ Z ) )
% 8.90/5.08 => ~ ! [X5: nat,Y4: nat] :
% 8.90/5.08 ( ( Z
% 8.90/5.08 = ( product_Pair_nat_nat @ X5 @ Y4 ) )
% 8.90/5.08 => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5274_case__prodE2,axiom,
% 8.90/5.08 ! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z: product_prod_int_int] :
% 8.90/5.08 ( ( Q @ ( produc4245557441103728435nt_int @ P @ Z ) )
% 8.90/5.08 => ~ ! [X5: int,Y4: int] :
% 8.90/5.08 ( ( Z
% 8.90/5.08 = ( product_Pair_int_int @ X5 @ Y4 ) )
% 8.90/5.08 => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5275_case__prodE2,axiom,
% 8.90/5.08 ! [Q: $o > $o,P: int > int > $o,Z: product_prod_int_int] :
% 8.90/5.08 ( ( Q @ ( produc4947309494688390418_int_o @ P @ Z ) )
% 8.90/5.08 => ~ ! [X5: int,Y4: int] :
% 8.90/5.08 ( ( Z
% 8.90/5.08 = ( product_Pair_int_int @ X5 @ Y4 ) )
% 8.90/5.08 => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5276_case__prod__eta,axiom,
% 8.90/5.08 ! [F: product_prod_nat_nat > nat] :
% 8.90/5.08 ( ( produc6842872674320459806at_nat
% 8.90/5.08 @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
% 8.90/5.08 = F ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5277_case__prod__eta,axiom,
% 8.90/5.08 ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 8.90/5.08 ( ( produc27273713700761075at_nat
% 8.90/5.08 @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
% 8.90/5.08 = F ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5278_case__prod__eta,axiom,
% 8.90/5.08 ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 8.90/5.08 ( ( produc8739625826339149834_nat_o
% 8.90/5.08 @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
% 8.90/5.08 = F ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5279_case__prod__eta,axiom,
% 8.90/5.08 ! [F: product_prod_int_int > product_prod_int_int] :
% 8.90/5.08 ( ( produc4245557441103728435nt_int
% 8.90/5.08 @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
% 8.90/5.08 = F ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5280_case__prod__eta,axiom,
% 8.90/5.08 ! [F: product_prod_int_int > $o] :
% 8.90/5.08 ( ( produc4947309494688390418_int_o
% 8.90/5.08 @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
% 8.90/5.08 = F ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5281_cond__case__prod__eta,axiom,
% 8.90/5.08 ! [F: nat > nat > nat,G: product_prod_nat_nat > nat] :
% 8.90/5.08 ( ! [X5: nat,Y4: nat] :
% 8.90/5.08 ( ( F @ X5 @ Y4 )
% 8.90/5.08 = ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
% 8.90/5.08 => ( ( produc6842872674320459806at_nat @ F )
% 8.90/5.08 = G ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5282_cond__case__prod__eta,axiom,
% 8.90/5.08 ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 8.90/5.08 ( ! [X5: nat,Y4: nat] :
% 8.90/5.08 ( ( F @ X5 @ Y4 )
% 8.90/5.08 = ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
% 8.90/5.08 => ( ( produc27273713700761075at_nat @ F )
% 8.90/5.08 = G ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5283_cond__case__prod__eta,axiom,
% 8.90/5.08 ! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 8.90/5.08 ( ! [X5: nat,Y4: nat] :
% 8.90/5.08 ( ( F @ X5 @ Y4 )
% 8.90/5.08 = ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
% 8.90/5.08 => ( ( produc8739625826339149834_nat_o @ F )
% 8.90/5.08 = G ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5284_cond__case__prod__eta,axiom,
% 8.90/5.08 ! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
% 8.90/5.08 ( ! [X5: int,Y4: int] :
% 8.90/5.08 ( ( F @ X5 @ Y4 )
% 8.90/5.08 = ( G @ ( product_Pair_int_int @ X5 @ Y4 ) ) )
% 8.90/5.08 => ( ( produc4245557441103728435nt_int @ F )
% 8.90/5.08 = G ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5285_cond__case__prod__eta,axiom,
% 8.90/5.08 ! [F: int > int > $o,G: product_prod_int_int > $o] :
% 8.90/5.08 ( ! [X5: int,Y4: int] :
% 8.90/5.08 ( ( F @ X5 @ Y4 )
% 8.90/5.08 = ( G @ ( product_Pair_int_int @ X5 @ Y4 ) ) )
% 8.90/5.08 => ( ( produc4947309494688390418_int_o @ F )
% 8.90/5.08 = G ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5286_zero__less__eq__of__bool,axiom,
% 8.90/5.08 ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5287_zero__less__eq__of__bool,axiom,
% 8.90/5.08 ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5288_zero__less__eq__of__bool,axiom,
% 8.90/5.08 ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5289_zero__less__eq__of__bool,axiom,
% 8.90/5.08 ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5290_zero__less__eq__of__bool,axiom,
% 8.90/5.08 ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5291_of__bool__less__eq__one,axiom,
% 8.90/5.08 ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5292_of__bool__less__eq__one,axiom,
% 8.90/5.08 ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5293_of__bool__less__eq__one,axiom,
% 8.90/5.08 ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5294_of__bool__less__eq__one,axiom,
% 8.90/5.08 ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5295_of__bool__less__eq__one,axiom,
% 8.90/5.08 ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5296_of__bool__def,axiom,
% 8.90/5.08 ( zero_n1201886186963655149omplex
% 8.90/5.08 = ( ^ [P4: $o] : ( if_complex @ P4 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5297_of__bool__def,axiom,
% 8.90/5.08 ( zero_n3304061248610475627l_real
% 8.90/5.08 = ( ^ [P4: $o] : ( if_real @ P4 @ one_one_real @ zero_zero_real ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5298_of__bool__def,axiom,
% 8.90/5.08 ( zero_n2052037380579107095ol_rat
% 8.90/5.08 = ( ^ [P4: $o] : ( if_rat @ P4 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5299_of__bool__def,axiom,
% 8.90/5.08 ( zero_n2687167440665602831ol_nat
% 8.90/5.08 = ( ^ [P4: $o] : ( if_nat @ P4 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5300_of__bool__def,axiom,
% 8.90/5.08 ( zero_n2684676970156552555ol_int
% 8.90/5.08 = ( ^ [P4: $o] : ( if_int @ P4 @ one_one_int @ zero_zero_int ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5301_of__bool__def,axiom,
% 8.90/5.08 ( zero_n356916108424825756nteger
% 8.90/5.08 = ( ^ [P4: $o] : ( if_Code_integer @ P4 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5302_split__of__bool,axiom,
% 8.90/5.08 ! [P: complex > $o,P6: $o] :
% 8.90/5.08 ( ( P @ ( zero_n1201886186963655149omplex @ P6 ) )
% 8.90/5.08 = ( ( P6
% 8.90/5.08 => ( P @ one_one_complex ) )
% 8.90/5.08 & ( ~ P6
% 8.90/5.08 => ( P @ zero_zero_complex ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5303_split__of__bool,axiom,
% 8.90/5.08 ! [P: real > $o,P6: $o] :
% 8.90/5.08 ( ( P @ ( zero_n3304061248610475627l_real @ P6 ) )
% 8.90/5.08 = ( ( P6
% 8.90/5.08 => ( P @ one_one_real ) )
% 8.90/5.08 & ( ~ P6
% 8.90/5.08 => ( P @ zero_zero_real ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5304_split__of__bool,axiom,
% 8.90/5.08 ! [P: rat > $o,P6: $o] :
% 8.90/5.08 ( ( P @ ( zero_n2052037380579107095ol_rat @ P6 ) )
% 8.90/5.08 = ( ( P6
% 8.90/5.08 => ( P @ one_one_rat ) )
% 8.90/5.08 & ( ~ P6
% 8.90/5.08 => ( P @ zero_zero_rat ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5305_split__of__bool,axiom,
% 8.90/5.08 ! [P: nat > $o,P6: $o] :
% 8.90/5.08 ( ( P @ ( zero_n2687167440665602831ol_nat @ P6 ) )
% 8.90/5.08 = ( ( P6
% 8.90/5.08 => ( P @ one_one_nat ) )
% 8.90/5.08 & ( ~ P6
% 8.90/5.08 => ( P @ zero_zero_nat ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5306_split__of__bool,axiom,
% 8.90/5.08 ! [P: int > $o,P6: $o] :
% 8.90/5.08 ( ( P @ ( zero_n2684676970156552555ol_int @ P6 ) )
% 8.90/5.08 = ( ( P6
% 8.90/5.08 => ( P @ one_one_int ) )
% 8.90/5.08 & ( ~ P6
% 8.90/5.08 => ( P @ zero_zero_int ) ) ) ) ).
% 8.90/5.08
% 8.90/5.08 thf(fact_5307_split__of__bool,axiom,
% 8.90/5.08 ! [P: code_integer > $o,P6: $o] :
% 8.90/5.08 ( ( P @ ( zero_n356916108424825756nteger @ P6 ) )
% 8.90/5.08 = ( ( P6
% 8.90/5.08 => ( P @ one_one_Code_integer ) )
% 8.90/5.08 & ( ~ P6
% 8.90/5.08 => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5308_split__of__bool__asm,axiom,
% 8.90/5.09 ! [P: complex > $o,P6: $o] :
% 8.90/5.09 ( ( P @ ( zero_n1201886186963655149omplex @ P6 ) )
% 8.90/5.09 = ( ~ ( ( P6
% 8.90/5.09 & ~ ( P @ one_one_complex ) )
% 8.90/5.09 | ( ~ P6
% 8.90/5.09 & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5309_split__of__bool__asm,axiom,
% 8.90/5.09 ! [P: real > $o,P6: $o] :
% 8.90/5.09 ( ( P @ ( zero_n3304061248610475627l_real @ P6 ) )
% 8.90/5.09 = ( ~ ( ( P6
% 8.90/5.09 & ~ ( P @ one_one_real ) )
% 8.90/5.09 | ( ~ P6
% 8.90/5.09 & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5310_split__of__bool__asm,axiom,
% 8.90/5.09 ! [P: rat > $o,P6: $o] :
% 8.90/5.09 ( ( P @ ( zero_n2052037380579107095ol_rat @ P6 ) )
% 8.90/5.09 = ( ~ ( ( P6
% 8.90/5.09 & ~ ( P @ one_one_rat ) )
% 8.90/5.09 | ( ~ P6
% 8.90/5.09 & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5311_split__of__bool__asm,axiom,
% 8.90/5.09 ! [P: nat > $o,P6: $o] :
% 8.90/5.09 ( ( P @ ( zero_n2687167440665602831ol_nat @ P6 ) )
% 8.90/5.09 = ( ~ ( ( P6
% 8.90/5.09 & ~ ( P @ one_one_nat ) )
% 8.90/5.09 | ( ~ P6
% 8.90/5.09 & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5312_split__of__bool__asm,axiom,
% 8.90/5.09 ! [P: int > $o,P6: $o] :
% 8.90/5.09 ( ( P @ ( zero_n2684676970156552555ol_int @ P6 ) )
% 8.90/5.09 = ( ~ ( ( P6
% 8.90/5.09 & ~ ( P @ one_one_int ) )
% 8.90/5.09 | ( ~ P6
% 8.90/5.09 & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5313_split__of__bool__asm,axiom,
% 8.90/5.09 ! [P: code_integer > $o,P6: $o] :
% 8.90/5.09 ( ( P @ ( zero_n356916108424825756nteger @ P6 ) )
% 8.90/5.09 = ( ~ ( ( P6
% 8.90/5.09 & ~ ( P @ one_one_Code_integer ) )
% 8.90/5.09 | ( ~ P6
% 8.90/5.09 & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5314_replicate__length__same,axiom,
% 8.90/5.09 ! [Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 8.90/5.09 ( ! [X5: vEBT_VEBT] :
% 8.90/5.09 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 8.90/5.09 => ( X5 = X2 ) )
% 8.90/5.09 => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ X2 )
% 8.90/5.09 = Xs2 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5315_replicate__length__same,axiom,
% 8.90/5.09 ! [Xs2: list_o,X2: $o] :
% 8.90/5.09 ( ! [X5: $o] :
% 8.90/5.09 ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
% 8.90/5.09 => ( X5 = X2 ) )
% 8.90/5.09 => ( ( replicate_o @ ( size_size_list_o @ Xs2 ) @ X2 )
% 8.90/5.09 = Xs2 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5316_replicate__length__same,axiom,
% 8.90/5.09 ! [Xs2: list_nat,X2: nat] :
% 8.90/5.09 ( ! [X5: nat] :
% 8.90/5.09 ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
% 8.90/5.09 => ( X5 = X2 ) )
% 8.90/5.09 => ( ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X2 )
% 8.90/5.09 = Xs2 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5317_replicate__length__same,axiom,
% 8.90/5.09 ! [Xs2: list_int,X2: int] :
% 8.90/5.09 ( ! [X5: int] :
% 8.90/5.09 ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
% 8.90/5.09 => ( X5 = X2 ) )
% 8.90/5.09 => ( ( replicate_int @ ( size_size_list_int @ Xs2 ) @ X2 )
% 8.90/5.09 = Xs2 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5318_replicate__eqI,axiom,
% 8.90/5.09 ! [Xs2: list_P6011104703257516679at_nat,N: nat,X2: product_prod_nat_nat] :
% 8.90/5.09 ( ( ( size_s5460976970255530739at_nat @ Xs2 )
% 8.90/5.09 = N )
% 8.90/5.09 => ( ! [Y4: product_prod_nat_nat] :
% 8.90/5.09 ( ( member8440522571783428010at_nat @ Y4 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
% 8.90/5.09 => ( Y4 = X2 ) )
% 8.90/5.09 => ( Xs2
% 8.90/5.09 = ( replic4235873036481779905at_nat @ N @ X2 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5319_replicate__eqI,axiom,
% 8.90/5.09 ! [Xs2: list_complex,N: nat,X2: complex] :
% 8.90/5.09 ( ( ( size_s3451745648224563538omplex @ Xs2 )
% 8.90/5.09 = N )
% 8.90/5.09 => ( ! [Y4: complex] :
% 8.90/5.09 ( ( member_complex @ Y4 @ ( set_complex2 @ Xs2 ) )
% 8.90/5.09 => ( Y4 = X2 ) )
% 8.90/5.09 => ( Xs2
% 8.90/5.09 = ( replicate_complex @ N @ X2 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5320_replicate__eqI,axiom,
% 8.90/5.09 ! [Xs2: list_real,N: nat,X2: real] :
% 8.90/5.09 ( ( ( size_size_list_real @ Xs2 )
% 8.90/5.09 = N )
% 8.90/5.09 => ( ! [Y4: real] :
% 8.90/5.09 ( ( member_real @ Y4 @ ( set_real2 @ Xs2 ) )
% 8.90/5.09 => ( Y4 = X2 ) )
% 8.90/5.09 => ( Xs2
% 8.90/5.09 = ( replicate_real @ N @ X2 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5321_replicate__eqI,axiom,
% 8.90/5.09 ! [Xs2: list_VEBT_VEBT,N: nat,X2: vEBT_VEBT] :
% 8.90/5.09 ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 8.90/5.09 = N )
% 8.90/5.09 => ( ! [Y4: vEBT_VEBT] :
% 8.90/5.09 ( ( member_VEBT_VEBT @ Y4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 8.90/5.09 => ( Y4 = X2 ) )
% 8.90/5.09 => ( Xs2
% 8.90/5.09 = ( replicate_VEBT_VEBT @ N @ X2 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5322_replicate__eqI,axiom,
% 8.90/5.09 ! [Xs2: list_o,N: nat,X2: $o] :
% 8.90/5.09 ( ( ( size_size_list_o @ Xs2 )
% 8.90/5.09 = N )
% 8.90/5.09 => ( ! [Y4: $o] :
% 8.90/5.09 ( ( member_o @ Y4 @ ( set_o2 @ Xs2 ) )
% 8.90/5.09 => ( Y4 = X2 ) )
% 8.90/5.09 => ( Xs2
% 8.90/5.09 = ( replicate_o @ N @ X2 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5323_replicate__eqI,axiom,
% 8.90/5.09 ! [Xs2: list_nat,N: nat,X2: nat] :
% 8.90/5.09 ( ( ( size_size_list_nat @ Xs2 )
% 8.90/5.09 = N )
% 8.90/5.09 => ( ! [Y4: nat] :
% 8.90/5.09 ( ( member_nat @ Y4 @ ( set_nat2 @ Xs2 ) )
% 8.90/5.09 => ( Y4 = X2 ) )
% 8.90/5.09 => ( Xs2
% 8.90/5.09 = ( replicate_nat @ N @ X2 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5324_replicate__eqI,axiom,
% 8.90/5.09 ! [Xs2: list_int,N: nat,X2: int] :
% 8.90/5.09 ( ( ( size_size_list_int @ Xs2 )
% 8.90/5.09 = N )
% 8.90/5.09 => ( ! [Y4: int] :
% 8.90/5.09 ( ( member_int @ Y4 @ ( set_int2 @ Xs2 ) )
% 8.90/5.09 => ( Y4 = X2 ) )
% 8.90/5.09 => ( Xs2
% 8.90/5.09 = ( replicate_int @ N @ X2 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5325_subset__decode__imp__le,axiom,
% 8.90/5.09 ! [M: nat,N: nat] :
% 8.90/5.09 ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 8.90/5.09 => ( ord_less_eq_nat @ M @ N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5326_of__bool__odd__eq__mod__2,axiom,
% 8.90/5.09 ! [A: nat] :
% 8.90/5.09 ( ( zero_n2687167440665602831ol_nat
% 8.90/5.09 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 8.90/5.09 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5327_of__bool__odd__eq__mod__2,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( zero_n2684676970156552555ol_int
% 8.90/5.09 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 8.90/5.09 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5328_of__bool__odd__eq__mod__2,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( zero_n356916108424825756nteger
% 8.90/5.09 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 8.90/5.09 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5329_signed__take__bit__int__less__exp,axiom,
% 8.90/5.09 ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5330_even__signed__take__bit__iff,axiom,
% 8.90/5.09 ! [M: nat,A: code_integer] :
% 8.90/5.09 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
% 8.90/5.09 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5331_even__signed__take__bit__iff,axiom,
% 8.90/5.09 ! [M: nat,A: int] :
% 8.90/5.09 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
% 8.90/5.09 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5332_bits__induct,axiom,
% 8.90/5.09 ! [P: nat > $o,A: nat] :
% 8.90/5.09 ( ! [A4: nat] :
% 8.90/5.09 ( ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.09 = A4 )
% 8.90/5.09 => ( P @ A4 ) )
% 8.90/5.09 => ( ! [A4: nat,B4: $o] :
% 8.90/5.09 ( ( P @ A4 )
% 8.90/5.09 => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.09 = A4 )
% 8.90/5.09 => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 8.90/5.09 => ( P @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5333_bits__induct,axiom,
% 8.90/5.09 ! [P: int > $o,A: int] :
% 8.90/5.09 ( ! [A4: int] :
% 8.90/5.09 ( ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.09 = A4 )
% 8.90/5.09 => ( P @ A4 ) )
% 8.90/5.09 => ( ! [A4: int,B4: $o] :
% 8.90/5.09 ( ( P @ A4 )
% 8.90/5.09 => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.09 = A4 )
% 8.90/5.09 => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 8.90/5.09 => ( P @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5334_bits__induct,axiom,
% 8.90/5.09 ! [P: code_integer > $o,A: code_integer] :
% 8.90/5.09 ( ! [A4: code_integer] :
% 8.90/5.09 ( ( ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.09 = A4 )
% 8.90/5.09 => ( P @ A4 ) )
% 8.90/5.09 => ( ! [A4: code_integer,B4: $o] :
% 8.90/5.09 ( ( P @ A4 )
% 8.90/5.09 => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B4 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.09 = A4 )
% 8.90/5.09 => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B4 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 8.90/5.09 => ( P @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5335_signed__take__bit__int__greater__eq__self__iff,axiom,
% 8.90/5.09 ! [K: int,N: nat] :
% 8.90/5.09 ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 8.90/5.09 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5336_signed__take__bit__int__less__self__iff,axiom,
% 8.90/5.09 ! [N: nat,K: int] :
% 8.90/5.09 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 8.90/5.09 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5337_exp__mod__exp,axiom,
% 8.90/5.09 ! [M: nat,N: nat] :
% 8.90/5.09 ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.09 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5338_exp__mod__exp,axiom,
% 8.90/5.09 ! [M: nat,N: nat] :
% 8.90/5.09 ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.09 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5339_exp__mod__exp,axiom,
% 8.90/5.09 ! [M: nat,N: nat] :
% 8.90/5.09 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.09 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5340_signed__take__bit__int__less__eq,axiom,
% 8.90/5.09 ! [N: nat,K: int] :
% 8.90/5.09 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 8.90/5.09 => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5341_option_Osize__gen_I1_J,axiom,
% 8.90/5.09 ! [X2: nat > nat] :
% 8.90/5.09 ( ( size_option_nat @ X2 @ none_nat )
% 8.90/5.09 = ( suc @ zero_zero_nat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5342_option_Osize__gen_I1_J,axiom,
% 8.90/5.09 ! [X2: product_prod_nat_nat > nat] :
% 8.90/5.09 ( ( size_o8335143837870341156at_nat @ X2 @ none_P5556105721700978146at_nat )
% 8.90/5.09 = ( suc @ zero_zero_nat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5343_option_Osize__gen_I1_J,axiom,
% 8.90/5.09 ! [X2: num > nat] :
% 8.90/5.09 ( ( size_option_num @ X2 @ none_num )
% 8.90/5.09 = ( suc @ zero_zero_nat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5344_divmod__step__nat__def,axiom,
% 8.90/5.09 ( unique5026877609467782581ep_nat
% 8.90/5.09 = ( ^ [L2: num] :
% 8.90/5.09 ( produc2626176000494625587at_nat
% 8.90/5.09 @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5345_set__decode__def,axiom,
% 8.90/5.09 ( nat_set_decode
% 8.90/5.09 = ( ^ [X: nat] :
% 8.90/5.09 ( collect_nat
% 8.90/5.09 @ ^ [N3: nat] :
% 8.90/5.09 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5346_exp__div__exp__eq,axiom,
% 8.90/5.09 ! [M: nat,N: nat] :
% 8.90/5.09 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.09 = ( times_times_nat
% 8.90/5.09 @ ( zero_n2687167440665602831ol_nat
% 8.90/5.09 @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 8.90/5.09 != zero_zero_nat )
% 8.90/5.09 & ( ord_less_eq_nat @ N @ M ) ) )
% 8.90/5.09 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5347_exp__div__exp__eq,axiom,
% 8.90/5.09 ! [M: nat,N: nat] :
% 8.90/5.09 ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.09 = ( times_times_int
% 8.90/5.09 @ ( zero_n2684676970156552555ol_int
% 8.90/5.09 @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 8.90/5.09 != zero_zero_int )
% 8.90/5.09 & ( ord_less_eq_nat @ N @ M ) ) )
% 8.90/5.09 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5348_exp__div__exp__eq,axiom,
% 8.90/5.09 ! [M: nat,N: nat] :
% 8.90/5.09 ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.09 = ( times_3573771949741848930nteger
% 8.90/5.09 @ ( zero_n356916108424825756nteger
% 8.90/5.09 @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 8.90/5.09 != zero_z3403309356797280102nteger )
% 8.90/5.09 & ( ord_less_eq_nat @ N @ M ) ) )
% 8.90/5.09 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5349_divmod__step__int__def,axiom,
% 8.90/5.09 ( unique5024387138958732305ep_int
% 8.90/5.09 = ( ^ [L2: num] :
% 8.90/5.09 ( produc4245557441103728435nt_int
% 8.90/5.09 @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5350_vebt__buildup_Osimps_I3_J,axiom,
% 8.90/5.09 ! [Va: nat] :
% 8.90/5.09 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 8.90/5.09 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 8.90/5.09 = ( 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 ) ) ) ) ) ) )
% 8.90/5.09 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 8.90/5.09 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 8.90/5.09 = ( 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 ) ) ) ) ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5351_divmod__step__def,axiom,
% 8.90/5.09 ( unique5026877609467782581ep_nat
% 8.90/5.09 = ( ^ [L2: num] :
% 8.90/5.09 ( produc2626176000494625587at_nat
% 8.90/5.09 @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5352_divmod__step__def,axiom,
% 8.90/5.09 ( unique5024387138958732305ep_int
% 8.90/5.09 = ( ^ [L2: num] :
% 8.90/5.09 ( produc4245557441103728435nt_int
% 8.90/5.09 @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5353_divmod__step__def,axiom,
% 8.90/5.09 ( unique4921790084139445826nteger
% 8.90/5.09 = ( ^ [L2: num] :
% 8.90/5.09 ( produc6916734918728496179nteger
% 8.90/5.09 @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5354_divmod__nat__if,axiom,
% 8.90/5.09 ( divmod_nat
% 8.90/5.09 = ( ^ [M3: nat,N3: nat] :
% 8.90/5.09 ( if_Pro6206227464963214023at_nat
% 8.90/5.09 @ ( ( N3 = zero_zero_nat )
% 8.90/5.09 | ( ord_less_nat @ M3 @ N3 ) )
% 8.90/5.09 @ ( product_Pair_nat_nat @ zero_zero_nat @ M3 )
% 8.90/5.09 @ ( produc2626176000494625587at_nat
% 8.90/5.09 @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
% 8.90/5.09 @ ( divmod_nat @ ( minus_minus_nat @ M3 @ N3 ) @ N3 ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5355_signed__take__bit__Suc__minus__bit1,axiom,
% 8.90/5.09 ! [N: nat,K: num] :
% 8.90/5.09 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 8.90/5.09 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5356_signed__take__bit__rec,axiom,
% 8.90/5.09 ( bit_ri6519982836138164636nteger
% 8.90/5.09 = ( ^ [N3: nat,A3: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5357_signed__take__bit__rec,axiom,
% 8.90/5.09 ( bit_ri631733984087533419it_int
% 8.90/5.09 = ( ^ [N3: nat,A3: int] : ( if_int @ ( N3 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5358_signed__take__bit__numeral__bit1,axiom,
% 8.90/5.09 ! [L: num,K: num] :
% 8.90/5.09 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 8.90/5.09 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5359_vebt__buildup_Opelims,axiom,
% 8.90/5.09 ! [X2: nat,Y3: vEBT_VEBT] :
% 8.90/5.09 ( ( ( vEBT_vebt_buildup @ X2 )
% 8.90/5.09 = Y3 )
% 8.90/5.09 => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X2 )
% 8.90/5.09 => ( ( ( X2 = zero_zero_nat )
% 8.90/5.09 => ( ( Y3
% 8.90/5.09 = ( vEBT_Leaf @ $false @ $false ) )
% 8.90/5.09 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 8.90/5.09 => ( ( ( X2
% 8.90/5.09 = ( suc @ zero_zero_nat ) )
% 8.90/5.09 => ( ( Y3
% 8.90/5.09 = ( vEBT_Leaf @ $false @ $false ) )
% 8.90/5.09 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 8.90/5.09 => ~ ! [Va3: nat] :
% 8.90/5.09 ( ( X2
% 8.90/5.09 = ( suc @ ( suc @ Va3 ) ) )
% 8.90/5.09 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 8.90/5.09 => ( Y3
% 8.90/5.09 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 8.90/5.09 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 8.90/5.09 => ( Y3
% 8.90/5.09 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 8.90/5.09 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5360_diff__shunt__var,axiom,
% 8.90/5.09 ! [X2: set_real,Y3: set_real] :
% 8.90/5.09 ( ( ( minus_minus_set_real @ X2 @ Y3 )
% 8.90/5.09 = bot_bot_set_real )
% 8.90/5.09 = ( ord_less_eq_set_real @ X2 @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5361_diff__shunt__var,axiom,
% 8.90/5.09 ! [X2: set_nat,Y3: set_nat] :
% 8.90/5.09 ( ( ( minus_minus_set_nat @ X2 @ Y3 )
% 8.90/5.09 = bot_bot_set_nat )
% 8.90/5.09 = ( ord_less_eq_set_nat @ X2 @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5362_diff__shunt__var,axiom,
% 8.90/5.09 ! [X2: set_int,Y3: set_int] :
% 8.90/5.09 ( ( ( minus_minus_set_int @ X2 @ Y3 )
% 8.90/5.09 = bot_bot_set_int )
% 8.90/5.09 = ( ord_less_eq_set_int @ X2 @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5363_add__scale__eq__noteq,axiom,
% 8.90/5.09 ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
% 8.90/5.09 ( ( R2 != zero_zero_complex )
% 8.90/5.09 => ( ( ( A = B )
% 8.90/5.09 & ( C != D ) )
% 8.90/5.09 => ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C ) )
% 8.90/5.09 != ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5364_add__scale__eq__noteq,axiom,
% 8.90/5.09 ! [R2: real,A: real,B: real,C: real,D: real] :
% 8.90/5.09 ( ( R2 != zero_zero_real )
% 8.90/5.09 => ( ( ( A = B )
% 8.90/5.09 & ( C != D ) )
% 8.90/5.09 => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
% 8.90/5.09 != ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5365_add__scale__eq__noteq,axiom,
% 8.90/5.09 ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
% 8.90/5.09 ( ( R2 != zero_zero_rat )
% 8.90/5.09 => ( ( ( A = B )
% 8.90/5.09 & ( C != D ) )
% 8.90/5.09 => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
% 8.90/5.09 != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5366_add__scale__eq__noteq,axiom,
% 8.90/5.09 ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
% 8.90/5.09 ( ( R2 != zero_zero_nat )
% 8.90/5.09 => ( ( ( A = B )
% 8.90/5.09 & ( C != D ) )
% 8.90/5.09 => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
% 8.90/5.09 != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5367_add__scale__eq__noteq,axiom,
% 8.90/5.09 ! [R2: int,A: int,B: int,C: int,D: int] :
% 8.90/5.09 ( ( R2 != zero_zero_int )
% 8.90/5.09 => ( ( ( A = B )
% 8.90/5.09 & ( C != D ) )
% 8.90/5.09 => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
% 8.90/5.09 != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5368_add_Oinverse__inverse,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 8.90/5.09 = A ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5369_add_Oinverse__inverse,axiom,
% 8.90/5.09 ! [A: real] :
% 8.90/5.09 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 8.90/5.09 = A ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5370_add_Oinverse__inverse,axiom,
% 8.90/5.09 ! [A: complex] :
% 8.90/5.09 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
% 8.90/5.09 = A ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5371_add_Oinverse__inverse,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
% 8.90/5.09 = A ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5372_add_Oinverse__inverse,axiom,
% 8.90/5.09 ! [A: rat] :
% 8.90/5.09 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
% 8.90/5.09 = A ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5373_neg__equal__iff__equal,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( ( uminus_uminus_int @ A )
% 8.90/5.09 = ( uminus_uminus_int @ B ) )
% 8.90/5.09 = ( A = B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5374_neg__equal__iff__equal,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( ( uminus_uminus_real @ A )
% 8.90/5.09 = ( uminus_uminus_real @ B ) )
% 8.90/5.09 = ( A = B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5375_neg__equal__iff__equal,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( ( uminus1482373934393186551omplex @ A )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ B ) )
% 8.90/5.09 = ( A = B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5376_neg__equal__iff__equal,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( ( uminus1351360451143612070nteger @ A )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ B ) )
% 8.90/5.09 = ( A = B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5377_neg__equal__iff__equal,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( ( uminus_uminus_rat @ A )
% 8.90/5.09 = ( uminus_uminus_rat @ B ) )
% 8.90/5.09 = ( A = B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5378_Compl__anti__mono,axiom,
% 8.90/5.09 ! [A2: set_int,B3: set_int] :
% 8.90/5.09 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 8.90/5.09 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B3 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5379_Compl__subset__Compl__iff,axiom,
% 8.90/5.09 ! [A2: set_int,B3: set_int] :
% 8.90/5.09 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B3 ) )
% 8.90/5.09 = ( ord_less_eq_set_int @ B3 @ A2 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5380_case__prodI2,axiom,
% 8.90/5.09 ! [P6: produc6271795597528267376eger_o,C: code_integer > $o > $o] :
% 8.90/5.09 ( ! [A4: code_integer,B4: $o] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( produc6677183202524767010eger_o @ A4 @ B4 ) )
% 8.90/5.09 => ( C @ A4 @ B4 ) )
% 8.90/5.09 => ( produc7828578312038201481er_o_o @ C @ P6 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5381_case__prodI2,axiom,
% 8.90/5.09 ! [P6: product_prod_num_num,C: num > num > $o] :
% 8.90/5.09 ( ! [A4: num,B4: num] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_num_num @ A4 @ B4 ) )
% 8.90/5.09 => ( C @ A4 @ B4 ) )
% 8.90/5.09 => ( produc5703948589228662326_num_o @ C @ P6 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5382_case__prodI2,axiom,
% 8.90/5.09 ! [P6: product_prod_nat_num,C: nat > num > $o] :
% 8.90/5.09 ( ! [A4: nat,B4: num] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_nat_num @ A4 @ B4 ) )
% 8.90/5.09 => ( C @ A4 @ B4 ) )
% 8.90/5.09 => ( produc4927758841916487424_num_o @ C @ P6 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5383_case__prodI2,axiom,
% 8.90/5.09 ! [P6: product_prod_nat_nat,C: nat > nat > $o] :
% 8.90/5.09 ( ! [A4: nat,B4: nat] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_nat_nat @ A4 @ B4 ) )
% 8.90/5.09 => ( C @ A4 @ B4 ) )
% 8.90/5.09 => ( produc6081775807080527818_nat_o @ C @ P6 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5384_case__prodI2,axiom,
% 8.90/5.09 ! [P6: product_prod_int_int,C: int > int > $o] :
% 8.90/5.09 ( ! [A4: int,B4: int] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_int_int @ A4 @ B4 ) )
% 8.90/5.09 => ( C @ A4 @ B4 ) )
% 8.90/5.09 => ( produc4947309494688390418_int_o @ C @ P6 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5385_case__prodI,axiom,
% 8.90/5.09 ! [F: code_integer > $o > $o,A: code_integer,B: $o] :
% 8.90/5.09 ( ( F @ A @ B )
% 8.90/5.09 => ( produc7828578312038201481er_o_o @ F @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5386_case__prodI,axiom,
% 8.90/5.09 ! [F: num > num > $o,A: num,B: num] :
% 8.90/5.09 ( ( F @ A @ B )
% 8.90/5.09 => ( produc5703948589228662326_num_o @ F @ ( product_Pair_num_num @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5387_case__prodI,axiom,
% 8.90/5.09 ! [F: nat > num > $o,A: nat,B: num] :
% 8.90/5.09 ( ( F @ A @ B )
% 8.90/5.09 => ( produc4927758841916487424_num_o @ F @ ( product_Pair_nat_num @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5388_case__prodI,axiom,
% 8.90/5.09 ! [F: nat > nat > $o,A: nat,B: nat] :
% 8.90/5.09 ( ( F @ A @ B )
% 8.90/5.09 => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5389_case__prodI,axiom,
% 8.90/5.09 ! [F: int > int > $o,A: int,B: int] :
% 8.90/5.09 ( ( F @ A @ B )
% 8.90/5.09 => ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5390_mem__case__prodI2,axiom,
% 8.90/5.09 ! [P6: produc6271795597528267376eger_o,Z: complex,C: code_integer > $o > set_complex] :
% 8.90/5.09 ( ! [A4: code_integer,B4: $o] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( produc6677183202524767010eger_o @ A4 @ B4 ) )
% 8.90/5.09 => ( member_complex @ Z @ ( C @ A4 @ B4 ) ) )
% 8.90/5.09 => ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ P6 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5391_mem__case__prodI2,axiom,
% 8.90/5.09 ! [P6: produc6271795597528267376eger_o,Z: real,C: code_integer > $o > set_real] :
% 8.90/5.09 ( ! [A4: code_integer,B4: $o] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( produc6677183202524767010eger_o @ A4 @ B4 ) )
% 8.90/5.09 => ( member_real @ Z @ ( C @ A4 @ B4 ) ) )
% 8.90/5.09 => ( member_real @ Z @ ( produc242741666403216561t_real @ C @ P6 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5392_mem__case__prodI2,axiom,
% 8.90/5.09 ! [P6: produc6271795597528267376eger_o,Z: nat,C: code_integer > $o > set_nat] :
% 8.90/5.09 ( ! [A4: code_integer,B4: $o] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( produc6677183202524767010eger_o @ A4 @ B4 ) )
% 8.90/5.09 => ( member_nat @ Z @ ( C @ A4 @ B4 ) ) )
% 8.90/5.09 => ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ P6 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5393_mem__case__prodI2,axiom,
% 8.90/5.09 ! [P6: produc6271795597528267376eger_o,Z: int,C: code_integer > $o > set_int] :
% 8.90/5.09 ( ! [A4: code_integer,B4: $o] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( produc6677183202524767010eger_o @ A4 @ B4 ) )
% 8.90/5.09 => ( member_int @ Z @ ( C @ A4 @ B4 ) ) )
% 8.90/5.09 => ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ P6 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5394_mem__case__prodI2,axiom,
% 8.90/5.09 ! [P6: product_prod_num_num,Z: complex,C: num > num > set_complex] :
% 8.90/5.09 ( ! [A4: num,B4: num] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_num_num @ A4 @ B4 ) )
% 8.90/5.09 => ( member_complex @ Z @ ( C @ A4 @ B4 ) ) )
% 8.90/5.09 => ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ P6 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5395_mem__case__prodI2,axiom,
% 8.90/5.09 ! [P6: product_prod_num_num,Z: real,C: num > num > set_real] :
% 8.90/5.09 ( ! [A4: num,B4: num] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_num_num @ A4 @ B4 ) )
% 8.90/5.09 => ( member_real @ Z @ ( C @ A4 @ B4 ) ) )
% 8.90/5.09 => ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ P6 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5396_mem__case__prodI2,axiom,
% 8.90/5.09 ! [P6: product_prod_num_num,Z: nat,C: num > num > set_nat] :
% 8.90/5.09 ( ! [A4: num,B4: num] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_num_num @ A4 @ B4 ) )
% 8.90/5.09 => ( member_nat @ Z @ ( C @ A4 @ B4 ) ) )
% 8.90/5.09 => ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ P6 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5397_mem__case__prodI2,axiom,
% 8.90/5.09 ! [P6: product_prod_num_num,Z: int,C: num > num > set_int] :
% 8.90/5.09 ( ! [A4: num,B4: num] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_num_num @ A4 @ B4 ) )
% 8.90/5.09 => ( member_int @ Z @ ( C @ A4 @ B4 ) ) )
% 8.90/5.09 => ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ P6 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5398_mem__case__prodI2,axiom,
% 8.90/5.09 ! [P6: product_prod_nat_num,Z: complex,C: nat > num > set_complex] :
% 8.90/5.09 ( ! [A4: nat,B4: num] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_nat_num @ A4 @ B4 ) )
% 8.90/5.09 => ( member_complex @ Z @ ( C @ A4 @ B4 ) ) )
% 8.90/5.09 => ( member_complex @ Z @ ( produc6231982587499038204omplex @ C @ P6 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5399_mem__case__prodI2,axiom,
% 8.90/5.09 ! [P6: product_prod_nat_num,Z: real,C: nat > num > set_real] :
% 8.90/5.09 ( ! [A4: nat,B4: num] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_nat_num @ A4 @ B4 ) )
% 8.90/5.09 => ( member_real @ Z @ ( C @ A4 @ B4 ) ) )
% 8.90/5.09 => ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ P6 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5400_mem__case__prodI,axiom,
% 8.90/5.09 ! [Z: complex,C: code_integer > $o > set_complex,A: code_integer,B: $o] :
% 8.90/5.09 ( ( member_complex @ Z @ ( C @ A @ B ) )
% 8.90/5.09 => ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5401_mem__case__prodI,axiom,
% 8.90/5.09 ! [Z: real,C: code_integer > $o > set_real,A: code_integer,B: $o] :
% 8.90/5.09 ( ( member_real @ Z @ ( C @ A @ B ) )
% 8.90/5.09 => ( member_real @ Z @ ( produc242741666403216561t_real @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5402_mem__case__prodI,axiom,
% 8.90/5.09 ! [Z: nat,C: code_integer > $o > set_nat,A: code_integer,B: $o] :
% 8.90/5.09 ( ( member_nat @ Z @ ( C @ A @ B ) )
% 8.90/5.09 => ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5403_mem__case__prodI,axiom,
% 8.90/5.09 ! [Z: int,C: code_integer > $o > set_int,A: code_integer,B: $o] :
% 8.90/5.09 ( ( member_int @ Z @ ( C @ A @ B ) )
% 8.90/5.09 => ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5404_mem__case__prodI,axiom,
% 8.90/5.09 ! [Z: complex,C: num > num > set_complex,A: num,B: num] :
% 8.90/5.09 ( ( member_complex @ Z @ ( C @ A @ B ) )
% 8.90/5.09 => ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5405_mem__case__prodI,axiom,
% 8.90/5.09 ! [Z: real,C: num > num > set_real,A: num,B: num] :
% 8.90/5.09 ( ( member_real @ Z @ ( C @ A @ B ) )
% 8.90/5.09 => ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5406_mem__case__prodI,axiom,
% 8.90/5.09 ! [Z: nat,C: num > num > set_nat,A: num,B: num] :
% 8.90/5.09 ( ( member_nat @ Z @ ( C @ A @ B ) )
% 8.90/5.09 => ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5407_mem__case__prodI,axiom,
% 8.90/5.09 ! [Z: int,C: num > num > set_int,A: num,B: num] :
% 8.90/5.09 ( ( member_int @ Z @ ( C @ A @ B ) )
% 8.90/5.09 => ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5408_mem__case__prodI,axiom,
% 8.90/5.09 ! [Z: complex,C: nat > num > set_complex,A: nat,B: num] :
% 8.90/5.09 ( ( member_complex @ Z @ ( C @ A @ B ) )
% 8.90/5.09 => ( member_complex @ Z @ ( produc6231982587499038204omplex @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5409_mem__case__prodI,axiom,
% 8.90/5.09 ! [Z: real,C: nat > num > set_real,A: nat,B: num] :
% 8.90/5.09 ( ( member_real @ Z @ ( C @ A @ B ) )
% 8.90/5.09 => ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5410_case__prodI2_H,axiom,
% 8.90/5.09 ! [P6: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X2: product_prod_nat_nat] :
% 8.90/5.09 ( ! [A4: nat,B4: nat] :
% 8.90/5.09 ( ( ( product_Pair_nat_nat @ A4 @ B4 )
% 8.90/5.09 = P6 )
% 8.90/5.09 => ( C @ A4 @ B4 @ X2 ) )
% 8.90/5.09 => ( produc8739625826339149834_nat_o @ C @ P6 @ X2 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5411_compl__le__compl__iff,axiom,
% 8.90/5.09 ! [X2: set_int,Y3: set_int] :
% 8.90/5.09 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ ( uminus1532241313380277803et_int @ Y3 ) )
% 8.90/5.09 = ( ord_less_eq_set_int @ Y3 @ X2 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5412_neg__le__iff__le,axiom,
% 8.90/5.09 ! [B: real,A: real] :
% 8.90/5.09 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 8.90/5.09 = ( ord_less_eq_real @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5413_neg__le__iff__le,axiom,
% 8.90/5.09 ! [B: code_integer,A: code_integer] :
% 8.90/5.09 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 8.90/5.09 = ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5414_neg__le__iff__le,axiom,
% 8.90/5.09 ! [B: rat,A: rat] :
% 8.90/5.09 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 8.90/5.09 = ( ord_less_eq_rat @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5415_neg__le__iff__le,axiom,
% 8.90/5.09 ! [B: int,A: int] :
% 8.90/5.09 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 8.90/5.09 = ( ord_less_eq_int @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5416_add_Oinverse__neutral,axiom,
% 8.90/5.09 ( ( uminus_uminus_int @ zero_zero_int )
% 8.90/5.09 = zero_zero_int ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5417_add_Oinverse__neutral,axiom,
% 8.90/5.09 ( ( uminus_uminus_real @ zero_zero_real )
% 8.90/5.09 = zero_zero_real ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5418_add_Oinverse__neutral,axiom,
% 8.90/5.09 ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 8.90/5.09 = zero_zero_complex ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5419_add_Oinverse__neutral,axiom,
% 8.90/5.09 ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 8.90/5.09 = zero_z3403309356797280102nteger ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5420_add_Oinverse__neutral,axiom,
% 8.90/5.09 ( ( uminus_uminus_rat @ zero_zero_rat )
% 8.90/5.09 = zero_zero_rat ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5421_neg__0__equal__iff__equal,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( zero_zero_int
% 8.90/5.09 = ( uminus_uminus_int @ A ) )
% 8.90/5.09 = ( zero_zero_int = A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5422_neg__0__equal__iff__equal,axiom,
% 8.90/5.09 ! [A: real] :
% 8.90/5.09 ( ( zero_zero_real
% 8.90/5.09 = ( uminus_uminus_real @ A ) )
% 8.90/5.09 = ( zero_zero_real = A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5423_neg__0__equal__iff__equal,axiom,
% 8.90/5.09 ! [A: complex] :
% 8.90/5.09 ( ( zero_zero_complex
% 8.90/5.09 = ( uminus1482373934393186551omplex @ A ) )
% 8.90/5.09 = ( zero_zero_complex = A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5424_neg__0__equal__iff__equal,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( zero_z3403309356797280102nteger
% 8.90/5.09 = ( uminus1351360451143612070nteger @ A ) )
% 8.90/5.09 = ( zero_z3403309356797280102nteger = A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5425_neg__0__equal__iff__equal,axiom,
% 8.90/5.09 ! [A: rat] :
% 8.90/5.09 ( ( zero_zero_rat
% 8.90/5.09 = ( uminus_uminus_rat @ A ) )
% 8.90/5.09 = ( zero_zero_rat = A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5426_neg__equal__0__iff__equal,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( ( uminus_uminus_int @ A )
% 8.90/5.09 = zero_zero_int )
% 8.90/5.09 = ( A = zero_zero_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5427_neg__equal__0__iff__equal,axiom,
% 8.90/5.09 ! [A: real] :
% 8.90/5.09 ( ( ( uminus_uminus_real @ A )
% 8.90/5.09 = zero_zero_real )
% 8.90/5.09 = ( A = zero_zero_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5428_neg__equal__0__iff__equal,axiom,
% 8.90/5.09 ! [A: complex] :
% 8.90/5.09 ( ( ( uminus1482373934393186551omplex @ A )
% 8.90/5.09 = zero_zero_complex )
% 8.90/5.09 = ( A = zero_zero_complex ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5429_neg__equal__0__iff__equal,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( ( uminus1351360451143612070nteger @ A )
% 8.90/5.09 = zero_z3403309356797280102nteger )
% 8.90/5.09 = ( A = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5430_neg__equal__0__iff__equal,axiom,
% 8.90/5.09 ! [A: rat] :
% 8.90/5.09 ( ( ( uminus_uminus_rat @ A )
% 8.90/5.09 = zero_zero_rat )
% 8.90/5.09 = ( A = zero_zero_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5431_equal__neg__zero,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( uminus_uminus_int @ A ) )
% 8.90/5.09 = ( A = zero_zero_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5432_equal__neg__zero,axiom,
% 8.90/5.09 ! [A: real] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( uminus_uminus_real @ A ) )
% 8.90/5.09 = ( A = zero_zero_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5433_equal__neg__zero,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( uminus1351360451143612070nteger @ A ) )
% 8.90/5.09 = ( A = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5434_equal__neg__zero,axiom,
% 8.90/5.09 ! [A: rat] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( uminus_uminus_rat @ A ) )
% 8.90/5.09 = ( A = zero_zero_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5435_neg__equal__zero,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( ( uminus_uminus_int @ A )
% 8.90/5.09 = A )
% 8.90/5.09 = ( A = zero_zero_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5436_neg__equal__zero,axiom,
% 8.90/5.09 ! [A: real] :
% 8.90/5.09 ( ( ( uminus_uminus_real @ A )
% 8.90/5.09 = A )
% 8.90/5.09 = ( A = zero_zero_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5437_neg__equal__zero,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( ( uminus1351360451143612070nteger @ A )
% 8.90/5.09 = A )
% 8.90/5.09 = ( A = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5438_neg__equal__zero,axiom,
% 8.90/5.09 ! [A: rat] :
% 8.90/5.09 ( ( ( uminus_uminus_rat @ A )
% 8.90/5.09 = A )
% 8.90/5.09 = ( A = zero_zero_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5439_neg__less__iff__less,axiom,
% 8.90/5.09 ! [B: int,A: int] :
% 8.90/5.09 ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 8.90/5.09 = ( ord_less_int @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5440_neg__less__iff__less,axiom,
% 8.90/5.09 ! [B: real,A: real] :
% 8.90/5.09 ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 8.90/5.09 = ( ord_less_real @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5441_neg__less__iff__less,axiom,
% 8.90/5.09 ! [B: code_integer,A: code_integer] :
% 8.90/5.09 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 8.90/5.09 = ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5442_neg__less__iff__less,axiom,
% 8.90/5.09 ! [B: rat,A: rat] :
% 8.90/5.09 ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 8.90/5.09 = ( ord_less_rat @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5443_neg__numeral__eq__iff,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 8.90/5.09 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.09 = ( M = N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5444_neg__numeral__eq__iff,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 8.90/5.09 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.90/5.09 = ( M = N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5445_neg__numeral__eq__iff,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 8.90/5.09 = ( M = N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5446_neg__numeral__eq__iff,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.90/5.09 = ( M = N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5447_neg__numeral__eq__iff,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 8.90/5.09 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.90/5.09 = ( M = N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5448_mult__minus__right,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
% 8.90/5.09 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5449_mult__minus__right,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
% 8.90/5.09 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5450_mult__minus__right,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5451_mult__minus__right,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5452_mult__minus__right,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
% 8.90/5.09 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5453_minus__mult__minus,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 8.90/5.09 = ( times_times_int @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5454_minus__mult__minus,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 8.90/5.09 = ( times_times_real @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5455_minus__mult__minus,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 8.90/5.09 = ( times_times_complex @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5456_minus__mult__minus,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 8.90/5.09 = ( times_3573771949741848930nteger @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5457_minus__mult__minus,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 8.90/5.09 = ( times_times_rat @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5458_mult__minus__left,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 8.90/5.09 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5459_mult__minus__left,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 8.90/5.09 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5460_mult__minus__left,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5461_mult__minus__left,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5462_mult__minus__left,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 8.90/5.09 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5463_add__minus__cancel,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 8.90/5.09 = B ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5464_add__minus__cancel,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
% 8.90/5.09 = B ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5465_add__minus__cancel,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
% 8.90/5.09 = B ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5466_add__minus__cancel,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
% 8.90/5.09 = B ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5467_add__minus__cancel,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
% 8.90/5.09 = B ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5468_minus__add__cancel,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 8.90/5.09 = B ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5469_minus__add__cancel,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
% 8.90/5.09 = B ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5470_minus__add__cancel,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
% 8.90/5.09 = B ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5471_minus__add__cancel,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 8.90/5.09 = B ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5472_minus__add__cancel,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
% 8.90/5.09 = B ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5473_minus__add__distrib,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 8.90/5.09 = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5474_minus__add__distrib,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 8.90/5.09 = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5475_minus__add__distrib,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 8.90/5.09 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5476_minus__add__distrib,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 8.90/5.09 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5477_minus__add__distrib,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 8.90/5.09 = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5478_minus__diff__eq,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 8.90/5.09 = ( minus_minus_int @ B @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5479_minus__diff__eq,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
% 8.90/5.09 = ( minus_minus_real @ B @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5480_minus__diff__eq,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
% 8.90/5.09 = ( minus_minus_complex @ B @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5481_minus__diff__eq,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
% 8.90/5.09 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5482_minus__diff__eq,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
% 8.90/5.09 = ( minus_minus_rat @ B @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5483_div__minus__minus,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 8.90/5.09 = ( divide_divide_int @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5484_div__minus__minus,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 8.90/5.09 = ( divide6298287555418463151nteger @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5485_dvd__minus__iff,axiom,
% 8.90/5.09 ! [X2: int,Y3: int] :
% 8.90/5.09 ( ( dvd_dvd_int @ X2 @ ( uminus_uminus_int @ Y3 ) )
% 8.90/5.09 = ( dvd_dvd_int @ X2 @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5486_dvd__minus__iff,axiom,
% 8.90/5.09 ! [X2: real,Y3: real] :
% 8.90/5.09 ( ( dvd_dvd_real @ X2 @ ( uminus_uminus_real @ Y3 ) )
% 8.90/5.09 = ( dvd_dvd_real @ X2 @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5487_dvd__minus__iff,axiom,
% 8.90/5.09 ! [X2: complex,Y3: complex] :
% 8.90/5.09 ( ( dvd_dvd_complex @ X2 @ ( uminus1482373934393186551omplex @ Y3 ) )
% 8.90/5.09 = ( dvd_dvd_complex @ X2 @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5488_dvd__minus__iff,axiom,
% 8.90/5.09 ! [X2: code_integer,Y3: code_integer] :
% 8.90/5.09 ( ( dvd_dvd_Code_integer @ X2 @ ( uminus1351360451143612070nteger @ Y3 ) )
% 8.90/5.09 = ( dvd_dvd_Code_integer @ X2 @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5489_dvd__minus__iff,axiom,
% 8.90/5.09 ! [X2: rat,Y3: rat] :
% 8.90/5.09 ( ( dvd_dvd_rat @ X2 @ ( uminus_uminus_rat @ Y3 ) )
% 8.90/5.09 = ( dvd_dvd_rat @ X2 @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5490_minus__dvd__iff,axiom,
% 8.90/5.09 ! [X2: int,Y3: int] :
% 8.90/5.09 ( ( dvd_dvd_int @ ( uminus_uminus_int @ X2 ) @ Y3 )
% 8.90/5.09 = ( dvd_dvd_int @ X2 @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5491_minus__dvd__iff,axiom,
% 8.90/5.09 ! [X2: real,Y3: real] :
% 8.90/5.09 ( ( dvd_dvd_real @ ( uminus_uminus_real @ X2 ) @ Y3 )
% 8.90/5.09 = ( dvd_dvd_real @ X2 @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5492_minus__dvd__iff,axiom,
% 8.90/5.09 ! [X2: complex,Y3: complex] :
% 8.90/5.09 ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X2 ) @ Y3 )
% 8.90/5.09 = ( dvd_dvd_complex @ X2 @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5493_minus__dvd__iff,axiom,
% 8.90/5.09 ! [X2: code_integer,Y3: code_integer] :
% 8.90/5.09 ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X2 ) @ Y3 )
% 8.90/5.09 = ( dvd_dvd_Code_integer @ X2 @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5494_minus__dvd__iff,axiom,
% 8.90/5.09 ! [X2: rat,Y3: rat] :
% 8.90/5.09 ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X2 ) @ Y3 )
% 8.90/5.09 = ( dvd_dvd_rat @ X2 @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5495_mod__minus__minus,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 8.90/5.09 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5496_mod__minus__minus,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5497_real__add__minus__iff,axiom,
% 8.90/5.09 ! [X2: real,A: real] :
% 8.90/5.09 ( ( ( plus_plus_real @ X2 @ ( uminus_uminus_real @ A ) )
% 8.90/5.09 = zero_zero_real )
% 8.90/5.09 = ( X2 = A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5498_neg__less__eq__nonneg,axiom,
% 8.90/5.09 ! [A: real] :
% 8.90/5.09 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 8.90/5.09 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5499_neg__less__eq__nonneg,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 8.90/5.09 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5500_neg__less__eq__nonneg,axiom,
% 8.90/5.09 ! [A: rat] :
% 8.90/5.09 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 8.90/5.09 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5501_neg__less__eq__nonneg,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 8.90/5.09 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5502_less__eq__neg__nonpos,axiom,
% 8.90/5.09 ! [A: real] :
% 8.90/5.09 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 8.90/5.09 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5503_less__eq__neg__nonpos,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 8.90/5.09 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5504_less__eq__neg__nonpos,axiom,
% 8.90/5.09 ! [A: rat] :
% 8.90/5.09 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 8.90/5.09 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5505_less__eq__neg__nonpos,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 8.90/5.09 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5506_neg__le__0__iff__le,axiom,
% 8.90/5.09 ! [A: real] :
% 8.90/5.09 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 8.90/5.09 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5507_neg__le__0__iff__le,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 8.90/5.09 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5508_neg__le__0__iff__le,axiom,
% 8.90/5.09 ! [A: rat] :
% 8.90/5.09 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 8.90/5.09 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5509_neg__le__0__iff__le,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 8.90/5.09 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5510_neg__0__le__iff__le,axiom,
% 8.90/5.09 ! [A: real] :
% 8.90/5.09 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 8.90/5.09 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5511_neg__0__le__iff__le,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 8.90/5.09 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5512_neg__0__le__iff__le,axiom,
% 8.90/5.09 ! [A: rat] :
% 8.90/5.09 ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 8.90/5.09 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5513_neg__0__le__iff__le,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 8.90/5.09 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5514_neg__less__0__iff__less,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 8.90/5.09 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5515_neg__less__0__iff__less,axiom,
% 8.90/5.09 ! [A: real] :
% 8.90/5.09 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 8.90/5.09 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5516_neg__less__0__iff__less,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 8.90/5.09 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5517_neg__less__0__iff__less,axiom,
% 8.90/5.09 ! [A: rat] :
% 8.90/5.09 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 8.90/5.09 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5518_neg__0__less__iff__less,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 8.90/5.09 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5519_neg__0__less__iff__less,axiom,
% 8.90/5.09 ! [A: real] :
% 8.90/5.09 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 8.90/5.09 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5520_neg__0__less__iff__less,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 8.90/5.09 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5521_neg__0__less__iff__less,axiom,
% 8.90/5.09 ! [A: rat] :
% 8.90/5.09 ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 8.90/5.09 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5522_neg__less__pos,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 8.90/5.09 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5523_neg__less__pos,axiom,
% 8.90/5.09 ! [A: real] :
% 8.90/5.09 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 8.90/5.09 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5524_neg__less__pos,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 8.90/5.09 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5525_neg__less__pos,axiom,
% 8.90/5.09 ! [A: rat] :
% 8.90/5.09 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 8.90/5.09 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5526_less__neg__neg,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 8.90/5.09 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5527_less__neg__neg,axiom,
% 8.90/5.09 ! [A: real] :
% 8.90/5.09 ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 8.90/5.09 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5528_less__neg__neg,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 8.90/5.09 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5529_less__neg__neg,axiom,
% 8.90/5.09 ! [A: rat] :
% 8.90/5.09 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 8.90/5.09 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5530_ab__left__minus,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 8.90/5.09 = zero_zero_int ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5531_ab__left__minus,axiom,
% 8.90/5.09 ! [A: real] :
% 8.90/5.09 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 8.90/5.09 = zero_zero_real ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5532_ab__left__minus,axiom,
% 8.90/5.09 ! [A: complex] :
% 8.90/5.09 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 8.90/5.09 = zero_zero_complex ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5533_ab__left__minus,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 8.90/5.09 = zero_z3403309356797280102nteger ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5534_ab__left__minus,axiom,
% 8.90/5.09 ! [A: rat] :
% 8.90/5.09 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 8.90/5.09 = zero_zero_rat ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5535_add_Oright__inverse,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 8.90/5.09 = zero_zero_int ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5536_add_Oright__inverse,axiom,
% 8.90/5.09 ! [A: real] :
% 8.90/5.09 ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 8.90/5.09 = zero_zero_real ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5537_add_Oright__inverse,axiom,
% 8.90/5.09 ! [A: complex] :
% 8.90/5.09 ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 8.90/5.09 = zero_zero_complex ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5538_add_Oright__inverse,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 8.90/5.09 = zero_z3403309356797280102nteger ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5539_add_Oright__inverse,axiom,
% 8.90/5.09 ! [A: rat] :
% 8.90/5.09 ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 8.90/5.09 = zero_zero_rat ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5540_diff__0,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( minus_minus_int @ zero_zero_int @ A )
% 8.90/5.09 = ( uminus_uminus_int @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5541_diff__0,axiom,
% 8.90/5.09 ! [A: real] :
% 8.90/5.09 ( ( minus_minus_real @ zero_zero_real @ A )
% 8.90/5.09 = ( uminus_uminus_real @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5542_diff__0,axiom,
% 8.90/5.09 ! [A: complex] :
% 8.90/5.09 ( ( minus_minus_complex @ zero_zero_complex @ A )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5543_diff__0,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5544_diff__0,axiom,
% 8.90/5.09 ! [A: rat] :
% 8.90/5.09 ( ( minus_minus_rat @ zero_zero_rat @ A )
% 8.90/5.09 = ( uminus_uminus_rat @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5545_verit__minus__simplify_I3_J,axiom,
% 8.90/5.09 ! [B: int] :
% 8.90/5.09 ( ( minus_minus_int @ zero_zero_int @ B )
% 8.90/5.09 = ( uminus_uminus_int @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5546_verit__minus__simplify_I3_J,axiom,
% 8.90/5.09 ! [B: real] :
% 8.90/5.09 ( ( minus_minus_real @ zero_zero_real @ B )
% 8.90/5.09 = ( uminus_uminus_real @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5547_verit__minus__simplify_I3_J,axiom,
% 8.90/5.09 ! [B: complex] :
% 8.90/5.09 ( ( minus_minus_complex @ zero_zero_complex @ B )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5548_verit__minus__simplify_I3_J,axiom,
% 8.90/5.09 ! [B: code_integer] :
% 8.90/5.09 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5549_verit__minus__simplify_I3_J,axiom,
% 8.90/5.09 ! [B: rat] :
% 8.90/5.09 ( ( minus_minus_rat @ zero_zero_rat @ B )
% 8.90/5.09 = ( uminus_uminus_rat @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5550_add__neg__numeral__simps_I3_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.09 = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5551_add__neg__numeral__simps_I3_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.90/5.09 = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5552_add__neg__numeral__simps_I3_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5553_add__neg__numeral__simps_I3_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5554_add__neg__numeral__simps_I3_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.90/5.09 = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5555_mult__minus1__right,axiom,
% 8.90/5.09 ! [Z: int] :
% 8.90/5.09 ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.09 = ( uminus_uminus_int @ Z ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5556_mult__minus1__right,axiom,
% 8.90/5.09 ! [Z: real] :
% 8.90/5.09 ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
% 8.90/5.09 = ( uminus_uminus_real @ Z ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5557_mult__minus1__right,axiom,
% 8.90/5.09 ! [Z: complex] :
% 8.90/5.09 ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5558_mult__minus1__right,axiom,
% 8.90/5.09 ! [Z: code_integer] :
% 8.90/5.09 ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5559_mult__minus1__right,axiom,
% 8.90/5.09 ! [Z: rat] :
% 8.90/5.09 ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.90/5.09 = ( uminus_uminus_rat @ Z ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5560_mult__minus1,axiom,
% 8.90/5.09 ! [Z: int] :
% 8.90/5.09 ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
% 8.90/5.09 = ( uminus_uminus_int @ Z ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5561_mult__minus1,axiom,
% 8.90/5.09 ! [Z: real] :
% 8.90/5.09 ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
% 8.90/5.09 = ( uminus_uminus_real @ Z ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5562_mult__minus1,axiom,
% 8.90/5.09 ! [Z: complex] :
% 8.90/5.09 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5563_mult__minus1,axiom,
% 8.90/5.09 ! [Z: code_integer] :
% 8.90/5.09 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5564_mult__minus1,axiom,
% 8.90/5.09 ! [Z: rat] :
% 8.90/5.09 ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
% 8.90/5.09 = ( uminus_uminus_rat @ Z ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5565_uminus__add__conv__diff,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 8.90/5.09 = ( minus_minus_int @ B @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5566_uminus__add__conv__diff,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
% 8.90/5.09 = ( minus_minus_real @ B @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5567_uminus__add__conv__diff,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 8.90/5.09 = ( minus_minus_complex @ B @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5568_uminus__add__conv__diff,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 8.90/5.09 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5569_uminus__add__conv__diff,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
% 8.90/5.09 = ( minus_minus_rat @ B @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5570_diff__minus__eq__add,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 8.90/5.09 = ( plus_plus_int @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5571_diff__minus__eq__add,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
% 8.90/5.09 = ( plus_plus_real @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5572_diff__minus__eq__add,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 8.90/5.09 = ( plus_plus_complex @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5573_diff__minus__eq__add,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 8.90/5.09 = ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5574_diff__minus__eq__add,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
% 8.90/5.09 = ( plus_plus_rat @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5575_divide__minus1,axiom,
% 8.90/5.09 ! [X2: real] :
% 8.90/5.09 ( ( divide_divide_real @ X2 @ ( uminus_uminus_real @ one_one_real ) )
% 8.90/5.09 = ( uminus_uminus_real @ X2 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5576_divide__minus1,axiom,
% 8.90/5.09 ! [X2: complex] :
% 8.90/5.09 ( ( divide1717551699836669952omplex @ X2 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ X2 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5577_divide__minus1,axiom,
% 8.90/5.09 ! [X2: rat] :
% 8.90/5.09 ( ( divide_divide_rat @ X2 @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.90/5.09 = ( uminus_uminus_rat @ X2 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5578_div__minus1__right,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.09 = ( uminus_uminus_int @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5579_div__minus1__right,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5580_minus__mod__self1,axiom,
% 8.90/5.09 ! [B: int,A: int] :
% 8.90/5.09 ( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
% 8.90/5.09 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5581_minus__mod__self1,axiom,
% 8.90/5.09 ! [B: code_integer,A: code_integer] :
% 8.90/5.09 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ B )
% 8.90/5.09 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5582_signed__take__bit__of__minus__1,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5583_signed__take__bit__of__minus__1,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.09 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5584_pred__numeral__simps_I1_J,axiom,
% 8.90/5.09 ( ( pred_numeral @ one )
% 8.90/5.09 = zero_zero_nat ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5585_Suc__eq__numeral,axiom,
% 8.90/5.09 ! [N: nat,K: num] :
% 8.90/5.09 ( ( ( suc @ N )
% 8.90/5.09 = ( numeral_numeral_nat @ K ) )
% 8.90/5.09 = ( N
% 8.90/5.09 = ( pred_numeral @ K ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5586_eq__numeral__Suc,axiom,
% 8.90/5.09 ! [K: num,N: nat] :
% 8.90/5.09 ( ( ( numeral_numeral_nat @ K )
% 8.90/5.09 = ( suc @ N ) )
% 8.90/5.09 = ( ( pred_numeral @ K )
% 8.90/5.09 = N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5587_dbl__simps_I1_J,axiom,
% 8.90/5.09 ! [K: num] :
% 8.90/5.09 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.90/5.09 = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5588_dbl__simps_I1_J,axiom,
% 8.90/5.09 ! [K: num] :
% 8.90/5.09 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 8.90/5.09 = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5589_dbl__simps_I1_J,axiom,
% 8.90/5.09 ! [K: num] :
% 8.90/5.09 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5590_dbl__simps_I1_J,axiom,
% 8.90/5.09 ! [K: num] :
% 8.90/5.09 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5591_dbl__simps_I1_J,axiom,
% 8.90/5.09 ! [K: num] :
% 8.90/5.09 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 8.90/5.09 = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5592_dbl__inc__simps_I4_J,axiom,
% 8.90/5.09 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.09 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5593_dbl__inc__simps_I4_J,axiom,
% 8.90/5.09 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
% 8.90/5.09 = ( uminus_uminus_real @ one_one_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5594_dbl__inc__simps_I4_J,axiom,
% 8.90/5.09 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5595_dbl__inc__simps_I4_J,axiom,
% 8.90/5.09 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5596_dbl__inc__simps_I4_J,axiom,
% 8.90/5.09 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.90/5.09 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5597_add__neg__numeral__special_I8_J,axiom,
% 8.90/5.09 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 8.90/5.09 = zero_zero_int ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5598_add__neg__numeral__special_I8_J,axiom,
% 8.90/5.09 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 8.90/5.09 = zero_zero_real ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5599_add__neg__numeral__special_I8_J,axiom,
% 8.90/5.09 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 8.90/5.09 = zero_zero_complex ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5600_add__neg__numeral__special_I8_J,axiom,
% 8.90/5.09 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 8.90/5.09 = zero_z3403309356797280102nteger ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5601_add__neg__numeral__special_I8_J,axiom,
% 8.90/5.09 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 8.90/5.09 = zero_zero_rat ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5602_add__neg__numeral__special_I7_J,axiom,
% 8.90/5.09 ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.09 = zero_zero_int ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5603_add__neg__numeral__special_I7_J,axiom,
% 8.90/5.09 ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 8.90/5.09 = zero_zero_real ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5604_add__neg__numeral__special_I7_J,axiom,
% 8.90/5.09 ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.90/5.09 = zero_zero_complex ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5605_add__neg__numeral__special_I7_J,axiom,
% 8.90/5.09 ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.90/5.09 = zero_z3403309356797280102nteger ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5606_add__neg__numeral__special_I7_J,axiom,
% 8.90/5.09 ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.90/5.09 = zero_zero_rat ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5607_diff__numeral__special_I12_J,axiom,
% 8.90/5.09 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.09 = zero_zero_int ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5608_diff__numeral__special_I12_J,axiom,
% 8.90/5.09 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 8.90/5.09 = zero_zero_real ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5609_diff__numeral__special_I12_J,axiom,
% 8.90/5.09 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.90/5.09 = zero_zero_complex ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5610_diff__numeral__special_I12_J,axiom,
% 8.90/5.09 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.90/5.09 = zero_z3403309356797280102nteger ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5611_diff__numeral__special_I12_J,axiom,
% 8.90/5.09 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.90/5.09 = zero_zero_rat ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5612_neg__one__eq__numeral__iff,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( ( uminus_uminus_int @ one_one_int )
% 8.90/5.09 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.09 = ( N = one ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5613_neg__one__eq__numeral__iff,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( ( uminus_uminus_real @ one_one_real )
% 8.90/5.09 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.90/5.09 = ( N = one ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5614_neg__one__eq__numeral__iff,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( ( uminus1482373934393186551omplex @ one_one_complex )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 8.90/5.09 = ( N = one ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5615_neg__one__eq__numeral__iff,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.90/5.09 = ( N = one ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5616_neg__one__eq__numeral__iff,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( ( uminus_uminus_rat @ one_one_rat )
% 8.90/5.09 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.90/5.09 = ( N = one ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5617_numeral__eq__neg__one__iff,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
% 8.90/5.09 = ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.09 = ( N = one ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5618_numeral__eq__neg__one__iff,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
% 8.90/5.09 = ( uminus_uminus_real @ one_one_real ) )
% 8.90/5.09 = ( N = one ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5619_numeral__eq__neg__one__iff,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.90/5.09 = ( N = one ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5620_numeral__eq__neg__one__iff,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.90/5.09 = ( N = one ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5621_numeral__eq__neg__one__iff,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
% 8.90/5.09 = ( uminus_uminus_rat @ one_one_rat ) )
% 8.90/5.09 = ( N = one ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5622_left__minus__one__mult__self,axiom,
% 8.90/5.09 ! [N: nat,A: int] :
% 8.90/5.09 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
% 8.90/5.09 = A ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5623_left__minus__one__mult__self,axiom,
% 8.90/5.09 ! [N: nat,A: real] :
% 8.90/5.09 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
% 8.90/5.09 = A ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5624_left__minus__one__mult__self,axiom,
% 8.90/5.09 ! [N: nat,A: complex] :
% 8.90/5.09 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
% 8.90/5.09 = A ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5625_left__minus__one__mult__self,axiom,
% 8.90/5.09 ! [N: nat,A: code_integer] :
% 8.90/5.09 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
% 8.90/5.09 = A ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5626_left__minus__one__mult__self,axiom,
% 8.90/5.09 ! [N: nat,A: rat] :
% 8.90/5.09 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ A ) )
% 8.90/5.09 = A ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5627_minus__one__mult__self,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
% 8.90/5.09 = one_one_int ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5628_minus__one__mult__self,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
% 8.90/5.09 = one_one_real ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5629_minus__one__mult__self,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
% 8.90/5.09 = one_one_complex ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5630_minus__one__mult__self,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
% 8.90/5.09 = one_one_Code_integer ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5631_minus__one__mult__self,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
% 8.90/5.09 = one_one_rat ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5632_mod__minus1__right,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.09 = zero_zero_int ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5633_mod__minus1__right,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.90/5.09 = zero_z3403309356797280102nteger ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5634_max__number__of_I2_J,axiom,
% 8.90/5.09 ! [U: num,V: num] :
% 8.90/5.09 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.90/5.09 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.90/5.09 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 8.90/5.09 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.90/5.09 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.90/5.09 = ( numeral_numeral_real @ U ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5635_max__number__of_I2_J,axiom,
% 8.90/5.09 ! [U: num,V: num] :
% 8.90/5.09 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 8.90/5.09 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 8.90/5.09 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 8.90/5.09 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 8.90/5.09 = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5636_max__number__of_I2_J,axiom,
% 8.90/5.09 ! [U: num,V: num] :
% 8.90/5.09 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.90/5.09 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.90/5.09 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 8.90/5.09 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.90/5.09 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.90/5.09 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5637_max__number__of_I2_J,axiom,
% 8.90/5.09 ! [U: num,V: num] :
% 8.90/5.09 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.90/5.09 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.90/5.09 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 8.90/5.09 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.90/5.09 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.90/5.09 = ( numeral_numeral_int @ U ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5638_max__number__of_I3_J,axiom,
% 8.90/5.09 ! [U: num,V: num] :
% 8.90/5.09 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 8.90/5.09 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 8.90/5.09 = ( numeral_numeral_real @ V ) ) )
% 8.90/5.09 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 8.90/5.09 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 8.90/5.09 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5639_max__number__of_I3_J,axiom,
% 8.90/5.09 ! [U: num,V: num] :
% 8.90/5.09 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 8.90/5.09 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 8.90/5.09 = ( numera6620942414471956472nteger @ V ) ) )
% 8.90/5.09 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 8.90/5.09 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5640_max__number__of_I3_J,axiom,
% 8.90/5.09 ! [U: num,V: num] :
% 8.90/5.09 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 8.90/5.09 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 8.90/5.09 = ( numeral_numeral_rat @ V ) ) )
% 8.90/5.09 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 8.90/5.09 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 8.90/5.09 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5641_max__number__of_I3_J,axiom,
% 8.90/5.09 ! [U: num,V: num] :
% 8.90/5.09 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 8.90/5.09 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 8.90/5.09 = ( numeral_numeral_int @ V ) ) )
% 8.90/5.09 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 8.90/5.09 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 8.90/5.09 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5642_max__number__of_I4_J,axiom,
% 8.90/5.09 ! [U: num,V: num] :
% 8.90/5.09 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.90/5.09 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.90/5.09 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 8.90/5.09 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.90/5.09 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.90/5.09 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5643_max__number__of_I4_J,axiom,
% 8.90/5.09 ! [U: num,V: num] :
% 8.90/5.09 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 8.90/5.09 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 8.90/5.09 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 8.90/5.09 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5644_max__number__of_I4_J,axiom,
% 8.90/5.09 ! [U: num,V: num] :
% 8.90/5.09 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.90/5.09 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.90/5.09 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 8.90/5.09 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.90/5.09 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.90/5.09 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5645_max__number__of_I4_J,axiom,
% 8.90/5.09 ! [U: num,V: num] :
% 8.90/5.09 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.90/5.09 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.90/5.09 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 8.90/5.09 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.90/5.09 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.90/5.09 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5646_semiring__norm_I168_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: int] :
% 8.90/5.09 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 8.90/5.09 = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5647_semiring__norm_I168_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: real] :
% 8.90/5.09 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 8.90/5.09 = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5648_semiring__norm_I168_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: complex] :
% 8.90/5.09 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 8.90/5.09 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5649_semiring__norm_I168_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: code_integer] :
% 8.90/5.09 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 8.90/5.09 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5650_semiring__norm_I168_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: rat] :
% 8.90/5.09 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 8.90/5.09 = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5651_diff__numeral__simps_I3_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.90/5.09 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5652_diff__numeral__simps_I3_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 8.90/5.09 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5653_diff__numeral__simps_I3_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5654_diff__numeral__simps_I3_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5655_diff__numeral__simps_I3_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 8.90/5.09 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5656_diff__numeral__simps_I2_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.09 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5657_diff__numeral__simps_I2_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.90/5.09 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5658_diff__numeral__simps_I2_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 8.90/5.09 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5659_diff__numeral__simps_I2_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.90/5.09 = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5660_diff__numeral__simps_I2_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.90/5.09 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5661_semiring__norm_I172_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: int] :
% 8.90/5.09 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 8.90/5.09 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5662_semiring__norm_I172_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: real] :
% 8.90/5.09 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 8.90/5.09 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5663_semiring__norm_I172_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: complex] :
% 8.90/5.09 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 8.90/5.09 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5664_semiring__norm_I172_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: code_integer] :
% 8.90/5.09 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 8.90/5.09 = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5665_semiring__norm_I172_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: rat] :
% 8.90/5.09 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 8.90/5.09 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5666_semiring__norm_I171_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: int] :
% 8.90/5.09 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 8.90/5.09 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5667_semiring__norm_I171_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: real] :
% 8.90/5.09 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 8.90/5.09 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5668_semiring__norm_I171_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: complex] :
% 8.90/5.09 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 8.90/5.09 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5669_semiring__norm_I171_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: code_integer] :
% 8.90/5.09 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 8.90/5.09 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5670_semiring__norm_I171_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: rat] :
% 8.90/5.09 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 8.90/5.09 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5671_semiring__norm_I170_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: int] :
% 8.90/5.09 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y3 ) )
% 8.90/5.09 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5672_semiring__norm_I170_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: real] :
% 8.90/5.09 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y3 ) )
% 8.90/5.09 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5673_semiring__norm_I170_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: complex] :
% 8.90/5.09 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y3 ) )
% 8.90/5.09 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5674_semiring__norm_I170_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: code_integer] :
% 8.90/5.09 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y3 ) )
% 8.90/5.09 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5675_semiring__norm_I170_J,axiom,
% 8.90/5.09 ! [V: num,W: num,Y3: rat] :
% 8.90/5.09 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y3 ) )
% 8.90/5.09 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5676_mult__neg__numeral__simps_I1_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.09 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5677_mult__neg__numeral__simps_I1_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.90/5.09 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5678_mult__neg__numeral__simps_I1_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 8.90/5.09 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5679_mult__neg__numeral__simps_I1_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.90/5.09 = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5680_mult__neg__numeral__simps_I1_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.90/5.09 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5681_mult__neg__numeral__simps_I2_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.90/5.09 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5682_mult__neg__numeral__simps_I2_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 8.90/5.09 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5683_mult__neg__numeral__simps_I2_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5684_mult__neg__numeral__simps_I2_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5685_mult__neg__numeral__simps_I2_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 8.90/5.09 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5686_mult__neg__numeral__simps_I3_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.09 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5687_mult__neg__numeral__simps_I3_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.90/5.09 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5688_mult__neg__numeral__simps_I3_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5689_mult__neg__numeral__simps_I3_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5690_mult__neg__numeral__simps_I3_J,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.90/5.09 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5691_less__numeral__Suc,axiom,
% 8.90/5.09 ! [K: num,N: nat] :
% 8.90/5.09 ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 8.90/5.09 = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5692_less__Suc__numeral,axiom,
% 8.90/5.09 ! [N: nat,K: num] :
% 8.90/5.09 ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 8.90/5.09 = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5693_pred__numeral__simps_I3_J,axiom,
% 8.90/5.09 ! [K: num] :
% 8.90/5.09 ( ( pred_numeral @ ( bit1 @ K ) )
% 8.90/5.09 = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5694_le__numeral__Suc,axiom,
% 8.90/5.09 ! [K: num,N: nat] :
% 8.90/5.09 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 8.90/5.09 = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5695_le__Suc__numeral,axiom,
% 8.90/5.09 ! [N: nat,K: num] :
% 8.90/5.09 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 8.90/5.09 = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5696_neg__numeral__le__iff,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.90/5.09 = ( ord_less_eq_num @ N @ M ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5697_neg__numeral__le__iff,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.90/5.09 = ( ord_less_eq_num @ N @ M ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5698_neg__numeral__le__iff,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.90/5.09 = ( ord_less_eq_num @ N @ M ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5699_neg__numeral__le__iff,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.09 = ( ord_less_eq_num @ N @ M ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5700_diff__Suc__numeral,axiom,
% 8.90/5.09 ! [N: nat,K: num] :
% 8.90/5.09 ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 8.90/5.09 = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5701_diff__numeral__Suc,axiom,
% 8.90/5.09 ! [K: num,N: nat] :
% 8.90/5.09 ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 8.90/5.09 = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5702_neg__numeral__less__iff,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.09 = ( ord_less_num @ N @ M ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5703_neg__numeral__less__iff,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.90/5.09 = ( ord_less_num @ N @ M ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5704_neg__numeral__less__iff,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.90/5.09 = ( ord_less_num @ N @ M ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5705_neg__numeral__less__iff,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.90/5.09 = ( ord_less_num @ N @ M ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5706_max__Suc__numeral,axiom,
% 8.90/5.09 ! [N: nat,K: num] :
% 8.90/5.09 ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 8.90/5.09 = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5707_max__numeral__Suc,axiom,
% 8.90/5.09 ! [K: num,N: nat] :
% 8.90/5.09 ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 8.90/5.09 = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5708_pred__numeral__simps_I2_J,axiom,
% 8.90/5.09 ! [K: num] :
% 8.90/5.09 ( ( pred_numeral @ ( bit0 @ K ) )
% 8.90/5.09 = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5709_not__neg__one__le__neg__numeral__iff,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 8.90/5.09 = ( M != one ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5710_not__neg__one__le__neg__numeral__iff,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 8.90/5.09 = ( M != one ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5711_not__neg__one__le__neg__numeral__iff,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 8.90/5.09 = ( M != one ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5712_not__neg__one__le__neg__numeral__iff,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 8.90/5.09 = ( M != one ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5713_neg__numeral__less__neg__one__iff,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.09 = ( M != one ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5714_neg__numeral__less__neg__one__iff,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 8.90/5.09 = ( M != one ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5715_neg__numeral__less__neg__one__iff,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.90/5.09 = ( M != one ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5716_neg__numeral__less__neg__one__iff,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.90/5.09 = ( M != one ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5717_le__divide__eq__numeral1_I2_J,axiom,
% 8.90/5.09 ! [A: real,B: real,W: num] :
% 8.90/5.09 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 8.90/5.09 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5718_le__divide__eq__numeral1_I2_J,axiom,
% 8.90/5.09 ! [A: rat,B: rat,W: num] :
% 8.90/5.09 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 8.90/5.09 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5719_divide__le__eq__numeral1_I2_J,axiom,
% 8.90/5.09 ! [B: real,W: num,A: real] :
% 8.90/5.09 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 8.90/5.09 = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5720_divide__le__eq__numeral1_I2_J,axiom,
% 8.90/5.09 ! [B: rat,W: num,A: rat] :
% 8.90/5.09 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 8.90/5.09 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5721_eq__divide__eq__numeral1_I2_J,axiom,
% 8.90/5.09 ! [A: real,B: real,W: num] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 8.90/5.09 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 8.90/5.09 != zero_zero_real )
% 8.90/5.09 => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 8.90/5.09 = B ) )
% 8.90/5.09 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 8.90/5.09 = zero_zero_real )
% 8.90/5.09 => ( A = zero_zero_real ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5722_eq__divide__eq__numeral1_I2_J,axiom,
% 8.90/5.09 ! [A: complex,B: complex,W: num] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
% 8.90/5.09 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 8.90/5.09 != zero_zero_complex )
% 8.90/5.09 => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 8.90/5.09 = B ) )
% 8.90/5.09 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 8.90/5.09 = zero_zero_complex )
% 8.90/5.09 => ( A = zero_zero_complex ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5723_eq__divide__eq__numeral1_I2_J,axiom,
% 8.90/5.09 ! [A: rat,B: rat,W: num] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 8.90/5.09 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 8.90/5.09 != zero_zero_rat )
% 8.90/5.09 => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 8.90/5.09 = B ) )
% 8.90/5.09 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 8.90/5.09 = zero_zero_rat )
% 8.90/5.09 => ( A = zero_zero_rat ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5724_divide__eq__eq__numeral1_I2_J,axiom,
% 8.90/5.09 ! [B: real,W: num,A: real] :
% 8.90/5.09 ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 8.90/5.09 = A )
% 8.90/5.09 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 8.90/5.09 != zero_zero_real )
% 8.90/5.09 => ( B
% 8.90/5.09 = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
% 8.90/5.09 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 8.90/5.09 = zero_zero_real )
% 8.90/5.09 => ( A = zero_zero_real ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5725_divide__eq__eq__numeral1_I2_J,axiom,
% 8.90/5.09 ! [B: complex,W: num,A: complex] :
% 8.90/5.09 ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 8.90/5.09 = A )
% 8.90/5.09 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 8.90/5.09 != zero_zero_complex )
% 8.90/5.09 => ( B
% 8.90/5.09 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
% 8.90/5.09 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 8.90/5.09 = zero_zero_complex )
% 8.90/5.09 => ( A = zero_zero_complex ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5726_divide__eq__eq__numeral1_I2_J,axiom,
% 8.90/5.09 ! [B: rat,W: num,A: rat] :
% 8.90/5.09 ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 8.90/5.09 = A )
% 8.90/5.09 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 8.90/5.09 != zero_zero_rat )
% 8.90/5.09 => ( B
% 8.90/5.09 = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
% 8.90/5.09 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 8.90/5.09 = zero_zero_rat )
% 8.90/5.09 => ( A = zero_zero_rat ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5727_less__divide__eq__numeral1_I2_J,axiom,
% 8.90/5.09 ! [A: real,B: real,W: num] :
% 8.90/5.09 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 8.90/5.09 = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5728_less__divide__eq__numeral1_I2_J,axiom,
% 8.90/5.09 ! [A: rat,B: rat,W: num] :
% 8.90/5.09 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 8.90/5.09 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5729_divide__less__eq__numeral1_I2_J,axiom,
% 8.90/5.09 ! [B: real,W: num,A: real] :
% 8.90/5.09 ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 8.90/5.09 = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5730_divide__less__eq__numeral1_I2_J,axiom,
% 8.90/5.09 ! [B: rat,W: num,A: rat] :
% 8.90/5.09 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 8.90/5.09 = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5731_power2__minus,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.09 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5732_power2__minus,axiom,
% 8.90/5.09 ! [A: real] :
% 8.90/5.09 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.09 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5733_power2__minus,axiom,
% 8.90/5.09 ! [A: complex] :
% 8.90/5.09 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.09 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5734_power2__minus,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.09 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5735_power2__minus,axiom,
% 8.90/5.09 ! [A: rat] :
% 8.90/5.09 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.09 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5736_add__neg__numeral__special_I9_J,axiom,
% 8.90/5.09 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.09 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5737_add__neg__numeral__special_I9_J,axiom,
% 8.90/5.09 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 8.90/5.09 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5738_add__neg__numeral__special_I9_J,axiom,
% 8.90/5.09 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5739_add__neg__numeral__special_I9_J,axiom,
% 8.90/5.09 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5740_add__neg__numeral__special_I9_J,axiom,
% 8.90/5.09 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.90/5.09 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5741_diff__numeral__special_I10_J,axiom,
% 8.90/5.09 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 8.90/5.09 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5742_diff__numeral__special_I10_J,axiom,
% 8.90/5.09 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 8.90/5.09 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5743_diff__numeral__special_I10_J,axiom,
% 8.90/5.09 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5744_diff__numeral__special_I10_J,axiom,
% 8.90/5.09 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5745_diff__numeral__special_I10_J,axiom,
% 8.90/5.09 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 8.90/5.09 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5746_diff__numeral__special_I11_J,axiom,
% 8.90/5.09 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.09 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5747_diff__numeral__special_I11_J,axiom,
% 8.90/5.09 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 8.90/5.09 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5748_diff__numeral__special_I11_J,axiom,
% 8.90/5.09 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.90/5.09 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5749_diff__numeral__special_I11_J,axiom,
% 8.90/5.09 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.90/5.09 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5750_diff__numeral__special_I11_J,axiom,
% 8.90/5.09 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.90/5.09 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5751_minus__1__div__2__eq,axiom,
% 8.90/5.09 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.09 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5752_minus__1__div__2__eq,axiom,
% 8.90/5.09 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5753_minus__1__mod__2__eq,axiom,
% 8.90/5.09 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.09 = one_one_int ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5754_minus__1__mod__2__eq,axiom,
% 8.90/5.09 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.09 = one_one_Code_integer ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5755_bits__minus__1__mod__2__eq,axiom,
% 8.90/5.09 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.09 = one_one_int ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5756_bits__minus__1__mod__2__eq,axiom,
% 8.90/5.09 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.09 = one_one_Code_integer ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5757_Power_Oring__1__class_Opower__minus__even,axiom,
% 8.90/5.09 ! [A: int,N: nat] :
% 8.90/5.09 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.09 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5758_Power_Oring__1__class_Opower__minus__even,axiom,
% 8.90/5.09 ! [A: real,N: nat] :
% 8.90/5.09 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.09 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5759_Power_Oring__1__class_Opower__minus__even,axiom,
% 8.90/5.09 ! [A: complex,N: nat] :
% 8.90/5.09 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.09 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5760_Power_Oring__1__class_Opower__minus__even,axiom,
% 8.90/5.09 ! [A: code_integer,N: nat] :
% 8.90/5.09 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.09 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5761_Power_Oring__1__class_Opower__minus__even,axiom,
% 8.90/5.09 ! [A: rat,N: nat] :
% 8.90/5.09 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.09 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5762_Parity_Oring__1__class_Opower__minus__even,axiom,
% 8.90/5.09 ! [N: nat,A: int] :
% 8.90/5.09 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 8.90/5.09 = ( power_power_int @ A @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5763_Parity_Oring__1__class_Opower__minus__even,axiom,
% 8.90/5.09 ! [N: nat,A: real] :
% 8.90/5.09 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 8.90/5.09 = ( power_power_real @ A @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5764_Parity_Oring__1__class_Opower__minus__even,axiom,
% 8.90/5.09 ! [N: nat,A: complex] :
% 8.90/5.09 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 8.90/5.09 = ( power_power_complex @ A @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5765_Parity_Oring__1__class_Opower__minus__even,axiom,
% 8.90/5.09 ! [N: nat,A: code_integer] :
% 8.90/5.09 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 8.90/5.09 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5766_Parity_Oring__1__class_Opower__minus__even,axiom,
% 8.90/5.09 ! [N: nat,A: rat] :
% 8.90/5.09 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 8.90/5.09 = ( power_power_rat @ A @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5767_power__minus__odd,axiom,
% 8.90/5.09 ! [N: nat,A: int] :
% 8.90/5.09 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 8.90/5.09 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5768_power__minus__odd,axiom,
% 8.90/5.09 ! [N: nat,A: real] :
% 8.90/5.09 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 8.90/5.09 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5769_power__minus__odd,axiom,
% 8.90/5.09 ! [N: nat,A: complex] :
% 8.90/5.09 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5770_power__minus__odd,axiom,
% 8.90/5.09 ! [N: nat,A: code_integer] :
% 8.90/5.09 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5771_power__minus__odd,axiom,
% 8.90/5.09 ! [N: nat,A: rat] :
% 8.90/5.09 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 8.90/5.09 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5772_diff__numeral__special_I3_J,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.09 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5773_diff__numeral__special_I3_J,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.90/5.09 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5774_diff__numeral__special_I3_J,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 8.90/5.09 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5775_diff__numeral__special_I3_J,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.90/5.09 = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5776_diff__numeral__special_I3_J,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.90/5.09 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5777_diff__numeral__special_I4_J,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
% 8.90/5.09 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5778_diff__numeral__special_I4_J,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
% 8.90/5.09 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5779_diff__numeral__special_I4_J,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5780_diff__numeral__special_I4_J,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5781_diff__numeral__special_I4_J,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
% 8.90/5.09 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5782_dbl__simps_I4_J,axiom,
% 8.90/5.09 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.09 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5783_dbl__simps_I4_J,axiom,
% 8.90/5.09 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
% 8.90/5.09 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5784_dbl__simps_I4_J,axiom,
% 8.90/5.09 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5785_dbl__simps_I4_J,axiom,
% 8.90/5.09 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5786_dbl__simps_I4_J,axiom,
% 8.90/5.09 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.90/5.09 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5787_power__minus1__even,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.09 = one_one_int ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5788_power__minus1__even,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.09 = one_one_real ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5789_power__minus1__even,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.09 = one_one_complex ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5790_power__minus1__even,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.09 = one_one_Code_integer ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5791_power__minus1__even,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.09 = one_one_rat ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5792_neg__one__even__power,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 8.90/5.09 = one_one_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5793_neg__one__even__power,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 8.90/5.09 = one_one_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5794_neg__one__even__power,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 8.90/5.09 = one_one_complex ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5795_neg__one__even__power,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 8.90/5.09 = one_one_Code_integer ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5796_neg__one__even__power,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 8.90/5.09 = one_one_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5797_neg__one__odd__power,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 8.90/5.09 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5798_neg__one__odd__power,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 8.90/5.09 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5799_neg__one__odd__power,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5800_neg__one__odd__power,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5801_neg__one__odd__power,axiom,
% 8.90/5.09 ! [N: nat] :
% 8.90/5.09 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.09 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 8.90/5.09 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5802_signed__take__bit__0,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5803_signed__take__bit__0,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 8.90/5.09 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5804_signed__take__bit__Suc__minus__bit0,axiom,
% 8.90/5.09 ! [N: nat,K: num] :
% 8.90/5.09 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 8.90/5.09 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5805_signed__take__bit__numeral__bit0,axiom,
% 8.90/5.09 ! [L: num,K: num] :
% 8.90/5.09 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 8.90/5.09 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5806_signed__take__bit__numeral__minus__bit0,axiom,
% 8.90/5.09 ! [L: num,K: num] :
% 8.90/5.09 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 8.90/5.09 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5807_signed__take__bit__numeral__minus__bit1,axiom,
% 8.90/5.09 ! [L: num,K: num] :
% 8.90/5.09 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 8.90/5.09 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5808_signed__take__bit__minus,axiom,
% 8.90/5.09 ! [N: nat,K: int] :
% 8.90/5.09 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
% 8.90/5.09 = ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5809_equation__minus__iff,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( uminus_uminus_int @ B ) )
% 8.90/5.09 = ( B
% 8.90/5.09 = ( uminus_uminus_int @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5810_equation__minus__iff,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( uminus_uminus_real @ B ) )
% 8.90/5.09 = ( B
% 8.90/5.09 = ( uminus_uminus_real @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5811_equation__minus__iff,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( uminus1482373934393186551omplex @ B ) )
% 8.90/5.09 = ( B
% 8.90/5.09 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5812_equation__minus__iff,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( uminus1351360451143612070nteger @ B ) )
% 8.90/5.09 = ( B
% 8.90/5.09 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5813_equation__minus__iff,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( uminus_uminus_rat @ B ) )
% 8.90/5.09 = ( B
% 8.90/5.09 = ( uminus_uminus_rat @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5814_minus__equation__iff,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( ( uminus_uminus_int @ A )
% 8.90/5.09 = B )
% 8.90/5.09 = ( ( uminus_uminus_int @ B )
% 8.90/5.09 = A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5815_minus__equation__iff,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( ( uminus_uminus_real @ A )
% 8.90/5.09 = B )
% 8.90/5.09 = ( ( uminus_uminus_real @ B )
% 8.90/5.09 = A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5816_minus__equation__iff,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( ( uminus1482373934393186551omplex @ A )
% 8.90/5.09 = B )
% 8.90/5.09 = ( ( uminus1482373934393186551omplex @ B )
% 8.90/5.09 = A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5817_minus__equation__iff,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( ( uminus1351360451143612070nteger @ A )
% 8.90/5.09 = B )
% 8.90/5.09 = ( ( uminus1351360451143612070nteger @ B )
% 8.90/5.09 = A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5818_minus__equation__iff,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( ( uminus_uminus_rat @ A )
% 8.90/5.09 = B )
% 8.90/5.09 = ( ( uminus_uminus_rat @ B )
% 8.90/5.09 = A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5819_compl__mono,axiom,
% 8.90/5.09 ! [X2: set_int,Y3: set_int] :
% 8.90/5.09 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 8.90/5.09 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ ( uminus1532241313380277803et_int @ X2 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5820_compl__le__swap1,axiom,
% 8.90/5.09 ! [Y3: set_int,X2: set_int] :
% 8.90/5.09 ( ( ord_less_eq_set_int @ Y3 @ ( uminus1532241313380277803et_int @ X2 ) )
% 8.90/5.09 => ( ord_less_eq_set_int @ X2 @ ( uminus1532241313380277803et_int @ Y3 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5821_compl__le__swap2,axiom,
% 8.90/5.09 ! [Y3: set_int,X2: set_int] :
% 8.90/5.09 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ X2 )
% 8.90/5.09 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5822_mem__case__prodE,axiom,
% 8.90/5.09 ! [Z: complex,C: code_integer > $o > set_complex,P6: produc6271795597528267376eger_o] :
% 8.90/5.09 ( ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ P6 ) )
% 8.90/5.09 => ~ ! [X5: code_integer,Y4: $o] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( produc6677183202524767010eger_o @ X5 @ Y4 ) )
% 8.90/5.09 => ~ ( member_complex @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5823_mem__case__prodE,axiom,
% 8.90/5.09 ! [Z: real,C: code_integer > $o > set_real,P6: produc6271795597528267376eger_o] :
% 8.90/5.09 ( ( member_real @ Z @ ( produc242741666403216561t_real @ C @ P6 ) )
% 8.90/5.09 => ~ ! [X5: code_integer,Y4: $o] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( produc6677183202524767010eger_o @ X5 @ Y4 ) )
% 8.90/5.09 => ~ ( member_real @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5824_mem__case__prodE,axiom,
% 8.90/5.09 ! [Z: nat,C: code_integer > $o > set_nat,P6: produc6271795597528267376eger_o] :
% 8.90/5.09 ( ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ P6 ) )
% 8.90/5.09 => ~ ! [X5: code_integer,Y4: $o] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( produc6677183202524767010eger_o @ X5 @ Y4 ) )
% 8.90/5.09 => ~ ( member_nat @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5825_mem__case__prodE,axiom,
% 8.90/5.09 ! [Z: int,C: code_integer > $o > set_int,P6: produc6271795597528267376eger_o] :
% 8.90/5.09 ( ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ P6 ) )
% 8.90/5.09 => ~ ! [X5: code_integer,Y4: $o] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( produc6677183202524767010eger_o @ X5 @ Y4 ) )
% 8.90/5.09 => ~ ( member_int @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5826_mem__case__prodE,axiom,
% 8.90/5.09 ! [Z: complex,C: num > num > set_complex,P6: product_prod_num_num] :
% 8.90/5.09 ( ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ P6 ) )
% 8.90/5.09 => ~ ! [X5: num,Y4: num] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_num_num @ X5 @ Y4 ) )
% 8.90/5.09 => ~ ( member_complex @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5827_mem__case__prodE,axiom,
% 8.90/5.09 ! [Z: real,C: num > num > set_real,P6: product_prod_num_num] :
% 8.90/5.09 ( ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ P6 ) )
% 8.90/5.09 => ~ ! [X5: num,Y4: num] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_num_num @ X5 @ Y4 ) )
% 8.90/5.09 => ~ ( member_real @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5828_mem__case__prodE,axiom,
% 8.90/5.09 ! [Z: nat,C: num > num > set_nat,P6: product_prod_num_num] :
% 8.90/5.09 ( ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ P6 ) )
% 8.90/5.09 => ~ ! [X5: num,Y4: num] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_num_num @ X5 @ Y4 ) )
% 8.90/5.09 => ~ ( member_nat @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5829_mem__case__prodE,axiom,
% 8.90/5.09 ! [Z: int,C: num > num > set_int,P6: product_prod_num_num] :
% 8.90/5.09 ( ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ P6 ) )
% 8.90/5.09 => ~ ! [X5: num,Y4: num] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_num_num @ X5 @ Y4 ) )
% 8.90/5.09 => ~ ( member_int @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5830_mem__case__prodE,axiom,
% 8.90/5.09 ! [Z: complex,C: nat > num > set_complex,P6: product_prod_nat_num] :
% 8.90/5.09 ( ( member_complex @ Z @ ( produc6231982587499038204omplex @ C @ P6 ) )
% 8.90/5.09 => ~ ! [X5: nat,Y4: num] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_nat_num @ X5 @ Y4 ) )
% 8.90/5.09 => ~ ( member_complex @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5831_mem__case__prodE,axiom,
% 8.90/5.09 ! [Z: real,C: nat > num > set_real,P6: product_prod_nat_num] :
% 8.90/5.09 ( ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ P6 ) )
% 8.90/5.09 => ~ ! [X5: nat,Y4: num] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_nat_num @ X5 @ Y4 ) )
% 8.90/5.09 => ~ ( member_real @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5832_le__minus__iff,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 8.90/5.09 = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5833_le__minus__iff,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 8.90/5.09 = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5834_le__minus__iff,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
% 8.90/5.09 = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5835_le__minus__iff,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 8.90/5.09 = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5836_minus__le__iff,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 8.90/5.09 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5837_minus__le__iff,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 8.90/5.09 = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5838_minus__le__iff,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 8.90/5.09 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5839_minus__le__iff,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 8.90/5.09 = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5840_le__imp__neg__le,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( ord_less_eq_real @ A @ B )
% 8.90/5.09 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5841_le__imp__neg__le,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( ord_le3102999989581377725nteger @ A @ B )
% 8.90/5.09 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5842_le__imp__neg__le,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.09 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5843_le__imp__neg__le,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( ord_less_eq_int @ A @ B )
% 8.90/5.09 => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5844_minus__less__iff,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
% 8.90/5.09 = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5845_minus__less__iff,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 8.90/5.09 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5846_minus__less__iff,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 8.90/5.09 = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5847_minus__less__iff,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
% 8.90/5.09 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5848_less__minus__iff,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
% 8.90/5.09 = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5849_less__minus__iff,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 8.90/5.09 = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5850_less__minus__iff,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 8.90/5.09 = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5851_less__minus__iff,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
% 8.90/5.09 = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5852_verit__negate__coefficient_I2_J,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( ord_less_int @ A @ B )
% 8.90/5.09 => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5853_verit__negate__coefficient_I2_J,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( ord_less_real @ A @ B )
% 8.90/5.09 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5854_verit__negate__coefficient_I2_J,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( ord_le6747313008572928689nteger @ A @ B )
% 8.90/5.09 => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5855_verit__negate__coefficient_I2_J,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( ord_less_rat @ A @ B )
% 8.90/5.09 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5856_numeral__neq__neg__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( numeral_numeral_int @ M )
% 8.90/5.09 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5857_numeral__neq__neg__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( numeral_numeral_real @ M )
% 8.90/5.09 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5858_numeral__neq__neg__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( numera6690914467698888265omplex @ M )
% 8.90/5.09 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5859_numeral__neq__neg__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( numera6620942414471956472nteger @ M )
% 8.90/5.09 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5860_numeral__neq__neg__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( numeral_numeral_rat @ M )
% 8.90/5.09 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5861_neg__numeral__neq__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 8.90/5.09 != ( numeral_numeral_int @ N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5862_neg__numeral__neq__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 8.90/5.09 != ( numeral_numeral_real @ N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5863_neg__numeral__neq__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 8.90/5.09 != ( numera6690914467698888265omplex @ N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5864_neg__numeral__neq__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 8.90/5.09 != ( numera6620942414471956472nteger @ N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5865_neg__numeral__neq__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 8.90/5.09 != ( numeral_numeral_rat @ N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5866_minus__mult__commute,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 8.90/5.09 = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5867_minus__mult__commute,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 8.90/5.09 = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5868_minus__mult__commute,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 8.90/5.09 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5869_minus__mult__commute,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 8.90/5.09 = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5870_minus__mult__commute,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 8.90/5.09 = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5871_square__eq__iff,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( ( times_times_int @ A @ A )
% 8.90/5.09 = ( times_times_int @ B @ B ) )
% 8.90/5.09 = ( ( A = B )
% 8.90/5.09 | ( A
% 8.90/5.09 = ( uminus_uminus_int @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5872_square__eq__iff,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( ( times_times_real @ A @ A )
% 8.90/5.09 = ( times_times_real @ B @ B ) )
% 8.90/5.09 = ( ( A = B )
% 8.90/5.09 | ( A
% 8.90/5.09 = ( uminus_uminus_real @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5873_square__eq__iff,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( ( times_times_complex @ A @ A )
% 8.90/5.09 = ( times_times_complex @ B @ B ) )
% 8.90/5.09 = ( ( A = B )
% 8.90/5.09 | ( A
% 8.90/5.09 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5874_square__eq__iff,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( ( times_3573771949741848930nteger @ A @ A )
% 8.90/5.09 = ( times_3573771949741848930nteger @ B @ B ) )
% 8.90/5.09 = ( ( A = B )
% 8.90/5.09 | ( A
% 8.90/5.09 = ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5875_square__eq__iff,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( ( times_times_rat @ A @ A )
% 8.90/5.09 = ( times_times_rat @ B @ B ) )
% 8.90/5.09 = ( ( A = B )
% 8.90/5.09 | ( A
% 8.90/5.09 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5876_one__neq__neg__one,axiom,
% 8.90/5.09 ( one_one_int
% 8.90/5.09 != ( uminus_uminus_int @ one_one_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5877_one__neq__neg__one,axiom,
% 8.90/5.09 ( one_one_real
% 8.90/5.09 != ( uminus_uminus_real @ one_one_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5878_one__neq__neg__one,axiom,
% 8.90/5.09 ( one_one_complex
% 8.90/5.09 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5879_one__neq__neg__one,axiom,
% 8.90/5.09 ( one_one_Code_integer
% 8.90/5.09 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5880_one__neq__neg__one,axiom,
% 8.90/5.09 ( one_one_rat
% 8.90/5.09 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5881_is__num__normalize_I8_J,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 8.90/5.09 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5882_is__num__normalize_I8_J,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 8.90/5.09 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5883_is__num__normalize_I8_J,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 8.90/5.09 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5884_is__num__normalize_I8_J,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 8.90/5.09 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5885_is__num__normalize_I8_J,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 8.90/5.09 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5886_group__cancel_Oneg1,axiom,
% 8.90/5.09 ! [A2: int,K: int,A: int] :
% 8.90/5.09 ( ( A2
% 8.90/5.09 = ( plus_plus_int @ K @ A ) )
% 8.90/5.09 => ( ( uminus_uminus_int @ A2 )
% 8.90/5.09 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5887_group__cancel_Oneg1,axiom,
% 8.90/5.09 ! [A2: real,K: real,A: real] :
% 8.90/5.09 ( ( A2
% 8.90/5.09 = ( plus_plus_real @ K @ A ) )
% 8.90/5.09 => ( ( uminus_uminus_real @ A2 )
% 8.90/5.09 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5888_group__cancel_Oneg1,axiom,
% 8.90/5.09 ! [A2: complex,K: complex,A: complex] :
% 8.90/5.09 ( ( A2
% 8.90/5.09 = ( plus_plus_complex @ K @ A ) )
% 8.90/5.09 => ( ( uminus1482373934393186551omplex @ A2 )
% 8.90/5.09 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5889_group__cancel_Oneg1,axiom,
% 8.90/5.09 ! [A2: code_integer,K: code_integer,A: code_integer] :
% 8.90/5.09 ( ( A2
% 8.90/5.09 = ( plus_p5714425477246183910nteger @ K @ A ) )
% 8.90/5.09 => ( ( uminus1351360451143612070nteger @ A2 )
% 8.90/5.09 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5890_group__cancel_Oneg1,axiom,
% 8.90/5.09 ! [A2: rat,K: rat,A: rat] :
% 8.90/5.09 ( ( A2
% 8.90/5.09 = ( plus_plus_rat @ K @ A ) )
% 8.90/5.09 => ( ( uminus_uminus_rat @ A2 )
% 8.90/5.09 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5891_add_Oinverse__distrib__swap,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 8.90/5.09 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5892_add_Oinverse__distrib__swap,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 8.90/5.09 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5893_add_Oinverse__distrib__swap,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 8.90/5.09 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5894_add_Oinverse__distrib__swap,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 8.90/5.09 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5895_add_Oinverse__distrib__swap,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 8.90/5.09 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5896_minus__diff__commute,axiom,
% 8.90/5.09 ! [B: int,A: int] :
% 8.90/5.09 ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 8.90/5.09 = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5897_minus__diff__commute,axiom,
% 8.90/5.09 ! [B: real,A: real] :
% 8.90/5.09 ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
% 8.90/5.09 = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5898_minus__diff__commute,axiom,
% 8.90/5.09 ! [B: complex,A: complex] :
% 8.90/5.09 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
% 8.90/5.09 = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5899_minus__diff__commute,axiom,
% 8.90/5.09 ! [B: code_integer,A: code_integer] :
% 8.90/5.09 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
% 8.90/5.09 = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5900_minus__diff__commute,axiom,
% 8.90/5.09 ! [B: rat,A: rat] :
% 8.90/5.09 ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
% 8.90/5.09 = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5901_minus__diff__minus,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 8.90/5.09 = ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5902_minus__diff__minus,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 8.90/5.09 = ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5903_minus__diff__minus,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5904_minus__diff__minus,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5905_minus__diff__minus,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 8.90/5.09 = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5906_minus__divide__left,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 8.90/5.09 = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5907_minus__divide__left,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.90/5.09 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5908_minus__divide__left,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 8.90/5.09 = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5909_minus__divide__divide,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 8.90/5.09 = ( divide_divide_real @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5910_minus__divide__divide,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 8.90/5.09 = ( divide1717551699836669952omplex @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5911_minus__divide__divide,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 8.90/5.09 = ( divide_divide_rat @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5912_minus__divide__right,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 8.90/5.09 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5913_minus__divide__right,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.90/5.09 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5914_minus__divide__right,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 8.90/5.09 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5915_div__minus__right,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 8.90/5.09 = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5916_div__minus__right,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 8.90/5.09 = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5917_mod__minus__eq,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
% 8.90/5.09 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5918_mod__minus__eq,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) @ B )
% 8.90/5.09 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5919_mod__minus__cong,axiom,
% 8.90/5.09 ! [A: int,B: int,A5: int] :
% 8.90/5.09 ( ( ( modulo_modulo_int @ A @ B )
% 8.90/5.09 = ( modulo_modulo_int @ A5 @ B ) )
% 8.90/5.09 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 8.90/5.09 = ( modulo_modulo_int @ ( uminus_uminus_int @ A5 ) @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5920_mod__minus__cong,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer,A5: code_integer] :
% 8.90/5.09 ( ( ( modulo364778990260209775nteger @ A @ B )
% 8.90/5.09 = ( modulo364778990260209775nteger @ A5 @ B ) )
% 8.90/5.09 => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 8.90/5.09 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A5 ) @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5921_mod__minus__right,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 8.90/5.09 = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5922_mod__minus__right,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5923_case__prodE,axiom,
% 8.90/5.09 ! [C: code_integer > $o > $o,P6: produc6271795597528267376eger_o] :
% 8.90/5.09 ( ( produc7828578312038201481er_o_o @ C @ P6 )
% 8.90/5.09 => ~ ! [X5: code_integer,Y4: $o] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( produc6677183202524767010eger_o @ X5 @ Y4 ) )
% 8.90/5.09 => ~ ( C @ X5 @ Y4 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5924_case__prodE,axiom,
% 8.90/5.09 ! [C: num > num > $o,P6: product_prod_num_num] :
% 8.90/5.09 ( ( produc5703948589228662326_num_o @ C @ P6 )
% 8.90/5.09 => ~ ! [X5: num,Y4: num] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_num_num @ X5 @ Y4 ) )
% 8.90/5.09 => ~ ( C @ X5 @ Y4 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5925_case__prodE,axiom,
% 8.90/5.09 ! [C: nat > num > $o,P6: product_prod_nat_num] :
% 8.90/5.09 ( ( produc4927758841916487424_num_o @ C @ P6 )
% 8.90/5.09 => ~ ! [X5: nat,Y4: num] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_nat_num @ X5 @ Y4 ) )
% 8.90/5.09 => ~ ( C @ X5 @ Y4 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5926_case__prodE,axiom,
% 8.90/5.09 ! [C: nat > nat > $o,P6: product_prod_nat_nat] :
% 8.90/5.09 ( ( produc6081775807080527818_nat_o @ C @ P6 )
% 8.90/5.09 => ~ ! [X5: nat,Y4: nat] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_nat_nat @ X5 @ Y4 ) )
% 8.90/5.09 => ~ ( C @ X5 @ Y4 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5927_case__prodE,axiom,
% 8.90/5.09 ! [C: int > int > $o,P6: product_prod_int_int] :
% 8.90/5.09 ( ( produc4947309494688390418_int_o @ C @ P6 )
% 8.90/5.09 => ~ ! [X5: int,Y4: int] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_int_int @ X5 @ Y4 ) )
% 8.90/5.09 => ~ ( C @ X5 @ Y4 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5928_case__prodD,axiom,
% 8.90/5.09 ! [F: code_integer > $o > $o,A: code_integer,B: $o] :
% 8.90/5.09 ( ( produc7828578312038201481er_o_o @ F @ ( produc6677183202524767010eger_o @ A @ B ) )
% 8.90/5.09 => ( F @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5929_case__prodD,axiom,
% 8.90/5.09 ! [F: num > num > $o,A: num,B: num] :
% 8.90/5.09 ( ( produc5703948589228662326_num_o @ F @ ( product_Pair_num_num @ A @ B ) )
% 8.90/5.09 => ( F @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5930_case__prodD,axiom,
% 8.90/5.09 ! [F: nat > num > $o,A: nat,B: num] :
% 8.90/5.09 ( ( produc4927758841916487424_num_o @ F @ ( product_Pair_nat_num @ A @ B ) )
% 8.90/5.09 => ( F @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5931_case__prodD,axiom,
% 8.90/5.09 ! [F: nat > nat > $o,A: nat,B: nat] :
% 8.90/5.09 ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 8.90/5.09 => ( F @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5932_case__prodD,axiom,
% 8.90/5.09 ! [F: int > int > $o,A: int,B: int] :
% 8.90/5.09 ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
% 8.90/5.09 => ( F @ A @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5933_case__prodE_H,axiom,
% 8.90/5.09 ! [C: nat > nat > product_prod_nat_nat > $o,P6: product_prod_nat_nat,Z: product_prod_nat_nat] :
% 8.90/5.09 ( ( produc8739625826339149834_nat_o @ C @ P6 @ Z )
% 8.90/5.09 => ~ ! [X5: nat,Y4: nat] :
% 8.90/5.09 ( ( P6
% 8.90/5.09 = ( product_Pair_nat_nat @ X5 @ Y4 ) )
% 8.90/5.09 => ~ ( C @ X5 @ Y4 @ Z ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5934_case__prodD_H,axiom,
% 8.90/5.09 ! [R: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
% 8.90/5.09 ( ( produc8739625826339149834_nat_o @ R @ ( product_Pair_nat_nat @ A @ B ) @ C )
% 8.90/5.09 => ( R @ A @ B @ C ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5935_neg__numeral__le__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5936_neg__numeral__le__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5937_neg__numeral__le__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5938_neg__numeral__le__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5939_not__numeral__le__neg__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5940_not__numeral__le__neg__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5941_not__numeral__le__neg__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5942_not__numeral__le__neg__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5943_zero__neq__neg__numeral,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( zero_zero_int
% 8.90/5.09 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5944_zero__neq__neg__numeral,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( zero_zero_real
% 8.90/5.09 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5945_zero__neq__neg__numeral,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( zero_zero_complex
% 8.90/5.09 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5946_zero__neq__neg__numeral,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( zero_z3403309356797280102nteger
% 8.90/5.09 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5947_zero__neq__neg__numeral,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( zero_zero_rat
% 8.90/5.09 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5948_not__numeral__less__neg__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5949_not__numeral__less__neg__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5950_not__numeral__less__neg__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5951_not__numeral__less__neg__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] :
% 8.90/5.09 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5952_neg__numeral__less__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5953_neg__numeral__less__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5954_neg__numeral__less__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5955_neg__numeral__less__numeral,axiom,
% 8.90/5.09 ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5956_le__minus__one__simps_I4_J,axiom,
% 8.90/5.09 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5957_le__minus__one__simps_I4_J,axiom,
% 8.90/5.09 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5958_le__minus__one__simps_I4_J,axiom,
% 8.90/5.09 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5959_le__minus__one__simps_I4_J,axiom,
% 8.90/5.09 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5960_le__minus__one__simps_I2_J,axiom,
% 8.90/5.09 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5961_le__minus__one__simps_I2_J,axiom,
% 8.90/5.09 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5962_le__minus__one__simps_I2_J,axiom,
% 8.90/5.09 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5963_le__minus__one__simps_I2_J,axiom,
% 8.90/5.09 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5964_zero__neq__neg__one,axiom,
% 8.90/5.09 ( zero_zero_int
% 8.90/5.09 != ( uminus_uminus_int @ one_one_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5965_zero__neq__neg__one,axiom,
% 8.90/5.09 ( zero_zero_real
% 8.90/5.09 != ( uminus_uminus_real @ one_one_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5966_zero__neq__neg__one,axiom,
% 8.90/5.09 ( zero_zero_complex
% 8.90/5.09 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5967_zero__neq__neg__one,axiom,
% 8.90/5.09 ( zero_z3403309356797280102nteger
% 8.90/5.09 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5968_zero__neq__neg__one,axiom,
% 8.90/5.09 ( zero_zero_rat
% 8.90/5.09 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5969_neg__eq__iff__add__eq__0,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( ( uminus_uminus_int @ A )
% 8.90/5.09 = B )
% 8.90/5.09 = ( ( plus_plus_int @ A @ B )
% 8.90/5.09 = zero_zero_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5970_neg__eq__iff__add__eq__0,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( ( uminus_uminus_real @ A )
% 8.90/5.09 = B )
% 8.90/5.09 = ( ( plus_plus_real @ A @ B )
% 8.90/5.09 = zero_zero_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5971_neg__eq__iff__add__eq__0,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( ( uminus1482373934393186551omplex @ A )
% 8.90/5.09 = B )
% 8.90/5.09 = ( ( plus_plus_complex @ A @ B )
% 8.90/5.09 = zero_zero_complex ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5972_neg__eq__iff__add__eq__0,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( ( uminus1351360451143612070nteger @ A )
% 8.90/5.09 = B )
% 8.90/5.09 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 8.90/5.09 = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5973_neg__eq__iff__add__eq__0,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( ( uminus_uminus_rat @ A )
% 8.90/5.09 = B )
% 8.90/5.09 = ( ( plus_plus_rat @ A @ B )
% 8.90/5.09 = zero_zero_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5974_eq__neg__iff__add__eq__0,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( uminus_uminus_int @ B ) )
% 8.90/5.09 = ( ( plus_plus_int @ A @ B )
% 8.90/5.09 = zero_zero_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5975_eq__neg__iff__add__eq__0,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( uminus_uminus_real @ B ) )
% 8.90/5.09 = ( ( plus_plus_real @ A @ B )
% 8.90/5.09 = zero_zero_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5976_eq__neg__iff__add__eq__0,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( uminus1482373934393186551omplex @ B ) )
% 8.90/5.09 = ( ( plus_plus_complex @ A @ B )
% 8.90/5.09 = zero_zero_complex ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5977_eq__neg__iff__add__eq__0,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( uminus1351360451143612070nteger @ B ) )
% 8.90/5.09 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 8.90/5.09 = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5978_eq__neg__iff__add__eq__0,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( uminus_uminus_rat @ B ) )
% 8.90/5.09 = ( ( plus_plus_rat @ A @ B )
% 8.90/5.09 = zero_zero_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5979_add_Oinverse__unique,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( ( plus_plus_int @ A @ B )
% 8.90/5.09 = zero_zero_int )
% 8.90/5.09 => ( ( uminus_uminus_int @ A )
% 8.90/5.09 = B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5980_add_Oinverse__unique,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( ( plus_plus_real @ A @ B )
% 8.90/5.09 = zero_zero_real )
% 8.90/5.09 => ( ( uminus_uminus_real @ A )
% 8.90/5.09 = B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5981_add_Oinverse__unique,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( ( plus_plus_complex @ A @ B )
% 8.90/5.09 = zero_zero_complex )
% 8.90/5.09 => ( ( uminus1482373934393186551omplex @ A )
% 8.90/5.09 = B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5982_add_Oinverse__unique,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 8.90/5.09 = zero_z3403309356797280102nteger )
% 8.90/5.09 => ( ( uminus1351360451143612070nteger @ A )
% 8.90/5.09 = B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5983_add_Oinverse__unique,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( ( plus_plus_rat @ A @ B )
% 8.90/5.09 = zero_zero_rat )
% 8.90/5.09 => ( ( uminus_uminus_rat @ A )
% 8.90/5.09 = B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5984_ab__group__add__class_Oab__left__minus,axiom,
% 8.90/5.09 ! [A: int] :
% 8.90/5.09 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 8.90/5.09 = zero_zero_int ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5985_ab__group__add__class_Oab__left__minus,axiom,
% 8.90/5.09 ! [A: real] :
% 8.90/5.09 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 8.90/5.09 = zero_zero_real ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5986_ab__group__add__class_Oab__left__minus,axiom,
% 8.90/5.09 ! [A: complex] :
% 8.90/5.09 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 8.90/5.09 = zero_zero_complex ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5987_ab__group__add__class_Oab__left__minus,axiom,
% 8.90/5.09 ! [A: code_integer] :
% 8.90/5.09 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 8.90/5.09 = zero_z3403309356797280102nteger ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5988_ab__group__add__class_Oab__left__minus,axiom,
% 8.90/5.09 ! [A: rat] :
% 8.90/5.09 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 8.90/5.09 = zero_zero_rat ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5989_add__eq__0__iff,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( ( plus_plus_int @ A @ B )
% 8.90/5.09 = zero_zero_int )
% 8.90/5.09 = ( B
% 8.90/5.09 = ( uminus_uminus_int @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5990_add__eq__0__iff,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( ( plus_plus_real @ A @ B )
% 8.90/5.09 = zero_zero_real )
% 8.90/5.09 = ( B
% 8.90/5.09 = ( uminus_uminus_real @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5991_add__eq__0__iff,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( ( plus_plus_complex @ A @ B )
% 8.90/5.09 = zero_zero_complex )
% 8.90/5.09 = ( B
% 8.90/5.09 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5992_add__eq__0__iff,axiom,
% 8.90/5.09 ! [A: code_integer,B: code_integer] :
% 8.90/5.09 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 8.90/5.09 = zero_z3403309356797280102nteger )
% 8.90/5.09 = ( B
% 8.90/5.09 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5993_add__eq__0__iff,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( ( plus_plus_rat @ A @ B )
% 8.90/5.09 = zero_zero_rat )
% 8.90/5.09 = ( B
% 8.90/5.09 = ( uminus_uminus_rat @ A ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5994_less__minus__one__simps_I4_J,axiom,
% 8.90/5.09 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5995_less__minus__one__simps_I4_J,axiom,
% 8.90/5.09 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5996_less__minus__one__simps_I4_J,axiom,
% 8.90/5.09 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5997_less__minus__one__simps_I4_J,axiom,
% 8.90/5.09 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5998_less__minus__one__simps_I2_J,axiom,
% 8.90/5.09 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 8.90/5.09
% 8.90/5.09 thf(fact_5999_less__minus__one__simps_I2_J,axiom,
% 8.90/5.09 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6000_less__minus__one__simps_I2_J,axiom,
% 8.90/5.09 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6001_less__minus__one__simps_I2_J,axiom,
% 8.90/5.09 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6002_numeral__times__minus__swap,axiom,
% 8.90/5.09 ! [W: num,X2: int] :
% 8.90/5.09 ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X2 ) )
% 8.90/5.09 = ( times_times_int @ X2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6003_numeral__times__minus__swap,axiom,
% 8.90/5.09 ! [W: num,X2: real] :
% 8.90/5.09 ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X2 ) )
% 8.90/5.09 = ( times_times_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6004_numeral__times__minus__swap,axiom,
% 8.90/5.09 ! [W: num,X2: complex] :
% 8.90/5.09 ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X2 ) )
% 8.90/5.09 = ( times_times_complex @ X2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6005_numeral__times__minus__swap,axiom,
% 8.90/5.09 ! [W: num,X2: code_integer] :
% 8.90/5.09 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X2 ) )
% 8.90/5.09 = ( times_3573771949741848930nteger @ X2 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6006_numeral__times__minus__swap,axiom,
% 8.90/5.09 ! [W: num,X2: rat] :
% 8.90/5.09 ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X2 ) )
% 8.90/5.09 = ( times_times_rat @ X2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6007_nonzero__minus__divide__divide,axiom,
% 8.90/5.09 ! [B: real,A: real] :
% 8.90/5.09 ( ( B != zero_zero_real )
% 8.90/5.09 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 8.90/5.09 = ( divide_divide_real @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6008_nonzero__minus__divide__divide,axiom,
% 8.90/5.09 ! [B: complex,A: complex] :
% 8.90/5.09 ( ( B != zero_zero_complex )
% 8.90/5.09 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 8.90/5.09 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6009_nonzero__minus__divide__divide,axiom,
% 8.90/5.09 ! [B: rat,A: rat] :
% 8.90/5.09 ( ( B != zero_zero_rat )
% 8.90/5.09 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 8.90/5.09 = ( divide_divide_rat @ A @ B ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6010_nonzero__minus__divide__right,axiom,
% 8.90/5.09 ! [B: real,A: real] :
% 8.90/5.09 ( ( B != zero_zero_real )
% 8.90/5.09 => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 8.90/5.09 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6011_nonzero__minus__divide__right,axiom,
% 8.90/5.09 ! [B: complex,A: complex] :
% 8.90/5.09 ( ( B != zero_zero_complex )
% 8.90/5.09 => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.90/5.09 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6012_nonzero__minus__divide__right,axiom,
% 8.90/5.09 ! [B: rat,A: rat] :
% 8.90/5.09 ( ( B != zero_zero_rat )
% 8.90/5.09 => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 8.90/5.09 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6013_numeral__neq__neg__one,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( numeral_numeral_int @ N )
% 8.90/5.09 != ( uminus_uminus_int @ one_one_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6014_numeral__neq__neg__one,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( numeral_numeral_real @ N )
% 8.90/5.09 != ( uminus_uminus_real @ one_one_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6015_numeral__neq__neg__one,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( numera6690914467698888265omplex @ N )
% 8.90/5.09 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6016_numeral__neq__neg__one,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( numera6620942414471956472nteger @ N )
% 8.90/5.09 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6017_numeral__neq__neg__one,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( ( numeral_numeral_rat @ N )
% 8.90/5.09 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6018_one__neq__neg__numeral,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( one_one_int
% 8.90/5.09 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6019_one__neq__neg__numeral,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( one_one_real
% 8.90/5.09 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6020_one__neq__neg__numeral,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( one_one_complex
% 8.90/5.09 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6021_one__neq__neg__numeral,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( one_one_Code_integer
% 8.90/5.09 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6022_one__neq__neg__numeral,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ( one_one_rat
% 8.90/5.09 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6023_square__eq__1__iff,axiom,
% 8.90/5.09 ! [X2: int] :
% 8.90/5.09 ( ( ( times_times_int @ X2 @ X2 )
% 8.90/5.09 = one_one_int )
% 8.90/5.09 = ( ( X2 = one_one_int )
% 8.90/5.09 | ( X2
% 8.90/5.09 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6024_square__eq__1__iff,axiom,
% 8.90/5.09 ! [X2: real] :
% 8.90/5.09 ( ( ( times_times_real @ X2 @ X2 )
% 8.90/5.09 = one_one_real )
% 8.90/5.09 = ( ( X2 = one_one_real )
% 8.90/5.09 | ( X2
% 8.90/5.09 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6025_square__eq__1__iff,axiom,
% 8.90/5.09 ! [X2: complex] :
% 8.90/5.09 ( ( ( times_times_complex @ X2 @ X2 )
% 8.90/5.09 = one_one_complex )
% 8.90/5.09 = ( ( X2 = one_one_complex )
% 8.90/5.09 | ( X2
% 8.90/5.09 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6026_square__eq__1__iff,axiom,
% 8.90/5.09 ! [X2: code_integer] :
% 8.90/5.09 ( ( ( times_3573771949741848930nteger @ X2 @ X2 )
% 8.90/5.09 = one_one_Code_integer )
% 8.90/5.09 = ( ( X2 = one_one_Code_integer )
% 8.90/5.09 | ( X2
% 8.90/5.09 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6027_square__eq__1__iff,axiom,
% 8.90/5.09 ! [X2: rat] :
% 8.90/5.09 ( ( ( times_times_rat @ X2 @ X2 )
% 8.90/5.09 = one_one_rat )
% 8.90/5.09 = ( ( X2 = one_one_rat )
% 8.90/5.09 | ( X2
% 8.90/5.09 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6028_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 8.90/5.09 ( minus_minus_int
% 8.90/5.09 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6029_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 8.90/5.09 ( minus_minus_real
% 8.90/5.09 = ( ^ [A3: real,B2: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6030_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 8.90/5.09 ( minus_minus_complex
% 8.90/5.09 = ( ^ [A3: complex,B2: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6031_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 8.90/5.09 ( minus_8373710615458151222nteger
% 8.90/5.09 = ( ^ [A3: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6032_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 8.90/5.09 ( minus_minus_rat
% 8.90/5.09 = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6033_diff__conv__add__uminus,axiom,
% 8.90/5.09 ( minus_minus_int
% 8.90/5.09 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6034_diff__conv__add__uminus,axiom,
% 8.90/5.09 ( minus_minus_real
% 8.90/5.09 = ( ^ [A3: real,B2: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6035_diff__conv__add__uminus,axiom,
% 8.90/5.09 ( minus_minus_complex
% 8.90/5.09 = ( ^ [A3: complex,B2: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6036_diff__conv__add__uminus,axiom,
% 8.90/5.09 ( minus_8373710615458151222nteger
% 8.90/5.09 = ( ^ [A3: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6037_diff__conv__add__uminus,axiom,
% 8.90/5.09 ( minus_minus_rat
% 8.90/5.09 = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6038_group__cancel_Osub2,axiom,
% 8.90/5.09 ! [B3: int,K: int,B: int,A: int] :
% 8.90/5.09 ( ( B3
% 8.90/5.09 = ( plus_plus_int @ K @ B ) )
% 8.90/5.09 => ( ( minus_minus_int @ A @ B3 )
% 8.90/5.09 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6039_group__cancel_Osub2,axiom,
% 8.90/5.09 ! [B3: real,K: real,B: real,A: real] :
% 8.90/5.09 ( ( B3
% 8.90/5.09 = ( plus_plus_real @ K @ B ) )
% 8.90/5.09 => ( ( minus_minus_real @ A @ B3 )
% 8.90/5.09 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6040_group__cancel_Osub2,axiom,
% 8.90/5.09 ! [B3: complex,K: complex,B: complex,A: complex] :
% 8.90/5.09 ( ( B3
% 8.90/5.09 = ( plus_plus_complex @ K @ B ) )
% 8.90/5.09 => ( ( minus_minus_complex @ A @ B3 )
% 8.90/5.09 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6041_group__cancel_Osub2,axiom,
% 8.90/5.09 ! [B3: code_integer,K: code_integer,B: code_integer,A: code_integer] :
% 8.90/5.09 ( ( B3
% 8.90/5.09 = ( plus_p5714425477246183910nteger @ K @ B ) )
% 8.90/5.09 => ( ( minus_8373710615458151222nteger @ A @ B3 )
% 8.90/5.09 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6042_group__cancel_Osub2,axiom,
% 8.90/5.09 ! [B3: rat,K: rat,B: rat,A: rat] :
% 8.90/5.09 ( ( B3
% 8.90/5.09 = ( plus_plus_rat @ K @ B ) )
% 8.90/5.09 => ( ( minus_minus_rat @ A @ B3 )
% 8.90/5.09 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6043_dvd__neg__div,axiom,
% 8.90/5.09 ! [B: int,A: int] :
% 8.90/5.09 ( ( dvd_dvd_int @ B @ A )
% 8.90/5.09 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 8.90/5.09 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6044_dvd__neg__div,axiom,
% 8.90/5.09 ! [B: real,A: real] :
% 8.90/5.09 ( ( dvd_dvd_real @ B @ A )
% 8.90/5.09 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
% 8.90/5.09 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6045_dvd__neg__div,axiom,
% 8.90/5.09 ! [B: complex,A: complex] :
% 8.90/5.09 ( ( dvd_dvd_complex @ B @ A )
% 8.90/5.09 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6046_dvd__neg__div,axiom,
% 8.90/5.09 ! [B: code_integer,A: code_integer] :
% 8.90/5.09 ( ( dvd_dvd_Code_integer @ B @ A )
% 8.90/5.09 => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6047_dvd__neg__div,axiom,
% 8.90/5.09 ! [B: rat,A: rat] :
% 8.90/5.09 ( ( dvd_dvd_rat @ B @ A )
% 8.90/5.09 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
% 8.90/5.09 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6048_dvd__div__neg,axiom,
% 8.90/5.09 ! [B: int,A: int] :
% 8.90/5.09 ( ( dvd_dvd_int @ B @ A )
% 8.90/5.09 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 8.90/5.09 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6049_dvd__div__neg,axiom,
% 8.90/5.09 ! [B: real,A: real] :
% 8.90/5.09 ( ( dvd_dvd_real @ B @ A )
% 8.90/5.09 => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
% 8.90/5.09 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6050_dvd__div__neg,axiom,
% 8.90/5.09 ! [B: complex,A: complex] :
% 8.90/5.09 ( ( dvd_dvd_complex @ B @ A )
% 8.90/5.09 => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6051_dvd__div__neg,axiom,
% 8.90/5.09 ! [B: code_integer,A: code_integer] :
% 8.90/5.09 ( ( dvd_dvd_Code_integer @ B @ A )
% 8.90/5.09 => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6052_dvd__div__neg,axiom,
% 8.90/5.09 ! [B: rat,A: rat] :
% 8.90/5.09 ( ( dvd_dvd_rat @ B @ A )
% 8.90/5.09 => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
% 8.90/5.09 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6053_subset__Compl__self__eq,axiom,
% 8.90/5.09 ! [A2: set_nat] :
% 8.90/5.09 ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
% 8.90/5.09 = ( A2 = bot_bot_set_nat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6054_subset__Compl__self__eq,axiom,
% 8.90/5.09 ! [A2: set_real] :
% 8.90/5.09 ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
% 8.90/5.09 = ( A2 = bot_bot_set_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6055_subset__Compl__self__eq,axiom,
% 8.90/5.09 ! [A2: set_int] :
% 8.90/5.09 ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
% 8.90/5.09 = ( A2 = bot_bot_set_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6056_real__minus__mult__self__le,axiom,
% 8.90/5.09 ! [U: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X2 @ X2 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6057_zmult__eq__1__iff,axiom,
% 8.90/5.09 ! [M: int,N: int] :
% 8.90/5.09 ( ( ( times_times_int @ M @ N )
% 8.90/5.09 = one_one_int )
% 8.90/5.09 = ( ( ( M = one_one_int )
% 8.90/5.09 & ( N = one_one_int ) )
% 8.90/5.09 | ( ( M
% 8.90/5.09 = ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.09 & ( N
% 8.90/5.09 = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6058_pos__zmult__eq__1__iff__lemma,axiom,
% 8.90/5.09 ! [M: int,N: int] :
% 8.90/5.09 ( ( ( times_times_int @ M @ N )
% 8.90/5.09 = one_one_int )
% 8.90/5.09 => ( ( M = one_one_int )
% 8.90/5.09 | ( M
% 8.90/5.09 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6059_minus__int__code_I2_J,axiom,
% 8.90/5.09 ! [L: int] :
% 8.90/5.09 ( ( minus_minus_int @ zero_zero_int @ L )
% 8.90/5.09 = ( uminus_uminus_int @ L ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6060_numeral__eq__Suc,axiom,
% 8.90/5.09 ( numeral_numeral_nat
% 8.90/5.09 = ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6061_minus__real__def,axiom,
% 8.90/5.09 ( minus_minus_real
% 8.90/5.09 = ( ^ [X: real,Y: real] : ( plus_plus_real @ X @ ( uminus_uminus_real @ Y ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6062_not__zero__le__neg__numeral,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6063_not__zero__le__neg__numeral,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6064_not__zero__le__neg__numeral,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6065_not__zero__le__neg__numeral,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6066_neg__numeral__le__zero,axiom,
% 8.90/5.09 ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6067_neg__numeral__le__zero,axiom,
% 8.90/5.09 ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6068_neg__numeral__le__zero,axiom,
% 8.90/5.09 ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6069_neg__numeral__le__zero,axiom,
% 8.90/5.09 ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6070_not__zero__less__neg__numeral,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6071_not__zero__less__neg__numeral,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6072_not__zero__less__neg__numeral,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6073_not__zero__less__neg__numeral,axiom,
% 8.90/5.09 ! [N: num] :
% 8.90/5.09 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6074_neg__numeral__less__zero,axiom,
% 8.90/5.09 ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6075_neg__numeral__less__zero,axiom,
% 8.90/5.09 ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6076_neg__numeral__less__zero,axiom,
% 8.90/5.09 ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6077_neg__numeral__less__zero,axiom,
% 8.90/5.09 ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6078_le__minus__one__simps_I3_J,axiom,
% 8.90/5.09 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6079_le__minus__one__simps_I3_J,axiom,
% 8.90/5.09 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6080_le__minus__one__simps_I3_J,axiom,
% 8.90/5.09 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6081_le__minus__one__simps_I3_J,axiom,
% 8.90/5.09 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6082_le__minus__one__simps_I1_J,axiom,
% 8.90/5.09 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6083_le__minus__one__simps_I1_J,axiom,
% 8.90/5.09 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6084_le__minus__one__simps_I1_J,axiom,
% 8.90/5.09 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6085_le__minus__one__simps_I1_J,axiom,
% 8.90/5.09 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6086_less__minus__one__simps_I1_J,axiom,
% 8.90/5.09 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6087_less__minus__one__simps_I1_J,axiom,
% 8.90/5.09 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6088_less__minus__one__simps_I1_J,axiom,
% 8.90/5.09 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6089_less__minus__one__simps_I1_J,axiom,
% 8.90/5.09 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6090_less__minus__one__simps_I3_J,axiom,
% 8.90/5.09 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6091_less__minus__one__simps_I3_J,axiom,
% 8.90/5.09 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6092_less__minus__one__simps_I3_J,axiom,
% 8.90/5.09 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6093_less__minus__one__simps_I3_J,axiom,
% 8.90/5.09 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6094_neg__numeral__le__one,axiom,
% 8.90/5.09 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6095_neg__numeral__le__one,axiom,
% 8.90/5.09 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6096_neg__numeral__le__one,axiom,
% 8.90/5.09 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6097_neg__numeral__le__one,axiom,
% 8.90/5.09 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6098_neg__one__le__numeral,axiom,
% 8.90/5.09 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6099_neg__one__le__numeral,axiom,
% 8.90/5.09 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6100_neg__one__le__numeral,axiom,
% 8.90/5.09 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6101_neg__one__le__numeral,axiom,
% 8.90/5.09 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6102_neg__numeral__le__neg__one,axiom,
% 8.90/5.09 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6103_neg__numeral__le__neg__one,axiom,
% 8.90/5.09 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6104_neg__numeral__le__neg__one,axiom,
% 8.90/5.09 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6105_neg__numeral__le__neg__one,axiom,
% 8.90/5.09 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6106_not__numeral__le__neg__one,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6107_not__numeral__le__neg__one,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6108_not__numeral__le__neg__one,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6109_not__numeral__le__neg__one,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6110_not__one__le__neg__numeral,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6111_not__one__le__neg__numeral,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6112_not__one__le__neg__numeral,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6113_not__one__le__neg__numeral,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6114_not__neg__one__less__neg__numeral,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6115_not__neg__one__less__neg__numeral,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6116_not__neg__one__less__neg__numeral,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6117_not__neg__one__less__neg__numeral,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6118_not__one__less__neg__numeral,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6119_not__one__less__neg__numeral,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6120_not__one__less__neg__numeral,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6121_not__one__less__neg__numeral,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6122_not__numeral__less__neg__one,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6123_not__numeral__less__neg__one,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6124_not__numeral__less__neg__one,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6125_not__numeral__less__neg__one,axiom,
% 8.90/5.09 ! [M: num] :
% 8.90/5.09 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6126_neg__one__less__numeral,axiom,
% 8.90/5.09 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6127_neg__one__less__numeral,axiom,
% 8.90/5.09 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6128_neg__one__less__numeral,axiom,
% 8.90/5.09 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6129_neg__one__less__numeral,axiom,
% 8.90/5.09 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6130_neg__numeral__less__one,axiom,
% 8.90/5.09 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6131_neg__numeral__less__one,axiom,
% 8.90/5.09 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6132_neg__numeral__less__one,axiom,
% 8.90/5.09 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6133_neg__numeral__less__one,axiom,
% 8.90/5.09 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6134_nonzero__neg__divide__eq__eq2,axiom,
% 8.90/5.09 ! [B: real,C: real,A: real] :
% 8.90/5.09 ( ( B != zero_zero_real )
% 8.90/5.09 => ( ( C
% 8.90/5.09 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
% 8.90/5.09 = ( ( times_times_real @ C @ B )
% 8.90/5.09 = ( uminus_uminus_real @ A ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6135_nonzero__neg__divide__eq__eq2,axiom,
% 8.90/5.09 ! [B: complex,C: complex,A: complex] :
% 8.90/5.09 ( ( B != zero_zero_complex )
% 8.90/5.09 => ( ( C
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 8.90/5.09 = ( ( times_times_complex @ C @ B )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6136_nonzero__neg__divide__eq__eq2,axiom,
% 8.90/5.09 ! [B: rat,C: rat,A: rat] :
% 8.90/5.09 ( ( B != zero_zero_rat )
% 8.90/5.09 => ( ( C
% 8.90/5.09 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
% 8.90/5.09 = ( ( times_times_rat @ C @ B )
% 8.90/5.09 = ( uminus_uminus_rat @ A ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6137_nonzero__neg__divide__eq__eq,axiom,
% 8.90/5.09 ! [B: real,A: real,C: real] :
% 8.90/5.09 ( ( B != zero_zero_real )
% 8.90/5.09 => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 8.90/5.09 = C )
% 8.90/5.09 = ( ( uminus_uminus_real @ A )
% 8.90/5.09 = ( times_times_real @ C @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6138_nonzero__neg__divide__eq__eq,axiom,
% 8.90/5.09 ! [B: complex,A: complex,C: complex] :
% 8.90/5.09 ( ( B != zero_zero_complex )
% 8.90/5.09 => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.90/5.09 = C )
% 8.90/5.09 = ( ( uminus1482373934393186551omplex @ A )
% 8.90/5.09 = ( times_times_complex @ C @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6139_nonzero__neg__divide__eq__eq,axiom,
% 8.90/5.09 ! [B: rat,A: rat,C: rat] :
% 8.90/5.09 ( ( B != zero_zero_rat )
% 8.90/5.09 => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 8.90/5.09 = C )
% 8.90/5.09 = ( ( uminus_uminus_rat @ A )
% 8.90/5.09 = ( times_times_rat @ C @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6140_minus__divide__eq__eq,axiom,
% 8.90/5.09 ! [B: real,C: real,A: real] :
% 8.90/5.09 ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
% 8.90/5.09 = A )
% 8.90/5.09 = ( ( ( C != zero_zero_real )
% 8.90/5.09 => ( ( uminus_uminus_real @ B )
% 8.90/5.09 = ( times_times_real @ A @ C ) ) )
% 8.90/5.09 & ( ( C = zero_zero_real )
% 8.90/5.09 => ( A = zero_zero_real ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6141_minus__divide__eq__eq,axiom,
% 8.90/5.09 ! [B: complex,C: complex,A: complex] :
% 8.90/5.09 ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
% 8.90/5.09 = A )
% 8.90/5.09 = ( ( ( C != zero_zero_complex )
% 8.90/5.09 => ( ( uminus1482373934393186551omplex @ B )
% 8.90/5.09 = ( times_times_complex @ A @ C ) ) )
% 8.90/5.09 & ( ( C = zero_zero_complex )
% 8.90/5.09 => ( A = zero_zero_complex ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6142_minus__divide__eq__eq,axiom,
% 8.90/5.09 ! [B: rat,C: rat,A: rat] :
% 8.90/5.09 ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
% 8.90/5.09 = A )
% 8.90/5.09 = ( ( ( C != zero_zero_rat )
% 8.90/5.09 => ( ( uminus_uminus_rat @ B )
% 8.90/5.09 = ( times_times_rat @ A @ C ) ) )
% 8.90/5.09 & ( ( C = zero_zero_rat )
% 8.90/5.09 => ( A = zero_zero_rat ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6143_eq__minus__divide__eq,axiom,
% 8.90/5.09 ! [A: real,B: real,C: real] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 8.90/5.09 = ( ( ( C != zero_zero_real )
% 8.90/5.09 => ( ( times_times_real @ A @ C )
% 8.90/5.09 = ( uminus_uminus_real @ B ) ) )
% 8.90/5.09 & ( ( C = zero_zero_real )
% 8.90/5.09 => ( A = zero_zero_real ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6144_eq__minus__divide__eq,axiom,
% 8.90/5.09 ! [A: complex,B: complex,C: complex] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
% 8.90/5.09 = ( ( ( C != zero_zero_complex )
% 8.90/5.09 => ( ( times_times_complex @ A @ C )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ B ) ) )
% 8.90/5.09 & ( ( C = zero_zero_complex )
% 8.90/5.09 => ( A = zero_zero_complex ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6145_eq__minus__divide__eq,axiom,
% 8.90/5.09 ! [A: rat,B: rat,C: rat] :
% 8.90/5.09 ( ( A
% 8.90/5.09 = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 8.90/5.09 = ( ( ( C != zero_zero_rat )
% 8.90/5.09 => ( ( times_times_rat @ A @ C )
% 8.90/5.09 = ( uminus_uminus_rat @ B ) ) )
% 8.90/5.09 & ( ( C = zero_zero_rat )
% 8.90/5.09 => ( A = zero_zero_rat ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6146_divide__eq__minus__1__iff,axiom,
% 8.90/5.09 ! [A: real,B: real] :
% 8.90/5.09 ( ( ( divide_divide_real @ A @ B )
% 8.90/5.09 = ( uminus_uminus_real @ one_one_real ) )
% 8.90/5.09 = ( ( B != zero_zero_real )
% 8.90/5.09 & ( A
% 8.90/5.09 = ( uminus_uminus_real @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6147_divide__eq__minus__1__iff,axiom,
% 8.90/5.09 ! [A: complex,B: complex] :
% 8.90/5.09 ( ( ( divide1717551699836669952omplex @ A @ B )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.90/5.09 = ( ( B != zero_zero_complex )
% 8.90/5.09 & ( A
% 8.90/5.09 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6148_divide__eq__minus__1__iff,axiom,
% 8.90/5.09 ! [A: rat,B: rat] :
% 8.90/5.09 ( ( ( divide_divide_rat @ A @ B )
% 8.90/5.09 = ( uminus_uminus_rat @ one_one_rat ) )
% 8.90/5.09 = ( ( B != zero_zero_rat )
% 8.90/5.09 & ( A
% 8.90/5.09 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6149_mult__1s__ring__1_I1_J,axiom,
% 8.90/5.09 ! [B: int] :
% 8.90/5.09 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
% 8.90/5.09 = ( uminus_uminus_int @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6150_mult__1s__ring__1_I1_J,axiom,
% 8.90/5.09 ! [B: real] :
% 8.90/5.09 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
% 8.90/5.09 = ( uminus_uminus_real @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6151_mult__1s__ring__1_I1_J,axiom,
% 8.90/5.09 ! [B: complex] :
% 8.90/5.09 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6152_mult__1s__ring__1_I1_J,axiom,
% 8.90/5.09 ! [B: code_integer] :
% 8.90/5.09 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6153_mult__1s__ring__1_I1_J,axiom,
% 8.90/5.09 ! [B: rat] :
% 8.90/5.09 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
% 8.90/5.09 = ( uminus_uminus_rat @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6154_mult__1s__ring__1_I2_J,axiom,
% 8.90/5.09 ! [B: int] :
% 8.90/5.09 ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
% 8.90/5.09 = ( uminus_uminus_int @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6155_mult__1s__ring__1_I2_J,axiom,
% 8.90/5.09 ! [B: real] :
% 8.90/5.09 ( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
% 8.90/5.09 = ( uminus_uminus_real @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6156_mult__1s__ring__1_I2_J,axiom,
% 8.90/5.09 ! [B: complex] :
% 8.90/5.09 ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6157_mult__1s__ring__1_I2_J,axiom,
% 8.90/5.09 ! [B: code_integer] :
% 8.90/5.09 ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6158_mult__1s__ring__1_I2_J,axiom,
% 8.90/5.09 ! [B: rat] :
% 8.90/5.09 ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
% 8.90/5.09 = ( uminus_uminus_rat @ B ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6159_uminus__numeral__One,axiom,
% 8.90/5.09 ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
% 8.90/5.09 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6160_uminus__numeral__One,axiom,
% 8.90/5.09 ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
% 8.90/5.09 = ( uminus_uminus_real @ one_one_real ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6161_uminus__numeral__One,axiom,
% 8.90/5.09 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6162_uminus__numeral__One,axiom,
% 8.90/5.09 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6163_uminus__numeral__One,axiom,
% 8.90/5.09 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
% 8.90/5.09 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6164_power__minus,axiom,
% 8.90/5.09 ! [A: int,N: nat] :
% 8.90/5.09 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 8.90/5.09 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6165_power__minus,axiom,
% 8.90/5.09 ! [A: real,N: nat] :
% 8.90/5.09 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 8.90/5.09 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6166_power__minus,axiom,
% 8.90/5.09 ! [A: complex,N: nat] :
% 8.90/5.09 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 8.90/5.09 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6167_power__minus,axiom,
% 8.90/5.09 ! [A: code_integer,N: nat] :
% 8.90/5.09 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 8.90/5.09 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6168_power__minus,axiom,
% 8.90/5.09 ! [A: rat,N: nat] :
% 8.90/5.09 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 8.90/5.09 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6169_power__minus__Bit0,axiom,
% 8.90/5.09 ! [X2: int,K: num] :
% 8.90/5.09 ( ( power_power_int @ ( uminus_uminus_int @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 8.90/5.09 = ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6170_power__minus__Bit0,axiom,
% 8.90/5.09 ! [X2: real,K: num] :
% 8.90/5.09 ( ( power_power_real @ ( uminus_uminus_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 8.90/5.09 = ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6171_power__minus__Bit0,axiom,
% 8.90/5.09 ! [X2: complex,K: num] :
% 8.90/5.09 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 8.90/5.09 = ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6172_power__minus__Bit0,axiom,
% 8.90/5.09 ! [X2: code_integer,K: num] :
% 8.90/5.09 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 8.90/5.09 = ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6173_power__minus__Bit0,axiom,
% 8.90/5.09 ! [X2: rat,K: num] :
% 8.90/5.09 ( ( power_power_rat @ ( uminus_uminus_rat @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 8.90/5.09 = ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6174_power__minus__Bit1,axiom,
% 8.90/5.09 ! [X2: int,K: num] :
% 8.90/5.09 ( ( power_power_int @ ( uminus_uminus_int @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 8.90/5.09 = ( uminus_uminus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6175_power__minus__Bit1,axiom,
% 8.90/5.09 ! [X2: real,K: num] :
% 8.90/5.09 ( ( power_power_real @ ( uminus_uminus_real @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 8.90/5.09 = ( uminus_uminus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6176_power__minus__Bit1,axiom,
% 8.90/5.09 ! [X2: complex,K: num] :
% 8.90/5.09 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6177_power__minus__Bit1,axiom,
% 8.90/5.09 ! [X2: code_integer,K: num] :
% 8.90/5.09 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 8.90/5.09 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6178_power__minus__Bit1,axiom,
% 8.90/5.09 ! [X2: rat,K: num] :
% 8.90/5.09 ( ( power_power_rat @ ( uminus_uminus_rat @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 8.90/5.09 = ( uminus_uminus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6179_real__0__less__add__iff,axiom,
% 8.90/5.09 ! [X2: real,Y3: real] :
% 8.90/5.09 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 8.90/5.09 = ( ord_less_real @ ( uminus_uminus_real @ X2 ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6180_real__add__less__0__iff,axiom,
% 8.90/5.09 ! [X2: real,Y3: real] :
% 8.90/5.09 ( ( ord_less_real @ ( plus_plus_real @ X2 @ Y3 ) @ zero_zero_real )
% 8.90/5.09 = ( ord_less_real @ Y3 @ ( uminus_uminus_real @ X2 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6181_real__0__le__add__iff,axiom,
% 8.90/5.09 ! [X2: real,Y3: real] :
% 8.90/5.09 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 8.90/5.09 = ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ Y3 ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6182_real__add__le__0__iff,axiom,
% 8.90/5.09 ! [X2: real,Y3: real] :
% 8.90/5.09 ( ( ord_less_eq_real @ ( plus_plus_real @ X2 @ Y3 ) @ zero_zero_real )
% 8.90/5.09 = ( ord_less_eq_real @ Y3 @ ( uminus_uminus_real @ X2 ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6183_zmod__zminus1__eq__if,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( ( ( modulo_modulo_int @ A @ B )
% 8.90/5.09 = zero_zero_int )
% 8.90/5.09 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 8.90/5.09 = zero_zero_int ) )
% 8.90/5.09 & ( ( ( modulo_modulo_int @ A @ B )
% 8.90/5.09 != zero_zero_int )
% 8.90/5.09 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 8.90/5.09 = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6184_zmod__zminus2__eq__if,axiom,
% 8.90/5.09 ! [A: int,B: int] :
% 8.90/5.09 ( ( ( ( modulo_modulo_int @ A @ B )
% 8.90/5.09 = zero_zero_int )
% 8.90/5.09 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 8.90/5.09 = zero_zero_int ) )
% 8.90/5.09 & ( ( ( modulo_modulo_int @ A @ B )
% 8.90/5.09 != zero_zero_int )
% 8.90/5.09 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 8.90/5.09 = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6185_pred__numeral__def,axiom,
% 8.90/5.09 ( pred_numeral
% 8.90/5.09 = ( ^ [K2: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K2 ) @ one_one_nat ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6186_less__minus__divide__eq,axiom,
% 8.90/5.09 ! [A: real,B: real,C: real] :
% 8.90/5.09 ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 8.90/5.09 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.09 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 8.90/5.09 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.09 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.09 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 8.90/5.09 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.09 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6187_less__minus__divide__eq,axiom,
% 8.90/5.09 ! [A: rat,B: rat,C: rat] :
% 8.90/5.09 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 8.90/5.09 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.09 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 8.90/5.09 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.09 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.09 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 8.90/5.09 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.09 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6188_minus__divide__less__eq,axiom,
% 8.90/5.09 ! [B: real,C: real,A: real] :
% 8.90/5.09 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 8.90/5.09 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.09 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 8.90/5.09 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.09 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.09 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 8.90/5.09 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.09 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6189_minus__divide__less__eq,axiom,
% 8.90/5.09 ! [B: rat,C: rat,A: rat] :
% 8.90/5.09 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 8.90/5.09 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.09 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 8.90/5.09 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.09 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.09 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 8.90/5.09 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.09 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6190_neg__less__minus__divide__eq,axiom,
% 8.90/5.09 ! [C: real,A: real,B: real] :
% 8.90/5.09 ( ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.09 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 8.90/5.09 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6191_neg__less__minus__divide__eq,axiom,
% 8.90/5.09 ! [C: rat,A: rat,B: rat] :
% 8.90/5.09 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.09 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 8.90/5.09 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6192_neg__minus__divide__less__eq,axiom,
% 8.90/5.09 ! [C: real,B: real,A: real] :
% 8.90/5.09 ( ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.09 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 8.90/5.09 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6193_neg__minus__divide__less__eq,axiom,
% 8.90/5.09 ! [C: rat,B: rat,A: rat] :
% 8.90/5.09 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.09 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 8.90/5.09 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6194_pos__less__minus__divide__eq,axiom,
% 8.90/5.09 ! [C: real,A: real,B: real] :
% 8.90/5.09 ( ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.09 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 8.90/5.09 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6195_pos__less__minus__divide__eq,axiom,
% 8.90/5.09 ! [C: rat,A: rat,B: rat] :
% 8.90/5.09 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.09 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 8.90/5.09 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6196_pos__minus__divide__less__eq,axiom,
% 8.90/5.09 ! [C: real,B: real,A: real] :
% 8.90/5.09 ( ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.09 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 8.90/5.09 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6197_pos__minus__divide__less__eq,axiom,
% 8.90/5.09 ! [C: rat,B: rat,A: rat] :
% 8.90/5.09 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.09 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 8.90/5.09 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6198_eq__divide__eq__numeral_I2_J,axiom,
% 8.90/5.09 ! [W: num,B: real,C: real] :
% 8.90/5.09 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 8.90/5.09 = ( divide_divide_real @ B @ C ) )
% 8.90/5.09 = ( ( ( C != zero_zero_real )
% 8.90/5.09 => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
% 8.90/5.09 = B ) )
% 8.90/5.09 & ( ( C = zero_zero_real )
% 8.90/5.09 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 8.90/5.09 = zero_zero_real ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6199_eq__divide__eq__numeral_I2_J,axiom,
% 8.90/5.09 ! [W: num,B: complex,C: complex] :
% 8.90/5.09 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 8.90/5.09 = ( divide1717551699836669952omplex @ B @ C ) )
% 8.90/5.09 = ( ( ( C != zero_zero_complex )
% 8.90/5.09 => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
% 8.90/5.09 = B ) )
% 8.90/5.09 & ( ( C = zero_zero_complex )
% 8.90/5.09 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 8.90/5.09 = zero_zero_complex ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6200_eq__divide__eq__numeral_I2_J,axiom,
% 8.90/5.09 ! [W: num,B: rat,C: rat] :
% 8.90/5.09 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 8.90/5.09 = ( divide_divide_rat @ B @ C ) )
% 8.90/5.09 = ( ( ( C != zero_zero_rat )
% 8.90/5.09 => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
% 8.90/5.09 = B ) )
% 8.90/5.09 & ( ( C = zero_zero_rat )
% 8.90/5.09 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 8.90/5.09 = zero_zero_rat ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6201_divide__eq__eq__numeral_I2_J,axiom,
% 8.90/5.09 ! [B: real,C: real,W: num] :
% 8.90/5.09 ( ( ( divide_divide_real @ B @ C )
% 8.90/5.09 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 8.90/5.09 = ( ( ( C != zero_zero_real )
% 8.90/5.09 => ( B
% 8.90/5.09 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 8.90/5.09 & ( ( C = zero_zero_real )
% 8.90/5.09 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 8.90/5.09 = zero_zero_real ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6202_divide__eq__eq__numeral_I2_J,axiom,
% 8.90/5.09 ! [B: complex,C: complex,W: num] :
% 8.90/5.09 ( ( ( divide1717551699836669952omplex @ B @ C )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 8.90/5.09 = ( ( ( C != zero_zero_complex )
% 8.90/5.09 => ( B
% 8.90/5.09 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
% 8.90/5.09 & ( ( C = zero_zero_complex )
% 8.90/5.09 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 8.90/5.09 = zero_zero_complex ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6203_divide__eq__eq__numeral_I2_J,axiom,
% 8.90/5.09 ! [B: rat,C: rat,W: num] :
% 8.90/5.09 ( ( ( divide_divide_rat @ B @ C )
% 8.90/5.09 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 8.90/5.09 = ( ( ( C != zero_zero_rat )
% 8.90/5.09 => ( B
% 8.90/5.09 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 8.90/5.09 & ( ( C = zero_zero_rat )
% 8.90/5.09 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 8.90/5.09 = zero_zero_rat ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6204_add__divide__eq__if__simps_I3_J,axiom,
% 8.90/5.09 ! [Z: real,A: real,B: real] :
% 8.90/5.09 ( ( ( Z = zero_zero_real )
% 8.90/5.09 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 8.90/5.09 = B ) )
% 8.90/5.09 & ( ( Z != zero_zero_real )
% 8.90/5.09 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 8.90/5.09 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6205_add__divide__eq__if__simps_I3_J,axiom,
% 8.90/5.09 ! [Z: complex,A: complex,B: complex] :
% 8.90/5.09 ( ( ( Z = zero_zero_complex )
% 8.90/5.09 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 8.90/5.09 = B ) )
% 8.90/5.09 & ( ( Z != zero_zero_complex )
% 8.90/5.09 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 8.90/5.09 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6206_add__divide__eq__if__simps_I3_J,axiom,
% 8.90/5.09 ! [Z: rat,A: rat,B: rat] :
% 8.90/5.09 ( ( ( Z = zero_zero_rat )
% 8.90/5.09 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 8.90/5.09 = B ) )
% 8.90/5.09 & ( ( Z != zero_zero_rat )
% 8.90/5.09 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 8.90/5.09 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6207_minus__divide__add__eq__iff,axiom,
% 8.90/5.09 ! [Z: real,X2: real,Y3: real] :
% 8.90/5.09 ( ( Z != zero_zero_real )
% 8.90/5.09 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X2 @ Z ) ) @ Y3 )
% 8.90/5.09 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6208_minus__divide__add__eq__iff,axiom,
% 8.90/5.09 ! [Z: complex,X2: complex,Y3: complex] :
% 8.90/5.09 ( ( Z != zero_zero_complex )
% 8.90/5.09 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X2 @ Z ) ) @ Y3 )
% 8.90/5.09 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6209_minus__divide__add__eq__iff,axiom,
% 8.90/5.09 ! [Z: rat,X2: rat,Y3: rat] :
% 8.90/5.09 ( ( Z != zero_zero_rat )
% 8.90/5.09 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X2 @ Z ) ) @ Y3 )
% 8.90/5.09 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X2 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6210_minus__divide__diff__eq__iff,axiom,
% 8.90/5.09 ! [Z: real,X2: real,Y3: real] :
% 8.90/5.09 ( ( Z != zero_zero_real )
% 8.90/5.09 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X2 @ Z ) ) @ Y3 )
% 8.90/5.09 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6211_minus__divide__diff__eq__iff,axiom,
% 8.90/5.09 ! [Z: complex,X2: complex,Y3: complex] :
% 8.90/5.09 ( ( Z != zero_zero_complex )
% 8.90/5.09 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X2 @ Z ) ) @ Y3 )
% 8.90/5.09 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6212_minus__divide__diff__eq__iff,axiom,
% 8.90/5.09 ! [Z: rat,X2: rat,Y3: rat] :
% 8.90/5.09 ( ( Z != zero_zero_rat )
% 8.90/5.09 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X2 @ Z ) ) @ Y3 )
% 8.90/5.09 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X2 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6213_add__divide__eq__if__simps_I5_J,axiom,
% 8.90/5.09 ! [Z: real,A: real,B: real] :
% 8.90/5.09 ( ( ( Z = zero_zero_real )
% 8.90/5.09 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 8.90/5.09 = ( uminus_uminus_real @ B ) ) )
% 8.90/5.09 & ( ( Z != zero_zero_real )
% 8.90/5.09 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 8.90/5.09 = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6214_add__divide__eq__if__simps_I5_J,axiom,
% 8.90/5.09 ! [Z: complex,A: complex,B: complex] :
% 8.90/5.09 ( ( ( Z = zero_zero_complex )
% 8.90/5.09 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 8.90/5.09 = ( uminus1482373934393186551omplex @ B ) ) )
% 8.90/5.09 & ( ( Z != zero_zero_complex )
% 8.90/5.09 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 8.90/5.09 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6215_add__divide__eq__if__simps_I5_J,axiom,
% 8.90/5.09 ! [Z: rat,A: rat,B: rat] :
% 8.90/5.09 ( ( ( Z = zero_zero_rat )
% 8.90/5.09 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 8.90/5.09 = ( uminus_uminus_rat @ B ) ) )
% 8.90/5.09 & ( ( Z != zero_zero_rat )
% 8.90/5.09 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 8.90/5.09 = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 8.90/5.09
% 8.90/5.09 thf(fact_6216_add__divide__eq__if__simps_I6_J,axiom,
% 8.90/5.09 ! [Z: real,A: real,B: real] :
% 8.90/5.09 ( ( ( Z = zero_zero_real )
% 8.90/5.09 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 8.90/5.10 = ( uminus_uminus_real @ B ) ) )
% 8.90/5.10 & ( ( Z != zero_zero_real )
% 8.90/5.10 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 8.90/5.10 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6217_add__divide__eq__if__simps_I6_J,axiom,
% 8.90/5.10 ! [Z: complex,A: complex,B: complex] :
% 8.90/5.10 ( ( ( Z = zero_zero_complex )
% 8.90/5.10 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 8.90/5.10 = ( uminus1482373934393186551omplex @ B ) ) )
% 8.90/5.10 & ( ( Z != zero_zero_complex )
% 8.90/5.10 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 8.90/5.10 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6218_add__divide__eq__if__simps_I6_J,axiom,
% 8.90/5.10 ! [Z: rat,A: rat,B: rat] :
% 8.90/5.10 ( ( ( Z = zero_zero_rat )
% 8.90/5.10 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 8.90/5.10 = ( uminus_uminus_rat @ B ) ) )
% 8.90/5.10 & ( ( Z != zero_zero_rat )
% 8.90/5.10 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 8.90/5.10 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6219_even__minus,axiom,
% 8.90/5.10 ! [A: int] :
% 8.90/5.10 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
% 8.90/5.10 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6220_even__minus,axiom,
% 8.90/5.10 ! [A: code_integer] :
% 8.90/5.10 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
% 8.90/5.10 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6221_power2__eq__iff,axiom,
% 8.90/5.10 ! [X2: int,Y3: int] :
% 8.90/5.10 ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.10 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.10 = ( ( X2 = Y3 )
% 8.90/5.10 | ( X2
% 8.90/5.10 = ( uminus_uminus_int @ Y3 ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6222_power2__eq__iff,axiom,
% 8.90/5.10 ! [X2: real,Y3: real] :
% 8.90/5.10 ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.10 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.10 = ( ( X2 = Y3 )
% 8.90/5.10 | ( X2
% 8.90/5.10 = ( uminus_uminus_real @ Y3 ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6223_power2__eq__iff,axiom,
% 8.90/5.10 ! [X2: complex,Y3: complex] :
% 8.90/5.10 ( ( ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.10 = ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.10 = ( ( X2 = Y3 )
% 8.90/5.10 | ( X2
% 8.90/5.10 = ( uminus1482373934393186551omplex @ Y3 ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6224_power2__eq__iff,axiom,
% 8.90/5.10 ! [X2: code_integer,Y3: code_integer] :
% 8.90/5.10 ( ( ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.10 = ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.10 = ( ( X2 = Y3 )
% 8.90/5.10 | ( X2
% 8.90/5.10 = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6225_power2__eq__iff,axiom,
% 8.90/5.10 ! [X2: rat,Y3: rat] :
% 8.90/5.10 ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.10 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.10 = ( ( X2 = Y3 )
% 8.90/5.10 | ( X2
% 8.90/5.10 = ( uminus_uminus_rat @ Y3 ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6226_verit__less__mono__div__int2,axiom,
% 8.90/5.10 ! [A2: int,B3: int,N: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ A2 @ B3 )
% 8.90/5.10 => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
% 8.90/5.10 => ( ord_less_eq_int @ ( divide_divide_int @ B3 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6227_div__eq__minus1,axiom,
% 8.90/5.10 ! [B: int] :
% 8.90/5.10 ( ( ord_less_int @ zero_zero_int @ B )
% 8.90/5.10 => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 8.90/5.10 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6228_pos__minus__divide__le__eq,axiom,
% 8.90/5.10 ! [C: real,B: real,A: real] :
% 8.90/5.10 ( ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.10 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 8.90/5.10 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6229_pos__minus__divide__le__eq,axiom,
% 8.90/5.10 ! [C: rat,B: rat,A: rat] :
% 8.90/5.10 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.10 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 8.90/5.10 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6230_pos__le__minus__divide__eq,axiom,
% 8.90/5.10 ! [C: real,A: real,B: real] :
% 8.90/5.10 ( ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.10 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 8.90/5.10 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6231_pos__le__minus__divide__eq,axiom,
% 8.90/5.10 ! [C: rat,A: rat,B: rat] :
% 8.90/5.10 ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.10 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 8.90/5.10 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6232_neg__minus__divide__le__eq,axiom,
% 8.90/5.10 ! [C: real,B: real,A: real] :
% 8.90/5.10 ( ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.10 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 8.90/5.10 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6233_neg__minus__divide__le__eq,axiom,
% 8.90/5.10 ! [C: rat,B: rat,A: rat] :
% 8.90/5.10 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.10 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 8.90/5.10 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6234_neg__le__minus__divide__eq,axiom,
% 8.90/5.10 ! [C: real,A: real,B: real] :
% 8.90/5.10 ( ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.10 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 8.90/5.10 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6235_neg__le__minus__divide__eq,axiom,
% 8.90/5.10 ! [C: rat,A: rat,B: rat] :
% 8.90/5.10 ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.10 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 8.90/5.10 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6236_minus__divide__le__eq,axiom,
% 8.90/5.10 ! [B: real,C: real,A: real] :
% 8.90/5.10 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 8.90/5.10 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.10 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 8.90/5.10 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.10 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.10 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 8.90/5.10 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6237_minus__divide__le__eq,axiom,
% 8.90/5.10 ! [B: rat,C: rat,A: rat] :
% 8.90/5.10 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 8.90/5.10 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.10 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 8.90/5.10 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.10 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.10 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 8.90/5.10 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6238_le__minus__divide__eq,axiom,
% 8.90/5.10 ! [A: real,B: real,C: real] :
% 8.90/5.10 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 8.90/5.10 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.10 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 8.90/5.10 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.10 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.10 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 8.90/5.10 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.10 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6239_le__minus__divide__eq,axiom,
% 8.90/5.10 ! [A: rat,B: rat,C: rat] :
% 8.90/5.10 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 8.90/5.10 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.10 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 8.90/5.10 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.10 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.10 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 8.90/5.10 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.10 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6240_less__divide__eq__numeral_I2_J,axiom,
% 8.90/5.10 ! [W: num,B: real,C: real] :
% 8.90/5.10 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 8.90/5.10 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.10 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 8.90/5.10 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.10 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.10 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 8.90/5.10 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.10 => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6241_less__divide__eq__numeral_I2_J,axiom,
% 8.90/5.10 ! [W: num,B: rat,C: rat] :
% 8.90/5.10 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 8.90/5.10 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.10 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 8.90/5.10 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.10 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.10 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 8.90/5.10 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.10 => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6242_divide__less__eq__numeral_I2_J,axiom,
% 8.90/5.10 ! [B: real,C: real,W: num] :
% 8.90/5.10 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 8.90/5.10 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.10 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 8.90/5.10 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.10 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.10 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 8.90/5.10 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.10 => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6243_divide__less__eq__numeral_I2_J,axiom,
% 8.90/5.10 ! [B: rat,C: rat,W: num] :
% 8.90/5.10 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 8.90/5.10 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.10 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 8.90/5.10 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.10 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.10 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 8.90/5.10 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.10 => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6244_power2__eq__1__iff,axiom,
% 8.90/5.10 ! [A: int] :
% 8.90/5.10 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.10 = one_one_int )
% 8.90/5.10 = ( ( A = one_one_int )
% 8.90/5.10 | ( A
% 8.90/5.10 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6245_power2__eq__1__iff,axiom,
% 8.90/5.10 ! [A: real] :
% 8.90/5.10 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.10 = one_one_real )
% 8.90/5.10 = ( ( A = one_one_real )
% 8.90/5.10 | ( A
% 8.90/5.10 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6246_power2__eq__1__iff,axiom,
% 8.90/5.10 ! [A: complex] :
% 8.90/5.10 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.10 = one_one_complex )
% 8.90/5.10 = ( ( A = one_one_complex )
% 8.90/5.10 | ( A
% 8.90/5.10 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6247_power2__eq__1__iff,axiom,
% 8.90/5.10 ! [A: code_integer] :
% 8.90/5.10 ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.10 = one_one_Code_integer )
% 8.90/5.10 = ( ( A = one_one_Code_integer )
% 8.90/5.10 | ( A
% 8.90/5.10 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6248_power2__eq__1__iff,axiom,
% 8.90/5.10 ! [A: rat] :
% 8.90/5.10 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.10 = one_one_rat )
% 8.90/5.10 = ( ( A = one_one_rat )
% 8.90/5.10 | ( A
% 8.90/5.10 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6249_uminus__power__if,axiom,
% 8.90/5.10 ! [N: nat,A: int] :
% 8.90/5.10 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 8.90/5.10 = ( power_power_int @ A @ N ) ) )
% 8.90/5.10 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 8.90/5.10 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6250_uminus__power__if,axiom,
% 8.90/5.10 ! [N: nat,A: real] :
% 8.90/5.10 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 8.90/5.10 = ( power_power_real @ A @ N ) ) )
% 8.90/5.10 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 8.90/5.10 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6251_uminus__power__if,axiom,
% 8.90/5.10 ! [N: nat,A: complex] :
% 8.90/5.10 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 8.90/5.10 = ( power_power_complex @ A @ N ) ) )
% 8.90/5.10 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 8.90/5.10 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6252_uminus__power__if,axiom,
% 8.90/5.10 ! [N: nat,A: code_integer] :
% 8.90/5.10 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 8.90/5.10 = ( power_8256067586552552935nteger @ A @ N ) ) )
% 8.90/5.10 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 8.90/5.10 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6253_uminus__power__if,axiom,
% 8.90/5.10 ! [N: nat,A: rat] :
% 8.90/5.10 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 8.90/5.10 = ( power_power_rat @ A @ N ) ) )
% 8.90/5.10 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 8.90/5.10 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6254_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 8.90/5.10 ! [K: nat,N: nat] :
% 8.90/5.10 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.10 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
% 8.90/5.10 = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6255_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 8.90/5.10 ! [K: nat,N: nat] :
% 8.90/5.10 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.10 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
% 8.90/5.10 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6256_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 8.90/5.10 ! [K: nat,N: nat] :
% 8.90/5.10 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.10 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
% 8.90/5.10 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6257_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 8.90/5.10 ! [K: nat,N: nat] :
% 8.90/5.10 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.10 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
% 8.90/5.10 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6258_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 8.90/5.10 ! [K: nat,N: nat] :
% 8.90/5.10 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.10 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
% 8.90/5.10 = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6259_realpow__square__minus__le,axiom,
% 8.90/5.10 ! [U: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6260_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 8.90/5.10 ! [N: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6261_signed__take__bit__int__less__eq__self__iff,axiom,
% 8.90/5.10 ! [N: nat,K: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 8.90/5.10 = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6262_signed__take__bit__int__greater__self__iff,axiom,
% 8.90/5.10 ! [K: int,N: nat] :
% 8.90/5.10 ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 8.90/5.10 = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6263_minus__mod__int__eq,axiom,
% 8.90/5.10 ! [L: int,K: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ zero_zero_int @ L )
% 8.90/5.10 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
% 8.90/5.10 = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6264_zmod__minus1,axiom,
% 8.90/5.10 ! [B: int] :
% 8.90/5.10 ( ( ord_less_int @ zero_zero_int @ B )
% 8.90/5.10 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 8.90/5.10 = ( minus_minus_int @ B @ one_one_int ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6265_zdiv__zminus2__eq__if,axiom,
% 8.90/5.10 ! [B: int,A: int] :
% 8.90/5.10 ( ( B != zero_zero_int )
% 8.90/5.10 => ( ( ( ( modulo_modulo_int @ A @ B )
% 8.90/5.10 = zero_zero_int )
% 8.90/5.10 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 8.90/5.10 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 8.90/5.10 & ( ( ( modulo_modulo_int @ A @ B )
% 8.90/5.10 != zero_zero_int )
% 8.90/5.10 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 8.90/5.10 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6266_zdiv__zminus1__eq__if,axiom,
% 8.90/5.10 ! [B: int,A: int] :
% 8.90/5.10 ( ( B != zero_zero_int )
% 8.90/5.10 => ( ( ( ( modulo_modulo_int @ A @ B )
% 8.90/5.10 = zero_zero_int )
% 8.90/5.10 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 8.90/5.10 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 8.90/5.10 & ( ( ( modulo_modulo_int @ A @ B )
% 8.90/5.10 != zero_zero_int )
% 8.90/5.10 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 8.90/5.10 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6267_zminus1__lemma,axiom,
% 8.90/5.10 ! [A: int,B: int,Q3: int,R2: int] :
% 8.90/5.10 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 8.90/5.10 => ( ( B != zero_zero_int )
% 8.90/5.10 => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q3 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q3 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R2 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6268_le__divide__eq__numeral_I2_J,axiom,
% 8.90/5.10 ! [W: num,B: real,C: real] :
% 8.90/5.10 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 8.90/5.10 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.10 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 8.90/5.10 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.10 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.10 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 8.90/5.10 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.10 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6269_le__divide__eq__numeral_I2_J,axiom,
% 8.90/5.10 ! [W: num,B: rat,C: rat] :
% 8.90/5.10 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 8.90/5.10 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.10 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 8.90/5.10 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.10 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.10 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 8.90/5.10 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.10 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6270_divide__le__eq__numeral_I2_J,axiom,
% 8.90/5.10 ! [B: real,C: real,W: num] :
% 8.90/5.10 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 8.90/5.10 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.10 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 8.90/5.10 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.10 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.10 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 8.90/5.10 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6271_divide__le__eq__numeral_I2_J,axiom,
% 8.90/5.10 ! [B: rat,C: rat,W: num] :
% 8.90/5.10 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 8.90/5.10 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.10 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 8.90/5.10 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 8.90/5.10 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.10 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 8.90/5.10 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6272_square__le__1,axiom,
% 8.90/5.10 ! [X2: real] :
% 8.90/5.10 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.10 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 8.90/5.10 => ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6273_square__le__1,axiom,
% 8.90/5.10 ! [X2: code_integer] :
% 8.90/5.10 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X2 )
% 8.90/5.10 => ( ( ord_le3102999989581377725nteger @ X2 @ one_one_Code_integer )
% 8.90/5.10 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6274_square__le__1,axiom,
% 8.90/5.10 ! [X2: rat] :
% 8.90/5.10 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X2 )
% 8.90/5.10 => ( ( ord_less_eq_rat @ X2 @ one_one_rat )
% 8.90/5.10 => ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6275_square__le__1,axiom,
% 8.90/5.10 ! [X2: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X2 )
% 8.90/5.10 => ( ( ord_less_eq_int @ X2 @ one_one_int )
% 8.90/5.10 => ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6276_minus__power__mult__self,axiom,
% 8.90/5.10 ! [A: int,N: nat] :
% 8.90/5.10 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 8.90/5.10 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6277_minus__power__mult__self,axiom,
% 8.90/5.10 ! [A: real,N: nat] :
% 8.90/5.10 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 8.90/5.10 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6278_minus__power__mult__self,axiom,
% 8.90/5.10 ! [A: complex,N: nat] :
% 8.90/5.10 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
% 8.90/5.10 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6279_minus__power__mult__self,axiom,
% 8.90/5.10 ! [A: code_integer,N: nat] :
% 8.90/5.10 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 8.90/5.10 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6280_minus__power__mult__self,axiom,
% 8.90/5.10 ! [A: rat,N: nat] :
% 8.90/5.10 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 8.90/5.10 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6281_minus__one__power__iff,axiom,
% 8.90/5.10 ! [N: nat] :
% 8.90/5.10 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 8.90/5.10 = one_one_int ) )
% 8.90/5.10 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 8.90/5.10 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6282_minus__one__power__iff,axiom,
% 8.90/5.10 ! [N: nat] :
% 8.90/5.10 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 8.90/5.10 = one_one_real ) )
% 8.90/5.10 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 8.90/5.10 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6283_minus__one__power__iff,axiom,
% 8.90/5.10 ! [N: nat] :
% 8.90/5.10 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 8.90/5.10 = one_one_complex ) )
% 8.90/5.10 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 8.90/5.10 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6284_minus__one__power__iff,axiom,
% 8.90/5.10 ! [N: nat] :
% 8.90/5.10 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 8.90/5.10 = one_one_Code_integer ) )
% 8.90/5.10 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 8.90/5.10 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6285_minus__one__power__iff,axiom,
% 8.90/5.10 ! [N: nat] :
% 8.90/5.10 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 8.90/5.10 = one_one_rat ) )
% 8.90/5.10 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.10 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 8.90/5.10 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6286_signed__take__bit__int__eq__self__iff,axiom,
% 8.90/5.10 ! [N: nat,K: int] :
% 8.90/5.10 ( ( ( bit_ri631733984087533419it_int @ N @ K )
% 8.90/5.10 = K )
% 8.90/5.10 = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 8.90/5.10 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6287_signed__take__bit__int__eq__self,axiom,
% 8.90/5.10 ! [N: nat,K: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 8.90/5.10 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.10 => ( ( bit_ri631733984087533419it_int @ N @ K )
% 8.90/5.10 = K ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6288_minus__1__div__exp__eq__int,axiom,
% 8.90/5.10 ! [N: nat] :
% 8.90/5.10 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.10 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6289_div__pos__neg__trivial,axiom,
% 8.90/5.10 ! [K: int,L: int] :
% 8.90/5.10 ( ( ord_less_int @ zero_zero_int @ K )
% 8.90/5.10 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 8.90/5.10 => ( ( divide_divide_int @ K @ L )
% 8.90/5.10 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6290_add__0__iff,axiom,
% 8.90/5.10 ! [B: complex,A: complex] :
% 8.90/5.10 ( ( B
% 8.90/5.10 = ( plus_plus_complex @ B @ A ) )
% 8.90/5.10 = ( A = zero_zero_complex ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6291_add__0__iff,axiom,
% 8.90/5.10 ! [B: real,A: real] :
% 8.90/5.10 ( ( B
% 8.90/5.10 = ( plus_plus_real @ B @ A ) )
% 8.90/5.10 = ( A = zero_zero_real ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6292_add__0__iff,axiom,
% 8.90/5.10 ! [B: rat,A: rat] :
% 8.90/5.10 ( ( B
% 8.90/5.10 = ( plus_plus_rat @ B @ A ) )
% 8.90/5.10 = ( A = zero_zero_rat ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6293_add__0__iff,axiom,
% 8.90/5.10 ! [B: nat,A: nat] :
% 8.90/5.10 ( ( B
% 8.90/5.10 = ( plus_plus_nat @ B @ A ) )
% 8.90/5.10 = ( A = zero_zero_nat ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6294_add__0__iff,axiom,
% 8.90/5.10 ! [B: int,A: int] :
% 8.90/5.10 ( ( B
% 8.90/5.10 = ( plus_plus_int @ B @ A ) )
% 8.90/5.10 = ( A = zero_zero_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6295_crossproduct__eq,axiom,
% 8.90/5.10 ! [W: real,Y3: real,X2: real,Z: real] :
% 8.90/5.10 ( ( ( plus_plus_real @ ( times_times_real @ W @ Y3 ) @ ( times_times_real @ X2 @ Z ) )
% 8.90/5.10 = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X2 @ Y3 ) ) )
% 8.90/5.10 = ( ( W = X2 )
% 8.90/5.10 | ( Y3 = Z ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6296_crossproduct__eq,axiom,
% 8.90/5.10 ! [W: rat,Y3: rat,X2: rat,Z: rat] :
% 8.90/5.10 ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y3 ) @ ( times_times_rat @ X2 @ Z ) )
% 8.90/5.10 = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X2 @ Y3 ) ) )
% 8.90/5.10 = ( ( W = X2 )
% 8.90/5.10 | ( Y3 = Z ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6297_crossproduct__eq,axiom,
% 8.90/5.10 ! [W: nat,Y3: nat,X2: nat,Z: nat] :
% 8.90/5.10 ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y3 ) @ ( times_times_nat @ X2 @ Z ) )
% 8.90/5.10 = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X2 @ Y3 ) ) )
% 8.90/5.10 = ( ( W = X2 )
% 8.90/5.10 | ( Y3 = Z ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6298_crossproduct__eq,axiom,
% 8.90/5.10 ! [W: int,Y3: int,X2: int,Z: int] :
% 8.90/5.10 ( ( ( plus_plus_int @ ( times_times_int @ W @ Y3 ) @ ( times_times_int @ X2 @ Z ) )
% 8.90/5.10 = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X2 @ Y3 ) ) )
% 8.90/5.10 = ( ( W = X2 )
% 8.90/5.10 | ( Y3 = Z ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6299_crossproduct__noteq,axiom,
% 8.90/5.10 ! [A: real,B: real,C: real,D: real] :
% 8.90/5.10 ( ( ( A != B )
% 8.90/5.10 & ( C != D ) )
% 8.90/5.10 = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
% 8.90/5.10 != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6300_crossproduct__noteq,axiom,
% 8.90/5.10 ! [A: rat,B: rat,C: rat,D: rat] :
% 8.90/5.10 ( ( ( A != B )
% 8.90/5.10 & ( C != D ) )
% 8.90/5.10 = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
% 8.90/5.10 != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6301_crossproduct__noteq,axiom,
% 8.90/5.10 ! [A: nat,B: nat,C: nat,D: nat] :
% 8.90/5.10 ( ( ( A != B )
% 8.90/5.10 & ( C != D ) )
% 8.90/5.10 = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
% 8.90/5.10 != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6302_crossproduct__noteq,axiom,
% 8.90/5.10 ! [A: int,B: int,C: int,D: int] :
% 8.90/5.10 ( ( ( A != B )
% 8.90/5.10 & ( C != D ) )
% 8.90/5.10 = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
% 8.90/5.10 != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6303_divmod__nat__def,axiom,
% 8.90/5.10 ( divmod_nat
% 8.90/5.10 = ( ^ [M3: nat,N3: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M3 @ N3 ) @ ( modulo_modulo_nat @ M3 @ N3 ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6304_power__minus1__odd,axiom,
% 8.90/5.10 ! [N: nat] :
% 8.90/5.10 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.90/5.10 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6305_power__minus1__odd,axiom,
% 8.90/5.10 ! [N: nat] :
% 8.90/5.10 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.90/5.10 = ( uminus_uminus_real @ one_one_real ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6306_power__minus1__odd,axiom,
% 8.90/5.10 ! [N: nat] :
% 8.90/5.10 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.90/5.10 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6307_power__minus1__odd,axiom,
% 8.90/5.10 ! [N: nat] :
% 8.90/5.10 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.90/5.10 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6308_power__minus1__odd,axiom,
% 8.90/5.10 ! [N: nat] :
% 8.90/5.10 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.90/5.10 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6309_int__bit__induct,axiom,
% 8.90/5.10 ! [P: int > $o,K: int] :
% 8.90/5.10 ( ( P @ zero_zero_int )
% 8.90/5.10 => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.10 => ( ! [K3: int] :
% 8.90/5.10 ( ( P @ K3 )
% 8.90/5.10 => ( ( K3 != zero_zero_int )
% 8.90/5.10 => ( P @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
% 8.90/5.10 => ( ! [K3: int] :
% 8.90/5.10 ( ( P @ K3 )
% 8.90/5.10 => ( ( K3
% 8.90/5.10 != ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.10 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
% 8.90/5.10 => ( P @ K ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6310_signed__take__bit__int__greater__eq,axiom,
% 8.90/5.10 ! [K: int,N: nat] :
% 8.90/5.10 ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 8.90/5.10 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6311_one__div__minus__numeral,axiom,
% 8.90/5.10 ! [N: num] :
% 8.90/5.10 ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.10 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6312_minus__one__div__numeral,axiom,
% 8.90/5.10 ! [N: num] :
% 8.90/5.10 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 8.90/5.10 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6313_numeral__div__minus__numeral,axiom,
% 8.90/5.10 ! [M: num,N: num] :
% 8.90/5.10 ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.10 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6314_minus__numeral__div__numeral,axiom,
% 8.90/5.10 ! [M: num,N: num] :
% 8.90/5.10 ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.90/5.10 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6315_dbl__dec__simps_I4_J,axiom,
% 8.90/5.10 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.10 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6316_dbl__dec__simps_I4_J,axiom,
% 8.90/5.10 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
% 8.90/5.10 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6317_dbl__dec__simps_I4_J,axiom,
% 8.90/5.10 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.90/5.10 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6318_dbl__dec__simps_I4_J,axiom,
% 8.90/5.10 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.90/5.10 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6319_dbl__dec__simps_I4_J,axiom,
% 8.90/5.10 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.90/5.10 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6320_of__int__code__if,axiom,
% 8.90/5.10 ( ring_1_of_int_int
% 8.90/5.10 = ( ^ [K2: int] :
% 8.90/5.10 ( if_int @ ( K2 = zero_zero_int ) @ zero_zero_int
% 8.90/5.10 @ ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K2 ) ) )
% 8.90/5.10 @ ( if_int
% 8.90/5.10 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.10 = zero_zero_int )
% 8.90/5.10 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 8.90/5.10 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6321_of__int__code__if,axiom,
% 8.90/5.10 ( ring_1_of_int_real
% 8.90/5.10 = ( ^ [K2: int] :
% 8.90/5.10 ( if_real @ ( K2 = zero_zero_int ) @ zero_zero_real
% 8.90/5.10 @ ( if_real @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K2 ) ) )
% 8.90/5.10 @ ( if_real
% 8.90/5.10 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.10 = zero_zero_int )
% 8.90/5.10 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 8.90/5.10 @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6322_of__int__code__if,axiom,
% 8.90/5.10 ( ring_17405671764205052669omplex
% 8.90/5.10 = ( ^ [K2: int] :
% 8.90/5.10 ( if_complex @ ( K2 = zero_zero_int ) @ zero_zero_complex
% 8.90/5.10 @ ( if_complex @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K2 ) ) )
% 8.90/5.10 @ ( if_complex
% 8.90/5.10 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.10 = zero_zero_int )
% 8.90/5.10 @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 8.90/5.10 @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6323_of__int__code__if,axiom,
% 8.90/5.10 ( ring_18347121197199848620nteger
% 8.90/5.10 = ( ^ [K2: int] :
% 8.90/5.10 ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 8.90/5.10 @ ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K2 ) ) )
% 8.90/5.10 @ ( if_Code_integer
% 8.90/5.10 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.10 = zero_zero_int )
% 8.90/5.10 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 8.90/5.10 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6324_of__int__code__if,axiom,
% 8.90/5.10 ( ring_1_of_int_rat
% 8.90/5.10 = ( ^ [K2: int] :
% 8.90/5.10 ( if_rat @ ( K2 = zero_zero_int ) @ zero_zero_rat
% 8.90/5.10 @ ( if_rat @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K2 ) ) )
% 8.90/5.10 @ ( if_rat
% 8.90/5.10 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.10 = zero_zero_int )
% 8.90/5.10 @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 8.90/5.10 @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6325_int__ge__less__than__def,axiom,
% 8.90/5.10 ( int_ge_less_than
% 8.90/5.10 = ( ^ [D2: int] :
% 8.90/5.10 ( collec213857154873943460nt_int
% 8.90/5.10 @ ( produc4947309494688390418_int_o
% 8.90/5.10 @ ^ [Z6: int,Z3: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ D2 @ Z6 )
% 8.90/5.10 & ( ord_less_int @ Z6 @ Z3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6326_dbl__dec__simps_I3_J,axiom,
% 8.90/5.10 ( ( neg_nu6511756317524482435omplex @ one_one_complex )
% 8.90/5.10 = one_one_complex ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6327_dbl__dec__simps_I3_J,axiom,
% 8.90/5.10 ( ( neg_nu6075765906172075777c_real @ one_one_real )
% 8.90/5.10 = one_one_real ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6328_dbl__dec__simps_I3_J,axiom,
% 8.90/5.10 ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
% 8.90/5.10 = one_one_rat ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6329_dbl__dec__simps_I3_J,axiom,
% 8.90/5.10 ( ( neg_nu3811975205180677377ec_int @ one_one_int )
% 8.90/5.10 = one_one_int ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6330_of__int__numeral,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
% 8.90/5.10 = ( numera6690914467698888265omplex @ K ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6331_of__int__numeral,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 8.90/5.10 = ( numeral_numeral_real @ K ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6332_of__int__numeral,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
% 8.90/5.10 = ( numeral_numeral_rat @ K ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6333_of__int__numeral,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 8.90/5.10 = ( numeral_numeral_int @ K ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6334_of__int__eq__numeral__iff,axiom,
% 8.90/5.10 ! [Z: int,N: num] :
% 8.90/5.10 ( ( ( ring_17405671764205052669omplex @ Z )
% 8.90/5.10 = ( numera6690914467698888265omplex @ N ) )
% 8.90/5.10 = ( Z
% 8.90/5.10 = ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6335_of__int__eq__numeral__iff,axiom,
% 8.90/5.10 ! [Z: int,N: num] :
% 8.90/5.10 ( ( ( ring_1_of_int_real @ Z )
% 8.90/5.10 = ( numeral_numeral_real @ N ) )
% 8.90/5.10 = ( Z
% 8.90/5.10 = ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6336_of__int__eq__numeral__iff,axiom,
% 8.90/5.10 ! [Z: int,N: num] :
% 8.90/5.10 ( ( ( ring_1_of_int_rat @ Z )
% 8.90/5.10 = ( numeral_numeral_rat @ N ) )
% 8.90/5.10 = ( Z
% 8.90/5.10 = ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6337_of__int__eq__numeral__iff,axiom,
% 8.90/5.10 ! [Z: int,N: num] :
% 8.90/5.10 ( ( ( ring_1_of_int_int @ Z )
% 8.90/5.10 = ( numeral_numeral_int @ N ) )
% 8.90/5.10 = ( Z
% 8.90/5.10 = ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6338_of__int__le__iff,axiom,
% 8.90/5.10 ! [W: int,Z: int] :
% 8.90/5.10 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 8.90/5.10 = ( ord_less_eq_int @ W @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6339_of__int__le__iff,axiom,
% 8.90/5.10 ! [W: int,Z: int] :
% 8.90/5.10 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 8.90/5.10 = ( ord_less_eq_int @ W @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6340_of__int__le__iff,axiom,
% 8.90/5.10 ! [W: int,Z: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 8.90/5.10 = ( ord_less_eq_int @ W @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6341_of__int__less__iff,axiom,
% 8.90/5.10 ! [W: int,Z: int] :
% 8.90/5.10 ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 8.90/5.10 = ( ord_less_int @ W @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6342_of__int__less__iff,axiom,
% 8.90/5.10 ! [W: int,Z: int] :
% 8.90/5.10 ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 8.90/5.10 = ( ord_less_int @ W @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6343_of__int__less__iff,axiom,
% 8.90/5.10 ! [W: int,Z: int] :
% 8.90/5.10 ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 8.90/5.10 = ( ord_less_int @ W @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6344_of__int__eq__1__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ( ring_17405671764205052669omplex @ Z )
% 8.90/5.10 = one_one_complex )
% 8.90/5.10 = ( Z = one_one_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6345_of__int__eq__1__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ( ring_1_of_int_int @ Z )
% 8.90/5.10 = one_one_int )
% 8.90/5.10 = ( Z = one_one_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6346_of__int__eq__1__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ( ring_1_of_int_real @ Z )
% 8.90/5.10 = one_one_real )
% 8.90/5.10 = ( Z = one_one_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6347_of__int__eq__1__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ( ring_1_of_int_rat @ Z )
% 8.90/5.10 = one_one_rat )
% 8.90/5.10 = ( Z = one_one_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6348_of__int__1,axiom,
% 8.90/5.10 ( ( ring_17405671764205052669omplex @ one_one_int )
% 8.90/5.10 = one_one_complex ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6349_of__int__1,axiom,
% 8.90/5.10 ( ( ring_1_of_int_int @ one_one_int )
% 8.90/5.10 = one_one_int ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6350_of__int__1,axiom,
% 8.90/5.10 ( ( ring_1_of_int_real @ one_one_int )
% 8.90/5.10 = one_one_real ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6351_of__int__1,axiom,
% 8.90/5.10 ( ( ring_1_of_int_rat @ one_one_int )
% 8.90/5.10 = one_one_rat ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6352_of__int__mult,axiom,
% 8.90/5.10 ! [W: int,Z: int] :
% 8.90/5.10 ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
% 8.90/5.10 = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6353_of__int__mult,axiom,
% 8.90/5.10 ! [W: int,Z: int] :
% 8.90/5.10 ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
% 8.90/5.10 = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6354_of__int__mult,axiom,
% 8.90/5.10 ! [W: int,Z: int] :
% 8.90/5.10 ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
% 8.90/5.10 = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6355_of__int__add,axiom,
% 8.90/5.10 ! [W: int,Z: int] :
% 8.90/5.10 ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
% 8.90/5.10 = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6356_of__int__add,axiom,
% 8.90/5.10 ! [W: int,Z: int] :
% 8.90/5.10 ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
% 8.90/5.10 = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6357_of__int__add,axiom,
% 8.90/5.10 ! [W: int,Z: int] :
% 8.90/5.10 ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
% 8.90/5.10 = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6358_of__int__diff,axiom,
% 8.90/5.10 ! [W: int,Z: int] :
% 8.90/5.10 ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
% 8.90/5.10 = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6359_of__int__diff,axiom,
% 8.90/5.10 ! [W: int,Z: int] :
% 8.90/5.10 ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
% 8.90/5.10 = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6360_of__int__diff,axiom,
% 8.90/5.10 ! [W: int,Z: int] :
% 8.90/5.10 ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
% 8.90/5.10 = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6361_of__int__power,axiom,
% 8.90/5.10 ! [Z: int,N: nat] :
% 8.90/5.10 ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N ) )
% 8.90/5.10 = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6362_of__int__power,axiom,
% 8.90/5.10 ! [Z: int,N: nat] :
% 8.90/5.10 ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
% 8.90/5.10 = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6363_of__int__power,axiom,
% 8.90/5.10 ! [Z: int,N: nat] :
% 8.90/5.10 ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
% 8.90/5.10 = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6364_of__int__power,axiom,
% 8.90/5.10 ! [Z: int,N: nat] :
% 8.90/5.10 ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N ) )
% 8.90/5.10 = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6365_of__int__eq__of__int__power__cancel__iff,axiom,
% 8.90/5.10 ! [B: int,W: nat,X2: int] :
% 8.90/5.10 ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
% 8.90/5.10 = ( ring_1_of_int_rat @ X2 ) )
% 8.90/5.10 = ( ( power_power_int @ B @ W )
% 8.90/5.10 = X2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6366_of__int__eq__of__int__power__cancel__iff,axiom,
% 8.90/5.10 ! [B: int,W: nat,X2: int] :
% 8.90/5.10 ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
% 8.90/5.10 = ( ring_1_of_int_real @ X2 ) )
% 8.90/5.10 = ( ( power_power_int @ B @ W )
% 8.90/5.10 = X2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6367_of__int__eq__of__int__power__cancel__iff,axiom,
% 8.90/5.10 ! [B: int,W: nat,X2: int] :
% 8.90/5.10 ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
% 8.90/5.10 = ( ring_1_of_int_int @ X2 ) )
% 8.90/5.10 = ( ( power_power_int @ B @ W )
% 8.90/5.10 = X2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6368_of__int__eq__of__int__power__cancel__iff,axiom,
% 8.90/5.10 ! [B: int,W: nat,X2: int] :
% 8.90/5.10 ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
% 8.90/5.10 = ( ring_17405671764205052669omplex @ X2 ) )
% 8.90/5.10 = ( ( power_power_int @ B @ W )
% 8.90/5.10 = X2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6369_of__int__power__eq__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: int,B: int,W: nat] :
% 8.90/5.10 ( ( ( ring_1_of_int_rat @ X2 )
% 8.90/5.10 = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 8.90/5.10 = ( X2
% 8.90/5.10 = ( power_power_int @ B @ W ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6370_of__int__power__eq__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: int,B: int,W: nat] :
% 8.90/5.10 ( ( ( ring_1_of_int_real @ X2 )
% 8.90/5.10 = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 8.90/5.10 = ( X2
% 8.90/5.10 = ( power_power_int @ B @ W ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6371_of__int__power__eq__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: int,B: int,W: nat] :
% 8.90/5.10 ( ( ( ring_1_of_int_int @ X2 )
% 8.90/5.10 = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 8.90/5.10 = ( X2
% 8.90/5.10 = ( power_power_int @ B @ W ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6372_of__int__power__eq__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: int,B: int,W: nat] :
% 8.90/5.10 ( ( ( ring_17405671764205052669omplex @ X2 )
% 8.90/5.10 = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
% 8.90/5.10 = ( X2
% 8.90/5.10 = ( power_power_int @ B @ W ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6373_dbl__dec__simps_I5_J,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
% 8.90/5.10 = ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6374_dbl__dec__simps_I5_J,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) )
% 8.90/5.10 = ( numeral_numeral_real @ ( bitM @ K ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6375_dbl__dec__simps_I5_J,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) )
% 8.90/5.10 = ( numeral_numeral_rat @ ( bitM @ K ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6376_dbl__dec__simps_I5_J,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
% 8.90/5.10 = ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6377_dbl__dec__simps_I2_J,axiom,
% 8.90/5.10 ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 8.90/5.10 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6378_dbl__dec__simps_I2_J,axiom,
% 8.90/5.10 ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 8.90/5.10 = ( uminus_uminus_real @ one_one_real ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6379_dbl__dec__simps_I2_J,axiom,
% 8.90/5.10 ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 8.90/5.10 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6380_dbl__dec__simps_I2_J,axiom,
% 8.90/5.10 ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 8.90/5.10 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6381_dbl__dec__simps_I2_J,axiom,
% 8.90/5.10 ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 8.90/5.10 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6382_dbl__dec__simps_I1_J,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.90/5.10 = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6383_dbl__dec__simps_I1_J,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 8.90/5.10 = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6384_dbl__dec__simps_I1_J,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 8.90/5.10 = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6385_dbl__dec__simps_I1_J,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 8.90/5.10 = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6386_dbl__dec__simps_I1_J,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 8.90/5.10 = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6387_dbl__inc__simps_I1_J,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.90/5.10 = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6388_dbl__inc__simps_I1_J,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 8.90/5.10 = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6389_dbl__inc__simps_I1_J,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 8.90/5.10 = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6390_dbl__inc__simps_I1_J,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 8.90/5.10 = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6391_dbl__inc__simps_I1_J,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 8.90/5.10 = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6392_of__int__le__0__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 8.90/5.10 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6393_of__int__le__0__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 8.90/5.10 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6394_of__int__le__0__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 8.90/5.10 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6395_of__int__0__le__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 8.90/5.10 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6396_of__int__0__le__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 8.90/5.10 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6397_of__int__0__le__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 8.90/5.10 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6398_of__int__less__0__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 8.90/5.10 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6399_of__int__less__0__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 8.90/5.10 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6400_of__int__less__0__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 8.90/5.10 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6401_of__int__0__less__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 8.90/5.10 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6402_of__int__0__less__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 8.90/5.10 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6403_of__int__0__less__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 8.90/5.10 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6404_of__int__le__numeral__iff,axiom,
% 8.90/5.10 ! [Z: int,N: num] :
% 8.90/5.10 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 8.90/5.10 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6405_of__int__le__numeral__iff,axiom,
% 8.90/5.10 ! [Z: int,N: num] :
% 8.90/5.10 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 8.90/5.10 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6406_of__int__le__numeral__iff,axiom,
% 8.90/5.10 ! [Z: int,N: num] :
% 8.90/5.10 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 8.90/5.10 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6407_of__int__numeral__le__iff,axiom,
% 8.90/5.10 ! [N: num,Z: int] :
% 8.90/5.10 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 8.90/5.10 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6408_of__int__numeral__le__iff,axiom,
% 8.90/5.10 ! [N: num,Z: int] :
% 8.90/5.10 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 8.90/5.10 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6409_of__int__numeral__le__iff,axiom,
% 8.90/5.10 ! [N: num,Z: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 8.90/5.10 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6410_of__int__less__numeral__iff,axiom,
% 8.90/5.10 ! [Z: int,N: num] :
% 8.90/5.10 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 8.90/5.10 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6411_of__int__less__numeral__iff,axiom,
% 8.90/5.10 ! [Z: int,N: num] :
% 8.90/5.10 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 8.90/5.10 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6412_of__int__less__numeral__iff,axiom,
% 8.90/5.10 ! [Z: int,N: num] :
% 8.90/5.10 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 8.90/5.10 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6413_of__int__numeral__less__iff,axiom,
% 8.90/5.10 ! [N: num,Z: int] :
% 8.90/5.10 ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 8.90/5.10 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6414_of__int__numeral__less__iff,axiom,
% 8.90/5.10 ! [N: num,Z: int] :
% 8.90/5.10 ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 8.90/5.10 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6415_of__int__numeral__less__iff,axiom,
% 8.90/5.10 ! [N: num,Z: int] :
% 8.90/5.10 ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 8.90/5.10 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6416_of__int__le__1__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 8.90/5.10 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6417_of__int__le__1__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 8.90/5.10 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6418_of__int__le__1__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 8.90/5.10 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6419_of__int__1__le__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 8.90/5.10 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6420_of__int__1__le__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 8.90/5.10 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6421_of__int__1__le__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 8.90/5.10 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6422_of__int__1__less__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 8.90/5.10 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6423_of__int__1__less__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 8.90/5.10 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6424_of__int__1__less__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 8.90/5.10 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6425_of__int__less__1__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 8.90/5.10 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6426_of__int__less__1__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 8.90/5.10 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6427_of__int__less__1__iff,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 8.90/5.10 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6428_of__int__eq__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [Y3: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ( ring_17405671764205052669omplex @ Y3 )
% 8.90/5.10 = ( power_power_complex @ ( numera6690914467698888265omplex @ X2 ) @ N ) )
% 8.90/5.10 = ( Y3
% 8.90/5.10 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6429_of__int__eq__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [Y3: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ( ring_1_of_int_real @ Y3 )
% 8.90/5.10 = ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
% 8.90/5.10 = ( Y3
% 8.90/5.10 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6430_of__int__eq__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [Y3: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ( ring_1_of_int_rat @ Y3 )
% 8.90/5.10 = ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) )
% 8.90/5.10 = ( Y3
% 8.90/5.10 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6431_of__int__eq__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [Y3: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ( ring_1_of_int_int @ Y3 )
% 8.90/5.10 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) )
% 8.90/5.10 = ( Y3
% 8.90/5.10 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6432_numeral__power__eq__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,Y3: int] :
% 8.90/5.10 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X2 ) @ N )
% 8.90/5.10 = ( ring_17405671764205052669omplex @ Y3 ) )
% 8.90/5.10 = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N )
% 8.90/5.10 = Y3 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6433_numeral__power__eq__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,Y3: int] :
% 8.90/5.10 ( ( ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N )
% 8.90/5.10 = ( ring_1_of_int_real @ Y3 ) )
% 8.90/5.10 = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N )
% 8.90/5.10 = Y3 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6434_numeral__power__eq__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,Y3: int] :
% 8.90/5.10 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N )
% 8.90/5.10 = ( ring_1_of_int_rat @ Y3 ) )
% 8.90/5.10 = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N )
% 8.90/5.10 = Y3 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6435_numeral__power__eq__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,Y3: int] :
% 8.90/5.10 ( ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N )
% 8.90/5.10 = ( ring_1_of_int_int @ Y3 ) )
% 8.90/5.10 = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N )
% 8.90/5.10 = Y3 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6436_of__int__power__le__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: int,B: int,W: nat] :
% 8.90/5.10 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X2 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 8.90/5.10 = ( ord_less_eq_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6437_of__int__power__le__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: int,B: int,W: nat] :
% 8.90/5.10 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X2 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 8.90/5.10 = ( ord_less_eq_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6438_of__int__power__le__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: int,B: int,W: nat] :
% 8.90/5.10 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X2 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 8.90/5.10 = ( ord_less_eq_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6439_of__int__le__of__int__power__cancel__iff,axiom,
% 8.90/5.10 ! [B: int,W: nat,X2: int] :
% 8.90/5.10 ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X2 ) )
% 8.90/5.10 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6440_of__int__le__of__int__power__cancel__iff,axiom,
% 8.90/5.10 ! [B: int,W: nat,X2: int] :
% 8.90/5.10 ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X2 ) )
% 8.90/5.10 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6441_of__int__le__of__int__power__cancel__iff,axiom,
% 8.90/5.10 ! [B: int,W: nat,X2: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X2 ) )
% 8.90/5.10 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6442_of__int__power__less__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: int,B: int,W: nat] :
% 8.90/5.10 ( ( ord_less_real @ ( ring_1_of_int_real @ X2 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 8.90/5.10 = ( ord_less_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6443_of__int__power__less__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: int,B: int,W: nat] :
% 8.90/5.10 ( ( ord_less_rat @ ( ring_1_of_int_rat @ X2 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 8.90/5.10 = ( ord_less_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6444_of__int__power__less__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: int,B: int,W: nat] :
% 8.90/5.10 ( ( ord_less_int @ ( ring_1_of_int_int @ X2 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 8.90/5.10 = ( ord_less_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6445_of__int__less__of__int__power__cancel__iff,axiom,
% 8.90/5.10 ! [B: int,W: nat,X2: int] :
% 8.90/5.10 ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X2 ) )
% 8.90/5.10 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6446_of__int__less__of__int__power__cancel__iff,axiom,
% 8.90/5.10 ! [B: int,W: nat,X2: int] :
% 8.90/5.10 ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X2 ) )
% 8.90/5.10 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6447_of__int__less__of__int__power__cancel__iff,axiom,
% 8.90/5.10 ! [B: int,W: nat,X2: int] :
% 8.90/5.10 ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X2 ) )
% 8.90/5.10 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6448_numeral__power__le__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,A: int] :
% 8.90/5.10 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 8.90/5.10 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6449_numeral__power__le__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,A: int] :
% 8.90/5.10 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 8.90/5.10 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6450_numeral__power__le__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,A: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 8.90/5.10 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6451_of__int__le__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [A: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
% 8.90/5.10 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6452_of__int__le__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [A: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) )
% 8.90/5.10 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6453_of__int__le__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [A: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) )
% 8.90/5.10 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6454_numeral__power__less__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,A: int] :
% 8.90/5.10 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 8.90/5.10 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6455_numeral__power__less__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,A: int] :
% 8.90/5.10 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 8.90/5.10 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6456_numeral__power__less__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,A: int] :
% 8.90/5.10 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 8.90/5.10 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6457_of__int__less__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [A: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
% 8.90/5.10 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6458_of__int__less__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [A: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) )
% 8.90/5.10 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6459_of__int__less__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [A: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) )
% 8.90/5.10 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6460_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [Y3: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ( ring_1_of_int_int @ Y3 )
% 8.90/5.10 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) )
% 8.90/5.10 = ( Y3
% 8.90/5.10 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6461_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [Y3: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ( ring_1_of_int_real @ Y3 )
% 8.90/5.10 = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) )
% 8.90/5.10 = ( Y3
% 8.90/5.10 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6462_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [Y3: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ( ring_17405671764205052669omplex @ Y3 )
% 8.90/5.10 = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X2 ) ) @ N ) )
% 8.90/5.10 = ( Y3
% 8.90/5.10 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6463_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [Y3: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ( ring_18347121197199848620nteger @ Y3 )
% 8.90/5.10 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) )
% 8.90/5.10 = ( Y3
% 8.90/5.10 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6464_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [Y3: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ( ring_1_of_int_rat @ Y3 )
% 8.90/5.10 = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) )
% 8.90/5.10 = ( Y3
% 8.90/5.10 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6465_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,Y3: int] :
% 8.90/5.10 ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N )
% 8.90/5.10 = ( ring_1_of_int_int @ Y3 ) )
% 8.90/5.10 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N )
% 8.90/5.10 = Y3 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6466_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,Y3: int] :
% 8.90/5.10 ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N )
% 8.90/5.10 = ( ring_1_of_int_real @ Y3 ) )
% 8.90/5.10 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N )
% 8.90/5.10 = Y3 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6467_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,Y3: int] :
% 8.90/5.10 ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X2 ) ) @ N )
% 8.90/5.10 = ( ring_17405671764205052669omplex @ Y3 ) )
% 8.90/5.10 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N )
% 8.90/5.10 = Y3 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6468_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,Y3: int] :
% 8.90/5.10 ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N )
% 8.90/5.10 = ( ring_18347121197199848620nteger @ Y3 ) )
% 8.90/5.10 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N )
% 8.90/5.10 = Y3 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6469_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,Y3: int] :
% 8.90/5.10 ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N )
% 8.90/5.10 = ( ring_1_of_int_rat @ Y3 ) )
% 8.90/5.10 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N )
% 8.90/5.10 = Y3 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6470_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [A: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) )
% 8.90/5.10 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6471_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [A: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) )
% 8.90/5.10 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6472_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [A: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) )
% 8.90/5.10 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6473_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [A: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) )
% 8.90/5.10 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6474_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,A: int] :
% 8.90/5.10 ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 8.90/5.10 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6475_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,A: int] :
% 8.90/5.10 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 8.90/5.10 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6476_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,A: int] :
% 8.90/5.10 ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 8.90/5.10 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6477_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,A: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 8.90/5.10 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6478_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [A: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) )
% 8.90/5.10 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6479_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [A: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) )
% 8.90/5.10 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6480_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [A: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) )
% 8.90/5.10 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6481_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 8.90/5.10 ! [A: int,X2: num,N: nat] :
% 8.90/5.10 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) )
% 8.90/5.10 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6482_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,A: int] :
% 8.90/5.10 ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 8.90/5.10 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6483_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,A: int] :
% 8.90/5.10 ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 8.90/5.10 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6484_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,A: int] :
% 8.90/5.10 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 8.90/5.10 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6485_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 8.90/5.10 ! [X2: num,N: nat,A: int] :
% 8.90/5.10 ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 8.90/5.10 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6486_Collect__case__prod__mono,axiom,
% 8.90/5.10 ! [A2: int > int > $o,B3: int > int > $o] :
% 8.90/5.10 ( ( ord_le6741204236512500942_int_o @ A2 @ B3 )
% 8.90/5.10 => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A2 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B3 ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6487_mult__of__int__commute,axiom,
% 8.90/5.10 ! [X2: int,Y3: real] :
% 8.90/5.10 ( ( times_times_real @ ( ring_1_of_int_real @ X2 ) @ Y3 )
% 8.90/5.10 = ( times_times_real @ Y3 @ ( ring_1_of_int_real @ X2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6488_mult__of__int__commute,axiom,
% 8.90/5.10 ! [X2: int,Y3: rat] :
% 8.90/5.10 ( ( times_times_rat @ ( ring_1_of_int_rat @ X2 ) @ Y3 )
% 8.90/5.10 = ( times_times_rat @ Y3 @ ( ring_1_of_int_rat @ X2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6489_mult__of__int__commute,axiom,
% 8.90/5.10 ! [X2: int,Y3: int] :
% 8.90/5.10 ( ( times_times_int @ ( ring_1_of_int_int @ X2 ) @ Y3 )
% 8.90/5.10 = ( times_times_int @ Y3 @ ( ring_1_of_int_int @ X2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6490_real__of__int__div4,axiom,
% 8.90/5.10 ! [N: int,X2: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X2 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6491_real__of__int__div,axiom,
% 8.90/5.10 ! [D: int,N: int] :
% 8.90/5.10 ( ( dvd_dvd_int @ D @ N )
% 8.90/5.10 => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
% 8.90/5.10 = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6492_of__int__nonneg,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6493_of__int__nonneg,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6494_of__int__nonneg,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 8.90/5.10 => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6495_of__int__pos,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_int @ zero_zero_int @ Z )
% 8.90/5.10 => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6496_of__int__pos,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_int @ zero_zero_int @ Z )
% 8.90/5.10 => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6497_of__int__pos,axiom,
% 8.90/5.10 ! [Z: int] :
% 8.90/5.10 ( ( ord_less_int @ zero_zero_int @ Z )
% 8.90/5.10 => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6498_of__int__neg__numeral,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.90/5.10 = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6499_of__int__neg__numeral,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.90/5.10 = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6500_of__int__neg__numeral,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.90/5.10 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6501_of__int__neg__numeral,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.90/5.10 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6502_of__int__neg__numeral,axiom,
% 8.90/5.10 ! [K: num] :
% 8.90/5.10 ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.90/5.10 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6503_int__le__real__less,axiom,
% 8.90/5.10 ( ord_less_eq_int
% 8.90/5.10 = ( ^ [N3: int,M3: int] : ( ord_less_real @ ( ring_1_of_int_real @ N3 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M3 ) @ one_one_real ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6504_int__less__real__le,axiom,
% 8.90/5.10 ( ord_less_int
% 8.90/5.10 = ( ^ [N3: int,M3: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) @ ( ring_1_of_int_real @ M3 ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6505_real__of__int__div__aux,axiom,
% 8.90/5.10 ! [X2: int,D: int] :
% 8.90/5.10 ( ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ D ) )
% 8.90/5.10 = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X2 @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X2 @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6506_real__of__int__div2,axiom,
% 8.90/5.10 ! [N: int,X2: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X2 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X2 ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6507_real__of__int__div3,axiom,
% 8.90/5.10 ! [N: int,X2: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X2 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X2 ) ) ) @ one_one_real ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6508_even__of__int__iff,axiom,
% 8.90/5.10 ! [K: int] :
% 8.90/5.10 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
% 8.90/5.10 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6509_even__of__int__iff,axiom,
% 8.90/5.10 ! [K: int] :
% 8.90/5.10 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
% 8.90/5.10 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6510_dbl__dec__def,axiom,
% 8.90/5.10 ( neg_nu6511756317524482435omplex
% 8.90/5.10 = ( ^ [X: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6511_dbl__dec__def,axiom,
% 8.90/5.10 ( neg_nu6075765906172075777c_real
% 8.90/5.10 = ( ^ [X: real] : ( minus_minus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6512_dbl__dec__def,axiom,
% 8.90/5.10 ( neg_nu3179335615603231917ec_rat
% 8.90/5.10 = ( ^ [X: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6513_dbl__dec__def,axiom,
% 8.90/5.10 ( neg_nu3811975205180677377ec_int
% 8.90/5.10 = ( ^ [X: int] : ( minus_minus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6514_floor__exists,axiom,
% 8.90/5.10 ! [X2: real] :
% 8.90/5.10 ? [Z2: int] :
% 8.90/5.10 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X2 )
% 8.90/5.10 & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6515_floor__exists,axiom,
% 8.90/5.10 ! [X2: rat] :
% 8.90/5.10 ? [Z2: int] :
% 8.90/5.10 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X2 )
% 8.90/5.10 & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6516_floor__exists1,axiom,
% 8.90/5.10 ! [X2: real] :
% 8.90/5.10 ? [X5: int] :
% 8.90/5.10 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X5 ) @ X2 )
% 8.90/5.10 & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ X5 @ one_one_int ) ) )
% 8.90/5.10 & ! [Y5: int] :
% 8.90/5.10 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X2 )
% 8.90/5.10 & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 8.90/5.10 => ( Y5 = X5 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6517_floor__exists1,axiom,
% 8.90/5.10 ! [X2: rat] :
% 8.90/5.10 ? [X5: int] :
% 8.90/5.10 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X5 ) @ X2 )
% 8.90/5.10 & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ X5 @ one_one_int ) ) )
% 8.90/5.10 & ! [Y5: int] :
% 8.90/5.10 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X2 )
% 8.90/5.10 & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 8.90/5.10 => ( Y5 = X5 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6518_int__ge__less__than2__def,axiom,
% 8.90/5.10 ( int_ge_less_than2
% 8.90/5.10 = ( ^ [D2: int] :
% 8.90/5.10 ( collec213857154873943460nt_int
% 8.90/5.10 @ ( produc4947309494688390418_int_o
% 8.90/5.10 @ ^ [Z6: int,Z3: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ D2 @ Z3 )
% 8.90/5.10 & ( ord_less_int @ Z6 @ Z3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6519_ln__one__minus__pos__lower__bound,axiom,
% 8.90/5.10 ! [X2: real] :
% 8.90/5.10 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.10 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.10 => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6520_artanh__def,axiom,
% 8.90/5.10 ( artanh_real
% 8.90/5.10 = ( ^ [X: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X ) @ ( minus_minus_real @ one_one_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6521_Sum__Icc__int,axiom,
% 8.90/5.10 ! [M: int,N: int] :
% 8.90/5.10 ( ( ord_less_eq_int @ M @ N )
% 8.90/5.10 => ( ( groups4538972089207619220nt_int
% 8.90/5.10 @ ^ [X: int] : X
% 8.90/5.10 @ ( set_or1266510415728281911st_int @ M @ N ) )
% 8.90/5.10 = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6522_ln__one,axiom,
% 8.90/5.10 ( ( ln_ln_real @ one_one_real )
% 8.90/5.10 = zero_zero_real ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6523_ln__le__cancel__iff,axiom,
% 8.90/5.10 ! [X2: real,Y3: real] :
% 8.90/5.10 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.10 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.90/5.10 => ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) )
% 8.90/5.10 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6524_ln__eq__zero__iff,axiom,
% 8.90/5.10 ! [X2: real] :
% 8.90/5.10 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.10 => ( ( ( ln_ln_real @ X2 )
% 8.90/5.10 = zero_zero_real )
% 8.90/5.10 = ( X2 = one_one_real ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6525_ln__gt__zero__iff,axiom,
% 8.90/5.10 ! [X2: real] :
% 8.90/5.10 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.10 => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
% 8.90/5.10 = ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6526_ln__less__zero__iff,axiom,
% 8.90/5.10 ! [X2: real] :
% 8.90/5.10 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.10 => ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
% 8.90/5.10 = ( ord_less_real @ X2 @ one_one_real ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6527_ln__le__zero__iff,axiom,
% 8.90/5.10 ! [X2: real] :
% 8.90/5.10 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.10 => ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
% 8.90/5.10 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6528_ln__ge__zero__iff,axiom,
% 8.90/5.10 ! [X2: real] :
% 8.90/5.10 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.10 => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
% 8.90/5.10 = ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6529_sum__mono,axiom,
% 8.90/5.10 ! [K5: set_complex,F: complex > rat,G: complex > rat] :
% 8.90/5.10 ( ! [I3: complex] :
% 8.90/5.10 ( ( member_complex @ I3 @ K5 )
% 8.90/5.10 => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ K5 ) @ ( groups5058264527183730370ex_rat @ G @ K5 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6530_sum__mono,axiom,
% 8.90/5.10 ! [K5: set_real,F: real > rat,G: real > rat] :
% 8.90/5.10 ( ! [I3: real] :
% 8.90/5.10 ( ( member_real @ I3 @ K5 )
% 8.90/5.10 => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6531_sum__mono,axiom,
% 8.90/5.10 ! [K5: set_nat,F: nat > rat,G: nat > rat] :
% 8.90/5.10 ( ! [I3: nat] :
% 8.90/5.10 ( ( member_nat @ I3 @ K5 )
% 8.90/5.10 => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6532_sum__mono,axiom,
% 8.90/5.10 ! [K5: set_int,F: int > rat,G: int > rat] :
% 8.90/5.10 ( ! [I3: int] :
% 8.90/5.10 ( ( member_int @ I3 @ K5 )
% 8.90/5.10 => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6533_sum__mono,axiom,
% 8.90/5.10 ! [K5: set_complex,F: complex > nat,G: complex > nat] :
% 8.90/5.10 ( ! [I3: complex] :
% 8.90/5.10 ( ( member_complex @ I3 @ K5 )
% 8.90/5.10 => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K5 ) @ ( groups5693394587270226106ex_nat @ G @ K5 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6534_sum__mono,axiom,
% 8.90/5.10 ! [K5: set_real,F: real > nat,G: real > nat] :
% 8.90/5.10 ( ! [I3: real] :
% 8.90/5.10 ( ( member_real @ I3 @ K5 )
% 8.90/5.10 => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6535_sum__mono,axiom,
% 8.90/5.10 ! [K5: set_int,F: int > nat,G: int > nat] :
% 8.90/5.10 ( ! [I3: int] :
% 8.90/5.10 ( ( member_int @ I3 @ K5 )
% 8.90/5.10 => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6536_sum__mono,axiom,
% 8.90/5.10 ! [K5: set_complex,F: complex > int,G: complex > int] :
% 8.90/5.10 ( ! [I3: complex] :
% 8.90/5.10 ( ( member_complex @ I3 @ K5 )
% 8.90/5.10 => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K5 ) @ ( groups5690904116761175830ex_int @ G @ K5 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6537_sum__mono,axiom,
% 8.90/5.10 ! [K5: set_real,F: real > int,G: real > int] :
% 8.90/5.10 ( ! [I3: real] :
% 8.90/5.10 ( ( member_real @ I3 @ K5 )
% 8.90/5.10 => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6538_sum__mono,axiom,
% 8.90/5.10 ! [K5: set_nat,F: nat > int,G: nat > int] :
% 8.90/5.10 ( ! [I3: nat] :
% 8.90/5.10 ( ( member_nat @ I3 @ K5 )
% 8.90/5.10 => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6539_sum__product,axiom,
% 8.90/5.10 ! [F: int > int,A2: set_int,G: int > int,B3: set_int] :
% 8.90/5.10 ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ B3 ) )
% 8.90/5.10 = ( groups4538972089207619220nt_int
% 8.90/5.10 @ ^ [I4: int] :
% 8.90/5.10 ( groups4538972089207619220nt_int
% 8.90/5.10 @ ^ [J3: int] : ( times_times_int @ ( F @ I4 ) @ ( G @ J3 ) )
% 8.90/5.10 @ B3 )
% 8.90/5.10 @ A2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6540_sum__product,axiom,
% 8.90/5.10 ! [F: complex > complex,A2: set_complex,G: complex > complex,B3: set_complex] :
% 8.90/5.10 ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ B3 ) )
% 8.90/5.10 = ( groups7754918857620584856omplex
% 8.90/5.10 @ ^ [I4: complex] :
% 8.90/5.10 ( groups7754918857620584856omplex
% 8.90/5.10 @ ^ [J3: complex] : ( times_times_complex @ ( F @ I4 ) @ ( G @ J3 ) )
% 8.90/5.10 @ B3 )
% 8.90/5.10 @ A2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6541_sum__product,axiom,
% 8.90/5.10 ! [F: nat > nat,A2: set_nat,G: nat > nat,B3: set_nat] :
% 8.90/5.10 ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B3 ) )
% 8.90/5.10 = ( groups3542108847815614940at_nat
% 8.90/5.10 @ ^ [I4: nat] :
% 8.90/5.10 ( groups3542108847815614940at_nat
% 8.90/5.10 @ ^ [J3: nat] : ( times_times_nat @ ( F @ I4 ) @ ( G @ J3 ) )
% 8.90/5.10 @ B3 )
% 8.90/5.10 @ A2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6542_sum__product,axiom,
% 8.90/5.10 ! [F: nat > real,A2: set_nat,G: nat > real,B3: set_nat] :
% 8.90/5.10 ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B3 ) )
% 8.90/5.10 = ( groups6591440286371151544t_real
% 8.90/5.10 @ ^ [I4: nat] :
% 8.90/5.10 ( groups6591440286371151544t_real
% 8.90/5.10 @ ^ [J3: nat] : ( times_times_real @ ( F @ I4 ) @ ( G @ J3 ) )
% 8.90/5.10 @ B3 )
% 8.90/5.10 @ A2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6543_sum__distrib__right,axiom,
% 8.90/5.10 ! [F: int > int,A2: set_int,R2: int] :
% 8.90/5.10 ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ R2 )
% 8.90/5.10 = ( groups4538972089207619220nt_int
% 8.90/5.10 @ ^ [N3: int] : ( times_times_int @ ( F @ N3 ) @ R2 )
% 8.90/5.10 @ A2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6544_sum__distrib__right,axiom,
% 8.90/5.10 ! [F: complex > complex,A2: set_complex,R2: complex] :
% 8.90/5.10 ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R2 )
% 8.90/5.10 = ( groups7754918857620584856omplex
% 8.90/5.10 @ ^ [N3: complex] : ( times_times_complex @ ( F @ N3 ) @ R2 )
% 8.90/5.10 @ A2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6545_sum__distrib__right,axiom,
% 8.90/5.10 ! [F: nat > nat,A2: set_nat,R2: nat] :
% 8.90/5.10 ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R2 )
% 8.90/5.10 = ( groups3542108847815614940at_nat
% 8.90/5.10 @ ^ [N3: nat] : ( times_times_nat @ ( F @ N3 ) @ R2 )
% 8.90/5.10 @ A2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6546_sum__distrib__right,axiom,
% 8.90/5.10 ! [F: nat > real,A2: set_nat,R2: real] :
% 8.90/5.10 ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R2 )
% 8.90/5.10 = ( groups6591440286371151544t_real
% 8.90/5.10 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ R2 )
% 8.90/5.10 @ A2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6547_sum__distrib__left,axiom,
% 8.90/5.10 ! [R2: int,F: int > int,A2: set_int] :
% 8.90/5.10 ( ( times_times_int @ R2 @ ( groups4538972089207619220nt_int @ F @ A2 ) )
% 8.90/5.10 = ( groups4538972089207619220nt_int
% 8.90/5.10 @ ^ [N3: int] : ( times_times_int @ R2 @ ( F @ N3 ) )
% 8.90/5.10 @ A2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6548_sum__distrib__left,axiom,
% 8.90/5.10 ! [R2: complex,F: complex > complex,A2: set_complex] :
% 8.90/5.10 ( ( times_times_complex @ R2 @ ( groups7754918857620584856omplex @ F @ A2 ) )
% 8.90/5.10 = ( groups7754918857620584856omplex
% 8.90/5.10 @ ^ [N3: complex] : ( times_times_complex @ R2 @ ( F @ N3 ) )
% 8.90/5.10 @ A2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6549_sum__distrib__left,axiom,
% 8.90/5.10 ! [R2: nat,F: nat > nat,A2: set_nat] :
% 8.90/5.10 ( ( times_times_nat @ R2 @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 8.90/5.10 = ( groups3542108847815614940at_nat
% 8.90/5.10 @ ^ [N3: nat] : ( times_times_nat @ R2 @ ( F @ N3 ) )
% 8.90/5.10 @ A2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6550_sum__distrib__left,axiom,
% 8.90/5.10 ! [R2: real,F: nat > real,A2: set_nat] :
% 8.90/5.10 ( ( times_times_real @ R2 @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 8.90/5.10 = ( groups6591440286371151544t_real
% 8.90/5.10 @ ^ [N3: nat] : ( times_times_real @ R2 @ ( F @ N3 ) )
% 8.90/5.10 @ A2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6551_sum_Odistrib,axiom,
% 8.90/5.10 ! [G: int > int,H2: int > int,A2: set_int] :
% 8.90/5.10 ( ( groups4538972089207619220nt_int
% 8.90/5.10 @ ^ [X: int] : ( plus_plus_int @ ( G @ X ) @ ( H2 @ X ) )
% 8.90/5.10 @ A2 )
% 8.90/5.10 = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H2 @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6552_sum_Odistrib,axiom,
% 8.90/5.10 ! [G: complex > complex,H2: complex > complex,A2: set_complex] :
% 8.90/5.10 ( ( groups7754918857620584856omplex
% 8.90/5.10 @ ^ [X: complex] : ( plus_plus_complex @ ( G @ X ) @ ( H2 @ X ) )
% 8.90/5.10 @ A2 )
% 8.90/5.10 = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6553_sum_Odistrib,axiom,
% 8.90/5.10 ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
% 8.90/5.10 ( ( groups3542108847815614940at_nat
% 8.90/5.10 @ ^ [X: nat] : ( plus_plus_nat @ ( G @ X ) @ ( H2 @ X ) )
% 8.90/5.10 @ A2 )
% 8.90/5.10 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6554_sum_Odistrib,axiom,
% 8.90/5.10 ! [G: nat > real,H2: nat > real,A2: set_nat] :
% 8.90/5.10 ( ( groups6591440286371151544t_real
% 8.90/5.10 @ ^ [X: nat] : ( plus_plus_real @ ( G @ X ) @ ( H2 @ X ) )
% 8.90/5.10 @ A2 )
% 8.90/5.10 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6555_sum__subtractf,axiom,
% 8.90/5.10 ! [F: int > int,G: int > int,A2: set_int] :
% 8.90/5.10 ( ( groups4538972089207619220nt_int
% 8.90/5.10 @ ^ [X: int] : ( minus_minus_int @ ( F @ X ) @ ( G @ X ) )
% 8.90/5.10 @ A2 )
% 8.90/5.10 = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6556_sum__subtractf,axiom,
% 8.90/5.10 ! [F: complex > complex,G: complex > complex,A2: set_complex] :
% 8.90/5.10 ( ( groups7754918857620584856omplex
% 8.90/5.10 @ ^ [X: complex] : ( minus_minus_complex @ ( F @ X ) @ ( G @ X ) )
% 8.90/5.10 @ A2 )
% 8.90/5.10 = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6557_sum__subtractf,axiom,
% 8.90/5.10 ! [F: nat > real,G: nat > real,A2: set_nat] :
% 8.90/5.10 ( ( groups6591440286371151544t_real
% 8.90/5.10 @ ^ [X: nat] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
% 8.90/5.10 @ A2 )
% 8.90/5.10 = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6558_sum__divide__distrib,axiom,
% 8.90/5.10 ! [F: complex > complex,A2: set_complex,R2: complex] :
% 8.90/5.10 ( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R2 )
% 8.90/5.10 = ( groups7754918857620584856omplex
% 8.90/5.10 @ ^ [N3: complex] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ R2 )
% 8.90/5.10 @ A2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6559_sum__divide__distrib,axiom,
% 8.90/5.10 ! [F: nat > real,A2: set_nat,R2: real] :
% 8.90/5.10 ( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R2 )
% 8.90/5.10 = ( groups6591440286371151544t_real
% 8.90/5.10 @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ R2 )
% 8.90/5.10 @ A2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6560_mod__sum__eq,axiom,
% 8.90/5.10 ! [F: int > int,A: int,A2: set_int] :
% 8.90/5.10 ( ( modulo_modulo_int
% 8.90/5.10 @ ( groups4538972089207619220nt_int
% 8.90/5.10 @ ^ [I4: int] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
% 8.90/5.10 @ A2 )
% 8.90/5.10 @ A )
% 8.90/5.10 = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ A ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6561_mod__sum__eq,axiom,
% 8.90/5.10 ! [F: nat > nat,A: nat,A2: set_nat] :
% 8.90/5.10 ( ( modulo_modulo_nat
% 8.90/5.10 @ ( groups3542108847815614940at_nat
% 8.90/5.10 @ ^ [I4: nat] : ( modulo_modulo_nat @ ( F @ I4 ) @ A )
% 8.90/5.10 @ A2 )
% 8.90/5.10 @ A )
% 8.90/5.10 = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ A ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6562_sum__nonneg,axiom,
% 8.90/5.10 ! [A2: set_complex,F: complex > real] :
% 8.90/5.10 ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6563_sum__nonneg,axiom,
% 8.90/5.10 ! [A2: set_real,F: real > real] :
% 8.90/5.10 ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6564_sum__nonneg,axiom,
% 8.90/5.10 ! [A2: set_int,F: int > real] :
% 8.90/5.10 ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6565_sum__nonneg,axiom,
% 8.90/5.10 ! [A2: set_complex,F: complex > rat] :
% 8.90/5.10 ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6566_sum__nonneg,axiom,
% 8.90/5.10 ! [A2: set_real,F: real > rat] :
% 8.90/5.10 ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6567_sum__nonneg,axiom,
% 8.90/5.10 ! [A2: set_nat,F: nat > rat] :
% 8.90/5.10 ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6568_sum__nonneg,axiom,
% 8.90/5.10 ! [A2: set_int,F: int > rat] :
% 8.90/5.10 ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6569_sum__nonneg,axiom,
% 8.90/5.10 ! [A2: set_complex,F: complex > nat] :
% 8.90/5.10 ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6570_sum__nonneg,axiom,
% 8.90/5.10 ! [A2: set_real,F: real > nat] :
% 8.90/5.10 ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6571_sum__nonneg,axiom,
% 8.90/5.10 ! [A2: set_int,F: int > nat] :
% 8.90/5.10 ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6572_sum__nonpos,axiom,
% 8.90/5.10 ! [A2: set_complex,F: complex > real] :
% 8.90/5.10 ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
% 8.90/5.10 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6573_sum__nonpos,axiom,
% 8.90/5.10 ! [A2: set_real,F: real > real] :
% 8.90/5.10 ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
% 8.90/5.10 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6574_sum__nonpos,axiom,
% 8.90/5.10 ! [A2: set_int,F: int > real] :
% 8.90/5.10 ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
% 8.90/5.10 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6575_sum__nonpos,axiom,
% 8.90/5.10 ! [A2: set_complex,F: complex > rat] :
% 8.90/5.10 ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 8.90/5.10 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6576_sum__nonpos,axiom,
% 8.90/5.10 ! [A2: set_real,F: real > rat] :
% 8.90/5.10 ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 8.90/5.10 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6577_sum__nonpos,axiom,
% 8.90/5.10 ! [A2: set_nat,F: nat > rat] :
% 8.90/5.10 ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 8.90/5.10 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6578_sum__nonpos,axiom,
% 8.90/5.10 ! [A2: set_int,F: int > rat] :
% 8.90/5.10 ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 8.90/5.10 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6579_sum__nonpos,axiom,
% 8.90/5.10 ! [A2: set_complex,F: complex > nat] :
% 8.90/5.10 ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
% 8.90/5.10 => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6580_sum__nonpos,axiom,
% 8.90/5.10 ! [A2: set_real,F: real > nat] :
% 8.90/5.10 ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
% 8.90/5.10 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6581_sum__nonpos,axiom,
% 8.90/5.10 ! [A2: set_int,F: int > nat] :
% 8.90/5.10 ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
% 8.90/5.10 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6582_sum__mono__inv,axiom,
% 8.90/5.10 ! [F: real > rat,I5: set_real,G: real > rat,I2: real] :
% 8.90/5.10 ( ( ( groups1300246762558778688al_rat @ F @ I5 )
% 8.90/5.10 = ( groups1300246762558778688al_rat @ G @ I5 ) )
% 8.90/5.10 => ( ! [I3: real] :
% 8.90/5.10 ( ( member_real @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ( member_real @ I2 @ I5 )
% 8.90/5.10 => ( ( finite_finite_real @ I5 )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = ( G @ I2 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6583_sum__mono__inv,axiom,
% 8.90/5.10 ! [F: nat > rat,I5: set_nat,G: nat > rat,I2: nat] :
% 8.90/5.10 ( ( ( groups2906978787729119204at_rat @ F @ I5 )
% 8.90/5.10 = ( groups2906978787729119204at_rat @ G @ I5 ) )
% 8.90/5.10 => ( ! [I3: nat] :
% 8.90/5.10 ( ( member_nat @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ( member_nat @ I2 @ I5 )
% 8.90/5.10 => ( ( finite_finite_nat @ I5 )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = ( G @ I2 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6584_sum__mono__inv,axiom,
% 8.90/5.10 ! [F: int > rat,I5: set_int,G: int > rat,I2: int] :
% 8.90/5.10 ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
% 8.90/5.10 = ( groups3906332499630173760nt_rat @ G @ I5 ) )
% 8.90/5.10 => ( ! [I3: int] :
% 8.90/5.10 ( ( member_int @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ( member_int @ I2 @ I5 )
% 8.90/5.10 => ( ( finite_finite_int @ I5 )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = ( G @ I2 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6585_sum__mono__inv,axiom,
% 8.90/5.10 ! [F: complex > rat,I5: set_complex,G: complex > rat,I2: complex] :
% 8.90/5.10 ( ( ( groups5058264527183730370ex_rat @ F @ I5 )
% 8.90/5.10 = ( groups5058264527183730370ex_rat @ G @ I5 ) )
% 8.90/5.10 => ( ! [I3: complex] :
% 8.90/5.10 ( ( member_complex @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ( member_complex @ I2 @ I5 )
% 8.90/5.10 => ( ( finite3207457112153483333omplex @ I5 )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = ( G @ I2 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6586_sum__mono__inv,axiom,
% 8.90/5.10 ! [F: real > nat,I5: set_real,G: real > nat,I2: real] :
% 8.90/5.10 ( ( ( groups1935376822645274424al_nat @ F @ I5 )
% 8.90/5.10 = ( groups1935376822645274424al_nat @ G @ I5 ) )
% 8.90/5.10 => ( ! [I3: real] :
% 8.90/5.10 ( ( member_real @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ( member_real @ I2 @ I5 )
% 8.90/5.10 => ( ( finite_finite_real @ I5 )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = ( G @ I2 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6587_sum__mono__inv,axiom,
% 8.90/5.10 ! [F: int > nat,I5: set_int,G: int > nat,I2: int] :
% 8.90/5.10 ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
% 8.90/5.10 = ( groups4541462559716669496nt_nat @ G @ I5 ) )
% 8.90/5.10 => ( ! [I3: int] :
% 8.90/5.10 ( ( member_int @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ( member_int @ I2 @ I5 )
% 8.90/5.10 => ( ( finite_finite_int @ I5 )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = ( G @ I2 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6588_sum__mono__inv,axiom,
% 8.90/5.10 ! [F: complex > nat,I5: set_complex,G: complex > nat,I2: complex] :
% 8.90/5.10 ( ( ( groups5693394587270226106ex_nat @ F @ I5 )
% 8.90/5.10 = ( groups5693394587270226106ex_nat @ G @ I5 ) )
% 8.90/5.10 => ( ! [I3: complex] :
% 8.90/5.10 ( ( member_complex @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ( member_complex @ I2 @ I5 )
% 8.90/5.10 => ( ( finite3207457112153483333omplex @ I5 )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = ( G @ I2 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6589_sum__mono__inv,axiom,
% 8.90/5.10 ! [F: real > int,I5: set_real,G: real > int,I2: real] :
% 8.90/5.10 ( ( ( groups1932886352136224148al_int @ F @ I5 )
% 8.90/5.10 = ( groups1932886352136224148al_int @ G @ I5 ) )
% 8.90/5.10 => ( ! [I3: real] :
% 8.90/5.10 ( ( member_real @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ( member_real @ I2 @ I5 )
% 8.90/5.10 => ( ( finite_finite_real @ I5 )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = ( G @ I2 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6590_sum__mono__inv,axiom,
% 8.90/5.10 ! [F: nat > int,I5: set_nat,G: nat > int,I2: nat] :
% 8.90/5.10 ( ( ( groups3539618377306564664at_int @ F @ I5 )
% 8.90/5.10 = ( groups3539618377306564664at_int @ G @ I5 ) )
% 8.90/5.10 => ( ! [I3: nat] :
% 8.90/5.10 ( ( member_nat @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ( member_nat @ I2 @ I5 )
% 8.90/5.10 => ( ( finite_finite_nat @ I5 )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = ( G @ I2 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6591_sum__mono__inv,axiom,
% 8.90/5.10 ! [F: complex > int,I5: set_complex,G: complex > int,I2: complex] :
% 8.90/5.10 ( ( ( groups5690904116761175830ex_int @ F @ I5 )
% 8.90/5.10 = ( groups5690904116761175830ex_int @ G @ I5 ) )
% 8.90/5.10 => ( ! [I3: complex] :
% 8.90/5.10 ( ( member_complex @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 8.90/5.10 => ( ( member_complex @ I2 @ I5 )
% 8.90/5.10 => ( ( finite3207457112153483333omplex @ I5 )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = ( G @ I2 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6592_sum__le__included,axiom,
% 8.90/5.10 ! [S: set_int,T: set_int,G: int > real,I2: int > int,F: int > real] :
% 8.90/5.10 ( ( finite_finite_int @ S )
% 8.90/5.10 => ( ( finite_finite_int @ T )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ T )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ S )
% 8.90/5.10 => ? [Xa: int] :
% 8.90/5.10 ( ( member_int @ Xa @ T )
% 8.90/5.10 & ( ( I2 @ Xa )
% 8.90/5.10 = X5 )
% 8.90/5.10 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 8.90/5.10 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6593_sum__le__included,axiom,
% 8.90/5.10 ! [S: set_int,T: set_complex,G: complex > real,I2: complex > int,F: int > real] :
% 8.90/5.10 ( ( finite_finite_int @ S )
% 8.90/5.10 => ( ( finite3207457112153483333omplex @ T )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ T )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ S )
% 8.90/5.10 => ? [Xa: complex] :
% 8.90/5.10 ( ( member_complex @ Xa @ T )
% 8.90/5.10 & ( ( I2 @ Xa )
% 8.90/5.10 = X5 )
% 8.90/5.10 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 8.90/5.10 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6594_sum__le__included,axiom,
% 8.90/5.10 ! [S: set_complex,T: set_int,G: int > real,I2: int > complex,F: complex > real] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ S )
% 8.90/5.10 => ( ( finite_finite_int @ T )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ T )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ S )
% 8.90/5.10 => ? [Xa: int] :
% 8.90/5.10 ( ( member_int @ Xa @ T )
% 8.90/5.10 & ( ( I2 @ Xa )
% 8.90/5.10 = X5 )
% 8.90/5.10 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 8.90/5.10 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6595_sum__le__included,axiom,
% 8.90/5.10 ! [S: set_complex,T: set_complex,G: complex > real,I2: complex > complex,F: complex > real] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ S )
% 8.90/5.10 => ( ( finite3207457112153483333omplex @ T )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ T )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ S )
% 8.90/5.10 => ? [Xa: complex] :
% 8.90/5.10 ( ( member_complex @ Xa @ T )
% 8.90/5.10 & ( ( I2 @ Xa )
% 8.90/5.10 = X5 )
% 8.90/5.10 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 8.90/5.10 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6596_sum__le__included,axiom,
% 8.90/5.10 ! [S: set_nat,T: set_nat,G: nat > rat,I2: nat > nat,F: nat > rat] :
% 8.90/5.10 ( ( finite_finite_nat @ S )
% 8.90/5.10 => ( ( finite_finite_nat @ T )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ T )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ S )
% 8.90/5.10 => ? [Xa: nat] :
% 8.90/5.10 ( ( member_nat @ Xa @ T )
% 8.90/5.10 & ( ( I2 @ Xa )
% 8.90/5.10 = X5 )
% 8.90/5.10 & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 8.90/5.10 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6597_sum__le__included,axiom,
% 8.90/5.10 ! [S: set_nat,T: set_int,G: int > rat,I2: int > nat,F: nat > rat] :
% 8.90/5.10 ( ( finite_finite_nat @ S )
% 8.90/5.10 => ( ( finite_finite_int @ T )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ T )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ S )
% 8.90/5.10 => ? [Xa: int] :
% 8.90/5.10 ( ( member_int @ Xa @ T )
% 8.90/5.10 & ( ( I2 @ Xa )
% 8.90/5.10 = X5 )
% 8.90/5.10 & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 8.90/5.10 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6598_sum__le__included,axiom,
% 8.90/5.10 ! [S: set_nat,T: set_complex,G: complex > rat,I2: complex > nat,F: nat > rat] :
% 8.90/5.10 ( ( finite_finite_nat @ S )
% 8.90/5.10 => ( ( finite3207457112153483333omplex @ T )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ T )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ S )
% 8.90/5.10 => ? [Xa: complex] :
% 8.90/5.10 ( ( member_complex @ Xa @ T )
% 8.90/5.10 & ( ( I2 @ Xa )
% 8.90/5.10 = X5 )
% 8.90/5.10 & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 8.90/5.10 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6599_sum__le__included,axiom,
% 8.90/5.10 ! [S: set_int,T: set_nat,G: nat > rat,I2: nat > int,F: int > rat] :
% 8.90/5.10 ( ( finite_finite_int @ S )
% 8.90/5.10 => ( ( finite_finite_nat @ T )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ T )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ S )
% 8.90/5.10 => ? [Xa: nat] :
% 8.90/5.10 ( ( member_nat @ Xa @ T )
% 8.90/5.10 & ( ( I2 @ Xa )
% 8.90/5.10 = X5 )
% 8.90/5.10 & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 8.90/5.10 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6600_sum__le__included,axiom,
% 8.90/5.10 ! [S: set_int,T: set_int,G: int > rat,I2: int > int,F: int > rat] :
% 8.90/5.10 ( ( finite_finite_int @ S )
% 8.90/5.10 => ( ( finite_finite_int @ T )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ T )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ S )
% 8.90/5.10 => ? [Xa: int] :
% 8.90/5.10 ( ( member_int @ Xa @ T )
% 8.90/5.10 & ( ( I2 @ Xa )
% 8.90/5.10 = X5 )
% 8.90/5.10 & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 8.90/5.10 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6601_sum__le__included,axiom,
% 8.90/5.10 ! [S: set_int,T: set_complex,G: complex > rat,I2: complex > int,F: int > rat] :
% 8.90/5.10 ( ( finite_finite_int @ S )
% 8.90/5.10 => ( ( finite3207457112153483333omplex @ T )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ T )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ S )
% 8.90/5.10 => ? [Xa: complex] :
% 8.90/5.10 ( ( member_complex @ Xa @ T )
% 8.90/5.10 & ( ( I2 @ Xa )
% 8.90/5.10 = X5 )
% 8.90/5.10 & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 8.90/5.10 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6602_sum__nonneg__eq__0__iff,axiom,
% 8.90/5.10 ! [A2: set_real,F: real > real] :
% 8.90/5.10 ( ( finite_finite_real @ A2 )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ( ( groups8097168146408367636l_real @ F @ A2 )
% 8.90/5.10 = zero_zero_real )
% 8.90/5.10 = ( ! [X: real] :
% 8.90/5.10 ( ( member_real @ X @ A2 )
% 8.90/5.10 => ( ( F @ X )
% 8.90/5.10 = zero_zero_real ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6603_sum__nonneg__eq__0__iff,axiom,
% 8.90/5.10 ! [A2: set_int,F: int > real] :
% 8.90/5.10 ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ( ( groups8778361861064173332t_real @ F @ A2 )
% 8.90/5.10 = zero_zero_real )
% 8.90/5.10 = ( ! [X: int] :
% 8.90/5.10 ( ( member_int @ X @ A2 )
% 8.90/5.10 => ( ( F @ X )
% 8.90/5.10 = zero_zero_real ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6604_sum__nonneg__eq__0__iff,axiom,
% 8.90/5.10 ! [A2: set_complex,F: complex > real] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ( ( groups5808333547571424918x_real @ F @ A2 )
% 8.90/5.10 = zero_zero_real )
% 8.90/5.10 = ( ! [X: complex] :
% 8.90/5.10 ( ( member_complex @ X @ A2 )
% 8.90/5.10 => ( ( F @ X )
% 8.90/5.10 = zero_zero_real ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6605_sum__nonneg__eq__0__iff,axiom,
% 8.90/5.10 ! [A2: set_real,F: real > rat] :
% 8.90/5.10 ( ( finite_finite_real @ A2 )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ( ( groups1300246762558778688al_rat @ F @ A2 )
% 8.90/5.10 = zero_zero_rat )
% 8.90/5.10 = ( ! [X: real] :
% 8.90/5.10 ( ( member_real @ X @ A2 )
% 8.90/5.10 => ( ( F @ X )
% 8.90/5.10 = zero_zero_rat ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6606_sum__nonneg__eq__0__iff,axiom,
% 8.90/5.10 ! [A2: set_nat,F: nat > rat] :
% 8.90/5.10 ( ( finite_finite_nat @ A2 )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ( ( groups2906978787729119204at_rat @ F @ A2 )
% 8.90/5.10 = zero_zero_rat )
% 8.90/5.10 = ( ! [X: nat] :
% 8.90/5.10 ( ( member_nat @ X @ A2 )
% 8.90/5.10 => ( ( F @ X )
% 8.90/5.10 = zero_zero_rat ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6607_sum__nonneg__eq__0__iff,axiom,
% 8.90/5.10 ! [A2: set_int,F: int > rat] :
% 8.90/5.10 ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
% 8.90/5.10 = zero_zero_rat )
% 8.90/5.10 = ( ! [X: int] :
% 8.90/5.10 ( ( member_int @ X @ A2 )
% 8.90/5.10 => ( ( F @ X )
% 8.90/5.10 = zero_zero_rat ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6608_sum__nonneg__eq__0__iff,axiom,
% 8.90/5.10 ! [A2: set_complex,F: complex > rat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
% 8.90/5.10 = zero_zero_rat )
% 8.90/5.10 = ( ! [X: complex] :
% 8.90/5.10 ( ( member_complex @ X @ A2 )
% 8.90/5.10 => ( ( F @ X )
% 8.90/5.10 = zero_zero_rat ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6609_sum__nonneg__eq__0__iff,axiom,
% 8.90/5.10 ! [A2: set_real,F: real > nat] :
% 8.90/5.10 ( ( finite_finite_real @ A2 )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ( ( groups1935376822645274424al_nat @ F @ A2 )
% 8.90/5.10 = zero_zero_nat )
% 8.90/5.10 = ( ! [X: real] :
% 8.90/5.10 ( ( member_real @ X @ A2 )
% 8.90/5.10 => ( ( F @ X )
% 8.90/5.10 = zero_zero_nat ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6610_sum__nonneg__eq__0__iff,axiom,
% 8.90/5.10 ! [A2: set_int,F: int > nat] :
% 8.90/5.10 ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 8.90/5.10 = zero_zero_nat )
% 8.90/5.10 = ( ! [X: int] :
% 8.90/5.10 ( ( member_int @ X @ A2 )
% 8.90/5.10 => ( ( F @ X )
% 8.90/5.10 = zero_zero_nat ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6611_sum__nonneg__eq__0__iff,axiom,
% 8.90/5.10 ! [A2: set_complex,F: complex > nat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 8.90/5.10 => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 8.90/5.10 = zero_zero_nat )
% 8.90/5.10 = ( ! [X: complex] :
% 8.90/5.10 ( ( member_complex @ X @ A2 )
% 8.90/5.10 => ( ( F @ X )
% 8.90/5.10 = zero_zero_nat ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6612_sum__strict__mono__ex1,axiom,
% 8.90/5.10 ! [A2: set_int,F: int > real,G: int > real] :
% 8.90/5.10 ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ? [X3: int] :
% 8.90/5.10 ( ( member_int @ X3 @ A2 )
% 8.90/5.10 & ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 8.90/5.10 => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6613_sum__strict__mono__ex1,axiom,
% 8.90/5.10 ! [A2: set_complex,F: complex > real,G: complex > real] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ? [X3: complex] :
% 8.90/5.10 ( ( member_complex @ X3 @ A2 )
% 8.90/5.10 & ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 8.90/5.10 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6614_sum__strict__mono__ex1,axiom,
% 8.90/5.10 ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 8.90/5.10 ( ( finite_finite_nat @ A2 )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ? [X3: nat] :
% 8.90/5.10 ( ( member_nat @ X3 @ A2 )
% 8.90/5.10 & ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 8.90/5.10 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6615_sum__strict__mono__ex1,axiom,
% 8.90/5.10 ! [A2: set_int,F: int > rat,G: int > rat] :
% 8.90/5.10 ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ? [X3: int] :
% 8.90/5.10 ( ( member_int @ X3 @ A2 )
% 8.90/5.10 & ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 8.90/5.10 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6616_sum__strict__mono__ex1,axiom,
% 8.90/5.10 ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ? [X3: complex] :
% 8.90/5.10 ( ( member_complex @ X3 @ A2 )
% 8.90/5.10 & ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 8.90/5.10 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6617_sum__strict__mono__ex1,axiom,
% 8.90/5.10 ! [A2: set_int,F: int > nat,G: int > nat] :
% 8.90/5.10 ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ? [X3: int] :
% 8.90/5.10 ( ( member_int @ X3 @ A2 )
% 8.90/5.10 & ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 8.90/5.10 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6618_sum__strict__mono__ex1,axiom,
% 8.90/5.10 ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ? [X3: complex] :
% 8.90/5.10 ( ( member_complex @ X3 @ A2 )
% 8.90/5.10 & ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 8.90/5.10 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6619_sum__strict__mono__ex1,axiom,
% 8.90/5.10 ! [A2: set_nat,F: nat > int,G: nat > int] :
% 8.90/5.10 ( ( finite_finite_nat @ A2 )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ? [X3: nat] :
% 8.90/5.10 ( ( member_nat @ X3 @ A2 )
% 8.90/5.10 & ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 8.90/5.10 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6620_sum__strict__mono__ex1,axiom,
% 8.90/5.10 ! [A2: set_complex,F: complex > int,G: complex > int] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ? [X3: complex] :
% 8.90/5.10 ( ( member_complex @ X3 @ A2 )
% 8.90/5.10 & ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 8.90/5.10 => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6621_sum__strict__mono__ex1,axiom,
% 8.90/5.10 ! [A2: set_int,F: int > int,G: int > int] :
% 8.90/5.10 ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ? [X3: int] :
% 8.90/5.10 ( ( member_int @ X3 @ A2 )
% 8.90/5.10 & ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 8.90/5.10 => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6622_sum_Orelated,axiom,
% 8.90/5.10 ! [R: complex > complex > $o,S3: set_nat,H2: nat > complex,G: nat > complex] :
% 8.90/5.10 ( ( R @ zero_zero_complex @ zero_zero_complex )
% 8.90/5.10 => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
% 8.90/5.10 ( ( ( R @ X15 @ X23 )
% 8.90/5.10 & ( R @ Y15 @ Y23 ) )
% 8.90/5.10 => ( R @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
% 8.90/5.10 => ( ( finite_finite_nat @ S3 )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ S3 )
% 8.90/5.10 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( R @ ( groups2073611262835488442omplex @ H2 @ S3 ) @ ( groups2073611262835488442omplex @ G @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6623_sum_Orelated,axiom,
% 8.90/5.10 ! [R: complex > complex > $o,S3: set_int,H2: int > complex,G: int > complex] :
% 8.90/5.10 ( ( R @ zero_zero_complex @ zero_zero_complex )
% 8.90/5.10 => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
% 8.90/5.10 ( ( ( R @ X15 @ X23 )
% 8.90/5.10 & ( R @ Y15 @ Y23 ) )
% 8.90/5.10 => ( R @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
% 8.90/5.10 => ( ( finite_finite_int @ S3 )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ S3 )
% 8.90/5.10 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( R @ ( groups3049146728041665814omplex @ H2 @ S3 ) @ ( groups3049146728041665814omplex @ G @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6624_sum_Orelated,axiom,
% 8.90/5.10 ! [R: real > real > $o,S3: set_int,H2: int > real,G: int > real] :
% 8.90/5.10 ( ( R @ zero_zero_real @ zero_zero_real )
% 8.90/5.10 => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
% 8.90/5.10 ( ( ( R @ X15 @ X23 )
% 8.90/5.10 & ( R @ Y15 @ Y23 ) )
% 8.90/5.10 => ( R @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
% 8.90/5.10 => ( ( finite_finite_int @ S3 )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ S3 )
% 8.90/5.10 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( R @ ( groups8778361861064173332t_real @ H2 @ S3 ) @ ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6625_sum_Orelated,axiom,
% 8.90/5.10 ! [R: real > real > $o,S3: set_complex,H2: complex > real,G: complex > real] :
% 8.90/5.10 ( ( R @ zero_zero_real @ zero_zero_real )
% 8.90/5.10 => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
% 8.90/5.10 ( ( ( R @ X15 @ X23 )
% 8.90/5.10 & ( R @ Y15 @ Y23 ) )
% 8.90/5.10 => ( R @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
% 8.90/5.10 => ( ( finite3207457112153483333omplex @ S3 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ S3 )
% 8.90/5.10 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( R @ ( groups5808333547571424918x_real @ H2 @ S3 ) @ ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6626_sum_Orelated,axiom,
% 8.90/5.10 ! [R: rat > rat > $o,S3: set_nat,H2: nat > rat,G: nat > rat] :
% 8.90/5.10 ( ( R @ zero_zero_rat @ zero_zero_rat )
% 8.90/5.10 => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
% 8.90/5.10 ( ( ( R @ X15 @ X23 )
% 8.90/5.10 & ( R @ Y15 @ Y23 ) )
% 8.90/5.10 => ( R @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 8.90/5.10 => ( ( finite_finite_nat @ S3 )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ S3 )
% 8.90/5.10 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( R @ ( groups2906978787729119204at_rat @ H2 @ S3 ) @ ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6627_sum_Orelated,axiom,
% 8.90/5.10 ! [R: rat > rat > $o,S3: set_int,H2: int > rat,G: int > rat] :
% 8.90/5.10 ( ( R @ zero_zero_rat @ zero_zero_rat )
% 8.90/5.10 => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
% 8.90/5.10 ( ( ( R @ X15 @ X23 )
% 8.90/5.10 & ( R @ Y15 @ Y23 ) )
% 8.90/5.10 => ( R @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 8.90/5.10 => ( ( finite_finite_int @ S3 )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ S3 )
% 8.90/5.10 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( R @ ( groups3906332499630173760nt_rat @ H2 @ S3 ) @ ( groups3906332499630173760nt_rat @ G @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6628_sum_Orelated,axiom,
% 8.90/5.10 ! [R: rat > rat > $o,S3: set_complex,H2: complex > rat,G: complex > rat] :
% 8.90/5.10 ( ( R @ zero_zero_rat @ zero_zero_rat )
% 8.90/5.10 => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
% 8.90/5.10 ( ( ( R @ X15 @ X23 )
% 8.90/5.10 & ( R @ Y15 @ Y23 ) )
% 8.90/5.10 => ( R @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 8.90/5.10 => ( ( finite3207457112153483333omplex @ S3 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ S3 )
% 8.90/5.10 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( R @ ( groups5058264527183730370ex_rat @ H2 @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6629_sum_Orelated,axiom,
% 8.90/5.10 ! [R: nat > nat > $o,S3: set_int,H2: int > nat,G: int > nat] :
% 8.90/5.10 ( ( R @ zero_zero_nat @ zero_zero_nat )
% 8.90/5.10 => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
% 8.90/5.10 ( ( ( R @ X15 @ X23 )
% 8.90/5.10 & ( R @ Y15 @ Y23 ) )
% 8.90/5.10 => ( R @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
% 8.90/5.10 => ( ( finite_finite_int @ S3 )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ S3 )
% 8.90/5.10 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( R @ ( groups4541462559716669496nt_nat @ H2 @ S3 ) @ ( groups4541462559716669496nt_nat @ G @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6630_sum_Orelated,axiom,
% 8.90/5.10 ! [R: nat > nat > $o,S3: set_complex,H2: complex > nat,G: complex > nat] :
% 8.90/5.10 ( ( R @ zero_zero_nat @ zero_zero_nat )
% 8.90/5.10 => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
% 8.90/5.10 ( ( ( R @ X15 @ X23 )
% 8.90/5.10 & ( R @ Y15 @ Y23 ) )
% 8.90/5.10 => ( R @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
% 8.90/5.10 => ( ( finite3207457112153483333omplex @ S3 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ S3 )
% 8.90/5.10 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( R @ ( groups5693394587270226106ex_nat @ H2 @ S3 ) @ ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6631_sum_Orelated,axiom,
% 8.90/5.10 ! [R: int > int > $o,S3: set_nat,H2: nat > int,G: nat > int] :
% 8.90/5.10 ( ( R @ zero_zero_int @ zero_zero_int )
% 8.90/5.10 => ( ! [X15: int,Y15: int,X23: int,Y23: int] :
% 8.90/5.10 ( ( ( R @ X15 @ X23 )
% 8.90/5.10 & ( R @ Y15 @ Y23 ) )
% 8.90/5.10 => ( R @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
% 8.90/5.10 => ( ( finite_finite_nat @ S3 )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ S3 )
% 8.90/5.10 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( R @ ( groups3539618377306564664at_int @ H2 @ S3 ) @ ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6632_sum__strict__mono,axiom,
% 8.90/5.10 ! [A2: set_complex,F: complex > real,G: complex > real] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ( A2 != bot_bot_set_complex )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6633_sum__strict__mono,axiom,
% 8.90/5.10 ! [A2: set_int,F: int > real,G: int > real] :
% 8.90/5.10 ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ( A2 != bot_bot_set_int )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6634_sum__strict__mono,axiom,
% 8.90/5.10 ! [A2: set_real,F: real > real,G: real > real] :
% 8.90/5.10 ( ( finite_finite_real @ A2 )
% 8.90/5.10 => ( ( A2 != bot_bot_set_real )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6635_sum__strict__mono,axiom,
% 8.90/5.10 ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ( A2 != bot_bot_set_complex )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6636_sum__strict__mono,axiom,
% 8.90/5.10 ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 8.90/5.10 ( ( finite_finite_nat @ A2 )
% 8.90/5.10 => ( ( A2 != bot_bot_set_nat )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6637_sum__strict__mono,axiom,
% 8.90/5.10 ! [A2: set_int,F: int > rat,G: int > rat] :
% 8.90/5.10 ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ( A2 != bot_bot_set_int )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6638_sum__strict__mono,axiom,
% 8.90/5.10 ! [A2: set_real,F: real > rat,G: real > rat] :
% 8.90/5.10 ( ( finite_finite_real @ A2 )
% 8.90/5.10 => ( ( A2 != bot_bot_set_real )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6639_sum__strict__mono,axiom,
% 8.90/5.10 ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ( A2 != bot_bot_set_complex )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6640_sum__strict__mono,axiom,
% 8.90/5.10 ! [A2: set_int,F: int > nat,G: int > nat] :
% 8.90/5.10 ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ( A2 != bot_bot_set_int )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6641_sum__strict__mono,axiom,
% 8.90/5.10 ! [A2: set_real,F: real > nat,G: real > nat] :
% 8.90/5.10 ( ( finite_finite_real @ A2 )
% 8.90/5.10 => ( ( A2 != bot_bot_set_real )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ A2 )
% 8.90/5.10 => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 8.90/5.10 => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6642_ln__bound,axiom,
% 8.90/5.10 ! [X2: real] :
% 8.90/5.10 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.10 => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6643_ln__gt__zero,axiom,
% 8.90/5.10 ! [X2: real] :
% 8.90/5.10 ( ( ord_less_real @ one_one_real @ X2 )
% 8.90/5.10 => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6644_ln__less__zero,axiom,
% 8.90/5.10 ! [X2: real] :
% 8.90/5.10 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.10 => ( ( ord_less_real @ X2 @ one_one_real )
% 8.90/5.10 => ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6645_ln__gt__zero__imp__gt__one,axiom,
% 8.90/5.10 ! [X2: real] :
% 8.90/5.10 ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
% 8.90/5.10 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.10 => ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6646_ln__ge__zero,axiom,
% 8.90/5.10 ! [X2: real] :
% 8.90/5.10 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6647_sum_Oreindex__bij__witness__not__neutral,axiom,
% 8.90/5.10 ! [S4: set_real,T4: set_real,S3: set_real,I2: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
% 8.90/5.10 ( ( finite_finite_real @ S4 )
% 8.90/5.10 => ( ( finite_finite_real @ T4 )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 8.90/5.10 => ( ( I2 @ ( J @ A4 ) )
% 8.90/5.10 = A4 ) )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 8.90/5.10 => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 8.90/5.10 => ( ( J @ ( I2 @ B4 ) )
% 8.90/5.10 = B4 ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 8.90/5.10 => ( member_real @ ( I2 @ B4 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ S4 )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ T4 )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ S3 )
% 8.90/5.10 => ( ( H2 @ ( J @ A4 ) )
% 8.90/5.10 = ( G @ A4 ) ) )
% 8.90/5.10 => ( ( groups5754745047067104278omplex @ G @ S3 )
% 8.90/5.10 = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6648_sum_Oreindex__bij__witness__not__neutral,axiom,
% 8.90/5.10 ! [S4: set_real,T4: set_int,S3: set_real,I2: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
% 8.90/5.10 ( ( finite_finite_real @ S4 )
% 8.90/5.10 => ( ( finite_finite_int @ T4 )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 8.90/5.10 => ( ( I2 @ ( J @ A4 ) )
% 8.90/5.10 = A4 ) )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 8.90/5.10 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 8.90/5.10 => ( ! [B4: int] :
% 8.90/5.10 ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 8.90/5.10 => ( ( J @ ( I2 @ B4 ) )
% 8.90/5.10 = B4 ) )
% 8.90/5.10 => ( ! [B4: int] :
% 8.90/5.10 ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 8.90/5.10 => ( member_real @ ( I2 @ B4 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ S4 )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ! [B4: int] :
% 8.90/5.10 ( ( member_int @ B4 @ T4 )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ S3 )
% 8.90/5.10 => ( ( H2 @ ( J @ A4 ) )
% 8.90/5.10 = ( G @ A4 ) ) )
% 8.90/5.10 => ( ( groups5754745047067104278omplex @ G @ S3 )
% 8.90/5.10 = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6649_sum_Oreindex__bij__witness__not__neutral,axiom,
% 8.90/5.10 ! [S4: set_int,T4: set_real,S3: set_int,I2: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
% 8.90/5.10 ( ( finite_finite_int @ S4 )
% 8.90/5.10 => ( ( finite_finite_real @ T4 )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 8.90/5.10 => ( ( I2 @ ( J @ A4 ) )
% 8.90/5.10 = A4 ) )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 8.90/5.10 => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 8.90/5.10 => ( ( J @ ( I2 @ B4 ) )
% 8.90/5.10 = B4 ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 8.90/5.10 => ( member_int @ ( I2 @ B4 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ S4 )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ T4 )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ S3 )
% 8.90/5.10 => ( ( H2 @ ( J @ A4 ) )
% 8.90/5.10 = ( G @ A4 ) ) )
% 8.90/5.10 => ( ( groups3049146728041665814omplex @ G @ S3 )
% 8.90/5.10 = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6650_sum_Oreindex__bij__witness__not__neutral,axiom,
% 8.90/5.10 ! [S4: set_int,T4: set_int,S3: set_int,I2: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
% 8.90/5.10 ( ( finite_finite_int @ S4 )
% 8.90/5.10 => ( ( finite_finite_int @ T4 )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 8.90/5.10 => ( ( I2 @ ( J @ A4 ) )
% 8.90/5.10 = A4 ) )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 8.90/5.10 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 8.90/5.10 => ( ! [B4: int] :
% 8.90/5.10 ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 8.90/5.10 => ( ( J @ ( I2 @ B4 ) )
% 8.90/5.10 = B4 ) )
% 8.90/5.10 => ( ! [B4: int] :
% 8.90/5.10 ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 8.90/5.10 => ( member_int @ ( I2 @ B4 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ S4 )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ! [B4: int] :
% 8.90/5.10 ( ( member_int @ B4 @ T4 )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ S3 )
% 8.90/5.10 => ( ( H2 @ ( J @ A4 ) )
% 8.90/5.10 = ( G @ A4 ) ) )
% 8.90/5.10 => ( ( groups3049146728041665814omplex @ G @ S3 )
% 8.90/5.10 = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6651_sum_Oreindex__bij__witness__not__neutral,axiom,
% 8.90/5.10 ! [S4: set_real,T4: set_real,S3: set_real,I2: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
% 8.90/5.10 ( ( finite_finite_real @ S4 )
% 8.90/5.10 => ( ( finite_finite_real @ T4 )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 8.90/5.10 => ( ( I2 @ ( J @ A4 ) )
% 8.90/5.10 = A4 ) )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 8.90/5.10 => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 8.90/5.10 => ( ( J @ ( I2 @ B4 ) )
% 8.90/5.10 = B4 ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 8.90/5.10 => ( member_real @ ( I2 @ B4 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ S4 )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ T4 )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ S3 )
% 8.90/5.10 => ( ( H2 @ ( J @ A4 ) )
% 8.90/5.10 = ( G @ A4 ) ) )
% 8.90/5.10 => ( ( groups8097168146408367636l_real @ G @ S3 )
% 8.90/5.10 = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6652_sum_Oreindex__bij__witness__not__neutral,axiom,
% 8.90/5.10 ! [S4: set_real,T4: set_int,S3: set_real,I2: int > real,J: real > int,T3: set_int,G: real > real,H2: int > real] :
% 8.90/5.10 ( ( finite_finite_real @ S4 )
% 8.90/5.10 => ( ( finite_finite_int @ T4 )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 8.90/5.10 => ( ( I2 @ ( J @ A4 ) )
% 8.90/5.10 = A4 ) )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 8.90/5.10 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 8.90/5.10 => ( ! [B4: int] :
% 8.90/5.10 ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 8.90/5.10 => ( ( J @ ( I2 @ B4 ) )
% 8.90/5.10 = B4 ) )
% 8.90/5.10 => ( ! [B4: int] :
% 8.90/5.10 ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 8.90/5.10 => ( member_real @ ( I2 @ B4 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ S4 )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [B4: int] :
% 8.90/5.10 ( ( member_int @ B4 @ T4 )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ S3 )
% 8.90/5.10 => ( ( H2 @ ( J @ A4 ) )
% 8.90/5.10 = ( G @ A4 ) ) )
% 8.90/5.10 => ( ( groups8097168146408367636l_real @ G @ S3 )
% 8.90/5.10 = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6653_sum_Oreindex__bij__witness__not__neutral,axiom,
% 8.90/5.10 ! [S4: set_real,T4: set_complex,S3: set_real,I2: complex > real,J: real > complex,T3: set_complex,G: real > real,H2: complex > real] :
% 8.90/5.10 ( ( finite_finite_real @ S4 )
% 8.90/5.10 => ( ( finite3207457112153483333omplex @ T4 )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 8.90/5.10 => ( ( I2 @ ( J @ A4 ) )
% 8.90/5.10 = A4 ) )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S4 ) )
% 8.90/5.10 => ( member_complex @ ( J @ A4 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 8.90/5.10 => ( ! [B4: complex] :
% 8.90/5.10 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 8.90/5.10 => ( ( J @ ( I2 @ B4 ) )
% 8.90/5.10 = B4 ) )
% 8.90/5.10 => ( ! [B4: complex] :
% 8.90/5.10 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 8.90/5.10 => ( member_real @ ( I2 @ B4 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ S4 )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [B4: complex] :
% 8.90/5.10 ( ( member_complex @ B4 @ T4 )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ S3 )
% 8.90/5.10 => ( ( H2 @ ( J @ A4 ) )
% 8.90/5.10 = ( G @ A4 ) ) )
% 8.90/5.10 => ( ( groups8097168146408367636l_real @ G @ S3 )
% 8.90/5.10 = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6654_sum_Oreindex__bij__witness__not__neutral,axiom,
% 8.90/5.10 ! [S4: set_int,T4: set_real,S3: set_int,I2: real > int,J: int > real,T3: set_real,G: int > real,H2: real > real] :
% 8.90/5.10 ( ( finite_finite_int @ S4 )
% 8.90/5.10 => ( ( finite_finite_real @ T4 )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 8.90/5.10 => ( ( I2 @ ( J @ A4 ) )
% 8.90/5.10 = A4 ) )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 8.90/5.10 => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 8.90/5.10 => ( ( J @ ( I2 @ B4 ) )
% 8.90/5.10 = B4 ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 8.90/5.10 => ( member_int @ ( I2 @ B4 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ S4 )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ T4 )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ S3 )
% 8.90/5.10 => ( ( H2 @ ( J @ A4 ) )
% 8.90/5.10 = ( G @ A4 ) ) )
% 8.90/5.10 => ( ( groups8778361861064173332t_real @ G @ S3 )
% 8.90/5.10 = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6655_sum_Oreindex__bij__witness__not__neutral,axiom,
% 8.90/5.10 ! [S4: set_int,T4: set_int,S3: set_int,I2: int > int,J: int > int,T3: set_int,G: int > real,H2: int > real] :
% 8.90/5.10 ( ( finite_finite_int @ S4 )
% 8.90/5.10 => ( ( finite_finite_int @ T4 )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 8.90/5.10 => ( ( I2 @ ( J @ A4 ) )
% 8.90/5.10 = A4 ) )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 8.90/5.10 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 8.90/5.10 => ( ! [B4: int] :
% 8.90/5.10 ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 8.90/5.10 => ( ( J @ ( I2 @ B4 ) )
% 8.90/5.10 = B4 ) )
% 8.90/5.10 => ( ! [B4: int] :
% 8.90/5.10 ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 8.90/5.10 => ( member_int @ ( I2 @ B4 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ S4 )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [B4: int] :
% 8.90/5.10 ( ( member_int @ B4 @ T4 )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ S3 )
% 8.90/5.10 => ( ( H2 @ ( J @ A4 ) )
% 8.90/5.10 = ( G @ A4 ) ) )
% 8.90/5.10 => ( ( groups8778361861064173332t_real @ G @ S3 )
% 8.90/5.10 = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6656_sum_Oreindex__bij__witness__not__neutral,axiom,
% 8.90/5.10 ! [S4: set_int,T4: set_complex,S3: set_int,I2: complex > int,J: int > complex,T3: set_complex,G: int > real,H2: complex > real] :
% 8.90/5.10 ( ( finite_finite_int @ S4 )
% 8.90/5.10 => ( ( finite3207457112153483333omplex @ T4 )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 8.90/5.10 => ( ( I2 @ ( J @ A4 ) )
% 8.90/5.10 = A4 ) )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S4 ) )
% 8.90/5.10 => ( member_complex @ ( J @ A4 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 8.90/5.10 => ( ! [B4: complex] :
% 8.90/5.10 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 8.90/5.10 => ( ( J @ ( I2 @ B4 ) )
% 8.90/5.10 = B4 ) )
% 8.90/5.10 => ( ! [B4: complex] :
% 8.90/5.10 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 8.90/5.10 => ( member_int @ ( I2 @ B4 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ S4 )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [B4: complex] :
% 8.90/5.10 ( ( member_complex @ B4 @ T4 )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [A4: int] :
% 8.90/5.10 ( ( member_int @ A4 @ S3 )
% 8.90/5.10 => ( ( H2 @ ( J @ A4 ) )
% 8.90/5.10 = ( G @ A4 ) ) )
% 8.90/5.10 => ( ( groups8778361861064173332t_real @ G @ S3 )
% 8.90/5.10 = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6657_sum__nonneg__leq__bound,axiom,
% 8.90/5.10 ! [S: set_real,F: real > real,B3: real,I2: real] :
% 8.90/5.10 ( ( finite_finite_real @ S )
% 8.90/5.10 => ( ! [I3: real] :
% 8.90/5.10 ( ( member_real @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups8097168146408367636l_real @ F @ S )
% 8.90/5.10 = B3 )
% 8.90/5.10 => ( ( member_real @ I2 @ S )
% 8.90/5.10 => ( ord_less_eq_real @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6658_sum__nonneg__leq__bound,axiom,
% 8.90/5.10 ! [S: set_int,F: int > real,B3: real,I2: int] :
% 8.90/5.10 ( ( finite_finite_int @ S )
% 8.90/5.10 => ( ! [I3: int] :
% 8.90/5.10 ( ( member_int @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups8778361861064173332t_real @ F @ S )
% 8.90/5.10 = B3 )
% 8.90/5.10 => ( ( member_int @ I2 @ S )
% 8.90/5.10 => ( ord_less_eq_real @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6659_sum__nonneg__leq__bound,axiom,
% 8.90/5.10 ! [S: set_complex,F: complex > real,B3: real,I2: complex] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ S )
% 8.90/5.10 => ( ! [I3: complex] :
% 8.90/5.10 ( ( member_complex @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups5808333547571424918x_real @ F @ S )
% 8.90/5.10 = B3 )
% 8.90/5.10 => ( ( member_complex @ I2 @ S )
% 8.90/5.10 => ( ord_less_eq_real @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6660_sum__nonneg__leq__bound,axiom,
% 8.90/5.10 ! [S: set_real,F: real > rat,B3: rat,I2: real] :
% 8.90/5.10 ( ( finite_finite_real @ S )
% 8.90/5.10 => ( ! [I3: real] :
% 8.90/5.10 ( ( member_real @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups1300246762558778688al_rat @ F @ S )
% 8.90/5.10 = B3 )
% 8.90/5.10 => ( ( member_real @ I2 @ S )
% 8.90/5.10 => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6661_sum__nonneg__leq__bound,axiom,
% 8.90/5.10 ! [S: set_nat,F: nat > rat,B3: rat,I2: nat] :
% 8.90/5.10 ( ( finite_finite_nat @ S )
% 8.90/5.10 => ( ! [I3: nat] :
% 8.90/5.10 ( ( member_nat @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups2906978787729119204at_rat @ F @ S )
% 8.90/5.10 = B3 )
% 8.90/5.10 => ( ( member_nat @ I2 @ S )
% 8.90/5.10 => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6662_sum__nonneg__leq__bound,axiom,
% 8.90/5.10 ! [S: set_int,F: int > rat,B3: rat,I2: int] :
% 8.90/5.10 ( ( finite_finite_int @ S )
% 8.90/5.10 => ( ! [I3: int] :
% 8.90/5.10 ( ( member_int @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups3906332499630173760nt_rat @ F @ S )
% 8.90/5.10 = B3 )
% 8.90/5.10 => ( ( member_int @ I2 @ S )
% 8.90/5.10 => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6663_sum__nonneg__leq__bound,axiom,
% 8.90/5.10 ! [S: set_complex,F: complex > rat,B3: rat,I2: complex] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ S )
% 8.90/5.10 => ( ! [I3: complex] :
% 8.90/5.10 ( ( member_complex @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups5058264527183730370ex_rat @ F @ S )
% 8.90/5.10 = B3 )
% 8.90/5.10 => ( ( member_complex @ I2 @ S )
% 8.90/5.10 => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6664_sum__nonneg__leq__bound,axiom,
% 8.90/5.10 ! [S: set_real,F: real > nat,B3: nat,I2: real] :
% 8.90/5.10 ( ( finite_finite_real @ S )
% 8.90/5.10 => ( ! [I3: real] :
% 8.90/5.10 ( ( member_real @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups1935376822645274424al_nat @ F @ S )
% 8.90/5.10 = B3 )
% 8.90/5.10 => ( ( member_real @ I2 @ S )
% 8.90/5.10 => ( ord_less_eq_nat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6665_sum__nonneg__leq__bound,axiom,
% 8.90/5.10 ! [S: set_int,F: int > nat,B3: nat,I2: int] :
% 8.90/5.10 ( ( finite_finite_int @ S )
% 8.90/5.10 => ( ! [I3: int] :
% 8.90/5.10 ( ( member_int @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups4541462559716669496nt_nat @ F @ S )
% 8.90/5.10 = B3 )
% 8.90/5.10 => ( ( member_int @ I2 @ S )
% 8.90/5.10 => ( ord_less_eq_nat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6666_sum__nonneg__leq__bound,axiom,
% 8.90/5.10 ! [S: set_complex,F: complex > nat,B3: nat,I2: complex] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ S )
% 8.90/5.10 => ( ! [I3: complex] :
% 8.90/5.10 ( ( member_complex @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups5693394587270226106ex_nat @ F @ S )
% 8.90/5.10 = B3 )
% 8.90/5.10 => ( ( member_complex @ I2 @ S )
% 8.90/5.10 => ( ord_less_eq_nat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6667_sum__nonneg__0,axiom,
% 8.90/5.10 ! [S: set_real,F: real > real,I2: real] :
% 8.90/5.10 ( ( finite_finite_real @ S )
% 8.90/5.10 => ( ! [I3: real] :
% 8.90/5.10 ( ( member_real @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups8097168146408367636l_real @ F @ S )
% 8.90/5.10 = zero_zero_real )
% 8.90/5.10 => ( ( member_real @ I2 @ S )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = zero_zero_real ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6668_sum__nonneg__0,axiom,
% 8.90/5.10 ! [S: set_int,F: int > real,I2: int] :
% 8.90/5.10 ( ( finite_finite_int @ S )
% 8.90/5.10 => ( ! [I3: int] :
% 8.90/5.10 ( ( member_int @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups8778361861064173332t_real @ F @ S )
% 8.90/5.10 = zero_zero_real )
% 8.90/5.10 => ( ( member_int @ I2 @ S )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = zero_zero_real ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6669_sum__nonneg__0,axiom,
% 8.90/5.10 ! [S: set_complex,F: complex > real,I2: complex] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ S )
% 8.90/5.10 => ( ! [I3: complex] :
% 8.90/5.10 ( ( member_complex @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups5808333547571424918x_real @ F @ S )
% 8.90/5.10 = zero_zero_real )
% 8.90/5.10 => ( ( member_complex @ I2 @ S )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = zero_zero_real ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6670_sum__nonneg__0,axiom,
% 8.90/5.10 ! [S: set_real,F: real > rat,I2: real] :
% 8.90/5.10 ( ( finite_finite_real @ S )
% 8.90/5.10 => ( ! [I3: real] :
% 8.90/5.10 ( ( member_real @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups1300246762558778688al_rat @ F @ S )
% 8.90/5.10 = zero_zero_rat )
% 8.90/5.10 => ( ( member_real @ I2 @ S )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = zero_zero_rat ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6671_sum__nonneg__0,axiom,
% 8.90/5.10 ! [S: set_nat,F: nat > rat,I2: nat] :
% 8.90/5.10 ( ( finite_finite_nat @ S )
% 8.90/5.10 => ( ! [I3: nat] :
% 8.90/5.10 ( ( member_nat @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups2906978787729119204at_rat @ F @ S )
% 8.90/5.10 = zero_zero_rat )
% 8.90/5.10 => ( ( member_nat @ I2 @ S )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = zero_zero_rat ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6672_sum__nonneg__0,axiom,
% 8.90/5.10 ! [S: set_int,F: int > rat,I2: int] :
% 8.90/5.10 ( ( finite_finite_int @ S )
% 8.90/5.10 => ( ! [I3: int] :
% 8.90/5.10 ( ( member_int @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups3906332499630173760nt_rat @ F @ S )
% 8.90/5.10 = zero_zero_rat )
% 8.90/5.10 => ( ( member_int @ I2 @ S )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = zero_zero_rat ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6673_sum__nonneg__0,axiom,
% 8.90/5.10 ! [S: set_complex,F: complex > rat,I2: complex] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ S )
% 8.90/5.10 => ( ! [I3: complex] :
% 8.90/5.10 ( ( member_complex @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups5058264527183730370ex_rat @ F @ S )
% 8.90/5.10 = zero_zero_rat )
% 8.90/5.10 => ( ( member_complex @ I2 @ S )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = zero_zero_rat ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6674_sum__nonneg__0,axiom,
% 8.90/5.10 ! [S: set_real,F: real > nat,I2: real] :
% 8.90/5.10 ( ( finite_finite_real @ S )
% 8.90/5.10 => ( ! [I3: real] :
% 8.90/5.10 ( ( member_real @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups1935376822645274424al_nat @ F @ S )
% 8.90/5.10 = zero_zero_nat )
% 8.90/5.10 => ( ( member_real @ I2 @ S )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = zero_zero_nat ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6675_sum__nonneg__0,axiom,
% 8.90/5.10 ! [S: set_int,F: int > nat,I2: int] :
% 8.90/5.10 ( ( finite_finite_int @ S )
% 8.90/5.10 => ( ! [I3: int] :
% 8.90/5.10 ( ( member_int @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups4541462559716669496nt_nat @ F @ S )
% 8.90/5.10 = zero_zero_nat )
% 8.90/5.10 => ( ( member_int @ I2 @ S )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = zero_zero_nat ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6676_sum__nonneg__0,axiom,
% 8.90/5.10 ! [S: set_complex,F: complex > nat,I2: complex] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ S )
% 8.90/5.10 => ( ! [I3: complex] :
% 8.90/5.10 ( ( member_complex @ I3 @ S )
% 8.90/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ( ( groups5693394587270226106ex_nat @ F @ S )
% 8.90/5.10 = zero_zero_nat )
% 8.90/5.10 => ( ( member_complex @ I2 @ S )
% 8.90/5.10 => ( ( F @ I2 )
% 8.90/5.10 = zero_zero_nat ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6677_sum_Osetdiff__irrelevant,axiom,
% 8.90/5.10 ! [A2: set_real,G: real > complex] :
% 8.90/5.10 ( ( finite_finite_real @ A2 )
% 8.90/5.10 => ( ( groups5754745047067104278omplex @ G
% 8.90/5.10 @ ( minus_minus_set_real @ A2
% 8.90/5.10 @ ( collect_real
% 8.90/5.10 @ ^ [X: real] :
% 8.90/5.10 ( ( G @ X )
% 8.90/5.10 = zero_zero_complex ) ) ) )
% 8.90/5.10 = ( groups5754745047067104278omplex @ G @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6678_sum_Osetdiff__irrelevant,axiom,
% 8.90/5.10 ! [A2: set_int,G: int > complex] :
% 8.90/5.10 ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ( groups3049146728041665814omplex @ G
% 8.90/5.10 @ ( minus_minus_set_int @ A2
% 8.90/5.10 @ ( collect_int
% 8.90/5.10 @ ^ [X: int] :
% 8.90/5.10 ( ( G @ X )
% 8.90/5.10 = zero_zero_complex ) ) ) )
% 8.90/5.10 = ( groups3049146728041665814omplex @ G @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6679_sum_Osetdiff__irrelevant,axiom,
% 8.90/5.10 ! [A2: set_real,G: real > real] :
% 8.90/5.10 ( ( finite_finite_real @ A2 )
% 8.90/5.10 => ( ( groups8097168146408367636l_real @ G
% 8.90/5.10 @ ( minus_minus_set_real @ A2
% 8.90/5.10 @ ( collect_real
% 8.90/5.10 @ ^ [X: real] :
% 8.90/5.10 ( ( G @ X )
% 8.90/5.10 = zero_zero_real ) ) ) )
% 8.90/5.10 = ( groups8097168146408367636l_real @ G @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6680_sum_Osetdiff__irrelevant,axiom,
% 8.90/5.10 ! [A2: set_int,G: int > real] :
% 8.90/5.10 ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ( groups8778361861064173332t_real @ G
% 8.90/5.10 @ ( minus_minus_set_int @ A2
% 8.90/5.10 @ ( collect_int
% 8.90/5.10 @ ^ [X: int] :
% 8.90/5.10 ( ( G @ X )
% 8.90/5.10 = zero_zero_real ) ) ) )
% 8.90/5.10 = ( groups8778361861064173332t_real @ G @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6681_sum_Osetdiff__irrelevant,axiom,
% 8.90/5.10 ! [A2: set_complex,G: complex > real] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ( groups5808333547571424918x_real @ G
% 8.90/5.10 @ ( minus_811609699411566653omplex @ A2
% 8.90/5.10 @ ( collect_complex
% 8.90/5.10 @ ^ [X: complex] :
% 8.90/5.10 ( ( G @ X )
% 8.90/5.10 = zero_zero_real ) ) ) )
% 8.90/5.10 = ( groups5808333547571424918x_real @ G @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6682_sum_Osetdiff__irrelevant,axiom,
% 8.90/5.10 ! [A2: set_real,G: real > rat] :
% 8.90/5.10 ( ( finite_finite_real @ A2 )
% 8.90/5.10 => ( ( groups1300246762558778688al_rat @ G
% 8.90/5.10 @ ( minus_minus_set_real @ A2
% 8.90/5.10 @ ( collect_real
% 8.90/5.10 @ ^ [X: real] :
% 8.90/5.10 ( ( G @ X )
% 8.90/5.10 = zero_zero_rat ) ) ) )
% 8.90/5.10 = ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6683_sum_Osetdiff__irrelevant,axiom,
% 8.90/5.10 ! [A2: set_int,G: int > rat] :
% 8.90/5.10 ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ( groups3906332499630173760nt_rat @ G
% 8.90/5.10 @ ( minus_minus_set_int @ A2
% 8.90/5.10 @ ( collect_int
% 8.90/5.10 @ ^ [X: int] :
% 8.90/5.10 ( ( G @ X )
% 8.90/5.10 = zero_zero_rat ) ) ) )
% 8.90/5.10 = ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6684_sum_Osetdiff__irrelevant,axiom,
% 8.90/5.10 ! [A2: set_complex,G: complex > rat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ( groups5058264527183730370ex_rat @ G
% 8.90/5.10 @ ( minus_811609699411566653omplex @ A2
% 8.90/5.10 @ ( collect_complex
% 8.90/5.10 @ ^ [X: complex] :
% 8.90/5.10 ( ( G @ X )
% 8.90/5.10 = zero_zero_rat ) ) ) )
% 8.90/5.10 = ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6685_sum_Osetdiff__irrelevant,axiom,
% 8.90/5.10 ! [A2: set_real,G: real > nat] :
% 8.90/5.10 ( ( finite_finite_real @ A2 )
% 8.90/5.10 => ( ( groups1935376822645274424al_nat @ G
% 8.90/5.10 @ ( minus_minus_set_real @ A2
% 8.90/5.10 @ ( collect_real
% 8.90/5.10 @ ^ [X: real] :
% 8.90/5.10 ( ( G @ X )
% 8.90/5.10 = zero_zero_nat ) ) ) )
% 8.90/5.10 = ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6686_sum_Osetdiff__irrelevant,axiom,
% 8.90/5.10 ! [A2: set_int,G: int > nat] :
% 8.90/5.10 ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ( groups4541462559716669496nt_nat @ G
% 8.90/5.10 @ ( minus_minus_set_int @ A2
% 8.90/5.10 @ ( collect_int
% 8.90/5.10 @ ^ [X: int] :
% 8.90/5.10 ( ( G @ X )
% 8.90/5.10 = zero_zero_nat ) ) ) )
% 8.90/5.10 = ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6687_sum__pos2,axiom,
% 8.90/5.10 ! [I5: set_real,I2: real,F: real > real] :
% 8.90/5.10 ( ( finite_finite_real @ I5 )
% 8.90/5.10 => ( ( member_real @ I2 @ I5 )
% 8.90/5.10 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 8.90/5.10 => ( ! [I3: real] :
% 8.90/5.10 ( ( member_real @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6688_sum__pos2,axiom,
% 8.90/5.10 ! [I5: set_int,I2: int,F: int > real] :
% 8.90/5.10 ( ( finite_finite_int @ I5 )
% 8.90/5.10 => ( ( member_int @ I2 @ I5 )
% 8.90/5.10 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 8.90/5.10 => ( ! [I3: int] :
% 8.90/5.10 ( ( member_int @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6689_sum__pos2,axiom,
% 8.90/5.10 ! [I5: set_complex,I2: complex,F: complex > real] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ I5 )
% 8.90/5.10 => ( ( member_complex @ I2 @ I5 )
% 8.90/5.10 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 8.90/5.10 => ( ! [I3: complex] :
% 8.90/5.10 ( ( member_complex @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6690_sum__pos2,axiom,
% 8.90/5.10 ! [I5: set_real,I2: real,F: real > rat] :
% 8.90/5.10 ( ( finite_finite_real @ I5 )
% 8.90/5.10 => ( ( member_real @ I2 @ I5 )
% 8.90/5.10 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
% 8.90/5.10 => ( ! [I3: real] :
% 8.90/5.10 ( ( member_real @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6691_sum__pos2,axiom,
% 8.90/5.10 ! [I5: set_nat,I2: nat,F: nat > rat] :
% 8.90/5.10 ( ( finite_finite_nat @ I5 )
% 8.90/5.10 => ( ( member_nat @ I2 @ I5 )
% 8.90/5.10 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
% 8.90/5.10 => ( ! [I3: nat] :
% 8.90/5.10 ( ( member_nat @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6692_sum__pos2,axiom,
% 8.90/5.10 ! [I5: set_int,I2: int,F: int > rat] :
% 8.90/5.10 ( ( finite_finite_int @ I5 )
% 8.90/5.10 => ( ( member_int @ I2 @ I5 )
% 8.90/5.10 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
% 8.90/5.10 => ( ! [I3: int] :
% 8.90/5.10 ( ( member_int @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6693_sum__pos2,axiom,
% 8.90/5.10 ! [I5: set_complex,I2: complex,F: complex > rat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ I5 )
% 8.90/5.10 => ( ( member_complex @ I2 @ I5 )
% 8.90/5.10 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
% 8.90/5.10 => ( ! [I3: complex] :
% 8.90/5.10 ( ( member_complex @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6694_sum__pos2,axiom,
% 8.90/5.10 ! [I5: set_real,I2: real,F: real > nat] :
% 8.90/5.10 ( ( finite_finite_real @ I5 )
% 8.90/5.10 => ( ( member_real @ I2 @ I5 )
% 8.90/5.10 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 8.90/5.10 => ( ! [I3: real] :
% 8.90/5.10 ( ( member_real @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6695_sum__pos2,axiom,
% 8.90/5.10 ! [I5: set_int,I2: int,F: int > nat] :
% 8.90/5.10 ( ( finite_finite_int @ I5 )
% 8.90/5.10 => ( ( member_int @ I2 @ I5 )
% 8.90/5.10 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 8.90/5.10 => ( ! [I3: int] :
% 8.90/5.10 ( ( member_int @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6696_sum__pos2,axiom,
% 8.90/5.10 ! [I5: set_complex,I2: complex,F: complex > nat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ I5 )
% 8.90/5.10 => ( ( member_complex @ I2 @ I5 )
% 8.90/5.10 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 8.90/5.10 => ( ! [I3: complex] :
% 8.90/5.10 ( ( member_complex @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6697_sum__pos,axiom,
% 8.90/5.10 ! [I5: set_complex,F: complex > real] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ I5 )
% 8.90/5.10 => ( ( I5 != bot_bot_set_complex )
% 8.90/5.10 => ( ! [I3: complex] :
% 8.90/5.10 ( ( member_complex @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6698_sum__pos,axiom,
% 8.90/5.10 ! [I5: set_int,F: int > real] :
% 8.90/5.10 ( ( finite_finite_int @ I5 )
% 8.90/5.10 => ( ( I5 != bot_bot_set_int )
% 8.90/5.10 => ( ! [I3: int] :
% 8.90/5.10 ( ( member_int @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6699_sum__pos,axiom,
% 8.90/5.10 ! [I5: set_real,F: real > real] :
% 8.90/5.10 ( ( finite_finite_real @ I5 )
% 8.90/5.10 => ( ( I5 != bot_bot_set_real )
% 8.90/5.10 => ( ! [I3: real] :
% 8.90/5.10 ( ( member_real @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6700_sum__pos,axiom,
% 8.90/5.10 ! [I5: set_complex,F: complex > rat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ I5 )
% 8.90/5.10 => ( ( I5 != bot_bot_set_complex )
% 8.90/5.10 => ( ! [I3: complex] :
% 8.90/5.10 ( ( member_complex @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6701_sum__pos,axiom,
% 8.90/5.10 ! [I5: set_nat,F: nat > rat] :
% 8.90/5.10 ( ( finite_finite_nat @ I5 )
% 8.90/5.10 => ( ( I5 != bot_bot_set_nat )
% 8.90/5.10 => ( ! [I3: nat] :
% 8.90/5.10 ( ( member_nat @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6702_sum__pos,axiom,
% 8.90/5.10 ! [I5: set_int,F: int > rat] :
% 8.90/5.10 ( ( finite_finite_int @ I5 )
% 8.90/5.10 => ( ( I5 != bot_bot_set_int )
% 8.90/5.10 => ( ! [I3: int] :
% 8.90/5.10 ( ( member_int @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6703_sum__pos,axiom,
% 8.90/5.10 ! [I5: set_real,F: real > rat] :
% 8.90/5.10 ( ( finite_finite_real @ I5 )
% 8.90/5.10 => ( ( I5 != bot_bot_set_real )
% 8.90/5.10 => ( ! [I3: real] :
% 8.90/5.10 ( ( member_real @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6704_sum__pos,axiom,
% 8.90/5.10 ! [I5: set_complex,F: complex > nat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ I5 )
% 8.90/5.10 => ( ( I5 != bot_bot_set_complex )
% 8.90/5.10 => ( ! [I3: complex] :
% 8.90/5.10 ( ( member_complex @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6705_sum__pos,axiom,
% 8.90/5.10 ! [I5: set_int,F: int > nat] :
% 8.90/5.10 ( ( finite_finite_int @ I5 )
% 8.90/5.10 => ( ( I5 != bot_bot_set_int )
% 8.90/5.10 => ( ! [I3: int] :
% 8.90/5.10 ( ( member_int @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6706_sum__pos,axiom,
% 8.90/5.10 ! [I5: set_real,F: real > nat] :
% 8.90/5.10 ( ( finite_finite_real @ I5 )
% 8.90/5.10 => ( ( I5 != bot_bot_set_real )
% 8.90/5.10 => ( ! [I3: real] :
% 8.90/5.10 ( ( member_real @ I3 @ I5 )
% 8.90/5.10 => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 8.90/5.10 => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6707_ln__ge__zero__imp__ge__one,axiom,
% 8.90/5.10 ! [X2: real] :
% 8.90/5.10 ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
% 8.90/5.10 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.10 => ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6708_sum_Omono__neutral__cong__right,axiom,
% 8.90/5.10 ! [T3: set_real,S3: set_real,G: real > complex,H2: real > complex] :
% 8.90/5.10 ( ( finite_finite_real @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups5754745047067104278omplex @ G @ T3 )
% 8.90/5.10 = ( groups5754745047067104278omplex @ H2 @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6709_sum_Omono__neutral__cong__right,axiom,
% 8.90/5.10 ! [T3: set_real,S3: set_real,G: real > real,H2: real > real] :
% 8.90/5.10 ( ( finite_finite_real @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups8097168146408367636l_real @ G @ T3 )
% 8.90/5.10 = ( groups8097168146408367636l_real @ H2 @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6710_sum_Omono__neutral__cong__right,axiom,
% 8.90/5.10 ! [T3: set_complex,S3: set_complex,G: complex > real,H2: complex > real] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ T3 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups5808333547571424918x_real @ G @ T3 )
% 8.90/5.10 = ( groups5808333547571424918x_real @ H2 @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6711_sum_Omono__neutral__cong__right,axiom,
% 8.90/5.10 ! [T3: set_real,S3: set_real,G: real > rat,H2: real > rat] :
% 8.90/5.10 ( ( finite_finite_real @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_rat ) )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups1300246762558778688al_rat @ G @ T3 )
% 8.90/5.10 = ( groups1300246762558778688al_rat @ H2 @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6712_sum_Omono__neutral__cong__right,axiom,
% 8.90/5.10 ! [T3: set_complex,S3: set_complex,G: complex > rat,H2: complex > rat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ T3 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_rat ) )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 8.90/5.10 = ( groups5058264527183730370ex_rat @ H2 @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6713_sum_Omono__neutral__cong__right,axiom,
% 8.90/5.10 ! [T3: set_real,S3: set_real,G: real > nat,H2: real > nat] :
% 8.90/5.10 ( ( finite_finite_real @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_nat ) )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups1935376822645274424al_nat @ G @ T3 )
% 8.90/5.10 = ( groups1935376822645274424al_nat @ H2 @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6714_sum_Omono__neutral__cong__right,axiom,
% 8.90/5.10 ! [T3: set_complex,S3: set_complex,G: complex > nat,H2: complex > nat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ T3 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_nat ) )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 8.90/5.10 = ( groups5693394587270226106ex_nat @ H2 @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6715_sum_Omono__neutral__cong__right,axiom,
% 8.90/5.10 ! [T3: set_real,S3: set_real,G: real > int,H2: real > int] :
% 8.90/5.10 ( ( finite_finite_real @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_int ) )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups1932886352136224148al_int @ G @ T3 )
% 8.90/5.10 = ( groups1932886352136224148al_int @ H2 @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6716_sum_Omono__neutral__cong__right,axiom,
% 8.90/5.10 ! [T3: set_complex,S3: set_complex,G: complex > int,H2: complex > int] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ T3 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_int ) )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 8.90/5.10 = ( groups5690904116761175830ex_int @ H2 @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6717_sum_Omono__neutral__cong__right,axiom,
% 8.90/5.10 ! [T3: set_nat,S3: set_nat,G: nat > complex,H2: nat > complex] :
% 8.90/5.10 ( ( finite_finite_nat @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups2073611262835488442omplex @ G @ T3 )
% 8.90/5.10 = ( groups2073611262835488442omplex @ H2 @ S3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6718_sum_Omono__neutral__cong__left,axiom,
% 8.90/5.10 ! [T3: set_real,S3: set_real,H2: real > complex,G: real > complex] :
% 8.90/5.10 ( ( finite_finite_real @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 8.90/5.10 => ( ( H2 @ X5 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups5754745047067104278omplex @ G @ S3 )
% 8.90/5.10 = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6719_sum_Omono__neutral__cong__left,axiom,
% 8.90/5.10 ! [T3: set_real,S3: set_real,H2: real > real,G: real > real] :
% 8.90/5.10 ( ( finite_finite_real @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 8.90/5.10 => ( ( H2 @ X5 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups8097168146408367636l_real @ G @ S3 )
% 8.90/5.10 = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6720_sum_Omono__neutral__cong__left,axiom,
% 8.90/5.10 ! [T3: set_complex,S3: set_complex,H2: complex > real,G: complex > real] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ T3 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 8.90/5.10 => ( ( H2 @ X5 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups5808333547571424918x_real @ G @ S3 )
% 8.90/5.10 = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6721_sum_Omono__neutral__cong__left,axiom,
% 8.90/5.10 ! [T3: set_real,S3: set_real,H2: real > rat,G: real > rat] :
% 8.90/5.10 ( ( finite_finite_real @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 8.90/5.10 => ( ( H2 @ X5 )
% 8.90/5.10 = zero_zero_rat ) )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups1300246762558778688al_rat @ G @ S3 )
% 8.90/5.10 = ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6722_sum_Omono__neutral__cong__left,axiom,
% 8.90/5.10 ! [T3: set_complex,S3: set_complex,H2: complex > rat,G: complex > rat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ T3 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 8.90/5.10 => ( ( H2 @ X5 )
% 8.90/5.10 = zero_zero_rat ) )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups5058264527183730370ex_rat @ G @ S3 )
% 8.90/5.10 = ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6723_sum_Omono__neutral__cong__left,axiom,
% 8.90/5.10 ! [T3: set_real,S3: set_real,H2: real > nat,G: real > nat] :
% 8.90/5.10 ( ( finite_finite_real @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 8.90/5.10 => ( ( H2 @ X5 )
% 8.90/5.10 = zero_zero_nat ) )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups1935376822645274424al_nat @ G @ S3 )
% 8.90/5.10 = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6724_sum_Omono__neutral__cong__left,axiom,
% 8.90/5.10 ! [T3: set_complex,S3: set_complex,H2: complex > nat,G: complex > nat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ T3 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 8.90/5.10 => ( ( H2 @ X5 )
% 8.90/5.10 = zero_zero_nat ) )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups5693394587270226106ex_nat @ G @ S3 )
% 8.90/5.10 = ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6725_sum_Omono__neutral__cong__left,axiom,
% 8.90/5.10 ! [T3: set_real,S3: set_real,H2: real > int,G: real > int] :
% 8.90/5.10 ( ( finite_finite_real @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 8.90/5.10 => ( ( H2 @ X5 )
% 8.90/5.10 = zero_zero_int ) )
% 8.90/5.10 => ( ! [X5: real] :
% 8.90/5.10 ( ( member_real @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups1932886352136224148al_int @ G @ S3 )
% 8.90/5.10 = ( groups1932886352136224148al_int @ H2 @ T3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6726_sum_Omono__neutral__cong__left,axiom,
% 8.90/5.10 ! [T3: set_complex,S3: set_complex,H2: complex > int,G: complex > int] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ T3 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 8.90/5.10 => ( ( H2 @ X5 )
% 8.90/5.10 = zero_zero_int ) )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups5690904116761175830ex_int @ G @ S3 )
% 8.90/5.10 = ( groups5690904116761175830ex_int @ H2 @ T3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6727_sum_Omono__neutral__cong__left,axiom,
% 8.90/5.10 ! [T3: set_nat,S3: set_nat,H2: nat > complex,G: nat > complex] :
% 8.90/5.10 ( ( finite_finite_nat @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 8.90/5.10 => ( ( H2 @ X5 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ S3 )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = ( H2 @ X5 ) ) )
% 8.90/5.10 => ( ( groups2073611262835488442omplex @ G @ S3 )
% 8.90/5.10 = ( groups2073611262835488442omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6728_sum_Omono__neutral__right,axiom,
% 8.90/5.10 ! [T3: set_complex,S3: set_complex,G: complex > real] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ T3 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ( groups5808333547571424918x_real @ G @ T3 )
% 8.90/5.10 = ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6729_sum_Omono__neutral__right,axiom,
% 8.90/5.10 ! [T3: set_complex,S3: set_complex,G: complex > rat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ T3 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_rat ) )
% 8.90/5.10 => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 8.90/5.10 = ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6730_sum_Omono__neutral__right,axiom,
% 8.90/5.10 ! [T3: set_complex,S3: set_complex,G: complex > nat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ T3 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_nat ) )
% 8.90/5.10 => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 8.90/5.10 = ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6731_sum_Omono__neutral__right,axiom,
% 8.90/5.10 ! [T3: set_complex,S3: set_complex,G: complex > int] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ T3 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_int ) )
% 8.90/5.10 => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 8.90/5.10 = ( groups5690904116761175830ex_int @ G @ S3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6732_sum_Omono__neutral__right,axiom,
% 8.90/5.10 ! [T3: set_nat,S3: set_nat,G: nat > complex] :
% 8.90/5.10 ( ( finite_finite_nat @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ( groups2073611262835488442omplex @ G @ T3 )
% 8.90/5.10 = ( groups2073611262835488442omplex @ G @ S3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6733_sum_Omono__neutral__right,axiom,
% 8.90/5.10 ! [T3: set_nat,S3: set_nat,G: nat > rat] :
% 8.90/5.10 ( ( finite_finite_nat @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_rat ) )
% 8.90/5.10 => ( ( groups2906978787729119204at_rat @ G @ T3 )
% 8.90/5.10 = ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6734_sum_Omono__neutral__right,axiom,
% 8.90/5.10 ! [T3: set_nat,S3: set_nat,G: nat > int] :
% 8.90/5.10 ( ( finite_finite_nat @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_int ) )
% 8.90/5.10 => ( ( groups3539618377306564664at_int @ G @ T3 )
% 8.90/5.10 = ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6735_sum_Omono__neutral__right,axiom,
% 8.90/5.10 ! [T3: set_int,S3: set_int,G: int > complex] :
% 8.90/5.10 ( ( finite_finite_int @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ( groups3049146728041665814omplex @ G @ T3 )
% 8.90/5.10 = ( groups3049146728041665814omplex @ G @ S3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6736_sum_Omono__neutral__right,axiom,
% 8.90/5.10 ! [T3: set_int,S3: set_int,G: int > real] :
% 8.90/5.10 ( ( finite_finite_int @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ( groups8778361861064173332t_real @ G @ T3 )
% 8.90/5.10 = ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6737_sum_Omono__neutral__right,axiom,
% 8.90/5.10 ! [T3: set_int,S3: set_int,G: int > rat] :
% 8.90/5.10 ( ( finite_finite_int @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_rat ) )
% 8.90/5.10 => ( ( groups3906332499630173760nt_rat @ G @ T3 )
% 8.90/5.10 = ( groups3906332499630173760nt_rat @ G @ S3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6738_sum_Omono__neutral__left,axiom,
% 8.90/5.10 ! [T3: set_complex,S3: set_complex,G: complex > real] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ T3 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ( groups5808333547571424918x_real @ G @ S3 )
% 8.90/5.10 = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6739_sum_Omono__neutral__left,axiom,
% 8.90/5.10 ! [T3: set_complex,S3: set_complex,G: complex > rat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ T3 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_rat ) )
% 8.90/5.10 => ( ( groups5058264527183730370ex_rat @ G @ S3 )
% 8.90/5.10 = ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6740_sum_Omono__neutral__left,axiom,
% 8.90/5.10 ! [T3: set_complex,S3: set_complex,G: complex > nat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ T3 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_nat ) )
% 8.90/5.10 => ( ( groups5693394587270226106ex_nat @ G @ S3 )
% 8.90/5.10 = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6741_sum_Omono__neutral__left,axiom,
% 8.90/5.10 ! [T3: set_complex,S3: set_complex,G: complex > int] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ T3 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: complex] :
% 8.90/5.10 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_int ) )
% 8.90/5.10 => ( ( groups5690904116761175830ex_int @ G @ S3 )
% 8.90/5.10 = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6742_sum_Omono__neutral__left,axiom,
% 8.90/5.10 ! [T3: set_nat,S3: set_nat,G: nat > complex] :
% 8.90/5.10 ( ( finite_finite_nat @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ( groups2073611262835488442omplex @ G @ S3 )
% 8.90/5.10 = ( groups2073611262835488442omplex @ G @ T3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6743_sum_Omono__neutral__left,axiom,
% 8.90/5.10 ! [T3: set_nat,S3: set_nat,G: nat > rat] :
% 8.90/5.10 ( ( finite_finite_nat @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_rat ) )
% 8.90/5.10 => ( ( groups2906978787729119204at_rat @ G @ S3 )
% 8.90/5.10 = ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6744_sum_Omono__neutral__left,axiom,
% 8.90/5.10 ! [T3: set_nat,S3: set_nat,G: nat > int] :
% 8.90/5.10 ( ( finite_finite_nat @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: nat] :
% 8.90/5.10 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_int ) )
% 8.90/5.10 => ( ( groups3539618377306564664at_int @ G @ S3 )
% 8.90/5.10 = ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6745_sum_Omono__neutral__left,axiom,
% 8.90/5.10 ! [T3: set_int,S3: set_int,G: int > complex] :
% 8.90/5.10 ( ( finite_finite_int @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ( groups3049146728041665814omplex @ G @ S3 )
% 8.90/5.10 = ( groups3049146728041665814omplex @ G @ T3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6746_sum_Omono__neutral__left,axiom,
% 8.90/5.10 ! [T3: set_int,S3: set_int,G: int > real] :
% 8.90/5.10 ( ( finite_finite_int @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ( groups8778361861064173332t_real @ G @ S3 )
% 8.90/5.10 = ( groups8778361861064173332t_real @ G @ T3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6747_sum_Omono__neutral__left,axiom,
% 8.90/5.10 ! [T3: set_int,S3: set_int,G: int > rat] :
% 8.90/5.10 ( ( finite_finite_int @ T3 )
% 8.90/5.10 => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 8.90/5.10 => ( ! [X5: int] :
% 8.90/5.10 ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 8.90/5.10 => ( ( G @ X5 )
% 8.90/5.10 = zero_zero_rat ) )
% 8.90/5.10 => ( ( groups3906332499630173760nt_rat @ G @ S3 )
% 8.90/5.10 = ( groups3906332499630173760nt_rat @ G @ T3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6748_sum_Osame__carrierI,axiom,
% 8.90/5.10 ! [C4: set_real,A2: set_real,B3: set_real,G: real > complex,H2: real > complex] :
% 8.90/5.10 ( ( finite_finite_real @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ( ( groups5754745047067104278omplex @ G @ C4 )
% 8.90/5.10 = ( groups5754745047067104278omplex @ H2 @ C4 ) )
% 8.90/5.10 => ( ( groups5754745047067104278omplex @ G @ A2 )
% 8.90/5.10 = ( groups5754745047067104278omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6749_sum_Osame__carrierI,axiom,
% 8.90/5.10 ! [C4: set_real,A2: set_real,B3: set_real,G: real > real,H2: real > real] :
% 8.90/5.10 ( ( finite_finite_real @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ( ( groups8097168146408367636l_real @ G @ C4 )
% 8.90/5.10 = ( groups8097168146408367636l_real @ H2 @ C4 ) )
% 8.90/5.10 => ( ( groups8097168146408367636l_real @ G @ A2 )
% 8.90/5.10 = ( groups8097168146408367636l_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6750_sum_Osame__carrierI,axiom,
% 8.90/5.10 ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ C4 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: complex] :
% 8.90/5.10 ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [B4: complex] :
% 8.90/5.10 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ( ( groups5808333547571424918x_real @ G @ C4 )
% 8.90/5.10 = ( groups5808333547571424918x_real @ H2 @ C4 ) )
% 8.90/5.10 => ( ( groups5808333547571424918x_real @ G @ A2 )
% 8.90/5.10 = ( groups5808333547571424918x_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6751_sum_Osame__carrierI,axiom,
% 8.90/5.10 ! [C4: set_real,A2: set_real,B3: set_real,G: real > rat,H2: real > rat] :
% 8.90/5.10 ( ( finite_finite_real @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_rat ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_rat ) )
% 8.90/5.10 => ( ( ( groups1300246762558778688al_rat @ G @ C4 )
% 8.90/5.10 = ( groups1300246762558778688al_rat @ H2 @ C4 ) )
% 8.90/5.10 => ( ( groups1300246762558778688al_rat @ G @ A2 )
% 8.90/5.10 = ( groups1300246762558778688al_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6752_sum_Osame__carrierI,axiom,
% 8.90/5.10 ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > rat,H2: complex > rat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ C4 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: complex] :
% 8.90/5.10 ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_rat ) )
% 8.90/5.10 => ( ! [B4: complex] :
% 8.90/5.10 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_rat ) )
% 8.90/5.10 => ( ( ( groups5058264527183730370ex_rat @ G @ C4 )
% 8.90/5.10 = ( groups5058264527183730370ex_rat @ H2 @ C4 ) )
% 8.90/5.10 => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 8.90/5.10 = ( groups5058264527183730370ex_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6753_sum_Osame__carrierI,axiom,
% 8.90/5.10 ! [C4: set_real,A2: set_real,B3: set_real,G: real > nat,H2: real > nat] :
% 8.90/5.10 ( ( finite_finite_real @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_nat ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_nat ) )
% 8.90/5.10 => ( ( ( groups1935376822645274424al_nat @ G @ C4 )
% 8.90/5.10 = ( groups1935376822645274424al_nat @ H2 @ C4 ) )
% 8.90/5.10 => ( ( groups1935376822645274424al_nat @ G @ A2 )
% 8.90/5.10 = ( groups1935376822645274424al_nat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6754_sum_Osame__carrierI,axiom,
% 8.90/5.10 ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > nat,H2: complex > nat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ C4 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: complex] :
% 8.90/5.10 ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_nat ) )
% 8.90/5.10 => ( ! [B4: complex] :
% 8.90/5.10 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_nat ) )
% 8.90/5.10 => ( ( ( groups5693394587270226106ex_nat @ G @ C4 )
% 8.90/5.10 = ( groups5693394587270226106ex_nat @ H2 @ C4 ) )
% 8.90/5.10 => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 8.90/5.10 = ( groups5693394587270226106ex_nat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6755_sum_Osame__carrierI,axiom,
% 8.90/5.10 ! [C4: set_real,A2: set_real,B3: set_real,G: real > int,H2: real > int] :
% 8.90/5.10 ( ( finite_finite_real @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_int ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_int ) )
% 8.90/5.10 => ( ( ( groups1932886352136224148al_int @ G @ C4 )
% 8.90/5.10 = ( groups1932886352136224148al_int @ H2 @ C4 ) )
% 8.90/5.10 => ( ( groups1932886352136224148al_int @ G @ A2 )
% 8.90/5.10 = ( groups1932886352136224148al_int @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6756_sum_Osame__carrierI,axiom,
% 8.90/5.10 ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > int,H2: complex > int] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ C4 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: complex] :
% 8.90/5.10 ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_int ) )
% 8.90/5.10 => ( ! [B4: complex] :
% 8.90/5.10 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_int ) )
% 8.90/5.10 => ( ( ( groups5690904116761175830ex_int @ G @ C4 )
% 8.90/5.10 = ( groups5690904116761175830ex_int @ H2 @ C4 ) )
% 8.90/5.10 => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 8.90/5.10 = ( groups5690904116761175830ex_int @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6757_sum_Osame__carrierI,axiom,
% 8.90/5.10 ! [C4: set_nat,A2: set_nat,B3: set_nat,G: nat > complex,H2: nat > complex] :
% 8.90/5.10 ( ( finite_finite_nat @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_nat @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_nat @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: nat] :
% 8.90/5.10 ( ( member_nat @ A4 @ ( minus_minus_set_nat @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ! [B4: nat] :
% 8.90/5.10 ( ( member_nat @ B4 @ ( minus_minus_set_nat @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ( ( groups2073611262835488442omplex @ G @ C4 )
% 8.90/5.10 = ( groups2073611262835488442omplex @ H2 @ C4 ) )
% 8.90/5.10 => ( ( groups2073611262835488442omplex @ G @ A2 )
% 8.90/5.10 = ( groups2073611262835488442omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6758_sum_Osame__carrier,axiom,
% 8.90/5.10 ! [C4: set_real,A2: set_real,B3: set_real,G: real > complex,H2: real > complex] :
% 8.90/5.10 ( ( finite_finite_real @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ( ( groups5754745047067104278omplex @ G @ A2 )
% 8.90/5.10 = ( groups5754745047067104278omplex @ H2 @ B3 ) )
% 8.90/5.10 = ( ( groups5754745047067104278omplex @ G @ C4 )
% 8.90/5.10 = ( groups5754745047067104278omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6759_sum_Osame__carrier,axiom,
% 8.90/5.10 ! [C4: set_real,A2: set_real,B3: set_real,G: real > real,H2: real > real] :
% 8.90/5.10 ( ( finite_finite_real @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ( ( groups8097168146408367636l_real @ G @ A2 )
% 8.90/5.10 = ( groups8097168146408367636l_real @ H2 @ B3 ) )
% 8.90/5.10 = ( ( groups8097168146408367636l_real @ G @ C4 )
% 8.90/5.10 = ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6760_sum_Osame__carrier,axiom,
% 8.90/5.10 ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ C4 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: complex] :
% 8.90/5.10 ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ! [B4: complex] :
% 8.90/5.10 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_real ) )
% 8.90/5.10 => ( ( ( groups5808333547571424918x_real @ G @ A2 )
% 8.90/5.10 = ( groups5808333547571424918x_real @ H2 @ B3 ) )
% 8.90/5.10 = ( ( groups5808333547571424918x_real @ G @ C4 )
% 8.90/5.10 = ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6761_sum_Osame__carrier,axiom,
% 8.90/5.10 ! [C4: set_real,A2: set_real,B3: set_real,G: real > rat,H2: real > rat] :
% 8.90/5.10 ( ( finite_finite_real @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_rat ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_rat ) )
% 8.90/5.10 => ( ( ( groups1300246762558778688al_rat @ G @ A2 )
% 8.90/5.10 = ( groups1300246762558778688al_rat @ H2 @ B3 ) )
% 8.90/5.10 = ( ( groups1300246762558778688al_rat @ G @ C4 )
% 8.90/5.10 = ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6762_sum_Osame__carrier,axiom,
% 8.90/5.10 ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > rat,H2: complex > rat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ C4 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: complex] :
% 8.90/5.10 ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_rat ) )
% 8.90/5.10 => ( ! [B4: complex] :
% 8.90/5.10 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_rat ) )
% 8.90/5.10 => ( ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 8.90/5.10 = ( groups5058264527183730370ex_rat @ H2 @ B3 ) )
% 8.90/5.10 = ( ( groups5058264527183730370ex_rat @ G @ C4 )
% 8.90/5.10 = ( groups5058264527183730370ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6763_sum_Osame__carrier,axiom,
% 8.90/5.10 ! [C4: set_real,A2: set_real,B3: set_real,G: real > nat,H2: real > nat] :
% 8.90/5.10 ( ( finite_finite_real @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_nat ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_nat ) )
% 8.90/5.10 => ( ( ( groups1935376822645274424al_nat @ G @ A2 )
% 8.90/5.10 = ( groups1935376822645274424al_nat @ H2 @ B3 ) )
% 8.90/5.10 = ( ( groups1935376822645274424al_nat @ G @ C4 )
% 8.90/5.10 = ( groups1935376822645274424al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6764_sum_Osame__carrier,axiom,
% 8.90/5.10 ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > nat,H2: complex > nat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ C4 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: complex] :
% 8.90/5.10 ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_nat ) )
% 8.90/5.10 => ( ! [B4: complex] :
% 8.90/5.10 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_nat ) )
% 8.90/5.10 => ( ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 8.90/5.10 = ( groups5693394587270226106ex_nat @ H2 @ B3 ) )
% 8.90/5.10 = ( ( groups5693394587270226106ex_nat @ G @ C4 )
% 8.90/5.10 = ( groups5693394587270226106ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6765_sum_Osame__carrier,axiom,
% 8.90/5.10 ! [C4: set_real,A2: set_real,B3: set_real,G: real > int,H2: real > int] :
% 8.90/5.10 ( ( finite_finite_real @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: real] :
% 8.90/5.10 ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_int ) )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_int ) )
% 8.90/5.10 => ( ( ( groups1932886352136224148al_int @ G @ A2 )
% 8.90/5.10 = ( groups1932886352136224148al_int @ H2 @ B3 ) )
% 8.90/5.10 = ( ( groups1932886352136224148al_int @ G @ C4 )
% 8.90/5.10 = ( groups1932886352136224148al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6766_sum_Osame__carrier,axiom,
% 8.90/5.10 ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > int,H2: complex > int] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ C4 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: complex] :
% 8.90/5.10 ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_int ) )
% 8.90/5.10 => ( ! [B4: complex] :
% 8.90/5.10 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_int ) )
% 8.90/5.10 => ( ( ( groups5690904116761175830ex_int @ G @ A2 )
% 8.90/5.10 = ( groups5690904116761175830ex_int @ H2 @ B3 ) )
% 8.90/5.10 = ( ( groups5690904116761175830ex_int @ G @ C4 )
% 8.90/5.10 = ( groups5690904116761175830ex_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6767_sum_Osame__carrier,axiom,
% 8.90/5.10 ! [C4: set_nat,A2: set_nat,B3: set_nat,G: nat > complex,H2: nat > complex] :
% 8.90/5.10 ( ( finite_finite_nat @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_nat @ A2 @ C4 )
% 8.90/5.10 => ( ( ord_less_eq_set_nat @ B3 @ C4 )
% 8.90/5.10 => ( ! [A4: nat] :
% 8.90/5.10 ( ( member_nat @ A4 @ ( minus_minus_set_nat @ C4 @ A2 ) )
% 8.90/5.10 => ( ( G @ A4 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ! [B4: nat] :
% 8.90/5.10 ( ( member_nat @ B4 @ ( minus_minus_set_nat @ C4 @ B3 ) )
% 8.90/5.10 => ( ( H2 @ B4 )
% 8.90/5.10 = zero_zero_complex ) )
% 8.90/5.10 => ( ( ( groups2073611262835488442omplex @ G @ A2 )
% 8.90/5.10 = ( groups2073611262835488442omplex @ H2 @ B3 ) )
% 8.90/5.10 = ( ( groups2073611262835488442omplex @ G @ C4 )
% 8.90/5.10 = ( groups2073611262835488442omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6768_sum_Osubset__diff,axiom,
% 8.90/5.10 ! [B3: set_complex,A2: set_complex,G: complex > real] :
% 8.90/5.10 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 8.90/5.10 => ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ( groups5808333547571424918x_real @ G @ A2 )
% 8.90/5.10 = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5808333547571424918x_real @ G @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6769_sum_Osubset__diff,axiom,
% 8.90/5.10 ! [B3: set_complex,A2: set_complex,G: complex > rat] :
% 8.90/5.10 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 8.90/5.10 => ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 8.90/5.10 = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5058264527183730370ex_rat @ G @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6770_sum_Osubset__diff,axiom,
% 8.90/5.10 ! [B3: set_complex,A2: set_complex,G: complex > nat] :
% 8.90/5.10 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 8.90/5.10 => ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 8.90/5.10 = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5693394587270226106ex_nat @ G @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6771_sum_Osubset__diff,axiom,
% 8.90/5.10 ! [B3: set_complex,A2: set_complex,G: complex > int] :
% 8.90/5.10 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 8.90/5.10 => ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 8.90/5.10 = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5690904116761175830ex_int @ G @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6772_sum_Osubset__diff,axiom,
% 8.90/5.10 ! [B3: set_nat,A2: set_nat,G: nat > rat] :
% 8.90/5.10 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 8.90/5.10 => ( ( finite_finite_nat @ A2 )
% 8.90/5.10 => ( ( groups2906978787729119204at_rat @ G @ A2 )
% 8.90/5.10 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups2906978787729119204at_rat @ G @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6773_sum_Osubset__diff,axiom,
% 8.90/5.10 ! [B3: set_nat,A2: set_nat,G: nat > int] :
% 8.90/5.10 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 8.90/5.10 => ( ( finite_finite_nat @ A2 )
% 8.90/5.10 => ( ( groups3539618377306564664at_int @ G @ A2 )
% 8.90/5.10 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups3539618377306564664at_int @ G @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6774_sum_Osubset__diff,axiom,
% 8.90/5.10 ! [B3: set_int,A2: set_int,G: int > real] :
% 8.90/5.10 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 8.90/5.10 => ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ( groups8778361861064173332t_real @ G @ A2 )
% 8.90/5.10 = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups8778361861064173332t_real @ G @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6775_sum_Osubset__diff,axiom,
% 8.90/5.10 ! [B3: set_int,A2: set_int,G: int > rat] :
% 8.90/5.10 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 8.90/5.10 => ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 8.90/5.10 = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups3906332499630173760nt_rat @ G @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6776_sum_Osubset__diff,axiom,
% 8.90/5.10 ! [B3: set_int,A2: set_int,G: int > nat] :
% 8.90/5.10 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 8.90/5.10 => ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 8.90/5.10 = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups4541462559716669496nt_nat @ G @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6777_sum_Osubset__diff,axiom,
% 8.90/5.10 ! [B3: set_int,A2: set_int,G: int > int] :
% 8.90/5.10 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 8.90/5.10 => ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ( groups4538972089207619220nt_int @ G @ A2 )
% 8.90/5.10 = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups4538972089207619220nt_int @ G @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6778_sum__diff,axiom,
% 8.90/5.10 ! [A2: set_complex,B3: set_complex,F: complex > real] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 8.90/5.10 => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 8.90/5.10 = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6779_sum__diff,axiom,
% 8.90/5.10 ! [A2: set_complex,B3: set_complex,F: complex > rat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 8.90/5.10 => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 8.90/5.10 = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6780_sum__diff,axiom,
% 8.90/5.10 ! [A2: set_complex,B3: set_complex,F: complex > int] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 8.90/5.10 => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 8.90/5.10 = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6781_sum__diff,axiom,
% 8.90/5.10 ! [A2: set_nat,B3: set_nat,F: nat > rat] :
% 8.90/5.10 ( ( finite_finite_nat @ A2 )
% 8.90/5.10 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 8.90/5.10 => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 8.90/5.10 = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6782_sum__diff,axiom,
% 8.90/5.10 ! [A2: set_nat,B3: set_nat,F: nat > int] :
% 8.90/5.10 ( ( finite_finite_nat @ A2 )
% 8.90/5.10 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 8.90/5.10 => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 8.90/5.10 = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6783_sum__diff,axiom,
% 8.90/5.10 ! [A2: set_int,B3: set_int,F: int > real] :
% 8.90/5.10 ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 8.90/5.10 => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
% 8.90/5.10 = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6784_sum__diff,axiom,
% 8.90/5.10 ! [A2: set_int,B3: set_int,F: int > rat] :
% 8.90/5.10 ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 8.90/5.10 => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
% 8.90/5.10 = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6785_sum__diff,axiom,
% 8.90/5.10 ! [A2: set_int,B3: set_int,F: int > int] :
% 8.90/5.10 ( ( finite_finite_int @ A2 )
% 8.90/5.10 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 8.90/5.10 => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
% 8.90/5.10 = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ F @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6786_sum__diff,axiom,
% 8.90/5.10 ! [A2: set_complex,B3: set_complex,F: complex > complex] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 8.90/5.10 => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 8.90/5.10 = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ F @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6787_sum__diff,axiom,
% 8.90/5.10 ! [A2: set_nat,B3: set_nat,F: nat > real] :
% 8.90/5.10 ( ( finite_finite_nat @ A2 )
% 8.90/5.10 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 8.90/5.10 => ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 8.90/5.10 = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6788_ln__add__one__self__le__self,axiom,
% 8.90/5.10 ! [X2: real] :
% 8.90/5.10 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.10 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6789_ln__mult,axiom,
% 8.90/5.10 ! [X2: real,Y3: real] :
% 8.90/5.10 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.10 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.90/5.10 => ( ( ln_ln_real @ ( times_times_real @ X2 @ Y3 ) )
% 8.90/5.10 = ( plus_plus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6790_ln__eq__minus__one,axiom,
% 8.90/5.10 ! [X2: real] :
% 8.90/5.10 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.10 => ( ( ( ln_ln_real @ X2 )
% 8.90/5.10 = ( minus_minus_real @ X2 @ one_one_real ) )
% 8.90/5.10 => ( X2 = one_one_real ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6791_ln__div,axiom,
% 8.90/5.10 ! [X2: real,Y3: real] :
% 8.90/5.10 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.10 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.90/5.10 => ( ( ln_ln_real @ ( divide_divide_real @ X2 @ Y3 ) )
% 8.90/5.10 = ( minus_minus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6792_sum__mono2,axiom,
% 8.90/5.10 ! [B3: set_real,A2: set_real,F: real > real] :
% 8.90/5.10 ( ( finite_finite_real @ B3 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A2 ) )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
% 8.90/5.10 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6793_sum__mono2,axiom,
% 8.90/5.10 ! [B3: set_complex,A2: set_complex,F: complex > real] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ B3 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 8.90/5.10 => ( ! [B4: complex] :
% 8.90/5.10 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 8.90/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
% 8.90/5.10 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6794_sum__mono2,axiom,
% 8.90/5.10 ! [B3: set_real,A2: set_real,F: real > rat] :
% 8.90/5.10 ( ( finite_finite_real @ B3 )
% 8.90/5.10 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 8.90/5.10 => ( ! [B4: real] :
% 8.90/5.10 ( ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A2 ) )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
% 8.90/5.10 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6795_sum__mono2,axiom,
% 8.90/5.10 ! [B3: set_complex,A2: set_complex,F: complex > rat] :
% 8.90/5.10 ( ( finite3207457112153483333omplex @ B3 )
% 8.90/5.10 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 8.90/5.10 => ( ! [B4: complex] :
% 8.90/5.10 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
% 8.90/5.10 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6796_sum__mono2,axiom,
% 8.90/5.10 ! [B3: set_nat,A2: set_nat,F: nat > rat] :
% 8.90/5.10 ( ( finite_finite_nat @ B3 )
% 8.90/5.10 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 8.90/5.10 => ( ! [B4: nat] :
% 8.90/5.10 ( ( member_nat @ B4 @ ( minus_minus_set_nat @ B3 @ A2 ) )
% 8.90/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
% 8.90/5.10 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B3 ) ) ) ) ) ).
% 8.90/5.10
% 8.90/5.10 thf(fact_6797_sum__mono2,axiom,
% 8.90/5.10 ! [B3: set_real,A2: set_real,F: real > nat] :
% 8.90/5.11 ( ( finite_finite_real @ B3 )
% 8.90/5.11 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 8.90/5.11 => ( ! [B4: real] :
% 8.90/5.11 ( ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A2 ) )
% 8.90/5.11 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B4 ) ) )
% 8.90/5.11 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B3 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6798_sum__mono2,axiom,
% 8.90/5.11 ! [B3: set_complex,A2: set_complex,F: complex > nat] :
% 8.90/5.11 ( ( finite3207457112153483333omplex @ B3 )
% 8.90/5.11 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 8.90/5.11 => ( ! [B4: complex] :
% 8.90/5.11 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 8.90/5.11 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B4 ) ) )
% 8.90/5.11 => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B3 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6799_sum__mono2,axiom,
% 8.90/5.11 ! [B3: set_real,A2: set_real,F: real > int] :
% 8.90/5.11 ( ( finite_finite_real @ B3 )
% 8.90/5.11 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 8.90/5.11 => ( ! [B4: real] :
% 8.90/5.11 ( ( member_real @ B4 @ ( minus_minus_set_real @ B3 @ A2 ) )
% 8.90/5.11 => ( ord_less_eq_int @ zero_zero_int @ ( F @ B4 ) ) )
% 8.90/5.11 => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B3 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6800_sum__mono2,axiom,
% 8.90/5.11 ! [B3: set_complex,A2: set_complex,F: complex > int] :
% 8.90/5.11 ( ( finite3207457112153483333omplex @ B3 )
% 8.90/5.11 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 8.90/5.11 => ( ! [B4: complex] :
% 8.90/5.11 ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 8.90/5.11 => ( ord_less_eq_int @ zero_zero_int @ ( F @ B4 ) ) )
% 8.90/5.11 => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B3 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6801_sum__mono2,axiom,
% 8.90/5.11 ! [B3: set_nat,A2: set_nat,F: nat > int] :
% 8.90/5.11 ( ( finite_finite_nat @ B3 )
% 8.90/5.11 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 8.90/5.11 => ( ! [B4: nat] :
% 8.90/5.11 ( ( member_nat @ B4 @ ( minus_minus_set_nat @ B3 @ A2 ) )
% 8.90/5.11 => ( ord_less_eq_int @ zero_zero_int @ ( F @ B4 ) ) )
% 8.90/5.11 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B3 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6802_ln__2__less__1,axiom,
% 8.90/5.11 ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6803_ln__le__minus__one,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.11 => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6804_ln__diff__le,axiom,
% 8.90/5.11 ! [X2: real,Y3: real] :
% 8.90/5.11 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.11 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.90/5.11 => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X2 @ Y3 ) @ Y3 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6805_ln__add__one__self__le__self2,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.11 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6806_sum__strict__mono2,axiom,
% 8.90/5.11 ! [B3: set_real,A2: set_real,B: real,F: real > real] :
% 8.90/5.11 ( ( finite_finite_real @ B3 )
% 8.90/5.11 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 8.90/5.11 => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
% 8.90/5.11 => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 8.90/5.11 => ( ! [X5: real] :
% 8.90/5.11 ( ( member_real @ X5 @ B3 )
% 8.90/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 8.90/5.11 => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B3 ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6807_sum__strict__mono2,axiom,
% 8.90/5.11 ! [B3: set_complex,A2: set_complex,B: complex,F: complex > real] :
% 8.90/5.11 ( ( finite3207457112153483333omplex @ B3 )
% 8.90/5.11 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 8.90/5.11 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 8.90/5.11 => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 8.90/5.11 => ( ! [X5: complex] :
% 8.90/5.11 ( ( member_complex @ X5 @ B3 )
% 8.90/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 8.90/5.11 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6808_sum__strict__mono2,axiom,
% 8.90/5.11 ! [B3: set_real,A2: set_real,B: real,F: real > rat] :
% 8.90/5.11 ( ( finite_finite_real @ B3 )
% 8.90/5.11 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 8.90/5.11 => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
% 8.90/5.11 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 8.90/5.11 => ( ! [X5: real] :
% 8.90/5.11 ( ( member_real @ X5 @ B3 )
% 8.90/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 8.90/5.11 => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B3 ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6809_sum__strict__mono2,axiom,
% 8.90/5.11 ! [B3: set_complex,A2: set_complex,B: complex,F: complex > rat] :
% 8.90/5.11 ( ( finite3207457112153483333omplex @ B3 )
% 8.90/5.11 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 8.90/5.11 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 8.90/5.11 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 8.90/5.11 => ( ! [X5: complex] :
% 8.90/5.11 ( ( member_complex @ X5 @ B3 )
% 8.90/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 8.90/5.11 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6810_sum__strict__mono2,axiom,
% 8.90/5.11 ! [B3: set_nat,A2: set_nat,B: nat,F: nat > rat] :
% 8.90/5.11 ( ( finite_finite_nat @ B3 )
% 8.90/5.11 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 8.90/5.11 => ( ( member_nat @ B @ ( minus_minus_set_nat @ B3 @ A2 ) )
% 8.90/5.11 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 8.90/5.11 => ( ! [X5: nat] :
% 8.90/5.11 ( ( member_nat @ X5 @ B3 )
% 8.90/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 8.90/5.11 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B3 ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6811_sum__strict__mono2,axiom,
% 8.90/5.11 ! [B3: set_real,A2: set_real,B: real,F: real > nat] :
% 8.90/5.11 ( ( finite_finite_real @ B3 )
% 8.90/5.11 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 8.90/5.11 => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
% 8.90/5.11 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 8.90/5.11 => ( ! [X5: real] :
% 8.90/5.11 ( ( member_real @ X5 @ B3 )
% 8.90/5.11 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 8.90/5.11 => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B3 ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6812_sum__strict__mono2,axiom,
% 8.90/5.11 ! [B3: set_complex,A2: set_complex,B: complex,F: complex > nat] :
% 8.90/5.11 ( ( finite3207457112153483333omplex @ B3 )
% 8.90/5.11 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 8.90/5.11 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 8.90/5.11 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 8.90/5.11 => ( ! [X5: complex] :
% 8.90/5.11 ( ( member_complex @ X5 @ B3 )
% 8.90/5.11 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 8.90/5.11 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B3 ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6813_sum__strict__mono2,axiom,
% 8.90/5.11 ! [B3: set_real,A2: set_real,B: real,F: real > int] :
% 8.90/5.11 ( ( finite_finite_real @ B3 )
% 8.90/5.11 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 8.90/5.11 => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
% 8.90/5.11 => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 8.90/5.11 => ( ! [X5: real] :
% 8.90/5.11 ( ( member_real @ X5 @ B3 )
% 8.90/5.11 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
% 8.90/5.11 => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B3 ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6814_sum__strict__mono2,axiom,
% 8.90/5.11 ! [B3: set_complex,A2: set_complex,B: complex,F: complex > int] :
% 8.90/5.11 ( ( finite3207457112153483333omplex @ B3 )
% 8.90/5.11 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 8.90/5.11 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 8.90/5.11 => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 8.90/5.11 => ( ! [X5: complex] :
% 8.90/5.11 ( ( member_complex @ X5 @ B3 )
% 8.90/5.11 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
% 8.90/5.11 => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B3 ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6815_sum__strict__mono2,axiom,
% 8.90/5.11 ! [B3: set_nat,A2: set_nat,B: nat,F: nat > int] :
% 8.90/5.11 ( ( finite_finite_nat @ B3 )
% 8.90/5.11 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 8.90/5.11 => ( ( member_nat @ B @ ( minus_minus_set_nat @ B3 @ A2 ) )
% 8.90/5.11 => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 8.90/5.11 => ( ! [X5: nat] :
% 8.90/5.11 ( ( member_nat @ X5 @ B3 )
% 8.90/5.11 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
% 8.90/5.11 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B3 ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6816_ln__one__minus__pos__upper__bound,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.11 => ( ( ord_less_real @ X2 @ one_one_real )
% 8.90/5.11 => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X2 ) ) @ ( uminus_uminus_real @ X2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6817_exists__least__lemma,axiom,
% 8.90/5.11 ! [P: nat > $o] :
% 8.90/5.11 ( ~ ( P @ zero_zero_nat )
% 8.90/5.11 => ( ? [X_1: nat] : ( P @ X_1 )
% 8.90/5.11 => ? [N2: nat] :
% 8.90/5.11 ( ~ ( P @ N2 )
% 8.90/5.11 & ( P @ ( suc @ N2 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6818_ex__le__of__int,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ? [Z2: int] : ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6819_ex__le__of__int,axiom,
% 8.90/5.11 ! [X2: rat] :
% 8.90/5.11 ? [Z2: int] : ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6820_ex__of__int__less,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ? [Z2: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X2 ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6821_ex__of__int__less,axiom,
% 8.90/5.11 ! [X2: rat] :
% 8.90/5.11 ? [Z2: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X2 ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6822_ex__less__of__int,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ? [Z2: int] : ( ord_less_real @ X2 @ ( ring_1_of_int_real @ Z2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6823_ex__less__of__int,axiom,
% 8.90/5.11 ! [X2: rat] :
% 8.90/5.11 ? [Z2: int] : ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ Z2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6824_ln__one__plus__pos__lower__bound,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.11 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 8.90/5.11 => ( ord_less_eq_real @ ( minus_minus_real @ X2 @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6825_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 8.90/5.11 => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 8.90/5.11 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6826_round__unique,axiom,
% 8.90/5.11 ! [X2: real,Y3: int] :
% 8.90/5.11 ( ( ord_less_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y3 ) )
% 8.90/5.11 => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y3 ) @ ( plus_plus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 8.90/5.11 => ( ( archim8280529875227126926d_real @ X2 )
% 8.90/5.11 = Y3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6827_round__unique,axiom,
% 8.90/5.11 ! [X2: rat,Y3: int] :
% 8.90/5.11 ( ( ord_less_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y3 ) )
% 8.90/5.11 => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y3 ) @ ( plus_plus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.11 => ( ( archim7778729529865785530nd_rat @ X2 )
% 8.90/5.11 = Y3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6828_tanh__ln__real,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.11 => ( ( tanh_real @ ( ln_ln_real @ X2 ) )
% 8.90/5.11 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6829_pred__subset__eq2,axiom,
% 8.90/5.11 ! [R: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
% 8.90/5.11 ( ( ord_le2162486998276636481er_o_o
% 8.90/5.11 @ ^ [X: code_integer,Y: $o] : ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X @ Y ) @ R )
% 8.90/5.11 @ ^ [X: code_integer,Y: $o] : ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X @ Y ) @ S3 ) )
% 8.90/5.11 = ( ord_le8980329558974975238eger_o @ R @ S3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6830_pred__subset__eq2,axiom,
% 8.90/5.11 ! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
% 8.90/5.11 ( ( ord_le6124364862034508274_num_o
% 8.90/5.11 @ ^ [X: num,Y: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X @ Y ) @ R )
% 8.90/5.11 @ ^ [X: num,Y: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X @ Y ) @ S3 ) )
% 8.90/5.11 = ( ord_le880128212290418581um_num @ R @ S3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6831_pred__subset__eq2,axiom,
% 8.90/5.11 ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
% 8.90/5.11 ( ( ord_le3404735783095501756_num_o
% 8.90/5.11 @ ^ [X: nat,Y: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ R )
% 8.90/5.11 @ ^ [X: nat,Y: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ S3 ) )
% 8.90/5.11 = ( ord_le8085105155179020875at_num @ R @ S3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6832_pred__subset__eq2,axiom,
% 8.90/5.11 ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
% 8.90/5.11 ( ( ord_le2646555220125990790_nat_o
% 8.90/5.11 @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
% 8.90/5.11 @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S3 ) )
% 8.90/5.11 = ( ord_le3146513528884898305at_nat @ R @ S3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6833_pred__subset__eq2,axiom,
% 8.90/5.11 ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
% 8.90/5.11 ( ( ord_le6741204236512500942_int_o
% 8.90/5.11 @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
% 8.90/5.11 @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S3 ) )
% 8.90/5.11 = ( ord_le2843351958646193337nt_int @ R @ S3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6834_abs__ln__one__plus__x__minus__x__bound,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.11 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6835_accp__subset,axiom,
% 8.90/5.11 ! [R1: product_prod_num_num > product_prod_num_num > $o,R22: product_prod_num_num > product_prod_num_num > $o] :
% 8.90/5.11 ( ( ord_le2556027599737686990_num_o @ R1 @ R22 )
% 8.90/5.11 => ( ord_le2239182809043710856_num_o @ ( accp_P3113834385874906142um_num @ R22 ) @ ( accp_P3113834385874906142um_num @ R1 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6836_accp__subset,axiom,
% 8.90/5.11 ! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 8.90/5.11 ( ( ord_le5604493270027003598_nat_o @ R1 @ R22 )
% 8.90/5.11 => ( ord_le704812498762024988_nat_o @ ( accp_P4275260045618599050at_nat @ R22 ) @ ( accp_P4275260045618599050at_nat @ R1 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6837_accp__subset,axiom,
% 8.90/5.11 ! [R1: product_prod_int_int > product_prod_int_int > $o,R22: product_prod_int_int > product_prod_int_int > $o] :
% 8.90/5.11 ( ( ord_le1598226405681992910_int_o @ R1 @ R22 )
% 8.90/5.11 => ( ord_le8369615600986905444_int_o @ ( accp_P1096762738010456898nt_int @ R22 ) @ ( accp_P1096762738010456898nt_int @ R1 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6838_accp__subset,axiom,
% 8.90/5.11 ! [R1: list_nat > list_nat > $o,R22: list_nat > list_nat > $o] :
% 8.90/5.11 ( ( ord_le6558929396352911974_nat_o @ R1 @ R22 )
% 8.90/5.11 => ( ord_le1520216061033275535_nat_o @ ( accp_list_nat @ R22 ) @ ( accp_list_nat @ R1 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6839_accp__subset,axiom,
% 8.90/5.11 ! [R1: nat > nat > $o,R22: nat > nat > $o] :
% 8.90/5.11 ( ( ord_le2646555220125990790_nat_o @ R1 @ R22 )
% 8.90/5.11 => ( ord_less_eq_nat_o @ ( accp_nat @ R22 ) @ ( accp_nat @ R1 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6840_abs__idempotent,axiom,
% 8.90/5.11 ! [A: real] :
% 8.90/5.11 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 8.90/5.11 = ( abs_abs_real @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6841_abs__idempotent,axiom,
% 8.90/5.11 ! [A: int] :
% 8.90/5.11 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 8.90/5.11 = ( abs_abs_int @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6842_abs__idempotent,axiom,
% 8.90/5.11 ! [A: code_integer] :
% 8.90/5.11 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 8.90/5.11 = ( abs_abs_Code_integer @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6843_abs__abs,axiom,
% 8.90/5.11 ! [A: real] :
% 8.90/5.11 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 8.90/5.11 = ( abs_abs_real @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6844_abs__abs,axiom,
% 8.90/5.11 ! [A: int] :
% 8.90/5.11 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 8.90/5.11 = ( abs_abs_int @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6845_abs__abs,axiom,
% 8.90/5.11 ! [A: code_integer] :
% 8.90/5.11 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 8.90/5.11 = ( abs_abs_Code_integer @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6846_abs__zero,axiom,
% 8.90/5.11 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 8.90/5.11 = zero_z3403309356797280102nteger ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6847_abs__zero,axiom,
% 8.90/5.11 ( ( abs_abs_real @ zero_zero_real )
% 8.90/5.11 = zero_zero_real ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6848_abs__zero,axiom,
% 8.90/5.11 ( ( abs_abs_rat @ zero_zero_rat )
% 8.90/5.11 = zero_zero_rat ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6849_abs__zero,axiom,
% 8.90/5.11 ( ( abs_abs_int @ zero_zero_int )
% 8.90/5.11 = zero_zero_int ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6850_abs__eq__0,axiom,
% 8.90/5.11 ! [A: code_integer] :
% 8.90/5.11 ( ( ( abs_abs_Code_integer @ A )
% 8.90/5.11 = zero_z3403309356797280102nteger )
% 8.90/5.11 = ( A = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6851_abs__eq__0,axiom,
% 8.90/5.11 ! [A: real] :
% 8.90/5.11 ( ( ( abs_abs_real @ A )
% 8.90/5.11 = zero_zero_real )
% 8.90/5.11 = ( A = zero_zero_real ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6852_abs__eq__0,axiom,
% 8.90/5.11 ! [A: rat] :
% 8.90/5.11 ( ( ( abs_abs_rat @ A )
% 8.90/5.11 = zero_zero_rat )
% 8.90/5.11 = ( A = zero_zero_rat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6853_abs__eq__0,axiom,
% 8.90/5.11 ! [A: int] :
% 8.90/5.11 ( ( ( abs_abs_int @ A )
% 8.90/5.11 = zero_zero_int )
% 8.90/5.11 = ( A = zero_zero_int ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6854_abs__0__eq,axiom,
% 8.90/5.11 ! [A: code_integer] :
% 8.90/5.11 ( ( zero_z3403309356797280102nteger
% 8.90/5.11 = ( abs_abs_Code_integer @ A ) )
% 8.90/5.11 = ( A = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6855_abs__0__eq,axiom,
% 8.90/5.11 ! [A: real] :
% 8.90/5.11 ( ( zero_zero_real
% 8.90/5.11 = ( abs_abs_real @ A ) )
% 8.90/5.11 = ( A = zero_zero_real ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6856_abs__0__eq,axiom,
% 8.90/5.11 ! [A: rat] :
% 8.90/5.11 ( ( zero_zero_rat
% 8.90/5.11 = ( abs_abs_rat @ A ) )
% 8.90/5.11 = ( A = zero_zero_rat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6857_abs__0__eq,axiom,
% 8.90/5.11 ! [A: int] :
% 8.90/5.11 ( ( zero_zero_int
% 8.90/5.11 = ( abs_abs_int @ A ) )
% 8.90/5.11 = ( A = zero_zero_int ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6858_abs__0,axiom,
% 8.90/5.11 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 8.90/5.11 = zero_z3403309356797280102nteger ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6859_abs__0,axiom,
% 8.90/5.11 ( ( abs_abs_complex @ zero_zero_complex )
% 8.90/5.11 = zero_zero_complex ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6860_abs__0,axiom,
% 8.90/5.11 ( ( abs_abs_real @ zero_zero_real )
% 8.90/5.11 = zero_zero_real ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6861_abs__0,axiom,
% 8.90/5.11 ( ( abs_abs_rat @ zero_zero_rat )
% 8.90/5.11 = zero_zero_rat ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6862_abs__0,axiom,
% 8.90/5.11 ( ( abs_abs_int @ zero_zero_int )
% 8.90/5.11 = zero_zero_int ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6863_abs__numeral,axiom,
% 8.90/5.11 ! [N: num] :
% 8.90/5.11 ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
% 8.90/5.11 = ( numera6620942414471956472nteger @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6864_abs__numeral,axiom,
% 8.90/5.11 ! [N: num] :
% 8.90/5.11 ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
% 8.90/5.11 = ( numeral_numeral_real @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6865_abs__numeral,axiom,
% 8.90/5.11 ! [N: num] :
% 8.90/5.11 ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
% 8.90/5.11 = ( numeral_numeral_rat @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6866_abs__numeral,axiom,
% 8.90/5.11 ! [N: num] :
% 8.90/5.11 ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
% 8.90/5.11 = ( numeral_numeral_int @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6867_abs__mult__self__eq,axiom,
% 8.90/5.11 ! [A: code_integer] :
% 8.90/5.11 ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 8.90/5.11 = ( times_3573771949741848930nteger @ A @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6868_abs__mult__self__eq,axiom,
% 8.90/5.11 ! [A: real] :
% 8.90/5.11 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
% 8.90/5.11 = ( times_times_real @ A @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6869_abs__mult__self__eq,axiom,
% 8.90/5.11 ! [A: rat] :
% 8.90/5.11 ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
% 8.90/5.11 = ( times_times_rat @ A @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6870_abs__mult__self__eq,axiom,
% 8.90/5.11 ! [A: int] :
% 8.90/5.11 ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
% 8.90/5.11 = ( times_times_int @ A @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6871_abs__1,axiom,
% 8.90/5.11 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 8.90/5.11 = one_one_Code_integer ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6872_abs__1,axiom,
% 8.90/5.11 ( ( abs_abs_complex @ one_one_complex )
% 8.90/5.11 = one_one_complex ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6873_abs__1,axiom,
% 8.90/5.11 ( ( abs_abs_real @ one_one_real )
% 8.90/5.11 = one_one_real ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6874_abs__1,axiom,
% 8.90/5.11 ( ( abs_abs_rat @ one_one_rat )
% 8.90/5.11 = one_one_rat ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6875_abs__1,axiom,
% 8.90/5.11 ( ( abs_abs_int @ one_one_int )
% 8.90/5.11 = one_one_int ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6876_abs__add__abs,axiom,
% 8.90/5.11 ! [A: code_integer,B: code_integer] :
% 8.90/5.11 ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
% 8.90/5.11 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6877_abs__add__abs,axiom,
% 8.90/5.11 ! [A: real,B: real] :
% 8.90/5.11 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 8.90/5.11 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6878_abs__add__abs,axiom,
% 8.90/5.11 ! [A: rat,B: rat] :
% 8.90/5.11 ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
% 8.90/5.11 = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6879_abs__add__abs,axiom,
% 8.90/5.11 ! [A: int,B: int] :
% 8.90/5.11 ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
% 8.90/5.11 = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6880_abs__divide,axiom,
% 8.90/5.11 ! [A: complex,B: complex] :
% 8.90/5.11 ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.90/5.11 = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6881_abs__divide,axiom,
% 8.90/5.11 ! [A: real,B: real] :
% 8.90/5.11 ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 8.90/5.11 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6882_abs__divide,axiom,
% 8.90/5.11 ! [A: rat,B: rat] :
% 8.90/5.11 ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 8.90/5.11 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6883_abs__minus__cancel,axiom,
% 8.90/5.11 ! [A: int] :
% 8.90/5.11 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 8.90/5.11 = ( abs_abs_int @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6884_abs__minus__cancel,axiom,
% 8.90/5.11 ! [A: real] :
% 8.90/5.11 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 8.90/5.11 = ( abs_abs_real @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6885_abs__minus__cancel,axiom,
% 8.90/5.11 ! [A: code_integer] :
% 8.90/5.11 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 8.90/5.11 = ( abs_abs_Code_integer @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6886_abs__minus__cancel,axiom,
% 8.90/5.11 ! [A: rat] :
% 8.90/5.11 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 8.90/5.11 = ( abs_abs_rat @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6887_abs__minus,axiom,
% 8.90/5.11 ! [A: int] :
% 8.90/5.11 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 8.90/5.11 = ( abs_abs_int @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6888_abs__minus,axiom,
% 8.90/5.11 ! [A: real] :
% 8.90/5.11 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 8.90/5.11 = ( abs_abs_real @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6889_abs__minus,axiom,
% 8.90/5.11 ! [A: complex] :
% 8.90/5.11 ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
% 8.90/5.11 = ( abs_abs_complex @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6890_abs__minus,axiom,
% 8.90/5.11 ! [A: code_integer] :
% 8.90/5.11 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 8.90/5.11 = ( abs_abs_Code_integer @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6891_abs__minus,axiom,
% 8.90/5.11 ! [A: rat] :
% 8.90/5.11 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 8.90/5.11 = ( abs_abs_rat @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6892_dvd__abs__iff,axiom,
% 8.90/5.11 ! [M: real,K: real] :
% 8.90/5.11 ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
% 8.90/5.11 = ( dvd_dvd_real @ M @ K ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6893_dvd__abs__iff,axiom,
% 8.90/5.11 ! [M: int,K: int] :
% 8.90/5.11 ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
% 8.90/5.11 = ( dvd_dvd_int @ M @ K ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6894_dvd__abs__iff,axiom,
% 8.90/5.11 ! [M: code_integer,K: code_integer] :
% 8.90/5.11 ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
% 8.90/5.11 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6895_abs__dvd__iff,axiom,
% 8.90/5.11 ! [M: real,K: real] :
% 8.90/5.11 ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
% 8.90/5.11 = ( dvd_dvd_real @ M @ K ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6896_abs__dvd__iff,axiom,
% 8.90/5.11 ! [M: int,K: int] :
% 8.90/5.11 ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
% 8.90/5.11 = ( dvd_dvd_int @ M @ K ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6897_abs__dvd__iff,axiom,
% 8.90/5.11 ! [M: code_integer,K: code_integer] :
% 8.90/5.11 ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
% 8.90/5.11 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6898_abs__bool__eq,axiom,
% 8.90/5.11 ! [P: $o] :
% 8.90/5.11 ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
% 8.90/5.11 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6899_abs__bool__eq,axiom,
% 8.90/5.11 ! [P: $o] :
% 8.90/5.11 ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 8.90/5.11 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6900_abs__bool__eq,axiom,
% 8.90/5.11 ! [P: $o] :
% 8.90/5.11 ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
% 8.90/5.11 = ( zero_n356916108424825756nteger @ P ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6901_tanh__real__le__iff,axiom,
% 8.90/5.11 ! [X2: real,Y3: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( tanh_real @ X2 ) @ ( tanh_real @ Y3 ) )
% 8.90/5.11 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6902_abs__of__nonneg,axiom,
% 8.90/5.11 ! [A: code_integer] :
% 8.90/5.11 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 8.90/5.11 => ( ( abs_abs_Code_integer @ A )
% 8.90/5.11 = A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6903_abs__of__nonneg,axiom,
% 8.90/5.11 ! [A: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.90/5.11 => ( ( abs_abs_real @ A )
% 8.90/5.11 = A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6904_abs__of__nonneg,axiom,
% 8.90/5.11 ! [A: rat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.90/5.11 => ( ( abs_abs_rat @ A )
% 8.90/5.11 = A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6905_abs__of__nonneg,axiom,
% 8.90/5.11 ! [A: int] :
% 8.90/5.11 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.90/5.11 => ( ( abs_abs_int @ A )
% 8.90/5.11 = A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6906_abs__le__self__iff,axiom,
% 8.90/5.11 ! [A: code_integer] :
% 8.90/5.11 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 8.90/5.11 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6907_abs__le__self__iff,axiom,
% 8.90/5.11 ! [A: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 8.90/5.11 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6908_abs__le__self__iff,axiom,
% 8.90/5.11 ! [A: rat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 8.90/5.11 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6909_abs__le__self__iff,axiom,
% 8.90/5.11 ! [A: int] :
% 8.90/5.11 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 8.90/5.11 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6910_abs__le__zero__iff,axiom,
% 8.90/5.11 ! [A: code_integer] :
% 8.90/5.11 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 8.90/5.11 = ( A = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6911_abs__le__zero__iff,axiom,
% 8.90/5.11 ! [A: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 8.90/5.11 = ( A = zero_zero_real ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6912_abs__le__zero__iff,axiom,
% 8.90/5.11 ! [A: rat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 8.90/5.11 = ( A = zero_zero_rat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6913_abs__le__zero__iff,axiom,
% 8.90/5.11 ! [A: int] :
% 8.90/5.11 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 8.90/5.11 = ( A = zero_zero_int ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6914_zero__less__abs__iff,axiom,
% 8.90/5.11 ! [A: code_integer] :
% 8.90/5.11 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 8.90/5.11 = ( A != zero_z3403309356797280102nteger ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6915_zero__less__abs__iff,axiom,
% 8.90/5.11 ! [A: real] :
% 8.90/5.11 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 8.90/5.11 = ( A != zero_zero_real ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6916_zero__less__abs__iff,axiom,
% 8.90/5.11 ! [A: rat] :
% 8.90/5.11 ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 8.90/5.11 = ( A != zero_zero_rat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6917_zero__less__abs__iff,axiom,
% 8.90/5.11 ! [A: int] :
% 8.90/5.11 ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 8.90/5.11 = ( A != zero_zero_int ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6918_abs__neg__numeral,axiom,
% 8.90/5.11 ! [N: num] :
% 8.90/5.11 ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.11 = ( numeral_numeral_int @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6919_abs__neg__numeral,axiom,
% 8.90/5.11 ! [N: num] :
% 8.90/5.11 ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.90/5.11 = ( numeral_numeral_real @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6920_abs__neg__numeral,axiom,
% 8.90/5.11 ! [N: num] :
% 8.90/5.11 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 8.90/5.11 = ( numera6620942414471956472nteger @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6921_abs__neg__numeral,axiom,
% 8.90/5.11 ! [N: num] :
% 8.90/5.11 ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.90/5.11 = ( numeral_numeral_rat @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6922_abs__neg__one,axiom,
% 8.90/5.11 ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.11 = one_one_int ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6923_abs__neg__one,axiom,
% 8.90/5.11 ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
% 8.90/5.11 = one_one_real ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6924_abs__neg__one,axiom,
% 8.90/5.11 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 8.90/5.11 = one_one_Code_integer ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6925_abs__neg__one,axiom,
% 8.90/5.11 ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.90/5.11 = one_one_rat ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6926_abs__power__minus,axiom,
% 8.90/5.11 ! [A: int,N: nat] :
% 8.90/5.11 ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 8.90/5.11 = ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6927_abs__power__minus,axiom,
% 8.90/5.11 ! [A: real,N: nat] :
% 8.90/5.11 ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 8.90/5.11 = ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6928_abs__power__minus,axiom,
% 8.90/5.11 ! [A: code_integer,N: nat] :
% 8.90/5.11 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 8.90/5.11 = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6929_abs__power__minus,axiom,
% 8.90/5.11 ! [A: rat,N: nat] :
% 8.90/5.11 ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 8.90/5.11 = ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6930_tanh__real__nonneg__iff,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X2 ) )
% 8.90/5.11 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6931_tanh__real__nonpos__iff,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( tanh_real @ X2 ) @ zero_zero_real )
% 8.90/5.11 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6932_round__numeral,axiom,
% 8.90/5.11 ! [N: num] :
% 8.90/5.11 ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
% 8.90/5.11 = ( numeral_numeral_int @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6933_round__numeral,axiom,
% 8.90/5.11 ! [N: num] :
% 8.90/5.11 ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
% 8.90/5.11 = ( numeral_numeral_int @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6934_round__1,axiom,
% 8.90/5.11 ( ( archim8280529875227126926d_real @ one_one_real )
% 8.90/5.11 = one_one_int ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6935_round__1,axiom,
% 8.90/5.11 ( ( archim7778729529865785530nd_rat @ one_one_rat )
% 8.90/5.11 = one_one_int ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6936_sum__abs,axiom,
% 8.90/5.11 ! [F: int > int,A2: set_int] :
% 8.90/5.11 ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
% 8.90/5.11 @ ( groups4538972089207619220nt_int
% 8.90/5.11 @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
% 8.90/5.11 @ A2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6937_sum__abs,axiom,
% 8.90/5.11 ! [F: nat > real,A2: set_nat] :
% 8.90/5.11 ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 8.90/5.11 @ ( groups6591440286371151544t_real
% 8.90/5.11 @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 8.90/5.11 @ A2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6938_divide__le__0__abs__iff,axiom,
% 8.90/5.11 ! [A: real,B: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
% 8.90/5.11 = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.90/5.11 | ( B = zero_zero_real ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6939_divide__le__0__abs__iff,axiom,
% 8.90/5.11 ! [A: rat,B: rat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
% 8.90/5.11 = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.90/5.11 | ( B = zero_zero_rat ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6940_zero__le__divide__abs__iff,axiom,
% 8.90/5.11 ! [A: real,B: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
% 8.90/5.11 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.90/5.11 | ( B = zero_zero_real ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6941_zero__le__divide__abs__iff,axiom,
% 8.90/5.11 ! [A: rat,B: rat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
% 8.90/5.11 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.90/5.11 | ( B = zero_zero_rat ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6942_abs__of__nonpos,axiom,
% 8.90/5.11 ! [A: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.90/5.11 => ( ( abs_abs_real @ A )
% 8.90/5.11 = ( uminus_uminus_real @ A ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6943_abs__of__nonpos,axiom,
% 8.90/5.11 ! [A: code_integer] :
% 8.90/5.11 ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
% 8.90/5.11 => ( ( abs_abs_Code_integer @ A )
% 8.90/5.11 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6944_abs__of__nonpos,axiom,
% 8.90/5.11 ! [A: rat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 8.90/5.11 => ( ( abs_abs_rat @ A )
% 8.90/5.11 = ( uminus_uminus_rat @ A ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6945_abs__of__nonpos,axiom,
% 8.90/5.11 ! [A: int] :
% 8.90/5.11 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 8.90/5.11 => ( ( abs_abs_int @ A )
% 8.90/5.11 = ( uminus_uminus_int @ A ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6946_artanh__minus__real,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 8.90/5.11 => ( ( artanh_real @ ( uminus_uminus_real @ X2 ) )
% 8.90/5.11 = ( uminus_uminus_real @ ( artanh_real @ X2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6947_sum__abs__ge__zero,axiom,
% 8.90/5.11 ! [F: int > int,A2: set_int] :
% 8.90/5.11 ( ord_less_eq_int @ zero_zero_int
% 8.90/5.11 @ ( groups4538972089207619220nt_int
% 8.90/5.11 @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
% 8.90/5.11 @ A2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6948_sum__abs__ge__zero,axiom,
% 8.90/5.11 ! [F: nat > real,A2: set_nat] :
% 8.90/5.11 ( ord_less_eq_real @ zero_zero_real
% 8.90/5.11 @ ( groups6591440286371151544t_real
% 8.90/5.11 @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 8.90/5.11 @ A2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6949_zero__less__power__abs__iff,axiom,
% 8.90/5.11 ! [A: code_integer,N: nat] :
% 8.90/5.11 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
% 8.90/5.11 = ( ( A != zero_z3403309356797280102nteger )
% 8.90/5.11 | ( N = zero_zero_nat ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6950_zero__less__power__abs__iff,axiom,
% 8.90/5.11 ! [A: real,N: nat] :
% 8.90/5.11 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 8.90/5.11 = ( ( A != zero_zero_real )
% 8.90/5.11 | ( N = zero_zero_nat ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6951_zero__less__power__abs__iff,axiom,
% 8.90/5.11 ! [A: rat,N: nat] :
% 8.90/5.11 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
% 8.90/5.11 = ( ( A != zero_zero_rat )
% 8.90/5.11 | ( N = zero_zero_nat ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6952_zero__less__power__abs__iff,axiom,
% 8.90/5.11 ! [A: int,N: nat] :
% 8.90/5.11 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
% 8.90/5.11 = ( ( A != zero_zero_int )
% 8.90/5.11 | ( N = zero_zero_nat ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6953_abs__power2,axiom,
% 8.90/5.11 ! [A: code_integer] :
% 8.90/5.11 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.11 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6954_abs__power2,axiom,
% 8.90/5.11 ! [A: real] :
% 8.90/5.11 ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.11 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6955_abs__power2,axiom,
% 8.90/5.11 ! [A: int] :
% 8.90/5.11 ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.11 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6956_power2__abs,axiom,
% 8.90/5.11 ! [A: code_integer] :
% 8.90/5.11 ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.11 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6957_power2__abs,axiom,
% 8.90/5.11 ! [A: real] :
% 8.90/5.11 ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.11 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6958_power2__abs,axiom,
% 8.90/5.11 ! [A: int] :
% 8.90/5.11 ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.11 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6959_sum_Ocl__ivl__Suc,axiom,
% 8.90/5.11 ! [N: nat,M: nat,G: nat > complex] :
% 8.90/5.11 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 8.90/5.11 => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 8.90/5.11 = zero_zero_complex ) )
% 8.90/5.11 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 8.90/5.11 => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 8.90/5.11 = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6960_sum_Ocl__ivl__Suc,axiom,
% 8.90/5.11 ! [N: nat,M: nat,G: nat > rat] :
% 8.90/5.11 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 8.90/5.11 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 8.90/5.11 = zero_zero_rat ) )
% 8.90/5.11 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 8.90/5.11 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 8.90/5.11 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6961_sum_Ocl__ivl__Suc,axiom,
% 8.90/5.11 ! [N: nat,M: nat,G: nat > int] :
% 8.90/5.11 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 8.90/5.11 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 8.90/5.11 = zero_zero_int ) )
% 8.90/5.11 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 8.90/5.11 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 8.90/5.11 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6962_sum_Ocl__ivl__Suc,axiom,
% 8.90/5.11 ! [N: nat,M: nat,G: nat > nat] :
% 8.90/5.11 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 8.90/5.11 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 8.90/5.11 = zero_zero_nat ) )
% 8.90/5.11 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 8.90/5.11 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 8.90/5.11 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6963_sum_Ocl__ivl__Suc,axiom,
% 8.90/5.11 ! [N: nat,M: nat,G: nat > real] :
% 8.90/5.11 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 8.90/5.11 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 8.90/5.11 = zero_zero_real ) )
% 8.90/5.11 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 8.90/5.11 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 8.90/5.11 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6964_round__neg__numeral,axiom,
% 8.90/5.11 ! [N: num] :
% 8.90/5.11 ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.90/5.11 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6965_round__neg__numeral,axiom,
% 8.90/5.11 ! [N: num] :
% 8.90/5.11 ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 8.90/5.11 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6966_sum__zero__power,axiom,
% 8.90/5.11 ! [A2: set_nat,C: nat > complex] :
% 8.90/5.11 ( ( ( ( finite_finite_nat @ A2 )
% 8.90/5.11 & ( member_nat @ zero_zero_nat @ A2 ) )
% 8.90/5.11 => ( ( groups2073611262835488442omplex
% 8.90/5.11 @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 8.90/5.11 @ A2 )
% 8.90/5.11 = ( C @ zero_zero_nat ) ) )
% 8.90/5.11 & ( ~ ( ( finite_finite_nat @ A2 )
% 8.90/5.11 & ( member_nat @ zero_zero_nat @ A2 ) )
% 8.90/5.11 => ( ( groups2073611262835488442omplex
% 8.90/5.11 @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 8.90/5.11 @ A2 )
% 8.90/5.11 = zero_zero_complex ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6967_sum__zero__power,axiom,
% 8.90/5.11 ! [A2: set_nat,C: nat > rat] :
% 8.90/5.11 ( ( ( ( finite_finite_nat @ A2 )
% 8.90/5.11 & ( member_nat @ zero_zero_nat @ A2 ) )
% 8.90/5.11 => ( ( groups2906978787729119204at_rat
% 8.90/5.11 @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 8.90/5.11 @ A2 )
% 8.90/5.11 = ( C @ zero_zero_nat ) ) )
% 8.90/5.11 & ( ~ ( ( finite_finite_nat @ A2 )
% 8.90/5.11 & ( member_nat @ zero_zero_nat @ A2 ) )
% 8.90/5.11 => ( ( groups2906978787729119204at_rat
% 8.90/5.11 @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 8.90/5.11 @ A2 )
% 8.90/5.11 = zero_zero_rat ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6968_sum__zero__power,axiom,
% 8.90/5.11 ! [A2: set_nat,C: nat > real] :
% 8.90/5.11 ( ( ( ( finite_finite_nat @ A2 )
% 8.90/5.11 & ( member_nat @ zero_zero_nat @ A2 ) )
% 8.90/5.11 => ( ( groups6591440286371151544t_real
% 8.90/5.11 @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 8.90/5.11 @ A2 )
% 8.90/5.11 = ( C @ zero_zero_nat ) ) )
% 8.90/5.11 & ( ~ ( ( finite_finite_nat @ A2 )
% 8.90/5.11 & ( member_nat @ zero_zero_nat @ A2 ) )
% 8.90/5.11 => ( ( groups6591440286371151544t_real
% 8.90/5.11 @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 8.90/5.11 @ A2 )
% 8.90/5.11 = zero_zero_real ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6969_power__even__abs__numeral,axiom,
% 8.90/5.11 ! [W: num,A: code_integer] :
% 8.90/5.11 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.11 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.11 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6970_power__even__abs__numeral,axiom,
% 8.90/5.11 ! [W: num,A: real] :
% 8.90/5.11 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.11 => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.11 = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6971_power__even__abs__numeral,axiom,
% 8.90/5.11 ! [W: num,A: int] :
% 8.90/5.11 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.11 => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
% 8.90/5.11 = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6972_sum__zero__power_H,axiom,
% 8.90/5.11 ! [A2: set_nat,C: nat > complex,D: nat > complex] :
% 8.90/5.11 ( ( ( ( finite_finite_nat @ A2 )
% 8.90/5.11 & ( member_nat @ zero_zero_nat @ A2 ) )
% 8.90/5.11 => ( ( groups2073611262835488442omplex
% 8.90/5.11 @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 8.90/5.11 @ A2 )
% 8.90/5.11 = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 8.90/5.11 & ( ~ ( ( finite_finite_nat @ A2 )
% 8.90/5.11 & ( member_nat @ zero_zero_nat @ A2 ) )
% 8.90/5.11 => ( ( groups2073611262835488442omplex
% 8.90/5.11 @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 8.90/5.11 @ A2 )
% 8.90/5.11 = zero_zero_complex ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6973_sum__zero__power_H,axiom,
% 8.90/5.11 ! [A2: set_nat,C: nat > rat,D: nat > rat] :
% 8.90/5.11 ( ( ( ( finite_finite_nat @ A2 )
% 8.90/5.11 & ( member_nat @ zero_zero_nat @ A2 ) )
% 8.90/5.11 => ( ( groups2906978787729119204at_rat
% 8.90/5.11 @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 8.90/5.11 @ A2 )
% 8.90/5.11 = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 8.90/5.11 & ( ~ ( ( finite_finite_nat @ A2 )
% 8.90/5.11 & ( member_nat @ zero_zero_nat @ A2 ) )
% 8.90/5.11 => ( ( groups2906978787729119204at_rat
% 8.90/5.11 @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 8.90/5.11 @ A2 )
% 8.90/5.11 = zero_zero_rat ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6974_sum__zero__power_H,axiom,
% 8.90/5.11 ! [A2: set_nat,C: nat > real,D: nat > real] :
% 8.90/5.11 ( ( ( ( finite_finite_nat @ A2 )
% 8.90/5.11 & ( member_nat @ zero_zero_nat @ A2 ) )
% 8.90/5.11 => ( ( groups6591440286371151544t_real
% 8.90/5.11 @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 8.90/5.11 @ A2 )
% 8.90/5.11 = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 8.90/5.11 & ( ~ ( ( finite_finite_nat @ A2 )
% 8.90/5.11 & ( member_nat @ zero_zero_nat @ A2 ) )
% 8.90/5.11 => ( ( groups6591440286371151544t_real
% 8.90/5.11 @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 8.90/5.11 @ A2 )
% 8.90/5.11 = zero_zero_real ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6975_abs__ge__self,axiom,
% 8.90/5.11 ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6976_abs__ge__self,axiom,
% 8.90/5.11 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6977_abs__ge__self,axiom,
% 8.90/5.11 ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6978_abs__ge__self,axiom,
% 8.90/5.11 ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6979_abs__le__D1,axiom,
% 8.90/5.11 ! [A: real,B: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 8.90/5.11 => ( ord_less_eq_real @ A @ B ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6980_abs__le__D1,axiom,
% 8.90/5.11 ! [A: code_integer,B: code_integer] :
% 8.90/5.11 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 8.90/5.11 => ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6981_abs__le__D1,axiom,
% 8.90/5.11 ! [A: rat,B: rat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 8.90/5.11 => ( ord_less_eq_rat @ A @ B ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6982_abs__le__D1,axiom,
% 8.90/5.11 ! [A: int,B: int] :
% 8.90/5.11 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 8.90/5.11 => ( ord_less_eq_int @ A @ B ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6983_abs__eq__0__iff,axiom,
% 8.90/5.11 ! [A: code_integer] :
% 8.90/5.11 ( ( ( abs_abs_Code_integer @ A )
% 8.90/5.11 = zero_z3403309356797280102nteger )
% 8.90/5.11 = ( A = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6984_abs__eq__0__iff,axiom,
% 8.90/5.11 ! [A: complex] :
% 8.90/5.11 ( ( ( abs_abs_complex @ A )
% 8.90/5.11 = zero_zero_complex )
% 8.90/5.11 = ( A = zero_zero_complex ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6985_abs__eq__0__iff,axiom,
% 8.90/5.11 ! [A: real] :
% 8.90/5.11 ( ( ( abs_abs_real @ A )
% 8.90/5.11 = zero_zero_real )
% 8.90/5.11 = ( A = zero_zero_real ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6986_abs__eq__0__iff,axiom,
% 8.90/5.11 ! [A: rat] :
% 8.90/5.11 ( ( ( abs_abs_rat @ A )
% 8.90/5.11 = zero_zero_rat )
% 8.90/5.11 = ( A = zero_zero_rat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6987_abs__eq__0__iff,axiom,
% 8.90/5.11 ! [A: int] :
% 8.90/5.11 ( ( ( abs_abs_int @ A )
% 8.90/5.11 = zero_zero_int )
% 8.90/5.11 = ( A = zero_zero_int ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6988_abs__mult,axiom,
% 8.90/5.11 ! [A: code_integer,B: code_integer] :
% 8.90/5.11 ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 8.90/5.11 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6989_abs__mult,axiom,
% 8.90/5.11 ! [A: real,B: real] :
% 8.90/5.11 ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 8.90/5.11 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6990_abs__mult,axiom,
% 8.90/5.11 ! [A: rat,B: rat] :
% 8.90/5.11 ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 8.90/5.11 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6991_abs__mult,axiom,
% 8.90/5.11 ! [A: int,B: int] :
% 8.90/5.11 ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 8.90/5.11 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6992_abs__one,axiom,
% 8.90/5.11 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 8.90/5.11 = one_one_Code_integer ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6993_abs__one,axiom,
% 8.90/5.11 ( ( abs_abs_real @ one_one_real )
% 8.90/5.11 = one_one_real ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6994_abs__one,axiom,
% 8.90/5.11 ( ( abs_abs_rat @ one_one_rat )
% 8.90/5.11 = one_one_rat ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6995_abs__one,axiom,
% 8.90/5.11 ( ( abs_abs_int @ one_one_int )
% 8.90/5.11 = one_one_int ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6996_abs__minus__commute,axiom,
% 8.90/5.11 ! [A: code_integer,B: code_integer] :
% 8.90/5.11 ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
% 8.90/5.11 = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6997_abs__minus__commute,axiom,
% 8.90/5.11 ! [A: real,B: real] :
% 8.90/5.11 ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 8.90/5.11 = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6998_abs__minus__commute,axiom,
% 8.90/5.11 ! [A: rat,B: rat] :
% 8.90/5.11 ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
% 8.90/5.11 = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_6999_abs__minus__commute,axiom,
% 8.90/5.11 ! [A: int,B: int] :
% 8.90/5.11 ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
% 8.90/5.11 = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7000_abs__eq__iff,axiom,
% 8.90/5.11 ! [X2: int,Y3: int] :
% 8.90/5.11 ( ( ( abs_abs_int @ X2 )
% 8.90/5.11 = ( abs_abs_int @ Y3 ) )
% 8.90/5.11 = ( ( X2 = Y3 )
% 8.90/5.11 | ( X2
% 8.90/5.11 = ( uminus_uminus_int @ Y3 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7001_abs__eq__iff,axiom,
% 8.90/5.11 ! [X2: real,Y3: real] :
% 8.90/5.11 ( ( ( abs_abs_real @ X2 )
% 8.90/5.11 = ( abs_abs_real @ Y3 ) )
% 8.90/5.11 = ( ( X2 = Y3 )
% 8.90/5.11 | ( X2
% 8.90/5.11 = ( uminus_uminus_real @ Y3 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7002_abs__eq__iff,axiom,
% 8.90/5.11 ! [X2: code_integer,Y3: code_integer] :
% 8.90/5.11 ( ( ( abs_abs_Code_integer @ X2 )
% 8.90/5.11 = ( abs_abs_Code_integer @ Y3 ) )
% 8.90/5.11 = ( ( X2 = Y3 )
% 8.90/5.11 | ( X2
% 8.90/5.11 = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7003_abs__eq__iff,axiom,
% 8.90/5.11 ! [X2: rat,Y3: rat] :
% 8.90/5.11 ( ( ( abs_abs_rat @ X2 )
% 8.90/5.11 = ( abs_abs_rat @ Y3 ) )
% 8.90/5.11 = ( ( X2 = Y3 )
% 8.90/5.11 | ( X2
% 8.90/5.11 = ( uminus_uminus_rat @ Y3 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7004_power__abs,axiom,
% 8.90/5.11 ! [A: code_integer,N: nat] :
% 8.90/5.11 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
% 8.90/5.11 = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7005_power__abs,axiom,
% 8.90/5.11 ! [A: real,N: nat] :
% 8.90/5.11 ( ( abs_abs_real @ ( power_power_real @ A @ N ) )
% 8.90/5.11 = ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7006_power__abs,axiom,
% 8.90/5.11 ! [A: int,N: nat] :
% 8.90/5.11 ( ( abs_abs_int @ ( power_power_int @ A @ N ) )
% 8.90/5.11 = ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7007_dvd__if__abs__eq,axiom,
% 8.90/5.11 ! [L: real,K: real] :
% 8.90/5.11 ( ( ( abs_abs_real @ L )
% 8.90/5.11 = ( abs_abs_real @ K ) )
% 8.90/5.11 => ( dvd_dvd_real @ L @ K ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7008_dvd__if__abs__eq,axiom,
% 8.90/5.11 ! [L: int,K: int] :
% 8.90/5.11 ( ( ( abs_abs_int @ L )
% 8.90/5.11 = ( abs_abs_int @ K ) )
% 8.90/5.11 => ( dvd_dvd_int @ L @ K ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7009_dvd__if__abs__eq,axiom,
% 8.90/5.11 ! [L: code_integer,K: code_integer] :
% 8.90/5.11 ( ( ( abs_abs_Code_integer @ L )
% 8.90/5.11 = ( abs_abs_Code_integer @ K ) )
% 8.90/5.11 => ( dvd_dvd_Code_integer @ L @ K ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7010_round__diff__minimal,axiom,
% 8.90/5.11 ! [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 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7011_round__diff__minimal,axiom,
% 8.90/5.11 ! [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 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7012_abs__ge__zero,axiom,
% 8.90/5.11 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7013_abs__ge__zero,axiom,
% 8.90/5.11 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7014_abs__ge__zero,axiom,
% 8.90/5.11 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7015_abs__ge__zero,axiom,
% 8.90/5.11 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7016_abs__of__pos,axiom,
% 8.90/5.11 ! [A: code_integer] :
% 8.90/5.11 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 8.90/5.11 => ( ( abs_abs_Code_integer @ A )
% 8.90/5.11 = A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7017_abs__of__pos,axiom,
% 8.90/5.11 ! [A: real] :
% 8.90/5.11 ( ( ord_less_real @ zero_zero_real @ A )
% 8.90/5.11 => ( ( abs_abs_real @ A )
% 8.90/5.11 = A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7018_abs__of__pos,axiom,
% 8.90/5.11 ! [A: rat] :
% 8.90/5.11 ( ( ord_less_rat @ zero_zero_rat @ A )
% 8.90/5.11 => ( ( abs_abs_rat @ A )
% 8.90/5.11 = A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7019_abs__of__pos,axiom,
% 8.90/5.11 ! [A: int] :
% 8.90/5.11 ( ( ord_less_int @ zero_zero_int @ A )
% 8.90/5.11 => ( ( abs_abs_int @ A )
% 8.90/5.11 = A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7020_abs__not__less__zero,axiom,
% 8.90/5.11 ! [A: code_integer] :
% 8.90/5.11 ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7021_abs__not__less__zero,axiom,
% 8.90/5.11 ! [A: real] :
% 8.90/5.11 ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7022_abs__not__less__zero,axiom,
% 8.90/5.11 ! [A: rat] :
% 8.90/5.11 ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7023_abs__not__less__zero,axiom,
% 8.90/5.11 ! [A: int] :
% 8.90/5.11 ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7024_abs__triangle__ineq,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7025_abs__triangle__ineq,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7026_abs__triangle__ineq,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7027_abs__triangle__ineq,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7028_abs__mult__less,axiom,
% 8.90/5.11 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 8.90/5.11 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 8.90/5.11 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
% 8.90/5.11 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7029_abs__mult__less,axiom,
% 8.90/5.11 ! [A: real,C: real,B: real,D: real] :
% 8.90/5.11 ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 8.90/5.11 => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
% 8.90/5.11 => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7030_abs__mult__less,axiom,
% 8.90/5.11 ! [A: rat,C: rat,B: rat,D: rat] :
% 8.90/5.11 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 8.90/5.11 => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
% 8.90/5.11 => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7031_abs__mult__less,axiom,
% 8.90/5.11 ! [A: int,C: int,B: int,D: int] :
% 8.90/5.11 ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 8.90/5.11 => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
% 8.90/5.11 => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7032_abs__triangle__ineq2,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7033_abs__triangle__ineq2,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7034_abs__triangle__ineq2,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7035_abs__triangle__ineq2,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7036_abs__triangle__ineq3,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7037_abs__triangle__ineq3,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7038_abs__triangle__ineq3,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7039_abs__triangle__ineq3,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7040_abs__triangle__ineq2__sym,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7041_abs__triangle__ineq2__sym,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7042_abs__triangle__ineq2__sym,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7043_abs__triangle__ineq2__sym,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7044_nonzero__abs__divide,axiom,
% 8.90/5.11 ! [B: real,A: real] :
% 8.90/5.11 ( ( B != zero_zero_real )
% 8.90/5.11 => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 8.90/5.11 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7045_nonzero__abs__divide,axiom,
% 8.90/5.11 ! [B: rat,A: rat] :
% 8.90/5.11 ( ( B != zero_zero_rat )
% 8.90/5.11 => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 8.90/5.11 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7046_abs__ge__minus__self,axiom,
% 8.90/5.11 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7047_abs__ge__minus__self,axiom,
% 8.90/5.11 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7048_abs__ge__minus__self,axiom,
% 8.90/5.11 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7049_abs__ge__minus__self,axiom,
% 8.90/5.11 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7050_abs__le__iff,axiom,
% 8.90/5.11 ! [A: real,B: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 8.90/5.11 = ( ( ord_less_eq_real @ A @ B )
% 8.90/5.11 & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7051_abs__le__iff,axiom,
% 8.90/5.11 ! [A: code_integer,B: code_integer] :
% 8.90/5.11 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 8.90/5.11 = ( ( ord_le3102999989581377725nteger @ A @ B )
% 8.90/5.11 & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7052_abs__le__iff,axiom,
% 8.90/5.11 ! [A: rat,B: rat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 8.90/5.11 = ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.11 & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7053_abs__le__iff,axiom,
% 8.90/5.11 ! [A: int,B: int] :
% 8.90/5.11 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 8.90/5.11 = ( ( ord_less_eq_int @ A @ B )
% 8.90/5.11 & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7054_abs__le__D2,axiom,
% 8.90/5.11 ! [A: real,B: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 8.90/5.11 => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7055_abs__le__D2,axiom,
% 8.90/5.11 ! [A: code_integer,B: code_integer] :
% 8.90/5.11 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 8.90/5.11 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7056_abs__le__D2,axiom,
% 8.90/5.11 ! [A: rat,B: rat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 8.90/5.11 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7057_abs__le__D2,axiom,
% 8.90/5.11 ! [A: int,B: int] :
% 8.90/5.11 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 8.90/5.11 => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7058_abs__leI,axiom,
% 8.90/5.11 ! [A: real,B: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ A @ B )
% 8.90/5.11 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 8.90/5.11 => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7059_abs__leI,axiom,
% 8.90/5.11 ! [A: code_integer,B: code_integer] :
% 8.90/5.11 ( ( ord_le3102999989581377725nteger @ A @ B )
% 8.90/5.11 => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 8.90/5.11 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7060_abs__leI,axiom,
% 8.90/5.11 ! [A: rat,B: rat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ A @ B )
% 8.90/5.11 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 8.90/5.11 => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7061_abs__leI,axiom,
% 8.90/5.11 ! [A: int,B: int] :
% 8.90/5.11 ( ( ord_less_eq_int @ A @ B )
% 8.90/5.11 => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 8.90/5.11 => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7062_abs__less__iff,axiom,
% 8.90/5.11 ! [A: int,B: int] :
% 8.90/5.11 ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
% 8.90/5.11 = ( ( ord_less_int @ A @ B )
% 8.90/5.11 & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7063_abs__less__iff,axiom,
% 8.90/5.11 ! [A: real,B: real] :
% 8.90/5.11 ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
% 8.90/5.11 = ( ( ord_less_real @ A @ B )
% 8.90/5.11 & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7064_abs__less__iff,axiom,
% 8.90/5.11 ! [A: code_integer,B: code_integer] :
% 8.90/5.11 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 8.90/5.11 = ( ( ord_le6747313008572928689nteger @ A @ B )
% 8.90/5.11 & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7065_abs__less__iff,axiom,
% 8.90/5.11 ! [A: rat,B: rat] :
% 8.90/5.11 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
% 8.90/5.11 = ( ( ord_less_rat @ A @ B )
% 8.90/5.11 & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7066_sum__cong__Suc,axiom,
% 8.90/5.11 ! [A2: set_nat,F: nat > nat,G: nat > nat] :
% 8.90/5.11 ( ~ ( member_nat @ zero_zero_nat @ A2 )
% 8.90/5.11 => ( ! [X5: nat] :
% 8.90/5.11 ( ( member_nat @ ( suc @ X5 ) @ A2 )
% 8.90/5.11 => ( ( F @ ( suc @ X5 ) )
% 8.90/5.11 = ( G @ ( suc @ X5 ) ) ) )
% 8.90/5.11 => ( ( groups3542108847815614940at_nat @ F @ A2 )
% 8.90/5.11 = ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7067_sum__cong__Suc,axiom,
% 8.90/5.11 ! [A2: set_nat,F: nat > real,G: nat > real] :
% 8.90/5.11 ( ~ ( member_nat @ zero_zero_nat @ A2 )
% 8.90/5.11 => ( ! [X5: nat] :
% 8.90/5.11 ( ( member_nat @ ( suc @ X5 ) @ A2 )
% 8.90/5.11 => ( ( F @ ( suc @ X5 ) )
% 8.90/5.11 = ( G @ ( suc @ X5 ) ) ) )
% 8.90/5.11 => ( ( groups6591440286371151544t_real @ F @ A2 )
% 8.90/5.11 = ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7068_sum__subtractf__nat,axiom,
% 8.90/5.11 ! [A2: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,F: product_prod_nat_nat > nat] :
% 8.90/5.11 ( ! [X5: product_prod_nat_nat] :
% 8.90/5.11 ( ( member8440522571783428010at_nat @ X5 @ A2 )
% 8.90/5.11 => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
% 8.90/5.11 => ( ( groups977919841031483927at_nat
% 8.90/5.11 @ ^ [X: product_prod_nat_nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 8.90/5.11 @ A2 )
% 8.90/5.11 = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( groups977919841031483927at_nat @ G @ A2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7069_sum__subtractf__nat,axiom,
% 8.90/5.11 ! [A2: set_complex,G: complex > nat,F: complex > nat] :
% 8.90/5.11 ( ! [X5: complex] :
% 8.90/5.11 ( ( member_complex @ X5 @ A2 )
% 8.90/5.11 => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
% 8.90/5.11 => ( ( groups5693394587270226106ex_nat
% 8.90/5.11 @ ^ [X: complex] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 8.90/5.11 @ A2 )
% 8.90/5.11 = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7070_sum__subtractf__nat,axiom,
% 8.90/5.11 ! [A2: set_real,G: real > nat,F: real > nat] :
% 8.90/5.11 ( ! [X5: real] :
% 8.90/5.11 ( ( member_real @ X5 @ A2 )
% 8.90/5.11 => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
% 8.90/5.11 => ( ( groups1935376822645274424al_nat
% 8.90/5.11 @ ^ [X: real] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 8.90/5.11 @ A2 )
% 8.90/5.11 = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7071_sum__subtractf__nat,axiom,
% 8.90/5.11 ! [A2: set_int,G: int > nat,F: int > nat] :
% 8.90/5.11 ( ! [X5: int] :
% 8.90/5.11 ( ( member_int @ X5 @ A2 )
% 8.90/5.11 => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
% 8.90/5.11 => ( ( groups4541462559716669496nt_nat
% 8.90/5.11 @ ^ [X: int] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 8.90/5.11 @ A2 )
% 8.90/5.11 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7072_sum__subtractf__nat,axiom,
% 8.90/5.11 ! [A2: set_nat,G: nat > nat,F: nat > nat] :
% 8.90/5.11 ( ! [X5: nat] :
% 8.90/5.11 ( ( member_nat @ X5 @ A2 )
% 8.90/5.11 => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
% 8.90/5.11 => ( ( groups3542108847815614940at_nat
% 8.90/5.11 @ ^ [X: nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 8.90/5.11 @ A2 )
% 8.90/5.11 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7073_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 8.90/5.11 ! [G: nat > nat,M: nat,N: nat] :
% 8.90/5.11 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 8.90/5.11 = ( groups3542108847815614940at_nat
% 8.90/5.11 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7074_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 8.90/5.11 ! [G: nat > real,M: nat,N: nat] :
% 8.90/5.11 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 8.90/5.11 = ( groups6591440286371151544t_real
% 8.90/5.11 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7075_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 8.90/5.11 ! [G: nat > nat,M: nat,K: nat,N: nat] :
% 8.90/5.11 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 8.90/5.11 = ( groups3542108847815614940at_nat
% 8.90/5.11 @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7076_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 8.90/5.11 ! [G: nat > real,M: nat,K: nat,N: nat] :
% 8.90/5.11 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 8.90/5.11 = ( groups6591440286371151544t_real
% 8.90/5.11 @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7077_tanh__real__lt__1,axiom,
% 8.90/5.11 ! [X2: real] : ( ord_less_real @ ( tanh_real @ X2 ) @ one_one_real ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7078_dense__eq0__I,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ( ! [E2: real] :
% 8.90/5.11 ( ( ord_less_real @ zero_zero_real @ E2 )
% 8.90/5.11 => ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ E2 ) )
% 8.90/5.11 => ( X2 = zero_zero_real ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7079_dense__eq0__I,axiom,
% 8.90/5.11 ! [X2: rat] :
% 8.90/5.11 ( ! [E2: rat] :
% 8.90/5.11 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 8.90/5.11 => ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ E2 ) )
% 8.90/5.11 => ( X2 = zero_zero_rat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7080_abs__mult__pos,axiom,
% 8.90/5.11 ! [X2: code_integer,Y3: code_integer] :
% 8.90/5.11 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X2 )
% 8.90/5.11 => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y3 ) @ X2 )
% 8.90/5.11 = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y3 @ X2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7081_abs__mult__pos,axiom,
% 8.90/5.11 ! [X2: real,Y3: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.11 => ( ( times_times_real @ ( abs_abs_real @ Y3 ) @ X2 )
% 8.90/5.11 = ( abs_abs_real @ ( times_times_real @ Y3 @ X2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7082_abs__mult__pos,axiom,
% 8.90/5.11 ! [X2: rat,Y3: rat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 8.90/5.11 => ( ( times_times_rat @ ( abs_abs_rat @ Y3 ) @ X2 )
% 8.90/5.11 = ( abs_abs_rat @ ( times_times_rat @ Y3 @ X2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7083_abs__mult__pos,axiom,
% 8.90/5.11 ! [X2: int,Y3: int] :
% 8.90/5.11 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.90/5.11 => ( ( times_times_int @ ( abs_abs_int @ Y3 ) @ X2 )
% 8.90/5.11 = ( abs_abs_int @ ( times_times_int @ Y3 @ X2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7084_abs__eq__mult,axiom,
% 8.90/5.11 ! [A: code_integer,B: code_integer] :
% 8.90/5.11 ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 8.90/5.11 | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 8.90/5.11 & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 8.90/5.11 | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
% 8.90/5.11 => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 8.90/5.11 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7085_abs__eq__mult,axiom,
% 8.90/5.11 ! [A: real,B: real] :
% 8.90/5.11 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.90/5.11 | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 8.90/5.11 & ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.90/5.11 | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 8.90/5.11 => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 8.90/5.11 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7086_abs__eq__mult,axiom,
% 8.90/5.11 ! [A: rat,B: rat] :
% 8.90/5.11 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.90/5.11 | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 8.90/5.11 & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.90/5.11 | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 8.90/5.11 => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 8.90/5.11 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7087_abs__eq__mult,axiom,
% 8.90/5.11 ! [A: int,B: int] :
% 8.90/5.11 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.90/5.11 | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 8.90/5.11 & ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.90/5.11 | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 8.90/5.11 => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 8.90/5.11 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7088_abs__minus__le__zero,axiom,
% 8.90/5.11 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7089_abs__minus__le__zero,axiom,
% 8.90/5.11 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7090_abs__minus__le__zero,axiom,
% 8.90/5.11 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7091_abs__minus__le__zero,axiom,
% 8.90/5.11 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7092_eq__abs__iff_H,axiom,
% 8.90/5.11 ! [A: real,B: real] :
% 8.90/5.11 ( ( A
% 8.90/5.11 = ( abs_abs_real @ B ) )
% 8.90/5.11 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.90/5.11 & ( ( B = A )
% 8.90/5.11 | ( B
% 8.90/5.11 = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7093_eq__abs__iff_H,axiom,
% 8.90/5.11 ! [A: code_integer,B: code_integer] :
% 8.90/5.11 ( ( A
% 8.90/5.11 = ( abs_abs_Code_integer @ B ) )
% 8.90/5.11 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 8.90/5.11 & ( ( B = A )
% 8.90/5.11 | ( B
% 8.90/5.11 = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7094_eq__abs__iff_H,axiom,
% 8.90/5.11 ! [A: rat,B: rat] :
% 8.90/5.11 ( ( A
% 8.90/5.11 = ( abs_abs_rat @ B ) )
% 8.90/5.11 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.90/5.11 & ( ( B = A )
% 8.90/5.11 | ( B
% 8.90/5.11 = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7095_eq__abs__iff_H,axiom,
% 8.90/5.11 ! [A: int,B: int] :
% 8.90/5.11 ( ( A
% 8.90/5.11 = ( abs_abs_int @ B ) )
% 8.90/5.11 = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 8.90/5.11 & ( ( B = A )
% 8.90/5.11 | ( B
% 8.90/5.11 = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7096_abs__eq__iff_H,axiom,
% 8.90/5.11 ! [A: real,B: real] :
% 8.90/5.11 ( ( ( abs_abs_real @ A )
% 8.90/5.11 = B )
% 8.90/5.11 = ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.90/5.11 & ( ( A = B )
% 8.90/5.11 | ( A
% 8.90/5.11 = ( uminus_uminus_real @ B ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7097_abs__eq__iff_H,axiom,
% 8.90/5.11 ! [A: code_integer,B: code_integer] :
% 8.90/5.11 ( ( ( abs_abs_Code_integer @ A )
% 8.90/5.11 = B )
% 8.90/5.11 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 8.90/5.11 & ( ( A = B )
% 8.90/5.11 | ( A
% 8.90/5.11 = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7098_abs__eq__iff_H,axiom,
% 8.90/5.11 ! [A: rat,B: rat] :
% 8.90/5.11 ( ( ( abs_abs_rat @ A )
% 8.90/5.11 = B )
% 8.90/5.11 = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.90/5.11 & ( ( A = B )
% 8.90/5.11 | ( A
% 8.90/5.11 = ( uminus_uminus_rat @ B ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7099_abs__eq__iff_H,axiom,
% 8.90/5.11 ! [A: int,B: int] :
% 8.90/5.11 ( ( ( abs_abs_int @ A )
% 8.90/5.11 = B )
% 8.90/5.11 = ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.90/5.11 & ( ( A = B )
% 8.90/5.11 | ( A
% 8.90/5.11 = ( uminus_uminus_int @ B ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7100_abs__div__pos,axiom,
% 8.90/5.11 ! [Y3: real,X2: real] :
% 8.90/5.11 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.90/5.11 => ( ( divide_divide_real @ ( abs_abs_real @ X2 ) @ Y3 )
% 8.90/5.11 = ( abs_abs_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7101_abs__div__pos,axiom,
% 8.90/5.11 ! [Y3: rat,X2: rat] :
% 8.90/5.11 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 8.90/5.11 => ( ( divide_divide_rat @ ( abs_abs_rat @ X2 ) @ Y3 )
% 8.90/5.11 = ( abs_abs_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7102_zero__le__power__abs,axiom,
% 8.90/5.11 ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7103_zero__le__power__abs,axiom,
% 8.90/5.11 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7104_zero__le__power__abs,axiom,
% 8.90/5.11 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7105_zero__le__power__abs,axiom,
% 8.90/5.11 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7106_abs__if__raw,axiom,
% 8.90/5.11 ( abs_abs_int
% 8.90/5.11 = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7107_abs__if__raw,axiom,
% 8.90/5.11 ( abs_abs_real
% 8.90/5.11 = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7108_abs__if__raw,axiom,
% 8.90/5.11 ( abs_abs_Code_integer
% 8.90/5.11 = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7109_abs__if__raw,axiom,
% 8.90/5.11 ( abs_abs_rat
% 8.90/5.11 = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7110_abs__if,axiom,
% 8.90/5.11 ( abs_abs_int
% 8.90/5.11 = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7111_abs__if,axiom,
% 8.90/5.11 ( abs_abs_real
% 8.90/5.11 = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7112_abs__if,axiom,
% 8.90/5.11 ( abs_abs_Code_integer
% 8.90/5.11 = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7113_abs__if,axiom,
% 8.90/5.11 ( abs_abs_rat
% 8.90/5.11 = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7114_abs__of__neg,axiom,
% 8.90/5.11 ! [A: int] :
% 8.90/5.11 ( ( ord_less_int @ A @ zero_zero_int )
% 8.90/5.11 => ( ( abs_abs_int @ A )
% 8.90/5.11 = ( uminus_uminus_int @ A ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7115_abs__of__neg,axiom,
% 8.90/5.11 ! [A: real] :
% 8.90/5.11 ( ( ord_less_real @ A @ zero_zero_real )
% 8.90/5.11 => ( ( abs_abs_real @ A )
% 8.90/5.11 = ( uminus_uminus_real @ A ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7116_abs__of__neg,axiom,
% 8.90/5.11 ! [A: code_integer] :
% 8.90/5.11 ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 8.90/5.11 => ( ( abs_abs_Code_integer @ A )
% 8.90/5.11 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7117_abs__of__neg,axiom,
% 8.90/5.11 ! [A: rat] :
% 8.90/5.11 ( ( ord_less_rat @ A @ zero_zero_rat )
% 8.90/5.11 => ( ( abs_abs_rat @ A )
% 8.90/5.11 = ( uminus_uminus_rat @ A ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7118_abs__diff__le__iff,axiom,
% 8.90/5.11 ! [X2: code_integer,A: code_integer,R2: code_integer] :
% 8.90/5.11 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X2 @ A ) ) @ R2 )
% 8.90/5.11 = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X2 )
% 8.90/5.11 & ( ord_le3102999989581377725nteger @ X2 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7119_abs__diff__le__iff,axiom,
% 8.90/5.11 ! [X2: real,A: real,R2: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ A ) ) @ R2 )
% 8.90/5.11 = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X2 )
% 8.90/5.11 & ( ord_less_eq_real @ X2 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7120_abs__diff__le__iff,axiom,
% 8.90/5.11 ! [X2: rat,A: rat,R2: rat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ A ) ) @ R2 )
% 8.90/5.11 = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X2 )
% 8.90/5.11 & ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7121_abs__diff__le__iff,axiom,
% 8.90/5.11 ! [X2: int,A: int,R2: int] :
% 8.90/5.11 ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ A ) ) @ R2 )
% 8.90/5.11 = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X2 )
% 8.90/5.11 & ( ord_less_eq_int @ X2 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7122_abs__diff__triangle__ineq,axiom,
% 8.90/5.11 ! [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 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7123_abs__diff__triangle__ineq,axiom,
% 8.90/5.11 ! [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 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7124_abs__diff__triangle__ineq,axiom,
% 8.90/5.11 ! [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 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7125_abs__diff__triangle__ineq,axiom,
% 8.90/5.11 ! [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 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7126_abs__triangle__ineq4,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7127_abs__triangle__ineq4,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7128_abs__triangle__ineq4,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7129_abs__triangle__ineq4,axiom,
% 8.90/5.11 ! [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 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7130_abs__diff__less__iff,axiom,
% 8.90/5.11 ! [X2: code_integer,A: code_integer,R2: code_integer] :
% 8.90/5.11 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X2 @ A ) ) @ R2 )
% 8.90/5.11 = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X2 )
% 8.90/5.11 & ( ord_le6747313008572928689nteger @ X2 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7131_abs__diff__less__iff,axiom,
% 8.90/5.11 ! [X2: real,A: real,R2: real] :
% 8.90/5.11 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ A ) ) @ R2 )
% 8.90/5.11 = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X2 )
% 8.90/5.11 & ( ord_less_real @ X2 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7132_abs__diff__less__iff,axiom,
% 8.90/5.11 ! [X2: rat,A: rat,R2: rat] :
% 8.90/5.11 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ A ) ) @ R2 )
% 8.90/5.11 = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X2 )
% 8.90/5.11 & ( ord_less_rat @ X2 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7133_abs__diff__less__iff,axiom,
% 8.90/5.11 ! [X2: int,A: int,R2: int] :
% 8.90/5.11 ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ A ) ) @ R2 )
% 8.90/5.11 = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X2 )
% 8.90/5.11 & ( ord_less_int @ X2 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7134_sum__eq__Suc0__iff,axiom,
% 8.90/5.11 ! [A2: set_int,F: int > nat] :
% 8.90/5.11 ( ( finite_finite_int @ A2 )
% 8.90/5.11 => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 8.90/5.11 = ( suc @ zero_zero_nat ) )
% 8.90/5.11 = ( ? [X: int] :
% 8.90/5.11 ( ( member_int @ X @ A2 )
% 8.90/5.11 & ( ( F @ X )
% 8.90/5.11 = ( suc @ zero_zero_nat ) )
% 8.90/5.11 & ! [Y: int] :
% 8.90/5.11 ( ( member_int @ Y @ A2 )
% 8.90/5.11 => ( ( X != Y )
% 8.90/5.11 => ( ( F @ Y )
% 8.90/5.11 = zero_zero_nat ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7135_sum__eq__Suc0__iff,axiom,
% 8.90/5.11 ! [A2: set_complex,F: complex > nat] :
% 8.90/5.11 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.11 => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 8.90/5.11 = ( suc @ zero_zero_nat ) )
% 8.90/5.11 = ( ? [X: complex] :
% 8.90/5.11 ( ( member_complex @ X @ A2 )
% 8.90/5.11 & ( ( F @ X )
% 8.90/5.11 = ( suc @ zero_zero_nat ) )
% 8.90/5.11 & ! [Y: complex] :
% 8.90/5.11 ( ( member_complex @ Y @ A2 )
% 8.90/5.11 => ( ( X != Y )
% 8.90/5.11 => ( ( F @ Y )
% 8.90/5.11 = zero_zero_nat ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7136_sum__eq__Suc0__iff,axiom,
% 8.90/5.11 ! [A2: set_nat,F: nat > nat] :
% 8.90/5.11 ( ( finite_finite_nat @ A2 )
% 8.90/5.11 => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 8.90/5.11 = ( suc @ zero_zero_nat ) )
% 8.90/5.11 = ( ? [X: nat] :
% 8.90/5.11 ( ( member_nat @ X @ A2 )
% 8.90/5.11 & ( ( F @ X )
% 8.90/5.11 = ( suc @ zero_zero_nat ) )
% 8.90/5.11 & ! [Y: nat] :
% 8.90/5.11 ( ( member_nat @ Y @ A2 )
% 8.90/5.11 => ( ( X != Y )
% 8.90/5.11 => ( ( F @ Y )
% 8.90/5.11 = zero_zero_nat ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7137_sum__SucD,axiom,
% 8.90/5.11 ! [F: nat > nat,A2: set_nat,N: nat] :
% 8.90/5.11 ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 8.90/5.11 = ( suc @ N ) )
% 8.90/5.11 => ? [X5: nat] :
% 8.90/5.11 ( ( member_nat @ X5 @ A2 )
% 8.90/5.11 & ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7138_sum__eq__1__iff,axiom,
% 8.90/5.11 ! [A2: set_int,F: int > nat] :
% 8.90/5.11 ( ( finite_finite_int @ A2 )
% 8.90/5.11 => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 8.90/5.11 = one_one_nat )
% 8.90/5.11 = ( ? [X: int] :
% 8.90/5.11 ( ( member_int @ X @ A2 )
% 8.90/5.11 & ( ( F @ X )
% 8.90/5.11 = one_one_nat )
% 8.90/5.11 & ! [Y: int] :
% 8.90/5.11 ( ( member_int @ Y @ A2 )
% 8.90/5.11 => ( ( X != Y )
% 8.90/5.11 => ( ( F @ Y )
% 8.90/5.11 = zero_zero_nat ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7139_sum__eq__1__iff,axiom,
% 8.90/5.11 ! [A2: set_complex,F: complex > nat] :
% 8.90/5.11 ( ( finite3207457112153483333omplex @ A2 )
% 8.90/5.11 => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 8.90/5.11 = one_one_nat )
% 8.90/5.11 = ( ? [X: complex] :
% 8.90/5.11 ( ( member_complex @ X @ A2 )
% 8.90/5.11 & ( ( F @ X )
% 8.90/5.11 = one_one_nat )
% 8.90/5.11 & ! [Y: complex] :
% 8.90/5.11 ( ( member_complex @ Y @ A2 )
% 8.90/5.11 => ( ( X != Y )
% 8.90/5.11 => ( ( F @ Y )
% 8.90/5.11 = zero_zero_nat ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7140_sum__eq__1__iff,axiom,
% 8.90/5.11 ! [A2: set_nat,F: nat > nat] :
% 8.90/5.11 ( ( finite_finite_nat @ A2 )
% 8.90/5.11 => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 8.90/5.11 = one_one_nat )
% 8.90/5.11 = ( ? [X: nat] :
% 8.90/5.11 ( ( member_nat @ X @ A2 )
% 8.90/5.11 & ( ( F @ X )
% 8.90/5.11 = one_one_nat )
% 8.90/5.11 & ! [Y: nat] :
% 8.90/5.11 ( ( member_nat @ Y @ A2 )
% 8.90/5.11 => ( ( X != Y )
% 8.90/5.11 => ( ( F @ Y )
% 8.90/5.11 = zero_zero_nat ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7141_round__mono,axiom,
% 8.90/5.11 ! [X2: rat,Y3: rat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 8.90/5.11 => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X2 ) @ ( archim7778729529865785530nd_rat @ Y3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7142_abs__real__def,axiom,
% 8.90/5.11 ( abs_abs_real
% 8.90/5.11 = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7143_lemma__interval__lt,axiom,
% 8.90/5.11 ! [A: real,X2: real,B: real] :
% 8.90/5.11 ( ( ord_less_real @ A @ X2 )
% 8.90/5.11 => ( ( ord_less_real @ X2 @ B )
% 8.90/5.11 => ? [D3: real] :
% 8.90/5.11 ( ( ord_less_real @ zero_zero_real @ D3 )
% 8.90/5.11 & ! [Y5: real] :
% 8.90/5.11 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y5 ) ) @ D3 )
% 8.90/5.11 => ( ( ord_less_real @ A @ Y5 )
% 8.90/5.11 & ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7144_sin__bound__lemma,axiom,
% 8.90/5.11 ! [X2: real,Y3: real,U: real,V: real] :
% 8.90/5.11 ( ( X2 = Y3 )
% 8.90/5.11 => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 8.90/5.11 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X2 @ U ) @ Y3 ) ) @ V ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7145_sum__power__add,axiom,
% 8.90/5.11 ! [X2: complex,M: nat,I5: set_nat] :
% 8.90/5.11 ( ( groups2073611262835488442omplex
% 8.90/5.11 @ ^ [I4: nat] : ( power_power_complex @ X2 @ ( plus_plus_nat @ M @ I4 ) )
% 8.90/5.11 @ I5 )
% 8.90/5.11 = ( times_times_complex @ ( power_power_complex @ X2 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ I5 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7146_sum__power__add,axiom,
% 8.90/5.11 ! [X2: rat,M: nat,I5: set_nat] :
% 8.90/5.11 ( ( groups2906978787729119204at_rat
% 8.90/5.11 @ ^ [I4: nat] : ( power_power_rat @ X2 @ ( plus_plus_nat @ M @ I4 ) )
% 8.90/5.11 @ I5 )
% 8.90/5.11 = ( times_times_rat @ ( power_power_rat @ X2 @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ I5 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7147_sum__power__add,axiom,
% 8.90/5.11 ! [X2: int,M: nat,I5: set_nat] :
% 8.90/5.11 ( ( groups3539618377306564664at_int
% 8.90/5.11 @ ^ [I4: nat] : ( power_power_int @ X2 @ ( plus_plus_nat @ M @ I4 ) )
% 8.90/5.11 @ I5 )
% 8.90/5.11 = ( times_times_int @ ( power_power_int @ X2 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ I5 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7148_sum__power__add,axiom,
% 8.90/5.11 ! [X2: real,M: nat,I5: set_nat] :
% 8.90/5.11 ( ( groups6591440286371151544t_real
% 8.90/5.11 @ ^ [I4: nat] : ( power_power_real @ X2 @ ( plus_plus_nat @ M @ I4 ) )
% 8.90/5.11 @ I5 )
% 8.90/5.11 = ( times_times_real @ ( power_power_real @ X2 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ I5 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7149_sum_OatLeastAtMost__rev,axiom,
% 8.90/5.11 ! [G: nat > nat,N: nat,M: nat] :
% 8.90/5.11 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 8.90/5.11 = ( groups3542108847815614940at_nat
% 8.90/5.11 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7150_sum_OatLeastAtMost__rev,axiom,
% 8.90/5.11 ! [G: nat > real,N: nat,M: nat] :
% 8.90/5.11 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 8.90/5.11 = ( groups6591440286371151544t_real
% 8.90/5.11 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7151_sum__nth__roots,axiom,
% 8.90/5.11 ! [N: nat,C: complex] :
% 8.90/5.11 ( ( ord_less_nat @ one_one_nat @ N )
% 8.90/5.11 => ( ( groups7754918857620584856omplex
% 8.90/5.11 @ ^ [X: complex] : X
% 8.90/5.11 @ ( collect_complex
% 8.90/5.11 @ ^ [Z3: complex] :
% 8.90/5.11 ( ( power_power_complex @ Z3 @ N )
% 8.90/5.11 = C ) ) )
% 8.90/5.11 = zero_zero_complex ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7152_sum__roots__unity,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( ord_less_nat @ one_one_nat @ N )
% 8.90/5.11 => ( ( groups7754918857620584856omplex
% 8.90/5.11 @ ^ [X: complex] : X
% 8.90/5.11 @ ( collect_complex
% 8.90/5.11 @ ^ [Z3: complex] :
% 8.90/5.11 ( ( power_power_complex @ Z3 @ N )
% 8.90/5.11 = one_one_complex ) ) )
% 8.90/5.11 = zero_zero_complex ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7153_tanh__real__gt__neg1,axiom,
% 8.90/5.11 ! [X2: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7154_abs__add__one__gt__zero,axiom,
% 8.90/5.11 ! [X2: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7155_abs__add__one__gt__zero,axiom,
% 8.90/5.11 ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7156_abs__add__one__gt__zero,axiom,
% 8.90/5.11 ! [X2: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7157_abs__add__one__gt__zero,axiom,
% 8.90/5.11 ! [X2: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7158_of__int__leD,axiom,
% 8.90/5.11 ! [N: int,X2: code_integer] :
% 8.90/5.11 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X2 )
% 8.90/5.11 => ( ( N = zero_zero_int )
% 8.90/5.11 | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7159_of__int__leD,axiom,
% 8.90/5.11 ! [N: int,X2: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X2 )
% 8.90/5.11 => ( ( N = zero_zero_int )
% 8.90/5.11 | ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7160_of__int__leD,axiom,
% 8.90/5.11 ! [N: int,X2: rat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X2 )
% 8.90/5.11 => ( ( N = zero_zero_int )
% 8.90/5.11 | ( ord_less_eq_rat @ one_one_rat @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7161_of__int__leD,axiom,
% 8.90/5.11 ! [N: int,X2: int] :
% 8.90/5.11 ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X2 )
% 8.90/5.11 => ( ( N = zero_zero_int )
% 8.90/5.11 | ( ord_less_eq_int @ one_one_int @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7162_of__int__lessD,axiom,
% 8.90/5.11 ! [N: int,X2: code_integer] :
% 8.90/5.11 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X2 )
% 8.90/5.11 => ( ( N = zero_zero_int )
% 8.90/5.11 | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7163_of__int__lessD,axiom,
% 8.90/5.11 ! [N: int,X2: real] :
% 8.90/5.11 ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X2 )
% 8.90/5.11 => ( ( N = zero_zero_int )
% 8.90/5.11 | ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7164_of__int__lessD,axiom,
% 8.90/5.11 ! [N: int,X2: rat] :
% 8.90/5.11 ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X2 )
% 8.90/5.11 => ( ( N = zero_zero_int )
% 8.90/5.11 | ( ord_less_rat @ one_one_rat @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7165_of__int__lessD,axiom,
% 8.90/5.11 ! [N: int,X2: int] :
% 8.90/5.11 ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X2 )
% 8.90/5.11 => ( ( N = zero_zero_int )
% 8.90/5.11 | ( ord_less_int @ one_one_int @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7166_sum__diff__nat,axiom,
% 8.90/5.11 ! [B3: set_complex,A2: set_complex,F: complex > nat] :
% 8.90/5.11 ( ( finite3207457112153483333omplex @ B3 )
% 8.90/5.11 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 8.90/5.11 => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 8.90/5.11 = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B3 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7167_sum__diff__nat,axiom,
% 8.90/5.11 ! [B3: set_int,A2: set_int,F: int > nat] :
% 8.90/5.11 ( ( finite_finite_int @ B3 )
% 8.90/5.11 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 8.90/5.11 => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
% 8.90/5.11 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B3 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7168_sum__diff__nat,axiom,
% 8.90/5.11 ! [B3: set_nat,A2: set_nat,F: nat > nat] :
% 8.90/5.11 ( ( finite_finite_nat @ B3 )
% 8.90/5.11 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 8.90/5.11 => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 8.90/5.11 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B3 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7169_sum__shift__lb__Suc0__0,axiom,
% 8.90/5.11 ! [F: nat > complex,K: nat] :
% 8.90/5.11 ( ( ( F @ zero_zero_nat )
% 8.90/5.11 = zero_zero_complex )
% 8.90/5.11 => ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 8.90/5.11 = ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7170_sum__shift__lb__Suc0__0,axiom,
% 8.90/5.11 ! [F: nat > rat,K: nat] :
% 8.90/5.11 ( ( ( F @ zero_zero_nat )
% 8.90/5.11 = zero_zero_rat )
% 8.90/5.11 => ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 8.90/5.11 = ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7171_sum__shift__lb__Suc0__0,axiom,
% 8.90/5.11 ! [F: nat > int,K: nat] :
% 8.90/5.11 ( ( ( F @ zero_zero_nat )
% 8.90/5.11 = zero_zero_int )
% 8.90/5.11 => ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 8.90/5.11 = ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7172_sum__shift__lb__Suc0__0,axiom,
% 8.90/5.11 ! [F: nat > nat,K: nat] :
% 8.90/5.11 ( ( ( F @ zero_zero_nat )
% 8.90/5.11 = zero_zero_nat )
% 8.90/5.11 => ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 8.90/5.11 = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7173_sum__shift__lb__Suc0__0,axiom,
% 8.90/5.11 ! [F: nat > real,K: nat] :
% 8.90/5.11 ( ( ( F @ zero_zero_nat )
% 8.90/5.11 = zero_zero_real )
% 8.90/5.11 => ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 8.90/5.11 = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7174_sum_OatLeast0__atMost__Suc,axiom,
% 8.90/5.11 ! [G: nat > rat,N: nat] :
% 8.90/5.11 ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 8.90/5.11 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7175_sum_OatLeast0__atMost__Suc,axiom,
% 8.90/5.11 ! [G: nat > int,N: nat] :
% 8.90/5.11 ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 8.90/5.11 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7176_sum_OatLeast0__atMost__Suc,axiom,
% 8.90/5.11 ! [G: nat > nat,N: nat] :
% 8.90/5.11 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 8.90/5.11 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7177_sum_OatLeast0__atMost__Suc,axiom,
% 8.90/5.11 ! [G: nat > real,N: nat] :
% 8.90/5.11 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 8.90/5.11 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7178_sum_OatLeast__Suc__atMost,axiom,
% 8.90/5.11 ! [M: nat,N: nat,G: nat > rat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7179_sum_OatLeast__Suc__atMost,axiom,
% 8.90/5.11 ! [M: nat,N: nat,G: nat > int] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7180_sum_OatLeast__Suc__atMost,axiom,
% 8.90/5.11 ! [M: nat,N: nat,G: nat > nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7181_sum_OatLeast__Suc__atMost,axiom,
% 8.90/5.11 ! [M: nat,N: nat,G: nat > real] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7182_sum_Onat__ivl__Suc_H,axiom,
% 8.90/5.11 ! [M: nat,N: nat,G: nat > rat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 8.90/5.11 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 8.90/5.11 = ( plus_plus_rat @ ( G @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7183_sum_Onat__ivl__Suc_H,axiom,
% 8.90/5.11 ! [M: nat,N: nat,G: nat > int] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 8.90/5.11 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 8.90/5.11 = ( plus_plus_int @ ( G @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7184_sum_Onat__ivl__Suc_H,axiom,
% 8.90/5.11 ! [M: nat,N: nat,G: nat > nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 8.90/5.11 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 8.90/5.11 = ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7185_sum_Onat__ivl__Suc_H,axiom,
% 8.90/5.11 ! [M: nat,N: nat,G: nat > real] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 8.90/5.11 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 8.90/5.11 = ( plus_plus_real @ ( G @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7186_lemma__interval,axiom,
% 8.90/5.11 ! [A: real,X2: real,B: real] :
% 8.90/5.11 ( ( ord_less_real @ A @ X2 )
% 8.90/5.11 => ( ( ord_less_real @ X2 @ B )
% 8.90/5.11 => ? [D3: real] :
% 8.90/5.11 ( ( ord_less_real @ zero_zero_real @ D3 )
% 8.90/5.11 & ! [Y5: real] :
% 8.90/5.11 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y5 ) ) @ D3 )
% 8.90/5.11 => ( ( ord_less_eq_real @ A @ Y5 )
% 8.90/5.11 & ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7187_sum_OSuc__reindex__ivl,axiom,
% 8.90/5.11 ! [M: nat,N: nat,G: nat > rat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 8.90/5.11 = ( plus_plus_rat @ ( G @ M )
% 8.90/5.11 @ ( groups2906978787729119204at_rat
% 8.90/5.11 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7188_sum_OSuc__reindex__ivl,axiom,
% 8.90/5.11 ! [M: nat,N: nat,G: nat > int] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 8.90/5.11 = ( plus_plus_int @ ( G @ M )
% 8.90/5.11 @ ( groups3539618377306564664at_int
% 8.90/5.11 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7189_sum_OSuc__reindex__ivl,axiom,
% 8.90/5.11 ! [M: nat,N: nat,G: nat > nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 8.90/5.11 = ( plus_plus_nat @ ( G @ M )
% 8.90/5.11 @ ( groups3542108847815614940at_nat
% 8.90/5.11 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7190_sum_OSuc__reindex__ivl,axiom,
% 8.90/5.11 ! [M: nat,N: nat,G: nat > real] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 8.90/5.11 = ( plus_plus_real @ ( G @ M )
% 8.90/5.11 @ ( groups6591440286371151544t_real
% 8.90/5.11 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7191_sum__Suc__diff,axiom,
% 8.90/5.11 ! [M: nat,N: nat,F: nat > rat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 8.90/5.11 => ( ( groups2906978787729119204at_rat
% 8.90/5.11 @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7192_sum__Suc__diff,axiom,
% 8.90/5.11 ! [M: nat,N: nat,F: nat > int] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 8.90/5.11 => ( ( groups3539618377306564664at_int
% 8.90/5.11 @ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7193_sum__Suc__diff,axiom,
% 8.90/5.11 ! [M: nat,N: nat,F: nat > real] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 8.90/5.11 => ( ( groups6591440286371151544t_real
% 8.90/5.11 @ ^ [I4: nat] : ( minus_minus_real @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7194_of__int__round__abs__le,axiom,
% 8.90/5.11 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) @ X2 ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7195_of__int__round__abs__le,axiom,
% 8.90/5.11 ! [X2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) @ X2 ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7196_round__unique_H,axiom,
% 8.90/5.11 ! [X2: real,N: int] :
% 8.90/5.11 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.11 => ( ( archim8280529875227126926d_real @ X2 )
% 8.90/5.11 = N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7197_round__unique_H,axiom,
% 8.90/5.11 ! [X2: rat,N: int] :
% 8.90/5.11 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 8.90/5.11 => ( ( archim7778729529865785530nd_rat @ X2 )
% 8.90/5.11 = N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7198_pred__equals__eq2,axiom,
% 8.90/5.11 ! [R: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
% 8.90/5.11 ( ( ( ^ [X: code_integer,Y: $o] : ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X @ Y ) @ R ) )
% 8.90/5.11 = ( ^ [X: code_integer,Y: $o] : ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X @ Y ) @ S3 ) ) )
% 8.90/5.11 = ( R = S3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7199_pred__equals__eq2,axiom,
% 8.90/5.11 ! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
% 8.90/5.11 ( ( ( ^ [X: num,Y: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X @ Y ) @ R ) )
% 8.90/5.11 = ( ^ [X: num,Y: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X @ Y ) @ S3 ) ) )
% 8.90/5.11 = ( R = S3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7200_pred__equals__eq2,axiom,
% 8.90/5.11 ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
% 8.90/5.11 ( ( ( ^ [X: nat,Y: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ R ) )
% 8.90/5.11 = ( ^ [X: nat,Y: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ S3 ) ) )
% 8.90/5.11 = ( R = S3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7201_pred__equals__eq2,axiom,
% 8.90/5.11 ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
% 8.90/5.11 ( ( ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) )
% 8.90/5.11 = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S3 ) ) )
% 8.90/5.11 = ( R = S3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7202_pred__equals__eq2,axiom,
% 8.90/5.11 ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
% 8.90/5.11 ( ( ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R ) )
% 8.90/5.11 = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S3 ) ) )
% 8.90/5.11 = ( R = S3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7203_bot__empty__eq2,axiom,
% 8.90/5.11 ( bot_bo4731626569425807221er_o_o
% 8.90/5.11 = ( ^ [X: code_integer,Y: $o] : ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X @ Y ) @ bot_bo5379713665208646970eger_o ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7204_bot__empty__eq2,axiom,
% 8.90/5.11 ( bot_bot_num_num_o
% 8.90/5.11 = ( ^ [X: num,Y: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X @ Y ) @ bot_bo9056780473022590049um_num ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7205_bot__empty__eq2,axiom,
% 8.90/5.11 ( bot_bot_nat_num_o
% 8.90/5.11 = ( ^ [X: nat,Y: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ bot_bo7038385379056416535at_num ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7206_bot__empty__eq2,axiom,
% 8.90/5.11 ( bot_bot_nat_nat_o
% 8.90/5.11 = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ bot_bo2099793752762293965at_nat ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7207_bot__empty__eq2,axiom,
% 8.90/5.11 ( bot_bot_int_int_o
% 8.90/5.11 = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ bot_bo1796632182523588997nt_int ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7208_abs__le__square__iff,axiom,
% 8.90/5.11 ! [X2: code_integer,Y3: code_integer] :
% 8.90/5.11 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ ( abs_abs_Code_integer @ Y3 ) )
% 8.90/5.11 = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7209_abs__le__square__iff,axiom,
% 8.90/5.11 ! [X2: real,Y3: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ Y3 ) )
% 8.90/5.11 = ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7210_abs__le__square__iff,axiom,
% 8.90/5.11 ! [X2: rat,Y3: rat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ ( abs_abs_rat @ Y3 ) )
% 8.90/5.11 = ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7211_abs__le__square__iff,axiom,
% 8.90/5.11 ! [X2: int,Y3: int] :
% 8.90/5.11 ( ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ ( abs_abs_int @ Y3 ) )
% 8.90/5.11 = ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7212_abs__square__eq__1,axiom,
% 8.90/5.11 ! [X2: code_integer] :
% 8.90/5.11 ( ( ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.11 = one_one_Code_integer )
% 8.90/5.11 = ( ( abs_abs_Code_integer @ X2 )
% 8.90/5.11 = one_one_Code_integer ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7213_abs__square__eq__1,axiom,
% 8.90/5.11 ! [X2: rat] :
% 8.90/5.11 ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.11 = one_one_rat )
% 8.90/5.11 = ( ( abs_abs_rat @ X2 )
% 8.90/5.11 = one_one_rat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7214_abs__square__eq__1,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.11 = one_one_real )
% 8.90/5.11 = ( ( abs_abs_real @ X2 )
% 8.90/5.11 = one_one_real ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7215_abs__square__eq__1,axiom,
% 8.90/5.11 ! [X2: int] :
% 8.90/5.11 ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.11 = one_one_int )
% 8.90/5.11 = ( ( abs_abs_int @ X2 )
% 8.90/5.11 = one_one_int ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7216_power__even__abs,axiom,
% 8.90/5.11 ! [N: nat,A: code_integer] :
% 8.90/5.11 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.11 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
% 8.90/5.11 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7217_power__even__abs,axiom,
% 8.90/5.11 ! [N: nat,A: real] :
% 8.90/5.11 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.11 => ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
% 8.90/5.11 = ( power_power_real @ A @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7218_power__even__abs,axiom,
% 8.90/5.11 ! [N: nat,A: int] :
% 8.90/5.11 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.11 => ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
% 8.90/5.11 = ( power_power_int @ A @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7219_sum_Oub__add__nat,axiom,
% 8.90/5.11 ! [M: nat,N: nat,G: nat > rat,P6: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 8.90/5.11 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 8.90/5.11 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7220_sum_Oub__add__nat,axiom,
% 8.90/5.11 ! [M: nat,N: nat,G: nat > int,P6: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 8.90/5.11 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 8.90/5.11 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7221_sum_Oub__add__nat,axiom,
% 8.90/5.11 ! [M: nat,N: nat,G: nat > nat,P6: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 8.90/5.11 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 8.90/5.11 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7222_sum_Oub__add__nat,axiom,
% 8.90/5.11 ! [M: nat,N: nat,G: nat > real,P6: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 8.90/5.11 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 8.90/5.11 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7223_abs__sqrt__wlog,axiom,
% 8.90/5.11 ! [P: code_integer > code_integer > $o,X2: code_integer] :
% 8.90/5.11 ( ! [X5: code_integer] :
% 8.90/5.11 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X5 )
% 8.90/5.11 => ( P @ X5 @ ( power_8256067586552552935nteger @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.11 => ( P @ ( abs_abs_Code_integer @ X2 ) @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7224_abs__sqrt__wlog,axiom,
% 8.90/5.11 ! [P: real > real > $o,X2: real] :
% 8.90/5.11 ( ! [X5: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 8.90/5.11 => ( P @ X5 @ ( power_power_real @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.11 => ( P @ ( abs_abs_real @ X2 ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7225_abs__sqrt__wlog,axiom,
% 8.90/5.11 ! [P: rat > rat > $o,X2: rat] :
% 8.90/5.11 ( ! [X5: rat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ zero_zero_rat @ X5 )
% 8.90/5.11 => ( P @ X5 @ ( power_power_rat @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.11 => ( P @ ( abs_abs_rat @ X2 ) @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7226_abs__sqrt__wlog,axiom,
% 8.90/5.11 ! [P: int > int > $o,X2: int] :
% 8.90/5.11 ( ! [X5: int] :
% 8.90/5.11 ( ( ord_less_eq_int @ zero_zero_int @ X5 )
% 8.90/5.11 => ( P @ X5 @ ( power_power_int @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.11 => ( P @ ( abs_abs_int @ X2 ) @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7227_power2__le__iff__abs__le,axiom,
% 8.90/5.11 ! [Y3: code_integer,X2: code_integer] :
% 8.90/5.11 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
% 8.90/5.11 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.11 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ Y3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7228_power2__le__iff__abs__le,axiom,
% 8.90/5.11 ! [Y3: real,X2: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.11 => ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.11 = ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ Y3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7229_power2__le__iff__abs__le,axiom,
% 8.90/5.11 ! [Y3: rat,X2: rat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 8.90/5.11 => ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.11 = ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ Y3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7230_power2__le__iff__abs__le,axiom,
% 8.90/5.11 ! [Y3: int,X2: int] :
% 8.90/5.11 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.90/5.11 => ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.11 = ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ Y3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7231_abs__square__le__1,axiom,
% 8.90/5.11 ! [X2: code_integer] :
% 8.90/5.11 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 8.90/5.11 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ one_one_Code_integer ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7232_abs__square__le__1,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 8.90/5.11 = ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7233_abs__square__le__1,axiom,
% 8.90/5.11 ! [X2: rat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 8.90/5.11 = ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ one_one_rat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7234_abs__square__le__1,axiom,
% 8.90/5.11 ! [X2: int] :
% 8.90/5.11 ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 8.90/5.11 = ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ one_one_int ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7235_abs__square__less__1,axiom,
% 8.90/5.11 ! [X2: code_integer] :
% 8.90/5.11 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 8.90/5.11 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X2 ) @ one_one_Code_integer ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7236_abs__square__less__1,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 8.90/5.11 = ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7237_abs__square__less__1,axiom,
% 8.90/5.11 ! [X2: rat] :
% 8.90/5.11 ( ( ord_less_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 8.90/5.11 = ( ord_less_rat @ ( abs_abs_rat @ X2 ) @ one_one_rat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7238_abs__square__less__1,axiom,
% 8.90/5.11 ! [X2: int] :
% 8.90/5.11 ( ( ord_less_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 8.90/5.11 = ( ord_less_int @ ( abs_abs_int @ X2 ) @ one_one_int ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7239_power__mono__even,axiom,
% 8.90/5.11 ! [N: nat,A: code_integer,B: code_integer] :
% 8.90/5.11 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.11 => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
% 8.90/5.11 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7240_power__mono__even,axiom,
% 8.90/5.11 ! [N: nat,A: real,B: real] :
% 8.90/5.11 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.11 => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
% 8.90/5.11 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7241_power__mono__even,axiom,
% 8.90/5.11 ! [N: nat,A: rat,B: rat] :
% 8.90/5.11 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.11 => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
% 8.90/5.11 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7242_power__mono__even,axiom,
% 8.90/5.11 ! [N: nat,A: int,B: int] :
% 8.90/5.11 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.11 => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
% 8.90/5.11 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7243_convex__sum__bound__le,axiom,
% 8.90/5.11 ! [I5: set_complex,X2: complex > code_integer,A: complex > code_integer,B: code_integer,Delta: code_integer] :
% 8.90/5.11 ( ! [I3: complex] :
% 8.90/5.11 ( ( member_complex @ I3 @ I5 )
% 8.90/5.11 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I3 ) ) )
% 8.90/5.11 => ( ( ( groups6621422865394947399nteger @ X2 @ I5 )
% 8.90/5.11 = one_one_Code_integer )
% 8.90/5.11 => ( ! [I3: complex] :
% 8.90/5.11 ( ( member_complex @ I3 @ I5 )
% 8.90/5.11 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 8.90/5.11 => ( ord_le3102999989581377725nteger
% 8.90/5.11 @ ( abs_abs_Code_integer
% 8.90/5.11 @ ( minus_8373710615458151222nteger
% 8.90/5.11 @ ( groups6621422865394947399nteger
% 8.90/5.11 @ ^ [I4: complex] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 8.90/5.11 @ I5 )
% 8.90/5.11 @ B ) )
% 8.90/5.11 @ Delta ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7244_convex__sum__bound__le,axiom,
% 8.90/5.11 ! [I5: set_real,X2: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
% 8.90/5.11 ( ! [I3: real] :
% 8.90/5.11 ( ( member_real @ I3 @ I5 )
% 8.90/5.11 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I3 ) ) )
% 8.90/5.11 => ( ( ( groups7713935264441627589nteger @ X2 @ I5 )
% 8.90/5.11 = one_one_Code_integer )
% 8.90/5.11 => ( ! [I3: real] :
% 8.90/5.11 ( ( member_real @ I3 @ I5 )
% 8.90/5.11 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 8.90/5.11 => ( ord_le3102999989581377725nteger
% 8.90/5.11 @ ( abs_abs_Code_integer
% 8.90/5.11 @ ( minus_8373710615458151222nteger
% 8.90/5.11 @ ( groups7713935264441627589nteger
% 8.90/5.11 @ ^ [I4: real] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 8.90/5.11 @ I5 )
% 8.90/5.11 @ B ) )
% 8.90/5.11 @ Delta ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7245_convex__sum__bound__le,axiom,
% 8.90/5.11 ! [I5: set_nat,X2: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
% 8.90/5.11 ( ! [I3: nat] :
% 8.90/5.11 ( ( member_nat @ I3 @ I5 )
% 8.90/5.11 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I3 ) ) )
% 8.90/5.11 => ( ( ( groups7501900531339628137nteger @ X2 @ I5 )
% 8.90/5.11 = one_one_Code_integer )
% 8.90/5.11 => ( ! [I3: nat] :
% 8.90/5.11 ( ( member_nat @ I3 @ I5 )
% 8.90/5.11 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 8.90/5.11 => ( ord_le3102999989581377725nteger
% 8.90/5.11 @ ( abs_abs_Code_integer
% 8.90/5.11 @ ( minus_8373710615458151222nteger
% 8.90/5.11 @ ( groups7501900531339628137nteger
% 8.90/5.11 @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 8.90/5.11 @ I5 )
% 8.90/5.11 @ B ) )
% 8.90/5.11 @ Delta ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7246_convex__sum__bound__le,axiom,
% 8.90/5.11 ! [I5: set_int,X2: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
% 8.90/5.11 ( ! [I3: int] :
% 8.90/5.11 ( ( member_int @ I3 @ I5 )
% 8.90/5.11 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I3 ) ) )
% 8.90/5.11 => ( ( ( groups7873554091576472773nteger @ X2 @ I5 )
% 8.90/5.11 = one_one_Code_integer )
% 8.90/5.11 => ( ! [I3: int] :
% 8.90/5.11 ( ( member_int @ I3 @ I5 )
% 8.90/5.11 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 8.90/5.11 => ( ord_le3102999989581377725nteger
% 8.90/5.11 @ ( abs_abs_Code_integer
% 8.90/5.11 @ ( minus_8373710615458151222nteger
% 8.90/5.11 @ ( groups7873554091576472773nteger
% 8.90/5.11 @ ^ [I4: int] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 8.90/5.11 @ I5 )
% 8.90/5.11 @ B ) )
% 8.90/5.11 @ Delta ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7247_convex__sum__bound__le,axiom,
% 8.90/5.11 ! [I5: set_complex,X2: complex > real,A: complex > real,B: real,Delta: real] :
% 8.90/5.11 ( ! [I3: complex] :
% 8.90/5.11 ( ( member_complex @ I3 @ I5 )
% 8.90/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I3 ) ) )
% 8.90/5.11 => ( ( ( groups5808333547571424918x_real @ X2 @ I5 )
% 8.90/5.11 = one_one_real )
% 8.90/5.11 => ( ! [I3: complex] :
% 8.90/5.11 ( ( member_complex @ I3 @ I5 )
% 8.90/5.11 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 8.90/5.11 => ( ord_less_eq_real
% 8.90/5.11 @ ( abs_abs_real
% 8.90/5.11 @ ( minus_minus_real
% 8.90/5.11 @ ( groups5808333547571424918x_real
% 8.90/5.11 @ ^ [I4: complex] : ( times_times_real @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 8.90/5.11 @ I5 )
% 8.90/5.11 @ B ) )
% 8.90/5.11 @ Delta ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7248_convex__sum__bound__le,axiom,
% 8.90/5.11 ! [I5: set_real,X2: real > real,A: real > real,B: real,Delta: real] :
% 8.90/5.11 ( ! [I3: real] :
% 8.90/5.11 ( ( member_real @ I3 @ I5 )
% 8.90/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I3 ) ) )
% 8.90/5.11 => ( ( ( groups8097168146408367636l_real @ X2 @ I5 )
% 8.90/5.11 = one_one_real )
% 8.90/5.11 => ( ! [I3: real] :
% 8.90/5.11 ( ( member_real @ I3 @ I5 )
% 8.90/5.11 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 8.90/5.11 => ( ord_less_eq_real
% 8.90/5.11 @ ( abs_abs_real
% 8.90/5.11 @ ( minus_minus_real
% 8.90/5.11 @ ( groups8097168146408367636l_real
% 8.90/5.11 @ ^ [I4: real] : ( times_times_real @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 8.90/5.11 @ I5 )
% 8.90/5.11 @ B ) )
% 8.90/5.11 @ Delta ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7249_convex__sum__bound__le,axiom,
% 8.90/5.11 ! [I5: set_int,X2: int > real,A: int > real,B: real,Delta: real] :
% 8.90/5.11 ( ! [I3: int] :
% 8.90/5.11 ( ( member_int @ I3 @ I5 )
% 8.90/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I3 ) ) )
% 8.90/5.11 => ( ( ( groups8778361861064173332t_real @ X2 @ I5 )
% 8.90/5.11 = one_one_real )
% 8.90/5.11 => ( ! [I3: int] :
% 8.90/5.11 ( ( member_int @ I3 @ I5 )
% 8.90/5.11 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 8.90/5.11 => ( ord_less_eq_real
% 8.90/5.11 @ ( abs_abs_real
% 8.90/5.11 @ ( minus_minus_real
% 8.90/5.11 @ ( groups8778361861064173332t_real
% 8.90/5.11 @ ^ [I4: int] : ( times_times_real @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 8.90/5.11 @ I5 )
% 8.90/5.11 @ B ) )
% 8.90/5.11 @ Delta ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7250_convex__sum__bound__le,axiom,
% 8.90/5.11 ! [I5: set_complex,X2: complex > rat,A: complex > rat,B: rat,Delta: rat] :
% 8.90/5.11 ( ! [I3: complex] :
% 8.90/5.11 ( ( member_complex @ I3 @ I5 )
% 8.90/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( X2 @ I3 ) ) )
% 8.90/5.11 => ( ( ( groups5058264527183730370ex_rat @ X2 @ I5 )
% 8.90/5.11 = one_one_rat )
% 8.90/5.11 => ( ! [I3: complex] :
% 8.90/5.11 ( ( member_complex @ I3 @ I5 )
% 8.90/5.11 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 8.90/5.11 => ( ord_less_eq_rat
% 8.90/5.11 @ ( abs_abs_rat
% 8.90/5.11 @ ( minus_minus_rat
% 8.90/5.11 @ ( groups5058264527183730370ex_rat
% 8.90/5.11 @ ^ [I4: complex] : ( times_times_rat @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 8.90/5.11 @ I5 )
% 8.90/5.11 @ B ) )
% 8.90/5.11 @ Delta ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7251_convex__sum__bound__le,axiom,
% 8.90/5.11 ! [I5: set_real,X2: real > rat,A: real > rat,B: rat,Delta: rat] :
% 8.90/5.11 ( ! [I3: real] :
% 8.90/5.11 ( ( member_real @ I3 @ I5 )
% 8.90/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( X2 @ I3 ) ) )
% 8.90/5.11 => ( ( ( groups1300246762558778688al_rat @ X2 @ I5 )
% 8.90/5.11 = one_one_rat )
% 8.90/5.11 => ( ! [I3: real] :
% 8.90/5.11 ( ( member_real @ I3 @ I5 )
% 8.90/5.11 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 8.90/5.11 => ( ord_less_eq_rat
% 8.90/5.11 @ ( abs_abs_rat
% 8.90/5.11 @ ( minus_minus_rat
% 8.90/5.11 @ ( groups1300246762558778688al_rat
% 8.90/5.11 @ ^ [I4: real] : ( times_times_rat @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 8.90/5.11 @ I5 )
% 8.90/5.11 @ B ) )
% 8.90/5.11 @ Delta ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7252_convex__sum__bound__le,axiom,
% 8.90/5.11 ! [I5: set_nat,X2: nat > rat,A: nat > rat,B: rat,Delta: rat] :
% 8.90/5.11 ( ! [I3: nat] :
% 8.90/5.11 ( ( member_nat @ I3 @ I5 )
% 8.90/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( X2 @ I3 ) ) )
% 8.90/5.11 => ( ( ( groups2906978787729119204at_rat @ X2 @ I5 )
% 8.90/5.11 = one_one_rat )
% 8.90/5.11 => ( ! [I3: nat] :
% 8.90/5.11 ( ( member_nat @ I3 @ I5 )
% 8.90/5.11 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 8.90/5.11 => ( ord_less_eq_rat
% 8.90/5.11 @ ( abs_abs_rat
% 8.90/5.11 @ ( minus_minus_rat
% 8.90/5.11 @ ( groups2906978787729119204at_rat
% 8.90/5.11 @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( X2 @ I4 ) )
% 8.90/5.11 @ I5 )
% 8.90/5.11 @ B ) )
% 8.90/5.11 @ Delta ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7253_sum__natinterval__diff,axiom,
% 8.90/5.11 ! [M: nat,N: nat,F: nat > complex] :
% 8.90/5.11 ( ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( groups2073611262835488442omplex
% 8.90/5.11 @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 8.90/5.11 & ( ~ ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( groups2073611262835488442omplex
% 8.90/5.11 @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = zero_zero_complex ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7254_sum__natinterval__diff,axiom,
% 8.90/5.11 ! [M: nat,N: nat,F: nat > rat] :
% 8.90/5.11 ( ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( groups2906978787729119204at_rat
% 8.90/5.11 @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 8.90/5.11 & ( ~ ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( groups2906978787729119204at_rat
% 8.90/5.11 @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = zero_zero_rat ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7255_sum__natinterval__diff,axiom,
% 8.90/5.11 ! [M: nat,N: nat,F: nat > int] :
% 8.90/5.11 ( ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( groups3539618377306564664at_int
% 8.90/5.11 @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 8.90/5.11 & ( ~ ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( groups3539618377306564664at_int
% 8.90/5.11 @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = zero_zero_int ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7256_sum__natinterval__diff,axiom,
% 8.90/5.11 ! [M: nat,N: nat,F: nat > real] :
% 8.90/5.11 ( ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( groups6591440286371151544t_real
% 8.90/5.11 @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 8.90/5.11 & ( ~ ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( groups6591440286371151544t_real
% 8.90/5.11 @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = zero_zero_real ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7257_sum__telescope_H_H,axiom,
% 8.90/5.11 ! [M: nat,N: nat,F: nat > rat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( groups2906978787729119204at_rat
% 8.90/5.11 @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 8.90/5.11 = ( minus_minus_rat @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7258_sum__telescope_H_H,axiom,
% 8.90/5.11 ! [M: nat,N: nat,F: nat > int] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( groups3539618377306564664at_int
% 8.90/5.11 @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 8.90/5.11 = ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7259_sum__telescope_H_H,axiom,
% 8.90/5.11 ! [M: nat,N: nat,F: nat > real] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( groups6591440286371151544t_real
% 8.90/5.11 @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 8.90/5.11 = ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7260_mask__eq__sum__exp,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
% 8.90/5.11 = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.11 @ ( collect_nat
% 8.90/5.11 @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7261_mask__eq__sum__exp,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
% 8.90/5.11 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.11 @ ( collect_nat
% 8.90/5.11 @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7262_sum__gp__multiplied,axiom,
% 8.90/5.11 ! [M: nat,N: nat,X2: complex] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 8.90/5.11 = ( minus_minus_complex @ ( power_power_complex @ X2 @ M ) @ ( power_power_complex @ X2 @ ( suc @ N ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7263_sum__gp__multiplied,axiom,
% 8.90/5.11 ! [M: nat,N: nat,X2: rat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 8.90/5.11 = ( minus_minus_rat @ ( power_power_rat @ X2 @ M ) @ ( power_power_rat @ X2 @ ( suc @ N ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7264_sum__gp__multiplied,axiom,
% 8.90/5.11 ! [M: nat,N: nat,X2: int] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 8.90/5.11 = ( minus_minus_int @ ( power_power_int @ X2 @ M ) @ ( power_power_int @ X2 @ ( suc @ N ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7265_sum__gp__multiplied,axiom,
% 8.90/5.11 ! [M: nat,N: nat,X2: real] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 8.90/5.11 = ( minus_minus_real @ ( power_power_real @ X2 @ M ) @ ( power_power_real @ X2 @ ( suc @ N ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7266_sum_Oin__pairs,axiom,
% 8.90/5.11 ! [G: nat > rat,M: nat,N: nat] :
% 8.90/5.11 ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 8.90/5.11 = ( groups2906978787729119204at_rat
% 8.90/5.11 @ ^ [I4: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7267_sum_Oin__pairs,axiom,
% 8.90/5.11 ! [G: nat > int,M: nat,N: nat] :
% 8.90/5.11 ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 8.90/5.11 = ( groups3539618377306564664at_int
% 8.90/5.11 @ ^ [I4: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7268_sum_Oin__pairs,axiom,
% 8.90/5.11 ! [G: nat > nat,M: nat,N: nat] :
% 8.90/5.11 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 8.90/5.11 = ( groups3542108847815614940at_nat
% 8.90/5.11 @ ^ [I4: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7269_sum_Oin__pairs,axiom,
% 8.90/5.11 ! [G: nat > real,M: nat,N: nat] :
% 8.90/5.11 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 8.90/5.11 = ( groups6591440286371151544t_real
% 8.90/5.11 @ ^ [I4: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7270_subrelI,axiom,
% 8.90/5.11 ! [R2: set_Pr448751882837621926eger_o,S: set_Pr448751882837621926eger_o] :
% 8.90/5.11 ( ! [X5: code_integer,Y4: $o] :
% 8.90/5.11 ( ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X5 @ Y4 ) @ R2 )
% 8.90/5.11 => ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X5 @ Y4 ) @ S ) )
% 8.90/5.11 => ( ord_le8980329558974975238eger_o @ R2 @ S ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7271_subrelI,axiom,
% 8.90/5.11 ! [R2: set_Pr8218934625190621173um_num,S: set_Pr8218934625190621173um_num] :
% 8.90/5.11 ( ! [X5: num,Y4: num] :
% 8.90/5.11 ( ( member7279096912039735102um_num @ ( product_Pair_num_num @ X5 @ Y4 ) @ R2 )
% 8.90/5.11 => ( member7279096912039735102um_num @ ( product_Pair_num_num @ X5 @ Y4 ) @ S ) )
% 8.90/5.11 => ( ord_le880128212290418581um_num @ R2 @ S ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7272_subrelI,axiom,
% 8.90/5.11 ! [R2: set_Pr6200539531224447659at_num,S: set_Pr6200539531224447659at_num] :
% 8.90/5.11 ( ! [X5: nat,Y4: num] :
% 8.90/5.11 ( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X5 @ Y4 ) @ R2 )
% 8.90/5.11 => ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X5 @ Y4 ) @ S ) )
% 8.90/5.11 => ( ord_le8085105155179020875at_num @ R2 @ S ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7273_subrelI,axiom,
% 8.90/5.11 ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
% 8.90/5.11 ( ! [X5: nat,Y4: nat] :
% 8.90/5.11 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ R2 )
% 8.90/5.11 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ S ) )
% 8.90/5.11 => ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7274_subrelI,axiom,
% 8.90/5.11 ! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
% 8.90/5.11 ( ! [X5: int,Y4: int] :
% 8.90/5.11 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ R2 )
% 8.90/5.11 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ S ) )
% 8.90/5.11 => ( ord_le2843351958646193337nt_int @ R2 @ S ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7275_accp__subset__induct,axiom,
% 8.90/5.11 ! [D4: product_prod_num_num > $o,R: product_prod_num_num > product_prod_num_num > $o,X2: product_prod_num_num,P: product_prod_num_num > $o] :
% 8.90/5.11 ( ( ord_le2239182809043710856_num_o @ D4 @ ( accp_P3113834385874906142um_num @ R ) )
% 8.90/5.11 => ( ! [X5: product_prod_num_num,Z2: product_prod_num_num] :
% 8.90/5.11 ( ( D4 @ X5 )
% 8.90/5.11 => ( ( R @ Z2 @ X5 )
% 8.90/5.11 => ( D4 @ Z2 ) ) )
% 8.90/5.11 => ( ( D4 @ X2 )
% 8.90/5.11 => ( ! [X5: product_prod_num_num] :
% 8.90/5.11 ( ( D4 @ X5 )
% 8.90/5.11 => ( ! [Z4: product_prod_num_num] :
% 8.90/5.11 ( ( R @ Z4 @ X5 )
% 8.90/5.11 => ( P @ Z4 ) )
% 8.90/5.11 => ( P @ X5 ) ) )
% 8.90/5.11 => ( P @ X2 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7276_accp__subset__induct,axiom,
% 8.90/5.11 ! [D4: product_prod_nat_nat > $o,R: product_prod_nat_nat > product_prod_nat_nat > $o,X2: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
% 8.90/5.11 ( ( ord_le704812498762024988_nat_o @ D4 @ ( accp_P4275260045618599050at_nat @ R ) )
% 8.90/5.11 => ( ! [X5: product_prod_nat_nat,Z2: product_prod_nat_nat] :
% 8.90/5.11 ( ( D4 @ X5 )
% 8.90/5.11 => ( ( R @ Z2 @ X5 )
% 8.90/5.11 => ( D4 @ Z2 ) ) )
% 8.90/5.11 => ( ( D4 @ X2 )
% 8.90/5.11 => ( ! [X5: product_prod_nat_nat] :
% 8.90/5.11 ( ( D4 @ X5 )
% 8.90/5.11 => ( ! [Z4: product_prod_nat_nat] :
% 8.90/5.11 ( ( R @ Z4 @ X5 )
% 8.90/5.11 => ( P @ Z4 ) )
% 8.90/5.11 => ( P @ X5 ) ) )
% 8.90/5.11 => ( P @ X2 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7277_accp__subset__induct,axiom,
% 8.90/5.11 ! [D4: product_prod_int_int > $o,R: product_prod_int_int > product_prod_int_int > $o,X2: product_prod_int_int,P: product_prod_int_int > $o] :
% 8.90/5.11 ( ( ord_le8369615600986905444_int_o @ D4 @ ( accp_P1096762738010456898nt_int @ R ) )
% 8.90/5.11 => ( ! [X5: product_prod_int_int,Z2: product_prod_int_int] :
% 8.90/5.11 ( ( D4 @ X5 )
% 8.90/5.11 => ( ( R @ Z2 @ X5 )
% 8.90/5.11 => ( D4 @ Z2 ) ) )
% 8.90/5.11 => ( ( D4 @ X2 )
% 8.90/5.11 => ( ! [X5: product_prod_int_int] :
% 8.90/5.11 ( ( D4 @ X5 )
% 8.90/5.11 => ( ! [Z4: product_prod_int_int] :
% 8.90/5.11 ( ( R @ Z4 @ X5 )
% 8.90/5.11 => ( P @ Z4 ) )
% 8.90/5.11 => ( P @ X5 ) ) )
% 8.90/5.11 => ( P @ X2 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7278_accp__subset__induct,axiom,
% 8.90/5.11 ! [D4: list_nat > $o,R: list_nat > list_nat > $o,X2: list_nat,P: list_nat > $o] :
% 8.90/5.11 ( ( ord_le1520216061033275535_nat_o @ D4 @ ( accp_list_nat @ R ) )
% 8.90/5.11 => ( ! [X5: list_nat,Z2: list_nat] :
% 8.90/5.11 ( ( D4 @ X5 )
% 8.90/5.11 => ( ( R @ Z2 @ X5 )
% 8.90/5.11 => ( D4 @ Z2 ) ) )
% 8.90/5.11 => ( ( D4 @ X2 )
% 8.90/5.11 => ( ! [X5: list_nat] :
% 8.90/5.11 ( ( D4 @ X5 )
% 8.90/5.11 => ( ! [Z4: list_nat] :
% 8.90/5.11 ( ( R @ Z4 @ X5 )
% 8.90/5.11 => ( P @ Z4 ) )
% 8.90/5.11 => ( P @ X5 ) ) )
% 8.90/5.11 => ( P @ X2 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7279_accp__subset__induct,axiom,
% 8.90/5.11 ! [D4: nat > $o,R: nat > nat > $o,X2: nat,P: nat > $o] :
% 8.90/5.11 ( ( ord_less_eq_nat_o @ D4 @ ( accp_nat @ R ) )
% 8.90/5.11 => ( ! [X5: nat,Z2: nat] :
% 8.90/5.11 ( ( D4 @ X5 )
% 8.90/5.11 => ( ( R @ Z2 @ X5 )
% 8.90/5.11 => ( D4 @ Z2 ) ) )
% 8.90/5.11 => ( ( D4 @ X2 )
% 8.90/5.11 => ( ! [X5: nat] :
% 8.90/5.11 ( ( D4 @ X5 )
% 8.90/5.11 => ( ! [Z4: nat] :
% 8.90/5.11 ( ( R @ Z4 @ X5 )
% 8.90/5.11 => ( P @ Z4 ) )
% 8.90/5.11 => ( P @ X5 ) ) )
% 8.90/5.11 => ( P @ X2 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7280_mask__eq__sum__exp__nat,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
% 8.90/5.11 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.11 @ ( collect_nat
% 8.90/5.11 @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7281_gauss__sum__nat,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( groups3542108847815614940at_nat
% 8.90/5.11 @ ^ [X: nat] : X
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 8.90/5.11 = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7282_arith__series__nat,axiom,
% 8.90/5.11 ! [A: nat,D: nat,N: nat] :
% 8.90/5.11 ( ( groups3542108847815614940at_nat
% 8.90/5.11 @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I4 @ D ) )
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 8.90/5.11 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7283_Sum__Icc__nat,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( groups3542108847815614940at_nat
% 8.90/5.11 @ ^ [X: nat] : X
% 8.90/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7284_pred__subset__eq,axiom,
% 8.90/5.11 ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
% 8.90/5.11 ( ( ord_le704812498762024988_nat_o
% 8.90/5.11 @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ R )
% 8.90/5.11 @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ S3 ) )
% 8.90/5.11 = ( ord_le3146513528884898305at_nat @ R @ S3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7285_pred__subset__eq,axiom,
% 8.90/5.11 ! [R: set_complex,S3: set_complex] :
% 8.90/5.11 ( ( ord_le4573692005234683329plex_o
% 8.90/5.11 @ ^ [X: complex] : ( member_complex @ X @ R )
% 8.90/5.11 @ ^ [X: complex] : ( member_complex @ X @ S3 ) )
% 8.90/5.11 = ( ord_le211207098394363844omplex @ R @ S3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7286_pred__subset__eq,axiom,
% 8.90/5.11 ! [R: set_real,S3: set_real] :
% 8.90/5.11 ( ( ord_less_eq_real_o
% 8.90/5.11 @ ^ [X: real] : ( member_real @ X @ R )
% 8.90/5.11 @ ^ [X: real] : ( member_real @ X @ S3 ) )
% 8.90/5.11 = ( ord_less_eq_set_real @ R @ S3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7287_pred__subset__eq,axiom,
% 8.90/5.11 ! [R: set_nat,S3: set_nat] :
% 8.90/5.11 ( ( ord_less_eq_nat_o
% 8.90/5.11 @ ^ [X: nat] : ( member_nat @ X @ R )
% 8.90/5.11 @ ^ [X: nat] : ( member_nat @ X @ S3 ) )
% 8.90/5.11 = ( ord_less_eq_set_nat @ R @ S3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7288_pred__subset__eq,axiom,
% 8.90/5.11 ! [R: set_int,S3: set_int] :
% 8.90/5.11 ( ( ord_less_eq_int_o
% 8.90/5.11 @ ^ [X: int] : ( member_int @ X @ R )
% 8.90/5.11 @ ^ [X: int] : ( member_int @ X @ S3 ) )
% 8.90/5.11 = ( ord_less_eq_set_int @ R @ S3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7289_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.11 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 8.90/5.11 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7290_of__int__round__le,axiom,
% 8.90/5.11 ! [X2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) @ ( plus_plus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7291_of__int__round__le,axiom,
% 8.90/5.11 ! [X2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) @ ( plus_plus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7292_of__int__round__ge,axiom,
% 8.90/5.11 ! [X2: real] : ( ord_less_eq_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7293_of__int__round__ge,axiom,
% 8.90/5.11 ! [X2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7294_of__int__round__gt,axiom,
% 8.90/5.11 ! [X2: real] : ( ord_less_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7295_of__int__round__gt,axiom,
% 8.90/5.11 ! [X2: rat] : ( ord_less_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7296_vebt__maxt_Opelims,axiom,
% 8.90/5.11 ! [X2: vEBT_VEBT,Y3: option_nat] :
% 8.90/5.11 ( ( ( vEBT_vebt_maxt @ X2 )
% 8.90/5.11 = Y3 )
% 8.90/5.11 => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X2 )
% 8.90/5.11 => ( ! [A4: $o,B4: $o] :
% 8.90/5.11 ( ( X2
% 8.90/5.11 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.90/5.11 => ( ( ( B4
% 8.90/5.11 => ( Y3
% 8.90/5.11 = ( some_nat @ one_one_nat ) ) )
% 8.90/5.11 & ( ~ B4
% 8.90/5.11 => ( ( A4
% 8.90/5.11 => ( Y3
% 8.90/5.11 = ( some_nat @ zero_zero_nat ) ) )
% 8.90/5.11 & ( ~ A4
% 8.90/5.11 => ( Y3 = none_nat ) ) ) ) )
% 8.90/5.11 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
% 8.90/5.11 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.90/5.11 ( ( X2
% 8.90/5.11 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 8.90/5.11 => ( ( Y3 = none_nat )
% 8.90/5.11 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 8.90/5.11 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 8.90/5.11 ( ( X2
% 8.90/5.11 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 8.90/5.11 => ( ( Y3
% 8.90/5.11 = ( some_nat @ Ma2 ) )
% 8.90/5.11 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7297_vebt__mint_Opelims,axiom,
% 8.90/5.11 ! [X2: vEBT_VEBT,Y3: option_nat] :
% 8.90/5.11 ( ( ( vEBT_vebt_mint @ X2 )
% 8.90/5.11 = Y3 )
% 8.90/5.11 => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X2 )
% 8.90/5.11 => ( ! [A4: $o,B4: $o] :
% 8.90/5.11 ( ( X2
% 8.90/5.11 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.90/5.11 => ( ( ( A4
% 8.90/5.11 => ( Y3
% 8.90/5.11 = ( some_nat @ zero_zero_nat ) ) )
% 8.90/5.11 & ( ~ A4
% 8.90/5.11 => ( ( B4
% 8.90/5.11 => ( Y3
% 8.90/5.11 = ( some_nat @ one_one_nat ) ) )
% 8.90/5.11 & ( ~ B4
% 8.90/5.11 => ( Y3 = none_nat ) ) ) ) )
% 8.90/5.11 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
% 8.90/5.11 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.90/5.11 ( ( X2
% 8.90/5.11 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 8.90/5.11 => ( ( Y3 = none_nat )
% 8.90/5.11 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 8.90/5.11 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 8.90/5.11 ( ( X2
% 8.90/5.11 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 8.90/5.11 => ( ( Y3
% 8.90/5.11 = ( some_nat @ Mi2 ) )
% 8.90/5.11 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7298_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
% 8.90/5.11 ! [X2: vEBT_VEBT,Y3: nat] :
% 8.90/5.11 ( ( ( vEBT_T_m_i_n_t @ X2 )
% 8.90/5.11 = Y3 )
% 8.90/5.11 => ( ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ X2 )
% 8.90/5.11 => ( ! [A4: $o,B4: $o] :
% 8.90/5.11 ( ( X2
% 8.90/5.11 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.90/5.11 => ( ( Y3
% 8.90/5.11 = ( plus_plus_nat @ one_one_nat @ ( if_nat @ A4 @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
% 8.90/5.11 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
% 8.90/5.11 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.90/5.11 ( ( X2
% 8.90/5.11 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 8.90/5.11 => ( ( Y3 = one_one_nat )
% 8.90/5.11 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 8.90/5.11 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 8.90/5.11 ( ( X2
% 8.90/5.11 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 8.90/5.11 => ( ( Y3 = one_one_nat )
% 8.90/5.11 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7299_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
% 8.90/5.11 ! [X2: vEBT_VEBT,Y3: nat] :
% 8.90/5.11 ( ( ( vEBT_T_m_a_x_t @ X2 )
% 8.90/5.11 = Y3 )
% 8.90/5.11 => ( ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ X2 )
% 8.90/5.11 => ( ! [A4: $o,B4: $o] :
% 8.90/5.11 ( ( X2
% 8.90/5.11 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.90/5.11 => ( ( Y3
% 8.90/5.11 = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B4 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
% 8.90/5.11 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
% 8.90/5.11 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 8.90/5.11 ( ( X2
% 8.90/5.11 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 8.90/5.11 => ( ( Y3 = one_one_nat )
% 8.90/5.11 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 8.90/5.11 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 8.90/5.11 ( ( X2
% 8.90/5.11 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 8.90/5.11 => ( ( Y3 = one_one_nat )
% 8.90/5.11 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7300_fold__atLeastAtMost__nat_Opinduct,axiom,
% 8.90/5.11 ! [A0: nat > num > num,A12: nat,A23: nat,A32: num,P: ( nat > num > num ) > nat > nat > num > $o] :
% 8.90/5.11 ( ( accp_P4916641582247091100at_num @ set_fo256927282339908995el_num @ ( produc851828971589881931at_num @ A0 @ ( produc1195630363706982562at_num @ A12 @ ( product_Pair_nat_num @ A23 @ A32 ) ) ) )
% 8.90/5.11 => ( ! [F2: nat > num > num,A4: nat,B4: nat,Acc: num] :
% 8.90/5.11 ( ( accp_P4916641582247091100at_num @ set_fo256927282339908995el_num @ ( produc851828971589881931at_num @ F2 @ ( produc1195630363706982562at_num @ A4 @ ( product_Pair_nat_num @ B4 @ Acc ) ) ) )
% 8.90/5.11 => ( ( ~ ( ord_less_nat @ B4 @ A4 )
% 8.90/5.11 => ( P @ F2 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B4 @ ( F2 @ A4 @ Acc ) ) )
% 8.90/5.11 => ( P @ F2 @ A4 @ B4 @ Acc ) ) )
% 8.90/5.11 => ( P @ A0 @ A12 @ A23 @ A32 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7301_fold__atLeastAtMost__nat_Opinduct,axiom,
% 8.90/5.11 ! [A0: nat > nat > nat,A12: nat,A23: nat,A32: nat,P: ( nat > nat > nat ) > nat > nat > nat > $o] :
% 8.90/5.11 ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ A0 @ ( produc487386426758144856at_nat @ A12 @ ( product_Pair_nat_nat @ A23 @ A32 ) ) ) )
% 8.90/5.11 => ( ! [F2: nat > nat > nat,A4: nat,B4: nat,Acc: nat] :
% 8.90/5.11 ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A4 @ ( product_Pair_nat_nat @ B4 @ Acc ) ) ) )
% 8.90/5.11 => ( ( ~ ( ord_less_nat @ B4 @ A4 )
% 8.90/5.11 => ( P @ F2 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B4 @ ( F2 @ A4 @ Acc ) ) )
% 8.90/5.11 => ( P @ F2 @ A4 @ B4 @ Acc ) ) )
% 8.90/5.11 => ( P @ A0 @ A12 @ A23 @ A32 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7302_VEBT__internal_Ooption__shift_Opelims,axiom,
% 8.90/5.11 ! [X2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
% 8.90/5.11 ( ( ( vEBT_V1502963449132264192at_nat @ X2 @ Xa2 @ Xb )
% 8.90/5.11 = Y3 )
% 8.90/5.11 => ( ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ Xa2 @ Xb ) ) )
% 8.90/5.11 => ( ( ( Xa2 = none_P5556105721700978146at_nat )
% 8.90/5.11 => ( ( Y3 = none_P5556105721700978146at_nat )
% 8.90/5.11 => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Xb ) ) ) ) )
% 8.90/5.11 => ( ! [V2: product_prod_nat_nat] :
% 8.90/5.11 ( ( Xa2
% 8.90/5.11 = ( some_P7363390416028606310at_nat @ V2 ) )
% 8.90/5.11 => ( ( Xb = none_P5556105721700978146at_nat )
% 8.90/5.11 => ( ( Y3 = none_P5556105721700978146at_nat )
% 8.90/5.11 => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) ) ) ) )
% 8.90/5.11 => ~ ! [A4: product_prod_nat_nat] :
% 8.90/5.11 ( ( Xa2
% 8.90/5.11 = ( some_P7363390416028606310at_nat @ A4 ) )
% 8.90/5.11 => ! [B4: product_prod_nat_nat] :
% 8.90/5.11 ( ( Xb
% 8.90/5.11 = ( some_P7363390416028606310at_nat @ B4 ) )
% 8.90/5.11 => ( ( Y3
% 8.90/5.11 = ( some_P7363390416028606310at_nat @ ( X2 @ A4 @ B4 ) ) )
% 8.90/5.11 => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A4 ) @ ( some_P7363390416028606310at_nat @ B4 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7303_VEBT__internal_Ooption__shift_Opelims,axiom,
% 8.90/5.11 ! [X2: num > num > num,Xa2: option_num,Xb: option_num,Y3: option_num] :
% 8.90/5.11 ( ( ( vEBT_V819420779217536731ft_num @ X2 @ Xa2 @ Xb )
% 8.90/5.11 = Y3 )
% 8.90/5.11 => ( ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ Xa2 @ Xb ) ) )
% 8.90/5.11 => ( ( ( Xa2 = none_num )
% 8.90/5.11 => ( ( Y3 = none_num )
% 8.90/5.11 => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ none_num @ Xb ) ) ) ) )
% 8.90/5.11 => ( ! [V2: num] :
% 8.90/5.11 ( ( Xa2
% 8.90/5.11 = ( some_num @ V2 ) )
% 8.90/5.11 => ( ( Xb = none_num )
% 8.90/5.11 => ( ( Y3 = none_num )
% 8.90/5.11 => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) ) ) ) )
% 8.90/5.11 => ~ ! [A4: num] :
% 8.90/5.11 ( ( Xa2
% 8.90/5.11 = ( some_num @ A4 ) )
% 8.90/5.11 => ! [B4: num] :
% 8.90/5.11 ( ( Xb
% 8.90/5.11 = ( some_num @ B4 ) )
% 8.90/5.11 => ( ( Y3
% 8.90/5.11 = ( some_num @ ( X2 @ A4 @ B4 ) ) )
% 8.90/5.11 => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ ( some_num @ A4 ) @ ( some_num @ B4 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7304_VEBT__internal_Ooption__shift_Opelims,axiom,
% 8.90/5.11 ! [X2: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y3: option_nat] :
% 8.90/5.11 ( ( ( vEBT_V4262088993061758097ft_nat @ X2 @ Xa2 @ Xb )
% 8.90/5.11 = Y3 )
% 8.90/5.11 => ( ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ Xa2 @ Xb ) ) )
% 8.90/5.11 => ( ( ( Xa2 = none_nat )
% 8.90/5.11 => ( ( Y3 = none_nat )
% 8.90/5.11 => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ none_nat @ Xb ) ) ) ) )
% 8.90/5.11 => ( ! [V2: nat] :
% 8.90/5.11 ( ( Xa2
% 8.90/5.11 = ( some_nat @ V2 ) )
% 8.90/5.11 => ( ( Xb = none_nat )
% 8.90/5.11 => ( ( Y3 = none_nat )
% 8.90/5.11 => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) ) ) ) )
% 8.90/5.11 => ~ ! [A4: nat] :
% 8.90/5.11 ( ( Xa2
% 8.90/5.11 = ( some_nat @ A4 ) )
% 8.90/5.11 => ! [B4: nat] :
% 8.90/5.11 ( ( Xb
% 8.90/5.11 = ( some_nat @ B4 ) )
% 8.90/5.11 => ( ( Y3
% 8.90/5.11 = ( some_nat @ ( X2 @ A4 @ B4 ) ) )
% 8.90/5.11 => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A4 ) @ ( some_nat @ B4 ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7305_zdvd1__eq,axiom,
% 8.90/5.11 ! [X2: int] :
% 8.90/5.11 ( ( dvd_dvd_int @ X2 @ one_one_int )
% 8.90/5.11 = ( ( abs_abs_int @ X2 )
% 8.90/5.11 = one_one_int ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7306_zabs__less__one__iff,axiom,
% 8.90/5.11 ! [Z: int] :
% 8.90/5.11 ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 8.90/5.11 = ( Z = zero_zero_int ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7307_abs__zmult__eq__1,axiom,
% 8.90/5.11 ! [M: int,N: int] :
% 8.90/5.11 ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
% 8.90/5.11 = one_one_int )
% 8.90/5.11 => ( ( abs_abs_int @ M )
% 8.90/5.11 = one_one_int ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7308_dvd__imp__le__int,axiom,
% 8.90/5.11 ! [I2: int,D: int] :
% 8.90/5.11 ( ( I2 != zero_zero_int )
% 8.90/5.11 => ( ( dvd_dvd_int @ D @ I2 )
% 8.90/5.11 => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7309_abs__mod__less,axiom,
% 8.90/5.11 ! [L: int,K: int] :
% 8.90/5.11 ( ( L != zero_zero_int )
% 8.90/5.11 => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7310_zdvd__mult__cancel1,axiom,
% 8.90/5.11 ! [M: int,N: int] :
% 8.90/5.11 ( ( M != zero_zero_int )
% 8.90/5.11 => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
% 8.90/5.11 = ( ( abs_abs_int @ N )
% 8.90/5.11 = one_one_int ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7311_even__add__abs__iff,axiom,
% 8.90/5.11 ! [K: int,L: int] :
% 8.90/5.11 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L ) ) )
% 8.90/5.11 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7312_even__abs__add__iff,axiom,
% 8.90/5.11 ! [K: int,L: int] :
% 8.90/5.11 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L ) )
% 8.90/5.11 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7313_nat__intermed__int__val,axiom,
% 8.90/5.11 ! [M: nat,N: nat,F: nat > int,K: int] :
% 8.90/5.11 ( ! [I3: nat] :
% 8.90/5.11 ( ( ( ord_less_eq_nat @ M @ I3 )
% 8.90/5.11 & ( ord_less_nat @ I3 @ N ) )
% 8.90/5.11 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 8.90/5.11 => ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.11 => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 8.90/5.11 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 8.90/5.11 => ? [I3: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ M @ I3 )
% 8.90/5.11 & ( ord_less_eq_nat @ I3 @ N )
% 8.90/5.11 & ( ( F @ I3 )
% 8.90/5.11 = K ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7314_decr__lemma,axiom,
% 8.90/5.11 ! [D: int,X2: int,Z: int] :
% 8.90/5.11 ( ( ord_less_int @ zero_zero_int @ D )
% 8.90/5.11 => ( ord_less_int @ ( minus_minus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7315_incr__lemma,axiom,
% 8.90/5.11 ! [D: int,Z: int,X2: int] :
% 8.90/5.11 ( ( ord_less_int @ zero_zero_int @ D )
% 8.90/5.11 => ( ord_less_int @ Z @ ( plus_plus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7316_nat__ivt__aux,axiom,
% 8.90/5.11 ! [N: nat,F: nat > int,K: int] :
% 8.90/5.11 ( ! [I3: nat] :
% 8.90/5.11 ( ( ord_less_nat @ I3 @ N )
% 8.90/5.11 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 8.90/5.11 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 8.90/5.11 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 8.90/5.11 => ? [I3: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ I3 @ N )
% 8.90/5.11 & ( ( F @ I3 )
% 8.90/5.11 = K ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7317_nat0__intermed__int__val,axiom,
% 8.90/5.11 ! [N: nat,F: nat > int,K: int] :
% 8.90/5.11 ( ! [I3: nat] :
% 8.90/5.11 ( ( ord_less_nat @ I3 @ N )
% 8.90/5.11 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 8.90/5.11 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 8.90/5.11 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 8.90/5.11 => ? [I3: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ I3 @ N )
% 8.90/5.11 & ( ( F @ I3 )
% 8.90/5.11 = K ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7318_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
% 8.90/5.11 ! [X2: vEBT_VEBT,Y3: nat] :
% 8.90/5.11 ( ( ( vEBT_T_m_i_n_N_u_l_l @ X2 )
% 8.90/5.11 = Y3 )
% 8.90/5.11 => ( ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ X2 )
% 8.90/5.11 => ( ( ( X2
% 8.90/5.11 = ( vEBT_Leaf @ $false @ $false ) )
% 8.90/5.11 => ( ( Y3 = one_one_nat )
% 8.90/5.11 => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
% 8.90/5.11 => ( ! [Uv2: $o] :
% 8.90/5.11 ( ( X2
% 8.90/5.11 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 8.90/5.11 => ( ( Y3 = one_one_nat )
% 8.90/5.11 => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
% 8.90/5.11 => ( ! [Uu2: $o] :
% 8.90/5.11 ( ( X2
% 8.90/5.11 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 8.90/5.11 => ( ( Y3 = one_one_nat )
% 8.90/5.11 => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
% 8.90/5.11 => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 8.90/5.11 ( ( X2
% 8.90/5.11 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 8.90/5.11 => ( ( Y3 = one_one_nat )
% 8.90/5.11 => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
% 8.90/5.11 => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.90/5.11 ( ( X2
% 8.90/5.11 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
% 8.90/5.11 => ( ( Y3 = one_one_nat )
% 8.90/5.11 => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7319_fold__atLeastAtMost__nat_Opelims,axiom,
% 8.90/5.11 ! [X2: nat > num > num,Xa2: nat,Xb: nat,Xc: num,Y3: num] :
% 8.90/5.11 ( ( ( set_fo8365102181078989356at_num @ X2 @ Xa2 @ Xb @ Xc )
% 8.90/5.11 = Y3 )
% 8.90/5.11 => ( ( accp_P4916641582247091100at_num @ set_fo256927282339908995el_num @ ( produc851828971589881931at_num @ X2 @ ( produc1195630363706982562at_num @ Xa2 @ ( product_Pair_nat_num @ Xb @ Xc ) ) ) )
% 8.90/5.11 => ~ ( ( ( ( ord_less_nat @ Xb @ Xa2 )
% 8.90/5.11 => ( Y3 = Xc ) )
% 8.90/5.11 & ( ~ ( ord_less_nat @ Xb @ Xa2 )
% 8.90/5.11 => ( Y3
% 8.90/5.11 = ( set_fo8365102181078989356at_num @ X2 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X2 @ Xa2 @ Xc ) ) ) ) )
% 8.90/5.11 => ~ ( accp_P4916641582247091100at_num @ set_fo256927282339908995el_num @ ( produc851828971589881931at_num @ X2 @ ( produc1195630363706982562at_num @ Xa2 @ ( product_Pair_nat_num @ Xb @ Xc ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7320_fold__atLeastAtMost__nat_Opelims,axiom,
% 8.90/5.11 ! [X2: nat > nat > nat,Xa2: nat,Xb: nat,Xc: nat,Y3: nat] :
% 8.90/5.11 ( ( ( set_fo2584398358068434914at_nat @ X2 @ Xa2 @ Xb @ Xc )
% 8.90/5.11 = Y3 )
% 8.90/5.11 => ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X2 @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) )
% 8.90/5.11 => ~ ( ( ( ( ord_less_nat @ Xb @ Xa2 )
% 8.90/5.11 => ( Y3 = Xc ) )
% 8.90/5.11 & ( ~ ( ord_less_nat @ Xb @ Xa2 )
% 8.90/5.11 => ( Y3
% 8.90/5.11 = ( set_fo2584398358068434914at_nat @ X2 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X2 @ Xa2 @ Xc ) ) ) ) )
% 8.90/5.11 => ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X2 @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7321_fold__atLeastAtMost__nat_Opsimps,axiom,
% 8.90/5.11 ! [F: nat > num > num,A: nat,B: nat,Acc2: num] :
% 8.90/5.11 ( ( accp_P4916641582247091100at_num @ set_fo256927282339908995el_num @ ( produc851828971589881931at_num @ F @ ( produc1195630363706982562at_num @ A @ ( product_Pair_nat_num @ B @ Acc2 ) ) ) )
% 8.90/5.11 => ( ( ( ord_less_nat @ B @ A )
% 8.90/5.11 => ( ( set_fo8365102181078989356at_num @ F @ A @ B @ Acc2 )
% 8.90/5.11 = Acc2 ) )
% 8.90/5.11 & ( ~ ( ord_less_nat @ B @ A )
% 8.90/5.11 => ( ( set_fo8365102181078989356at_num @ F @ A @ B @ Acc2 )
% 8.90/5.11 = ( set_fo8365102181078989356at_num @ F @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F @ A @ Acc2 ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7322_fold__atLeastAtMost__nat_Opsimps,axiom,
% 8.90/5.11 ! [F: nat > nat > nat,A: nat,B: nat,Acc2: nat] :
% 8.90/5.11 ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc2 ) ) ) )
% 8.90/5.11 => ( ( ( ord_less_nat @ B @ A )
% 8.90/5.11 => ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc2 )
% 8.90/5.11 = Acc2 ) )
% 8.90/5.11 & ( ~ ( ord_less_nat @ B @ A )
% 8.90/5.11 => ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc2 )
% 8.90/5.11 = ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F @ A @ Acc2 ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7323_in__measure,axiom,
% 8.90/5.11 ! [X2: num,Y3: num,F: num > nat] :
% 8.90/5.11 ( ( member7279096912039735102um_num @ ( product_Pair_num_num @ X2 @ Y3 ) @ ( measure_num @ F ) )
% 8.90/5.11 = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7324_in__measure,axiom,
% 8.90/5.11 ! [X2: nat,Y3: nat,F: nat > nat] :
% 8.90/5.11 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( measure_nat @ F ) )
% 8.90/5.11 = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7325_in__measure,axiom,
% 8.90/5.11 ! [X2: int,Y3: int,F: int > nat] :
% 8.90/5.11 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ ( measure_int @ F ) )
% 8.90/5.11 = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7326_sum__gp,axiom,
% 8.90/5.11 ! [N: nat,M: nat,X2: complex] :
% 8.90/5.11 ( ( ( ord_less_nat @ N @ M )
% 8.90/5.11 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = zero_zero_complex ) )
% 8.90/5.11 & ( ~ ( ord_less_nat @ N @ M )
% 8.90/5.11 => ( ( ( X2 = one_one_complex )
% 8.90/5.11 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 8.90/5.11 & ( ( X2 != one_one_complex )
% 8.90/5.11 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X2 @ M ) @ ( power_power_complex @ X2 @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7327_sum__gp,axiom,
% 8.90/5.11 ! [N: nat,M: nat,X2: rat] :
% 8.90/5.11 ( ( ( ord_less_nat @ N @ M )
% 8.90/5.11 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = zero_zero_rat ) )
% 8.90/5.11 & ( ~ ( ord_less_nat @ N @ M )
% 8.90/5.11 => ( ( ( X2 = one_one_rat )
% 8.90/5.11 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 8.90/5.11 & ( ( X2 != one_one_rat )
% 8.90/5.11 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X2 @ M ) @ ( power_power_rat @ X2 @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7328_sum__gp,axiom,
% 8.90/5.11 ! [N: nat,M: nat,X2: real] :
% 8.90/5.11 ( ( ( ord_less_nat @ N @ M )
% 8.90/5.11 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = zero_zero_real ) )
% 8.90/5.11 & ( ~ ( ord_less_nat @ N @ M )
% 8.90/5.11 => ( ( ( X2 = one_one_real )
% 8.90/5.11 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 8.90/5.11 & ( ( X2 != one_one_real )
% 8.90/5.11 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 8.90/5.11 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ M ) @ ( power_power_real @ X2 @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7329_arctan__double,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 8.90/5.11 => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X2 ) )
% 8.90/5.11 = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7330_of__nat__eq__iff,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ( semiri1314217659103216013at_int @ M )
% 8.90/5.11 = ( semiri1314217659103216013at_int @ N ) )
% 8.90/5.11 = ( M = N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7331_of__nat__eq__iff,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ( semiri5074537144036343181t_real @ M )
% 8.90/5.11 = ( semiri5074537144036343181t_real @ N ) )
% 8.90/5.11 = ( M = N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7332_of__nat__eq__iff,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ( semiri1316708129612266289at_nat @ M )
% 8.90/5.11 = ( semiri1316708129612266289at_nat @ N ) )
% 8.90/5.11 = ( M = N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7333_of__nat__eq__iff,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ( semiri681578069525770553at_rat @ M )
% 8.90/5.11 = ( semiri681578069525770553at_rat @ N ) )
% 8.90/5.11 = ( M = N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7334_int__eq__iff__numeral,axiom,
% 8.90/5.11 ! [M: nat,V: num] :
% 8.90/5.11 ( ( ( semiri1314217659103216013at_int @ M )
% 8.90/5.11 = ( numeral_numeral_int @ V ) )
% 8.90/5.11 = ( M
% 8.90/5.11 = ( numeral_numeral_nat @ V ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7335_abs__of__nat,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 8.90/5.11 = ( semiri4939895301339042750nteger @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7336_abs__of__nat,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
% 8.90/5.11 = ( semiri1314217659103216013at_int @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7337_abs__of__nat,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
% 8.90/5.11 = ( semiri5074537144036343181t_real @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7338_abs__of__nat,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
% 8.90/5.11 = ( semiri681578069525770553at_rat @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7339_negative__zle,axiom,
% 8.90/5.11 ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7340_of__nat__eq__0__iff,axiom,
% 8.90/5.11 ! [M: nat] :
% 8.90/5.11 ( ( ( semiri8010041392384452111omplex @ M )
% 8.90/5.11 = zero_zero_complex )
% 8.90/5.11 = ( M = zero_zero_nat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7341_of__nat__eq__0__iff,axiom,
% 8.90/5.11 ! [M: nat] :
% 8.90/5.11 ( ( ( semiri1314217659103216013at_int @ M )
% 8.90/5.11 = zero_zero_int )
% 8.90/5.11 = ( M = zero_zero_nat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7342_of__nat__eq__0__iff,axiom,
% 8.90/5.11 ! [M: nat] :
% 8.90/5.11 ( ( ( semiri5074537144036343181t_real @ M )
% 8.90/5.11 = zero_zero_real )
% 8.90/5.11 = ( M = zero_zero_nat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7343_of__nat__eq__0__iff,axiom,
% 8.90/5.11 ! [M: nat] :
% 8.90/5.11 ( ( ( semiri1316708129612266289at_nat @ M )
% 8.90/5.11 = zero_zero_nat )
% 8.90/5.11 = ( M = zero_zero_nat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7344_of__nat__eq__0__iff,axiom,
% 8.90/5.11 ! [M: nat] :
% 8.90/5.11 ( ( ( semiri681578069525770553at_rat @ M )
% 8.90/5.11 = zero_zero_rat )
% 8.90/5.11 = ( M = zero_zero_nat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7345_of__nat__0__eq__iff,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( zero_zero_complex
% 8.90/5.11 = ( semiri8010041392384452111omplex @ N ) )
% 8.90/5.11 = ( zero_zero_nat = N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7346_of__nat__0__eq__iff,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( zero_zero_int
% 8.90/5.11 = ( semiri1314217659103216013at_int @ N ) )
% 8.90/5.11 = ( zero_zero_nat = N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7347_of__nat__0__eq__iff,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( zero_zero_real
% 8.90/5.11 = ( semiri5074537144036343181t_real @ N ) )
% 8.90/5.11 = ( zero_zero_nat = N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7348_of__nat__0__eq__iff,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( zero_zero_nat
% 8.90/5.11 = ( semiri1316708129612266289at_nat @ N ) )
% 8.90/5.11 = ( zero_zero_nat = N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7349_of__nat__0__eq__iff,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( zero_zero_rat
% 8.90/5.11 = ( semiri681578069525770553at_rat @ N ) )
% 8.90/5.11 = ( zero_zero_nat = N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7350_of__nat__0,axiom,
% 8.90/5.11 ( ( semiri8010041392384452111omplex @ zero_zero_nat )
% 8.90/5.11 = zero_zero_complex ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7351_of__nat__0,axiom,
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 8.90/5.11 = zero_zero_int ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7352_of__nat__0,axiom,
% 8.90/5.11 ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 8.90/5.11 = zero_zero_real ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7353_of__nat__0,axiom,
% 8.90/5.11 ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 8.90/5.11 = zero_zero_nat ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7354_of__nat__0,axiom,
% 8.90/5.11 ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 8.90/5.11 = zero_zero_rat ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7355_of__nat__less__iff,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 8.90/5.11 = ( ord_less_nat @ M @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7356_of__nat__less__iff,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 8.90/5.11 = ( ord_less_nat @ M @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7357_of__nat__less__iff,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 8.90/5.11 = ( ord_less_nat @ M @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7358_of__nat__less__iff,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 8.90/5.11 = ( ord_less_nat @ M @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7359_of__nat__numeral,axiom,
% 8.90/5.11 ! [N: num] :
% 8.90/5.11 ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
% 8.90/5.11 = ( numera6690914467698888265omplex @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7360_of__nat__numeral,axiom,
% 8.90/5.11 ! [N: num] :
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 8.90/5.11 = ( numeral_numeral_int @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7361_of__nat__numeral,axiom,
% 8.90/5.11 ! [N: num] :
% 8.90/5.11 ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
% 8.90/5.11 = ( numeral_numeral_real @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7362_of__nat__numeral,axiom,
% 8.90/5.11 ! [N: num] :
% 8.90/5.11 ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
% 8.90/5.11 = ( numeral_numeral_nat @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7363_of__nat__numeral,axiom,
% 8.90/5.11 ! [N: num] :
% 8.90/5.11 ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
% 8.90/5.11 = ( numeral_numeral_rat @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7364_of__nat__le__iff,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 8.90/5.11 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7365_of__nat__le__iff,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 8.90/5.11 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7366_of__nat__le__iff,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 8.90/5.11 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7367_of__nat__le__iff,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 8.90/5.11 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7368_of__nat__add,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
% 8.90/5.11 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7369_of__nat__add,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
% 8.90/5.11 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7370_of__nat__add,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
% 8.90/5.11 = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7371_of__nat__add,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
% 8.90/5.11 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7372_of__nat__eq__1__iff,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( ( semiri8010041392384452111omplex @ N )
% 8.90/5.11 = one_one_complex )
% 8.90/5.11 = ( N = one_one_nat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7373_of__nat__eq__1__iff,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( ( semiri1314217659103216013at_int @ N )
% 8.90/5.11 = one_one_int )
% 8.90/5.11 = ( N = one_one_nat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7374_of__nat__eq__1__iff,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( ( semiri5074537144036343181t_real @ N )
% 8.90/5.11 = one_one_real )
% 8.90/5.11 = ( N = one_one_nat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7375_of__nat__eq__1__iff,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( ( semiri1316708129612266289at_nat @ N )
% 8.90/5.11 = one_one_nat )
% 8.90/5.11 = ( N = one_one_nat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7376_of__nat__eq__1__iff,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( ( semiri681578069525770553at_rat @ N )
% 8.90/5.11 = one_one_rat )
% 8.90/5.11 = ( N = one_one_nat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7377_of__nat__1__eq__iff,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( one_one_complex
% 8.90/5.11 = ( semiri8010041392384452111omplex @ N ) )
% 8.90/5.11 = ( N = one_one_nat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7378_of__nat__1__eq__iff,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( one_one_int
% 8.90/5.11 = ( semiri1314217659103216013at_int @ N ) )
% 8.90/5.11 = ( N = one_one_nat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7379_of__nat__1__eq__iff,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( one_one_real
% 8.90/5.11 = ( semiri5074537144036343181t_real @ N ) )
% 8.90/5.11 = ( N = one_one_nat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7380_of__nat__1__eq__iff,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( one_one_nat
% 8.90/5.11 = ( semiri1316708129612266289at_nat @ N ) )
% 8.90/5.11 = ( N = one_one_nat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7381_of__nat__1__eq__iff,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( one_one_rat
% 8.90/5.11 = ( semiri681578069525770553at_rat @ N ) )
% 8.90/5.11 = ( N = one_one_nat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7382_of__nat__1,axiom,
% 8.90/5.11 ( ( semiri8010041392384452111omplex @ one_one_nat )
% 8.90/5.11 = one_one_complex ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7383_of__nat__1,axiom,
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 8.90/5.11 = one_one_int ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7384_of__nat__1,axiom,
% 8.90/5.11 ( ( semiri5074537144036343181t_real @ one_one_nat )
% 8.90/5.11 = one_one_real ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7385_of__nat__1,axiom,
% 8.90/5.11 ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 8.90/5.11 = one_one_nat ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7386_of__nat__1,axiom,
% 8.90/5.11 ( ( semiri681578069525770553at_rat @ one_one_nat )
% 8.90/5.11 = one_one_rat ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7387_of__nat__mult,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
% 8.90/5.11 = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7388_of__nat__mult,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
% 8.90/5.11 = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7389_of__nat__mult,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
% 8.90/5.11 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7390_of__nat__mult,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
% 8.90/5.11 = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7391_of__nat__power,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
% 8.90/5.11 = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7392_of__nat__power,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
% 8.90/5.11 = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7393_of__nat__power,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
% 8.90/5.11 = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7394_of__nat__power,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
% 8.90/5.11 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7395_of__nat__power,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N ) )
% 8.90/5.11 = ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7396_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 8.90/5.11 ! [B: nat,W: nat,X2: nat] :
% 8.90/5.11 ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
% 8.90/5.11 = ( semiri8010041392384452111omplex @ X2 ) )
% 8.90/5.11 = ( ( power_power_nat @ B @ W )
% 8.90/5.11 = X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7397_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 8.90/5.11 ! [B: nat,W: nat,X2: nat] :
% 8.90/5.11 ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
% 8.90/5.11 = ( semiri1314217659103216013at_int @ X2 ) )
% 8.90/5.11 = ( ( power_power_nat @ B @ W )
% 8.90/5.11 = X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7398_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 8.90/5.11 ! [B: nat,W: nat,X2: nat] :
% 8.90/5.11 ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
% 8.90/5.11 = ( semiri5074537144036343181t_real @ X2 ) )
% 8.90/5.11 = ( ( power_power_nat @ B @ W )
% 8.90/5.11 = X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7399_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 8.90/5.11 ! [B: nat,W: nat,X2: nat] :
% 8.90/5.11 ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
% 8.90/5.11 = ( semiri1316708129612266289at_nat @ X2 ) )
% 8.90/5.11 = ( ( power_power_nat @ B @ W )
% 8.90/5.11 = X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7400_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 8.90/5.11 ! [B: nat,W: nat,X2: nat] :
% 8.90/5.11 ( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W )
% 8.90/5.11 = ( semiri681578069525770553at_rat @ X2 ) )
% 8.90/5.11 = ( ( power_power_nat @ B @ W )
% 8.90/5.11 = X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7401_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,B: nat,W: nat] :
% 8.90/5.11 ( ( ( semiri8010041392384452111omplex @ X2 )
% 8.90/5.11 = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
% 8.90/5.11 = ( X2
% 8.90/5.11 = ( power_power_nat @ B @ W ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7402_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,B: nat,W: nat] :
% 8.90/5.11 ( ( ( semiri1314217659103216013at_int @ X2 )
% 8.90/5.11 = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 8.90/5.11 = ( X2
% 8.90/5.11 = ( power_power_nat @ B @ W ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7403_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,B: nat,W: nat] :
% 8.90/5.11 ( ( ( semiri5074537144036343181t_real @ X2 )
% 8.90/5.11 = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 8.90/5.11 = ( X2
% 8.90/5.11 = ( power_power_nat @ B @ W ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7404_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,B: nat,W: nat] :
% 8.90/5.11 ( ( ( semiri1316708129612266289at_nat @ X2 )
% 8.90/5.11 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 8.90/5.11 = ( X2
% 8.90/5.11 = ( power_power_nat @ B @ W ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7405_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,B: nat,W: nat] :
% 8.90/5.11 ( ( ( semiri681578069525770553at_rat @ X2 )
% 8.90/5.11 = ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 8.90/5.11 = ( X2
% 8.90/5.11 = ( power_power_nat @ B @ W ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7406_negative__zless,axiom,
% 8.90/5.11 ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7407_arctan__le__zero__iff,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( arctan @ X2 ) @ zero_zero_real )
% 8.90/5.11 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7408_zero__le__arctan__iff,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X2 ) )
% 8.90/5.11 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7409_of__nat__of__bool,axiom,
% 8.90/5.11 ! [P: $o] :
% 8.90/5.11 ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
% 8.90/5.11 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7410_of__nat__of__bool,axiom,
% 8.90/5.11 ! [P: $o] :
% 8.90/5.11 ( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 8.90/5.11 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7411_of__nat__of__bool,axiom,
% 8.90/5.11 ! [P: $o] :
% 8.90/5.11 ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 8.90/5.11 = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7412_of__nat__of__bool,axiom,
% 8.90/5.11 ! [P: $o] :
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
% 8.90/5.11 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7413_of__nat__of__bool,axiom,
% 8.90/5.11 ! [P: $o] :
% 8.90/5.11 ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
% 8.90/5.11 = ( zero_n356916108424825756nteger @ P ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7414_of__nat__le__0__iff,axiom,
% 8.90/5.11 ! [M: nat] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 8.90/5.11 = ( M = zero_zero_nat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7415_of__nat__le__0__iff,axiom,
% 8.90/5.11 ! [M: nat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 8.90/5.11 = ( M = zero_zero_nat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7416_of__nat__le__0__iff,axiom,
% 8.90/5.11 ! [M: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 8.90/5.11 = ( M = zero_zero_nat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7417_of__nat__le__0__iff,axiom,
% 8.90/5.11 ! [M: nat] :
% 8.90/5.11 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 8.90/5.11 = ( M = zero_zero_nat ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7418_of__nat__Suc,axiom,
% 8.90/5.11 ! [M: nat] :
% 8.90/5.11 ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 8.90/5.11 = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7419_of__nat__Suc,axiom,
% 8.90/5.11 ! [M: nat] :
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 8.90/5.11 = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7420_of__nat__Suc,axiom,
% 8.90/5.11 ! [M: nat] :
% 8.90/5.11 ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 8.90/5.11 = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7421_of__nat__Suc,axiom,
% 8.90/5.11 ! [M: nat] :
% 8.90/5.11 ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 8.90/5.11 = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7422_of__nat__Suc,axiom,
% 8.90/5.11 ! [M: nat] :
% 8.90/5.11 ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 8.90/5.11 = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7423_numeral__less__real__of__nat__iff,axiom,
% 8.90/5.11 ! [W: num,N: nat] :
% 8.90/5.11 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
% 8.90/5.11 = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7424_real__of__nat__less__numeral__iff,axiom,
% 8.90/5.11 ! [N: nat,W: num] :
% 8.90/5.11 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
% 8.90/5.11 = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7425_numeral__le__real__of__nat__iff,axiom,
% 8.90/5.11 ! [N: num,M: nat] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
% 8.90/5.11 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7426_of__nat__0__less__iff,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
% 8.90/5.11 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7427_of__nat__0__less__iff,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
% 8.90/5.11 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7428_of__nat__0__less__iff,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
% 8.90/5.11 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7429_of__nat__0__less__iff,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
% 8.90/5.11 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7430_of__nat__less__of__nat__power__cancel__iff,axiom,
% 8.90/5.11 ! [B: nat,W: nat,X2: nat] :
% 8.90/5.11 ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X2 ) )
% 8.90/5.11 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7431_of__nat__less__of__nat__power__cancel__iff,axiom,
% 8.90/5.11 ! [B: nat,W: nat,X2: nat] :
% 8.90/5.11 ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X2 ) )
% 8.90/5.11 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7432_of__nat__less__of__nat__power__cancel__iff,axiom,
% 8.90/5.11 ! [B: nat,W: nat,X2: nat] :
% 8.90/5.11 ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X2 ) )
% 8.90/5.11 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7433_of__nat__less__of__nat__power__cancel__iff,axiom,
% 8.90/5.11 ! [B: nat,W: nat,X2: nat] :
% 8.90/5.11 ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X2 ) )
% 8.90/5.11 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7434_of__nat__power__less__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,B: nat,W: nat] :
% 8.90/5.11 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 8.90/5.11 = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7435_of__nat__power__less__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,B: nat,W: nat] :
% 8.90/5.11 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 8.90/5.11 = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7436_of__nat__power__less__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,B: nat,W: nat] :
% 8.90/5.11 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 8.90/5.11 = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7437_of__nat__power__less__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,B: nat,W: nat] :
% 8.90/5.11 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 8.90/5.11 = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7438_numeral__power__eq__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [X2: num,N: nat,Y3: nat] :
% 8.90/5.11 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X2 ) @ N )
% 8.90/5.11 = ( semiri8010041392384452111omplex @ Y3 ) )
% 8.90/5.11 = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
% 8.90/5.11 = Y3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7439_numeral__power__eq__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [X2: num,N: nat,Y3: nat] :
% 8.90/5.11 ( ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N )
% 8.90/5.11 = ( semiri1314217659103216013at_int @ Y3 ) )
% 8.90/5.11 = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
% 8.90/5.11 = Y3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7440_numeral__power__eq__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [X2: num,N: nat,Y3: nat] :
% 8.90/5.11 ( ( ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N )
% 8.90/5.11 = ( semiri5074537144036343181t_real @ Y3 ) )
% 8.90/5.11 = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
% 8.90/5.11 = Y3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7441_numeral__power__eq__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [X2: num,N: nat,Y3: nat] :
% 8.90/5.11 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
% 8.90/5.11 = ( semiri1316708129612266289at_nat @ Y3 ) )
% 8.90/5.11 = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
% 8.90/5.11 = Y3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7442_numeral__power__eq__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [X2: num,N: nat,Y3: nat] :
% 8.90/5.11 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N )
% 8.90/5.11 = ( semiri681578069525770553at_rat @ Y3 ) )
% 8.90/5.11 = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
% 8.90/5.11 = Y3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7443_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 8.90/5.11 ! [Y3: nat,X2: num,N: nat] :
% 8.90/5.11 ( ( ( semiri8010041392384452111omplex @ Y3 )
% 8.90/5.11 = ( power_power_complex @ ( numera6690914467698888265omplex @ X2 ) @ N ) )
% 8.90/5.11 = ( Y3
% 8.90/5.11 = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7444_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 8.90/5.11 ! [Y3: nat,X2: num,N: nat] :
% 8.90/5.11 ( ( ( semiri1314217659103216013at_int @ Y3 )
% 8.90/5.11 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) )
% 8.90/5.11 = ( Y3
% 8.90/5.11 = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7445_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 8.90/5.11 ! [Y3: nat,X2: num,N: nat] :
% 8.90/5.11 ( ( ( semiri5074537144036343181t_real @ Y3 )
% 8.90/5.11 = ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
% 8.90/5.11 = ( Y3
% 8.90/5.11 = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7446_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 8.90/5.11 ! [Y3: nat,X2: num,N: nat] :
% 8.90/5.11 ( ( ( semiri1316708129612266289at_nat @ Y3 )
% 8.90/5.11 = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) )
% 8.90/5.11 = ( Y3
% 8.90/5.11 = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7447_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 8.90/5.11 ! [Y3: nat,X2: num,N: nat] :
% 8.90/5.11 ( ( ( semiri681578069525770553at_rat @ Y3 )
% 8.90/5.11 = ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) )
% 8.90/5.11 = ( Y3
% 8.90/5.11 = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7448_of__nat__power__le__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,B: nat,W: nat] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 8.90/5.11 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7449_of__nat__power__le__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,B: nat,W: nat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 8.90/5.11 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7450_of__nat__power__le__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,B: nat,W: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 8.90/5.11 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7451_of__nat__power__le__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,B: nat,W: nat] :
% 8.90/5.11 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 8.90/5.11 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7452_of__nat__le__of__nat__power__cancel__iff,axiom,
% 8.90/5.11 ! [B: nat,W: nat,X2: nat] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X2 ) )
% 8.90/5.11 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7453_of__nat__le__of__nat__power__cancel__iff,axiom,
% 8.90/5.11 ! [B: nat,W: nat,X2: nat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X2 ) )
% 8.90/5.11 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7454_of__nat__le__of__nat__power__cancel__iff,axiom,
% 8.90/5.11 ! [B: nat,W: nat,X2: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X2 ) )
% 8.90/5.11 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7455_of__nat__le__of__nat__power__cancel__iff,axiom,
% 8.90/5.11 ! [B: nat,W: nat,X2: nat] :
% 8.90/5.11 ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X2 ) )
% 8.90/5.11 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7456_of__nat__zero__less__power__iff,axiom,
% 8.90/5.11 ! [X2: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X2 ) @ N ) )
% 8.90/5.11 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 8.90/5.11 | ( N = zero_zero_nat ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7457_of__nat__zero__less__power__iff,axiom,
% 8.90/5.11 ! [X2: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X2 ) @ N ) )
% 8.90/5.11 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 8.90/5.11 | ( N = zero_zero_nat ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7458_of__nat__zero__less__power__iff,axiom,
% 8.90/5.11 ! [X2: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ N ) )
% 8.90/5.11 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 8.90/5.11 | ( N = zero_zero_nat ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7459_of__nat__zero__less__power__iff,axiom,
% 8.90/5.11 ! [X2: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X2 ) @ N ) )
% 8.90/5.11 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 8.90/5.11 | ( N = zero_zero_nat ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7460_even__of__nat,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
% 8.90/5.11 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7461_even__of__nat,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
% 8.90/5.11 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7462_even__of__nat,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
% 8.90/5.11 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7463_numeral__power__less__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [I2: num,N: nat,X2: nat] :
% 8.90/5.11 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) @ ( semiri1314217659103216013at_int @ X2 ) )
% 8.90/5.11 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7464_numeral__power__less__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [I2: num,N: nat,X2: nat] :
% 8.90/5.11 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) @ ( semiri5074537144036343181t_real @ X2 ) )
% 8.90/5.11 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7465_numeral__power__less__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [I2: num,N: nat,X2: nat] :
% 8.90/5.11 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ ( semiri1316708129612266289at_nat @ X2 ) )
% 8.90/5.11 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7466_numeral__power__less__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [I2: num,N: nat,X2: nat] :
% 8.90/5.11 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) @ ( semiri681578069525770553at_rat @ X2 ) )
% 8.90/5.11 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7467_of__nat__less__numeral__power__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,I2: num,N: nat] :
% 8.90/5.11 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) )
% 8.90/5.11 = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7468_of__nat__less__numeral__power__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,I2: num,N: nat] :
% 8.90/5.11 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) )
% 8.90/5.11 = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7469_of__nat__less__numeral__power__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,I2: num,N: nat] :
% 8.90/5.11 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) )
% 8.90/5.11 = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7470_of__nat__less__numeral__power__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,I2: num,N: nat] :
% 8.90/5.11 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) )
% 8.90/5.11 = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7471_numeral__power__le__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [I2: num,N: nat,X2: nat] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) @ ( semiri5074537144036343181t_real @ X2 ) )
% 8.90/5.11 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7472_numeral__power__le__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [I2: num,N: nat,X2: nat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) @ ( semiri681578069525770553at_rat @ X2 ) )
% 8.90/5.11 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7473_numeral__power__le__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [I2: num,N: nat,X2: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ ( semiri1316708129612266289at_nat @ X2 ) )
% 8.90/5.11 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7474_numeral__power__le__of__nat__cancel__iff,axiom,
% 8.90/5.11 ! [I2: num,N: nat,X2: nat] :
% 8.90/5.11 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) @ ( semiri1314217659103216013at_int @ X2 ) )
% 8.90/5.11 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7475_of__nat__le__numeral__power__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,I2: num,N: nat] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) )
% 8.90/5.11 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7476_of__nat__le__numeral__power__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,I2: num,N: nat] :
% 8.90/5.11 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) )
% 8.90/5.11 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7477_of__nat__le__numeral__power__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,I2: num,N: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) )
% 8.90/5.11 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7478_of__nat__le__numeral__power__cancel__iff,axiom,
% 8.90/5.11 ! [X2: nat,I2: num,N: nat] :
% 8.90/5.11 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) )
% 8.90/5.11 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7479_real__arch__simple,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ? [N2: nat] : ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7480_real__arch__simple,axiom,
% 8.90/5.11 ! [X2: rat] :
% 8.90/5.11 ? [N2: nat] : ( ord_less_eq_rat @ X2 @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7481_reals__Archimedean2,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ? [N2: nat] : ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7482_reals__Archimedean2,axiom,
% 8.90/5.11 ! [X2: rat] :
% 8.90/5.11 ? [N2: nat] : ( ord_less_rat @ X2 @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7483_mult__of__nat__commute,axiom,
% 8.90/5.11 ! [X2: nat,Y3: int] :
% 8.90/5.11 ( ( times_times_int @ ( semiri1314217659103216013at_int @ X2 ) @ Y3 )
% 8.90/5.11 = ( times_times_int @ Y3 @ ( semiri1314217659103216013at_int @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7484_mult__of__nat__commute,axiom,
% 8.90/5.11 ! [X2: nat,Y3: real] :
% 8.90/5.11 ( ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ Y3 )
% 8.90/5.11 = ( times_times_real @ Y3 @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7485_mult__of__nat__commute,axiom,
% 8.90/5.11 ! [X2: nat,Y3: nat] :
% 8.90/5.11 ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ Y3 )
% 8.90/5.11 = ( times_times_nat @ Y3 @ ( semiri1316708129612266289at_nat @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7486_mult__of__nat__commute,axiom,
% 8.90/5.11 ! [X2: nat,Y3: rat] :
% 8.90/5.11 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X2 ) @ Y3 )
% 8.90/5.11 = ( times_times_rat @ Y3 @ ( semiri681578069525770553at_rat @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7487_arctan__monotone_H,axiom,
% 8.90/5.11 ! [X2: real,Y3: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ X2 @ Y3 )
% 8.90/5.11 => ( ord_less_eq_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7488_arctan__le__iff,axiom,
% 8.90/5.11 ! [X2: real,Y3: real] :
% 8.90/5.11 ( ( ord_less_eq_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) )
% 8.90/5.11 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7489_int__diff__cases,axiom,
% 8.90/5.11 ! [Z: int] :
% 8.90/5.11 ~ ! [M4: nat,N2: nat] :
% 8.90/5.11 ( Z
% 8.90/5.11 != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7490_of__nat__less__of__int__iff,axiom,
% 8.90/5.11 ! [N: nat,X2: int] :
% 8.90/5.11 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X2 ) )
% 8.90/5.11 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7491_of__nat__less__of__int__iff,axiom,
% 8.90/5.11 ! [N: nat,X2: int] :
% 8.90/5.11 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X2 ) )
% 8.90/5.11 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7492_of__nat__less__of__int__iff,axiom,
% 8.90/5.11 ! [N: nat,X2: int] :
% 8.90/5.11 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X2 ) )
% 8.90/5.11 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X2 ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7493_of__nat__0__le__iff,axiom,
% 8.90/5.11 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7494_of__nat__0__le__iff,axiom,
% 8.90/5.11 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7495_of__nat__0__le__iff,axiom,
% 8.90/5.11 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7496_of__nat__0__le__iff,axiom,
% 8.90/5.11 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7497_of__nat__less__0__iff,axiom,
% 8.90/5.11 ! [M: nat] :
% 8.90/5.11 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7498_of__nat__less__0__iff,axiom,
% 8.90/5.11 ! [M: nat] :
% 8.90/5.11 ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7499_of__nat__less__0__iff,axiom,
% 8.90/5.11 ! [M: nat] :
% 8.90/5.11 ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7500_of__nat__less__0__iff,axiom,
% 8.90/5.11 ! [M: nat] :
% 8.90/5.11 ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7501_of__nat__neq__0,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
% 8.90/5.11 != zero_zero_complex ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7502_of__nat__neq__0,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 8.90/5.11 != zero_zero_int ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7503_of__nat__neq__0,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
% 8.90/5.11 != zero_zero_real ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7504_of__nat__neq__0,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
% 8.90/5.11 != zero_zero_nat ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7505_of__nat__neq__0,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
% 8.90/5.11 != zero_zero_rat ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7506_div__mult2__eq_H,axiom,
% 8.90/5.11 ! [A: int,M: nat,N: nat] :
% 8.90/5.11 ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 8.90/5.11 = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7507_div__mult2__eq_H,axiom,
% 8.90/5.11 ! [A: nat,M: nat,N: nat] :
% 8.90/5.11 ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 8.90/5.11 = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7508_less__imp__of__nat__less,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_nat @ M @ N )
% 8.90/5.11 => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7509_less__imp__of__nat__less,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_nat @ M @ N )
% 8.90/5.11 => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7510_less__imp__of__nat__less,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_nat @ M @ N )
% 8.90/5.11 => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7511_less__imp__of__nat__less,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_nat @ M @ N )
% 8.90/5.11 => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7512_of__nat__less__imp__less,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 8.90/5.11 => ( ord_less_nat @ M @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7513_of__nat__less__imp__less,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 8.90/5.11 => ( ord_less_nat @ M @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7514_of__nat__less__imp__less,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 8.90/5.11 => ( ord_less_nat @ M @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7515_of__nat__less__imp__less,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 8.90/5.11 => ( ord_less_nat @ M @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7516_of__nat__mono,axiom,
% 8.90/5.11 ! [I2: nat,J: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ I2 @ J )
% 8.90/5.11 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I2 ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7517_of__nat__mono,axiom,
% 8.90/5.11 ! [I2: nat,J: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ I2 @ J )
% 8.90/5.11 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I2 ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7518_of__nat__mono,axiom,
% 8.90/5.11 ! [I2: nat,J: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ I2 @ J )
% 8.90/5.11 => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7519_of__nat__mono,axiom,
% 8.90/5.11 ! [I2: nat,J: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ I2 @ J )
% 8.90/5.11 => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7520_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
% 8.90/5.11 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7521_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
% 8.90/5.11 = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7522_of__nat__dvd__iff,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
% 8.90/5.11 = ( dvd_dvd_nat @ M @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7523_of__nat__dvd__iff,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 8.90/5.11 = ( dvd_dvd_nat @ M @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7524_of__nat__dvd__iff,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 8.90/5.11 = ( dvd_dvd_nat @ M @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7525_int__ops_I3_J,axiom,
% 8.90/5.11 ! [N: num] :
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 8.90/5.11 = ( numeral_numeral_int @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7526_int__cases,axiom,
% 8.90/5.11 ! [Z: int] :
% 8.90/5.11 ( ! [N2: nat] :
% 8.90/5.11 ( Z
% 8.90/5.11 != ( semiri1314217659103216013at_int @ N2 ) )
% 8.90/5.11 => ~ ! [N2: nat] :
% 8.90/5.11 ( Z
% 8.90/5.11 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7527_int__of__nat__induct,axiom,
% 8.90/5.11 ! [P: int > $o,Z: int] :
% 8.90/5.11 ( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
% 8.90/5.11 => ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
% 8.90/5.11 => ( P @ Z ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7528_nat__int__comparison_I2_J,axiom,
% 8.90/5.11 ( ord_less_nat
% 8.90/5.11 = ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7529_nat__int__comparison_I3_J,axiom,
% 8.90/5.11 ( ord_less_eq_nat
% 8.90/5.11 = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7530_zle__int,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 8.90/5.11 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7531_zero__le__imp__eq__int,axiom,
% 8.90/5.11 ! [K: int] :
% 8.90/5.11 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.90/5.11 => ? [N2: nat] :
% 8.90/5.11 ( K
% 8.90/5.11 = ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7532_nonneg__int__cases,axiom,
% 8.90/5.11 ! [K: int] :
% 8.90/5.11 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.90/5.11 => ~ ! [N2: nat] :
% 8.90/5.11 ( K
% 8.90/5.11 != ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7533_of__nat__mod,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
% 8.90/5.11 = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7534_of__nat__mod,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
% 8.90/5.11 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7535_of__nat__mod,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
% 8.90/5.11 = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7536_int__ops_I2_J,axiom,
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 8.90/5.11 = one_one_int ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7537_int__plus,axiom,
% 8.90/5.11 ! [N: nat,M: nat] :
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
% 8.90/5.11 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7538_int__ops_I5_J,axiom,
% 8.90/5.11 ! [A: nat,B: nat] :
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
% 8.90/5.11 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7539_zadd__int__left,axiom,
% 8.90/5.11 ! [M: nat,N: nat,Z: int] :
% 8.90/5.11 ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
% 8.90/5.11 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7540_int__ops_I7_J,axiom,
% 8.90/5.11 ! [A: nat,B: nat] :
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
% 8.90/5.11 = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7541_zle__iff__zadd,axiom,
% 8.90/5.11 ( ord_less_eq_int
% 8.90/5.11 = ( ^ [W3: int,Z3: int] :
% 8.90/5.11 ? [N3: nat] :
% 8.90/5.11 ( Z3
% 8.90/5.11 = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7542_zdiv__int,axiom,
% 8.90/5.11 ! [A: nat,B: nat] :
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
% 8.90/5.11 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7543_of__nat__max,axiom,
% 8.90/5.11 ! [X2: nat,Y3: nat] :
% 8.90/5.11 ( ( semiri4216267220026989637d_enat @ ( ord_max_nat @ X2 @ Y3 ) )
% 8.90/5.11 = ( ord_ma741700101516333627d_enat @ ( semiri4216267220026989637d_enat @ X2 ) @ ( semiri4216267220026989637d_enat @ Y3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7544_of__nat__max,axiom,
% 8.90/5.11 ! [X2: nat,Y3: nat] :
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X2 @ Y3 ) )
% 8.90/5.11 = ( ord_max_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7545_of__nat__max,axiom,
% 8.90/5.11 ! [X2: nat,Y3: nat] :
% 8.90/5.11 ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X2 @ Y3 ) )
% 8.90/5.11 = ( ord_max_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( semiri5074537144036343181t_real @ Y3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7546_of__nat__max,axiom,
% 8.90/5.11 ! [X2: nat,Y3: nat] :
% 8.90/5.11 ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X2 @ Y3 ) )
% 8.90/5.11 = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( semiri1316708129612266289at_nat @ Y3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7547_of__nat__max,axiom,
% 8.90/5.11 ! [X2: nat,Y3: nat] :
% 8.90/5.11 ( ( semiri681578069525770553at_rat @ ( ord_max_nat @ X2 @ Y3 ) )
% 8.90/5.11 = ( ord_max_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( semiri681578069525770553at_rat @ Y3 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7548_nat__less__as__int,axiom,
% 8.90/5.11 ( ord_less_nat
% 8.90/5.11 = ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7549_nat__leq__as__int,axiom,
% 8.90/5.11 ( ord_less_eq_nat
% 8.90/5.11 = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7550_ex__less__of__nat__mult,axiom,
% 8.90/5.11 ! [X2: real,Y3: real] :
% 8.90/5.11 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.11 => ? [N2: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7551_ex__less__of__nat__mult,axiom,
% 8.90/5.11 ! [X2: rat,Y3: rat] :
% 8.90/5.11 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 8.90/5.11 => ? [N2: nat] : ( ord_less_rat @ Y3 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7552_of__nat__diff,axiom,
% 8.90/5.11 ! [N: nat,M: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.11 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
% 8.90/5.11 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7553_of__nat__diff,axiom,
% 8.90/5.11 ! [N: nat,M: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.11 => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
% 8.90/5.11 = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7554_of__nat__diff,axiom,
% 8.90/5.11 ! [N: nat,M: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.11 => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
% 8.90/5.11 = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7555_of__nat__diff,axiom,
% 8.90/5.11 ! [N: nat,M: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.11 => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
% 8.90/5.11 = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7556_reals__Archimedean3,axiom,
% 8.90/5.11 ! [X2: real] :
% 8.90/5.11 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.11 => ! [Y5: real] :
% 8.90/5.11 ? [N2: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7557_int__cases4,axiom,
% 8.90/5.11 ! [M: int] :
% 8.90/5.11 ( ! [N2: nat] :
% 8.90/5.11 ( M
% 8.90/5.11 != ( semiri1314217659103216013at_int @ N2 ) )
% 8.90/5.11 => ~ ! [N2: nat] :
% 8.90/5.11 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 8.90/5.11 => ( M
% 8.90/5.11 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7558_real__of__nat__div4,axiom,
% 8.90/5.11 ! [N: nat,X2: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7559_int__ops_I4_J,axiom,
% 8.90/5.11 ! [A: nat] :
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
% 8.90/5.11 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7560_int__Suc,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 8.90/5.11 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7561_int__zle__neg,axiom,
% 8.90/5.11 ! [N: nat,M: nat] :
% 8.90/5.11 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
% 8.90/5.11 = ( ( N = zero_zero_nat )
% 8.90/5.11 & ( M = zero_zero_nat ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7562_zless__iff__Suc__zadd,axiom,
% 8.90/5.11 ( ord_less_int
% 8.90/5.11 = ( ^ [W3: int,Z3: int] :
% 8.90/5.11 ? [N3: nat] :
% 8.90/5.11 ( Z3
% 8.90/5.11 = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7563_nonpos__int__cases,axiom,
% 8.90/5.11 ! [K: int] :
% 8.90/5.11 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 8.90/5.11 => ~ ! [N2: nat] :
% 8.90/5.11 ( K
% 8.90/5.11 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7564_negative__zle__0,axiom,
% 8.90/5.11 ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7565_real__of__nat__div,axiom,
% 8.90/5.11 ! [D: nat,N: nat] :
% 8.90/5.11 ( ( dvd_dvd_nat @ D @ N )
% 8.90/5.11 => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
% 8.90/5.11 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7566_mod__mult2__eq_H,axiom,
% 8.90/5.11 ! [A: code_integer,M: nat,N: nat] :
% 8.90/5.11 ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
% 8.90/5.11 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7567_mod__mult2__eq_H,axiom,
% 8.90/5.11 ! [A: int,M: nat,N: nat] :
% 8.90/5.11 ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 8.90/5.11 = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7568_mod__mult2__eq_H,axiom,
% 8.90/5.11 ! [A: nat,M: nat,N: nat] :
% 8.90/5.11 ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 8.90/5.11 = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7569_field__char__0__class_Oof__nat__div,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
% 8.90/5.11 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7570_field__char__0__class_Oof__nat__div,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
% 8.90/5.11 = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7571_field__char__0__class_Oof__nat__div,axiom,
% 8.90/5.11 ! [M: nat,N: nat] :
% 8.90/5.11 ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
% 8.90/5.11 = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7572_zero__less__imp__eq__int,axiom,
% 8.90/5.11 ! [K: int] :
% 8.90/5.11 ( ( ord_less_int @ zero_zero_int @ K )
% 8.90/5.11 => ? [N2: nat] :
% 8.90/5.11 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 8.90/5.11 & ( K
% 8.90/5.11 = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7573_pos__int__cases,axiom,
% 8.90/5.11 ! [K: int] :
% 8.90/5.11 ( ( ord_less_int @ zero_zero_int @ K )
% 8.90/5.11 => ~ ! [N2: nat] :
% 8.90/5.11 ( ( K
% 8.90/5.11 = ( semiri1314217659103216013at_int @ N2 ) )
% 8.90/5.11 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7574_int__cases3,axiom,
% 8.90/5.11 ! [K: int] :
% 8.90/5.11 ( ( K != zero_zero_int )
% 8.90/5.11 => ( ! [N2: nat] :
% 8.90/5.11 ( ( K
% 8.90/5.11 = ( semiri1314217659103216013at_int @ N2 ) )
% 8.90/5.11 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 8.90/5.11 => ~ ! [N2: nat] :
% 8.90/5.11 ( ( K
% 8.90/5.11 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 8.90/5.11 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7575_nat__less__real__le,axiom,
% 8.90/5.11 ( ord_less_nat
% 8.90/5.11 = ( ^ [N3: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7576_nat__le__real__less,axiom,
% 8.90/5.11 ( ord_less_eq_nat
% 8.90/5.11 = ( ^ [N3: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7577_zmult__zless__mono2__lemma,axiom,
% 8.90/5.11 ! [I2: int,J: int,K: nat] :
% 8.90/5.11 ( ( ord_less_int @ I2 @ J )
% 8.90/5.11 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.90/5.11 => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7578_not__zle__0__negative,axiom,
% 8.90/5.11 ! [N: nat] :
% 8.90/5.11 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7579_negative__zless__0,axiom,
% 8.90/5.11 ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7580_negD,axiom,
% 8.90/5.11 ! [X2: int] :
% 8.90/5.11 ( ( ord_less_int @ X2 @ zero_zero_int )
% 8.90/5.11 => ? [N2: nat] :
% 8.90/5.11 ( X2
% 8.90/5.11 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7581_int__ops_I6_J,axiom,
% 8.90/5.11 ! [A: nat,B: nat] :
% 8.90/5.11 ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 8.90/5.11 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 8.90/5.11 = zero_zero_int ) )
% 8.90/5.11 & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 8.90/5.11 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 8.90/5.11 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7582_real__of__nat__div__aux,axiom,
% 8.90/5.11 ! [X2: nat,D: nat] :
% 8.90/5.11 ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( semiri5074537144036343181t_real @ D ) )
% 8.90/5.11 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X2 @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X2 @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7583_nat__approx__posE,axiom,
% 8.90/5.11 ! [E: real] :
% 8.90/5.11 ( ( ord_less_real @ zero_zero_real @ E )
% 8.90/5.11 => ~ ! [N2: nat] :
% 8.90/5.11 ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ E ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7584_nat__approx__posE,axiom,
% 8.90/5.11 ! [E: rat] :
% 8.90/5.11 ( ( ord_less_rat @ zero_zero_rat @ E )
% 8.90/5.11 => ~ ! [N2: nat] :
% 8.90/5.11 ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ E ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7585_of__nat__less__two__power,axiom,
% 8.90/5.11 ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7586_of__nat__less__two__power,axiom,
% 8.90/5.11 ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7587_of__nat__less__two__power,axiom,
% 8.90/5.11 ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7588_inverse__of__nat__le,axiom,
% 8.90/5.11 ! [N: nat,M: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.11 => ( ( N != zero_zero_nat )
% 8.90/5.11 => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7589_inverse__of__nat__le,axiom,
% 8.90/5.11 ! [N: nat,M: nat] :
% 8.90/5.11 ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.11 => ( ( N != zero_zero_nat )
% 8.90/5.11 => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7590_fold__atLeastAtMost__nat_Osimps,axiom,
% 8.90/5.11 ( set_fo2584398358068434914at_nat
% 8.90/5.11 = ( ^ [F3: nat > nat > nat,A3: nat,B2: nat,Acc3: nat] : ( if_nat @ ( ord_less_nat @ B2 @ A3 ) @ Acc3 @ ( set_fo2584398358068434914at_nat @ F3 @ ( plus_plus_nat @ A3 @ one_one_nat ) @ B2 @ ( F3 @ A3 @ Acc3 ) ) ) ) ) ).
% 8.90/5.11
% 8.90/5.11 thf(fact_7591_fold__atLeastAtMost__nat_Oelims,axiom,
% 8.90/5.11 ! [X2: nat > nat > nat,Xa2: nat,Xb: nat,Xc: nat,Y3: nat] :
% 8.90/5.11 ( ( ( set_fo2584398358068434914at_nat @ X2 @ Xa2 @ Xb @ Xc )
% 8.90/5.11 = Y3 )
% 8.90/5.11 => ( ( ( ord_less_nat @ Xb @ Xa2 )
% 8.90/5.11 => ( Y3 = Xc ) )
% 8.90/5.11 & ( ~ ( ord_less_nat @ Xb @ Xa2 )
% 8.90/5.12 => ( Y3
% 8.90/5.12 = ( set_fo2584398358068434914at_nat @ X2 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X2 @ Xa2 @ Xc ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7592_real__archimedian__rdiv__eq__0,axiom,
% 8.90/5.12 ! [X2: real,C: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 8.90/5.12 => ( ! [M4: nat] :
% 8.90/5.12 ( ( ord_less_nat @ zero_zero_nat @ M4 )
% 8.90/5.12 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X2 ) @ C ) )
% 8.90/5.12 => ( X2 = zero_zero_real ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7593_neg__int__cases,axiom,
% 8.90/5.12 ! [K: int] :
% 8.90/5.12 ( ( ord_less_int @ K @ zero_zero_int )
% 8.90/5.12 => ~ ! [N2: nat] :
% 8.90/5.12 ( ( K
% 8.90/5.12 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 8.90/5.12 => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7594_zdiff__int__split,axiom,
% 8.90/5.12 ! [P: int > $o,X2: nat,Y3: nat] :
% 8.90/5.12 ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X2 @ Y3 ) ) )
% 8.90/5.12 = ( ( ( ord_less_eq_nat @ Y3 @ X2 )
% 8.90/5.12 => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) )
% 8.90/5.12 & ( ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.12 => ( P @ zero_zero_int ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7595_real__of__nat__div2,axiom,
% 8.90/5.12 ! [N: nat,X2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7596_real__of__nat__div3,axiom,
% 8.90/5.12 ! [N: nat,X2: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) ) @ one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7597_ln__realpow,axiom,
% 8.90/5.12 ! [X2: real,N: nat] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ln_ln_real @ ( power_power_real @ X2 @ N ) )
% 8.90/5.12 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7598_linear__plus__1__le__power,axiom,
% 8.90/5.12 ! [X2: real,N: nat] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X2 @ one_one_real ) @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7599_Bernoulli__inequality,axiom,
% 8.90/5.12 ! [X2: real,N: nat] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.12 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X2 ) @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7600_sum__atLeastAtMost__code,axiom,
% 8.90/5.12 ! [F: nat > complex,A: nat,B: nat] :
% 8.90/5.12 ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 8.90/5.12 = ( set_fo1517530859248394432omplex
% 8.90/5.12 @ ^ [A3: nat] : ( plus_plus_complex @ ( F @ A3 ) )
% 8.90/5.12 @ A
% 8.90/5.12 @ B
% 8.90/5.12 @ zero_zero_complex ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7601_sum__atLeastAtMost__code,axiom,
% 8.90/5.12 ! [F: nat > rat,A: nat,B: nat] :
% 8.90/5.12 ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 8.90/5.12 = ( set_fo1949268297981939178at_rat
% 8.90/5.12 @ ^ [A3: nat] : ( plus_plus_rat @ ( F @ A3 ) )
% 8.90/5.12 @ A
% 8.90/5.12 @ B
% 8.90/5.12 @ zero_zero_rat ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7602_sum__atLeastAtMost__code,axiom,
% 8.90/5.12 ! [F: nat > int,A: nat,B: nat] :
% 8.90/5.12 ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 8.90/5.12 = ( set_fo2581907887559384638at_int
% 8.90/5.12 @ ^ [A3: nat] : ( plus_plus_int @ ( F @ A3 ) )
% 8.90/5.12 @ A
% 8.90/5.12 @ B
% 8.90/5.12 @ zero_zero_int ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7603_sum__atLeastAtMost__code,axiom,
% 8.90/5.12 ! [F: nat > nat,A: nat,B: nat] :
% 8.90/5.12 ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 8.90/5.12 = ( set_fo2584398358068434914at_nat
% 8.90/5.12 @ ^ [A3: nat] : ( plus_plus_nat @ ( F @ A3 ) )
% 8.90/5.12 @ A
% 8.90/5.12 @ B
% 8.90/5.12 @ zero_zero_nat ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7604_sum__atLeastAtMost__code,axiom,
% 8.90/5.12 ! [F: nat > real,A: nat,B: nat] :
% 8.90/5.12 ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 8.90/5.12 = ( set_fo3111899725591712190t_real
% 8.90/5.12 @ ^ [A3: nat] : ( plus_plus_real @ ( F @ A3 ) )
% 8.90/5.12 @ A
% 8.90/5.12 @ B
% 8.90/5.12 @ zero_zero_real ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7605_arctan__add,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 8.90/5.12 => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 8.90/5.12 => ( ( plus_plus_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) )
% 8.90/5.12 = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7606_double__gauss__sum,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 8.90/5.12 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7607_double__gauss__sum,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 8.90/5.12 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7608_double__gauss__sum,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 8.90/5.12 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7609_double__gauss__sum,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 8.90/5.12 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7610_double__gauss__sum,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 8.90/5.12 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7611_double__arith__series,axiom,
% 8.90/5.12 ! [A: complex,D: complex,N: nat] :
% 8.90/5.12 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 8.90/5.12 @ ( groups2073611262835488442omplex
% 8.90/5.12 @ ^ [I4: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I4 ) @ D ) )
% 8.90/5.12 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 8.90/5.12 = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ D ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7612_double__arith__series,axiom,
% 8.90/5.12 ! [A: int,D: int,N: nat] :
% 8.90/5.12 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 8.90/5.12 @ ( groups3539618377306564664at_int
% 8.90/5.12 @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 8.90/5.12 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 8.90/5.12 = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7613_double__arith__series,axiom,
% 8.90/5.12 ! [A: rat,D: rat,N: nat] :
% 8.90/5.12 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 8.90/5.12 @ ( groups2906978787729119204at_rat
% 8.90/5.12 @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I4 ) @ D ) )
% 8.90/5.12 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 8.90/5.12 = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7614_double__arith__series,axiom,
% 8.90/5.12 ! [A: nat,D: nat,N: nat] :
% 8.90/5.12 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
% 8.90/5.12 @ ( groups3542108847815614940at_nat
% 8.90/5.12 @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 8.90/5.12 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 8.90/5.12 = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7615_double__arith__series,axiom,
% 8.90/5.12 ! [A: real,D: real,N: nat] :
% 8.90/5.12 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 8.90/5.12 @ ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [I4: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I4 ) @ D ) )
% 8.90/5.12 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 8.90/5.12 = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ D ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7616_Bernoulli__inequality__even,axiom,
% 8.90/5.12 ! [N: nat,X2: real] :
% 8.90/5.12 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.12 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X2 ) @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7617_gauss__sum,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 8.90/5.12 = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7618_gauss__sum,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 8.90/5.12 = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7619_arith__series,axiom,
% 8.90/5.12 ! [A: int,D: int,N: nat] :
% 8.90/5.12 ( ( groups3539618377306564664at_int
% 8.90/5.12 @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 8.90/5.12 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 8.90/5.12 = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7620_arith__series,axiom,
% 8.90/5.12 ! [A: nat,D: nat,N: nat] :
% 8.90/5.12 ( ( groups3542108847815614940at_nat
% 8.90/5.12 @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 8.90/5.12 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 8.90/5.12 = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7621_double__gauss__sum__from__Suc__0,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 8.90/5.12 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7622_double__gauss__sum__from__Suc__0,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 8.90/5.12 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7623_double__gauss__sum__from__Suc__0,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 8.90/5.12 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7624_double__gauss__sum__from__Suc__0,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 8.90/5.12 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7625_double__gauss__sum__from__Suc__0,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 8.90/5.12 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7626_sum__gp__offset,axiom,
% 8.90/5.12 ! [X2: complex,M: nat,N: nat] :
% 8.90/5.12 ( ( ( X2 = one_one_complex )
% 8.90/5.12 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 8.90/5.12 = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) )
% 8.90/5.12 & ( ( X2 != one_one_complex )
% 8.90/5.12 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 8.90/5.12 = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X2 @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7627_sum__gp__offset,axiom,
% 8.90/5.12 ! [X2: rat,M: nat,N: nat] :
% 8.90/5.12 ( ( ( X2 = one_one_rat )
% 8.90/5.12 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 8.90/5.12 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
% 8.90/5.12 & ( ( X2 != one_one_rat )
% 8.90/5.12 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 8.90/5.12 = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X2 @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7628_sum__gp__offset,axiom,
% 8.90/5.12 ! [X2: real,M: nat,N: nat] :
% 8.90/5.12 ( ( ( X2 = one_one_real )
% 8.90/5.12 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 8.90/5.12 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) )
% 8.90/5.12 & ( ( X2 != one_one_real )
% 8.90/5.12 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 8.90/5.12 = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X2 @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( suc @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7629_gauss__sum__from__Suc__0,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 8.90/5.12 = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7630_gauss__sum__from__Suc__0,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 8.90/5.12 = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7631_of__nat__code__if,axiom,
% 8.90/5.12 ( semiri8010041392384452111omplex
% 8.90/5.12 = ( ^ [N3: nat] :
% 8.90/5.12 ( if_complex @ ( N3 = zero_zero_nat ) @ zero_zero_complex
% 8.90/5.12 @ ( produc1917071388513777916omplex
% 8.90/5.12 @ ^ [M3: nat,Q4: nat] : ( if_complex @ ( Q4 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ one_one_complex ) )
% 8.90/5.12 @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7632_of__nat__code__if,axiom,
% 8.90/5.12 ( semiri1314217659103216013at_int
% 8.90/5.12 = ( ^ [N3: nat] :
% 8.90/5.12 ( if_int @ ( N3 = zero_zero_nat ) @ zero_zero_int
% 8.90/5.12 @ ( produc6840382203811409530at_int
% 8.90/5.12 @ ^ [M3: nat,Q4: nat] : ( if_int @ ( Q4 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ one_one_int ) )
% 8.90/5.12 @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7633_of__nat__code__if,axiom,
% 8.90/5.12 ( semiri5074537144036343181t_real
% 8.90/5.12 = ( ^ [N3: nat] :
% 8.90/5.12 ( if_real @ ( N3 = zero_zero_nat ) @ zero_zero_real
% 8.90/5.12 @ ( produc1703576794950452218t_real
% 8.90/5.12 @ ^ [M3: nat,Q4: nat] : ( if_real @ ( Q4 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ one_one_real ) )
% 8.90/5.12 @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7634_of__nat__code__if,axiom,
% 8.90/5.12 ( semiri681578069525770553at_rat
% 8.90/5.12 = ( ^ [N3: nat] :
% 8.90/5.12 ( if_rat @ ( N3 = zero_zero_nat ) @ zero_zero_rat
% 8.90/5.12 @ ( produc6207742614233964070at_rat
% 8.90/5.12 @ ^ [M3: nat,Q4: nat] : ( if_rat @ ( Q4 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ one_one_rat ) )
% 8.90/5.12 @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7635_of__nat__code__if,axiom,
% 8.90/5.12 ( semiri1316708129612266289at_nat
% 8.90/5.12 = ( ^ [N3: nat] :
% 8.90/5.12 ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat
% 8.90/5.12 @ ( produc6842872674320459806at_nat
% 8.90/5.12 @ ^ [M3: nat,Q4: nat] : ( if_nat @ ( Q4 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ one_one_nat ) )
% 8.90/5.12 @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7636_height__double__log__univ__size,axiom,
% 8.90/5.12 ! [U: real,Deg: nat,T: vEBT_VEBT] :
% 8.90/5.12 ( ( U
% 8.90/5.12 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Deg ) )
% 8.90/5.12 => ( ( vEBT_invar_vebt @ T @ Deg )
% 8.90/5.12 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_height @ T ) ) @ ( plus_plus_real @ one_one_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7637_monoseq__arctan__series,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 8.90/5.12 => ( topolo6980174941875973593q_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7638_lemma__termdiff3,axiom,
% 8.90/5.12 ! [H2: real,Z: real,K5: real,N: nat] :
% 8.90/5.12 ( ( H2 != zero_zero_real )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K5 )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7639_lemma__termdiff3,axiom,
% 8.90/5.12 ! [H2: complex,Z: complex,K5: real,N: nat] :
% 8.90/5.12 ( ( H2 != zero_zero_complex )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K5 )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7640_ln__series,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.90/5.12 => ( ( ln_ln_real @ X2 )
% 8.90/5.12 = ( suminf_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X2 @ one_one_real ) @ ( suc @ N3 ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7641_lemma__termdiff2,axiom,
% 8.90/5.12 ! [H2: complex,Z: complex,N: nat] :
% 8.90/5.12 ( ( H2 != zero_zero_complex )
% 8.90/5.12 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 8.90/5.12 = ( times_times_complex @ H2
% 8.90/5.12 @ ( groups2073611262835488442omplex
% 8.90/5.12 @ ^ [P4: nat] :
% 8.90/5.12 ( groups2073611262835488442omplex
% 8.90/5.12 @ ^ [Q4: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q4 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P4 ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7642_lemma__termdiff2,axiom,
% 8.90/5.12 ! [H2: rat,Z: rat,N: nat] :
% 8.90/5.12 ( ( H2 != zero_zero_rat )
% 8.90/5.12 => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ N ) @ ( power_power_rat @ Z @ N ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 8.90/5.12 = ( times_times_rat @ H2
% 8.90/5.12 @ ( groups2906978787729119204at_rat
% 8.90/5.12 @ ^ [P4: nat] :
% 8.90/5.12 ( groups2906978787729119204at_rat
% 8.90/5.12 @ ^ [Q4: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ Q4 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P4 ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7643_lemma__termdiff2,axiom,
% 8.90/5.12 ! [H2: real,Z: real,N: nat] :
% 8.90/5.12 ( ( H2 != zero_zero_real )
% 8.90/5.12 => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 8.90/5.12 = ( times_times_real @ H2
% 8.90/5.12 @ ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [P4: nat] :
% 8.90/5.12 ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [Q4: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q4 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P4 ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7644_lessThan__iff,axiom,
% 8.90/5.12 ! [I2: rat,K: rat] :
% 8.90/5.12 ( ( member_rat @ I2 @ ( set_ord_lessThan_rat @ K ) )
% 8.90/5.12 = ( ord_less_rat @ I2 @ K ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7645_lessThan__iff,axiom,
% 8.90/5.12 ! [I2: num,K: num] :
% 8.90/5.12 ( ( member_num @ I2 @ ( set_ord_lessThan_num @ K ) )
% 8.90/5.12 = ( ord_less_num @ I2 @ K ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7646_lessThan__iff,axiom,
% 8.90/5.12 ! [I2: nat,K: nat] :
% 8.90/5.12 ( ( member_nat @ I2 @ ( set_ord_lessThan_nat @ K ) )
% 8.90/5.12 = ( ord_less_nat @ I2 @ K ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7647_lessThan__iff,axiom,
% 8.90/5.12 ! [I2: int,K: int] :
% 8.90/5.12 ( ( member_int @ I2 @ ( set_ord_lessThan_int @ K ) )
% 8.90/5.12 = ( ord_less_int @ I2 @ K ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7648_lessThan__iff,axiom,
% 8.90/5.12 ! [I2: real,K: real] :
% 8.90/5.12 ( ( member_real @ I2 @ ( set_or5984915006950818249n_real @ K ) )
% 8.90/5.12 = ( ord_less_real @ I2 @ K ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7649_lessThan__subset__iff,axiom,
% 8.90/5.12 ! [X2: rat,Y3: rat] :
% 8.90/5.12 ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X2 ) @ ( set_ord_lessThan_rat @ Y3 ) )
% 8.90/5.12 = ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7650_lessThan__subset__iff,axiom,
% 8.90/5.12 ! [X2: num,Y3: num] :
% 8.90/5.12 ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X2 ) @ ( set_ord_lessThan_num @ Y3 ) )
% 8.90/5.12 = ( ord_less_eq_num @ X2 @ Y3 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7651_lessThan__subset__iff,axiom,
% 8.90/5.12 ! [X2: nat,Y3: nat] :
% 8.90/5.12 ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X2 ) @ ( set_ord_lessThan_nat @ Y3 ) )
% 8.90/5.12 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7652_lessThan__subset__iff,axiom,
% 8.90/5.12 ! [X2: int,Y3: int] :
% 8.90/5.12 ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X2 ) @ ( set_ord_lessThan_int @ Y3 ) )
% 8.90/5.12 = ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7653_lessThan__subset__iff,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X2 ) @ ( set_or5984915006950818249n_real @ Y3 ) )
% 8.90/5.12 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7654_log__one,axiom,
% 8.90/5.12 ! [A: real] :
% 8.90/5.12 ( ( log @ A @ one_one_real )
% 8.90/5.12 = zero_zero_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7655_sum_OlessThan__Suc,axiom,
% 8.90/5.12 ! [G: nat > rat,N: nat] :
% 8.90/5.12 ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 8.90/5.12 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7656_sum_OlessThan__Suc,axiom,
% 8.90/5.12 ! [G: nat > int,N: nat] :
% 8.90/5.12 ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 8.90/5.12 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7657_sum_OlessThan__Suc,axiom,
% 8.90/5.12 ! [G: nat > nat,N: nat] :
% 8.90/5.12 ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 8.90/5.12 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7658_sum_OlessThan__Suc,axiom,
% 8.90/5.12 ! [G: nat > real,N: nat] :
% 8.90/5.12 ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 8.90/5.12 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7659_log__eq__one,axiom,
% 8.90/5.12 ! [A: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ A )
% 8.90/5.12 => ( ( A != one_one_real )
% 8.90/5.12 => ( ( log @ A @ A )
% 8.90/5.12 = one_one_real ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7660_log__less__cancel__iff,axiom,
% 8.90/5.12 ! [A: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ A )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.90/5.12 => ( ( ord_less_real @ ( log @ A @ X2 ) @ ( log @ A @ Y3 ) )
% 8.90/5.12 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7661_log__less__one__cancel__iff,axiom,
% 8.90/5.12 ! [A: real,X2: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ A )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ ( log @ A @ X2 ) @ one_one_real )
% 8.90/5.12 = ( ord_less_real @ X2 @ A ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7662_one__less__log__cancel__iff,axiom,
% 8.90/5.12 ! [A: real,X2: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ A )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ one_one_real @ ( log @ A @ X2 ) )
% 8.90/5.12 = ( ord_less_real @ A @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7663_log__less__zero__cancel__iff,axiom,
% 8.90/5.12 ! [A: real,X2: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ A )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ ( log @ A @ X2 ) @ zero_zero_real )
% 8.90/5.12 = ( ord_less_real @ X2 @ one_one_real ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7664_zero__less__log__cancel__iff,axiom,
% 8.90/5.12 ! [A: real,X2: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ A )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X2 ) )
% 8.90/5.12 = ( ord_less_real @ one_one_real @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7665_zero__le__log__cancel__iff,axiom,
% 8.90/5.12 ! [A: real,X2: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ A )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X2 ) )
% 8.90/5.12 = ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7666_log__le__zero__cancel__iff,axiom,
% 8.90/5.12 ! [A: real,X2: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ A )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( log @ A @ X2 ) @ zero_zero_real )
% 8.90/5.12 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7667_one__le__log__cancel__iff,axiom,
% 8.90/5.12 ! [A: real,X2: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ A )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X2 ) )
% 8.90/5.12 = ( ord_less_eq_real @ A @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7668_log__le__one__cancel__iff,axiom,
% 8.90/5.12 ! [A: real,X2: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ A )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( log @ A @ X2 ) @ one_one_real )
% 8.90/5.12 = ( ord_less_eq_real @ X2 @ A ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7669_log__le__cancel__iff,axiom,
% 8.90/5.12 ! [A: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ A )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( log @ A @ X2 ) @ ( log @ A @ Y3 ) )
% 8.90/5.12 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7670_powser__zero,axiom,
% 8.90/5.12 ! [F: nat > complex] :
% 8.90/5.12 ( ( suminf_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) )
% 8.90/5.12 = ( F @ zero_zero_nat ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7671_powser__zero,axiom,
% 8.90/5.12 ! [F: nat > real] :
% 8.90/5.12 ( ( suminf_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) )
% 8.90/5.12 = ( F @ zero_zero_nat ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7672_log__pow__cancel,axiom,
% 8.90/5.12 ! [A: real,B: nat] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ A )
% 8.90/5.12 => ( ( A != one_one_real )
% 8.90/5.12 => ( ( log @ A @ ( power_power_real @ A @ B ) )
% 8.90/5.12 = ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7673_lessThan__def,axiom,
% 8.90/5.12 ( set_ord_lessThan_rat
% 8.90/5.12 = ( ^ [U2: rat] :
% 8.90/5.12 ( collect_rat
% 8.90/5.12 @ ^ [X: rat] : ( ord_less_rat @ X @ U2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7674_lessThan__def,axiom,
% 8.90/5.12 ( set_ord_lessThan_num
% 8.90/5.12 = ( ^ [U2: num] :
% 8.90/5.12 ( collect_num
% 8.90/5.12 @ ^ [X: num] : ( ord_less_num @ X @ U2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7675_lessThan__def,axiom,
% 8.90/5.12 ( set_ord_lessThan_nat
% 8.90/5.12 = ( ^ [U2: nat] :
% 8.90/5.12 ( collect_nat
% 8.90/5.12 @ ^ [X: nat] : ( ord_less_nat @ X @ U2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7676_lessThan__def,axiom,
% 8.90/5.12 ( set_ord_lessThan_int
% 8.90/5.12 = ( ^ [U2: int] :
% 8.90/5.12 ( collect_int
% 8.90/5.12 @ ^ [X: int] : ( ord_less_int @ X @ U2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7677_lessThan__def,axiom,
% 8.90/5.12 ( set_or5984915006950818249n_real
% 8.90/5.12 = ( ^ [U2: real] :
% 8.90/5.12 ( collect_real
% 8.90/5.12 @ ^ [X: real] : ( ord_less_real @ X @ U2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7678_lessThan__strict__subset__iff,axiom,
% 8.90/5.12 ! [M: rat,N: rat] :
% 8.90/5.12 ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N ) )
% 8.90/5.12 = ( ord_less_rat @ M @ N ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7679_lessThan__strict__subset__iff,axiom,
% 8.90/5.12 ! [M: num,N: num] :
% 8.90/5.12 ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
% 8.90/5.12 = ( ord_less_num @ M @ N ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7680_lessThan__strict__subset__iff,axiom,
% 8.90/5.12 ! [M: nat,N: nat] :
% 8.90/5.12 ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.12 = ( ord_less_nat @ M @ N ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7681_lessThan__strict__subset__iff,axiom,
% 8.90/5.12 ! [M: int,N: int] :
% 8.90/5.12 ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
% 8.90/5.12 = ( ord_less_int @ M @ N ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7682_lessThan__strict__subset__iff,axiom,
% 8.90/5.12 ! [M: real,N: real] :
% 8.90/5.12 ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
% 8.90/5.12 = ( ord_less_real @ M @ N ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7683_complex__mod__minus__le__complex__mod,axiom,
% 8.90/5.12 ! [X2: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X2 ) ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7684_complex__mod__triangle__ineq2,axiom,
% 8.90/5.12 ! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7685_log__of__power__eq,axiom,
% 8.90/5.12 ! [M: nat,B: real,N: nat] :
% 8.90/5.12 ( ( ( semiri5074537144036343181t_real @ M )
% 8.90/5.12 = ( power_power_real @ B @ N ) )
% 8.90/5.12 => ( ( ord_less_real @ one_one_real @ B )
% 8.90/5.12 => ( ( semiri5074537144036343181t_real @ N )
% 8.90/5.12 = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7686_less__log__of__power,axiom,
% 8.90/5.12 ! [B: real,N: nat,M: real] :
% 8.90/5.12 ( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
% 8.90/5.12 => ( ( ord_less_real @ one_one_real @ B )
% 8.90/5.12 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7687_sum_Onat__diff__reindex,axiom,
% 8.90/5.12 ! [G: nat > nat,N: nat] :
% 8.90/5.12 ( ( groups3542108847815614940at_nat
% 8.90/5.12 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.12 = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7688_sum_Onat__diff__reindex,axiom,
% 8.90/5.12 ! [G: nat > real,N: nat] :
% 8.90/5.12 ( ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.12 = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7689_sum__diff__distrib,axiom,
% 8.90/5.12 ! [Q: int > nat,P: int > nat,N: int] :
% 8.90/5.12 ( ! [X5: int] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
% 8.90/5.12 => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N ) ) )
% 8.90/5.12 = ( groups4541462559716669496nt_nat
% 8.90/5.12 @ ^ [X: int] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 8.90/5.12 @ ( set_ord_lessThan_int @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7690_sum__diff__distrib,axiom,
% 8.90/5.12 ! [Q: real > nat,P: real > nat,N: real] :
% 8.90/5.12 ( ! [X5: real] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
% 8.90/5.12 => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N ) ) )
% 8.90/5.12 = ( groups1935376822645274424al_nat
% 8.90/5.12 @ ^ [X: real] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 8.90/5.12 @ ( set_or5984915006950818249n_real @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7691_sum__diff__distrib,axiom,
% 8.90/5.12 ! [Q: nat > nat,P: nat > nat,N: nat] :
% 8.90/5.12 ( ! [X5: nat] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
% 8.90/5.12 => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
% 8.90/5.12 = ( groups3542108847815614940at_nat
% 8.90/5.12 @ ^ [X: nat] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7692_log__base__change,axiom,
% 8.90/5.12 ! [A: real,B: real,X2: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ A )
% 8.90/5.12 => ( ( A != one_one_real )
% 8.90/5.12 => ( ( log @ B @ X2 )
% 8.90/5.12 = ( divide_divide_real @ ( log @ A @ X2 ) @ ( log @ A @ B ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7693_le__log__of__power,axiom,
% 8.90/5.12 ! [B: real,N: nat,M: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
% 8.90/5.12 => ( ( ord_less_real @ one_one_real @ B )
% 8.90/5.12 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7694_log__base__pow,axiom,
% 8.90/5.12 ! [A: real,N: nat,X2: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ A )
% 8.90/5.12 => ( ( log @ ( power_power_real @ A @ N ) @ X2 )
% 8.90/5.12 = ( divide_divide_real @ ( log @ A @ X2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7695_log__nat__power,axiom,
% 8.90/5.12 ! [X2: real,B: real,N: nat] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( log @ B @ ( power_power_real @ X2 @ N ) )
% 8.90/5.12 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7696_sum_OlessThan__Suc__shift,axiom,
% 8.90/5.12 ! [G: nat > rat,N: nat] :
% 8.90/5.12 ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 8.90/5.12 = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 8.90/5.12 @ ( groups2906978787729119204at_rat
% 8.90/5.12 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7697_sum_OlessThan__Suc__shift,axiom,
% 8.90/5.12 ! [G: nat > int,N: nat] :
% 8.90/5.12 ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 8.90/5.12 = ( plus_plus_int @ ( G @ zero_zero_nat )
% 8.90/5.12 @ ( groups3539618377306564664at_int
% 8.90/5.12 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7698_sum_OlessThan__Suc__shift,axiom,
% 8.90/5.12 ! [G: nat > nat,N: nat] :
% 8.90/5.12 ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 8.90/5.12 = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 8.90/5.12 @ ( groups3542108847815614940at_nat
% 8.90/5.12 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7699_sum_OlessThan__Suc__shift,axiom,
% 8.90/5.12 ! [G: nat > real,N: nat] :
% 8.90/5.12 ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 8.90/5.12 = ( plus_plus_real @ ( G @ zero_zero_nat )
% 8.90/5.12 @ ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7700_sum__lessThan__telescope,axiom,
% 8.90/5.12 ! [F: nat > rat,M: nat] :
% 8.90/5.12 ( ( groups2906978787729119204at_rat
% 8.90/5.12 @ ^ [N3: nat] : ( minus_minus_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ M ) )
% 8.90/5.12 = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7701_sum__lessThan__telescope,axiom,
% 8.90/5.12 ! [F: nat > int,M: nat] :
% 8.90/5.12 ( ( groups3539618377306564664at_int
% 8.90/5.12 @ ^ [N3: nat] : ( minus_minus_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ M ) )
% 8.90/5.12 = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7702_sum__lessThan__telescope,axiom,
% 8.90/5.12 ! [F: nat > real,M: nat] :
% 8.90/5.12 ( ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [N3: nat] : ( minus_minus_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ M ) )
% 8.90/5.12 = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7703_sum__lessThan__telescope_H,axiom,
% 8.90/5.12 ! [F: nat > rat,M: nat] :
% 8.90/5.12 ( ( groups2906978787729119204at_rat
% 8.90/5.12 @ ^ [N3: nat] : ( minus_minus_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ M ) )
% 8.90/5.12 = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7704_sum__lessThan__telescope_H,axiom,
% 8.90/5.12 ! [F: nat > int,M: nat] :
% 8.90/5.12 ( ( groups3539618377306564664at_int
% 8.90/5.12 @ ^ [N3: nat] : ( minus_minus_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ M ) )
% 8.90/5.12 = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7705_sum__lessThan__telescope_H,axiom,
% 8.90/5.12 ! [F: nat > real,M: nat] :
% 8.90/5.12 ( ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ M ) )
% 8.90/5.12 = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7706_sumr__diff__mult__const2,axiom,
% 8.90/5.12 ! [F: nat > int,N: nat,R2: int] :
% 8.90/5.12 ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ R2 ) )
% 8.90/5.12 = ( groups3539618377306564664at_int
% 8.90/5.12 @ ^ [I4: nat] : ( minus_minus_int @ ( F @ I4 ) @ R2 )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7707_sumr__diff__mult__const2,axiom,
% 8.90/5.12 ! [F: nat > rat,N: nat,R2: rat] :
% 8.90/5.12 ( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ R2 ) )
% 8.90/5.12 = ( groups2906978787729119204at_rat
% 8.90/5.12 @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ I4 ) @ R2 )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7708_sumr__diff__mult__const2,axiom,
% 8.90/5.12 ! [F: nat > real,N: nat,R2: real] :
% 8.90/5.12 ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ R2 ) )
% 8.90/5.12 = ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [I4: nat] : ( minus_minus_real @ ( F @ I4 ) @ R2 )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7709_sum_OatLeast1__atMost__eq,axiom,
% 8.90/5.12 ! [G: nat > nat,N: nat] :
% 8.90/5.12 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 8.90/5.12 = ( groups3542108847815614940at_nat
% 8.90/5.12 @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7710_sum_OatLeast1__atMost__eq,axiom,
% 8.90/5.12 ! [G: nat > real,N: nat] :
% 8.90/5.12 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 8.90/5.12 = ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7711_log2__of__power__eq,axiom,
% 8.90/5.12 ! [M: nat,N: nat] :
% 8.90/5.12 ( ( M
% 8.90/5.12 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.12 => ( ( semiri5074537144036343181t_real @ N )
% 8.90/5.12 = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7712_log__of__power__less,axiom,
% 8.90/5.12 ! [M: nat,B: real,N: nat] :
% 8.90/5.12 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 8.90/5.12 => ( ( ord_less_real @ one_one_real @ B )
% 8.90/5.12 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.90/5.12 => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7713_log__mult,axiom,
% 8.90/5.12 ! [A: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ A )
% 8.90/5.12 => ( ( A != one_one_real )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.90/5.12 => ( ( log @ A @ ( times_times_real @ X2 @ Y3 ) )
% 8.90/5.12 = ( plus_plus_real @ ( log @ A @ X2 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7714_log__divide,axiom,
% 8.90/5.12 ! [A: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ A )
% 8.90/5.12 => ( ( A != one_one_real )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.90/5.12 => ( ( log @ A @ ( divide_divide_real @ X2 @ Y3 ) )
% 8.90/5.12 = ( minus_minus_real @ ( log @ A @ X2 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7715_one__diff__power__eq,axiom,
% 8.90/5.12 ! [X2: complex,N: nat] :
% 8.90/5.12 ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ N ) )
% 8.90/5.12 = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7716_one__diff__power__eq,axiom,
% 8.90/5.12 ! [X2: rat,N: nat] :
% 8.90/5.12 ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N ) )
% 8.90/5.12 = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7717_one__diff__power__eq,axiom,
% 8.90/5.12 ! [X2: int,N: nat] :
% 8.90/5.12 ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X2 @ N ) )
% 8.90/5.12 = ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7718_one__diff__power__eq,axiom,
% 8.90/5.12 ! [X2: real,N: nat] :
% 8.90/5.12 ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N ) )
% 8.90/5.12 = ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7719_power__diff__1__eq,axiom,
% 8.90/5.12 ! [X2: complex,N: nat] :
% 8.90/5.12 ( ( minus_minus_complex @ ( power_power_complex @ X2 @ N ) @ one_one_complex )
% 8.90/5.12 = ( times_times_complex @ ( minus_minus_complex @ X2 @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7720_power__diff__1__eq,axiom,
% 8.90/5.12 ! [X2: rat,N: nat] :
% 8.90/5.12 ( ( minus_minus_rat @ ( power_power_rat @ X2 @ N ) @ one_one_rat )
% 8.90/5.12 = ( times_times_rat @ ( minus_minus_rat @ X2 @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7721_power__diff__1__eq,axiom,
% 8.90/5.12 ! [X2: int,N: nat] :
% 8.90/5.12 ( ( minus_minus_int @ ( power_power_int @ X2 @ N ) @ one_one_int )
% 8.90/5.12 = ( times_times_int @ ( minus_minus_int @ X2 @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7722_power__diff__1__eq,axiom,
% 8.90/5.12 ! [X2: real,N: nat] :
% 8.90/5.12 ( ( minus_minus_real @ ( power_power_real @ X2 @ N ) @ one_one_real )
% 8.90/5.12 = ( times_times_real @ ( minus_minus_real @ X2 @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7723_geometric__sum,axiom,
% 8.90/5.12 ! [X2: complex,N: nat] :
% 8.90/5.12 ( ( X2 != one_one_complex )
% 8.90/5.12 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.12 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X2 @ N ) @ one_one_complex ) @ ( minus_minus_complex @ X2 @ one_one_complex ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7724_geometric__sum,axiom,
% 8.90/5.12 ! [X2: rat,N: nat] :
% 8.90/5.12 ( ( X2 != one_one_rat )
% 8.90/5.12 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.12 = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X2 @ N ) @ one_one_rat ) @ ( minus_minus_rat @ X2 @ one_one_rat ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7725_geometric__sum,axiom,
% 8.90/5.12 ! [X2: real,N: nat] :
% 8.90/5.12 ( ( X2 != one_one_real )
% 8.90/5.12 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.12 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ N ) @ one_one_real ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7726_log__of__power__le,axiom,
% 8.90/5.12 ! [M: nat,B: real,N: nat] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 8.90/5.12 => ( ( ord_less_real @ one_one_real @ B )
% 8.90/5.12 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.90/5.12 => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7727_log__eq__div__ln__mult__log,axiom,
% 8.90/5.12 ! [A: real,B: real,X2: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ A )
% 8.90/5.12 => ( ( A != one_one_real )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ B )
% 8.90/5.12 => ( ( B != one_one_real )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( log @ A @ X2 )
% 8.90/5.12 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X2 ) ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7728_monoseq__realpow,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 8.90/5.12 => ( topolo6980174941875973593q_real @ ( power_power_real @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7729_sum__gp__strict,axiom,
% 8.90/5.12 ! [X2: complex,N: nat] :
% 8.90/5.12 ( ( ( X2 = one_one_complex )
% 8.90/5.12 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.12 = ( semiri8010041392384452111omplex @ N ) ) )
% 8.90/5.12 & ( ( X2 != one_one_complex )
% 8.90/5.12 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.12 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ N ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7730_sum__gp__strict,axiom,
% 8.90/5.12 ! [X2: rat,N: nat] :
% 8.90/5.12 ( ( ( X2 = one_one_rat )
% 8.90/5.12 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.12 = ( semiri681578069525770553at_rat @ N ) ) )
% 8.90/5.12 & ( ( X2 != one_one_rat )
% 8.90/5.12 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.12 = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7731_sum__gp__strict,axiom,
% 8.90/5.12 ! [X2: real,N: nat] :
% 8.90/5.12 ( ( ( X2 = one_one_real )
% 8.90/5.12 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.12 = ( semiri5074537144036343181t_real @ N ) ) )
% 8.90/5.12 & ( ( X2 != one_one_real )
% 8.90/5.12 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.12 = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7732_lemma__termdiff1,axiom,
% 8.90/5.12 ! [Z: complex,H2: complex,M: nat] :
% 8.90/5.12 ( ( groups2073611262835488442omplex
% 8.90/5.12 @ ^ [P4: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_complex @ Z @ P4 ) ) @ ( power_power_complex @ Z @ M ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ M ) )
% 8.90/5.12 = ( groups2073611262835488442omplex
% 8.90/5.12 @ ^ [P4: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P4 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7733_lemma__termdiff1,axiom,
% 8.90/5.12 ! [Z: rat,H2: rat,M: nat] :
% 8.90/5.12 ( ( groups2906978787729119204at_rat
% 8.90/5.12 @ ^ [P4: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_rat @ Z @ P4 ) ) @ ( power_power_rat @ Z @ M ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ M ) )
% 8.90/5.12 = ( groups2906978787729119204at_rat
% 8.90/5.12 @ ^ [P4: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P4 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7734_lemma__termdiff1,axiom,
% 8.90/5.12 ! [Z: int,H2: int,M: nat] :
% 8.90/5.12 ( ( groups3539618377306564664at_int
% 8.90/5.12 @ ^ [P4: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_int @ Z @ P4 ) ) @ ( power_power_int @ Z @ M ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ M ) )
% 8.90/5.12 = ( groups3539618377306564664at_int
% 8.90/5.12 @ ^ [P4: nat] : ( times_times_int @ ( power_power_int @ Z @ P4 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7735_lemma__termdiff1,axiom,
% 8.90/5.12 ! [Z: real,H2: real,M: nat] :
% 8.90/5.12 ( ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [P4: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_real @ Z @ P4 ) ) @ ( power_power_real @ Z @ M ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ M ) )
% 8.90/5.12 = ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [P4: nat] : ( times_times_real @ ( power_power_real @ Z @ P4 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7736_diff__power__eq__sum,axiom,
% 8.90/5.12 ! [X2: complex,N: nat,Y3: complex] :
% 8.90/5.12 ( ( minus_minus_complex @ ( power_power_complex @ X2 @ ( suc @ N ) ) @ ( power_power_complex @ Y3 @ ( suc @ N ) ) )
% 8.90/5.12 = ( times_times_complex @ ( minus_minus_complex @ X2 @ Y3 )
% 8.90/5.12 @ ( groups2073611262835488442omplex
% 8.90/5.12 @ ^ [P4: nat] : ( times_times_complex @ ( power_power_complex @ X2 @ P4 ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ P4 ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7737_diff__power__eq__sum,axiom,
% 8.90/5.12 ! [X2: rat,N: nat,Y3: rat] :
% 8.90/5.12 ( ( minus_minus_rat @ ( power_power_rat @ X2 @ ( suc @ N ) ) @ ( power_power_rat @ Y3 @ ( suc @ N ) ) )
% 8.90/5.12 = ( times_times_rat @ ( minus_minus_rat @ X2 @ Y3 )
% 8.90/5.12 @ ( groups2906978787729119204at_rat
% 8.90/5.12 @ ^ [P4: nat] : ( times_times_rat @ ( power_power_rat @ X2 @ P4 ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ P4 ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7738_diff__power__eq__sum,axiom,
% 8.90/5.12 ! [X2: int,N: nat,Y3: int] :
% 8.90/5.12 ( ( minus_minus_int @ ( power_power_int @ X2 @ ( suc @ N ) ) @ ( power_power_int @ Y3 @ ( suc @ N ) ) )
% 8.90/5.12 = ( times_times_int @ ( minus_minus_int @ X2 @ Y3 )
% 8.90/5.12 @ ( groups3539618377306564664at_int
% 8.90/5.12 @ ^ [P4: nat] : ( times_times_int @ ( power_power_int @ X2 @ P4 ) @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ P4 ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7739_diff__power__eq__sum,axiom,
% 8.90/5.12 ! [X2: real,N: nat,Y3: real] :
% 8.90/5.12 ( ( minus_minus_real @ ( power_power_real @ X2 @ ( suc @ N ) ) @ ( power_power_real @ Y3 @ ( suc @ N ) ) )
% 8.90/5.12 = ( times_times_real @ ( minus_minus_real @ X2 @ Y3 )
% 8.90/5.12 @ ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [P4: nat] : ( times_times_real @ ( power_power_real @ X2 @ P4 ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ P4 ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7740_power__diff__sumr2,axiom,
% 8.90/5.12 ! [X2: complex,N: nat,Y3: complex] :
% 8.90/5.12 ( ( minus_minus_complex @ ( power_power_complex @ X2 @ N ) @ ( power_power_complex @ Y3 @ N ) )
% 8.90/5.12 = ( times_times_complex @ ( minus_minus_complex @ X2 @ Y3 )
% 8.90/5.12 @ ( groups2073611262835488442omplex
% 8.90/5.12 @ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_complex @ X2 @ I4 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7741_power__diff__sumr2,axiom,
% 8.90/5.12 ! [X2: rat,N: nat,Y3: rat] :
% 8.90/5.12 ( ( minus_minus_rat @ ( power_power_rat @ X2 @ N ) @ ( power_power_rat @ Y3 @ N ) )
% 8.90/5.12 = ( times_times_rat @ ( minus_minus_rat @ X2 @ Y3 )
% 8.90/5.12 @ ( groups2906978787729119204at_rat
% 8.90/5.12 @ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_rat @ X2 @ I4 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7742_power__diff__sumr2,axiom,
% 8.90/5.12 ! [X2: int,N: nat,Y3: int] :
% 8.90/5.12 ( ( minus_minus_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y3 @ N ) )
% 8.90/5.12 = ( times_times_int @ ( minus_minus_int @ X2 @ Y3 )
% 8.90/5.12 @ ( groups3539618377306564664at_int
% 8.90/5.12 @ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_int @ X2 @ I4 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7743_power__diff__sumr2,axiom,
% 8.90/5.12 ! [X2: real,N: nat,Y3: real] :
% 8.90/5.12 ( ( minus_minus_real @ ( power_power_real @ X2 @ N ) @ ( power_power_real @ Y3 @ N ) )
% 8.90/5.12 = ( times_times_real @ ( minus_minus_real @ X2 @ Y3 )
% 8.90/5.12 @ ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_real @ X2 @ I4 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7744_less__log2__of__power,axiom,
% 8.90/5.12 ! [N: nat,M: nat] :
% 8.90/5.12 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 8.90/5.12 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7745_le__log2__of__power,axiom,
% 8.90/5.12 ! [N: nat,M: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 8.90/5.12 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7746_real__sum__nat__ivl__bounded2,axiom,
% 8.90/5.12 ! [N: nat,F: nat > rat,K5: rat,K: nat] :
% 8.90/5.12 ( ! [P7: nat] :
% 8.90/5.12 ( ( ord_less_nat @ P7 @ N )
% 8.90/5.12 => ( ord_less_eq_rat @ ( F @ P7 ) @ K5 ) )
% 8.90/5.12 => ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
% 8.90/5.12 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K5 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7747_real__sum__nat__ivl__bounded2,axiom,
% 8.90/5.12 ! [N: nat,F: nat > int,K5: int,K: nat] :
% 8.90/5.12 ( ! [P7: nat] :
% 8.90/5.12 ( ( ord_less_nat @ P7 @ N )
% 8.90/5.12 => ( ord_less_eq_int @ ( F @ P7 ) @ K5 ) )
% 8.90/5.12 => ( ( ord_less_eq_int @ zero_zero_int @ K5 )
% 8.90/5.12 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K5 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7748_real__sum__nat__ivl__bounded2,axiom,
% 8.90/5.12 ! [N: nat,F: nat > nat,K5: nat,K: nat] :
% 8.90/5.12 ( ! [P7: nat] :
% 8.90/5.12 ( ( ord_less_nat @ P7 @ N )
% 8.90/5.12 => ( ord_less_eq_nat @ ( F @ P7 ) @ K5 ) )
% 8.90/5.12 => ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
% 8.90/5.12 => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K5 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7749_real__sum__nat__ivl__bounded2,axiom,
% 8.90/5.12 ! [N: nat,F: nat > real,K5: real,K: nat] :
% 8.90/5.12 ( ! [P7: nat] :
% 8.90/5.12 ( ( ord_less_nat @ P7 @ N )
% 8.90/5.12 => ( ord_less_eq_real @ ( F @ P7 ) @ K5 ) )
% 8.90/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ K5 )
% 8.90/5.12 => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K5 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7750_one__diff__power__eq_H,axiom,
% 8.90/5.12 ! [X2: complex,N: nat] :
% 8.90/5.12 ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ N ) )
% 8.90/5.12 = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X2 )
% 8.90/5.12 @ ( groups2073611262835488442omplex
% 8.90/5.12 @ ^ [I4: nat] : ( power_power_complex @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7751_one__diff__power__eq_H,axiom,
% 8.90/5.12 ! [X2: rat,N: nat] :
% 8.90/5.12 ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N ) )
% 8.90/5.12 = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 )
% 8.90/5.12 @ ( groups2906978787729119204at_rat
% 8.90/5.12 @ ^ [I4: nat] : ( power_power_rat @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7752_one__diff__power__eq_H,axiom,
% 8.90/5.12 ! [X2: int,N: nat] :
% 8.90/5.12 ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X2 @ N ) )
% 8.90/5.12 = ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 )
% 8.90/5.12 @ ( groups3539618377306564664at_int
% 8.90/5.12 @ ^ [I4: nat] : ( power_power_int @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7753_one__diff__power__eq_H,axiom,
% 8.90/5.12 ! [X2: real,N: nat] :
% 8.90/5.12 ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N ) )
% 8.90/5.12 = ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 )
% 8.90/5.12 @ ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [I4: nat] : ( power_power_real @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7754_log2__of__power__less,axiom,
% 8.90/5.12 ! [M: nat,N: nat] :
% 8.90/5.12 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.12 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.90/5.12 => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7755_sum__split__even__odd,axiom,
% 8.90/5.12 ! [F: nat > real,G: nat > real,N: nat] :
% 8.90/5.12 ( ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( F @ I4 ) @ ( G @ I4 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.90/5.12 = ( plus_plus_real
% 8.90/5.12 @ ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [I4: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.12 @ ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ one_one_nat ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7756_log2__of__power__le,axiom,
% 8.90/5.12 ! [M: nat,N: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.12 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.90/5.12 => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7757_insert__bound__size__univ,axiom,
% 8.90/5.12 ! [T: vEBT_VEBT,N: nat,U: real,X2: nat] :
% 8.90/5.12 ( ( vEBT_invar_vebt @ T @ N )
% 8.90/5.12 => ( ( U
% 8.90/5.12 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.12 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_i_n_s_e_r_t @ T @ X2 ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7758_member__bound__size__univ,axiom,
% 8.90/5.12 ! [T: vEBT_VEBT,N: nat,U: real,X2: nat] :
% 8.90/5.12 ( ( vEBT_invar_vebt @ T @ N )
% 8.90/5.12 => ( ( U
% 8.90/5.12 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.12 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_m_e_m_b_e_r @ T @ X2 ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7759_arctan__series,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 8.90/5.12 => ( ( arctan @ X2 )
% 8.90/5.12 = ( suminf_real
% 8.90/5.12 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7760_norm__divide__numeral,axiom,
% 8.90/5.12 ! [A: real,W: num] :
% 8.90/5.12 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
% 8.90/5.12 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7761_norm__divide__numeral,axiom,
% 8.90/5.12 ! [A: complex,W: num] :
% 8.90/5.12 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 8.90/5.12 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7762_norm__mult__numeral2,axiom,
% 8.90/5.12 ! [A: real,W: num] :
% 8.90/5.12 ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
% 8.90/5.12 = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7763_norm__mult__numeral2,axiom,
% 8.90/5.12 ! [A: complex,W: num] :
% 8.90/5.12 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 8.90/5.12 = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7764_norm__mult__numeral1,axiom,
% 8.90/5.12 ! [W: num,A: real] :
% 8.90/5.12 ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 8.90/5.12 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7765_norm__mult__numeral1,axiom,
% 8.90/5.12 ! [W: num,A: complex] :
% 8.90/5.12 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 8.90/5.12 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7766_norm__neg__numeral,axiom,
% 8.90/5.12 ! [W: num] :
% 8.90/5.12 ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 8.90/5.12 = ( numeral_numeral_real @ W ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7767_norm__neg__numeral,axiom,
% 8.90/5.12 ! [W: num] :
% 8.90/5.12 ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 8.90/5.12 = ( numeral_numeral_real @ W ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7768_norm__le__zero__iff,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X2 ) @ zero_zero_real )
% 8.90/5.12 = ( X2 = zero_zero_real ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7769_norm__le__zero__iff,axiom,
% 8.90/5.12 ! [X2: complex] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ zero_zero_real )
% 8.90/5.12 = ( X2 = zero_zero_complex ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7770_norm__one,axiom,
% 8.90/5.12 ( ( real_V7735802525324610683m_real @ one_one_real )
% 8.90/5.12 = one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7771_norm__one,axiom,
% 8.90/5.12 ( ( real_V1022390504157884413omplex @ one_one_complex )
% 8.90/5.12 = one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7772_norm__numeral,axiom,
% 8.90/5.12 ! [W: num] :
% 8.90/5.12 ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
% 8.90/5.12 = ( numeral_numeral_real @ W ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7773_norm__numeral,axiom,
% 8.90/5.12 ! [W: num] :
% 8.90/5.12 ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
% 8.90/5.12 = ( numeral_numeral_real @ W ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7774_norm__minus__commute,axiom,
% 8.90/5.12 ! [A: real,B: real] :
% 8.90/5.12 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) )
% 8.90/5.12 = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7775_norm__minus__commute,axiom,
% 8.90/5.12 ! [A: complex,B: complex] :
% 8.90/5.12 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) )
% 8.90/5.12 = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ A ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7776_norm__ge__zero,axiom,
% 8.90/5.12 ! [X2: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7777_norm__mult,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( real_V7735802525324610683m_real @ ( times_times_real @ X2 @ Y3 ) )
% 8.90/5.12 = ( times_times_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7778_norm__mult,axiom,
% 8.90/5.12 ! [X2: complex,Y3: complex] :
% 8.90/5.12 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X2 @ Y3 ) )
% 8.90/5.12 = ( times_times_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7779_norm__divide,axiom,
% 8.90/5.12 ! [A: real,B: real] :
% 8.90/5.12 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 8.90/5.12 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7780_norm__divide,axiom,
% 8.90/5.12 ! [A: complex,B: complex] :
% 8.90/5.12 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.90/5.12 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7781_sum__norm__le,axiom,
% 8.90/5.12 ! [S3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > complex,G: product_prod_nat_nat > real] :
% 8.90/5.12 ( ! [X5: product_prod_nat_nat] :
% 8.90/5.12 ( ( member8440522571783428010at_nat @ X5 @ S3 )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups6381953495645901045omplex @ F @ S3 ) ) @ ( groups4567486121110086003t_real @ G @ S3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7782_sum__norm__le,axiom,
% 8.90/5.12 ! [S3: set_real,F: real > complex,G: real > real] :
% 8.90/5.12 ( ! [X5: real] :
% 8.90/5.12 ( ( member_real @ X5 @ S3 )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S3 ) ) @ ( groups8097168146408367636l_real @ G @ S3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7783_sum__norm__le,axiom,
% 8.90/5.12 ! [S3: set_int,F: int > complex,G: int > real] :
% 8.90/5.12 ( ! [X5: int] :
% 8.90/5.12 ( ( member_int @ X5 @ S3 )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S3 ) ) @ ( groups8778361861064173332t_real @ G @ S3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7784_sum__norm__le,axiom,
% 8.90/5.12 ! [S3: set_nat,F: nat > complex,G: nat > real] :
% 8.90/5.12 ( ! [X5: nat] :
% 8.90/5.12 ( ( member_nat @ X5 @ S3 )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S3 ) ) @ ( groups6591440286371151544t_real @ G @ S3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7785_sum__norm__le,axiom,
% 8.90/5.12 ! [S3: set_complex,F: complex > complex,G: complex > real] :
% 8.90/5.12 ( ! [X5: complex] :
% 8.90/5.12 ( ( member_complex @ X5 @ S3 )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S3 ) ) @ ( groups5808333547571424918x_real @ G @ S3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7786_sum__norm__le,axiom,
% 8.90/5.12 ! [S3: set_nat,F: nat > real,G: nat > real] :
% 8.90/5.12 ( ! [X5: nat] :
% 8.90/5.12 ( ( member_nat @ X5 @ S3 )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S3 ) ) @ ( groups6591440286371151544t_real @ G @ S3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7787_norm__power,axiom,
% 8.90/5.12 ! [X2: real,N: nat] :
% 8.90/5.12 ( ( real_V7735802525324610683m_real @ ( power_power_real @ X2 @ N ) )
% 8.90/5.12 = ( power_power_real @ ( real_V7735802525324610683m_real @ X2 ) @ N ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7788_norm__power,axiom,
% 8.90/5.12 ! [X2: complex,N: nat] :
% 8.90/5.12 ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X2 @ N ) )
% 8.90/5.12 = ( power_power_real @ ( real_V1022390504157884413omplex @ X2 ) @ N ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7789_norm__sum,axiom,
% 8.90/5.12 ! [F: nat > complex,A2: set_nat] :
% 8.90/5.12 ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A2 ) )
% 8.90/5.12 @ ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [I4: nat] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
% 8.90/5.12 @ A2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7790_norm__sum,axiom,
% 8.90/5.12 ! [F: complex > complex,A2: set_complex] :
% 8.90/5.12 ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A2 ) )
% 8.90/5.12 @ ( groups5808333547571424918x_real
% 8.90/5.12 @ ^ [I4: complex] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
% 8.90/5.12 @ A2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7791_norm__sum,axiom,
% 8.90/5.12 ! [F: nat > real,A2: set_nat] :
% 8.90/5.12 ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 8.90/5.12 @ ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [I4: nat] : ( real_V7735802525324610683m_real @ ( F @ I4 ) )
% 8.90/5.12 @ A2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7792_norm__uminus__minus,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X2 ) @ Y3 ) )
% 8.90/5.12 = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7793_norm__uminus__minus,axiom,
% 8.90/5.12 ! [X2: complex,Y3: complex] :
% 8.90/5.12 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X2 ) @ Y3 ) )
% 8.90/5.12 = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7794_nonzero__norm__divide,axiom,
% 8.90/5.12 ! [B: real,A: real] :
% 8.90/5.12 ( ( B != zero_zero_real )
% 8.90/5.12 => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 8.90/5.12 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7795_nonzero__norm__divide,axiom,
% 8.90/5.12 ! [B: complex,A: complex] :
% 8.90/5.12 ( ( B != zero_zero_complex )
% 8.90/5.12 => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.90/5.12 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7796_power__eq__imp__eq__norm,axiom,
% 8.90/5.12 ! [W: real,N: nat,Z: real] :
% 8.90/5.12 ( ( ( power_power_real @ W @ N )
% 8.90/5.12 = ( power_power_real @ Z @ N ) )
% 8.90/5.12 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.12 => ( ( real_V7735802525324610683m_real @ W )
% 8.90/5.12 = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7797_power__eq__imp__eq__norm,axiom,
% 8.90/5.12 ! [W: complex,N: nat,Z: complex] :
% 8.90/5.12 ( ( ( power_power_complex @ W @ N )
% 8.90/5.12 = ( power_power_complex @ Z @ N ) )
% 8.90/5.12 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.12 => ( ( real_V1022390504157884413omplex @ W )
% 8.90/5.12 = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7798_norm__mult__less,axiom,
% 8.90/5.12 ! [X2: real,R2: real,Y3: real,S: real] :
% 8.90/5.12 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ R2 )
% 8.90/5.12 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
% 8.90/5.12 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X2 @ Y3 ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7799_norm__mult__less,axiom,
% 8.90/5.12 ! [X2: complex,R2: real,Y3: complex,S: real] :
% 8.90/5.12 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ R2 )
% 8.90/5.12 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
% 8.90/5.12 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X2 @ Y3 ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7800_norm__mult__ineq,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X2 @ Y3 ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7801_norm__mult__ineq,axiom,
% 8.90/5.12 ! [X2: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X2 @ Y3 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7802_norm__triangle__lt,axiom,
% 8.90/5.12 ! [X2: real,Y3: real,E: real] :
% 8.90/5.12 ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 8.90/5.12 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ E ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7803_norm__triangle__lt,axiom,
% 8.90/5.12 ! [X2: complex,Y3: complex,E: real] :
% 8.90/5.12 ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 8.90/5.12 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ E ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7804_norm__add__less,axiom,
% 8.90/5.12 ! [X2: real,R2: real,Y3: real,S: real] :
% 8.90/5.12 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ R2 )
% 8.90/5.12 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
% 8.90/5.12 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7805_norm__add__less,axiom,
% 8.90/5.12 ! [X2: complex,R2: real,Y3: complex,S: real] :
% 8.90/5.12 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ R2 )
% 8.90/5.12 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
% 8.90/5.12 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7806_norm__add__leD,axiom,
% 8.90/5.12 ! [A: real,B: real,C: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7807_norm__add__leD,axiom,
% 8.90/5.12 ! [A: complex,B: complex,C: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7808_norm__triangle__le,axiom,
% 8.90/5.12 ! [X2: real,Y3: real,E: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ E ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7809_norm__triangle__le,axiom,
% 8.90/5.12 ! [X2: complex,Y3: complex,E: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ E ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7810_norm__triangle__ineq,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7811_norm__triangle__ineq,axiom,
% 8.90/5.12 ! [X2: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7812_norm__triangle__mono,axiom,
% 8.90/5.12 ! [A: real,R2: real,B: real,S: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7813_norm__triangle__mono,axiom,
% 8.90/5.12 ! [A: complex,R2: real,B: complex,S: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7814_norm__power__ineq,axiom,
% 8.90/5.12 ! [X2: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X2 @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X2 ) @ N ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7815_norm__power__ineq,axiom,
% 8.90/5.12 ! [X2: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X2 @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X2 ) @ N ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7816_norm__diff__triangle__less,axiom,
% 8.90/5.12 ! [X2: real,Y3: real,E1: real,Z: real,E22: real] :
% 8.90/5.12 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ E1 )
% 8.90/5.12 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
% 8.90/5.12 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7817_norm__diff__triangle__less,axiom,
% 8.90/5.12 ! [X2: complex,Y3: complex,E1: real,Z: complex,E22: real] :
% 8.90/5.12 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) @ E1 )
% 8.90/5.12 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
% 8.90/5.12 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7818_norm__triangle__sub,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y3 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7819_norm__triangle__sub,axiom,
% 8.90/5.12 ! [X2: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y3 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7820_norm__triangle__ineq4,axiom,
% 8.90/5.12 ! [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 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7821_norm__triangle__ineq4,axiom,
% 8.90/5.12 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7822_norm__diff__triangle__le,axiom,
% 8.90/5.12 ! [X2: real,Y3: real,E1: real,Z: real,E22: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ E1 )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7823_norm__diff__triangle__le,axiom,
% 8.90/5.12 ! [X2: complex,Y3: complex,E1: real,Z: complex,E22: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) @ E1 )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7824_norm__triangle__le__diff,axiom,
% 8.90/5.12 ! [X2: real,Y3: real,E: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ E ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7825_norm__triangle__le__diff,axiom,
% 8.90/5.12 ! [X2: complex,Y3: complex,E: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) @ E ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7826_norm__diff__ineq,axiom,
% 8.90/5.12 ! [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 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7827_norm__diff__ineq,axiom,
% 8.90/5.12 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7828_norm__triangle__ineq2,axiom,
% 8.90/5.12 ! [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 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7829_norm__triangle__ineq2,axiom,
% 8.90/5.12 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7830_power__eq__1__iff,axiom,
% 8.90/5.12 ! [W: real,N: nat] :
% 8.90/5.12 ( ( ( power_power_real @ W @ N )
% 8.90/5.12 = one_one_real )
% 8.90/5.12 => ( ( ( real_V7735802525324610683m_real @ W )
% 8.90/5.12 = one_one_real )
% 8.90/5.12 | ( N = zero_zero_nat ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7831_power__eq__1__iff,axiom,
% 8.90/5.12 ! [W: complex,N: nat] :
% 8.90/5.12 ( ( ( power_power_complex @ W @ N )
% 8.90/5.12 = one_one_complex )
% 8.90/5.12 => ( ( ( real_V1022390504157884413omplex @ W )
% 8.90/5.12 = one_one_real )
% 8.90/5.12 | ( N = zero_zero_nat ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7832_norm__diff__triangle__ineq,axiom,
% 8.90/5.12 ! [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 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7833_norm__diff__triangle__ineq,axiom,
% 8.90/5.12 ! [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 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7834_norm__triangle__ineq3,axiom,
% 8.90/5.12 ! [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 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7835_norm__triangle__ineq3,axiom,
% 8.90/5.12 ! [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 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7836_square__norm__one,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.12 = one_one_real )
% 8.90/5.12 => ( ( real_V7735802525324610683m_real @ X2 )
% 8.90/5.12 = one_one_real ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7837_square__norm__one,axiom,
% 8.90/5.12 ! [X2: complex] :
% 8.90/5.12 ( ( ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.12 = one_one_complex )
% 8.90/5.12 => ( ( real_V1022390504157884413omplex @ X2 )
% 8.90/5.12 = one_one_real ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7838_norm__power__diff,axiom,
% 8.90/5.12 ! [Z: real,W: real,M: nat] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
% 8.90/5.12 => ( 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 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7839_norm__power__diff,axiom,
% 8.90/5.12 ! [Z: complex,W: complex,M: nat] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
% 8.90/5.12 => ( 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 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7840_suminf__geometric,axiom,
% 8.90/5.12 ! [C: real] :
% 8.90/5.12 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 8.90/5.12 => ( ( suminf_real @ ( power_power_real @ C ) )
% 8.90/5.12 = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7841_suminf__geometric,axiom,
% 8.90/5.12 ! [C: complex] :
% 8.90/5.12 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 8.90/5.12 => ( ( suminf_complex @ ( power_power_complex @ C ) )
% 8.90/5.12 = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7842_sum__bounds__lt__plus1,axiom,
% 8.90/5.12 ! [F: nat > nat,Mm: nat] :
% 8.90/5.12 ( ( groups3542108847815614940at_nat
% 8.90/5.12 @ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ Mm ) )
% 8.90/5.12 = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7843_sum__bounds__lt__plus1,axiom,
% 8.90/5.12 ! [F: nat > real,Mm: nat] :
% 8.90/5.12 ( ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ Mm ) )
% 8.90/5.12 = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7844_heigt__uplog__rel,axiom,
% 8.90/5.12 ! [T: vEBT_VEBT,N: nat] :
% 8.90/5.12 ( ( vEBT_invar_vebt @ T @ N )
% 8.90/5.12 => ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ T ) )
% 8.90/5.12 = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7845_sumr__cos__zero__one,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ zero_zero_real @ M3 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 8.90/5.12 = one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7846_log__ceil__idem,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 8.90/5.12 => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) )
% 8.90/5.12 = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7847_of__int__ceiling__cancel,axiom,
% 8.90/5.12 ! [X2: rat] :
% 8.90/5.12 ( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) )
% 8.90/5.12 = X2 )
% 8.90/5.12 = ( ? [N3: int] :
% 8.90/5.12 ( X2
% 8.90/5.12 = ( ring_1_of_int_rat @ N3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7848_of__int__ceiling__cancel,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) )
% 8.90/5.12 = X2 )
% 8.90/5.12 = ( ? [N3: int] :
% 8.90/5.12 ( X2
% 8.90/5.12 = ( ring_1_of_int_real @ N3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7849_ceiling__numeral,axiom,
% 8.90/5.12 ! [V: num] :
% 8.90/5.12 ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
% 8.90/5.12 = ( numeral_numeral_int @ V ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7850_ceiling__numeral,axiom,
% 8.90/5.12 ! [V: num] :
% 8.90/5.12 ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
% 8.90/5.12 = ( numeral_numeral_int @ V ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7851_ceiling__one,axiom,
% 8.90/5.12 ( ( archim2889992004027027881ng_rat @ one_one_rat )
% 8.90/5.12 = one_one_int ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7852_ceiling__one,axiom,
% 8.90/5.12 ( ( archim7802044766580827645g_real @ one_one_real )
% 8.90/5.12 = one_one_int ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7853_cos__coeff__0,axiom,
% 8.90/5.12 ( ( cos_coeff @ zero_zero_nat )
% 8.90/5.12 = one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7854_ceiling__add__of__int,axiom,
% 8.90/5.12 ! [X2: rat,Z: int] :
% 8.90/5.12 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) )
% 8.90/5.12 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ Z ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7855_ceiling__add__of__int,axiom,
% 8.90/5.12 ! [X2: real,Z: int] :
% 8.90/5.12 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X2 @ ( ring_1_of_int_real @ Z ) ) )
% 8.90/5.12 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X2 ) @ Z ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7856_ceiling__diff__of__int,axiom,
% 8.90/5.12 ! [X2: rat,Z: int] :
% 8.90/5.12 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) )
% 8.90/5.12 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ Z ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7857_ceiling__diff__of__int,axiom,
% 8.90/5.12 ! [X2: real,Z: int] :
% 8.90/5.12 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X2 @ ( ring_1_of_int_real @ Z ) ) )
% 8.90/5.12 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X2 ) @ Z ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7858_ceiling__le__zero,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ zero_zero_int )
% 8.90/5.12 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7859_ceiling__le__zero,axiom,
% 8.90/5.12 ! [X2: rat] :
% 8.90/5.12 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ zero_zero_int )
% 8.90/5.12 = ( ord_less_eq_rat @ X2 @ zero_zero_rat ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7860_zero__less__ceiling,axiom,
% 8.90/5.12 ! [X2: rat] :
% 8.90/5.12 ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X2 ) )
% 8.90/5.12 = ( ord_less_rat @ zero_zero_rat @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7861_zero__less__ceiling,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X2 ) )
% 8.90/5.12 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7862_ceiling__le__numeral,axiom,
% 8.90/5.12 ! [X2: real,V: num] :
% 8.90/5.12 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ ( numeral_numeral_int @ V ) )
% 8.90/5.12 = ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ V ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7863_ceiling__le__numeral,axiom,
% 8.90/5.12 ! [X2: rat,V: num] :
% 8.90/5.12 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( numeral_numeral_int @ V ) )
% 8.90/5.12 = ( ord_less_eq_rat @ X2 @ ( numeral_numeral_rat @ V ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7864_ceiling__less__one,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ one_one_int )
% 8.90/5.12 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7865_ceiling__less__one,axiom,
% 8.90/5.12 ! [X2: rat] :
% 8.90/5.12 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ one_one_int )
% 8.90/5.12 = ( ord_less_eq_rat @ X2 @ zero_zero_rat ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7866_one__le__ceiling,axiom,
% 8.90/5.12 ! [X2: rat] :
% 8.90/5.12 ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X2 ) )
% 8.90/5.12 = ( ord_less_rat @ zero_zero_rat @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7867_one__le__ceiling,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X2 ) )
% 8.90/5.12 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7868_numeral__less__ceiling,axiom,
% 8.90/5.12 ! [V: num,X2: real] :
% 8.90/5.12 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X2 ) )
% 8.90/5.12 = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7869_numeral__less__ceiling,axiom,
% 8.90/5.12 ! [V: num,X2: rat] :
% 8.90/5.12 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X2 ) )
% 8.90/5.12 = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7870_ceiling__le__one,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ one_one_int )
% 8.90/5.12 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7871_ceiling__le__one,axiom,
% 8.90/5.12 ! [X2: rat] :
% 8.90/5.12 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ one_one_int )
% 8.90/5.12 = ( ord_less_eq_rat @ X2 @ one_one_rat ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7872_one__less__ceiling,axiom,
% 8.90/5.12 ! [X2: rat] :
% 8.90/5.12 ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X2 ) )
% 8.90/5.12 = ( ord_less_rat @ one_one_rat @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7873_one__less__ceiling,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X2 ) )
% 8.90/5.12 = ( ord_less_real @ one_one_real @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7874_ceiling__add__numeral,axiom,
% 8.90/5.12 ! [X2: real,V: num] :
% 8.90/5.12 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X2 @ ( numeral_numeral_real @ V ) ) )
% 8.90/5.12 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X2 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7875_ceiling__add__numeral,axiom,
% 8.90/5.12 ! [X2: rat,V: num] :
% 8.90/5.12 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X2 @ ( numeral_numeral_rat @ V ) ) )
% 8.90/5.12 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7876_ceiling__neg__numeral,axiom,
% 8.90/5.12 ! [V: num] :
% 8.90/5.12 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.90/5.12 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7877_ceiling__neg__numeral,axiom,
% 8.90/5.12 ! [V: num] :
% 8.90/5.12 ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.90/5.12 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7878_ceiling__add__one,axiom,
% 8.90/5.12 ! [X2: rat] :
% 8.90/5.12 ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X2 @ one_one_rat ) )
% 8.90/5.12 = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ one_one_int ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7879_ceiling__add__one,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X2 @ one_one_real ) )
% 8.90/5.12 = ( plus_plus_int @ ( archim7802044766580827645g_real @ X2 ) @ one_one_int ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7880_ceiling__diff__numeral,axiom,
% 8.90/5.12 ! [X2: real,V: num] :
% 8.90/5.12 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X2 @ ( numeral_numeral_real @ V ) ) )
% 8.90/5.12 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X2 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7881_ceiling__diff__numeral,axiom,
% 8.90/5.12 ! [X2: rat,V: num] :
% 8.90/5.12 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X2 @ ( numeral_numeral_rat @ V ) ) )
% 8.90/5.12 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7882_ceiling__diff__one,axiom,
% 8.90/5.12 ! [X2: rat] :
% 8.90/5.12 ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X2 @ one_one_rat ) )
% 8.90/5.12 = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ one_one_int ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7883_ceiling__diff__one,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X2 @ one_one_real ) )
% 8.90/5.12 = ( minus_minus_int @ ( archim7802044766580827645g_real @ X2 ) @ one_one_int ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7884_ceiling__numeral__power,axiom,
% 8.90/5.12 ! [X2: num,N: nat] :
% 8.90/5.12 ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
% 8.90/5.12 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7885_ceiling__numeral__power,axiom,
% 8.90/5.12 ! [X2: num,N: nat] :
% 8.90/5.12 ( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) )
% 8.90/5.12 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7886_ceiling__less__zero,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ zero_zero_int )
% 8.90/5.12 = ( ord_less_eq_real @ X2 @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7887_ceiling__less__zero,axiom,
% 8.90/5.12 ! [X2: rat] :
% 8.90/5.12 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ zero_zero_int )
% 8.90/5.12 = ( ord_less_eq_rat @ X2 @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7888_zero__le__ceiling,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X2 ) )
% 8.90/5.12 = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7889_zero__le__ceiling,axiom,
% 8.90/5.12 ! [X2: rat] :
% 8.90/5.12 ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X2 ) )
% 8.90/5.12 = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7890_ceiling__divide__eq__div__numeral,axiom,
% 8.90/5.12 ! [A: num,B: num] :
% 8.90/5.12 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 8.90/5.12 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7891_ceiling__less__numeral,axiom,
% 8.90/5.12 ! [X2: real,V: num] :
% 8.90/5.12 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ ( numeral_numeral_int @ V ) )
% 8.90/5.12 = ( ord_less_eq_real @ X2 @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7892_ceiling__less__numeral,axiom,
% 8.90/5.12 ! [X2: rat,V: num] :
% 8.90/5.12 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( numeral_numeral_int @ V ) )
% 8.90/5.12 = ( ord_less_eq_rat @ X2 @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7893_numeral__le__ceiling,axiom,
% 8.90/5.12 ! [V: num,X2: real] :
% 8.90/5.12 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X2 ) )
% 8.90/5.12 = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7894_numeral__le__ceiling,axiom,
% 8.90/5.12 ! [V: num,X2: rat] :
% 8.90/5.12 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X2 ) )
% 8.90/5.12 = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7895_ceiling__le__neg__numeral,axiom,
% 8.90/5.12 ! [X2: real,V: num] :
% 8.90/5.12 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.90/5.12 = ( ord_less_eq_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7896_ceiling__le__neg__numeral,axiom,
% 8.90/5.12 ! [X2: rat,V: num] :
% 8.90/5.12 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.90/5.12 = ( ord_less_eq_rat @ X2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7897_neg__numeral__less__ceiling,axiom,
% 8.90/5.12 ! [V: num,X2: real] :
% 8.90/5.12 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X2 ) )
% 8.90/5.12 = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7898_neg__numeral__less__ceiling,axiom,
% 8.90/5.12 ! [V: num,X2: rat] :
% 8.90/5.12 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X2 ) )
% 8.90/5.12 = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7899_ceiling__minus__divide__eq__div__numeral,axiom,
% 8.90/5.12 ! [A: num,B: num] :
% 8.90/5.12 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 8.90/5.12 = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7900_ceiling__less__neg__numeral,axiom,
% 8.90/5.12 ! [X2: real,V: num] :
% 8.90/5.12 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.90/5.12 = ( ord_less_eq_real @ X2 @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7901_ceiling__less__neg__numeral,axiom,
% 8.90/5.12 ! [X2: rat,V: num] :
% 8.90/5.12 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.90/5.12 = ( ord_less_eq_rat @ X2 @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7902_neg__numeral__le__ceiling,axiom,
% 8.90/5.12 ! [V: num,X2: real] :
% 8.90/5.12 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X2 ) )
% 8.90/5.12 = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7903_neg__numeral__le__ceiling,axiom,
% 8.90/5.12 ! [V: num,X2: rat] :
% 8.90/5.12 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X2 ) )
% 8.90/5.12 = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7904_ceiling__mono,axiom,
% 8.90/5.12 ! [Y3: real,X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ Y3 @ X2 )
% 8.90/5.12 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y3 ) @ ( archim7802044766580827645g_real @ X2 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7905_ceiling__mono,axiom,
% 8.90/5.12 ! [Y3: rat,X2: rat] :
% 8.90/5.12 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 8.90/5.12 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y3 ) @ ( archim2889992004027027881ng_rat @ X2 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7906_le__of__int__ceiling,axiom,
% 8.90/5.12 ! [X2: real] : ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7907_le__of__int__ceiling,axiom,
% 8.90/5.12 ! [X2: rat] : ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7908_ceiling__less__cancel,axiom,
% 8.90/5.12 ! [X2: rat,Y3: rat] :
% 8.90/5.12 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( archim2889992004027027881ng_rat @ Y3 ) )
% 8.90/5.12 => ( ord_less_rat @ X2 @ Y3 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7909_ceiling__less__cancel,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ ( archim7802044766580827645g_real @ Y3 ) )
% 8.90/5.12 => ( ord_less_real @ X2 @ Y3 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7910_ceiling__ge__round,axiom,
% 8.90/5.12 ! [X2: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X2 ) @ ( archim7802044766580827645g_real @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7911_ceiling__le__iff,axiom,
% 8.90/5.12 ! [X2: real,Z: int] :
% 8.90/5.12 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ Z )
% 8.90/5.12 = ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7912_ceiling__le__iff,axiom,
% 8.90/5.12 ! [X2: rat,Z: int] :
% 8.90/5.12 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ Z )
% 8.90/5.12 = ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7913_ceiling__le,axiom,
% 8.90/5.12 ! [X2: real,A: int] :
% 8.90/5.12 ( ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ A ) )
% 8.90/5.12 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ A ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7914_ceiling__le,axiom,
% 8.90/5.12 ! [X2: rat,A: int] :
% 8.90/5.12 ( ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ A ) )
% 8.90/5.12 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ A ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7915_less__ceiling__iff,axiom,
% 8.90/5.12 ! [Z: int,X2: rat] :
% 8.90/5.12 ( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X2 ) )
% 8.90/5.12 = ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7916_less__ceiling__iff,axiom,
% 8.90/5.12 ! [Z: int,X2: real] :
% 8.90/5.12 ( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X2 ) )
% 8.90/5.12 = ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7917_ceiling__add__le,axiom,
% 8.90/5.12 ! [X2: rat,Y3: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X2 @ Y3 ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( archim2889992004027027881ng_rat @ Y3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7918_ceiling__add__le,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X2 ) @ ( archim7802044766580827645g_real @ Y3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7919_of__int__ceiling__le__add__one,axiom,
% 8.90/5.12 ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7920_of__int__ceiling__le__add__one,axiom,
% 8.90/5.12 ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7921_of__int__ceiling__diff__one__le,axiom,
% 8.90/5.12 ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7922_of__int__ceiling__diff__one__le,axiom,
% 8.90/5.12 ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7923_ceiling__divide__eq__div,axiom,
% 8.90/5.12 ! [A: int,B: int] :
% 8.90/5.12 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) )
% 8.90/5.12 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7924_ceiling__divide__eq__div,axiom,
% 8.90/5.12 ! [A: int,B: int] :
% 8.90/5.12 ( ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) )
% 8.90/5.12 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7925_ceiling__correct,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) @ one_one_real ) @ X2 )
% 8.90/5.12 & ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7926_ceiling__correct,axiom,
% 8.90/5.12 ! [X2: rat] :
% 8.90/5.12 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) ) @ one_one_rat ) @ X2 )
% 8.90/5.12 & ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7927_ceiling__unique,axiom,
% 8.90/5.12 ! [Z: int,X2: real] :
% 8.90/5.12 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z ) )
% 8.90/5.12 => ( ( archim7802044766580827645g_real @ X2 )
% 8.90/5.12 = Z ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7928_ceiling__unique,axiom,
% 8.90/5.12 ! [Z: int,X2: rat] :
% 8.90/5.12 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z ) )
% 8.90/5.12 => ( ( archim2889992004027027881ng_rat @ X2 )
% 8.90/5.12 = Z ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7929_ceiling__eq__iff,axiom,
% 8.90/5.12 ! [X2: real,A: int] :
% 8.90/5.12 ( ( ( archim7802044766580827645g_real @ X2 )
% 8.90/5.12 = A )
% 8.90/5.12 = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X2 )
% 8.90/5.12 & ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ A ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7930_ceiling__eq__iff,axiom,
% 8.90/5.12 ! [X2: rat,A: int] :
% 8.90/5.12 ( ( ( archim2889992004027027881ng_rat @ X2 )
% 8.90/5.12 = A )
% 8.90/5.12 = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X2 )
% 8.90/5.12 & ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ A ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7931_ceiling__split,axiom,
% 8.90/5.12 ! [P: int > $o,T: real] :
% 8.90/5.12 ( ( P @ ( archim7802044766580827645g_real @ T ) )
% 8.90/5.12 = ( ! [I4: int] :
% 8.90/5.12 ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) @ T )
% 8.90/5.12 & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I4 ) ) )
% 8.90/5.12 => ( P @ I4 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7932_ceiling__split,axiom,
% 8.90/5.12 ! [P: int > $o,T: rat] :
% 8.90/5.12 ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
% 8.90/5.12 = ( ! [I4: int] :
% 8.90/5.12 ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) @ T )
% 8.90/5.12 & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I4 ) ) )
% 8.90/5.12 => ( P @ I4 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7933_mult__ceiling__le,axiom,
% 8.90/5.12 ! [A: real,B: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.90/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 8.90/5.12 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7934_mult__ceiling__le,axiom,
% 8.90/5.12 ! [A: rat,B: rat] :
% 8.90/5.12 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 8.90/5.12 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 8.90/5.12 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7935_ceiling__less__iff,axiom,
% 8.90/5.12 ! [X2: real,Z: int] :
% 8.90/5.12 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ Z )
% 8.90/5.12 = ( ord_less_eq_real @ X2 @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7936_ceiling__less__iff,axiom,
% 8.90/5.12 ! [X2: rat,Z: int] :
% 8.90/5.12 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ Z )
% 8.90/5.12 = ( ord_less_eq_rat @ X2 @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7937_le__ceiling__iff,axiom,
% 8.90/5.12 ! [Z: int,X2: rat] :
% 8.90/5.12 ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X2 ) )
% 8.90/5.12 = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7938_le__ceiling__iff,axiom,
% 8.90/5.12 ! [Z: int,X2: real] :
% 8.90/5.12 ( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X2 ) )
% 8.90/5.12 = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7939_ceiling__divide__upper,axiom,
% 8.90/5.12 ! [Q3: real,P6: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ Q3 )
% 8.90/5.12 => ( ord_less_eq_real @ P6 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P6 @ Q3 ) ) ) @ Q3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7940_ceiling__divide__upper,axiom,
% 8.90/5.12 ! [Q3: rat,P6: rat] :
% 8.90/5.12 ( ( ord_less_rat @ zero_zero_rat @ Q3 )
% 8.90/5.12 => ( ord_less_eq_rat @ P6 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P6 @ Q3 ) ) ) @ Q3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7941_ceiling__divide__lower,axiom,
% 8.90/5.12 ! [Q3: real,P6: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ Q3 )
% 8.90/5.12 => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P6 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) @ P6 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7942_ceiling__divide__lower,axiom,
% 8.90/5.12 ! [Q3: rat,P6: rat] :
% 8.90/5.12 ( ( ord_less_rat @ zero_zero_rat @ Q3 )
% 8.90/5.12 => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P6 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) @ P6 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7943_ceiling__eq,axiom,
% 8.90/5.12 ! [N: int,X2: real] :
% 8.90/5.12 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 8.90/5.12 => ( ( archim7802044766580827645g_real @ X2 )
% 8.90/5.12 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7944_ceiling__eq,axiom,
% 8.90/5.12 ! [N: int,X2: rat] :
% 8.90/5.12 ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
% 8.90/5.12 => ( ( archim2889992004027027881ng_rat @ X2 )
% 8.90/5.12 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7945_ceiling__log2__div2,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.12 => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 8.90/5.12 = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7946_ceiling__log__nat__eq__if,axiom,
% 8.90/5.12 ! [B: nat,N: nat,K: nat] :
% 8.90/5.12 ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 8.90/5.12 => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 8.90/5.12 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 8.90/5.12 => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 8.90/5.12 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7947_ceiling__log__nat__eq__powr__iff,axiom,
% 8.90/5.12 ! [B: nat,K: nat,N: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 8.90/5.12 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.90/5.12 => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 8.90/5.12 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
% 8.90/5.12 = ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 8.90/5.12 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7948_pi__series,axiom,
% 8.90/5.12 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.90/5.12 = ( suminf_real
% 8.90/5.12 @ ^ [K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7949_summable__arctan__series,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 8.90/5.12 => ( summable_real
% 8.90/5.12 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7950_finite__nat__bounded,axiom,
% 8.90/5.12 ! [S3: set_nat] :
% 8.90/5.12 ( ( finite_finite_nat @ S3 )
% 8.90/5.12 => ? [K3: nat] : ( ord_less_eq_set_nat @ S3 @ ( set_ord_lessThan_nat @ K3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7951_finite__nat__iff__bounded,axiom,
% 8.90/5.12 ( finite_finite_nat
% 8.90/5.12 = ( ^ [S5: set_nat] :
% 8.90/5.12 ? [K2: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_lessThan_nat @ K2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7952_infinite__int__iff__unbounded__le,axiom,
% 8.90/5.12 ! [S3: set_int] :
% 8.90/5.12 ( ( ~ ( finite_finite_int @ S3 ) )
% 8.90/5.12 = ( ! [M3: int] :
% 8.90/5.12 ? [N3: int] :
% 8.90/5.12 ( ( ord_less_eq_int @ M3 @ ( abs_abs_int @ N3 ) )
% 8.90/5.12 & ( member_int @ N3 @ S3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7953_summable__iff__shift,axiom,
% 8.90/5.12 ! [F: nat > real,K: nat] :
% 8.90/5.12 ( ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) )
% 8.90/5.12 = ( summable_real @ F ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7954_summable__cmult__iff,axiom,
% 8.90/5.12 ! [C: complex,F: nat > complex] :
% 8.90/5.12 ( ( summable_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) ) )
% 8.90/5.12 = ( ( C = zero_zero_complex )
% 8.90/5.12 | ( summable_complex @ F ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7955_summable__cmult__iff,axiom,
% 8.90/5.12 ! [C: real,F: nat > real] :
% 8.90/5.12 ( ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) )
% 8.90/5.12 = ( ( C = zero_zero_real )
% 8.90/5.12 | ( summable_real @ F ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7956_summable__divide__iff,axiom,
% 8.90/5.12 ! [F: nat > complex,C: complex] :
% 8.90/5.12 ( ( summable_complex
% 8.90/5.12 @ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C ) )
% 8.90/5.12 = ( ( C = zero_zero_complex )
% 8.90/5.12 | ( summable_complex @ F ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7957_summable__divide__iff,axiom,
% 8.90/5.12 ! [F: nat > real,C: real] :
% 8.90/5.12 ( ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C ) )
% 8.90/5.12 = ( ( C = zero_zero_real )
% 8.90/5.12 | ( summable_real @ F ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7958_summable__geometric__iff,axiom,
% 8.90/5.12 ! [C: real] :
% 8.90/5.12 ( ( summable_real @ ( power_power_real @ C ) )
% 8.90/5.12 = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7959_summable__geometric__iff,axiom,
% 8.90/5.12 ! [C: complex] :
% 8.90/5.12 ( ( summable_complex @ ( power_power_complex @ C ) )
% 8.90/5.12 = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7960_summable__comparison__test,axiom,
% 8.90/5.12 ! [F: nat > real,G: nat > real] :
% 8.90/5.12 ( ? [N7: nat] :
% 8.90/5.12 ! [N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ N7 @ N2 )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
% 8.90/5.12 => ( ( summable_real @ G )
% 8.90/5.12 => ( summable_real @ F ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7961_summable__comparison__test,axiom,
% 8.90/5.12 ! [F: nat > complex,G: nat > real] :
% 8.90/5.12 ( ? [N7: nat] :
% 8.90/5.12 ! [N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ N7 @ N2 )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
% 8.90/5.12 => ( ( summable_real @ G )
% 8.90/5.12 => ( summable_complex @ F ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7962_summable__comparison__test_H,axiom,
% 8.90/5.12 ! [G: nat > real,N5: nat,F: nat > real] :
% 8.90/5.12 ( ( summable_real @ G )
% 8.90/5.12 => ( ! [N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ N5 @ N2 )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
% 8.90/5.12 => ( summable_real @ F ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7963_summable__comparison__test_H,axiom,
% 8.90/5.12 ! [G: nat > real,N5: nat,F: nat > complex] :
% 8.90/5.12 ( ( summable_real @ G )
% 8.90/5.12 => ( ! [N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ N5 @ N2 )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
% 8.90/5.12 => ( summable_complex @ F ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7964_summable__mult2,axiom,
% 8.90/5.12 ! [F: nat > real,C: real] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7965_summable__mult,axiom,
% 8.90/5.12 ! [F: nat > real,C: real] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7966_summable__add,axiom,
% 8.90/5.12 ! [F: nat > real,G: nat > real] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ( summable_real @ G )
% 8.90/5.12 => ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( plus_plus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7967_summable__add,axiom,
% 8.90/5.12 ! [F: nat > nat,G: nat > nat] :
% 8.90/5.12 ( ( summable_nat @ F )
% 8.90/5.12 => ( ( summable_nat @ G )
% 8.90/5.12 => ( summable_nat
% 8.90/5.12 @ ^ [N3: nat] : ( plus_plus_nat @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7968_summable__add,axiom,
% 8.90/5.12 ! [F: nat > int,G: nat > int] :
% 8.90/5.12 ( ( summable_int @ F )
% 8.90/5.12 => ( ( summable_int @ G )
% 8.90/5.12 => ( summable_int
% 8.90/5.12 @ ^ [N3: nat] : ( plus_plus_int @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7969_summable__diff,axiom,
% 8.90/5.12 ! [F: nat > real,G: nat > real] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ( summable_real @ G )
% 8.90/5.12 => ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7970_summable__divide,axiom,
% 8.90/5.12 ! [F: nat > complex,C: complex] :
% 8.90/5.12 ( ( summable_complex @ F )
% 8.90/5.12 => ( summable_complex
% 8.90/5.12 @ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7971_summable__divide,axiom,
% 8.90/5.12 ! [F: nat > real,C: real] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7972_summable__Suc__iff,axiom,
% 8.90/5.12 ! [F: nat > real] :
% 8.90/5.12 ( ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
% 8.90/5.12 = ( summable_real @ F ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7973_summable__ignore__initial__segment,axiom,
% 8.90/5.12 ! [F: nat > real,K: nat] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7974_powser__insidea,axiom,
% 8.90/5.12 ! [F: nat > real,X2: real,Z: real] :
% 8.90/5.12 ( ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X2 @ N3 ) ) )
% 8.90/5.12 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X2 ) )
% 8.90/5.12 => ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7975_powser__insidea,axiom,
% 8.90/5.12 ! [F: nat > complex,X2: complex,Z: complex] :
% 8.90/5.12 ( ( summable_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ X2 @ N3 ) ) )
% 8.90/5.12 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X2 ) )
% 8.90/5.12 => ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7976_suminf__le,axiom,
% 8.90/5.12 ! [F: nat > real,G: nat > real] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 8.90/5.12 => ( ( summable_real @ F )
% 8.90/5.12 => ( ( summable_real @ G )
% 8.90/5.12 => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7977_suminf__le,axiom,
% 8.90/5.12 ! [F: nat > nat,G: nat > nat] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( G @ N2 ) )
% 8.90/5.12 => ( ( summable_nat @ F )
% 8.90/5.12 => ( ( summable_nat @ G )
% 8.90/5.12 => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7978_suminf__le,axiom,
% 8.90/5.12 ! [F: nat > int,G: nat > int] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( G @ N2 ) )
% 8.90/5.12 => ( ( summable_int @ F )
% 8.90/5.12 => ( ( summable_int @ G )
% 8.90/5.12 => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7979_summable__mult__D,axiom,
% 8.90/5.12 ! [C: complex,F: nat > complex] :
% 8.90/5.12 ( ( summable_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) ) )
% 8.90/5.12 => ( ( C != zero_zero_complex )
% 8.90/5.12 => ( summable_complex @ F ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7980_summable__mult__D,axiom,
% 8.90/5.12 ! [C: real,F: nat > real] :
% 8.90/5.12 ( ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) )
% 8.90/5.12 => ( ( C != zero_zero_real )
% 8.90/5.12 => ( summable_real @ F ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7981_summable__zero__power,axiom,
% 8.90/5.12 summable_real @ ( power_power_real @ zero_zero_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7982_summable__zero__power,axiom,
% 8.90/5.12 summable_int @ ( power_power_int @ zero_zero_int ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7983_summable__zero__power,axiom,
% 8.90/5.12 summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7984_pi__ge__zero,axiom,
% 8.90/5.12 ord_less_eq_real @ zero_zero_real @ pi ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7985_suminf__mult,axiom,
% 8.90/5.12 ! [F: nat > real,C: real] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ( suminf_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) )
% 8.90/5.12 = ( times_times_real @ C @ ( suminf_real @ F ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7986_suminf__mult2,axiom,
% 8.90/5.12 ! [F: nat > real,C: real] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ( times_times_real @ ( suminf_real @ F ) @ C )
% 8.90/5.12 = ( suminf_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7987_suminf__add,axiom,
% 8.90/5.12 ! [F: nat > real,G: nat > real] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ( summable_real @ G )
% 8.90/5.12 => ( ( plus_plus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
% 8.90/5.12 = ( suminf_real
% 8.90/5.12 @ ^ [N3: nat] : ( plus_plus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7988_suminf__add,axiom,
% 8.90/5.12 ! [F: nat > nat,G: nat > nat] :
% 8.90/5.12 ( ( summable_nat @ F )
% 8.90/5.12 => ( ( summable_nat @ G )
% 8.90/5.12 => ( ( plus_plus_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) )
% 8.90/5.12 = ( suminf_nat
% 8.90/5.12 @ ^ [N3: nat] : ( plus_plus_nat @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7989_suminf__add,axiom,
% 8.90/5.12 ! [F: nat > int,G: nat > int] :
% 8.90/5.12 ( ( summable_int @ F )
% 8.90/5.12 => ( ( summable_int @ G )
% 8.90/5.12 => ( ( plus_plus_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) )
% 8.90/5.12 = ( suminf_int
% 8.90/5.12 @ ^ [N3: nat] : ( plus_plus_int @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7990_suminf__diff,axiom,
% 8.90/5.12 ! [F: nat > real,G: nat > real] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ( summable_real @ G )
% 8.90/5.12 => ( ( minus_minus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
% 8.90/5.12 = ( suminf_real
% 8.90/5.12 @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7991_suminf__divide,axiom,
% 8.90/5.12 ! [F: nat > complex,C: complex] :
% 8.90/5.12 ( ( summable_complex @ F )
% 8.90/5.12 => ( ( suminf_complex
% 8.90/5.12 @ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C ) )
% 8.90/5.12 = ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7992_suminf__divide,axiom,
% 8.90/5.12 ! [F: nat > real,C: real] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ( suminf_real
% 8.90/5.12 @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C ) )
% 8.90/5.12 = ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7993_suminf__eq__zero__iff,axiom,
% 8.90/5.12 ! [F: nat > real] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
% 8.90/5.12 => ( ( ( suminf_real @ F )
% 8.90/5.12 = zero_zero_real )
% 8.90/5.12 = ( ! [N3: nat] :
% 8.90/5.12 ( ( F @ N3 )
% 8.90/5.12 = zero_zero_real ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7994_suminf__eq__zero__iff,axiom,
% 8.90/5.12 ! [F: nat > nat] :
% 8.90/5.12 ( ( summable_nat @ F )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
% 8.90/5.12 => ( ( ( suminf_nat @ F )
% 8.90/5.12 = zero_zero_nat )
% 8.90/5.12 = ( ! [N3: nat] :
% 8.90/5.12 ( ( F @ N3 )
% 8.90/5.12 = zero_zero_nat ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7995_suminf__eq__zero__iff,axiom,
% 8.90/5.12 ! [F: nat > int] :
% 8.90/5.12 ( ( summable_int @ F )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
% 8.90/5.12 => ( ( ( suminf_int @ F )
% 8.90/5.12 = zero_zero_int )
% 8.90/5.12 = ( ! [N3: nat] :
% 8.90/5.12 ( ( F @ N3 )
% 8.90/5.12 = zero_zero_int ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7996_suminf__nonneg,axiom,
% 8.90/5.12 ! [F: nat > real] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
% 8.90/5.12 => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7997_suminf__nonneg,axiom,
% 8.90/5.12 ! [F: nat > nat] :
% 8.90/5.12 ( ( summable_nat @ F )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
% 8.90/5.12 => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7998_suminf__nonneg,axiom,
% 8.90/5.12 ! [F: nat > int] :
% 8.90/5.12 ( ( summable_int @ F )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
% 8.90/5.12 => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_7999_suminf__pos,axiom,
% 8.90/5.12 ! [F: nat > real] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N2 ) )
% 8.90/5.12 => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8000_suminf__pos,axiom,
% 8.90/5.12 ! [F: nat > nat] :
% 8.90/5.12 ( ( summable_nat @ F )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N2 ) )
% 8.90/5.12 => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8001_suminf__pos,axiom,
% 8.90/5.12 ! [F: nat > int] :
% 8.90/5.12 ( ( summable_int @ F )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N2 ) )
% 8.90/5.12 => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8002_summable__0__powser,axiom,
% 8.90/5.12 ! [F: nat > complex] :
% 8.90/5.12 ( summable_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8003_summable__0__powser,axiom,
% 8.90/5.12 ! [F: nat > real] :
% 8.90/5.12 ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8004_summable__zero__power_H,axiom,
% 8.90/5.12 ! [F: nat > complex] :
% 8.90/5.12 ( summable_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8005_summable__zero__power_H,axiom,
% 8.90/5.12 ! [F: nat > real] :
% 8.90/5.12 ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8006_summable__zero__power_H,axiom,
% 8.90/5.12 ! [F: nat > int] :
% 8.90/5.12 ( summable_int
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_int @ ( F @ N3 ) @ ( power_power_int @ zero_zero_int @ N3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8007_powser__split__head_I3_J,axiom,
% 8.90/5.12 ! [F: nat > complex,Z: complex] :
% 8.90/5.12 ( ( summable_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
% 8.90/5.12 => ( summable_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8008_powser__split__head_I3_J,axiom,
% 8.90/5.12 ! [F: nat > real,Z: real] :
% 8.90/5.12 ( ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
% 8.90/5.12 => ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8009_summable__powser__split__head,axiom,
% 8.90/5.12 ! [F: nat > complex,Z: complex] :
% 8.90/5.12 ( ( summable_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) )
% 8.90/5.12 = ( summable_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8010_summable__powser__split__head,axiom,
% 8.90/5.12 ! [F: nat > real,Z: real] :
% 8.90/5.12 ( ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) )
% 8.90/5.12 = ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8011_summable__powser__ignore__initial__segment,axiom,
% 8.90/5.12 ! [F: nat > complex,M: nat,Z: complex] :
% 8.90/5.12 ( ( summable_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N3 @ M ) ) @ ( power_power_complex @ Z @ N3 ) ) )
% 8.90/5.12 = ( summable_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8012_summable__powser__ignore__initial__segment,axiom,
% 8.90/5.12 ! [F: nat > real,M: nat,Z: real] :
% 8.90/5.12 ( ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N3 @ M ) ) @ ( power_power_real @ Z @ N3 ) ) )
% 8.90/5.12 = ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8013_summable__norm__comparison__test,axiom,
% 8.90/5.12 ! [F: nat > complex,G: nat > real] :
% 8.90/5.12 ( ? [N7: nat] :
% 8.90/5.12 ! [N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ N7 @ N2 )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
% 8.90/5.12 => ( ( summable_real @ G )
% 8.90/5.12 => ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8014_summable__rabs__comparison__test,axiom,
% 8.90/5.12 ! [F: nat > real,G: nat > real] :
% 8.90/5.12 ( ? [N7: nat] :
% 8.90/5.12 ! [N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ N7 @ N2 )
% 8.90/5.12 => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
% 8.90/5.12 => ( ( summable_real @ G )
% 8.90/5.12 => ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8015_summable__rabs,axiom,
% 8.90/5.12 ! [F: nat > real] :
% 8.90/5.12 ( ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) )
% 8.90/5.12 => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
% 8.90/5.12 @ ( suminf_real
% 8.90/5.12 @ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8016_suminf__pos__iff,axiom,
% 8.90/5.12 ! [F: nat > real] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
% 8.90/5.12 = ( ? [I4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8017_suminf__pos__iff,axiom,
% 8.90/5.12 ! [F: nat > nat] :
% 8.90/5.12 ( ( summable_nat @ F )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
% 8.90/5.12 => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
% 8.90/5.12 = ( ? [I4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8018_suminf__pos__iff,axiom,
% 8.90/5.12 ! [F: nat > int] :
% 8.90/5.12 ( ( summable_int @ F )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
% 8.90/5.12 => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
% 8.90/5.12 = ( ? [I4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I4 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8019_suminf__pos2,axiom,
% 8.90/5.12 ! [F: nat > real,I2: nat] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 8.90/5.12 => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8020_suminf__pos2,axiom,
% 8.90/5.12 ! [F: nat > nat,I2: nat] :
% 8.90/5.12 ( ( summable_nat @ F )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
% 8.90/5.12 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 8.90/5.12 => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8021_suminf__pos2,axiom,
% 8.90/5.12 ! [F: nat > int,I2: nat] :
% 8.90/5.12 ( ( summable_int @ F )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
% 8.90/5.12 => ( ( ord_less_int @ zero_zero_int @ ( F @ I2 ) )
% 8.90/5.12 => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8022_suminf__le__const,axiom,
% 8.90/5.12 ! [F: nat > int,X2: int] :
% 8.90/5.12 ( ( summable_int @ F )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X2 )
% 8.90/5.12 => ( ord_less_eq_int @ ( suminf_int @ F ) @ X2 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8023_suminf__le__const,axiom,
% 8.90/5.12 ! [F: nat > nat,X2: nat] :
% 8.90/5.12 ( ( summable_nat @ F )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X2 )
% 8.90/5.12 => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X2 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8024_suminf__le__const,axiom,
% 8.90/5.12 ! [F: nat > real,X2: real] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X2 )
% 8.90/5.12 => ( ord_less_eq_real @ ( suminf_real @ F ) @ X2 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8025_powser__inside,axiom,
% 8.90/5.12 ! [F: nat > real,X2: real,Z: real] :
% 8.90/5.12 ( ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X2 @ N3 ) ) )
% 8.90/5.12 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X2 ) )
% 8.90/5.12 => ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8026_powser__inside,axiom,
% 8.90/5.12 ! [F: nat > complex,X2: complex,Z: complex] :
% 8.90/5.12 ( ( summable_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ X2 @ N3 ) ) )
% 8.90/5.12 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X2 ) )
% 8.90/5.12 => ( summable_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8027_summableI__nonneg__bounded,axiom,
% 8.90/5.12 ! [F: nat > int,X2: int] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X2 )
% 8.90/5.12 => ( summable_int @ F ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8028_summableI__nonneg__bounded,axiom,
% 8.90/5.12 ! [F: nat > nat,X2: nat] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X2 )
% 8.90/5.12 => ( summable_nat @ F ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8029_summableI__nonneg__bounded,axiom,
% 8.90/5.12 ! [F: nat > real,X2: real] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X2 )
% 8.90/5.12 => ( summable_real @ F ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8030_summable__geometric,axiom,
% 8.90/5.12 ! [C: real] :
% 8.90/5.12 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 8.90/5.12 => ( summable_real @ ( power_power_real @ C ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8031_summable__geometric,axiom,
% 8.90/5.12 ! [C: complex] :
% 8.90/5.12 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 8.90/5.12 => ( summable_complex @ ( power_power_complex @ C ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8032_complete__algebra__summable__geometric,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ one_one_real )
% 8.90/5.12 => ( summable_real @ ( power_power_real @ X2 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8033_complete__algebra__summable__geometric,axiom,
% 8.90/5.12 ! [X2: complex] :
% 8.90/5.12 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ one_one_real )
% 8.90/5.12 => ( summable_complex @ ( power_power_complex @ X2 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8034_suminf__split__head,axiom,
% 8.90/5.12 ! [F: nat > real] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ( suminf_real
% 8.90/5.12 @ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
% 8.90/5.12 = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8035_pi__less__4,axiom,
% 8.90/5.12 ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8036_pi__ge__two,axiom,
% 8.90/5.12 ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8037_pi__half__neq__two,axiom,
% 8.90/5.12 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.90/5.12 != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8038_summable__norm,axiom,
% 8.90/5.12 ! [F: nat > real] :
% 8.90/5.12 ( ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( real_V7735802525324610683m_real @ ( F @ N3 ) ) )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
% 8.90/5.12 @ ( suminf_real
% 8.90/5.12 @ ^ [N3: nat] : ( real_V7735802525324610683m_real @ ( F @ N3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8039_summable__norm,axiom,
% 8.90/5.12 ! [F: nat > complex] :
% 8.90/5.12 ( ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( F @ N3 ) ) )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
% 8.90/5.12 @ ( suminf_real
% 8.90/5.12 @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8040_sum__le__suminf,axiom,
% 8.90/5.12 ! [F: nat > int,I5: set_nat] :
% 8.90/5.12 ( ( summable_int @ F )
% 8.90/5.12 => ( ( finite_finite_nat @ I5 )
% 8.90/5.12 => ( ! [N2: nat] :
% 8.90/5.12 ( ( member_nat @ N2 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 8.90/5.12 => ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) ) )
% 8.90/5.12 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8041_sum__le__suminf,axiom,
% 8.90/5.12 ! [F: nat > nat,I5: set_nat] :
% 8.90/5.12 ( ( summable_nat @ F )
% 8.90/5.12 => ( ( finite_finite_nat @ I5 )
% 8.90/5.12 => ( ! [N2: nat] :
% 8.90/5.12 ( ( member_nat @ N2 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 8.90/5.12 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) ) )
% 8.90/5.12 => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8042_sum__le__suminf,axiom,
% 8.90/5.12 ! [F: nat > real,I5: set_nat] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ( finite_finite_nat @ I5 )
% 8.90/5.12 => ( ! [N2: nat] :
% 8.90/5.12 ( ( member_nat @ N2 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 8.90/5.12 => ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) ) )
% 8.90/5.12 => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8043_suminf__split__initial__segment,axiom,
% 8.90/5.12 ! [F: nat > real,K: nat] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ( suminf_real @ F )
% 8.90/5.12 = ( plus_plus_real
% 8.90/5.12 @ ( suminf_real
% 8.90/5.12 @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) )
% 8.90/5.12 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8044_suminf__minus__initial__segment,axiom,
% 8.90/5.12 ! [F: nat > real,K: nat] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ( suminf_real
% 8.90/5.12 @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) )
% 8.90/5.12 = ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8045_pi__half__neq__zero,axiom,
% 8.90/5.12 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.90/5.12 != zero_zero_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8046_pi__half__less__two,axiom,
% 8.90/5.12 ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8047_pi__half__le__two,axiom,
% 8.90/5.12 ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8048_sum__less__suminf,axiom,
% 8.90/5.12 ! [F: nat > int,N: nat] :
% 8.90/5.12 ( ( summable_int @ F )
% 8.90/5.12 => ( ! [M4: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ N @ M4 )
% 8.90/5.12 => ( ord_less_int @ zero_zero_int @ ( F @ M4 ) ) )
% 8.90/5.12 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8049_sum__less__suminf,axiom,
% 8.90/5.12 ! [F: nat > nat,N: nat] :
% 8.90/5.12 ( ( summable_nat @ F )
% 8.90/5.12 => ( ! [M4: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ N @ M4 )
% 8.90/5.12 => ( ord_less_nat @ zero_zero_nat @ ( F @ M4 ) ) )
% 8.90/5.12 => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8050_sum__less__suminf,axiom,
% 8.90/5.12 ! [F: nat > real,N: nat] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ! [M4: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ N @ M4 )
% 8.90/5.12 => ( ord_less_real @ zero_zero_real @ ( F @ M4 ) ) )
% 8.90/5.12 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8051_powser__split__head_I1_J,axiom,
% 8.90/5.12 ! [F: nat > complex,Z: complex] :
% 8.90/5.12 ( ( summable_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
% 8.90/5.12 => ( ( suminf_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
% 8.90/5.12 = ( plus_plus_complex @ ( F @ zero_zero_nat )
% 8.90/5.12 @ ( times_times_complex
% 8.90/5.12 @ ( suminf_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) )
% 8.90/5.12 @ Z ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8052_powser__split__head_I1_J,axiom,
% 8.90/5.12 ! [F: nat > real,Z: real] :
% 8.90/5.12 ( ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
% 8.90/5.12 => ( ( suminf_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
% 8.90/5.12 = ( plus_plus_real @ ( F @ zero_zero_nat )
% 8.90/5.12 @ ( times_times_real
% 8.90/5.12 @ ( suminf_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) )
% 8.90/5.12 @ Z ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8053_powser__split__head_I2_J,axiom,
% 8.90/5.12 ! [F: nat > complex,Z: complex] :
% 8.90/5.12 ( ( summable_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
% 8.90/5.12 => ( ( times_times_complex
% 8.90/5.12 @ ( suminf_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) )
% 8.90/5.12 @ Z )
% 8.90/5.12 = ( minus_minus_complex
% 8.90/5.12 @ ( suminf_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
% 8.90/5.12 @ ( F @ zero_zero_nat ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8054_powser__split__head_I2_J,axiom,
% 8.90/5.12 ! [F: nat > real,Z: real] :
% 8.90/5.12 ( ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
% 8.90/5.12 => ( ( times_times_real
% 8.90/5.12 @ ( suminf_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) )
% 8.90/5.12 @ Z )
% 8.90/5.12 = ( minus_minus_real
% 8.90/5.12 @ ( suminf_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
% 8.90/5.12 @ ( F @ zero_zero_nat ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8055_summable__partial__sum__bound,axiom,
% 8.90/5.12 ! [F: nat > complex,E: real] :
% 8.90/5.12 ( ( summable_complex @ F )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ E )
% 8.90/5.12 => ~ ! [N8: nat] :
% 8.90/5.12 ~ ! [M2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ N8 @ M2 )
% 8.90/5.12 => ! [N9: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M2 @ N9 ) ) ) @ E ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8056_summable__partial__sum__bound,axiom,
% 8.90/5.12 ! [F: nat > real,E: real] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ E )
% 8.90/5.12 => ~ ! [N8: nat] :
% 8.90/5.12 ~ ! [M2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ N8 @ M2 )
% 8.90/5.12 => ! [N9: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M2 @ N9 ) ) ) @ E ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8057_suminf__exist__split,axiom,
% 8.90/5.12 ! [R2: real,F: nat > real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ R2 )
% 8.90/5.12 => ( ( summable_real @ F )
% 8.90/5.12 => ? [N8: nat] :
% 8.90/5.12 ! [N9: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ N8 @ N9 )
% 8.90/5.12 => ( ord_less_real
% 8.90/5.12 @ ( real_V7735802525324610683m_real
% 8.90/5.12 @ ( suminf_real
% 8.90/5.12 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N9 ) ) ) )
% 8.90/5.12 @ R2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8058_suminf__exist__split,axiom,
% 8.90/5.12 ! [R2: real,F: nat > complex] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ R2 )
% 8.90/5.12 => ( ( summable_complex @ F )
% 8.90/5.12 => ? [N8: nat] :
% 8.90/5.12 ! [N9: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ N8 @ N9 )
% 8.90/5.12 => ( ord_less_real
% 8.90/5.12 @ ( real_V1022390504157884413omplex
% 8.90/5.12 @ ( suminf_complex
% 8.90/5.12 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N9 ) ) ) )
% 8.90/5.12 @ R2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8059_summable__power__series,axiom,
% 8.90/5.12 ! [F: nat > real,Z: real] :
% 8.90/5.12 ( ! [I3: nat] : ( ord_less_eq_real @ ( F @ I3 ) @ one_one_real )
% 8.90/5.12 => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 8.90/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Z )
% 8.90/5.12 => ( ( ord_less_real @ Z @ one_one_real )
% 8.90/5.12 => ( summable_real
% 8.90/5.12 @ ^ [I4: nat] : ( times_times_real @ ( F @ I4 ) @ ( power_power_real @ Z @ I4 ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8060_Abel__lemma,axiom,
% 8.90/5.12 ! [R2: real,R0: real,A: nat > complex,M7: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 8.90/5.12 => ( ( ord_less_real @ R2 @ R0 )
% 8.90/5.12 => ( ! [N2: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N2 ) ) @ ( power_power_real @ R0 @ N2 ) ) @ M7 )
% 8.90/5.12 => ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N3 ) ) @ ( power_power_real @ R2 @ N3 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8061_pi__half__gt__zero,axiom,
% 8.90/5.12 ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8062_pi__half__ge__zero,axiom,
% 8.90/5.12 ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8063_m2pi__less__pi,axiom,
% 8.90/5.12 ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8064_summable__ratio__test,axiom,
% 8.90/5.12 ! [C: real,N5: nat,F: nat > real] :
% 8.90/5.12 ( ( ord_less_real @ C @ one_one_real )
% 8.90/5.12 => ( ! [N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ N5 @ N2 )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N2 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) ) ) )
% 8.90/5.12 => ( summable_real @ F ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8065_summable__ratio__test,axiom,
% 8.90/5.12 ! [C: real,N5: nat,F: nat > complex] :
% 8.90/5.12 ( ( ord_less_real @ C @ one_one_real )
% 8.90/5.12 => ( ! [N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ N5 @ N2 )
% 8.90/5.12 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N2 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) )
% 8.90/5.12 => ( summable_complex @ F ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8066_sum__less__suminf2,axiom,
% 8.90/5.12 ! [F: nat > int,N: nat,I2: nat] :
% 8.90/5.12 ( ( summable_int @ F )
% 8.90/5.12 => ( ! [M4: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ N @ M4 )
% 8.90/5.12 => ( ord_less_eq_int @ zero_zero_int @ ( F @ M4 ) ) )
% 8.90/5.12 => ( ( ord_less_eq_nat @ N @ I2 )
% 8.90/5.12 => ( ( ord_less_int @ zero_zero_int @ ( F @ I2 ) )
% 8.90/5.12 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8067_sum__less__suminf2,axiom,
% 8.90/5.12 ! [F: nat > nat,N: nat,I2: nat] :
% 8.90/5.12 ( ( summable_nat @ F )
% 8.90/5.12 => ( ! [M4: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ N @ M4 )
% 8.90/5.12 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M4 ) ) )
% 8.90/5.12 => ( ( ord_less_eq_nat @ N @ I2 )
% 8.90/5.12 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 8.90/5.12 => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8068_sum__less__suminf2,axiom,
% 8.90/5.12 ! [F: nat > real,N: nat,I2: nat] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ! [M4: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ N @ M4 )
% 8.90/5.12 => ( ord_less_eq_real @ zero_zero_real @ ( F @ M4 ) ) )
% 8.90/5.12 => ( ( ord_less_eq_nat @ N @ I2 )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 8.90/5.12 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8069_arctan__ubound,axiom,
% 8.90/5.12 ! [Y3: real] : ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8070_arctan__one,axiom,
% 8.90/5.12 ( ( arctan @ one_one_real )
% 8.90/5.12 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8071_minus__pi__half__less__zero,axiom,
% 8.90/5.12 ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8072_arctan__bounded,axiom,
% 8.90/5.12 ! [Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 8.90/5.12 & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8073_arctan__lbound,axiom,
% 8.90/5.12 ! [Y3: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8074_infinite__nat__iff__unbounded,axiom,
% 8.90/5.12 ! [S3: set_nat] :
% 8.90/5.12 ( ( ~ ( finite_finite_nat @ S3 ) )
% 8.90/5.12 = ( ! [M3: nat] :
% 8.90/5.12 ? [N3: nat] :
% 8.90/5.12 ( ( ord_less_nat @ M3 @ N3 )
% 8.90/5.12 & ( member_nat @ N3 @ S3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8075_unbounded__k__infinite,axiom,
% 8.90/5.12 ! [K: nat,S3: set_nat] :
% 8.90/5.12 ( ! [M4: nat] :
% 8.90/5.12 ( ( ord_less_nat @ K @ M4 )
% 8.90/5.12 => ? [N9: nat] :
% 8.90/5.12 ( ( ord_less_nat @ M4 @ N9 )
% 8.90/5.12 & ( member_nat @ N9 @ S3 ) ) )
% 8.90/5.12 => ~ ( finite_finite_nat @ S3 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8076_infinite__nat__iff__unbounded__le,axiom,
% 8.90/5.12 ! [S3: set_nat] :
% 8.90/5.12 ( ( ~ ( finite_finite_nat @ S3 ) )
% 8.90/5.12 = ( ! [M3: nat] :
% 8.90/5.12 ? [N3: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M3 @ N3 )
% 8.90/5.12 & ( member_nat @ N3 @ S3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8077_machin__Euler,axiom,
% 8.90/5.12 ( ( 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 ) ) ) ) ) ) ) ) ) ) )
% 8.90/5.12 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8078_machin,axiom,
% 8.90/5.12 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.90/5.12 = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8079_sum__pos__lt__pair,axiom,
% 8.90/5.12 ! [F: nat > real,K: nat] :
% 8.90/5.12 ( ( summable_real @ F )
% 8.90/5.12 => ( ! [D3: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) @ one_one_nat ) ) ) ) )
% 8.90/5.12 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8080_sin__cos__npi,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.12 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8081_cos__pi__eq__zero,axiom,
% 8.90/5.12 ! [M: nat] :
% 8.90/5.12 ( ( 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 ) ) ) )
% 8.90/5.12 = zero_zero_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8082_ceiling__log__eq__powr__iff,axiom,
% 8.90/5.12 ! [X2: real,B: real,K: nat] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ one_one_real @ B )
% 8.90/5.12 => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X2 ) )
% 8.90/5.12 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 8.90/5.12 = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X2 )
% 8.90/5.12 & ( ord_less_eq_real @ X2 @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8083_geometric__deriv__sums,axiom,
% 8.90/5.12 ! [Z: real] :
% 8.90/5.12 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 8.90/5.12 => ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) )
% 8.90/5.12 @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8084_geometric__deriv__sums,axiom,
% 8.90/5.12 ! [Z: complex] :
% 8.90/5.12 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 8.90/5.12 => ( sums_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) )
% 8.90/5.12 @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8085_monoseq__def,axiom,
% 8.90/5.12 ( topolo6980174941875973593q_real
% 8.90/5.12 = ( ^ [X4: nat > real] :
% 8.90/5.12 ( ! [M3: nat,N3: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M3 @ N3 )
% 8.90/5.12 => ( ord_less_eq_real @ ( X4 @ M3 ) @ ( X4 @ N3 ) ) )
% 8.90/5.12 | ! [M3: nat,N3: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M3 @ N3 )
% 8.90/5.12 => ( ord_less_eq_real @ ( X4 @ N3 ) @ ( X4 @ M3 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8086_monoseq__def,axiom,
% 8.90/5.12 ( topolo3100542954746470799et_int
% 8.90/5.12 = ( ^ [X4: nat > set_int] :
% 8.90/5.12 ( ! [M3: nat,N3: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M3 @ N3 )
% 8.90/5.12 => ( ord_less_eq_set_int @ ( X4 @ M3 ) @ ( X4 @ N3 ) ) )
% 8.90/5.12 | ! [M3: nat,N3: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M3 @ N3 )
% 8.90/5.12 => ( ord_less_eq_set_int @ ( X4 @ N3 ) @ ( X4 @ M3 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8087_monoseq__def,axiom,
% 8.90/5.12 ( topolo4267028734544971653eq_rat
% 8.90/5.12 = ( ^ [X4: nat > rat] :
% 8.90/5.12 ( ! [M3: nat,N3: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M3 @ N3 )
% 8.90/5.12 => ( ord_less_eq_rat @ ( X4 @ M3 ) @ ( X4 @ N3 ) ) )
% 8.90/5.12 | ! [M3: nat,N3: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M3 @ N3 )
% 8.90/5.12 => ( ord_less_eq_rat @ ( X4 @ N3 ) @ ( X4 @ M3 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8088_monoseq__def,axiom,
% 8.90/5.12 ( topolo1459490580787246023eq_num
% 8.90/5.12 = ( ^ [X4: nat > num] :
% 8.90/5.12 ( ! [M3: nat,N3: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M3 @ N3 )
% 8.90/5.12 => ( ord_less_eq_num @ ( X4 @ M3 ) @ ( X4 @ N3 ) ) )
% 8.90/5.12 | ! [M3: nat,N3: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M3 @ N3 )
% 8.90/5.12 => ( ord_less_eq_num @ ( X4 @ N3 ) @ ( X4 @ M3 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8089_monoseq__def,axiom,
% 8.90/5.12 ( topolo4902158794631467389eq_nat
% 8.90/5.12 = ( ^ [X4: nat > nat] :
% 8.90/5.12 ( ! [M3: nat,N3: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M3 @ N3 )
% 8.90/5.12 => ( ord_less_eq_nat @ ( X4 @ M3 ) @ ( X4 @ N3 ) ) )
% 8.90/5.12 | ! [M3: nat,N3: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M3 @ N3 )
% 8.90/5.12 => ( ord_less_eq_nat @ ( X4 @ N3 ) @ ( X4 @ M3 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8090_monoseq__def,axiom,
% 8.90/5.12 ( topolo4899668324122417113eq_int
% 8.90/5.12 = ( ^ [X4: nat > int] :
% 8.90/5.12 ( ! [M3: nat,N3: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M3 @ N3 )
% 8.90/5.12 => ( ord_less_eq_int @ ( X4 @ M3 ) @ ( X4 @ N3 ) ) )
% 8.90/5.12 | ! [M3: nat,N3: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M3 @ N3 )
% 8.90/5.12 => ( ord_less_eq_int @ ( X4 @ N3 ) @ ( X4 @ M3 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8091_powr__one__eq__one,axiom,
% 8.90/5.12 ! [A: real] :
% 8.90/5.12 ( ( powr_real @ one_one_real @ A )
% 8.90/5.12 = one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8092_cos__zero,axiom,
% 8.90/5.12 ( ( cos_complex @ zero_zero_complex )
% 8.90/5.12 = one_one_complex ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8093_cos__zero,axiom,
% 8.90/5.12 ( ( cos_real @ zero_zero_real )
% 8.90/5.12 = one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8094_powr__zero__eq__one,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ( X2 = zero_zero_real )
% 8.90/5.12 => ( ( powr_real @ X2 @ zero_zero_real )
% 8.90/5.12 = zero_zero_real ) )
% 8.90/5.12 & ( ( X2 != zero_zero_real )
% 8.90/5.12 => ( ( powr_real @ X2 @ zero_zero_real )
% 8.90/5.12 = one_one_real ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8095_powr__nonneg__iff,axiom,
% 8.90/5.12 ! [A: real,X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( powr_real @ A @ X2 ) @ zero_zero_real )
% 8.90/5.12 = ( A = zero_zero_real ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8096_powr__less__cancel__iff,axiom,
% 8.90/5.12 ! [X2: real,A: real,B: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) )
% 8.90/5.12 = ( ord_less_real @ A @ B ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8097_sin__pi__minus,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( sin_real @ ( minus_minus_real @ pi @ X2 ) )
% 8.90/5.12 = ( sin_real @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8098_powr__eq__one__iff,axiom,
% 8.90/5.12 ! [A: real,X2: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ A )
% 8.90/5.12 => ( ( ( powr_real @ A @ X2 )
% 8.90/5.12 = one_one_real )
% 8.90/5.12 = ( X2 = zero_zero_real ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8099_powr__one__gt__zero__iff,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ( powr_real @ X2 @ one_one_real )
% 8.90/5.12 = X2 )
% 8.90/5.12 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8100_powr__one,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( powr_real @ X2 @ one_one_real )
% 8.90/5.12 = X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8101_powr__le__cancel__iff,axiom,
% 8.90/5.12 ! [X2: real,A: real,B: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) )
% 8.90/5.12 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8102_numeral__powr__numeral__real,axiom,
% 8.90/5.12 ! [M: num,N: num] :
% 8.90/5.12 ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 8.90/5.12 = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8103_cos__pi,axiom,
% 8.90/5.12 ( ( cos_real @ pi )
% 8.90/5.12 = ( uminus_uminus_real @ one_one_real ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8104_cos__minus__pi,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( cos_real @ ( minus_minus_real @ X2 @ pi ) )
% 8.90/5.12 = ( uminus_uminus_real @ ( cos_real @ X2 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8105_cos__pi__minus,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( cos_real @ ( minus_minus_real @ pi @ X2 ) )
% 8.90/5.12 = ( uminus_uminus_real @ ( cos_real @ X2 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8106_sin__minus__pi,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( sin_real @ ( minus_minus_real @ X2 @ pi ) )
% 8.90/5.12 = ( uminus_uminus_real @ ( sin_real @ X2 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8107_sin__cos__squared__add3,axiom,
% 8.90/5.12 ! [X2: complex] :
% 8.90/5.12 ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X2 ) @ ( cos_complex @ X2 ) ) @ ( times_times_complex @ ( sin_complex @ X2 ) @ ( sin_complex @ X2 ) ) )
% 8.90/5.12 = one_one_complex ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8108_sin__cos__squared__add3,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ X2 ) ) @ ( times_times_real @ ( sin_real @ X2 ) @ ( sin_real @ X2 ) ) )
% 8.90/5.12 = one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8109_log__powr__cancel,axiom,
% 8.90/5.12 ! [A: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ A )
% 8.90/5.12 => ( ( A != one_one_real )
% 8.90/5.12 => ( ( log @ A @ ( powr_real @ A @ Y3 ) )
% 8.90/5.12 = Y3 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8110_powr__log__cancel,axiom,
% 8.90/5.12 ! [A: real,X2: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ A )
% 8.90/5.12 => ( ( A != one_one_real )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( powr_real @ A @ ( log @ A @ X2 ) )
% 8.90/5.12 = X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8111_sin__npi2,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 8.90/5.12 = zero_zero_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8112_sin__npi,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 8.90/5.12 = zero_zero_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8113_sin__npi__int,axiom,
% 8.90/5.12 ! [N: int] :
% 8.90/5.12 ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 8.90/5.12 = zero_zero_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8114_powser__sums__zero__iff,axiom,
% 8.90/5.12 ! [A: nat > complex,X2: complex] :
% 8.90/5.12 ( ( sums_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( A @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) )
% 8.90/5.12 @ X2 )
% 8.90/5.12 = ( ( A @ zero_zero_nat )
% 8.90/5.12 = X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8115_powser__sums__zero__iff,axiom,
% 8.90/5.12 ! [A: nat > real,X2: real] :
% 8.90/5.12 ( ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( A @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) )
% 8.90/5.12 @ X2 )
% 8.90/5.12 = ( ( A @ zero_zero_nat )
% 8.90/5.12 = X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8116_powr__numeral,axiom,
% 8.90/5.12 ! [X2: real,N: num] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( powr_real @ X2 @ ( numeral_numeral_real @ N ) )
% 8.90/5.12 = ( power_power_real @ X2 @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8117_cos__pi__half,axiom,
% 8.90/5.12 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.12 = zero_zero_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8118_sin__two__pi,axiom,
% 8.90/5.12 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 8.90/5.12 = zero_zero_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8119_sin__pi__half,axiom,
% 8.90/5.12 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.12 = one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8120_cos__two__pi,axiom,
% 8.90/5.12 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 8.90/5.12 = one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8121_cos__periodic,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( cos_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 8.90/5.12 = ( cos_real @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8122_sin__periodic,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( sin_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 8.90/5.12 = ( sin_real @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8123_cos__2pi__minus,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X2 ) )
% 8.90/5.12 = ( cos_real @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8124_cos__npi,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 8.90/5.12 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8125_cos__npi2,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 8.90/5.12 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8126_sin__cos__squared__add2,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.12 = one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8127_sin__cos__squared__add2,axiom,
% 8.90/5.12 ! [X2: complex] :
% 8.90/5.12 ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.12 = one_one_complex ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8128_sin__cos__squared__add,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.12 = one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8129_sin__cos__squared__add,axiom,
% 8.90/5.12 ! [X2: complex] :
% 8.90/5.12 ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.12 = one_one_complex ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8130_sin__2npi,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 8.90/5.12 = zero_zero_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8131_cos__2npi,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 8.90/5.12 = one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8132_sin__2pi__minus,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X2 ) )
% 8.90/5.12 = ( uminus_uminus_real @ ( sin_real @ X2 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8133_sin__int__2pin,axiom,
% 8.90/5.12 ! [N: int] :
% 8.90/5.12 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 8.90/5.12 = zero_zero_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8134_cos__int__2pin,axiom,
% 8.90/5.12 ! [N: int] :
% 8.90/5.12 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 8.90/5.12 = one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8135_cos__3over2__pi,axiom,
% 8.90/5.12 ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 8.90/5.12 = zero_zero_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8136_square__powr__half,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( powr_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.12 = ( abs_abs_real @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8137_sin__3over2__pi,axiom,
% 8.90/5.12 ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 8.90/5.12 = ( uminus_uminus_real @ one_one_real ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8138_cos__npi__int,axiom,
% 8.90/5.12 ! [N: int] :
% 8.90/5.12 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 8.90/5.12 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 8.90/5.12 = one_one_real ) )
% 8.90/5.12 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 8.90/5.12 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 8.90/5.12 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8139_sin__add,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( sin_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 8.90/5.12 = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X2 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8140_polar__Ex,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ? [R3: real,A4: real] :
% 8.90/5.12 ( ( X2
% 8.90/5.12 = ( times_times_real @ R3 @ ( cos_real @ A4 ) ) )
% 8.90/5.12 & ( Y3
% 8.90/5.12 = ( times_times_real @ R3 @ ( sin_real @ A4 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8141_sin__diff,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( sin_real @ ( minus_minus_real @ X2 @ Y3 ) )
% 8.90/5.12 = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X2 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8142_cos__one__sin__zero,axiom,
% 8.90/5.12 ! [X2: complex] :
% 8.90/5.12 ( ( ( cos_complex @ X2 )
% 8.90/5.12 = one_one_complex )
% 8.90/5.12 => ( ( sin_complex @ X2 )
% 8.90/5.12 = zero_zero_complex ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8143_cos__one__sin__zero,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ( cos_real @ X2 )
% 8.90/5.12 = one_one_real )
% 8.90/5.12 => ( ( sin_real @ X2 )
% 8.90/5.12 = zero_zero_real ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8144_powr__powr,axiom,
% 8.90/5.12 ! [X2: real,A: real,B: real] :
% 8.90/5.12 ( ( powr_real @ ( powr_real @ X2 @ A ) @ B )
% 8.90/5.12 = ( powr_real @ X2 @ ( times_times_real @ A @ B ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8145_sums__le,axiom,
% 8.90/5.12 ! [F: nat > real,G: nat > real,S: real,T: real] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 8.90/5.12 => ( ( sums_real @ F @ S )
% 8.90/5.12 => ( ( sums_real @ G @ T )
% 8.90/5.12 => ( ord_less_eq_real @ S @ T ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8146_sums__le,axiom,
% 8.90/5.12 ! [F: nat > nat,G: nat > nat,S: nat,T: nat] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( G @ N2 ) )
% 8.90/5.12 => ( ( sums_nat @ F @ S )
% 8.90/5.12 => ( ( sums_nat @ G @ T )
% 8.90/5.12 => ( ord_less_eq_nat @ S @ T ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8147_sums__le,axiom,
% 8.90/5.12 ! [F: nat > int,G: nat > int,S: int,T: int] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( G @ N2 ) )
% 8.90/5.12 => ( ( sums_int @ F @ S )
% 8.90/5.12 => ( ( sums_int @ G @ T )
% 8.90/5.12 => ( ord_less_eq_int @ S @ T ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8148_sums__mult,axiom,
% 8.90/5.12 ! [F: nat > real,A: real,C: real] :
% 8.90/5.12 ( ( sums_real @ F @ A )
% 8.90/5.12 => ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) )
% 8.90/5.12 @ ( times_times_real @ C @ A ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8149_sums__mult2,axiom,
% 8.90/5.12 ! [F: nat > real,A: real,C: real] :
% 8.90/5.12 ( ( sums_real @ F @ A )
% 8.90/5.12 => ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C )
% 8.90/5.12 @ ( times_times_real @ A @ C ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8150_sums__add,axiom,
% 8.90/5.12 ! [F: nat > real,A: real,G: nat > real,B: real] :
% 8.90/5.12 ( ( sums_real @ F @ A )
% 8.90/5.12 => ( ( sums_real @ G @ B )
% 8.90/5.12 => ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( plus_plus_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 8.90/5.12 @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8151_sums__add,axiom,
% 8.90/5.12 ! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
% 8.90/5.12 ( ( sums_nat @ F @ A )
% 8.90/5.12 => ( ( sums_nat @ G @ B )
% 8.90/5.12 => ( sums_nat
% 8.90/5.12 @ ^ [N3: nat] : ( plus_plus_nat @ ( F @ N3 ) @ ( G @ N3 ) )
% 8.90/5.12 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8152_sums__add,axiom,
% 8.90/5.12 ! [F: nat > int,A: int,G: nat > int,B: int] :
% 8.90/5.12 ( ( sums_int @ F @ A )
% 8.90/5.12 => ( ( sums_int @ G @ B )
% 8.90/5.12 => ( sums_int
% 8.90/5.12 @ ^ [N3: nat] : ( plus_plus_int @ ( F @ N3 ) @ ( G @ N3 ) )
% 8.90/5.12 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8153_sums__diff,axiom,
% 8.90/5.12 ! [F: nat > real,A: real,G: nat > real,B: real] :
% 8.90/5.12 ( ( sums_real @ F @ A )
% 8.90/5.12 => ( ( sums_real @ G @ B )
% 8.90/5.12 => ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 8.90/5.12 @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8154_sums__divide,axiom,
% 8.90/5.12 ! [F: nat > complex,A: complex,C: complex] :
% 8.90/5.12 ( ( sums_complex @ F @ A )
% 8.90/5.12 => ( sums_complex
% 8.90/5.12 @ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C )
% 8.90/5.12 @ ( divide1717551699836669952omplex @ A @ C ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8155_sums__divide,axiom,
% 8.90/5.12 ! [F: nat > real,A: real,C: real] :
% 8.90/5.12 ( ( sums_real @ F @ A )
% 8.90/5.12 => ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C )
% 8.90/5.12 @ ( divide_divide_real @ A @ C ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8156_cos__diff,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( cos_real @ ( minus_minus_real @ X2 @ Y3 ) )
% 8.90/5.12 = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X2 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8157_cos__add,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( cos_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 8.90/5.12 = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X2 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8158_sin__zero__norm__cos__one,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ( sin_real @ X2 )
% 8.90/5.12 = zero_zero_real )
% 8.90/5.12 => ( ( real_V7735802525324610683m_real @ ( cos_real @ X2 ) )
% 8.90/5.12 = one_one_real ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8159_sin__zero__norm__cos__one,axiom,
% 8.90/5.12 ! [X2: complex] :
% 8.90/5.12 ( ( ( sin_complex @ X2 )
% 8.90/5.12 = zero_zero_complex )
% 8.90/5.12 => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X2 ) )
% 8.90/5.12 = one_one_real ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8160_sin__zero__abs__cos__one,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ( sin_real @ X2 )
% 8.90/5.12 = zero_zero_real )
% 8.90/5.12 => ( ( abs_abs_real @ ( cos_real @ X2 ) )
% 8.90/5.12 = one_one_real ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8161_sin__double,axiom,
% 8.90/5.12 ! [X2: complex] :
% 8.90/5.12 ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) )
% 8.90/5.12 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X2 ) ) @ ( cos_complex @ X2 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8162_sin__double,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) )
% 8.90/5.12 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X2 ) ) @ ( cos_real @ X2 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8163_sincos__principal__value,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ? [Y4: real] :
% 8.90/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
% 8.90/5.12 & ( ord_less_eq_real @ Y4 @ pi )
% 8.90/5.12 & ( ( sin_real @ Y4 )
% 8.90/5.12 = ( sin_real @ X2 ) )
% 8.90/5.12 & ( ( cos_real @ Y4 )
% 8.90/5.12 = ( cos_real @ X2 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8164_powr__mono2,axiom,
% 8.90/5.12 ! [A: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.90/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 8.90/5.12 => ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8165_powr__ge__pzero,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X2 @ Y3 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8166_powr__less__cancel,axiom,
% 8.90/5.12 ! [X2: real,A: real,B: real] :
% 8.90/5.12 ( ( ord_less_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) )
% 8.90/5.12 => ( ( ord_less_real @ one_one_real @ X2 )
% 8.90/5.12 => ( ord_less_real @ A @ B ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8167_powr__less__mono,axiom,
% 8.90/5.12 ! [A: real,B: real,X2: real] :
% 8.90/5.12 ( ( ord_less_real @ A @ B )
% 8.90/5.12 => ( ( ord_less_real @ one_one_real @ X2 )
% 8.90/5.12 => ( ord_less_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8168_powr__mono,axiom,
% 8.90/5.12 ! [A: real,B: real,X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ A @ B )
% 8.90/5.12 => ( ( ord_less_eq_real @ one_one_real @ X2 )
% 8.90/5.12 => ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8169_sin__x__le__x,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ord_less_eq_real @ ( sin_real @ X2 ) @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8170_sin__le__one,axiom,
% 8.90/5.12 ! [X2: real] : ( ord_less_eq_real @ ( sin_real @ X2 ) @ one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8171_cos__le__one,axiom,
% 8.90/5.12 ! [X2: real] : ( ord_less_eq_real @ ( cos_real @ X2 ) @ one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8172_abs__sin__x__le__abs__x,axiom,
% 8.90/5.12 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X2 ) ) @ ( abs_abs_real @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8173_sums__mult__iff,axiom,
% 8.90/5.12 ! [C: complex,F: nat > complex,D: complex] :
% 8.90/5.12 ( ( C != zero_zero_complex )
% 8.90/5.12 => ( ( sums_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) )
% 8.90/5.12 @ ( times_times_complex @ C @ D ) )
% 8.90/5.12 = ( sums_complex @ F @ D ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8174_sums__mult__iff,axiom,
% 8.90/5.12 ! [C: real,F: nat > real,D: real] :
% 8.90/5.12 ( ( C != zero_zero_real )
% 8.90/5.12 => ( ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) )
% 8.90/5.12 @ ( times_times_real @ C @ D ) )
% 8.90/5.12 = ( sums_real @ F @ D ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8175_sums__mult2__iff,axiom,
% 8.90/5.12 ! [C: complex,F: nat > complex,D: complex] :
% 8.90/5.12 ( ( C != zero_zero_complex )
% 8.90/5.12 => ( ( sums_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ C )
% 8.90/5.12 @ ( times_times_complex @ D @ C ) )
% 8.90/5.12 = ( sums_complex @ F @ D ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8176_sums__mult2__iff,axiom,
% 8.90/5.12 ! [C: real,F: nat > real,D: real] :
% 8.90/5.12 ( ( C != zero_zero_real )
% 8.90/5.12 => ( ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C )
% 8.90/5.12 @ ( times_times_real @ D @ C ) )
% 8.90/5.12 = ( sums_real @ F @ D ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8177_sin__cos__le1,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X2 ) @ ( sin_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) ) ) @ one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8178_sin__squared__eq,axiom,
% 8.90/5.12 ! [X2: complex] :
% 8.90/5.12 ( ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.12 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8179_sin__squared__eq,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.12 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8180_cos__squared__eq,axiom,
% 8.90/5.12 ! [X2: complex] :
% 8.90/5.12 ( ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.12 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8181_cos__squared__eq,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.12 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8182_powr__mono2_H,axiom,
% 8.90/5.12 ! [A: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 8.90/5.12 => ( ord_less_eq_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X2 @ A ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8183_powr__less__mono2,axiom,
% 8.90/5.12 ! [A: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ A )
% 8.90/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.12 => ( ord_less_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8184_powr__inj,axiom,
% 8.90/5.12 ! [A: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ A )
% 8.90/5.12 => ( ( A != one_one_real )
% 8.90/5.12 => ( ( ( powr_real @ A @ X2 )
% 8.90/5.12 = ( powr_real @ A @ Y3 ) )
% 8.90/5.12 = ( X2 = Y3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8185_gr__one__powr,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.90/5.12 => ( ord_less_real @ one_one_real @ ( powr_real @ X2 @ Y3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8186_ge__one__powr__ge__zero,axiom,
% 8.90/5.12 ! [X2: real,A: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.90/5.12 => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X2 @ A ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8187_powr__mono__both,axiom,
% 8.90/5.12 ! [A: real,B: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.90/5.12 => ( ( ord_less_eq_real @ A @ B )
% 8.90/5.12 => ( ( ord_less_eq_real @ one_one_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 8.90/5.12 => ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y3 @ B ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8188_powr__le1,axiom,
% 8.90/5.12 ! [A: real,X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 8.90/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 8.90/5.12 => ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ one_one_real ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8189_powr__divide,axiom,
% 8.90/5.12 ! [X2: real,Y3: real,A: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.12 => ( ( powr_real @ ( divide_divide_real @ X2 @ Y3 ) @ A )
% 8.90/5.12 = ( divide_divide_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8190_powr__mult,axiom,
% 8.90/5.12 ! [X2: real,Y3: real,A: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.12 => ( ( powr_real @ ( times_times_real @ X2 @ Y3 ) @ A )
% 8.90/5.12 = ( times_times_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8191_sums__mult__D,axiom,
% 8.90/5.12 ! [C: complex,F: nat > complex,A: complex] :
% 8.90/5.12 ( ( sums_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) )
% 8.90/5.12 @ A )
% 8.90/5.12 => ( ( C != zero_zero_complex )
% 8.90/5.12 => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8192_sums__mult__D,axiom,
% 8.90/5.12 ! [C: real,F: nat > real,A: real] :
% 8.90/5.12 ( ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) )
% 8.90/5.12 @ A )
% 8.90/5.12 => ( ( C != zero_zero_real )
% 8.90/5.12 => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8193_sums__Suc__imp,axiom,
% 8.90/5.12 ! [F: nat > complex,S: complex] :
% 8.90/5.12 ( ( ( F @ zero_zero_nat )
% 8.90/5.12 = zero_zero_complex )
% 8.90/5.12 => ( ( sums_complex
% 8.90/5.12 @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
% 8.90/5.12 @ S )
% 8.90/5.12 => ( sums_complex @ F @ S ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8194_sums__Suc__imp,axiom,
% 8.90/5.12 ! [F: nat > real,S: real] :
% 8.90/5.12 ( ( ( F @ zero_zero_nat )
% 8.90/5.12 = zero_zero_real )
% 8.90/5.12 => ( ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
% 8.90/5.12 @ S )
% 8.90/5.12 => ( sums_real @ F @ S ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8195_sums__Suc__iff,axiom,
% 8.90/5.12 ! [F: nat > real,S: real] :
% 8.90/5.12 ( ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
% 8.90/5.12 @ S )
% 8.90/5.12 = ( sums_real @ F @ ( plus_plus_real @ S @ ( F @ zero_zero_nat ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8196_sums__Suc,axiom,
% 8.90/5.12 ! [F: nat > real,L: real] :
% 8.90/5.12 ( ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
% 8.90/5.12 @ L )
% 8.90/5.12 => ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8197_sums__Suc,axiom,
% 8.90/5.12 ! [F: nat > nat,L: nat] :
% 8.90/5.12 ( ( sums_nat
% 8.90/5.12 @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
% 8.90/5.12 @ L )
% 8.90/5.12 => ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8198_sums__Suc,axiom,
% 8.90/5.12 ! [F: nat > int,L: int] :
% 8.90/5.12 ( ( sums_int
% 8.90/5.12 @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
% 8.90/5.12 @ L )
% 8.90/5.12 => ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8199_divide__powr__uminus,axiom,
% 8.90/5.12 ! [A: real,B: real,C: real] :
% 8.90/5.12 ( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
% 8.90/5.12 = ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8200_sums__zero__iff__shift,axiom,
% 8.90/5.12 ! [N: nat,F: nat > complex,S: complex] :
% 8.90/5.12 ( ! [I3: nat] :
% 8.90/5.12 ( ( ord_less_nat @ I3 @ N )
% 8.90/5.12 => ( ( F @ I3 )
% 8.90/5.12 = zero_zero_complex ) )
% 8.90/5.12 => ( ( sums_complex
% 8.90/5.12 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 8.90/5.12 @ S )
% 8.90/5.12 = ( sums_complex @ F @ S ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8201_sums__zero__iff__shift,axiom,
% 8.90/5.12 ! [N: nat,F: nat > real,S: real] :
% 8.90/5.12 ( ! [I3: nat] :
% 8.90/5.12 ( ( ord_less_nat @ I3 @ N )
% 8.90/5.12 => ( ( F @ I3 )
% 8.90/5.12 = zero_zero_real ) )
% 8.90/5.12 => ( ( sums_real
% 8.90/5.12 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 8.90/5.12 @ S )
% 8.90/5.12 = ( sums_real @ F @ S ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8202_sin__x__ge__neg__x,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ ( sin_real @ X2 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8203_sin__ge__zero,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ pi )
% 8.90/5.12 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8204_log__powr,axiom,
% 8.90/5.12 ! [X2: real,B: real,Y3: real] :
% 8.90/5.12 ( ( X2 != zero_zero_real )
% 8.90/5.12 => ( ( log @ B @ ( powr_real @ X2 @ Y3 ) )
% 8.90/5.12 = ( times_times_real @ Y3 @ ( log @ B @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8205_ln__powr,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( X2 != zero_zero_real )
% 8.90/5.12 => ( ( ln_ln_real @ ( powr_real @ X2 @ Y3 ) )
% 8.90/5.12 = ( times_times_real @ Y3 @ ( ln_ln_real @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8206_sin__ge__minus__one,axiom,
% 8.90/5.12 ! [X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8207_cos__monotone__0__pi__le,axiom,
% 8.90/5.12 ! [Y3: real,X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.12 => ( ( ord_less_eq_real @ Y3 @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ pi )
% 8.90/5.12 => ( ord_less_eq_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8208_cos__mono__le__eq,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ pi )
% 8.90/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.12 => ( ( ord_less_eq_real @ Y3 @ pi )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) )
% 8.90/5.12 = ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8209_cos__inj__pi,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ pi )
% 8.90/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.12 => ( ( ord_less_eq_real @ Y3 @ pi )
% 8.90/5.12 => ( ( ( cos_real @ X2 )
% 8.90/5.12 = ( cos_real @ Y3 ) )
% 8.90/5.12 => ( X2 = Y3 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8210_cos__ge__minus__one,axiom,
% 8.90/5.12 ! [X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8211_abs__sin__le__one,axiom,
% 8.90/5.12 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X2 ) ) @ one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8212_abs__cos__le__one,axiom,
% 8.90/5.12 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X2 ) ) @ one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8213_sin__times__sin,axiom,
% 8.90/5.12 ! [W: complex,Z: complex] :
% 8.90/5.12 ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 8.90/5.12 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8214_sin__times__sin,axiom,
% 8.90/5.12 ! [W: real,Z: real] :
% 8.90/5.12 ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 8.90/5.12 = ( 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 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8215_sin__times__cos,axiom,
% 8.90/5.12 ! [W: complex,Z: complex] :
% 8.90/5.12 ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
% 8.90/5.12 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8216_sin__times__cos,axiom,
% 8.90/5.12 ! [W: real,Z: real] :
% 8.90/5.12 ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
% 8.90/5.12 = ( 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 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8217_cos__times__sin,axiom,
% 8.90/5.12 ! [W: complex,Z: complex] :
% 8.90/5.12 ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
% 8.90/5.12 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8218_cos__times__sin,axiom,
% 8.90/5.12 ! [W: real,Z: real] :
% 8.90/5.12 ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
% 8.90/5.12 = ( 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 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8219_sin__plus__sin,axiom,
% 8.90/5.12 ! [W: complex,Z: complex] :
% 8.90/5.12 ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 8.90/5.12 = ( 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 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8220_sin__plus__sin,axiom,
% 8.90/5.12 ! [W: real,Z: real] :
% 8.90/5.12 ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 8.90/5.12 = ( 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 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8221_sin__diff__sin,axiom,
% 8.90/5.12 ! [W: complex,Z: complex] :
% 8.90/5.12 ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 8.90/5.12 = ( 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 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8222_sin__diff__sin,axiom,
% 8.90/5.12 ! [W: real,Z: real] :
% 8.90/5.12 ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 8.90/5.12 = ( 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 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8223_cos__diff__cos,axiom,
% 8.90/5.12 ! [W: complex,Z: complex] :
% 8.90/5.12 ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 8.90/5.12 = ( 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 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8224_cos__diff__cos,axiom,
% 8.90/5.12 ! [W: real,Z: real] :
% 8.90/5.12 ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 8.90/5.12 = ( 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 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8225_cos__double,axiom,
% 8.90/5.12 ! [X2: complex] :
% 8.90/5.12 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) )
% 8.90/5.12 = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8226_cos__double,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) )
% 8.90/5.12 = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8227_powr__add,axiom,
% 8.90/5.12 ! [X2: real,A: real,B: real] :
% 8.90/5.12 ( ( powr_real @ X2 @ ( plus_plus_real @ A @ B ) )
% 8.90/5.12 = ( times_times_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8228_powr__diff,axiom,
% 8.90/5.12 ! [W: real,Z1: real,Z22: real] :
% 8.90/5.12 ( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z22 ) )
% 8.90/5.12 = ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z22 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8229_cos__double__sin,axiom,
% 8.90/5.12 ! [W: complex] :
% 8.90/5.12 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 8.90/5.12 = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8230_cos__double__sin,axiom,
% 8.90/5.12 ! [W: real] :
% 8.90/5.12 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 8.90/5.12 = ( 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 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8231_powser__sums__if,axiom,
% 8.90/5.12 ! [M: nat,Z: complex] :
% 8.90/5.12 ( sums_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( if_complex @ ( N3 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N3 ) )
% 8.90/5.12 @ ( power_power_complex @ Z @ M ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8232_powser__sums__if,axiom,
% 8.90/5.12 ! [M: nat,Z: real] :
% 8.90/5.12 ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( if_real @ ( N3 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N3 ) )
% 8.90/5.12 @ ( power_power_real @ Z @ M ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8233_powser__sums__if,axiom,
% 8.90/5.12 ! [M: nat,Z: int] :
% 8.90/5.12 ( sums_int
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_int @ ( if_int @ ( N3 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N3 ) )
% 8.90/5.12 @ ( power_power_int @ Z @ M ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8234_powser__sums__zero,axiom,
% 8.90/5.12 ! [A: nat > complex] :
% 8.90/5.12 ( sums_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( A @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) )
% 8.90/5.12 @ ( A @ zero_zero_nat ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8235_powser__sums__zero,axiom,
% 8.90/5.12 ! [A: nat > real] :
% 8.90/5.12 ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( A @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) )
% 8.90/5.12 @ ( A @ zero_zero_nat ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8236_cos__two__neq__zero,axiom,
% 8.90/5.12 ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.90/5.12 != zero_zero_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8237_powr__realpow,axiom,
% 8.90/5.12 ! [X2: real,N: nat] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( powr_real @ X2 @ ( semiri5074537144036343181t_real @ N ) )
% 8.90/5.12 = ( power_power_real @ X2 @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8238_powr__less__iff,axiom,
% 8.90/5.12 ! [B: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ B )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ ( powr_real @ B @ Y3 ) @ X2 )
% 8.90/5.12 = ( ord_less_real @ Y3 @ ( log @ B @ X2 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8239_less__powr__iff,axiom,
% 8.90/5.12 ! [B: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ B )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ X2 @ ( powr_real @ B @ Y3 ) )
% 8.90/5.12 = ( ord_less_real @ ( log @ B @ X2 ) @ Y3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8240_log__less__iff,axiom,
% 8.90/5.12 ! [B: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ B )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ ( log @ B @ X2 ) @ Y3 )
% 8.90/5.12 = ( ord_less_real @ X2 @ ( powr_real @ B @ Y3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8241_less__log__iff,axiom,
% 8.90/5.12 ! [B: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ B )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ Y3 @ ( log @ B @ X2 ) )
% 8.90/5.12 = ( ord_less_real @ ( powr_real @ B @ Y3 ) @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8242_cos__mono__less__eq,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ pi )
% 8.90/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.12 => ( ( ord_less_eq_real @ Y3 @ pi )
% 8.90/5.12 => ( ( ord_less_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) )
% 8.90/5.12 = ( ord_less_real @ Y3 @ X2 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8243_cos__monotone__0__pi,axiom,
% 8.90/5.12 ! [Y3: real,X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.12 => ( ( ord_less_real @ Y3 @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ pi )
% 8.90/5.12 => ( ord_less_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8244_sums__iff__shift,axiom,
% 8.90/5.12 ! [F: nat > real,N: nat,S: real] :
% 8.90/5.12 ( ( sums_real
% 8.90/5.12 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 8.90/5.12 @ S )
% 8.90/5.12 = ( sums_real @ F @ ( plus_plus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8245_sums__iff__shift_H,axiom,
% 8.90/5.12 ! [F: nat > real,N: nat,S: real] :
% 8.90/5.12 ( ( sums_real
% 8.90/5.12 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 8.90/5.12 @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) )
% 8.90/5.12 = ( sums_real @ F @ S ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8246_sums__split__initial__segment,axiom,
% 8.90/5.12 ! [F: nat > real,S: real,N: nat] :
% 8.90/5.12 ( ( sums_real @ F @ S )
% 8.90/5.12 => ( sums_real
% 8.90/5.12 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 8.90/5.12 @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8247_sums__If__finite__set_H,axiom,
% 8.90/5.12 ! [G: nat > real,S3: real,A2: set_nat,S4: real,F: nat > real] :
% 8.90/5.12 ( ( sums_real @ G @ S3 )
% 8.90/5.12 => ( ( finite_finite_nat @ A2 )
% 8.90/5.12 => ( ( S4
% 8.90/5.12 = ( plus_plus_real @ S3
% 8.90/5.12 @ ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 8.90/5.12 @ A2 ) ) )
% 8.90/5.12 => ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( if_real @ ( member_nat @ N3 @ A2 ) @ ( F @ N3 ) @ ( G @ N3 ) )
% 8.90/5.12 @ S4 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8248_cos__monotone__minus__pi__0_H,axiom,
% 8.90/5.12 ! [Y3: real,X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 8.90/5.12 => ( ( ord_less_eq_real @ Y3 @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 8.90/5.12 => ( ord_less_eq_real @ ( cos_real @ Y3 ) @ ( cos_real @ X2 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8249_sin__zero__iff__int2,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ( sin_real @ X2 )
% 8.90/5.12 = zero_zero_real )
% 8.90/5.12 = ( ? [I4: int] :
% 8.90/5.12 ( X2
% 8.90/5.12 = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ pi ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8250_sincos__total__pi,axiom,
% 8.90/5.12 ! [Y3: real,X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.12 => ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.12 = one_one_real )
% 8.90/5.12 => ? [T5: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 8.90/5.12 & ( ord_less_eq_real @ T5 @ pi )
% 8.90/5.12 & ( X2
% 8.90/5.12 = ( cos_real @ T5 ) )
% 8.90/5.12 & ( Y3
% 8.90/5.12 = ( sin_real @ T5 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8251_sin__expansion__lemma,axiom,
% 8.90/5.12 ! [X2: real,M: nat] :
% 8.90/5.12 ( ( sin_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 8.90/5.12 = ( cos_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8252_powr__minus__divide,axiom,
% 8.90/5.12 ! [X2: real,A: real] :
% 8.90/5.12 ( ( powr_real @ X2 @ ( uminus_uminus_real @ A ) )
% 8.90/5.12 = ( divide_divide_real @ one_one_real @ ( powr_real @ X2 @ A ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8253_cos__expansion__lemma,axiom,
% 8.90/5.12 ! [X2: real,M: nat] :
% 8.90/5.12 ( ( cos_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 8.90/5.12 = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8254_powr__neg__one,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( powr_real @ X2 @ ( uminus_uminus_real @ one_one_real ) )
% 8.90/5.12 = ( divide_divide_real @ one_one_real @ X2 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8255_sin__gt__zero__02,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.90/5.12 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8256_powr__mult__base,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( times_times_real @ X2 @ ( powr_real @ X2 @ Y3 ) )
% 8.90/5.12 = ( powr_real @ X2 @ ( plus_plus_real @ one_one_real @ Y3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8257_le__log__iff,axiom,
% 8.90/5.12 ! [B: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ B )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ Y3 @ ( log @ B @ X2 ) )
% 8.90/5.12 = ( ord_less_eq_real @ ( powr_real @ B @ Y3 ) @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8258_log__le__iff,axiom,
% 8.90/5.12 ! [B: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ B )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( log @ B @ X2 ) @ Y3 )
% 8.90/5.12 = ( ord_less_eq_real @ X2 @ ( powr_real @ B @ Y3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8259_le__powr__iff,axiom,
% 8.90/5.12 ! [B: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ B )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ ( powr_real @ B @ Y3 ) )
% 8.90/5.12 = ( ord_less_eq_real @ ( log @ B @ X2 ) @ Y3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8260_powr__le__iff,axiom,
% 8.90/5.12 ! [B: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ B )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( powr_real @ B @ Y3 ) @ X2 )
% 8.90/5.12 = ( ord_less_eq_real @ Y3 @ ( log @ B @ X2 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8261_cos__two__less__zero,axiom,
% 8.90/5.12 ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8262_cos__two__le__zero,axiom,
% 8.90/5.12 ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8263_cos__is__zero,axiom,
% 8.90/5.12 ? [X5: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 8.90/5.12 & ( ord_less_eq_real @ X5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.90/5.12 & ( ( cos_real @ X5 )
% 8.90/5.12 = zero_zero_real )
% 8.90/5.12 & ! [Y5: real] :
% 8.90/5.12 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 8.90/5.12 & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.90/5.12 & ( ( cos_real @ Y5 )
% 8.90/5.12 = zero_zero_real ) )
% 8.90/5.12 => ( Y5 = X5 ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8264_cos__monotone__minus__pi__0,axiom,
% 8.90/5.12 ! [Y3: real,X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 8.90/5.12 => ( ( ord_less_real @ Y3 @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 8.90/5.12 => ( ord_less_real @ ( cos_real @ Y3 ) @ ( cos_real @ X2 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8265_cos__total,axiom,
% 8.90/5.12 ! [Y3: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 8.90/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.90/5.12 => ? [X5: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 8.90/5.12 & ( ord_less_eq_real @ X5 @ pi )
% 8.90/5.12 & ( ( cos_real @ X5 )
% 8.90/5.12 = Y3 )
% 8.90/5.12 & ! [Y5: real] :
% 8.90/5.12 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 8.90/5.12 & ( ord_less_eq_real @ Y5 @ pi )
% 8.90/5.12 & ( ( cos_real @ Y5 )
% 8.90/5.12 = Y3 ) )
% 8.90/5.12 => ( Y5 = X5 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8266_sincos__total__pi__half,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.12 => ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.12 = one_one_real )
% 8.90/5.12 => ? [T5: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 8.90/5.12 & ( ord_less_eq_real @ T5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.12 & ( X2
% 8.90/5.12 = ( cos_real @ T5 ) )
% 8.90/5.12 & ( Y3
% 8.90/5.12 = ( sin_real @ T5 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8267_sincos__total__2pi__le,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.12 = one_one_real )
% 8.90/5.12 => ? [T5: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 8.90/5.12 & ( ord_less_eq_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 8.90/5.12 & ( X2
% 8.90/5.12 = ( cos_real @ T5 ) )
% 8.90/5.12 & ( Y3
% 8.90/5.12 = ( sin_real @ T5 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8268_sincos__total__2pi,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.12 = one_one_real )
% 8.90/5.12 => ~ ! [T5: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 8.90/5.12 => ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 8.90/5.12 => ( ( X2
% 8.90/5.12 = ( cos_real @ T5 ) )
% 8.90/5.12 => ( Y3
% 8.90/5.12 != ( sin_real @ T5 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8269_ln__powr__bound,axiom,
% 8.90/5.12 ! [X2: real,A: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.90/5.12 => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( divide_divide_real @ ( powr_real @ X2 @ A ) @ A ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8270_ln__powr__bound2,axiom,
% 8.90/5.12 ! [X2: real,A: real] :
% 8.90/5.12 ( ( ord_less_real @ one_one_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.90/5.12 => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X2 ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8271_log__add__eq__powr,axiom,
% 8.90/5.12 ! [B: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ B )
% 8.90/5.12 => ( ( B != one_one_real )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( plus_plus_real @ ( log @ B @ X2 ) @ Y3 )
% 8.90/5.12 = ( log @ B @ ( times_times_real @ X2 @ ( powr_real @ B @ Y3 ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8272_add__log__eq__powr,axiom,
% 8.90/5.12 ! [B: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ B )
% 8.90/5.12 => ( ( B != one_one_real )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( plus_plus_real @ Y3 @ ( log @ B @ X2 ) )
% 8.90/5.12 = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y3 ) @ X2 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8273_minus__log__eq__powr,axiom,
% 8.90/5.12 ! [B: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ B )
% 8.90/5.12 => ( ( B != one_one_real )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( minus_minus_real @ Y3 @ ( log @ B @ X2 ) )
% 8.90/5.12 = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y3 ) @ X2 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8274_sin__pi__divide__n__ge__0,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( N != zero_zero_nat )
% 8.90/5.12 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8275_cos__times__cos,axiom,
% 8.90/5.12 ! [W: complex,Z: complex] :
% 8.90/5.12 ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 8.90/5.12 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8276_cos__times__cos,axiom,
% 8.90/5.12 ! [W: real,Z: real] :
% 8.90/5.12 ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 8.90/5.12 = ( 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 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8277_cos__plus__cos,axiom,
% 8.90/5.12 ! [W: complex,Z: complex] :
% 8.90/5.12 ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 8.90/5.12 = ( 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 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8278_cos__plus__cos,axiom,
% 8.90/5.12 ! [W: real,Z: real] :
% 8.90/5.12 ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 8.90/5.12 = ( 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 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8279_geometric__sums,axiom,
% 8.90/5.12 ! [C: real] :
% 8.90/5.12 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 8.90/5.12 => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8280_geometric__sums,axiom,
% 8.90/5.12 ! [C: complex] :
% 8.90/5.12 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 8.90/5.12 => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8281_power__half__series,axiom,
% 8.90/5.12 ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N3 ) )
% 8.90/5.12 @ one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8282_sin__gt__zero2,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.12 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8283_sin__lt__zero,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_real @ pi @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 8.90/5.12 => ( ord_less_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8284_cos__double__less__one,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.90/5.12 => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_real ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8285_sin__30,axiom,
% 8.90/5.12 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 8.90/5.12 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8286_cos__gt__zero,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.12 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8287_sin__inj__pi,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 8.90/5.12 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.12 => ( ( ( sin_real @ X2 )
% 8.90/5.12 = ( sin_real @ Y3 ) )
% 8.90/5.12 => ( X2 = Y3 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8288_sin__mono__le__eq,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 8.90/5.12 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( sin_real @ X2 ) @ ( sin_real @ Y3 ) )
% 8.90/5.12 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8289_sin__monotone__2pi__le,axiom,
% 8.90/5.12 ! [Y3: real,X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 8.90/5.12 => ( ( ord_less_eq_real @ Y3 @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.12 => ( ord_less_eq_real @ ( sin_real @ Y3 ) @ ( sin_real @ X2 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8290_log__minus__eq__powr,axiom,
% 8.90/5.12 ! [B: real,X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ B )
% 8.90/5.12 => ( ( B != one_one_real )
% 8.90/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( minus_minus_real @ ( log @ B @ X2 ) @ Y3 )
% 8.90/5.12 = ( log @ B @ ( times_times_real @ X2 @ ( powr_real @ B @ ( uminus_uminus_real @ Y3 ) ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8291_cos__60,axiom,
% 8.90/5.12 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 8.90/5.12 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8292_cos__one__2pi__int,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ( cos_real @ X2 )
% 8.90/5.12 = one_one_real )
% 8.90/5.12 = ( ? [X: int] :
% 8.90/5.12 ( X2
% 8.90/5.12 = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8293_cos__double__cos,axiom,
% 8.90/5.12 ! [W: complex] :
% 8.90/5.12 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 8.90/5.12 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8294_cos__double__cos,axiom,
% 8.90/5.12 ! [W: real] :
% 8.90/5.12 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 8.90/5.12 = ( 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 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8295_cos__treble__cos,axiom,
% 8.90/5.12 ! [X2: complex] :
% 8.90/5.12 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X2 ) )
% 8.90/5.12 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8296_cos__treble__cos,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X2 ) )
% 8.90/5.12 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8297_sums__if_H,axiom,
% 8.90/5.12 ! [G: nat > real,X2: real] :
% 8.90/5.12 ( ( sums_real @ G @ X2 )
% 8.90/5.12 => ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.12 @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8298_sums__if,axiom,
% 8.90/5.12 ! [G: nat > real,X2: real,F: nat > real,Y3: real] :
% 8.90/5.12 ( ( sums_real @ G @ X2 )
% 8.90/5.12 => ( ( sums_real @ F @ Y3 )
% 8.90/5.12 => ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( F @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.12 @ ( plus_plus_real @ X2 @ Y3 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8299_powr__neg__numeral,axiom,
% 8.90/5.12 ! [X2: real,N: num] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( powr_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 8.90/5.12 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8300_sin__le__zero,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ pi @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 8.90/5.12 => ( ord_less_eq_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8301_sin__less__zero,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 8.90/5.12 => ( ord_less_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8302_sin__mono__less__eq,axiom,
% 8.90/5.12 ! [X2: real,Y3: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.12 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 8.90/5.12 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.12 => ( ( ord_less_real @ ( sin_real @ X2 ) @ ( sin_real @ Y3 ) )
% 8.90/5.12 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8303_sin__monotone__2pi,axiom,
% 8.90/5.12 ! [Y3: real,X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 8.90/5.12 => ( ( ord_less_real @ Y3 @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.12 => ( ord_less_real @ ( sin_real @ Y3 ) @ ( sin_real @ X2 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8304_sin__total,axiom,
% 8.90/5.12 ! [Y3: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 8.90/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.90/5.12 => ? [X5: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
% 8.90/5.12 & ( ord_less_eq_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.12 & ( ( sin_real @ X5 )
% 8.90/5.12 = Y3 )
% 8.90/5.12 & ! [Y5: real] :
% 8.90/5.12 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 8.90/5.12 & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.12 & ( ( sin_real @ Y5 )
% 8.90/5.12 = Y3 ) )
% 8.90/5.12 => ( Y5 = X5 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8305_cos__gt__zero__pi,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 8.90/5.12 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.12 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8306_cos__ge__zero,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 8.90/5.12 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.12 => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8307_cos__one__2pi,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ( cos_real @ X2 )
% 8.90/5.12 = one_one_real )
% 8.90/5.12 = ( ? [X: nat] :
% 8.90/5.12 ( X2
% 8.90/5.12 = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 8.90/5.12 | ? [X: nat] :
% 8.90/5.12 ( X2
% 8.90/5.12 = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8308_sin__pi__divide__n__gt__0,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.12 => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8309_sin__zero__iff__int,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ( sin_real @ X2 )
% 8.90/5.12 = zero_zero_real )
% 8.90/5.12 = ( ? [I4: int] :
% 8.90/5.12 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
% 8.90/5.12 & ( X2
% 8.90/5.12 = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8310_cos__zero__iff__int,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ( cos_real @ X2 )
% 8.90/5.12 = zero_zero_real )
% 8.90/5.12 = ( ? [I4: int] :
% 8.90/5.12 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
% 8.90/5.12 & ( X2
% 8.90/5.12 = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8311_sin__zero__lemma,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ( sin_real @ X2 )
% 8.90/5.12 = zero_zero_real )
% 8.90/5.12 => ? [N2: nat] :
% 8.90/5.12 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 8.90/5.12 & ( X2
% 8.90/5.12 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8312_sin__zero__iff,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ( sin_real @ X2 )
% 8.90/5.12 = zero_zero_real )
% 8.90/5.12 = ( ? [N3: nat] :
% 8.90/5.12 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 8.90/5.12 & ( X2
% 8.90/5.12 = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 8.90/5.12 | ? [N3: nat] :
% 8.90/5.12 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 8.90/5.12 & ( X2
% 8.90/5.12 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8313_cos__zero__lemma,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ( cos_real @ X2 )
% 8.90/5.12 = zero_zero_real )
% 8.90/5.12 => ? [N2: nat] :
% 8.90/5.12 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 8.90/5.12 & ( X2
% 8.90/5.12 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8314_cos__zero__iff,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ( cos_real @ X2 )
% 8.90/5.12 = zero_zero_real )
% 8.90/5.12 = ( ? [N3: nat] :
% 8.90/5.12 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 8.90/5.12 & ( X2
% 8.90/5.12 = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 8.90/5.12 | ? [N3: nat] :
% 8.90/5.12 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 8.90/5.12 & ( X2
% 8.90/5.12 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8315_mono__SucI1,axiom,
% 8.90/5.12 ! [X8: nat > real] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_real @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 8.90/5.12 => ( topolo6980174941875973593q_real @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8316_mono__SucI1,axiom,
% 8.90/5.12 ! [X8: nat > set_int] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_set_int @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 8.90/5.12 => ( topolo3100542954746470799et_int @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8317_mono__SucI1,axiom,
% 8.90/5.12 ! [X8: nat > rat] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_rat @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 8.90/5.12 => ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8318_mono__SucI1,axiom,
% 8.90/5.12 ! [X8: nat > num] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_num @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 8.90/5.12 => ( topolo1459490580787246023eq_num @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8319_mono__SucI1,axiom,
% 8.90/5.12 ! [X8: nat > nat] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_nat @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 8.90/5.12 => ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8320_mono__SucI1,axiom,
% 8.90/5.12 ! [X8: nat > int] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_int @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 8.90/5.12 => ( topolo4899668324122417113eq_int @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8321_mono__SucI2,axiom,
% 8.90/5.12 ! [X8: nat > real] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_real @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) )
% 8.90/5.12 => ( topolo6980174941875973593q_real @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8322_mono__SucI2,axiom,
% 8.90/5.12 ! [X8: nat > set_int] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_set_int @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) )
% 8.90/5.12 => ( topolo3100542954746470799et_int @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8323_mono__SucI2,axiom,
% 8.90/5.12 ! [X8: nat > rat] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_rat @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) )
% 8.90/5.12 => ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8324_mono__SucI2,axiom,
% 8.90/5.12 ! [X8: nat > num] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_num @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) )
% 8.90/5.12 => ( topolo1459490580787246023eq_num @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8325_mono__SucI2,axiom,
% 8.90/5.12 ! [X8: nat > nat] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_nat @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) )
% 8.90/5.12 => ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8326_mono__SucI2,axiom,
% 8.90/5.12 ! [X8: nat > int] :
% 8.90/5.12 ( ! [N2: nat] : ( ord_less_eq_int @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) )
% 8.90/5.12 => ( topolo4899668324122417113eq_int @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8327_monoseq__Suc,axiom,
% 8.90/5.12 ( topolo6980174941875973593q_real
% 8.90/5.12 = ( ^ [X4: nat > real] :
% 8.90/5.12 ( ! [N3: nat] : ( ord_less_eq_real @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
% 8.90/5.12 | ! [N3: nat] : ( ord_less_eq_real @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8328_monoseq__Suc,axiom,
% 8.90/5.12 ( topolo3100542954746470799et_int
% 8.90/5.12 = ( ^ [X4: nat > set_int] :
% 8.90/5.12 ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
% 8.90/5.12 | ! [N3: nat] : ( ord_less_eq_set_int @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8329_monoseq__Suc,axiom,
% 8.90/5.12 ( topolo4267028734544971653eq_rat
% 8.90/5.12 = ( ^ [X4: nat > rat] :
% 8.90/5.12 ( ! [N3: nat] : ( ord_less_eq_rat @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
% 8.90/5.12 | ! [N3: nat] : ( ord_less_eq_rat @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8330_monoseq__Suc,axiom,
% 8.90/5.12 ( topolo1459490580787246023eq_num
% 8.90/5.12 = ( ^ [X4: nat > num] :
% 8.90/5.12 ( ! [N3: nat] : ( ord_less_eq_num @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
% 8.90/5.12 | ! [N3: nat] : ( ord_less_eq_num @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8331_monoseq__Suc,axiom,
% 8.90/5.12 ( topolo4902158794631467389eq_nat
% 8.90/5.12 = ( ^ [X4: nat > nat] :
% 8.90/5.12 ( ! [N3: nat] : ( ord_less_eq_nat @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
% 8.90/5.12 | ! [N3: nat] : ( ord_less_eq_nat @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8332_monoseq__Suc,axiom,
% 8.90/5.12 ( topolo4899668324122417113eq_int
% 8.90/5.12 = ( ^ [X4: nat > int] :
% 8.90/5.12 ( ! [N3: nat] : ( ord_less_eq_int @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
% 8.90/5.12 | ! [N3: nat] : ( ord_less_eq_int @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8333_monoI1,axiom,
% 8.90/5.12 ! [X8: nat > real] :
% 8.90/5.12 ( ! [M4: nat,N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M4 @ N2 )
% 8.90/5.12 => ( ord_less_eq_real @ ( X8 @ M4 ) @ ( X8 @ N2 ) ) )
% 8.90/5.12 => ( topolo6980174941875973593q_real @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8334_monoI1,axiom,
% 8.90/5.12 ! [X8: nat > set_int] :
% 8.90/5.12 ( ! [M4: nat,N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M4 @ N2 )
% 8.90/5.12 => ( ord_less_eq_set_int @ ( X8 @ M4 ) @ ( X8 @ N2 ) ) )
% 8.90/5.12 => ( topolo3100542954746470799et_int @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8335_monoI1,axiom,
% 8.90/5.12 ! [X8: nat > rat] :
% 8.90/5.12 ( ! [M4: nat,N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M4 @ N2 )
% 8.90/5.12 => ( ord_less_eq_rat @ ( X8 @ M4 ) @ ( X8 @ N2 ) ) )
% 8.90/5.12 => ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8336_monoI1,axiom,
% 8.90/5.12 ! [X8: nat > num] :
% 8.90/5.12 ( ! [M4: nat,N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M4 @ N2 )
% 8.90/5.12 => ( ord_less_eq_num @ ( X8 @ M4 ) @ ( X8 @ N2 ) ) )
% 8.90/5.12 => ( topolo1459490580787246023eq_num @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8337_monoI1,axiom,
% 8.90/5.12 ! [X8: nat > nat] :
% 8.90/5.12 ( ! [M4: nat,N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M4 @ N2 )
% 8.90/5.12 => ( ord_less_eq_nat @ ( X8 @ M4 ) @ ( X8 @ N2 ) ) )
% 8.90/5.12 => ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8338_monoI1,axiom,
% 8.90/5.12 ! [X8: nat > int] :
% 8.90/5.12 ( ! [M4: nat,N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M4 @ N2 )
% 8.90/5.12 => ( ord_less_eq_int @ ( X8 @ M4 ) @ ( X8 @ N2 ) ) )
% 8.90/5.12 => ( topolo4899668324122417113eq_int @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8339_monoI2,axiom,
% 8.90/5.12 ! [X8: nat > real] :
% 8.90/5.12 ( ! [M4: nat,N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M4 @ N2 )
% 8.90/5.12 => ( ord_less_eq_real @ ( X8 @ N2 ) @ ( X8 @ M4 ) ) )
% 8.90/5.12 => ( topolo6980174941875973593q_real @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8340_monoI2,axiom,
% 8.90/5.12 ! [X8: nat > set_int] :
% 8.90/5.12 ( ! [M4: nat,N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M4 @ N2 )
% 8.90/5.12 => ( ord_less_eq_set_int @ ( X8 @ N2 ) @ ( X8 @ M4 ) ) )
% 8.90/5.12 => ( topolo3100542954746470799et_int @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8341_monoI2,axiom,
% 8.90/5.12 ! [X8: nat > rat] :
% 8.90/5.12 ( ! [M4: nat,N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M4 @ N2 )
% 8.90/5.12 => ( ord_less_eq_rat @ ( X8 @ N2 ) @ ( X8 @ M4 ) ) )
% 8.90/5.12 => ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8342_monoI2,axiom,
% 8.90/5.12 ! [X8: nat > num] :
% 8.90/5.12 ( ! [M4: nat,N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M4 @ N2 )
% 8.90/5.12 => ( ord_less_eq_num @ ( X8 @ N2 ) @ ( X8 @ M4 ) ) )
% 8.90/5.12 => ( topolo1459490580787246023eq_num @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8343_monoI2,axiom,
% 8.90/5.12 ! [X8: nat > nat] :
% 8.90/5.12 ( ! [M4: nat,N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M4 @ N2 )
% 8.90/5.12 => ( ord_less_eq_nat @ ( X8 @ N2 ) @ ( X8 @ M4 ) ) )
% 8.90/5.12 => ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8344_monoI2,axiom,
% 8.90/5.12 ! [X8: nat > int] :
% 8.90/5.12 ( ! [M4: nat,N2: nat] :
% 8.90/5.12 ( ( ord_less_eq_nat @ M4 @ N2 )
% 8.90/5.12 => ( ord_less_eq_int @ ( X8 @ N2 ) @ ( X8 @ M4 ) ) )
% 8.90/5.12 => ( topolo4899668324122417113eq_int @ X8 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8345_Maclaurin__cos__expansion2,axiom,
% 8.90/5.12 ! [X2: real,N: nat] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.12 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.12 => ? [T5: real] :
% 8.90/5.12 ( ( ord_less_real @ zero_zero_real @ T5 )
% 8.90/5.12 & ( ord_less_real @ T5 @ X2 )
% 8.90/5.12 & ( ( cos_real @ X2 )
% 8.90/5.12 = ( plus_plus_real
% 8.90/5.12 @ ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.12 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8346_Maclaurin__minus__cos__expansion,axiom,
% 8.90/5.12 ! [N: nat,X2: real] :
% 8.90/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.12 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 8.90/5.12 => ? [T5: real] :
% 8.90/5.12 ( ( ord_less_real @ X2 @ T5 )
% 8.90/5.12 & ( ord_less_real @ T5 @ zero_zero_real )
% 8.90/5.12 & ( ( cos_real @ X2 )
% 8.90/5.12 = ( plus_plus_real
% 8.90/5.12 @ ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.12 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8347_Maclaurin__cos__expansion,axiom,
% 8.90/5.12 ! [X2: real,N: nat] :
% 8.90/5.12 ? [T5: real] :
% 8.90/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X2 ) )
% 8.90/5.12 & ( ( cos_real @ X2 )
% 8.90/5.12 = ( plus_plus_real
% 8.90/5.12 @ ( groups6591440286371151544t_real
% 8.90/5.12 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
% 8.90/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.12 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8348_diffs__equiv,axiom,
% 8.90/5.12 ! [C: nat > complex,X2: complex] :
% 8.90/5.12 ( ( summable_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X2 @ N3 ) ) )
% 8.90/5.12 => ( sums_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N3 ) @ ( C @ N3 ) ) @ ( power_power_complex @ X2 @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) )
% 8.90/5.12 @ ( suminf_complex
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X2 @ N3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8349_diffs__equiv,axiom,
% 8.90/5.12 ! [C: nat > real,X2: real] :
% 8.90/5.12 ( ( summable_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X2 @ N3 ) ) )
% 8.90/5.12 => ( sums_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( C @ N3 ) ) @ ( power_power_real @ X2 @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) )
% 8.90/5.12 @ ( suminf_real
% 8.90/5.12 @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X2 @ N3 ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8350_tan__double,axiom,
% 8.90/5.12 ! [X2: complex] :
% 8.90/5.12 ( ( ( cos_complex @ X2 )
% 8.90/5.12 != zero_zero_complex )
% 8.90/5.12 => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) )
% 8.90/5.12 != zero_zero_complex )
% 8.90/5.12 => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) )
% 8.90/5.12 = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X2 ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8351_tan__double,axiom,
% 8.90/5.12 ! [X2: real] :
% 8.90/5.12 ( ( ( cos_real @ X2 )
% 8.90/5.12 != zero_zero_real )
% 8.90/5.12 => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) )
% 8.90/5.12 != zero_zero_real )
% 8.90/5.12 => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) )
% 8.90/5.12 = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X2 ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8352_fact__0,axiom,
% 8.90/5.12 ( ( semiri5044797733671781792omplex @ zero_zero_nat )
% 8.90/5.12 = one_one_complex ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8353_fact__0,axiom,
% 8.90/5.12 ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
% 8.90/5.12 = one_one_rat ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8354_fact__0,axiom,
% 8.90/5.12 ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
% 8.90/5.12 = one_one_int ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8355_fact__0,axiom,
% 8.90/5.12 ( ( semiri2265585572941072030t_real @ zero_zero_nat )
% 8.90/5.12 = one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8356_fact__0,axiom,
% 8.90/5.12 ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
% 8.90/5.12 = one_one_nat ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8357_fact__1,axiom,
% 8.90/5.12 ( ( semiri5044797733671781792omplex @ one_one_nat )
% 8.90/5.12 = one_one_complex ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8358_fact__1,axiom,
% 8.90/5.12 ( ( semiri773545260158071498ct_rat @ one_one_nat )
% 8.90/5.12 = one_one_rat ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8359_fact__1,axiom,
% 8.90/5.12 ( ( semiri1406184849735516958ct_int @ one_one_nat )
% 8.90/5.12 = one_one_int ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8360_fact__1,axiom,
% 8.90/5.12 ( ( semiri2265585572941072030t_real @ one_one_nat )
% 8.90/5.12 = one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8361_fact__1,axiom,
% 8.90/5.12 ( ( semiri1408675320244567234ct_nat @ one_one_nat )
% 8.90/5.12 = one_one_nat ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8362_fact__Suc__0,axiom,
% 8.90/5.12 ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
% 8.90/5.12 = one_one_complex ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8363_fact__Suc__0,axiom,
% 8.90/5.12 ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
% 8.90/5.12 = one_one_rat ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8364_fact__Suc__0,axiom,
% 8.90/5.12 ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
% 8.90/5.12 = one_one_int ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8365_fact__Suc__0,axiom,
% 8.90/5.12 ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
% 8.90/5.12 = one_one_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8366_fact__Suc__0,axiom,
% 8.90/5.12 ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
% 8.90/5.12 = one_one_nat ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8367_fact__Suc,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
% 8.90/5.12 = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8368_fact__Suc,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
% 8.90/5.12 = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8369_fact__Suc,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
% 8.90/5.12 = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8370_fact__Suc,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
% 8.90/5.12 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8371_tan__npi,axiom,
% 8.90/5.12 ! [N: nat] :
% 8.90/5.12 ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 8.90/5.12 = zero_zero_real ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8372_tan__periodic__n,axiom,
% 8.90/5.12 ! [X2: real,N: num] :
% 8.90/5.12 ( ( tan_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
% 8.90/5.12 = ( tan_real @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8373_tan__periodic__nat,axiom,
% 8.90/5.12 ! [X2: real,N: nat] :
% 8.90/5.12 ( ( tan_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
% 8.90/5.12 = ( tan_real @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8374_tan__periodic__int,axiom,
% 8.90/5.12 ! [X2: real,I2: int] :
% 8.90/5.12 ( ( tan_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ pi ) ) )
% 8.90/5.12 = ( tan_real @ X2 ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8375_fact__2,axiom,
% 8.90/5.12 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.12 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8376_fact__2,axiom,
% 8.90/5.12 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.12 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 8.90/5.12
% 8.90/5.12 thf(fact_8377_fact__2,axiom,
% 8.90/5.12 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.13 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8378_fact__2,axiom,
% 8.90/5.13 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.13 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8379_fact__2,axiom,
% 8.90/5.13 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.13 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8380_tan__periodic,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( tan_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 8.90/5.13 = ( tan_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8381_fact__ge__zero,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8382_fact__ge__zero,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8383_fact__ge__zero,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8384_fact__ge__zero,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8385_fact__not__neg,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8386_fact__not__neg,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8387_fact__not__neg,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8388_fact__not__neg,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8389_fact__gt__zero,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8390_fact__gt__zero,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8391_fact__gt__zero,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8392_fact__gt__zero,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8393_fact__ge__1,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8394_fact__ge__1,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8395_fact__ge__1,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8396_fact__ge__1,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8397_fact__mono,axiom,
% 8.90/5.13 ! [M: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.13 => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8398_fact__mono,axiom,
% 8.90/5.13 ! [M: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.13 => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8399_fact__mono,axiom,
% 8.90/5.13 ! [M: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.13 => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8400_fact__mono,axiom,
% 8.90/5.13 ! [M: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.13 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8401_fact__dvd,axiom,
% 8.90/5.13 ! [N: nat,M: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.13 => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8402_fact__dvd,axiom,
% 8.90/5.13 ! [N: nat,M: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.13 => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8403_fact__dvd,axiom,
% 8.90/5.13 ! [N: nat,M: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.13 => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8404_fact__dvd,axiom,
% 8.90/5.13 ! [N: nat,M: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.13 => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8405_fact__less__mono,axiom,
% 8.90/5.13 ! [M: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.90/5.13 => ( ( ord_less_nat @ M @ N )
% 8.90/5.13 => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8406_fact__less__mono,axiom,
% 8.90/5.13 ! [M: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.90/5.13 => ( ( ord_less_nat @ M @ N )
% 8.90/5.13 => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8407_fact__less__mono,axiom,
% 8.90/5.13 ! [M: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.90/5.13 => ( ( ord_less_nat @ M @ N )
% 8.90/5.13 => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8408_fact__less__mono,axiom,
% 8.90/5.13 ! [M: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.90/5.13 => ( ( ord_less_nat @ M @ N )
% 8.90/5.13 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8409_fact__fact__dvd__fact,axiom,
% 8.90/5.13 ! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8410_fact__fact__dvd__fact,axiom,
% 8.90/5.13 ! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8411_fact__fact__dvd__fact,axiom,
% 8.90/5.13 ! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8412_fact__fact__dvd__fact,axiom,
% 8.90/5.13 ! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8413_fact__fact__dvd__fact,axiom,
% 8.90/5.13 ! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8414_fact__mod,axiom,
% 8.90/5.13 ! [M: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.13 => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
% 8.90/5.13 = zero_zero_int ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8415_fact__mod,axiom,
% 8.90/5.13 ! [M: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.13 => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
% 8.90/5.13 = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8416_fact__mod,axiom,
% 8.90/5.13 ! [M: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.13 => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
% 8.90/5.13 = zero_zero_nat ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8417_fact__le__power,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8418_fact__le__power,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8419_fact__le__power,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8420_fact__le__power,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8421_tan__def,axiom,
% 8.90/5.13 ( tan_complex
% 8.90/5.13 = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X ) @ ( cos_complex @ X ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8422_tan__def,axiom,
% 8.90/5.13 ( tan_real
% 8.90/5.13 = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ X ) @ ( cos_real @ X ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8423_choose__dvd,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8424_choose__dvd,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8425_choose__dvd,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8426_choose__dvd,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8427_choose__dvd,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8428_diffs__def,axiom,
% 8.90/5.13 ( diffs_int
% 8.90/5.13 = ( ^ [C2: nat > int,N3: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) @ ( C2 @ ( suc @ N3 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8429_diffs__def,axiom,
% 8.90/5.13 ( diffs_real
% 8.90/5.13 = ( ^ [C2: nat > real,N3: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) @ ( C2 @ ( suc @ N3 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8430_diffs__def,axiom,
% 8.90/5.13 ( diffs_rat
% 8.90/5.13 = ( ^ [C2: nat > rat,N3: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) @ ( C2 @ ( suc @ N3 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8431_fact__numeral,axiom,
% 8.90/5.13 ! [K: num] :
% 8.90/5.13 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
% 8.90/5.13 = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8432_fact__numeral,axiom,
% 8.90/5.13 ! [K: num] :
% 8.90/5.13 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
% 8.90/5.13 = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8433_fact__numeral,axiom,
% 8.90/5.13 ! [K: num] :
% 8.90/5.13 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
% 8.90/5.13 = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8434_fact__numeral,axiom,
% 8.90/5.13 ! [K: num] :
% 8.90/5.13 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
% 8.90/5.13 = ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8435_fact__numeral,axiom,
% 8.90/5.13 ! [K: num] :
% 8.90/5.13 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
% 8.90/5.13 = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8436_termdiff__converges__all,axiom,
% 8.90/5.13 ! [C: nat > complex,X2: complex] :
% 8.90/5.13 ( ! [X5: complex] :
% 8.90/5.13 ( summable_complex
% 8.90/5.13 @ ^ [N3: nat] : ( times_times_complex @ ( C @ N3 ) @ ( power_power_complex @ X5 @ N3 ) ) )
% 8.90/5.13 => ( summable_complex
% 8.90/5.13 @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X2 @ N3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8437_termdiff__converges__all,axiom,
% 8.90/5.13 ! [C: nat > real,X2: real] :
% 8.90/5.13 ( ! [X5: real] :
% 8.90/5.13 ( summable_real
% 8.90/5.13 @ ^ [N3: nat] : ( times_times_real @ ( C @ N3 ) @ ( power_power_real @ X5 @ N3 ) ) )
% 8.90/5.13 => ( summable_real
% 8.90/5.13 @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X2 @ N3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8438_square__fact__le__2__fact,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8439_tan__45,axiom,
% 8.90/5.13 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 8.90/5.13 = one_one_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8440_fact__num__eq__if,axiom,
% 8.90/5.13 ( semiri5044797733671781792omplex
% 8.90/5.13 = ( ^ [M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M3 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8441_fact__num__eq__if,axiom,
% 8.90/5.13 ( semiri1406184849735516958ct_int
% 8.90/5.13 = ( ^ [M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8442_fact__num__eq__if,axiom,
% 8.90/5.13 ( semiri773545260158071498ct_rat
% 8.90/5.13 = ( ^ [M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M3 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8443_fact__num__eq__if,axiom,
% 8.90/5.13 ( semiri2265585572941072030t_real
% 8.90/5.13 = ( ^ [M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8444_fact__num__eq__if,axiom,
% 8.90/5.13 ( semiri1408675320244567234ct_nat
% 8.90/5.13 = ( ^ [M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8445_fact__code,axiom,
% 8.90/5.13 ( semiri1406184849735516958ct_int
% 8.90/5.13 = ( ^ [N3: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8446_fact__code,axiom,
% 8.90/5.13 ( semiri773545260158071498ct_rat
% 8.90/5.13 = ( ^ [N3: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8447_fact__code,axiom,
% 8.90/5.13 ( semiri2265585572941072030t_real
% 8.90/5.13 = ( ^ [N3: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8448_fact__code,axiom,
% 8.90/5.13 ( semiri1408675320244567234ct_nat
% 8.90/5.13 = ( ^ [N3: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8449_fact__reduce,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( semiri1406184849735516958ct_int @ N )
% 8.90/5.13 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8450_fact__reduce,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( semiri773545260158071498ct_rat @ N )
% 8.90/5.13 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8451_fact__reduce,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( semiri2265585572941072030t_real @ N )
% 8.90/5.13 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8452_fact__reduce,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( semiri1408675320244567234ct_nat @ N )
% 8.90/5.13 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8453_lemma__tan__total,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 8.90/5.13 => ? [X5: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ X5 )
% 8.90/5.13 & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 & ( ord_less_real @ Y3 @ ( tan_real @ X5 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8454_tan__gt__zero,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ord_less_real @ zero_zero_real @ ( tan_real @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8455_lemma__tan__total1,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ? [X5: real] :
% 8.90/5.13 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
% 8.90/5.13 & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 & ( ( tan_real @ X5 )
% 8.90/5.13 = Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8456_tan__mono__lt__eq,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 8.90/5.13 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ( ord_less_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) )
% 8.90/5.13 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8457_tan__monotone_H,axiom,
% 8.90/5.13 ! [Y3: real,X2: real] :
% 8.90/5.13 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 8.90/5.13 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ( ord_less_real @ Y3 @ X2 )
% 8.90/5.13 = ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X2 ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8458_tan__monotone,axiom,
% 8.90/5.13 ! [Y3: real,X2: real] :
% 8.90/5.13 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 8.90/5.13 => ( ( ord_less_real @ Y3 @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X2 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8459_tan__total,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ? [X5: real] :
% 8.90/5.13 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
% 8.90/5.13 & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 & ( ( tan_real @ X5 )
% 8.90/5.13 = Y3 )
% 8.90/5.13 & ! [Y5: real] :
% 8.90/5.13 ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 8.90/5.13 & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 & ( ( tan_real @ Y5 )
% 8.90/5.13 = Y3 ) )
% 8.90/5.13 => ( Y5 = X5 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8460_tan__minus__45,axiom,
% 8.90/5.13 ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
% 8.90/5.13 = ( uminus_uminus_real @ one_one_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8461_tan__inverse,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y3 ) )
% 8.90/5.13 = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8462_add__tan__eq,axiom,
% 8.90/5.13 ! [X2: complex,Y3: complex] :
% 8.90/5.13 ( ( ( cos_complex @ X2 )
% 8.90/5.13 != zero_zero_complex )
% 8.90/5.13 => ( ( ( cos_complex @ Y3 )
% 8.90/5.13 != zero_zero_complex )
% 8.90/5.13 => ( ( plus_plus_complex @ ( tan_complex @ X2 ) @ ( tan_complex @ Y3 ) )
% 8.90/5.13 = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X2 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8463_add__tan__eq,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ( cos_real @ X2 )
% 8.90/5.13 != zero_zero_real )
% 8.90/5.13 => ( ( ( cos_real @ Y3 )
% 8.90/5.13 != zero_zero_real )
% 8.90/5.13 => ( ( plus_plus_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) )
% 8.90/5.13 = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8464_termdiff__converges,axiom,
% 8.90/5.13 ! [X2: real,K5: real,C: nat > real] :
% 8.90/5.13 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ K5 )
% 8.90/5.13 => ( ! [X5: real] :
% 8.90/5.13 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X5 ) @ K5 )
% 8.90/5.13 => ( summable_real
% 8.90/5.13 @ ^ [N3: nat] : ( times_times_real @ ( C @ N3 ) @ ( power_power_real @ X5 @ N3 ) ) ) )
% 8.90/5.13 => ( summable_real
% 8.90/5.13 @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X2 @ N3 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8465_termdiff__converges,axiom,
% 8.90/5.13 ! [X2: complex,K5: real,C: nat > complex] :
% 8.90/5.13 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ K5 )
% 8.90/5.13 => ( ! [X5: complex] :
% 8.90/5.13 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X5 ) @ K5 )
% 8.90/5.13 => ( summable_complex
% 8.90/5.13 @ ^ [N3: nat] : ( times_times_complex @ ( C @ N3 ) @ ( power_power_complex @ X5 @ N3 ) ) ) )
% 8.90/5.13 => ( summable_complex
% 8.90/5.13 @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X2 @ N3 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8466_tan__pos__pi2__le,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8467_tan__total__pos,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.13 => ? [X5: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 8.90/5.13 & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 & ( ( tan_real @ X5 )
% 8.90/5.13 = Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8468_tan__less__zero,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 8.90/5.13 => ( ord_less_real @ ( tan_real @ X2 ) @ zero_zero_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8469_tan__mono__le__eq,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 8.90/5.13 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) )
% 8.90/5.13 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8470_tan__mono__le,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 8.90/5.13 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ord_less_eq_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8471_tan__bound__pi2,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 8.90/5.13 => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X2 ) ) @ one_one_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8472_arctan__unique,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ( ( tan_real @ X2 )
% 8.90/5.13 = Y3 )
% 8.90/5.13 => ( ( arctan @ Y3 )
% 8.90/5.13 = X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8473_arctan__tan,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ( arctan @ ( tan_real @ X2 ) )
% 8.90/5.13 = X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8474_arctan,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 8.90/5.13 & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 & ( ( tan_real @ ( arctan @ Y3 ) )
% 8.90/5.13 = Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8475_Maclaurin__zero,axiom,
% 8.90/5.13 ! [X2: real,N: nat,Diff: nat > complex > real] :
% 8.90/5.13 ( ( X2 = zero_zero_real )
% 8.90/5.13 => ( ( N != zero_zero_nat )
% 8.90/5.13 => ( ( groups6591440286371151544t_real
% 8.90/5.13 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
% 8.90/5.13 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.13 = ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8476_Maclaurin__zero,axiom,
% 8.90/5.13 ! [X2: real,N: nat,Diff: nat > real > real] :
% 8.90/5.13 ( ( X2 = zero_zero_real )
% 8.90/5.13 => ( ( N != zero_zero_nat )
% 8.90/5.13 => ( ( groups6591440286371151544t_real
% 8.90/5.13 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
% 8.90/5.13 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.13 = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8477_Maclaurin__zero,axiom,
% 8.90/5.13 ! [X2: real,N: nat,Diff: nat > rat > real] :
% 8.90/5.13 ( ( X2 = zero_zero_real )
% 8.90/5.13 => ( ( N != zero_zero_nat )
% 8.90/5.13 => ( ( groups6591440286371151544t_real
% 8.90/5.13 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
% 8.90/5.13 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.13 = ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8478_Maclaurin__zero,axiom,
% 8.90/5.13 ! [X2: real,N: nat,Diff: nat > nat > real] :
% 8.90/5.13 ( ( X2 = zero_zero_real )
% 8.90/5.13 => ( ( N != zero_zero_nat )
% 8.90/5.13 => ( ( groups6591440286371151544t_real
% 8.90/5.13 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
% 8.90/5.13 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.13 = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8479_Maclaurin__zero,axiom,
% 8.90/5.13 ! [X2: real,N: nat,Diff: nat > int > real] :
% 8.90/5.13 ( ( X2 = zero_zero_real )
% 8.90/5.13 => ( ( N != zero_zero_nat )
% 8.90/5.13 => ( ( groups6591440286371151544t_real
% 8.90/5.13 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
% 8.90/5.13 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.13 = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8480_Maclaurin__lemma,axiom,
% 8.90/5.13 ! [H2: real,F: real > real,J: nat > real,N: nat] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ H2 )
% 8.90/5.13 => ? [B8: real] :
% 8.90/5.13 ( ( F @ H2 )
% 8.90/5.13 = ( plus_plus_real
% 8.90/5.13 @ ( groups6591440286371151544t_real
% 8.90/5.13 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 8.90/5.13 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.13 @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8481_lemma__tan__add1,axiom,
% 8.90/5.13 ! [X2: complex,Y3: complex] :
% 8.90/5.13 ( ( ( cos_complex @ X2 )
% 8.90/5.13 != zero_zero_complex )
% 8.90/5.13 => ( ( ( cos_complex @ Y3 )
% 8.90/5.13 != zero_zero_complex )
% 8.90/5.13 => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X2 ) @ ( tan_complex @ Y3 ) ) )
% 8.90/5.13 = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X2 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8482_lemma__tan__add1,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ( cos_real @ X2 )
% 8.90/5.13 != zero_zero_real )
% 8.90/5.13 => ( ( ( cos_real @ Y3 )
% 8.90/5.13 != zero_zero_real )
% 8.90/5.13 => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) ) )
% 8.90/5.13 = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8483_tan__diff,axiom,
% 8.90/5.13 ! [X2: complex,Y3: complex] :
% 8.90/5.13 ( ( ( cos_complex @ X2 )
% 8.90/5.13 != zero_zero_complex )
% 8.90/5.13 => ( ( ( cos_complex @ Y3 )
% 8.90/5.13 != zero_zero_complex )
% 8.90/5.13 => ( ( ( cos_complex @ ( minus_minus_complex @ X2 @ Y3 ) )
% 8.90/5.13 != zero_zero_complex )
% 8.90/5.13 => ( ( tan_complex @ ( minus_minus_complex @ X2 @ Y3 ) )
% 8.90/5.13 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X2 ) @ ( tan_complex @ Y3 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X2 ) @ ( tan_complex @ Y3 ) ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8484_tan__diff,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ( cos_real @ X2 )
% 8.90/5.13 != zero_zero_real )
% 8.90/5.13 => ( ( ( cos_real @ Y3 )
% 8.90/5.13 != zero_zero_real )
% 8.90/5.13 => ( ( ( cos_real @ ( minus_minus_real @ X2 @ Y3 ) )
% 8.90/5.13 != zero_zero_real )
% 8.90/5.13 => ( ( tan_real @ ( minus_minus_real @ X2 @ Y3 ) )
% 8.90/5.13 = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8485_tan__add,axiom,
% 8.90/5.13 ! [X2: complex,Y3: complex] :
% 8.90/5.13 ( ( ( cos_complex @ X2 )
% 8.90/5.13 != zero_zero_complex )
% 8.90/5.13 => ( ( ( cos_complex @ Y3 )
% 8.90/5.13 != zero_zero_complex )
% 8.90/5.13 => ( ( ( cos_complex @ ( plus_plus_complex @ X2 @ Y3 ) )
% 8.90/5.13 != zero_zero_complex )
% 8.90/5.13 => ( ( tan_complex @ ( plus_plus_complex @ X2 @ Y3 ) )
% 8.90/5.13 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X2 ) @ ( tan_complex @ Y3 ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X2 ) @ ( tan_complex @ Y3 ) ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8486_tan__add,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ( cos_real @ X2 )
% 8.90/5.13 != zero_zero_real )
% 8.90/5.13 => ( ( ( cos_real @ Y3 )
% 8.90/5.13 != zero_zero_real )
% 8.90/5.13 => ( ( ( cos_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 8.90/5.13 != zero_zero_real )
% 8.90/5.13 => ( ( tan_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 8.90/5.13 = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8487_tan__total__pi4,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 8.90/5.13 => ? [Z2: real] :
% 8.90/5.13 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z2 )
% 8.90/5.13 & ( ord_less_real @ Z2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 8.90/5.13 & ( ( tan_real @ Z2 )
% 8.90/5.13 = X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8488_cos__paired,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( sums_real
% 8.90/5.13 @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_real @ X2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 8.90/5.13 @ ( cos_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8489_tan__half,axiom,
% 8.90/5.13 ( tan_complex
% 8.90/5.13 = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ one_one_complex ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8490_tan__half,axiom,
% 8.90/5.13 ( tan_real
% 8.90/5.13 = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8491_cos__coeff__def,axiom,
% 8.90/5.13 ( cos_coeff
% 8.90/5.13 = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ zero_zero_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8492_sin__paired,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( sums_real
% 8.90/5.13 @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 8.90/5.13 @ ( sin_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8493_Maclaurin__sin__expansion3,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ? [T5: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ T5 )
% 8.90/5.13 & ( ord_less_real @ T5 @ X2 )
% 8.90/5.13 & ( ( sin_real @ X2 )
% 8.90/5.13 = ( plus_plus_real
% 8.90/5.13 @ ( groups6591440286371151544t_real
% 8.90/5.13 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
% 8.90/5.13 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.13 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8494_Maclaurin__sin__expansion4,axiom,
% 8.90/5.13 ! [X2: real,N: nat] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ? [T5: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ T5 )
% 8.90/5.13 & ( ord_less_eq_real @ T5 @ X2 )
% 8.90/5.13 & ( ( sin_real @ X2 )
% 8.90/5.13 = ( plus_plus_real
% 8.90/5.13 @ ( groups6591440286371151544t_real
% 8.90/5.13 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
% 8.90/5.13 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.13 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8495_Maclaurin__sin__expansion2,axiom,
% 8.90/5.13 ! [X2: real,N: nat] :
% 8.90/5.13 ? [T5: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X2 ) )
% 8.90/5.13 & ( ( sin_real @ X2 )
% 8.90/5.13 = ( plus_plus_real
% 8.90/5.13 @ ( groups6591440286371151544t_real
% 8.90/5.13 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
% 8.90/5.13 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.13 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8496_Maclaurin__sin__expansion,axiom,
% 8.90/5.13 ! [X2: real,N: nat] :
% 8.90/5.13 ? [T5: real] :
% 8.90/5.13 ( ( sin_real @ X2 )
% 8.90/5.13 = ( plus_plus_real
% 8.90/5.13 @ ( groups6591440286371151544t_real
% 8.90/5.13 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
% 8.90/5.13 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.13 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8497_sin__coeff__def,axiom,
% 8.90/5.13 ( sin_coeff
% 8.90/5.13 = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8498_fact__ge__self,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8499_fact__mono__nat,axiom,
% 8.90/5.13 ! [M: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.13 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8500_fact__less__mono__nat,axiom,
% 8.90/5.13 ! [M: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.90/5.13 => ( ( ord_less_nat @ M @ N )
% 8.90/5.13 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8501_fact__ge__Suc__0__nat,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8502_dvd__fact,axiom,
% 8.90/5.13 ! [M: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 8.90/5.13 => ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.13 => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8503_fact__diff__Suc,axiom,
% 8.90/5.13 ! [N: nat,M: nat] :
% 8.90/5.13 ( ( ord_less_nat @ N @ ( suc @ M ) )
% 8.90/5.13 => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
% 8.90/5.13 = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8504_fact__div__fact__le__pow,axiom,
% 8.90/5.13 ! [R2: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ R2 @ N )
% 8.90/5.13 => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8505_sin__coeff__Suc,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( sin_coeff @ ( suc @ N ) )
% 8.90/5.13 = ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8506_cos__coeff__Suc,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( cos_coeff @ ( suc @ N ) )
% 8.90/5.13 = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8507_sin__tan,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ( sin_real @ X2 )
% 8.90/5.13 = ( divide_divide_real @ ( tan_real @ X2 ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8508_cos__tan,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ( cos_real @ X2 )
% 8.90/5.13 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8509_arcosh__def,axiom,
% 8.90/5.13 ( arcosh_real
% 8.90/5.13 = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8510_complex__unimodular__polar,axiom,
% 8.90/5.13 ! [Z: complex] :
% 8.90/5.13 ( ( ( real_V1022390504157884413omplex @ Z )
% 8.90/5.13 = one_one_real )
% 8.90/5.13 => ~ ! [T5: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 8.90/5.13 => ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 8.90/5.13 => ( Z
% 8.90/5.13 != ( complex2 @ ( cos_real @ T5 ) @ ( sin_real @ T5 ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8511_Maclaurin__exp__lt,axiom,
% 8.90/5.13 ! [X2: real,N: nat] :
% 8.90/5.13 ( ( X2 != zero_zero_real )
% 8.90/5.13 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ? [T5: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
% 8.90/5.13 & ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X2 ) )
% 8.90/5.13 & ( ( exp_real @ X2 )
% 8.90/5.13 = ( plus_plus_real
% 8.90/5.13 @ ( groups6591440286371151544t_real
% 8.90/5.13 @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X2 @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
% 8.90/5.13 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.13 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8512_real__sqrt__eq__iff,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ( sqrt @ X2 )
% 8.90/5.13 = ( sqrt @ Y3 ) )
% 8.90/5.13 = ( X2 = Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8513_real__sqrt__eq__zero__cancel__iff,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ( sqrt @ X2 )
% 8.90/5.13 = zero_zero_real )
% 8.90/5.13 = ( X2 = zero_zero_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8514_real__sqrt__zero,axiom,
% 8.90/5.13 ( ( sqrt @ zero_zero_real )
% 8.90/5.13 = zero_zero_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8515_real__sqrt__less__iff,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) )
% 8.90/5.13 = ( ord_less_real @ X2 @ Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8516_real__sqrt__le__iff,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) )
% 8.90/5.13 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8517_real__sqrt__eq__1__iff,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ( sqrt @ X2 )
% 8.90/5.13 = one_one_real )
% 8.90/5.13 = ( X2 = one_one_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8518_real__sqrt__one,axiom,
% 8.90/5.13 ( ( sqrt @ one_one_real )
% 8.90/5.13 = one_one_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8519_exp__le__cancel__iff,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) )
% 8.90/5.13 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8520_exp__zero,axiom,
% 8.90/5.13 ( ( exp_complex @ zero_zero_complex )
% 8.90/5.13 = one_one_complex ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8521_exp__zero,axiom,
% 8.90/5.13 ( ( exp_real @ zero_zero_real )
% 8.90/5.13 = one_one_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8522_of__real__eq__1__iff,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ( real_V1803761363581548252l_real @ X2 )
% 8.90/5.13 = one_one_real )
% 8.90/5.13 = ( X2 = one_one_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8523_of__real__eq__1__iff,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ( real_V4546457046886955230omplex @ X2 )
% 8.90/5.13 = one_one_complex )
% 8.90/5.13 = ( X2 = one_one_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8524_of__real__1,axiom,
% 8.90/5.13 ( ( real_V1803761363581548252l_real @ one_one_real )
% 8.90/5.13 = one_one_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8525_of__real__1,axiom,
% 8.90/5.13 ( ( real_V4546457046886955230omplex @ one_one_real )
% 8.90/5.13 = one_one_complex ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8526_of__real__numeral,axiom,
% 8.90/5.13 ! [W: num] :
% 8.90/5.13 ( ( real_V1803761363581548252l_real @ ( numeral_numeral_real @ W ) )
% 8.90/5.13 = ( numeral_numeral_real @ W ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8527_of__real__numeral,axiom,
% 8.90/5.13 ! [W: num] :
% 8.90/5.13 ( ( real_V4546457046886955230omplex @ ( numeral_numeral_real @ W ) )
% 8.90/5.13 = ( numera6690914467698888265omplex @ W ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8528_of__real__mult,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( real_V1803761363581548252l_real @ ( times_times_real @ X2 @ Y3 ) )
% 8.90/5.13 = ( times_times_real @ ( real_V1803761363581548252l_real @ X2 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8529_of__real__mult,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( real_V4546457046886955230omplex @ ( times_times_real @ X2 @ Y3 ) )
% 8.90/5.13 = ( times_times_complex @ ( real_V4546457046886955230omplex @ X2 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8530_real__sqrt__lt__0__iff,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ ( sqrt @ X2 ) @ zero_zero_real )
% 8.90/5.13 = ( ord_less_real @ X2 @ zero_zero_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8531_real__sqrt__gt__0__iff,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 8.90/5.13 = ( ord_less_real @ zero_zero_real @ Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8532_of__real__divide,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X2 @ Y3 ) )
% 8.90/5.13 = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X2 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8533_of__real__divide,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X2 @ Y3 ) )
% 8.90/5.13 = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X2 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8534_real__sqrt__ge__0__iff,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 8.90/5.13 = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8535_real__sqrt__le__0__iff,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ zero_zero_real )
% 8.90/5.13 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8536_of__real__add,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 8.90/5.13 = ( plus_plus_real @ ( real_V1803761363581548252l_real @ X2 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8537_of__real__add,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X2 @ Y3 ) )
% 8.90/5.13 = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X2 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8538_of__real__power,axiom,
% 8.90/5.13 ! [X2: real,N: nat] :
% 8.90/5.13 ( ( real_V1803761363581548252l_real @ ( power_power_real @ X2 @ N ) )
% 8.90/5.13 = ( power_power_real @ ( real_V1803761363581548252l_real @ X2 ) @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8539_of__real__power,axiom,
% 8.90/5.13 ! [X2: real,N: nat] :
% 8.90/5.13 ( ( real_V4546457046886955230omplex @ ( power_power_real @ X2 @ N ) )
% 8.90/5.13 = ( power_power_complex @ ( real_V4546457046886955230omplex @ X2 ) @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8540_real__sqrt__gt__1__iff,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_real @ one_one_real @ ( sqrt @ Y3 ) )
% 8.90/5.13 = ( ord_less_real @ one_one_real @ Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8541_real__sqrt__lt__1__iff,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ ( sqrt @ X2 ) @ one_one_real )
% 8.90/5.13 = ( ord_less_real @ X2 @ one_one_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8542_real__sqrt__ge__1__iff,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y3 ) )
% 8.90/5.13 = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8543_real__sqrt__le__1__iff,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ one_one_real )
% 8.90/5.13 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8544_of__real__diff,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( real_V1803761363581548252l_real @ ( minus_minus_real @ X2 @ Y3 ) )
% 8.90/5.13 = ( minus_minus_real @ ( real_V1803761363581548252l_real @ X2 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8545_of__real__diff,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( real_V4546457046886955230omplex @ ( minus_minus_real @ X2 @ Y3 ) )
% 8.90/5.13 = ( minus_minus_complex @ ( real_V4546457046886955230omplex @ X2 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8546_exp__eq__one__iff,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ( exp_real @ X2 )
% 8.90/5.13 = one_one_real )
% 8.90/5.13 = ( X2 = zero_zero_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8547_real__sqrt__abs2,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( sqrt @ ( times_times_real @ X2 @ X2 ) )
% 8.90/5.13 = ( abs_abs_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8548_real__sqrt__mult__self,axiom,
% 8.90/5.13 ! [A: real] :
% 8.90/5.13 ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
% 8.90/5.13 = ( abs_abs_real @ A ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8549_real__sqrt__four,axiom,
% 8.90/5.13 ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 8.90/5.13 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8550_one__less__exp__iff,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ one_one_real @ ( exp_real @ X2 ) )
% 8.90/5.13 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8551_exp__less__one__iff,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ ( exp_real @ X2 ) @ one_one_real )
% 8.90/5.13 = ( ord_less_real @ X2 @ zero_zero_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8552_exp__le__one__iff,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( exp_real @ X2 ) @ one_one_real )
% 8.90/5.13 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8553_one__le__exp__iff,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8554_of__real__neg__numeral,axiom,
% 8.90/5.13 ! [W: num] :
% 8.90/5.13 ( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 8.90/5.13 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8555_of__real__neg__numeral,axiom,
% 8.90/5.13 ! [W: num] :
% 8.90/5.13 ( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 8.90/5.13 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8556_cos__of__real__pi,axiom,
% 8.90/5.13 ( ( cos_real @ ( real_V1803761363581548252l_real @ pi ) )
% 8.90/5.13 = ( uminus_uminus_real @ one_one_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8557_cos__of__real__pi,axiom,
% 8.90/5.13 ( ( cos_complex @ ( real_V4546457046886955230omplex @ pi ) )
% 8.90/5.13 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8558_norm__cos__sin,axiom,
% 8.90/5.13 ! [T: real] :
% 8.90/5.13 ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
% 8.90/5.13 = one_one_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8559_real__sqrt__abs,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( sqrt @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.13 = ( abs_abs_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8560_real__sqrt__pow2,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( power_power_real @ ( sqrt @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.13 = X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8561_real__sqrt__pow2__iff,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ( power_power_real @ ( sqrt @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.13 = X2 )
% 8.90/5.13 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8562_real__sqrt__sum__squares__mult__squared__eq,axiom,
% 8.90/5.13 ! [X2: real,Y3: real,Xa2: real,Ya: real] :
% 8.90/5.13 ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.13 = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8563_norm__of__real__add1,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X2 ) @ one_one_real ) )
% 8.90/5.13 = ( abs_abs_real @ ( plus_plus_real @ X2 @ one_one_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8564_norm__of__real__add1,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X2 ) @ one_one_complex ) )
% 8.90/5.13 = ( abs_abs_real @ ( plus_plus_real @ X2 @ one_one_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8565_norm__of__real__addn,axiom,
% 8.90/5.13 ! [X2: real,B: num] :
% 8.90/5.13 ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X2 ) @ ( numeral_numeral_real @ B ) ) )
% 8.90/5.13 = ( abs_abs_real @ ( plus_plus_real @ X2 @ ( numeral_numeral_real @ B ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8566_norm__of__real__addn,axiom,
% 8.90/5.13 ! [X2: real,B: num] :
% 8.90/5.13 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X2 ) @ ( numera6690914467698888265omplex @ B ) ) )
% 8.90/5.13 = ( abs_abs_real @ ( plus_plus_real @ X2 @ ( numeral_numeral_real @ B ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8567_cos__of__real__pi__half,axiom,
% 8.90/5.13 ( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 = zero_zero_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8568_cos__of__real__pi__half,axiom,
% 8.90/5.13 ( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 8.90/5.13 = zero_zero_complex ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8569_sin__of__real__pi__half,axiom,
% 8.90/5.13 ( ( sin_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 = one_one_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8570_sin__of__real__pi__half,axiom,
% 8.90/5.13 ( ( sin_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 8.90/5.13 = one_one_complex ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8571_real__sqrt__minus,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( sqrt @ ( uminus_uminus_real @ X2 ) )
% 8.90/5.13 = ( uminus_uminus_real @ ( sqrt @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8572_real__sqrt__less__mono,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.13 => ( ord_less_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8573_real__sqrt__le__mono,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ X2 @ Y3 )
% 8.90/5.13 => ( ord_less_eq_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8574_complex__of__real__mult__Complex,axiom,
% 8.90/5.13 ! [R2: real,X2: real,Y3: real] :
% 8.90/5.13 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X2 @ Y3 ) )
% 8.90/5.13 = ( complex2 @ ( times_times_real @ R2 @ X2 ) @ ( times_times_real @ R2 @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8575_Complex__mult__complex__of__real,axiom,
% 8.90/5.13 ! [X2: real,Y3: real,R2: real] :
% 8.90/5.13 ( ( times_times_complex @ ( complex2 @ X2 @ Y3 ) @ ( real_V4546457046886955230omplex @ R2 ) )
% 8.90/5.13 = ( complex2 @ ( times_times_real @ X2 @ R2 ) @ ( times_times_real @ Y3 @ R2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8576_real__sqrt__power,axiom,
% 8.90/5.13 ! [X2: real,K: nat] :
% 8.90/5.13 ( ( sqrt @ ( power_power_real @ X2 @ K ) )
% 8.90/5.13 = ( power_power_real @ ( sqrt @ X2 ) @ K ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8577_real__sqrt__mult,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( sqrt @ ( times_times_real @ X2 @ Y3 ) )
% 8.90/5.13 = ( times_times_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8578_real__sqrt__divide,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( sqrt @ ( divide_divide_real @ X2 @ Y3 ) )
% 8.90/5.13 = ( divide_divide_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8579_norm__exp,axiom,
% 8.90/5.13 ! [X2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X2 ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8580_norm__exp,axiom,
% 8.90/5.13 ! [X2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X2 ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8581_exp__times__arg__commute,axiom,
% 8.90/5.13 ! [A2: complex] :
% 8.90/5.13 ( ( times_times_complex @ ( exp_complex @ A2 ) @ A2 )
% 8.90/5.13 = ( times_times_complex @ A2 @ ( exp_complex @ A2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8582_exp__times__arg__commute,axiom,
% 8.90/5.13 ! [A2: real] :
% 8.90/5.13 ( ( times_times_real @ ( exp_real @ A2 ) @ A2 )
% 8.90/5.13 = ( times_times_real @ A2 @ ( exp_real @ A2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8583_complex__diff,axiom,
% 8.90/5.13 ! [A: real,B: real,C: real,D: real] :
% 8.90/5.13 ( ( minus_minus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 8.90/5.13 = ( complex2 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8584_real__sqrt__gt__zero,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ord_less_real @ zero_zero_real @ ( sqrt @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8585_real__sqrt__ge__zero,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8586_real__sqrt__eq__zero__cancel,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ( sqrt @ X2 )
% 8.90/5.13 = zero_zero_real )
% 8.90/5.13 => ( X2 = zero_zero_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8587_not__exp__le__zero,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ~ ( ord_less_eq_real @ ( exp_real @ X2 ) @ zero_zero_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8588_exp__ge__zero,axiom,
% 8.90/5.13 ! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8589_real__sqrt__ge__one,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 8.90/5.13 => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8590_mult__exp__exp,axiom,
% 8.90/5.13 ! [X2: complex,Y3: complex] :
% 8.90/5.13 ( ( times_times_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ Y3 ) )
% 8.90/5.13 = ( exp_complex @ ( plus_plus_complex @ X2 @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8591_mult__exp__exp,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( times_times_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) )
% 8.90/5.13 = ( exp_real @ ( plus_plus_real @ X2 @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8592_exp__add__commuting,axiom,
% 8.90/5.13 ! [X2: complex,Y3: complex] :
% 8.90/5.13 ( ( ( times_times_complex @ X2 @ Y3 )
% 8.90/5.13 = ( times_times_complex @ Y3 @ X2 ) )
% 8.90/5.13 => ( ( exp_complex @ ( plus_plus_complex @ X2 @ Y3 ) )
% 8.90/5.13 = ( times_times_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8593_exp__add__commuting,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ( times_times_real @ X2 @ Y3 )
% 8.90/5.13 = ( times_times_real @ Y3 @ X2 ) )
% 8.90/5.13 => ( ( exp_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 8.90/5.13 = ( times_times_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8594_exp__diff,axiom,
% 8.90/5.13 ! [X2: complex,Y3: complex] :
% 8.90/5.13 ( ( exp_complex @ ( minus_minus_complex @ X2 @ Y3 ) )
% 8.90/5.13 = ( divide1717551699836669952omplex @ ( exp_complex @ X2 ) @ ( exp_complex @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8595_exp__diff,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( exp_real @ ( minus_minus_real @ X2 @ Y3 ) )
% 8.90/5.13 = ( divide_divide_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8596_Complex__eq__numeral,axiom,
% 8.90/5.13 ! [A: real,B: real,W: num] :
% 8.90/5.13 ( ( ( complex2 @ A @ B )
% 8.90/5.13 = ( numera6690914467698888265omplex @ W ) )
% 8.90/5.13 = ( ( A
% 8.90/5.13 = ( numeral_numeral_real @ W ) )
% 8.90/5.13 & ( B = zero_zero_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8597_complex__norm,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( real_V1022390504157884413omplex @ ( complex2 @ X2 @ Y3 ) )
% 8.90/5.13 = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8598_nonzero__of__real__divide,axiom,
% 8.90/5.13 ! [Y3: real,X2: real] :
% 8.90/5.13 ( ( Y3 != zero_zero_real )
% 8.90/5.13 => ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X2 @ Y3 ) )
% 8.90/5.13 = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X2 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8599_nonzero__of__real__divide,axiom,
% 8.90/5.13 ! [Y3: real,X2: real] :
% 8.90/5.13 ( ( Y3 != zero_zero_real )
% 8.90/5.13 => ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X2 @ Y3 ) )
% 8.90/5.13 = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X2 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8600_exp__gt__one,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ord_less_real @ one_one_real @ ( exp_real @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8601_real__div__sqrt,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( divide_divide_real @ X2 @ ( sqrt @ X2 ) )
% 8.90/5.13 = ( sqrt @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8602_sqrt__add__le__add__sqrt,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.13 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8603_exp__ge__add__one__self,axiom,
% 8.90/5.13 ! [X2: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( exp_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8604_le__real__sqrt__sumsq,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] : ( ord_less_eq_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8605_exp__minus__inverse,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( times_times_real @ ( exp_real @ X2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) )
% 8.90/5.13 = one_one_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8606_exp__minus__inverse,axiom,
% 8.90/5.13 ! [X2: complex] :
% 8.90/5.13 ( ( times_times_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X2 ) ) )
% 8.90/5.13 = one_one_complex ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8607_exp__of__nat__mult,axiom,
% 8.90/5.13 ! [N: nat,X2: complex] :
% 8.90/5.13 ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ X2 ) )
% 8.90/5.13 = ( power_power_complex @ ( exp_complex @ X2 ) @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8608_exp__of__nat__mult,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) )
% 8.90/5.13 = ( power_power_real @ ( exp_real @ X2 ) @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8609_exp__of__nat2__mult,axiom,
% 8.90/5.13 ! [X2: complex,N: nat] :
% 8.90/5.13 ( ( exp_complex @ ( times_times_complex @ X2 @ ( semiri8010041392384452111omplex @ N ) ) )
% 8.90/5.13 = ( power_power_complex @ ( exp_complex @ X2 ) @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8610_exp__of__nat2__mult,axiom,
% 8.90/5.13 ! [X2: real,N: nat] :
% 8.90/5.13 ( ( exp_real @ ( times_times_real @ X2 @ ( semiri5074537144036343181t_real @ N ) ) )
% 8.90/5.13 = ( power_power_real @ ( exp_real @ X2 ) @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8611_Complex__eq__neg__numeral,axiom,
% 8.90/5.13 ! [A: real,B: real,W: num] :
% 8.90/5.13 ( ( ( complex2 @ A @ B )
% 8.90/5.13 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 8.90/5.13 = ( ( A
% 8.90/5.13 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 8.90/5.13 & ( B = zero_zero_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8612_log__ln,axiom,
% 8.90/5.13 ( ln_ln_real
% 8.90/5.13 = ( log @ ( exp_real @ one_one_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8613_complex__mult,axiom,
% 8.90/5.13 ! [A: real,B: real,C: real,D: real] :
% 8.90/5.13 ( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 8.90/5.13 = ( 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 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8614_one__complex_Ocode,axiom,
% 8.90/5.13 ( one_one_complex
% 8.90/5.13 = ( complex2 @ one_one_real @ zero_zero_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8615_Complex__eq__1,axiom,
% 8.90/5.13 ! [A: real,B: real] :
% 8.90/5.13 ( ( ( complex2 @ A @ B )
% 8.90/5.13 = one_one_complex )
% 8.90/5.13 = ( ( A = one_one_real )
% 8.90/5.13 & ( B = zero_zero_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8616_norm__less__p1,axiom,
% 8.90/5.13 ! [X2: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X2 ) ) @ one_one_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8617_norm__less__p1,axiom,
% 8.90/5.13 ! [X2: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X2 ) ) @ one_one_complex ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8618_sqrt2__less__2,axiom,
% 8.90/5.13 ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8619_exp__ge__add__one__self__aux,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( exp_real @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8620_lemma__exp__total,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ one_one_real @ Y3 )
% 8.90/5.13 => ? [X5: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 8.90/5.13 & ( ord_less_eq_real @ X5 @ ( minus_minus_real @ Y3 @ one_one_real ) )
% 8.90/5.13 & ( ( exp_real @ X5 )
% 8.90/5.13 = Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8621_ln__ge__iff,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ ( ln_ln_real @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_real @ ( exp_real @ Y3 ) @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8622_ln__x__over__x__mono,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 8.90/5.13 => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y3 ) @ Y3 ) @ ( divide_divide_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8623_powr__def,axiom,
% 8.90/5.13 ( powr_real
% 8.90/5.13 = ( ^ [X: real,A3: real] : ( if_real @ ( X = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A3 @ ( ln_ln_real @ X ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8624_Complex__eq__neg__1,axiom,
% 8.90/5.13 ! [A: real,B: real] :
% 8.90/5.13 ( ( ( complex2 @ A @ B )
% 8.90/5.13 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 8.90/5.13 = ( ( A
% 8.90/5.13 = ( uminus_uminus_real @ one_one_real ) )
% 8.90/5.13 & ( B = zero_zero_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8625_real__less__rsqrt,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 8.90/5.13 => ( ord_less_real @ X2 @ ( sqrt @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8626_sqrt__le__D,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ Y3 )
% 8.90/5.13 => ( ord_less_eq_real @ X2 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8627_real__le__rsqrt,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 8.90/5.13 => ( ord_less_eq_real @ X2 @ ( sqrt @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8628_exp__le,axiom,
% 8.90/5.13 ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8629_norm__of__real__diff,axiom,
% 8.90/5.13 ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( real_V1803761363581548252l_real @ B ) @ ( real_V1803761363581548252l_real @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8630_norm__of__real__diff,axiom,
% 8.90/5.13 ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( real_V4546457046886955230omplex @ B ) @ ( real_V4546457046886955230omplex @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8631_exp__divide__power__eq,axiom,
% 8.90/5.13 ! [N: nat,X2: complex] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X2 @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
% 8.90/5.13 = ( exp_complex @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8632_exp__divide__power__eq,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
% 8.90/5.13 = ( exp_real @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8633_tanh__altdef,axiom,
% 8.90/5.13 ( tanh_real
% 8.90/5.13 = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8634_tanh__altdef,axiom,
% 8.90/5.13 ( tanh_complex
% 8.90/5.13 = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8635_real__le__lsqrt,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ X2 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ord_less_eq_real @ ( sqrt @ X2 ) @ Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8636_real__sqrt__unique,axiom,
% 8.90/5.13 ! [Y3: real,X2: real] :
% 8.90/5.13 ( ( ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.13 = X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.13 => ( ( sqrt @ X2 )
% 8.90/5.13 = Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8637_lemma__real__divide__sqrt__less,axiom,
% 8.90/5.13 ! [U: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ U )
% 8.90/5.13 => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8638_real__sqrt__sum__squares__eq__cancel,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.13 = X2 )
% 8.90/5.13 => ( Y3 = zero_zero_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8639_real__sqrt__sum__squares__eq__cancel2,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.13 = Y3 )
% 8.90/5.13 => ( X2 = zero_zero_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8640_real__sqrt__sum__squares__ge1,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] : ( ord_less_eq_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8641_real__sqrt__sum__squares__ge2,axiom,
% 8.90/5.13 ! [Y3: real,X2: real] : ( ord_less_eq_real @ Y3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8642_real__sqrt__sum__squares__triangle__ineq,axiom,
% 8.90/5.13 ! [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 ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8643_exp__half__le2,axiom,
% 8.90/5.13 ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8644_sqrt__ge__absD,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( sqrt @ Y3 ) )
% 8.90/5.13 => ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8645_cos__45,axiom,
% 8.90/5.13 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 8.90/5.13 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8646_sin__45,axiom,
% 8.90/5.13 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 8.90/5.13 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8647_tan__60,axiom,
% 8.90/5.13 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 8.90/5.13 = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8648_cos__int__times__real,axiom,
% 8.90/5.13 ! [M: int,X2: real] :
% 8.90/5.13 ( ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X2 ) ) )
% 8.90/5.13 = ( real_V1803761363581548252l_real @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8649_cos__int__times__real,axiom,
% 8.90/5.13 ! [M: int,X2: real] :
% 8.90/5.13 ( ( cos_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X2 ) ) )
% 8.90/5.13 = ( real_V4546457046886955230omplex @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8650_sin__int__times__real,axiom,
% 8.90/5.13 ! [M: int,X2: real] :
% 8.90/5.13 ( ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X2 ) ) )
% 8.90/5.13 = ( real_V1803761363581548252l_real @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8651_sin__int__times__real,axiom,
% 8.90/5.13 ! [M: int,X2: real] :
% 8.90/5.13 ( ( sin_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X2 ) ) )
% 8.90/5.13 = ( real_V4546457046886955230omplex @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8652_exp__double,axiom,
% 8.90/5.13 ! [Z: complex] :
% 8.90/5.13 ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) )
% 8.90/5.13 = ( power_power_complex @ ( exp_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8653_exp__double,axiom,
% 8.90/5.13 ! [Z: real] :
% 8.90/5.13 ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) )
% 8.90/5.13 = ( power_power_real @ ( exp_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8654_real__less__lsqrt,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ord_less_real @ ( sqrt @ X2 ) @ Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8655_sqrt__sum__squares__le__sum,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.13 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X2 @ Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8656_sqrt__even__pow2,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.13 => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.13 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8657_real__sqrt__ge__abs1,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8658_real__sqrt__ge__abs2,axiom,
% 8.90/5.13 ! [Y3: real,X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8659_sqrt__sum__squares__le__sum__abs,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8660_ln__sqrt,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ln_ln_real @ ( sqrt @ X2 ) )
% 8.90/5.13 = ( divide_divide_real @ ( ln_ln_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8661_cos__30,axiom,
% 8.90/5.13 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 8.90/5.13 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8662_sin__60,axiom,
% 8.90/5.13 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 8.90/5.13 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8663_exp__bound__half,axiom,
% 8.90/5.13 ! [Z: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8664_exp__bound__half,axiom,
% 8.90/5.13 ! [Z: complex] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8665_sin__cos__eq,axiom,
% 8.90/5.13 ( sin_real
% 8.90/5.13 = ( ^ [X: real] : ( cos_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8666_sin__cos__eq,axiom,
% 8.90/5.13 ( sin_complex
% 8.90/5.13 = ( ^ [X: complex] : ( cos_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8667_cos__sin__eq,axiom,
% 8.90/5.13 ( cos_real
% 8.90/5.13 = ( ^ [X: real] : ( sin_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8668_cos__sin__eq,axiom,
% 8.90/5.13 ( cos_complex
% 8.90/5.13 = ( ^ [X: complex] : ( sin_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8669_arsinh__real__aux,axiom,
% 8.90/5.13 ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8670_real__sqrt__power__even,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.13 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( power_power_real @ ( sqrt @ X2 ) @ N )
% 8.90/5.13 = ( power_power_real @ X2 @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8671_exp__bound,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 8.90/5.13 => ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8672_real__sqrt__sum__squares__mult__ge__zero,axiom,
% 8.90/5.13 ! [X2: real,Y3: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8673_arith__geo__mean__sqrt,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.13 => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X2 @ Y3 ) ) @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8674_powr__half__sqrt,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( powr_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 = ( sqrt @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8675_tan__30,axiom,
% 8.90/5.13 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 8.90/5.13 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8676_minus__sin__cos__eq,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( uminus_uminus_real @ ( sin_real @ X2 ) )
% 8.90/5.13 = ( cos_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8677_minus__sin__cos__eq,axiom,
% 8.90/5.13 ! [X2: complex] :
% 8.90/5.13 ( ( uminus1482373934393186551omplex @ ( sin_complex @ X2 ) )
% 8.90/5.13 = ( cos_complex @ ( plus_plus_complex @ X2 @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8678_real__exp__bound__lemma,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8679_cos__x__y__le__one,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8680_real__sqrt__sum__squares__less,axiom,
% 8.90/5.13 ! [X2: real,U: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 8.90/5.13 => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 8.90/5.13 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8681_arcosh__real__def,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 8.90/5.13 => ( ( arcosh_real @ X2 )
% 8.90/5.13 = ( ln_ln_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8682_exp__ge__one__plus__x__over__n__power__n,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X2 )
% 8.90/5.13 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8683_exp__ge__one__minus__x__over__n__power__n,axiom,
% 8.90/5.13 ! [X2: real,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ N ) )
% 8.90/5.13 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8684_cos__arctan,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( cos_real @ ( arctan @ X2 ) )
% 8.90/5.13 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8685_sin__arctan,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( sin_real @ ( arctan @ X2 ) )
% 8.90/5.13 = ( divide_divide_real @ X2 @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8686_exp__bound__lemma,axiom,
% 8.90/5.13 ! [Z: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( 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 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8687_exp__bound__lemma,axiom,
% 8.90/5.13 ! [Z: complex] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( 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 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8688_Maclaurin__exp__le,axiom,
% 8.90/5.13 ! [X2: real,N: nat] :
% 8.90/5.13 ? [T5: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X2 ) )
% 8.90/5.13 & ( ( exp_real @ X2 )
% 8.90/5.13 = ( plus_plus_real
% 8.90/5.13 @ ( groups6591440286371151544t_real
% 8.90/5.13 @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X2 @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
% 8.90/5.13 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.13 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8689_sqrt__sum__squares__half__less,axiom,
% 8.90/5.13 ! [X2: real,U: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_real @ X2 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.13 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8690_exp__lower__Taylor__quadratic,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( divide_divide_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8691_log__base__10__eq2,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X2 )
% 8.90/5.13 = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8692_sin__cos__sqrt,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X2 ) )
% 8.90/5.13 => ( ( sin_real @ X2 )
% 8.90/5.13 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8693_arctan__half,axiom,
% 8.90/5.13 ( arctan
% 8.90/5.13 = ( ^ [X: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8694_tanh__real__altdef,axiom,
% 8.90/5.13 ( tanh_real
% 8.90/5.13 = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8695_log__base__10__eq1,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X2 )
% 8.90/5.13 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8696_arsinh__def,axiom,
% 8.90/5.13 ( arsinh_real
% 8.90/5.13 = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8697_arsinh__real__def,axiom,
% 8.90/5.13 ( arsinh_real
% 8.90/5.13 = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8698_binomial__code,axiom,
% 8.90/5.13 ( binomial
% 8.90/5.13 = ( ^ [N3: nat,K2: nat] : ( if_nat @ ( ord_less_nat @ N3 @ K2 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( binomial @ N3 @ ( minus_minus_nat @ N3 @ K2 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N3 @ K2 ) @ one_one_nat ) @ N3 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K2 ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8699_cos__arcsin,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 8.90/5.13 => ( ( cos_real @ ( arcsin @ X2 ) )
% 8.90/5.13 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8700_sin__arccos__abs,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 8.90/5.13 => ( ( sin_real @ ( arccos @ Y3 ) )
% 8.90/5.13 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8701_binomial__Suc__n,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( binomial @ ( suc @ N ) @ N )
% 8.90/5.13 = ( suc @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8702_binomial__n__n,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( binomial @ N @ N )
% 8.90/5.13 = one_one_nat ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8703_binomial__1,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
% 8.90/5.13 = N ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8704_binomial__0__Suc,axiom,
% 8.90/5.13 ! [K: nat] :
% 8.90/5.13 ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
% 8.90/5.13 = zero_zero_nat ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8705_binomial__eq__0__iff,axiom,
% 8.90/5.13 ! [N: nat,K: nat] :
% 8.90/5.13 ( ( ( binomial @ N @ K )
% 8.90/5.13 = zero_zero_nat )
% 8.90/5.13 = ( ord_less_nat @ N @ K ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8706_binomial__Suc__Suc,axiom,
% 8.90/5.13 ! [N: nat,K: nat] :
% 8.90/5.13 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 8.90/5.13 = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8707_binomial__n__0,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( binomial @ N @ zero_zero_nat )
% 8.90/5.13 = one_one_nat ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8708_arccos__1,axiom,
% 8.90/5.13 ( ( arccos @ one_one_real )
% 8.90/5.13 = zero_zero_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8709_zero__less__binomial__iff,axiom,
% 8.90/5.13 ! [N: nat,K: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
% 8.90/5.13 = ( ord_less_eq_nat @ K @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8710_arccos__minus__1,axiom,
% 8.90/5.13 ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
% 8.90/5.13 = pi ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8711_cos__arccos,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.90/5.13 => ( ( cos_real @ ( arccos @ Y3 ) )
% 8.90/5.13 = Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8712_sin__arcsin,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.90/5.13 => ( ( sin_real @ ( arcsin @ Y3 ) )
% 8.90/5.13 = Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8713_arccos__0,axiom,
% 8.90/5.13 ( ( arccos @ zero_zero_real )
% 8.90/5.13 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8714_arcsin__1,axiom,
% 8.90/5.13 ( ( arcsin @ one_one_real )
% 8.90/5.13 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8715_arcsin__minus__1,axiom,
% 8.90/5.13 ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
% 8.90/5.13 = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8716_complex__exp__exists,axiom,
% 8.90/5.13 ! [Z: complex] :
% 8.90/5.13 ? [A4: complex,R3: real] :
% 8.90/5.13 ( Z
% 8.90/5.13 = ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( exp_complex @ A4 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8717_choose__one,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( binomial @ N @ one_one_nat )
% 8.90/5.13 = N ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8718_binomial__eq__0,axiom,
% 8.90/5.13 ! [N: nat,K: nat] :
% 8.90/5.13 ( ( ord_less_nat @ N @ K )
% 8.90/5.13 => ( ( binomial @ N @ K )
% 8.90/5.13 = zero_zero_nat ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8719_Suc__times__binomial,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
% 8.90/5.13 = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8720_Suc__times__binomial__eq,axiom,
% 8.90/5.13 ! [N: nat,K: nat] :
% 8.90/5.13 ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
% 8.90/5.13 = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8721_binomial__symmetric,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( ( binomial @ N @ K )
% 8.90/5.13 = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8722_choose__mult__lemma,axiom,
% 8.90/5.13 ! [M: nat,R2: nat,K: nat] :
% 8.90/5.13 ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
% 8.90/5.13 = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8723_binomial__le__pow,axiom,
% 8.90/5.13 ! [R2: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ R2 @ N )
% 8.90/5.13 => ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8724_zero__less__binomial,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8725_Suc__times__binomial__add,axiom,
% 8.90/5.13 ! [A: nat,B: nat] :
% 8.90/5.13 ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
% 8.90/5.13 = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8726_binomial__Suc__Suc__eq__times,axiom,
% 8.90/5.13 ! [N: nat,K: nat] :
% 8.90/5.13 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 8.90/5.13 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8727_choose__mult,axiom,
% 8.90/5.13 ! [K: nat,M: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ M )
% 8.90/5.13 => ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.13 => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 8.90/5.13 = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8728_binomial__absorb__comp,axiom,
% 8.90/5.13 ! [N: nat,K: nat] :
% 8.90/5.13 ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
% 8.90/5.13 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8729_arccos__le__arccos,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.90/5.13 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( arccos @ X2 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8730_arccos__eq__iff,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 8.90/5.13 & ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real ) )
% 8.90/5.13 => ( ( ( arccos @ X2 )
% 8.90/5.13 = ( arccos @ Y3 ) )
% 8.90/5.13 = ( X2 = Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8731_arccos__le__mono,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( arccos @ X2 ) @ ( arccos @ Y3 ) )
% 8.90/5.13 = ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8732_arcsin__le__arcsin,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.90/5.13 => ( ord_less_eq_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8733_arcsin__minus,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 8.90/5.13 => ( ( arcsin @ ( uminus_uminus_real @ X2 ) )
% 8.90/5.13 = ( uminus_uminus_real @ ( arcsin @ X2 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8734_arcsin__le__mono,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) )
% 8.90/5.13 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8735_arcsin__eq__iff,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 8.90/5.13 => ( ( ( arcsin @ X2 )
% 8.90/5.13 = ( arcsin @ Y3 ) )
% 8.90/5.13 = ( X2 = Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8736_binomial__absorption,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
% 8.90/5.13 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8737_binomial__fact__lemma,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 8.90/5.13 = ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8738_binomial__ge__n__over__k__pow__k,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8739_binomial__ge__n__over__k__pow__k,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8740_binomial__maximum_H,axiom,
% 8.90/5.13 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8741_binomial__mono,axiom,
% 8.90/5.13 ! [K: nat,K6: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ K6 )
% 8.90/5.13 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 8.90/5.13 => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8742_binomial__antimono,axiom,
% 8.90/5.13 ! [K: nat,K6: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ K6 )
% 8.90/5.13 => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 8.90/5.13 => ( ( ord_less_eq_nat @ K6 @ N )
% 8.90/5.13 => ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8743_binomial__maximum,axiom,
% 8.90/5.13 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8744_binomial__le__pow2,axiom,
% 8.90/5.13 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8745_choose__reduce__nat,axiom,
% 8.90/5.13 ! [N: nat,K: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.90/5.13 => ( ( binomial @ N @ K )
% 8.90/5.13 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8746_arccos__lbound,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.90/5.13 => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8747_arccos__less__arccos,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.90/5.13 => ( ord_less_real @ ( arccos @ Y3 ) @ ( arccos @ X2 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8748_times__binomial__minus1__eq,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.90/5.13 => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
% 8.90/5.13 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8749_arccos__less__mono,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 8.90/5.13 => ( ( ord_less_real @ ( arccos @ X2 ) @ ( arccos @ Y3 ) )
% 8.90/5.13 = ( ord_less_real @ Y3 @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8750_arccos__ubound,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.90/5.13 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8751_arccos__cos,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ X2 @ pi )
% 8.90/5.13 => ( ( arccos @ ( cos_real @ X2 ) )
% 8.90/5.13 = X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8752_arcsin__less__arcsin,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.90/5.13 => ( ord_less_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8753_arcsin__less__mono,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 8.90/5.13 => ( ( ord_less_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) )
% 8.90/5.13 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8754_cos__arccos__abs,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 8.90/5.13 => ( ( cos_real @ ( arccos @ Y3 ) )
% 8.90/5.13 = Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8755_arccos__cos__eq__abs,axiom,
% 8.90/5.13 ! [Theta: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
% 8.90/5.13 => ( ( arccos @ ( cos_real @ Theta ) )
% 8.90/5.13 = ( abs_abs_real @ Theta ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8756_binomial__altdef__nat,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( ( binomial @ N @ K )
% 8.90/5.13 = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8757_binomial__less__binomial__Suc,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8758_binomial__strict__antimono,axiom,
% 8.90/5.13 ! [K: nat,K6: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ K @ K6 )
% 8.90/5.13 => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 8.90/5.13 => ( ( ord_less_eq_nat @ K6 @ N )
% 8.90/5.13 => ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8759_binomial__strict__mono,axiom,
% 8.90/5.13 ! [K: nat,K6: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ K @ K6 )
% 8.90/5.13 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 8.90/5.13 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8760_central__binomial__odd,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.13 => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.13 = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8761_binomial__addition__formula,axiom,
% 8.90/5.13 ! [N: nat,K: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( binomial @ N @ ( suc @ K ) )
% 8.90/5.13 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8762_binomial__fact,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 8.90/5.13 = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8763_binomial__fact,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 8.90/5.13 = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8764_binomial__fact,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 8.90/5.13 = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8765_fact__binomial,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
% 8.90/5.13 = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8766_fact__binomial,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
% 8.90/5.13 = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8767_fact__binomial,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
% 8.90/5.13 = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8768_arccos__lt__bounded,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 8.90/5.13 => ( ( ord_less_real @ Y3 @ one_one_real )
% 8.90/5.13 => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 8.90/5.13 & ( ord_less_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8769_arccos__bounded,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.90/5.13 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 8.90/5.13 & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8770_sin__arccos__nonzero,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ one_one_real )
% 8.90/5.13 => ( ( sin_real @ ( arccos @ X2 ) )
% 8.90/5.13 != zero_zero_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8771_arccos__cos2,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X2 )
% 8.90/5.13 => ( ( arccos @ ( cos_real @ X2 ) )
% 8.90/5.13 = ( uminus_uminus_real @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8772_arccos__minus,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 8.90/5.13 => ( ( arccos @ ( uminus_uminus_real @ X2 ) )
% 8.90/5.13 = ( minus_minus_real @ pi @ ( arccos @ X2 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8773_cos__arcsin__nonzero,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ one_one_real )
% 8.90/5.13 => ( ( cos_real @ ( arcsin @ X2 ) )
% 8.90/5.13 != zero_zero_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8774_choose__two,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.13 = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8775_arccos,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.90/5.13 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 8.90/5.13 & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi )
% 8.90/5.13 & ( ( cos_real @ ( arccos @ Y3 ) )
% 8.90/5.13 = Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8776_arccos__minus__abs,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 8.90/5.13 => ( ( arccos @ ( uminus_uminus_real @ X2 ) )
% 8.90/5.13 = ( minus_minus_real @ pi @ ( arccos @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8777_arccos__le__pi2,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.90/5.13 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8778_arcsin__lt__bounded,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 8.90/5.13 => ( ( ord_less_real @ Y3 @ one_one_real )
% 8.90/5.13 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 8.90/5.13 & ( ord_less_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8779_arcsin__lbound,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.90/5.13 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8780_arcsin__ubound,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.90/5.13 => ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8781_arcsin__bounded,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 8.90/5.13 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8782_arcsin__sin,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ( arcsin @ ( sin_real @ X2 ) )
% 8.90/5.13 = X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8783_arcsin,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 8.90/5.13 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 & ( ( sin_real @ ( arcsin @ Y3 ) )
% 8.90/5.13 = Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8784_arcsin__pi,axiom,
% 8.90/5.13 ! [Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 8.90/5.13 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ pi )
% 8.90/5.13 & ( ( sin_real @ ( arcsin @ Y3 ) )
% 8.90/5.13 = Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8785_arcsin__le__iff,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( arcsin @ X2 ) @ Y3 )
% 8.90/5.13 = ( ord_less_eq_real @ X2 @ ( sin_real @ Y3 ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8786_le__arcsin__iff,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ ( arcsin @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_real @ ( sin_real @ Y3 ) @ X2 ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8787_arccos__cos__eq__abs__2pi,axiom,
% 8.90/5.13 ! [Theta: real] :
% 8.90/5.13 ~ ! [K3: int] :
% 8.90/5.13 ( ( arccos @ ( cos_real @ Theta ) )
% 8.90/5.13 != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8788_central__binomial__lower__bound,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8789_sin__arccos,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 8.90/5.13 => ( ( sin_real @ ( arccos @ X2 ) )
% 8.90/5.13 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8790_choose__odd__sum,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 8.90/5.13 @ ( groups2073611262835488442omplex
% 8.90/5.13 @ ^ [I4: nat] :
% 8.90/5.13 ( if_complex
% 8.90/5.13 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 8.90/5.13 @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) )
% 8.90/5.13 @ zero_zero_complex )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) ) )
% 8.90/5.13 = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8791_choose__odd__sum,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 8.90/5.13 @ ( groups3539618377306564664at_int
% 8.90/5.13 @ ^ [I4: nat] :
% 8.90/5.13 ( if_int
% 8.90/5.13 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 8.90/5.13 @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) )
% 8.90/5.13 @ zero_zero_int )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) ) )
% 8.90/5.13 = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8792_choose__odd__sum,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 8.90/5.13 @ ( groups2906978787729119204at_rat
% 8.90/5.13 @ ^ [I4: nat] :
% 8.90/5.13 ( if_rat
% 8.90/5.13 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 8.90/5.13 @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) )
% 8.90/5.13 @ zero_zero_rat )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) ) )
% 8.90/5.13 = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8793_choose__odd__sum,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 8.90/5.13 @ ( groups6591440286371151544t_real
% 8.90/5.13 @ ^ [I4: nat] :
% 8.90/5.13 ( if_real
% 8.90/5.13 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 8.90/5.13 @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) )
% 8.90/5.13 @ zero_zero_real )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) ) )
% 8.90/5.13 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8794_choose__even__sum,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 8.90/5.13 @ ( groups2073611262835488442omplex
% 8.90/5.13 @ ^ [I4: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) @ zero_zero_complex )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) ) )
% 8.90/5.13 = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8795_choose__even__sum,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 8.90/5.13 @ ( groups3539618377306564664at_int
% 8.90/5.13 @ ^ [I4: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) @ zero_zero_int )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) ) )
% 8.90/5.13 = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8796_choose__even__sum,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 8.90/5.13 @ ( groups2906978787729119204at_rat
% 8.90/5.13 @ ^ [I4: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) @ zero_zero_rat )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) ) )
% 8.90/5.13 = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8797_choose__even__sum,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 8.90/5.13 @ ( groups6591440286371151544t_real
% 8.90/5.13 @ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) @ zero_zero_real )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) ) )
% 8.90/5.13 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8798_pochhammer__double,axiom,
% 8.90/5.13 ! [Z: complex,N: nat] :
% 8.90/5.13 ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.13 = ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8799_pochhammer__double,axiom,
% 8.90/5.13 ! [Z: real,N: nat] :
% 8.90/5.13 ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.13 = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8800_pochhammer__double,axiom,
% 8.90/5.13 ! [Z: rat,N: nat] :
% 8.90/5.13 ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.13 = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8801_floor__log__nat__eq__powr__iff,axiom,
% 8.90/5.13 ! [B: nat,K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 8.90/5.13 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 8.90/5.13 => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 8.90/5.13 = ( semiri1314217659103216013at_int @ N ) )
% 8.90/5.13 = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 8.90/5.13 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8802_of__nat__code,axiom,
% 8.90/5.13 ( semiri8010041392384452111omplex
% 8.90/5.13 = ( ^ [N3: nat] :
% 8.90/5.13 ( semiri2816024913162550771omplex
% 8.90/5.13 @ ^ [I4: complex] : ( plus_plus_complex @ I4 @ one_one_complex )
% 8.90/5.13 @ N3
% 8.90/5.13 @ zero_zero_complex ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8803_of__nat__code,axiom,
% 8.90/5.13 ( semiri1314217659103216013at_int
% 8.90/5.13 = ( ^ [N3: nat] :
% 8.90/5.13 ( semiri8420488043553186161ux_int
% 8.90/5.13 @ ^ [I4: int] : ( plus_plus_int @ I4 @ one_one_int )
% 8.90/5.13 @ N3
% 8.90/5.13 @ zero_zero_int ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8804_of__nat__code,axiom,
% 8.90/5.13 ( semiri5074537144036343181t_real
% 8.90/5.13 = ( ^ [N3: nat] :
% 8.90/5.13 ( semiri7260567687927622513x_real
% 8.90/5.13 @ ^ [I4: real] : ( plus_plus_real @ I4 @ one_one_real )
% 8.90/5.13 @ N3
% 8.90/5.13 @ zero_zero_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8805_of__nat__code,axiom,
% 8.90/5.13 ( semiri1316708129612266289at_nat
% 8.90/5.13 = ( ^ [N3: nat] :
% 8.90/5.13 ( semiri8422978514062236437ux_nat
% 8.90/5.13 @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ one_one_nat )
% 8.90/5.13 @ N3
% 8.90/5.13 @ zero_zero_nat ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8806_of__nat__code,axiom,
% 8.90/5.13 ( semiri681578069525770553at_rat
% 8.90/5.13 = ( ^ [N3: nat] :
% 8.90/5.13 ( semiri7787848453975740701ux_rat
% 8.90/5.13 @ ^ [I4: rat] : ( plus_plus_rat @ I4 @ one_one_rat )
% 8.90/5.13 @ N3
% 8.90/5.13 @ zero_zero_rat ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8807_atMost__iff,axiom,
% 8.90/5.13 ! [I2: real,K: real] :
% 8.90/5.13 ( ( member_real @ I2 @ ( set_ord_atMost_real @ K ) )
% 8.90/5.13 = ( ord_less_eq_real @ I2 @ K ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8808_atMost__iff,axiom,
% 8.90/5.13 ! [I2: set_int,K: set_int] :
% 8.90/5.13 ( ( member_set_int @ I2 @ ( set_or58775011639299419et_int @ K ) )
% 8.90/5.13 = ( ord_less_eq_set_int @ I2 @ K ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8809_atMost__iff,axiom,
% 8.90/5.13 ! [I2: rat,K: rat] :
% 8.90/5.13 ( ( member_rat @ I2 @ ( set_ord_atMost_rat @ K ) )
% 8.90/5.13 = ( ord_less_eq_rat @ I2 @ K ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8810_atMost__iff,axiom,
% 8.90/5.13 ! [I2: num,K: num] :
% 8.90/5.13 ( ( member_num @ I2 @ ( set_ord_atMost_num @ K ) )
% 8.90/5.13 = ( ord_less_eq_num @ I2 @ K ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8811_atMost__iff,axiom,
% 8.90/5.13 ! [I2: nat,K: nat] :
% 8.90/5.13 ( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K ) )
% 8.90/5.13 = ( ord_less_eq_nat @ I2 @ K ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8812_atMost__iff,axiom,
% 8.90/5.13 ! [I2: int,K: int] :
% 8.90/5.13 ( ( member_int @ I2 @ ( set_ord_atMost_int @ K ) )
% 8.90/5.13 = ( ord_less_eq_int @ I2 @ K ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8813_of__int__floor__cancel,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X2 ) )
% 8.90/5.13 = X2 )
% 8.90/5.13 = ( ? [N3: int] :
% 8.90/5.13 ( X2
% 8.90/5.13 = ( ring_1_of_int_real @ N3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8814_of__int__floor__cancel,axiom,
% 8.90/5.13 ! [X2: rat] :
% 8.90/5.13 ( ( ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X2 ) )
% 8.90/5.13 = X2 )
% 8.90/5.13 = ( ? [N3: int] :
% 8.90/5.13 ( X2
% 8.90/5.13 = ( ring_1_of_int_rat @ N3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8815_atMost__subset__iff,axiom,
% 8.90/5.13 ! [X2: set_int,Y3: set_int] :
% 8.90/5.13 ( ( ord_le4403425263959731960et_int @ ( set_or58775011639299419et_int @ X2 ) @ ( set_or58775011639299419et_int @ Y3 ) )
% 8.90/5.13 = ( ord_less_eq_set_int @ X2 @ Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8816_atMost__subset__iff,axiom,
% 8.90/5.13 ! [X2: rat,Y3: rat] :
% 8.90/5.13 ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X2 ) @ ( set_ord_atMost_rat @ Y3 ) )
% 8.90/5.13 = ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8817_atMost__subset__iff,axiom,
% 8.90/5.13 ! [X2: num,Y3: num] :
% 8.90/5.13 ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X2 ) @ ( set_ord_atMost_num @ Y3 ) )
% 8.90/5.13 = ( ord_less_eq_num @ X2 @ Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8818_atMost__subset__iff,axiom,
% 8.90/5.13 ! [X2: nat,Y3: nat] :
% 8.90/5.13 ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X2 ) @ ( set_ord_atMost_nat @ Y3 ) )
% 8.90/5.13 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8819_atMost__subset__iff,axiom,
% 8.90/5.13 ! [X2: int,Y3: int] :
% 8.90/5.13 ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X2 ) @ ( set_ord_atMost_int @ Y3 ) )
% 8.90/5.13 = ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8820_floor__numeral,axiom,
% 8.90/5.13 ! [V: num] :
% 8.90/5.13 ( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V ) )
% 8.90/5.13 = ( numeral_numeral_int @ V ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8821_floor__numeral,axiom,
% 8.90/5.13 ! [V: num] :
% 8.90/5.13 ( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
% 8.90/5.13 = ( numeral_numeral_int @ V ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8822_floor__one,axiom,
% 8.90/5.13 ( ( archim6058952711729229775r_real @ one_one_real )
% 8.90/5.13 = one_one_int ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8823_floor__one,axiom,
% 8.90/5.13 ( ( archim3151403230148437115or_rat @ one_one_rat )
% 8.90/5.13 = one_one_int ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8824_pochhammer__0,axiom,
% 8.90/5.13 ! [A: complex] :
% 8.90/5.13 ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
% 8.90/5.13 = one_one_complex ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8825_pochhammer__0,axiom,
% 8.90/5.13 ! [A: real] :
% 8.90/5.13 ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
% 8.90/5.13 = one_one_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8826_pochhammer__0,axiom,
% 8.90/5.13 ! [A: rat] :
% 8.90/5.13 ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
% 8.90/5.13 = one_one_rat ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8827_pochhammer__0,axiom,
% 8.90/5.13 ! [A: nat] :
% 8.90/5.13 ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
% 8.90/5.13 = one_one_nat ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8828_pochhammer__0,axiom,
% 8.90/5.13 ! [A: int] :
% 8.90/5.13 ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
% 8.90/5.13 = one_one_int ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8829_Icc__subset__Iic__iff,axiom,
% 8.90/5.13 ! [L: set_int,H2: set_int,H3: set_int] :
% 8.90/5.13 ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ L @ H2 ) @ ( set_or58775011639299419et_int @ H3 ) )
% 8.90/5.13 = ( ~ ( ord_less_eq_set_int @ L @ H2 )
% 8.90/5.13 | ( ord_less_eq_set_int @ H2 @ H3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8830_Icc__subset__Iic__iff,axiom,
% 8.90/5.13 ! [L: rat,H2: rat,H3: rat] :
% 8.90/5.13 ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H2 ) @ ( set_ord_atMost_rat @ H3 ) )
% 8.90/5.13 = ( ~ ( ord_less_eq_rat @ L @ H2 )
% 8.90/5.13 | ( ord_less_eq_rat @ H2 @ H3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8831_Icc__subset__Iic__iff,axiom,
% 8.90/5.13 ! [L: num,H2: num,H3: num] :
% 8.90/5.13 ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L @ H2 ) @ ( set_ord_atMost_num @ H3 ) )
% 8.90/5.13 = ( ~ ( ord_less_eq_num @ L @ H2 )
% 8.90/5.13 | ( ord_less_eq_num @ H2 @ H3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8832_Icc__subset__Iic__iff,axiom,
% 8.90/5.13 ! [L: nat,H2: nat,H3: nat] :
% 8.90/5.13 ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H2 ) @ ( set_ord_atMost_nat @ H3 ) )
% 8.90/5.13 = ( ~ ( ord_less_eq_nat @ L @ H2 )
% 8.90/5.13 | ( ord_less_eq_nat @ H2 @ H3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8833_Icc__subset__Iic__iff,axiom,
% 8.90/5.13 ! [L: int,H2: int,H3: int] :
% 8.90/5.13 ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H2 ) @ ( set_ord_atMost_int @ H3 ) )
% 8.90/5.13 = ( ~ ( ord_less_eq_int @ L @ H2 )
% 8.90/5.13 | ( ord_less_eq_int @ H2 @ H3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8834_Icc__subset__Iic__iff,axiom,
% 8.90/5.13 ! [L: real,H2: real,H3: real] :
% 8.90/5.13 ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H2 ) @ ( set_ord_atMost_real @ H3 ) )
% 8.90/5.13 = ( ~ ( ord_less_eq_real @ L @ H2 )
% 8.90/5.13 | ( ord_less_eq_real @ H2 @ H3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8835_sum_OatMost__Suc,axiom,
% 8.90/5.13 ! [G: nat > rat,N: nat] :
% 8.90/5.13 ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 8.90/5.13 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8836_sum_OatMost__Suc,axiom,
% 8.90/5.13 ! [G: nat > int,N: nat] :
% 8.90/5.13 ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 8.90/5.13 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8837_sum_OatMost__Suc,axiom,
% 8.90/5.13 ! [G: nat > nat,N: nat] :
% 8.90/5.13 ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 8.90/5.13 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8838_sum_OatMost__Suc,axiom,
% 8.90/5.13 ! [G: nat > real,N: nat] :
% 8.90/5.13 ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 8.90/5.13 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8839_floor__diff__of__int,axiom,
% 8.90/5.13 ! [X2: real,Z: int] :
% 8.90/5.13 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X2 @ ( ring_1_of_int_real @ Z ) ) )
% 8.90/5.13 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X2 ) @ Z ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8840_floor__diff__of__int,axiom,
% 8.90/5.13 ! [X2: rat,Z: int] :
% 8.90/5.13 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) )
% 8.90/5.13 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X2 ) @ Z ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8841_zero__le__floor,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8842_zero__le__floor,axiom,
% 8.90/5.13 ! [X2: rat] :
% 8.90/5.13 ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_rat @ zero_zero_rat @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8843_floor__less__zero,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ zero_zero_int )
% 8.90/5.13 = ( ord_less_real @ X2 @ zero_zero_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8844_floor__less__zero,axiom,
% 8.90/5.13 ! [X2: rat] :
% 8.90/5.13 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X2 ) @ zero_zero_int )
% 8.90/5.13 = ( ord_less_rat @ X2 @ zero_zero_rat ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8845_numeral__le__floor,axiom,
% 8.90/5.13 ! [V: num,X2: real] :
% 8.90/5.13 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8846_numeral__le__floor,axiom,
% 8.90/5.13 ! [V: num,X2: rat] :
% 8.90/5.13 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8847_zero__less__floor,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_real @ one_one_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8848_zero__less__floor,axiom,
% 8.90/5.13 ! [X2: rat] :
% 8.90/5.13 ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_rat @ one_one_rat @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8849_floor__le__zero,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ zero_zero_int )
% 8.90/5.13 = ( ord_less_real @ X2 @ one_one_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8850_floor__le__zero,axiom,
% 8.90/5.13 ! [X2: rat] :
% 8.90/5.13 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ zero_zero_int )
% 8.90/5.13 = ( ord_less_rat @ X2 @ one_one_rat ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8851_floor__less__numeral,axiom,
% 8.90/5.13 ! [X2: real,V: num] :
% 8.90/5.13 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ ( numeral_numeral_int @ V ) )
% 8.90/5.13 = ( ord_less_real @ X2 @ ( numeral_numeral_real @ V ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8852_floor__less__numeral,axiom,
% 8.90/5.13 ! [X2: rat,V: num] :
% 8.90/5.13 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( numeral_numeral_int @ V ) )
% 8.90/5.13 = ( ord_less_rat @ X2 @ ( numeral_numeral_rat @ V ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8853_one__le__floor,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_real @ one_one_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8854_one__le__floor,axiom,
% 8.90/5.13 ! [X2: rat] :
% 8.90/5.13 ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_rat @ one_one_rat @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8855_floor__less__one,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ one_one_int )
% 8.90/5.13 = ( ord_less_real @ X2 @ one_one_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8856_floor__less__one,axiom,
% 8.90/5.13 ! [X2: rat] :
% 8.90/5.13 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X2 ) @ one_one_int )
% 8.90/5.13 = ( ord_less_rat @ X2 @ one_one_rat ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8857_floor__neg__numeral,axiom,
% 8.90/5.13 ! [V: num] :
% 8.90/5.13 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 8.90/5.13 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8858_floor__neg__numeral,axiom,
% 8.90/5.13 ! [V: num] :
% 8.90/5.13 ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 8.90/5.13 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8859_floor__diff__numeral,axiom,
% 8.90/5.13 ! [X2: real,V: num] :
% 8.90/5.13 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X2 @ ( numeral_numeral_real @ V ) ) )
% 8.90/5.13 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X2 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8860_floor__diff__numeral,axiom,
% 8.90/5.13 ! [X2: rat,V: num] :
% 8.90/5.13 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X2 @ ( numeral_numeral_rat @ V ) ) )
% 8.90/5.13 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8861_floor__diff__one,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X2 @ one_one_real ) )
% 8.90/5.13 = ( minus_minus_int @ ( archim6058952711729229775r_real @ X2 ) @ one_one_int ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8862_floor__diff__one,axiom,
% 8.90/5.13 ! [X2: rat] :
% 8.90/5.13 ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X2 @ one_one_rat ) )
% 8.90/5.13 = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X2 ) @ one_one_int ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8863_floor__numeral__power,axiom,
% 8.90/5.13 ! [X2: num,N: nat] :
% 8.90/5.13 ( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
% 8.90/5.13 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8864_floor__numeral__power,axiom,
% 8.90/5.13 ! [X2: num,N: nat] :
% 8.90/5.13 ( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) )
% 8.90/5.13 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8865_floor__divide__eq__div__numeral,axiom,
% 8.90/5.13 ! [A: num,B: num] :
% 8.90/5.13 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 8.90/5.13 = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8866_numeral__less__floor,axiom,
% 8.90/5.13 ! [V: num,X2: real] :
% 8.90/5.13 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8867_numeral__less__floor,axiom,
% 8.90/5.13 ! [V: num,X2: rat] :
% 8.90/5.13 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8868_floor__le__numeral,axiom,
% 8.90/5.13 ! [X2: real,V: num] :
% 8.90/5.13 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ ( numeral_numeral_int @ V ) )
% 8.90/5.13 = ( ord_less_real @ X2 @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8869_floor__le__numeral,axiom,
% 8.90/5.13 ! [X2: rat,V: num] :
% 8.90/5.13 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( numeral_numeral_int @ V ) )
% 8.90/5.13 = ( ord_less_rat @ X2 @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8870_one__less__floor,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8871_one__less__floor,axiom,
% 8.90/5.13 ! [X2: rat] :
% 8.90/5.13 ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8872_floor__le__one,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ one_one_int )
% 8.90/5.13 = ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8873_floor__le__one,axiom,
% 8.90/5.13 ! [X2: rat] :
% 8.90/5.13 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ one_one_int )
% 8.90/5.13 = ( ord_less_rat @ X2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8874_neg__numeral__le__floor,axiom,
% 8.90/5.13 ! [V: num,X2: real] :
% 8.90/5.13 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8875_neg__numeral__le__floor,axiom,
% 8.90/5.13 ! [V: num,X2: rat] :
% 8.90/5.13 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8876_floor__less__neg__numeral,axiom,
% 8.90/5.13 ! [X2: real,V: num] :
% 8.90/5.13 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.90/5.13 = ( ord_less_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8877_floor__less__neg__numeral,axiom,
% 8.90/5.13 ! [X2: rat,V: num] :
% 8.90/5.13 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.90/5.13 = ( ord_less_rat @ X2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8878_floor__one__divide__eq__div__numeral,axiom,
% 8.90/5.13 ! [B: num] :
% 8.90/5.13 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
% 8.90/5.13 = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8879_floor__minus__divide__eq__div__numeral,axiom,
% 8.90/5.13 ! [A: num,B: num] :
% 8.90/5.13 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 8.90/5.13 = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8880_neg__numeral__less__floor,axiom,
% 8.90/5.13 ! [V: num,X2: real] :
% 8.90/5.13 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8881_neg__numeral__less__floor,axiom,
% 8.90/5.13 ! [V: num,X2: rat] :
% 8.90/5.13 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8882_floor__le__neg__numeral,axiom,
% 8.90/5.13 ! [X2: real,V: num] :
% 8.90/5.13 ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.90/5.13 = ( ord_less_real @ X2 @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8883_floor__le__neg__numeral,axiom,
% 8.90/5.13 ! [X2: rat,V: num] :
% 8.90/5.13 ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 8.90/5.13 = ( ord_less_rat @ X2 @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8884_floor__minus__one__divide__eq__div__numeral,axiom,
% 8.90/5.13 ! [B: num] :
% 8.90/5.13 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
% 8.90/5.13 = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8885_atMost__def,axiom,
% 8.90/5.13 ( set_ord_atMost_real
% 8.90/5.13 = ( ^ [U2: real] :
% 8.90/5.13 ( collect_real
% 8.90/5.13 @ ^ [X: real] : ( ord_less_eq_real @ X @ U2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8886_atMost__def,axiom,
% 8.90/5.13 ( set_or58775011639299419et_int
% 8.90/5.13 = ( ^ [U2: set_int] :
% 8.90/5.13 ( collect_set_int
% 8.90/5.13 @ ^ [X: set_int] : ( ord_less_eq_set_int @ X @ U2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8887_atMost__def,axiom,
% 8.90/5.13 ( set_ord_atMost_rat
% 8.90/5.13 = ( ^ [U2: rat] :
% 8.90/5.13 ( collect_rat
% 8.90/5.13 @ ^ [X: rat] : ( ord_less_eq_rat @ X @ U2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8888_atMost__def,axiom,
% 8.90/5.13 ( set_ord_atMost_num
% 8.90/5.13 = ( ^ [U2: num] :
% 8.90/5.13 ( collect_num
% 8.90/5.13 @ ^ [X: num] : ( ord_less_eq_num @ X @ U2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8889_atMost__def,axiom,
% 8.90/5.13 ( set_ord_atMost_nat
% 8.90/5.13 = ( ^ [U2: nat] :
% 8.90/5.13 ( collect_nat
% 8.90/5.13 @ ^ [X: nat] : ( ord_less_eq_nat @ X @ U2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8890_atMost__def,axiom,
% 8.90/5.13 ( set_ord_atMost_int
% 8.90/5.13 = ( ^ [U2: int] :
% 8.90/5.13 ( collect_int
% 8.90/5.13 @ ^ [X: int] : ( ord_less_eq_int @ X @ U2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8891_floor__mono,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ X2 @ Y3 )
% 8.90/5.13 => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim6058952711729229775r_real @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8892_floor__mono,axiom,
% 8.90/5.13 ! [X2: rat,Y3: rat] :
% 8.90/5.13 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 8.90/5.13 => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8893_of__int__floor__le,axiom,
% 8.90/5.13 ! [X2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X2 ) ) @ X2 ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8894_of__int__floor__le,axiom,
% 8.90/5.13 ! [X2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X2 ) ) @ X2 ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8895_floor__less__cancel,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim6058952711729229775r_real @ Y3 ) )
% 8.90/5.13 => ( ord_less_real @ X2 @ Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8896_floor__less__cancel,axiom,
% 8.90/5.13 ! [X2: rat,Y3: rat] :
% 8.90/5.13 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( archim3151403230148437115or_rat @ Y3 ) )
% 8.90/5.13 => ( ord_less_rat @ X2 @ Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8897_floor__le__ceiling,axiom,
% 8.90/5.13 ! [X2: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim7802044766580827645g_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8898_floor__le__ceiling,axiom,
% 8.90/5.13 ! [X2: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( archim2889992004027027881ng_rat @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8899_lessThan__Suc__atMost,axiom,
% 8.90/5.13 ! [K: nat] :
% 8.90/5.13 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 8.90/5.13 = ( set_ord_atMost_nat @ K ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8900_pochhammer__pos,axiom,
% 8.90/5.13 ! [X2: real,N: nat] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X2 @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8901_pochhammer__pos,axiom,
% 8.90/5.13 ! [X2: rat,N: nat] :
% 8.90/5.13 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 8.90/5.13 => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X2 @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8902_pochhammer__pos,axiom,
% 8.90/5.13 ! [X2: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 8.90/5.13 => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X2 @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8903_pochhammer__pos,axiom,
% 8.90/5.13 ! [X2: int,N: nat] :
% 8.90/5.13 ( ( ord_less_int @ zero_zero_int @ X2 )
% 8.90/5.13 => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X2 @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8904_pochhammer__neq__0__mono,axiom,
% 8.90/5.13 ! [A: complex,M: nat,N: nat] :
% 8.90/5.13 ( ( ( comm_s2602460028002588243omplex @ A @ M )
% 8.90/5.13 != zero_zero_complex )
% 8.90/5.13 => ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.13 => ( ( comm_s2602460028002588243omplex @ A @ N )
% 8.90/5.13 != zero_zero_complex ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8905_pochhammer__neq__0__mono,axiom,
% 8.90/5.13 ! [A: real,M: nat,N: nat] :
% 8.90/5.13 ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 8.90/5.13 != zero_zero_real )
% 8.90/5.13 => ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.13 => ( ( comm_s7457072308508201937r_real @ A @ N )
% 8.90/5.13 != zero_zero_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8906_pochhammer__neq__0__mono,axiom,
% 8.90/5.13 ! [A: rat,M: nat,N: nat] :
% 8.90/5.13 ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 8.90/5.13 != zero_zero_rat )
% 8.90/5.13 => ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.13 => ( ( comm_s4028243227959126397er_rat @ A @ N )
% 8.90/5.13 != zero_zero_rat ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8907_pochhammer__eq__0__mono,axiom,
% 8.90/5.13 ! [A: complex,N: nat,M: nat] :
% 8.90/5.13 ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 8.90/5.13 = zero_zero_complex )
% 8.90/5.13 => ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.13 => ( ( comm_s2602460028002588243omplex @ A @ M )
% 8.90/5.13 = zero_zero_complex ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8908_pochhammer__eq__0__mono,axiom,
% 8.90/5.13 ! [A: real,N: nat,M: nat] :
% 8.90/5.13 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 8.90/5.13 = zero_zero_real )
% 8.90/5.13 => ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.13 => ( ( comm_s7457072308508201937r_real @ A @ M )
% 8.90/5.13 = zero_zero_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8909_pochhammer__eq__0__mono,axiom,
% 8.90/5.13 ! [A: rat,N: nat,M: nat] :
% 8.90/5.13 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 8.90/5.13 = zero_zero_rat )
% 8.90/5.13 => ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.13 => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 8.90/5.13 = zero_zero_rat ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8910_floor__le__round,axiom,
% 8.90/5.13 ! [X2: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim8280529875227126926d_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8911_floor__le__round,axiom,
% 8.90/5.13 ! [X2: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( archim7778729529865785530nd_rat @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8912_pochhammer__fact,axiom,
% 8.90/5.13 ( semiri5044797733671781792omplex
% 8.90/5.13 = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8913_pochhammer__fact,axiom,
% 8.90/5.13 ( semiri773545260158071498ct_rat
% 8.90/5.13 = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8914_pochhammer__fact,axiom,
% 8.90/5.13 ( semiri1406184849735516958ct_int
% 8.90/5.13 = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8915_pochhammer__fact,axiom,
% 8.90/5.13 ( semiri2265585572941072030t_real
% 8.90/5.13 = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8916_pochhammer__fact,axiom,
% 8.90/5.13 ( semiri1408675320244567234ct_nat
% 8.90/5.13 = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8917_not__Iic__le__Icc,axiom,
% 8.90/5.13 ! [H2: int,L3: int,H3: int] :
% 8.90/5.13 ~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H2 ) @ ( set_or1266510415728281911st_int @ L3 @ H3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8918_not__Iic__le__Icc,axiom,
% 8.90/5.13 ! [H2: real,L3: real,H3: real] :
% 8.90/5.13 ~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H2 ) @ ( set_or1222579329274155063t_real @ L3 @ H3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8919_finite__nat__iff__bounded__le,axiom,
% 8.90/5.13 ( finite_finite_nat
% 8.90/5.13 = ( ^ [S5: set_nat] :
% 8.90/5.13 ? [K2: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_atMost_nat @ K2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8920_le__floor__iff,axiom,
% 8.90/5.13 ! [Z: int,X2: real] :
% 8.90/5.13 ( ( ord_less_eq_int @ Z @ ( archim6058952711729229775r_real @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8921_le__floor__iff,axiom,
% 8.90/5.13 ! [Z: int,X2: rat] :
% 8.90/5.13 ( ( ord_less_eq_int @ Z @ ( archim3151403230148437115or_rat @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8922_floor__less__iff,axiom,
% 8.90/5.13 ! [X2: real,Z: int] :
% 8.90/5.13 ( ( ord_less_int @ ( archim6058952711729229775r_real @ X2 ) @ Z )
% 8.90/5.13 = ( ord_less_real @ X2 @ ( ring_1_of_int_real @ Z ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8923_floor__less__iff,axiom,
% 8.90/5.13 ! [X2: rat,Z: int] :
% 8.90/5.13 ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X2 ) @ Z )
% 8.90/5.13 = ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8924_le__floor__add,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X2 ) @ ( archim6058952711729229775r_real @ Y3 ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X2 @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8925_le__floor__add,axiom,
% 8.90/5.13 ! [X2: rat,Y3: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X2 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X2 @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8926_floor__add__int,axiom,
% 8.90/5.13 ! [X2: real,Z: int] :
% 8.90/5.13 ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X2 ) @ Z )
% 8.90/5.13 = ( archim6058952711729229775r_real @ ( plus_plus_real @ X2 @ ( ring_1_of_int_real @ Z ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8927_floor__add__int,axiom,
% 8.90/5.13 ! [X2: rat,Z: int] :
% 8.90/5.13 ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X2 ) @ Z )
% 8.90/5.13 = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8928_int__add__floor,axiom,
% 8.90/5.13 ! [Z: int,X2: real] :
% 8.90/5.13 ( ( plus_plus_int @ Z @ ( archim6058952711729229775r_real @ X2 ) )
% 8.90/5.13 = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8929_int__add__floor,axiom,
% 8.90/5.13 ! [Z: int,X2: rat] :
% 8.90/5.13 ( ( plus_plus_int @ Z @ ( archim3151403230148437115or_rat @ X2 ) )
% 8.90/5.13 = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8930_floor__divide__of__int__eq,axiom,
% 8.90/5.13 ! [K: int,L: int] :
% 8.90/5.13 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( ring_1_of_int_real @ K ) @ ( ring_1_of_int_real @ L ) ) )
% 8.90/5.13 = ( divide_divide_int @ K @ L ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8931_floor__divide__of__int__eq,axiom,
% 8.90/5.13 ! [K: int,L: int] :
% 8.90/5.13 ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ K ) @ ( ring_1_of_int_rat @ L ) ) )
% 8.90/5.13 = ( divide_divide_int @ K @ L ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8932_floor__power,axiom,
% 8.90/5.13 ! [X2: real,N: nat] :
% 8.90/5.13 ( ( X2
% 8.90/5.13 = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X2 ) ) )
% 8.90/5.13 => ( ( archim6058952711729229775r_real @ ( power_power_real @ X2 @ N ) )
% 8.90/5.13 = ( power_power_int @ ( archim6058952711729229775r_real @ X2 ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8933_floor__power,axiom,
% 8.90/5.13 ! [X2: rat,N: nat] :
% 8.90/5.13 ( ( X2
% 8.90/5.13 = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X2 ) ) )
% 8.90/5.13 => ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X2 @ N ) )
% 8.90/5.13 = ( power_power_int @ ( archim3151403230148437115or_rat @ X2 ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8934_pochhammer__nonneg,axiom,
% 8.90/5.13 ! [X2: real,N: nat] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X2 @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8935_pochhammer__nonneg,axiom,
% 8.90/5.13 ! [X2: rat,N: nat] :
% 8.90/5.13 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 8.90/5.13 => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X2 @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8936_pochhammer__nonneg,axiom,
% 8.90/5.13 ! [X2: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 8.90/5.13 => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X2 @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8937_pochhammer__nonneg,axiom,
% 8.90/5.13 ! [X2: int,N: nat] :
% 8.90/5.13 ( ( ord_less_int @ zero_zero_int @ X2 )
% 8.90/5.13 => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X2 @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8938_pochhammer__binomial__sum,axiom,
% 8.90/5.13 ! [A: int,B: int,N: nat] :
% 8.90/5.13 ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B ) @ N )
% 8.90/5.13 = ( groups3539618377306564664at_int
% 8.90/5.13 @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K2 ) ) @ ( comm_s4660882817536571857er_int @ A @ K2 ) ) @ ( comm_s4660882817536571857er_int @ B @ ( minus_minus_nat @ N @ K2 ) ) )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8939_pochhammer__binomial__sum,axiom,
% 8.90/5.13 ! [A: rat,B: rat,N: nat] :
% 8.90/5.13 ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ B ) @ N )
% 8.90/5.13 = ( groups2906978787729119204at_rat
% 8.90/5.13 @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ A @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ B @ ( minus_minus_nat @ N @ K2 ) ) )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8940_pochhammer__binomial__sum,axiom,
% 8.90/5.13 ! [A: real,B: real,N: nat] :
% 8.90/5.13 ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ B ) @ N )
% 8.90/5.13 = ( groups6591440286371151544t_real
% 8.90/5.13 @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K2 ) ) @ ( comm_s7457072308508201937r_real @ A @ K2 ) ) @ ( comm_s7457072308508201937r_real @ B @ ( minus_minus_nat @ N @ K2 ) ) )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8941_pochhammer__0__left,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ( N = zero_zero_nat )
% 8.90/5.13 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 8.90/5.13 = one_one_complex ) )
% 8.90/5.13 & ( ( N != zero_zero_nat )
% 8.90/5.13 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 8.90/5.13 = zero_zero_complex ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8942_pochhammer__0__left,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ( N = zero_zero_nat )
% 8.90/5.13 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 8.90/5.13 = one_one_real ) )
% 8.90/5.13 & ( ( N != zero_zero_nat )
% 8.90/5.13 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 8.90/5.13 = zero_zero_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8943_pochhammer__0__left,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ( N = zero_zero_nat )
% 8.90/5.13 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 8.90/5.13 = one_one_rat ) )
% 8.90/5.13 & ( ( N != zero_zero_nat )
% 8.90/5.13 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 8.90/5.13 = zero_zero_rat ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8944_pochhammer__0__left,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ( N = zero_zero_nat )
% 8.90/5.13 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 8.90/5.13 = one_one_nat ) )
% 8.90/5.13 & ( ( N != zero_zero_nat )
% 8.90/5.13 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 8.90/5.13 = zero_zero_nat ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8945_pochhammer__0__left,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ( N = zero_zero_nat )
% 8.90/5.13 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 8.90/5.13 = one_one_int ) )
% 8.90/5.13 & ( ( N != zero_zero_nat )
% 8.90/5.13 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 8.90/5.13 = zero_zero_int ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8946_Iic__subset__Iio__iff,axiom,
% 8.90/5.13 ! [A: rat,B: rat] :
% 8.90/5.13 ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A ) @ ( set_ord_lessThan_rat @ B ) )
% 8.90/5.13 = ( ord_less_rat @ A @ B ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8947_Iic__subset__Iio__iff,axiom,
% 8.90/5.13 ! [A: num,B: num] :
% 8.90/5.13 ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ A ) @ ( set_ord_lessThan_num @ B ) )
% 8.90/5.13 = ( ord_less_num @ A @ B ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8948_Iic__subset__Iio__iff,axiom,
% 8.90/5.13 ! [A: nat,B: nat] :
% 8.90/5.13 ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
% 8.90/5.13 = ( ord_less_nat @ A @ B ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8949_Iic__subset__Iio__iff,axiom,
% 8.90/5.13 ! [A: int,B: int] :
% 8.90/5.13 ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B ) )
% 8.90/5.13 = ( ord_less_int @ A @ B ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8950_Iic__subset__Iio__iff,axiom,
% 8.90/5.13 ! [A: real,B: real] :
% 8.90/5.13 ( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ A ) @ ( set_or5984915006950818249n_real @ B ) )
% 8.90/5.13 = ( ord_less_real @ A @ B ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8951_sum__choose__upper,axiom,
% 8.90/5.13 ! [M: nat,N: nat] :
% 8.90/5.13 ( ( groups3542108847815614940at_nat
% 8.90/5.13 @ ^ [K2: nat] : ( binomial @ K2 @ M )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) )
% 8.90/5.13 = ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8952_one__add__floor,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X2 ) @ one_one_int )
% 8.90/5.13 = ( archim6058952711729229775r_real @ ( plus_plus_real @ X2 @ one_one_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8953_one__add__floor,axiom,
% 8.90/5.13 ! [X2: rat] :
% 8.90/5.13 ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X2 ) @ one_one_int )
% 8.90/5.13 = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X2 @ one_one_rat ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8954_floor__divide__of__nat__eq,axiom,
% 8.90/5.13 ! [M: nat,N: nat] :
% 8.90/5.13 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 8.90/5.13 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8955_floor__divide__of__nat__eq,axiom,
% 8.90/5.13 ! [M: nat,N: nat] :
% 8.90/5.13 ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) )
% 8.90/5.13 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8956_ceiling__altdef,axiom,
% 8.90/5.13 ( archim7802044766580827645g_real
% 8.90/5.13 = ( ^ [X: real] :
% 8.90/5.13 ( if_int
% 8.90/5.13 @ ( X
% 8.90/5.13 = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) )
% 8.90/5.13 @ ( archim6058952711729229775r_real @ X )
% 8.90/5.13 @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8957_ceiling__altdef,axiom,
% 8.90/5.13 ( archim2889992004027027881ng_rat
% 8.90/5.13 = ( ^ [X: rat] :
% 8.90/5.13 ( if_int
% 8.90/5.13 @ ( X
% 8.90/5.13 = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) )
% 8.90/5.13 @ ( archim3151403230148437115or_rat @ X )
% 8.90/5.13 @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8958_ceiling__diff__floor__le__1,axiom,
% 8.90/5.13 ! [X2: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim7802044766580827645g_real @ X2 ) @ ( archim6058952711729229775r_real @ X2 ) ) @ one_one_int ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8959_ceiling__diff__floor__le__1,axiom,
% 8.90/5.13 ! [X2: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( archim3151403230148437115or_rat @ X2 ) ) @ one_one_int ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8960_floor__eq,axiom,
% 8.90/5.13 ! [N: int,X2: real] :
% 8.90/5.13 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 8.90/5.13 => ( ( archim6058952711729229775r_real @ X2 )
% 8.90/5.13 = N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8961_real__of__int__floor__add__one__gt,axiom,
% 8.90/5.13 ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8962_real__of__int__floor__add__one__ge,axiom,
% 8.90/5.13 ! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8963_real__of__int__floor__gt__diff__one,axiom,
% 8.90/5.13 ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8964_real__of__int__floor__ge__diff__one,axiom,
% 8.90/5.13 ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8965_pochhammer__rec,axiom,
% 8.90/5.13 ! [A: complex,N: nat] :
% 8.90/5.13 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 8.90/5.13 = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8966_pochhammer__rec,axiom,
% 8.90/5.13 ! [A: real,N: nat] :
% 8.90/5.13 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 8.90/5.13 = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8967_pochhammer__rec,axiom,
% 8.90/5.13 ! [A: rat,N: nat] :
% 8.90/5.13 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 8.90/5.13 = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8968_pochhammer__rec,axiom,
% 8.90/5.13 ! [A: nat,N: nat] :
% 8.90/5.13 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 8.90/5.13 = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8969_pochhammer__rec,axiom,
% 8.90/5.13 ! [A: int,N: nat] :
% 8.90/5.13 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 8.90/5.13 = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8970_pochhammer__Suc,axiom,
% 8.90/5.13 ! [A: int,N: nat] :
% 8.90/5.13 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 8.90/5.13 = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8971_pochhammer__Suc,axiom,
% 8.90/5.13 ! [A: real,N: nat] :
% 8.90/5.13 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 8.90/5.13 = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8972_pochhammer__Suc,axiom,
% 8.90/5.13 ! [A: nat,N: nat] :
% 8.90/5.13 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 8.90/5.13 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8973_pochhammer__Suc,axiom,
% 8.90/5.13 ! [A: rat,N: nat] :
% 8.90/5.13 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 8.90/5.13 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8974_pochhammer__rec_H,axiom,
% 8.90/5.13 ! [Z: int,N: nat] :
% 8.90/5.13 ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
% 8.90/5.13 = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8975_pochhammer__rec_H,axiom,
% 8.90/5.13 ! [Z: real,N: nat] :
% 8.90/5.13 ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) )
% 8.90/5.13 = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8976_pochhammer__rec_H,axiom,
% 8.90/5.13 ! [Z: nat,N: nat] :
% 8.90/5.13 ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
% 8.90/5.13 = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8977_pochhammer__rec_H,axiom,
% 8.90/5.13 ! [Z: rat,N: nat] :
% 8.90/5.13 ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
% 8.90/5.13 = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8978_pochhammer__eq__0__iff,axiom,
% 8.90/5.13 ! [A: complex,N: nat] :
% 8.90/5.13 ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 8.90/5.13 = zero_zero_complex )
% 8.90/5.13 = ( ? [K2: nat] :
% 8.90/5.13 ( ( ord_less_nat @ K2 @ N )
% 8.90/5.13 & ( A
% 8.90/5.13 = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K2 ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8979_pochhammer__eq__0__iff,axiom,
% 8.90/5.13 ! [A: real,N: nat] :
% 8.90/5.13 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 8.90/5.13 = zero_zero_real )
% 8.90/5.13 = ( ? [K2: nat] :
% 8.90/5.13 ( ( ord_less_nat @ K2 @ N )
% 8.90/5.13 & ( A
% 8.90/5.13 = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K2 ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8980_pochhammer__eq__0__iff,axiom,
% 8.90/5.13 ! [A: rat,N: nat] :
% 8.90/5.13 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 8.90/5.13 = zero_zero_rat )
% 8.90/5.13 = ( ? [K2: nat] :
% 8.90/5.13 ( ( ord_less_nat @ K2 @ N )
% 8.90/5.13 & ( A
% 8.90/5.13 = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K2 ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8981_pochhammer__of__nat__eq__0__iff,axiom,
% 8.90/5.13 ! [N: nat,K: nat] :
% 8.90/5.13 ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 8.90/5.13 = zero_zero_complex )
% 8.90/5.13 = ( ord_less_nat @ N @ K ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8982_pochhammer__of__nat__eq__0__iff,axiom,
% 8.90/5.13 ! [N: nat,K: nat] :
% 8.90/5.13 ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 8.90/5.13 = zero_z3403309356797280102nteger )
% 8.90/5.13 = ( ord_less_nat @ N @ K ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8983_pochhammer__of__nat__eq__0__iff,axiom,
% 8.90/5.13 ! [N: nat,K: nat] :
% 8.90/5.13 ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 8.90/5.13 = zero_zero_int )
% 8.90/5.13 = ( ord_less_nat @ N @ K ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8984_pochhammer__of__nat__eq__0__iff,axiom,
% 8.90/5.13 ! [N: nat,K: nat] :
% 8.90/5.13 ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 8.90/5.13 = zero_zero_real )
% 8.90/5.13 = ( ord_less_nat @ N @ K ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8985_pochhammer__of__nat__eq__0__iff,axiom,
% 8.90/5.13 ! [N: nat,K: nat] :
% 8.90/5.13 ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 8.90/5.13 = zero_zero_rat )
% 8.90/5.13 = ( ord_less_nat @ N @ K ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8986_pochhammer__of__nat__eq__0__lemma,axiom,
% 8.90/5.13 ! [N: nat,K: nat] :
% 8.90/5.13 ( ( ord_less_nat @ N @ K )
% 8.90/5.13 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 8.90/5.13 = zero_zero_complex ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8987_pochhammer__of__nat__eq__0__lemma,axiom,
% 8.90/5.13 ! [N: nat,K: nat] :
% 8.90/5.13 ( ( ord_less_nat @ N @ K )
% 8.90/5.13 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 8.90/5.13 = zero_z3403309356797280102nteger ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8988_pochhammer__of__nat__eq__0__lemma,axiom,
% 8.90/5.13 ! [N: nat,K: nat] :
% 8.90/5.13 ( ( ord_less_nat @ N @ K )
% 8.90/5.13 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 8.90/5.13 = zero_zero_int ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8989_pochhammer__of__nat__eq__0__lemma,axiom,
% 8.90/5.13 ! [N: nat,K: nat] :
% 8.90/5.13 ( ( ord_less_nat @ N @ K )
% 8.90/5.13 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 8.90/5.13 = zero_zero_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8990_pochhammer__of__nat__eq__0__lemma,axiom,
% 8.90/5.13 ! [N: nat,K: nat] :
% 8.90/5.13 ( ( ord_less_nat @ N @ K )
% 8.90/5.13 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 8.90/5.13 = zero_zero_rat ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8991_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 8.90/5.13 != zero_zero_complex ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8992_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 8.90/5.13 != zero_z3403309356797280102nteger ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8993_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 8.90/5.13 != zero_zero_int ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8994_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 8.90/5.13 != zero_zero_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8995_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 8.90/5.13 ! [K: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ K @ N )
% 8.90/5.13 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 8.90/5.13 != zero_zero_rat ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8996_sum_OatMost__Suc__shift,axiom,
% 8.90/5.13 ! [G: nat > int,N: nat] :
% 8.90/5.13 ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 8.90/5.13 = ( plus_plus_int @ ( G @ zero_zero_nat )
% 8.90/5.13 @ ( groups3539618377306564664at_int
% 8.90/5.13 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8997_sum_OatMost__Suc__shift,axiom,
% 8.90/5.13 ! [G: nat > nat,N: nat] :
% 8.90/5.13 ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 8.90/5.13 = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 8.90/5.13 @ ( groups3542108847815614940at_nat
% 8.90/5.13 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8998_sum_OatMost__Suc__shift,axiom,
% 8.90/5.13 ! [G: nat > real,N: nat] :
% 8.90/5.13 ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 8.90/5.13 = ( plus_plus_real @ ( G @ zero_zero_nat )
% 8.90/5.13 @ ( groups6591440286371151544t_real
% 8.90/5.13 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_8999_sum__choose__lower,axiom,
% 8.90/5.13 ! [R2: nat,N: nat] :
% 8.90/5.13 ( ( groups3542108847815614940at_nat
% 8.90/5.13 @ ^ [K2: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K2 ) @ K2 )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) )
% 8.90/5.13 = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N ) ) @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9000_choose__rising__sum_I2_J,axiom,
% 8.90/5.13 ! [N: nat,M: nat] :
% 8.90/5.13 ( ( groups3542108847815614940at_nat
% 8.90/5.13 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 8.90/5.13 @ ( set_ord_atMost_nat @ M ) )
% 8.90/5.13 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9001_choose__rising__sum_I1_J,axiom,
% 8.90/5.13 ! [N: nat,M: nat] :
% 8.90/5.13 ( ( groups3542108847815614940at_nat
% 8.90/5.13 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 8.90/5.13 @ ( set_ord_atMost_nat @ M ) )
% 8.90/5.13 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9002_floor__eq2,axiom,
% 8.90/5.13 ! [N: int,X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 8.90/5.13 => ( ( archim6058952711729229775r_real @ X2 )
% 8.90/5.13 = N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9003_floor__divide__real__eq__div,axiom,
% 8.90/5.13 ! [B: int,A: real] :
% 8.90/5.13 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 8.90/5.13 => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
% 8.90/5.13 = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9004_sum__choose__diagonal,axiom,
% 8.90/5.13 ! [M: nat,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.13 => ( ( groups3542108847815614940at_nat
% 8.90/5.13 @ ^ [K2: nat] : ( binomial @ ( minus_minus_nat @ N @ K2 ) @ ( minus_minus_nat @ M @ K2 ) )
% 8.90/5.13 @ ( set_ord_atMost_nat @ M ) )
% 8.90/5.13 = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9005_vandermonde,axiom,
% 8.90/5.13 ! [M: nat,N: nat,R2: nat] :
% 8.90/5.13 ( ( groups3542108847815614940at_nat
% 8.90/5.13 @ ^ [K2: nat] : ( times_times_nat @ ( binomial @ M @ K2 ) @ ( binomial @ N @ ( minus_minus_nat @ R2 @ K2 ) ) )
% 8.90/5.13 @ ( set_ord_atMost_nat @ R2 ) )
% 8.90/5.13 = ( binomial @ ( plus_plus_nat @ M @ N ) @ R2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9006_choose__row__sum,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
% 8.90/5.13 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9007_binomial,axiom,
% 8.90/5.13 ! [A: nat,B: nat,N: nat] :
% 8.90/5.13 ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
% 8.90/5.13 = ( groups3542108847815614940at_nat
% 8.90/5.13 @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K2 ) ) )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9008_polynomial__product__nat,axiom,
% 8.90/5.13 ! [M: nat,A: nat > nat,N: nat,B: nat > nat,X2: nat] :
% 8.90/5.13 ( ! [I3: nat] :
% 8.90/5.13 ( ( ord_less_nat @ M @ I3 )
% 8.90/5.13 => ( ( A @ I3 )
% 8.90/5.13 = zero_zero_nat ) )
% 8.90/5.13 => ( ! [J2: nat] :
% 8.90/5.13 ( ( ord_less_nat @ N @ J2 )
% 8.90/5.13 => ( ( B @ J2 )
% 8.90/5.13 = zero_zero_nat ) )
% 8.90/5.13 => ( ( times_times_nat
% 8.90/5.13 @ ( groups3542108847815614940at_nat
% 8.90/5.13 @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( power_power_nat @ X2 @ I4 ) )
% 8.90/5.13 @ ( set_ord_atMost_nat @ M ) )
% 8.90/5.13 @ ( groups3542108847815614940at_nat
% 8.90/5.13 @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X2 @ J3 ) )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) ) )
% 8.90/5.13 = ( groups3542108847815614940at_nat
% 8.90/5.13 @ ^ [R5: nat] :
% 8.90/5.13 ( times_times_nat
% 8.90/5.13 @ ( groups3542108847815614940at_nat
% 8.90/5.13 @ ^ [K2: nat] : ( times_times_nat @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
% 8.90/5.13 @ ( set_ord_atMost_nat @ R5 ) )
% 8.90/5.13 @ ( power_power_nat @ X2 @ R5 ) )
% 8.90/5.13 @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9009_choose__square__sum,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( groups3542108847815614940at_nat
% 8.90/5.13 @ ^ [K2: nat] : ( power_power_nat @ ( binomial @ N @ K2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) )
% 8.90/5.13 = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9010_floor__log__eq__powr__iff,axiom,
% 8.90/5.13 ! [X2: real,B: real,K: int] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ one_one_real @ B )
% 8.90/5.13 => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X2 ) )
% 8.90/5.13 = K )
% 8.90/5.13 = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X2 )
% 8.90/5.13 & ( ord_less_real @ X2 @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9011_binomial__r__part__sum,axiom,
% 8.90/5.13 ! [M: nat] :
% 8.90/5.13 ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
% 8.90/5.13 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9012_choose__linear__sum,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( groups3542108847815614940at_nat
% 8.90/5.13 @ ^ [I4: nat] : ( times_times_nat @ I4 @ ( binomial @ N @ I4 ) )
% 8.90/5.13 @ ( set_ord_atMost_nat @ N ) )
% 8.90/5.13 = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9013_floor__log2__div2,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.13 => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 8.90/5.13 = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9014_floor__log__nat__eq__if,axiom,
% 8.90/5.13 ! [B: nat,N: nat,K: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 8.90/5.13 => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 8.90/5.13 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 8.90/5.13 => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 8.90/5.13 = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9015_of__nat__id,axiom,
% 8.90/5.13 ( semiri1316708129612266289at_nat
% 8.90/5.13 = ( ^ [N3: nat] : N3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9016_real__scaleR__def,axiom,
% 8.90/5.13 real_V1485227260804924795R_real = times_times_real ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9017_complex__scaleR,axiom,
% 8.90/5.13 ! [R2: real,A: real,B: real] :
% 8.90/5.13 ( ( real_V2046097035970521341omplex @ R2 @ ( complex2 @ A @ B ) )
% 8.90/5.13 = ( complex2 @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ B ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9018_fact__eq__fact__times,axiom,
% 8.90/5.13 ! [N: nat,M: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.13 => ( ( semiri1408675320244567234ct_nat @ M )
% 8.90/5.13 = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
% 8.90/5.13 @ ( groups708209901874060359at_nat
% 8.90/5.13 @ ^ [X: nat] : X
% 8.90/5.13 @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9019_fact__div__fact,axiom,
% 8.90/5.13 ! [N: nat,M: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.13 => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
% 8.90/5.13 = ( groups708209901874060359at_nat
% 8.90/5.13 @ ^ [X: nat] : X
% 8.90/5.13 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9020_exp__two__pi__i_H,axiom,
% 8.90/5.13 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
% 8.90/5.13 = one_one_complex ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9021_exp__two__pi__i,axiom,
% 8.90/5.13 ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
% 8.90/5.13 = one_one_complex ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9022_norm__ii,axiom,
% 8.90/5.13 ( ( real_V1022390504157884413omplex @ imaginary_unit )
% 8.90/5.13 = one_one_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9023_divide__i,axiom,
% 8.90/5.13 ! [X2: complex] :
% 8.90/5.13 ( ( divide1717551699836669952omplex @ X2 @ imaginary_unit )
% 8.90/5.13 = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9024_complex__i__mult__minus,axiom,
% 8.90/5.13 ! [X2: complex] :
% 8.90/5.13 ( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X2 ) )
% 8.90/5.13 = ( uminus1482373934393186551omplex @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9025_i__squared,axiom,
% 8.90/5.13 ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
% 8.90/5.13 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9026_divide__numeral__i,axiom,
% 8.90/5.13 ! [Z: complex,N: num] :
% 8.90/5.13 ( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
% 8.90/5.13 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9027_power2__i,axiom,
% 8.90/5.13 ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.13 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9028_exp__pi__i,axiom,
% 8.90/5.13 ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
% 8.90/5.13 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9029_exp__pi__i_H,axiom,
% 8.90/5.13 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
% 8.90/5.13 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9030_i__even__power,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.13 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9031_complex__i__not__one,axiom,
% 8.90/5.13 imaginary_unit != one_one_complex ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9032_complex__i__not__numeral,axiom,
% 8.90/5.13 ! [W: num] :
% 8.90/5.13 ( imaginary_unit
% 8.90/5.13 != ( numera6690914467698888265omplex @ W ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9033_i__times__eq__iff,axiom,
% 8.90/5.13 ! [W: complex,Z: complex] :
% 8.90/5.13 ( ( ( times_times_complex @ imaginary_unit @ W )
% 8.90/5.13 = Z )
% 8.90/5.13 = ( W
% 8.90/5.13 = ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9034_complex__i__not__neg__numeral,axiom,
% 8.90/5.13 ! [W: num] :
% 8.90/5.13 ( imaginary_unit
% 8.90/5.13 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9035_imaginary__unit_Ocode,axiom,
% 8.90/5.13 ( imaginary_unit
% 8.90/5.13 = ( complex2 @ zero_zero_real @ one_one_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9036_Complex__eq__i,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ( complex2 @ X2 @ Y3 )
% 8.90/5.13 = imaginary_unit )
% 8.90/5.13 = ( ( X2 = zero_zero_real )
% 8.90/5.13 & ( Y3 = one_one_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9037_Complex__mult__i,axiom,
% 8.90/5.13 ! [A: real,B: real] :
% 8.90/5.13 ( ( times_times_complex @ ( complex2 @ A @ B ) @ imaginary_unit )
% 8.90/5.13 = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9038_i__mult__Complex,axiom,
% 8.90/5.13 ! [A: real,B: real] :
% 8.90/5.13 ( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B ) )
% 8.90/5.13 = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9039_prod__int__plus__eq,axiom,
% 8.90/5.13 ! [I2: nat,J: nat] :
% 8.90/5.13 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ ( plus_plus_nat @ I2 @ J ) ) )
% 8.90/5.13 = ( groups1705073143266064639nt_int
% 8.90/5.13 @ ^ [X: int] : X
% 8.90/5.13 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I2 @ J ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9040_complex__of__real__i,axiom,
% 8.90/5.13 ! [R2: real] :
% 8.90/5.13 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ imaginary_unit )
% 8.90/5.13 = ( complex2 @ zero_zero_real @ R2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9041_i__complex__of__real,axiom,
% 8.90/5.13 ! [R2: real] :
% 8.90/5.13 ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R2 ) )
% 8.90/5.13 = ( complex2 @ zero_zero_real @ R2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9042_Complex__eq,axiom,
% 8.90/5.13 ( complex2
% 8.90/5.13 = ( ^ [A3: real,B2: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A3 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B2 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9043_complex__split__polar,axiom,
% 8.90/5.13 ! [Z: complex] :
% 8.90/5.13 ? [R3: real,A4: real] :
% 8.90/5.13 ( Z
% 8.90/5.13 = ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A4 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A4 ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9044_cmod__unit__one,axiom,
% 8.90/5.13 ! [A: real] :
% 8.90/5.13 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
% 8.90/5.13 = one_one_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9045_cmod__complex__polar,axiom,
% 8.90/5.13 ! [R2: real,A: real] :
% 8.90/5.13 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
% 8.90/5.13 = ( abs_abs_real @ R2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9046_Arg__minus__ii,axiom,
% 8.90/5.13 ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
% 8.90/5.13 = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9047_csqrt__ii,axiom,
% 8.90/5.13 ( ( csqrt @ imaginary_unit )
% 8.90/5.13 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9048_Arg__ii,axiom,
% 8.90/5.13 ( ( arg @ imaginary_unit )
% 8.90/5.13 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9049_cis__minus__pi__half,axiom,
% 8.90/5.13 ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 8.90/5.13 = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9050_csqrt__1,axiom,
% 8.90/5.13 ( ( csqrt @ one_one_complex )
% 8.90/5.13 = one_one_complex ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9051_csqrt__eq__1,axiom,
% 8.90/5.13 ! [Z: complex] :
% 8.90/5.13 ( ( ( csqrt @ Z )
% 8.90/5.13 = one_one_complex )
% 8.90/5.13 = ( Z = one_one_complex ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9052_norm__cis,axiom,
% 8.90/5.13 ! [A: real] :
% 8.90/5.13 ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
% 8.90/5.13 = one_one_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9053_cis__zero,axiom,
% 8.90/5.13 ( ( cis @ zero_zero_real )
% 8.90/5.13 = one_one_complex ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9054_cis__pi,axiom,
% 8.90/5.13 ( ( cis @ pi )
% 8.90/5.13 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9055_power2__csqrt,axiom,
% 8.90/5.13 ! [Z: complex] :
% 8.90/5.13 ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.13 = Z ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9056_cis__pi__half,axiom,
% 8.90/5.13 ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 = imaginary_unit ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9057_cis__2pi,axiom,
% 8.90/5.13 ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 8.90/5.13 = one_one_complex ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9058_cis__divide,axiom,
% 8.90/5.13 ! [A: real,B: real] :
% 8.90/5.13 ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B ) )
% 8.90/5.13 = ( cis @ ( minus_minus_real @ A @ B ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9059_cis__mult,axiom,
% 8.90/5.13 ! [A: real,B: real] :
% 8.90/5.13 ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
% 8.90/5.13 = ( cis @ ( plus_plus_real @ A @ B ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9060_DeMoivre,axiom,
% 8.90/5.13 ! [A: real,N: nat] :
% 8.90/5.13 ( ( power_power_complex @ ( cis @ A ) @ N )
% 8.90/5.13 = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9061_cis__conv__exp,axiom,
% 8.90/5.13 ( cis
% 8.90/5.13 = ( ^ [B2: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B2 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9062_of__real__sqrt,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( real_V4546457046886955230omplex @ ( sqrt @ X2 ) )
% 8.90/5.13 = ( csqrt @ ( real_V4546457046886955230omplex @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9063_Arg__bounded,axiom,
% 8.90/5.13 ! [Z: complex] :
% 8.90/5.13 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 8.90/5.13 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9064_bij__betw__roots__unity,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( bij_betw_nat_complex
% 8.90/5.13 @ ^ [K2: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 8.90/5.13 @ ( set_ord_lessThan_nat @ N )
% 8.90/5.13 @ ( collect_complex
% 8.90/5.13 @ ^ [Z3: complex] :
% 8.90/5.13 ( ( power_power_complex @ Z3 @ N )
% 8.90/5.13 = one_one_complex ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9065_Maclaurin__sin__bound,axiom,
% 8.90/5.13 ! [X2: real,N: nat] :
% 8.90/5.13 ( ord_less_eq_real
% 8.90/5.13 @ ( abs_abs_real
% 8.90/5.13 @ ( minus_minus_real @ ( sin_real @ X2 )
% 8.90/5.13 @ ( groups6591440286371151544t_real
% 8.90/5.13 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
% 8.90/5.13 @ ( set_ord_lessThan_nat @ N ) ) ) )
% 8.90/5.13 @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X2 ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9066_real__sqrt__inverse,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( sqrt @ ( inverse_inverse_real @ X2 ) )
% 8.90/5.13 = ( inverse_inverse_real @ ( sqrt @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9067_divide__real__def,axiom,
% 8.90/5.13 ( divide_divide_real
% 8.90/5.13 = ( ^ [X: real,Y: real] : ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9068_inverse__powr,axiom,
% 8.90/5.13 ! [Y3: real,A: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.13 => ( ( powr_real @ ( inverse_inverse_real @ Y3 ) @ A )
% 8.90/5.13 = ( inverse_inverse_real @ ( powr_real @ Y3 @ A ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9069_forall__pos__mono__1,axiom,
% 8.90/5.13 ! [P: real > $o,E: real] :
% 8.90/5.13 ( ! [D3: real,E2: real] :
% 8.90/5.13 ( ( ord_less_real @ D3 @ E2 )
% 8.90/5.13 => ( ( P @ D3 )
% 8.90/5.13 => ( P @ E2 ) ) )
% 8.90/5.13 => ( ! [N2: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
% 8.90/5.13 => ( ( ord_less_real @ zero_zero_real @ E )
% 8.90/5.13 => ( P @ E ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9070_forall__pos__mono,axiom,
% 8.90/5.13 ! [P: real > $o,E: real] :
% 8.90/5.13 ( ! [D3: real,E2: real] :
% 8.90/5.13 ( ( ord_less_real @ D3 @ E2 )
% 8.90/5.13 => ( ( P @ D3 )
% 8.90/5.13 => ( P @ E2 ) ) )
% 8.90/5.13 => ( ! [N2: nat] :
% 8.90/5.13 ( ( N2 != zero_zero_nat )
% 8.90/5.13 => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
% 8.90/5.13 => ( ( ord_less_real @ zero_zero_real @ E )
% 8.90/5.13 => ( P @ E ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9071_real__arch__inverse,axiom,
% 8.90/5.13 ! [E: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ E )
% 8.90/5.13 = ( ? [N3: nat] :
% 8.90/5.13 ( ( N3 != zero_zero_nat )
% 8.90/5.13 & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) )
% 8.90/5.13 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ E ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9072_sqrt__divide__self__eq,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( divide_divide_real @ ( sqrt @ X2 ) @ X2 )
% 8.90/5.13 = ( inverse_inverse_real @ ( sqrt @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9073_log__inverse,axiom,
% 8.90/5.13 ! [A: real,X2: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ A )
% 8.90/5.13 => ( ( A != one_one_real )
% 8.90/5.13 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( log @ A @ ( inverse_inverse_real @ X2 ) )
% 8.90/5.13 = ( uminus_uminus_real @ ( log @ A @ X2 ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9074_exp__plus__inverse__exp,axiom,
% 8.90/5.13 ! [X2: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9075_plus__inverse__ge__2,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9076_real__inv__sqrt__pow2,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.13 = ( inverse_inverse_real @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9077_tan__cot,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) )
% 8.90/5.13 = ( inverse_inverse_real @ ( tan_real @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9078_real__le__x__sinh,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ord_less_eq_real @ X2 @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9079_real__le__abs__sinh,axiom,
% 8.90/5.13 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9080_VEBT__internal_OminNull_Opelims_I1_J,axiom,
% 8.90/5.13 ! [X2: vEBT_VEBT,Y3: $o] :
% 8.90/5.13 ( ( ( vEBT_VEBT_minNull @ X2 )
% 8.90/5.13 = Y3 )
% 8.90/5.13 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
% 8.90/5.13 => ( ( ( X2
% 8.90/5.13 = ( vEBT_Leaf @ $false @ $false ) )
% 8.90/5.13 => ( Y3
% 8.90/5.13 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
% 8.90/5.13 => ( ! [Uv2: $o] :
% 8.90/5.13 ( ( X2
% 8.90/5.13 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 8.90/5.13 => ( ~ Y3
% 8.90/5.13 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
% 8.90/5.13 => ( ! [Uu2: $o] :
% 8.90/5.13 ( ( X2
% 8.90/5.13 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 8.90/5.13 => ( ~ Y3
% 8.90/5.13 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
% 8.90/5.13 => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 8.90/5.13 ( ( X2
% 8.90/5.13 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 8.90/5.13 => ( Y3
% 8.90/5.13 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
% 8.90/5.13 => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.90/5.13 ( ( X2
% 8.90/5.13 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
% 8.90/5.13 => ( ~ Y3
% 8.90/5.13 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9081_cot__less__zero,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 8.90/5.13 => ( ord_less_real @ ( cot_real @ X2 ) @ zero_zero_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9082_sinh__real__le__iff,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( sinh_real @ X2 ) @ ( sinh_real @ Y3 ) )
% 8.90/5.13 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9083_sinh__real__nonpos__iff,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( sinh_real @ X2 ) @ zero_zero_real )
% 8.90/5.13 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9084_sinh__real__nonneg__iff,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X2 ) )
% 8.90/5.13 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9085_cot__npi,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 8.90/5.13 = zero_zero_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9086_cot__periodic,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( cot_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 8.90/5.13 = ( cot_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9087_sinh__le__cosh__real,axiom,
% 8.90/5.13 ! [X2: real] : ( ord_less_eq_real @ ( sinh_real @ X2 ) @ ( cosh_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9088_cosh__real__nonneg,axiom,
% 8.90/5.13 ! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9089_cosh__real__nonneg__le__iff,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
% 8.90/5.13 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9090_cosh__real__nonpos__le__iff,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
% 8.90/5.13 = ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9091_cosh__real__ge__1,axiom,
% 8.90/5.13 ! [X2: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9092_divide__complex__def,axiom,
% 8.90/5.13 ( divide1717551699836669952omplex
% 8.90/5.13 = ( ^ [X: complex,Y: complex] : ( times_times_complex @ X @ ( invers8013647133539491842omplex @ Y ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9093_cosh__real__strict__mono,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.13 => ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9094_cosh__real__nonneg__less__iff,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.13 => ( ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
% 8.90/5.13 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9095_cosh__real__nonpos__less__iff,axiom,
% 8.90/5.13 ! [X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 8.90/5.13 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 8.90/5.13 => ( ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
% 8.90/5.13 = ( ord_less_real @ Y3 @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9096_arcosh__cosh__real,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( arcosh_real @ ( cosh_real @ X2 ) )
% 8.90/5.13 = X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9097_complex__inverse,axiom,
% 8.90/5.13 ! [A: real,B: real] :
% 8.90/5.13 ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
% 8.90/5.13 = ( 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 ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9098_VEBT__internal_OminNull_Opelims_I3_J,axiom,
% 8.90/5.13 ! [X2: vEBT_VEBT] :
% 8.90/5.13 ( ~ ( vEBT_VEBT_minNull @ X2 )
% 8.90/5.13 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
% 8.90/5.13 => ( ! [Uv2: $o] :
% 8.90/5.13 ( ( X2
% 8.90/5.13 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 8.90/5.13 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
% 8.90/5.13 => ( ! [Uu2: $o] :
% 8.90/5.13 ( ( X2
% 8.90/5.13 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 8.90/5.13 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
% 8.90/5.13 => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 8.90/5.13 ( ( X2
% 8.90/5.13 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
% 8.90/5.13 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9099_VEBT__internal_OminNull_Opelims_I2_J,axiom,
% 8.90/5.13 ! [X2: vEBT_VEBT] :
% 8.90/5.13 ( ( vEBT_VEBT_minNull @ X2 )
% 8.90/5.13 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
% 8.90/5.13 => ( ( ( X2
% 8.90/5.13 = ( vEBT_Leaf @ $false @ $false ) )
% 8.90/5.13 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
% 8.90/5.13 => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 8.90/5.13 ( ( X2
% 8.90/5.13 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 8.90/5.13 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9100_cosh__ln__real,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( cosh_real @ ( ln_ln_real @ X2 ) )
% 8.90/5.13 = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9101_cot__gt__zero,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 => ( ord_less_real @ zero_zero_real @ ( cot_real @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9102_sinh__ln__real,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( sinh_real @ ( ln_ln_real @ X2 ) )
% 8.90/5.13 = ( divide_divide_real @ ( minus_minus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9103_tan__cot_H,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) )
% 8.90/5.13 = ( cot_real @ X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9104_bij__betw__nth__root__unity,axiom,
% 8.90/5.13 ! [C: complex,N: nat] :
% 8.90/5.13 ( ( C != zero_zero_complex )
% 8.90/5.13 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 8.90/5.13 @ ( collect_complex
% 8.90/5.13 @ ^ [Z3: complex] :
% 8.90/5.13 ( ( power_power_complex @ Z3 @ N )
% 8.90/5.13 = one_one_complex ) )
% 8.90/5.13 @ ( collect_complex
% 8.90/5.13 @ ^ [Z3: complex] :
% 8.90/5.13 ( ( power_power_complex @ Z3 @ N )
% 8.90/5.13 = C ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9105_upto_Opinduct,axiom,
% 8.90/5.13 ! [A0: int,A12: int,P: int > int > $o] :
% 8.90/5.13 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 8.90/5.13 => ( ! [I3: int,J2: int] :
% 8.90/5.13 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
% 8.90/5.13 => ( ( ( ord_less_eq_int @ I3 @ J2 )
% 8.90/5.13 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
% 8.90/5.13 => ( P @ I3 @ J2 ) ) )
% 8.90/5.13 => ( P @ A0 @ A12 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9106_arctan__def,axiom,
% 8.90/5.13 ( arctan
% 8.90/5.13 = ( ^ [Y: real] :
% 8.90/5.13 ( the_real
% 8.90/5.13 @ ^ [X: real] :
% 8.90/5.13 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 8.90/5.13 & ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 & ( ( tan_real @ X )
% 8.90/5.13 = Y ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9107_arcsin__def,axiom,
% 8.90/5.13 ( arcsin
% 8.90/5.13 = ( ^ [Y: real] :
% 8.90/5.13 ( the_real
% 8.90/5.13 @ ^ [X: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 8.90/5.13 & ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.13 & ( ( sin_real @ X )
% 8.90/5.13 = Y ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9108_real__root__zero,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( root @ N @ zero_zero_real )
% 8.90/5.13 = zero_zero_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9109_real__root__Suc__0,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( root @ ( suc @ zero_zero_nat ) @ X2 )
% 8.90/5.13 = X2 ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9110_real__root__eq__iff,axiom,
% 8.90/5.13 ! [N: nat,X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ( root @ N @ X2 )
% 8.90/5.13 = ( root @ N @ Y3 ) )
% 8.90/5.13 = ( X2 = Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9111_root__0,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( root @ zero_zero_nat @ X2 )
% 8.90/5.13 = zero_zero_real ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9112_real__root__eq__0__iff,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ( root @ N @ X2 )
% 8.90/5.13 = zero_zero_real )
% 8.90/5.13 = ( X2 = zero_zero_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9113_real__root__less__iff,axiom,
% 8.90/5.13 ! [N: nat,X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) )
% 8.90/5.13 = ( ord_less_real @ X2 @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9114_real__root__le__iff,axiom,
% 8.90/5.13 ! [N: nat,X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) )
% 8.90/5.13 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9115_real__root__one,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( root @ N @ one_one_real )
% 8.90/5.13 = one_one_real ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9116_real__root__eq__1__iff,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ( root @ N @ X2 )
% 8.90/5.13 = one_one_real )
% 8.90/5.13 = ( X2 = one_one_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9117_real__root__lt__0__iff,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_real @ ( root @ N @ X2 ) @ zero_zero_real )
% 8.90/5.13 = ( ord_less_real @ X2 @ zero_zero_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9118_real__root__gt__0__iff,axiom,
% 8.90/5.13 ! [N: nat,Y3: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 8.90/5.13 = ( ord_less_real @ zero_zero_real @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9119_real__root__ge__0__iff,axiom,
% 8.90/5.13 ! [N: nat,Y3: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 8.90/5.13 = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9120_real__root__le__0__iff,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ zero_zero_real )
% 8.90/5.13 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9121_real__root__gt__1__iff,axiom,
% 8.90/5.13 ! [N: nat,Y3: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y3 ) )
% 8.90/5.13 = ( ord_less_real @ one_one_real @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9122_real__root__lt__1__iff,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_real @ ( root @ N @ X2 ) @ one_one_real )
% 8.90/5.13 = ( ord_less_real @ X2 @ one_one_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9123_real__root__ge__1__iff,axiom,
% 8.90/5.13 ! [N: nat,Y3: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y3 ) )
% 8.90/5.13 = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9124_real__root__le__1__iff,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ one_one_real )
% 8.90/5.13 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9125_real__root__pow__pos2,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( power_power_real @ ( root @ N @ X2 ) @ N )
% 8.90/5.13 = X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9126_real__root__inverse,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( root @ N @ ( inverse_inverse_real @ X2 ) )
% 8.90/5.13 = ( inverse_inverse_real @ ( root @ N @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9127_real__root__divide,axiom,
% 8.90/5.13 ! [N: nat,X2: real,Y3: real] :
% 8.90/5.13 ( ( root @ N @ ( divide_divide_real @ X2 @ Y3 ) )
% 8.90/5.13 = ( divide_divide_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9128_real__root__mult,axiom,
% 8.90/5.13 ! [N: nat,X2: real,Y3: real] :
% 8.90/5.13 ( ( root @ N @ ( times_times_real @ X2 @ Y3 ) )
% 8.90/5.13 = ( times_times_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9129_real__root__mult__exp,axiom,
% 8.90/5.13 ! [M: nat,N: nat,X2: real] :
% 8.90/5.13 ( ( root @ ( times_times_nat @ M @ N ) @ X2 )
% 8.90/5.13 = ( root @ M @ ( root @ N @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9130_real__root__minus,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( root @ N @ ( uminus_uminus_real @ X2 ) )
% 8.90/5.13 = ( uminus_uminus_real @ ( root @ N @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9131_real__root__commute,axiom,
% 8.90/5.13 ! [M: nat,N: nat,X2: real] :
% 8.90/5.13 ( ( root @ M @ ( root @ N @ X2 ) )
% 8.90/5.13 = ( root @ N @ ( root @ M @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9132_real__root__pos__pos__le,axiom,
% 8.90/5.13 ! [X2: real,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9133_real__root__less__mono,axiom,
% 8.90/5.13 ! [N: nat,X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.13 => ( ord_less_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9134_real__root__le__mono,axiom,
% 8.90/5.13 ! [N: nat,X2: real,Y3: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 8.90/5.13 => ( ord_less_eq_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9135_real__root__power,axiom,
% 8.90/5.13 ! [N: nat,X2: real,K: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( root @ N @ ( power_power_real @ X2 @ K ) )
% 8.90/5.13 = ( power_power_real @ ( root @ N @ X2 ) @ K ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9136_real__root__abs,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( root @ N @ ( abs_abs_real @ X2 ) )
% 8.90/5.13 = ( abs_abs_real @ ( root @ N @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9137_ln__neg__is__const,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 8.90/5.13 => ( ( ln_ln_real @ X2 )
% 8.90/5.13 = ( the_real
% 8.90/5.13 @ ^ [X: real] : $false ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9138_real__root__gt__zero,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ord_less_real @ zero_zero_real @ ( root @ N @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9139_real__root__strict__decreasing,axiom,
% 8.90/5.13 ! [N: nat,N5: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_nat @ N @ N5 )
% 8.90/5.13 => ( ( ord_less_real @ one_one_real @ X2 )
% 8.90/5.13 => ( ord_less_real @ ( root @ N5 @ X2 ) @ ( root @ N @ X2 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9140_sqrt__def,axiom,
% 8.90/5.13 ( sqrt
% 8.90/5.13 = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9141_root__abs__power,axiom,
% 8.90/5.13 ! [N: nat,Y3: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y3 @ N ) ) )
% 8.90/5.13 = ( abs_abs_real @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9142_real__root__pos__pos,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9143_real__root__strict__increasing,axiom,
% 8.90/5.13 ! [N: nat,N5: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_nat @ N @ N5 )
% 8.90/5.13 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ one_one_real )
% 8.90/5.13 => ( ord_less_real @ ( root @ N @ X2 ) @ ( root @ N5 @ X2 ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9144_real__root__decreasing,axiom,
% 8.90/5.13 ! [N: nat,N5: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_eq_nat @ N @ N5 )
% 8.90/5.13 => ( ( ord_less_eq_real @ one_one_real @ X2 )
% 8.90/5.13 => ( ord_less_eq_real @ ( root @ N5 @ X2 ) @ ( root @ N @ X2 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9145_real__root__pow__pos,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( power_power_real @ ( root @ N @ X2 ) @ N )
% 8.90/5.13 = X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9146_real__root__power__cancel,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( root @ N @ ( power_power_real @ X2 @ N ) )
% 8.90/5.13 = X2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9147_real__root__pos__unique,axiom,
% 8.90/5.13 ! [N: nat,Y3: real,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 8.90/5.13 => ( ( ( power_power_real @ Y3 @ N )
% 8.90/5.13 = X2 )
% 8.90/5.13 => ( ( root @ N @ X2 )
% 8.90/5.13 = Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9148_odd__real__root__pow,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.13 => ( ( power_power_real @ ( root @ N @ X2 ) @ N )
% 8.90/5.13 = X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9149_odd__real__root__unique,axiom,
% 8.90/5.13 ! [N: nat,Y3: real,X2: real] :
% 8.90/5.13 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.13 => ( ( ( power_power_real @ Y3 @ N )
% 8.90/5.13 = X2 )
% 8.90/5.13 => ( ( root @ N @ X2 )
% 8.90/5.13 = Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9150_odd__real__root__power__cancel,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.13 => ( ( root @ N @ ( power_power_real @ X2 @ N ) )
% 8.90/5.13 = X2 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9151_real__root__increasing,axiom,
% 8.90/5.13 ! [N: nat,N5: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_eq_nat @ N @ N5 )
% 8.90/5.13 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 8.90/5.13 => ( ord_less_eq_real @ ( root @ N @ X2 ) @ ( root @ N5 @ X2 ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9152_arccos__def,axiom,
% 8.90/5.13 ( arccos
% 8.90/5.13 = ( ^ [Y: real] :
% 8.90/5.13 ( the_real
% 8.90/5.13 @ ^ [X: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.90/5.13 & ( ord_less_eq_real @ X @ pi )
% 8.90/5.13 & ( ( cos_real @ X )
% 8.90/5.13 = Y ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9153_log__root,axiom,
% 8.90/5.13 ! [N: nat,A: real,B: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_real @ zero_zero_real @ A )
% 8.90/5.13 => ( ( log @ B @ ( root @ N @ A ) )
% 8.90/5.13 = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9154_log__base__root,axiom,
% 8.90/5.13 ! [N: nat,B: real,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_real @ zero_zero_real @ B )
% 8.90/5.13 => ( ( log @ ( root @ N @ B ) @ X2 )
% 8.90/5.13 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X2 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9155_ln__root,axiom,
% 8.90/5.13 ! [N: nat,B: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_real @ zero_zero_real @ B )
% 8.90/5.13 => ( ( ln_ln_real @ ( root @ N @ B ) )
% 8.90/5.13 = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9156_root__powr__inverse,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.13 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( root @ N @ X2 )
% 8.90/5.13 = ( powr_real @ X2 @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9157_pi__half,axiom,
% 8.90/5.13 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.90/5.13 = ( the_real
% 8.90/5.13 @ ^ [X: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.90/5.13 & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.90/5.13 & ( ( cos_real @ X )
% 8.90/5.13 = zero_zero_real ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9158_pi__def,axiom,
% 8.90/5.13 ( pi
% 8.90/5.13 = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 8.90/5.13 @ ( the_real
% 8.90/5.13 @ ^ [X: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 8.90/5.13 & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 8.90/5.13 & ( ( cos_real @ X )
% 8.90/5.13 = zero_zero_real ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9159_modulo__int__unfold,axiom,
% 8.90/5.13 ! [L: int,K: int,N: nat,M: nat] :
% 8.90/5.13 ( ( ( ( ( sgn_sgn_int @ L )
% 8.90/5.13 = zero_zero_int )
% 8.90/5.13 | ( ( sgn_sgn_int @ K )
% 8.90/5.13 = zero_zero_int )
% 8.90/5.13 | ( N = zero_zero_nat ) )
% 8.90/5.13 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 8.90/5.13 = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 8.90/5.13 & ( ~ ( ( ( sgn_sgn_int @ L )
% 8.90/5.13 = zero_zero_int )
% 8.90/5.13 | ( ( sgn_sgn_int @ K )
% 8.90/5.13 = zero_zero_int )
% 8.90/5.13 | ( N = zero_zero_nat ) )
% 8.90/5.13 => ( ( ( ( sgn_sgn_int @ K )
% 8.90/5.13 = ( sgn_sgn_int @ L ) )
% 8.90/5.13 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 8.90/5.13 = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
% 8.90/5.13 & ( ( ( sgn_sgn_int @ K )
% 8.90/5.13 != ( sgn_sgn_int @ L ) )
% 8.90/5.13 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 8.90/5.13 = ( times_times_int @ ( sgn_sgn_int @ L )
% 8.90/5.13 @ ( minus_minus_int
% 8.90/5.13 @ ( semiri1314217659103216013at_int
% 8.90/5.13 @ ( times_times_nat @ N
% 8.90/5.13 @ ( zero_n2687167440665602831ol_nat
% 8.90/5.13 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
% 8.90/5.13 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9160_even__set__encode__iff,axiom,
% 8.90/5.13 ! [A2: set_nat] :
% 8.90/5.13 ( ( finite_finite_nat @ A2 )
% 8.90/5.13 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
% 8.90/5.13 = ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9161_powr__int,axiom,
% 8.90/5.13 ! [X2: real,I2: int] :
% 8.90/5.13 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.13 => ( ( ( ord_less_eq_int @ zero_zero_int @ I2 )
% 8.90/5.13 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ I2 ) )
% 8.90/5.13 = ( power_power_real @ X2 @ ( nat2 @ I2 ) ) ) )
% 8.90/5.13 & ( ~ ( ord_less_eq_int @ zero_zero_int @ I2 )
% 8.90/5.13 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ I2 ) )
% 8.90/5.13 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X2 @ ( nat2 @ ( uminus_uminus_int @ I2 ) ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9162_mask__nat__positive__iff,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 8.90/5.13 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9163_nat__numeral,axiom,
% 8.90/5.13 ! [K: num] :
% 8.90/5.13 ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 8.90/5.13 = ( numeral_numeral_nat @ K ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9164_nat__1,axiom,
% 8.90/5.13 ( ( nat2 @ one_one_int )
% 8.90/5.13 = ( suc @ zero_zero_nat ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9165_nat__0__iff,axiom,
% 8.90/5.13 ! [I2: int] :
% 8.90/5.13 ( ( ( nat2 @ I2 )
% 8.90/5.13 = zero_zero_nat )
% 8.90/5.13 = ( ord_less_eq_int @ I2 @ zero_zero_int ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9166_nat__le__0,axiom,
% 8.90/5.13 ! [Z: int] :
% 8.90/5.13 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 8.90/5.13 => ( ( nat2 @ Z )
% 8.90/5.13 = zero_zero_nat ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9167_zless__nat__conj,axiom,
% 8.90/5.13 ! [W: int,Z: int] :
% 8.90/5.13 ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 8.90/5.13 = ( ( ord_less_int @ zero_zero_int @ Z )
% 8.90/5.13 & ( ord_less_int @ W @ Z ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9168_nat__neg__numeral,axiom,
% 8.90/5.13 ! [K: num] :
% 8.90/5.13 ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 8.90/5.13 = zero_zero_nat ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9169_int__nat__eq,axiom,
% 8.90/5.13 ! [Z: int] :
% 8.90/5.13 ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 8.90/5.13 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 8.90/5.13 = Z ) )
% 8.90/5.13 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 8.90/5.13 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 8.90/5.13 = zero_zero_int ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9170_dvd__mult__sgn__iff,axiom,
% 8.90/5.13 ! [L: int,K: int,R2: int] :
% 8.90/5.13 ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
% 8.90/5.13 = ( ( dvd_dvd_int @ L @ K )
% 8.90/5.13 | ( R2 = zero_zero_int ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9171_dvd__sgn__mult__iff,axiom,
% 8.90/5.13 ! [L: int,R2: int,K: int] :
% 8.90/5.13 ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
% 8.90/5.13 = ( ( dvd_dvd_int @ L @ K )
% 8.90/5.13 | ( R2 = zero_zero_int ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9172_mult__sgn__dvd__iff,axiom,
% 8.90/5.13 ! [L: int,R2: int,K: int] :
% 8.90/5.13 ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K )
% 8.90/5.13 = ( ( dvd_dvd_int @ L @ K )
% 8.90/5.13 & ( ( R2 = zero_zero_int )
% 8.90/5.13 => ( K = zero_zero_int ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9173_sgn__mult__dvd__iff,axiom,
% 8.90/5.13 ! [R2: int,L: int,K: int] :
% 8.90/5.13 ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K )
% 8.90/5.13 = ( ( dvd_dvd_int @ L @ K )
% 8.90/5.13 & ( ( R2 = zero_zero_int )
% 8.90/5.13 => ( K = zero_zero_int ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9174_zero__less__nat__eq,axiom,
% 8.90/5.13 ! [Z: int] :
% 8.90/5.13 ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
% 8.90/5.13 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9175_diff__nat__numeral,axiom,
% 8.90/5.13 ! [V: num,V3: num] :
% 8.90/5.13 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
% 8.90/5.13 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9176_nat__eq__numeral__power__cancel__iff,axiom,
% 8.90/5.13 ! [Y3: int,X2: num,N: nat] :
% 8.90/5.13 ( ( ( nat2 @ Y3 )
% 8.90/5.13 = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) )
% 8.90/5.13 = ( Y3
% 8.90/5.13 = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9177_numeral__power__eq__nat__cancel__iff,axiom,
% 8.90/5.13 ! [X2: num,N: nat,Y3: int] :
% 8.90/5.13 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
% 8.90/5.13 = ( nat2 @ Y3 ) )
% 8.90/5.13 = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N )
% 8.90/5.13 = Y3 ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9178_nat__ceiling__le__eq,axiom,
% 8.90/5.13 ! [X2: real,A: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X2 ) ) @ A )
% 8.90/5.13 = ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9179_one__less__nat__eq,axiom,
% 8.90/5.13 ! [Z: int] :
% 8.90/5.13 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
% 8.90/5.13 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9180_nat__numeral__diff__1,axiom,
% 8.90/5.13 ! [V: num] :
% 8.90/5.13 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
% 8.90/5.13 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9181_nat__less__numeral__power__cancel__iff,axiom,
% 8.90/5.13 ! [A: int,X2: num,N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) )
% 8.90/5.13 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9182_numeral__power__less__nat__cancel__iff,axiom,
% 8.90/5.13 ! [X2: num,N: nat,A: int] :
% 8.90/5.13 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) @ ( nat2 @ A ) )
% 8.90/5.13 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9183_nat__le__numeral__power__cancel__iff,axiom,
% 8.90/5.13 ! [A: int,X2: num,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) )
% 8.90/5.13 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9184_numeral__power__le__nat__cancel__iff,axiom,
% 8.90/5.13 ! [X2: num,N: nat,A: int] :
% 8.90/5.13 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) @ ( nat2 @ A ) )
% 8.90/5.13 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9185_less__eq__mask,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9186_nat__mask__eq,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
% 8.90/5.13 = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9187_nat__numeral__as__int,axiom,
% 8.90/5.13 ( numeral_numeral_nat
% 8.90/5.13 = ( ^ [I4: num] : ( nat2 @ ( numeral_numeral_int @ I4 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9188_nat__mono,axiom,
% 8.90/5.13 ! [X2: int,Y3: int] :
% 8.90/5.13 ( ( ord_less_eq_int @ X2 @ Y3 )
% 8.90/5.13 => ( ord_less_eq_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9189_int__sgnE,axiom,
% 8.90/5.13 ! [K: int] :
% 8.90/5.13 ~ ! [N2: nat,L4: int] :
% 8.90/5.13 ( K
% 8.90/5.13 != ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9190_ex__nat,axiom,
% 8.90/5.13 ( ( ^ [P2: nat > $o] :
% 8.90/5.13 ? [X6: nat] : ( P2 @ X6 ) )
% 8.90/5.13 = ( ^ [P3: nat > $o] :
% 8.90/5.13 ? [X: int] :
% 8.90/5.13 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.90/5.13 & ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9191_all__nat,axiom,
% 8.90/5.13 ( ( ^ [P2: nat > $o] :
% 8.90/5.13 ! [X6: nat] : ( P2 @ X6 ) )
% 8.90/5.13 = ( ^ [P3: nat > $o] :
% 8.90/5.13 ! [X: int] :
% 8.90/5.13 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 8.90/5.13 => ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9192_eq__nat__nat__iff,axiom,
% 8.90/5.13 ! [Z: int,Z7: int] :
% 8.90/5.13 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 8.90/5.13 => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 8.90/5.13 => ( ( ( nat2 @ Z )
% 8.90/5.13 = ( nat2 @ Z7 ) )
% 8.90/5.13 = ( Z = Z7 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9193_nat__one__as__int,axiom,
% 8.90/5.13 ( one_one_nat
% 8.90/5.13 = ( nat2 @ one_one_int ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9194_unset__bit__nat__def,axiom,
% 8.90/5.13 ( bit_se4205575877204974255it_nat
% 8.90/5.13 = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M3 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9195_mask__nonnegative__int,axiom,
% 8.90/5.13 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9196_not__mask__negative__int,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9197_nat__mono__iff,axiom,
% 8.90/5.13 ! [Z: int,W: int] :
% 8.90/5.13 ( ( ord_less_int @ zero_zero_int @ Z )
% 8.90/5.13 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 8.90/5.13 = ( ord_less_int @ W @ Z ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9198_zless__nat__eq__int__zless,axiom,
% 8.90/5.13 ! [M: nat,Z: int] :
% 8.90/5.13 ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
% 8.90/5.13 = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9199_nat__le__iff,axiom,
% 8.90/5.13 ! [X2: int,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_nat @ ( nat2 @ X2 ) @ N )
% 8.90/5.13 = ( ord_less_eq_int @ X2 @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9200_int__eq__iff,axiom,
% 8.90/5.13 ! [M: nat,Z: int] :
% 8.90/5.13 ( ( ( semiri1314217659103216013at_int @ M )
% 8.90/5.13 = Z )
% 8.90/5.13 = ( ( M
% 8.90/5.13 = ( nat2 @ Z ) )
% 8.90/5.13 & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9201_nat__0__le,axiom,
% 8.90/5.13 ! [Z: int] :
% 8.90/5.13 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 8.90/5.13 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 8.90/5.13 = Z ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9202_nat__int__add,axiom,
% 8.90/5.13 ! [A: nat,B: nat] :
% 8.90/5.13 ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
% 8.90/5.13 = ( plus_plus_nat @ A @ B ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9203_sgn__mod,axiom,
% 8.90/5.13 ! [L: int,K: int] :
% 8.90/5.13 ( ( L != zero_zero_int )
% 8.90/5.13 => ( ~ ( dvd_dvd_int @ L @ K )
% 8.90/5.13 => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
% 8.90/5.13 = ( sgn_sgn_int @ L ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9204_int__minus,axiom,
% 8.90/5.13 ! [N: nat,M: nat] :
% 8.90/5.13 ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
% 8.90/5.13 = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9205_nat__abs__mult__distrib,axiom,
% 8.90/5.13 ! [W: int,Z: int] :
% 8.90/5.13 ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
% 8.90/5.13 = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9206_real__nat__ceiling__ge,axiom,
% 8.90/5.13 ! [X2: real] : ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X2 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9207_nat__plus__as__int,axiom,
% 8.90/5.13 ( plus_plus_nat
% 8.90/5.13 = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9208_nat__times__as__int,axiom,
% 8.90/5.13 ( times_times_nat
% 8.90/5.13 = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9209_less__mask,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.90/5.13 => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9210_nat__minus__as__int,axiom,
% 8.90/5.13 ( minus_minus_nat
% 8.90/5.13 = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9211_nat__div__as__int,axiom,
% 8.90/5.13 ( divide_divide_nat
% 8.90/5.13 = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9212_zsgn__def,axiom,
% 8.90/5.13 ( sgn_sgn_int
% 8.90/5.13 = ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9213_modulo__int__def,axiom,
% 8.90/5.13 ( modulo_modulo_int
% 8.90/5.13 = ( ^ [K2: int,L2: int] :
% 8.90/5.13 ( if_int @ ( L2 = zero_zero_int ) @ K2
% 8.90/5.13 @ ( if_int
% 8.90/5.13 @ ( ( sgn_sgn_int @ K2 )
% 8.90/5.13 = ( sgn_sgn_int @ L2 ) )
% 8.90/5.13 @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) )
% 8.90/5.13 @ ( times_times_int @ ( sgn_sgn_int @ L2 )
% 8.90/5.13 @ ( minus_minus_int
% 8.90/5.13 @ ( times_times_int @ ( abs_abs_int @ L2 )
% 8.90/5.13 @ ( zero_n2684676970156552555ol_int
% 8.90/5.13 @ ~ ( dvd_dvd_int @ L2 @ K2 ) ) )
% 8.90/5.13 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9214_nat__less__eq__zless,axiom,
% 8.90/5.13 ! [W: int,Z: int] :
% 8.90/5.13 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 8.90/5.13 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 8.90/5.13 = ( ord_less_int @ W @ Z ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9215_nat__le__eq__zle,axiom,
% 8.90/5.13 ! [W: int,Z: int] :
% 8.90/5.13 ( ( ( ord_less_int @ zero_zero_int @ W )
% 8.90/5.13 | ( ord_less_eq_int @ zero_zero_int @ Z ) )
% 8.90/5.13 => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 8.90/5.13 = ( ord_less_eq_int @ W @ Z ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9216_nat__eq__iff,axiom,
% 8.90/5.13 ! [W: int,M: nat] :
% 8.90/5.13 ( ( ( nat2 @ W )
% 8.90/5.13 = M )
% 8.90/5.13 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 8.90/5.13 => ( W
% 8.90/5.13 = ( semiri1314217659103216013at_int @ M ) ) )
% 8.90/5.13 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 8.90/5.13 => ( M = zero_zero_nat ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9217_nat__eq__iff2,axiom,
% 8.90/5.13 ! [M: nat,W: int] :
% 8.90/5.13 ( ( M
% 8.90/5.13 = ( nat2 @ W ) )
% 8.90/5.13 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 8.90/5.13 => ( W
% 8.90/5.13 = ( semiri1314217659103216013at_int @ M ) ) )
% 8.90/5.13 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 8.90/5.13 => ( M = zero_zero_nat ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9218_le__nat__iff,axiom,
% 8.90/5.13 ! [K: int,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.90/5.13 => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 8.90/5.13 = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9219_nat__add__distrib,axiom,
% 8.90/5.13 ! [Z: int,Z7: int] :
% 8.90/5.13 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 8.90/5.13 => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 8.90/5.13 => ( ( nat2 @ ( plus_plus_int @ Z @ Z7 ) )
% 8.90/5.13 = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9220_div__sgn__abs__cancel,axiom,
% 8.90/5.13 ! [V: int,K: int,L: int] :
% 8.90/5.13 ( ( V != zero_zero_int )
% 8.90/5.13 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
% 8.90/5.13 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9221_Suc__as__int,axiom,
% 8.90/5.13 ( suc
% 8.90/5.13 = ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9222_nat__mult__distrib,axiom,
% 8.90/5.13 ! [Z: int,Z7: int] :
% 8.90/5.13 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 8.90/5.13 => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 8.90/5.13 = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9223_nat__diff__distrib_H,axiom,
% 8.90/5.13 ! [X2: int,Y3: int] :
% 8.90/5.13 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.90/5.13 => ( ( nat2 @ ( minus_minus_int @ X2 @ Y3 ) )
% 8.90/5.13 = ( minus_minus_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9224_nat__diff__distrib,axiom,
% 8.90/5.13 ! [Z7: int,Z: int] :
% 8.90/5.13 ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 8.90/5.13 => ( ( ord_less_eq_int @ Z7 @ Z )
% 8.90/5.13 => ( ( nat2 @ ( minus_minus_int @ Z @ Z7 ) )
% 8.90/5.13 = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9225_nat__abs__triangle__ineq,axiom,
% 8.90/5.13 ! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9226_nat__div__distrib_H,axiom,
% 8.90/5.13 ! [Y3: int,X2: int] :
% 8.90/5.13 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.90/5.13 => ( ( nat2 @ ( divide_divide_int @ X2 @ Y3 ) )
% 8.90/5.13 = ( divide_divide_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9227_nat__div__distrib,axiom,
% 8.90/5.13 ! [X2: int,Y3: int] :
% 8.90/5.13 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.90/5.13 => ( ( nat2 @ ( divide_divide_int @ X2 @ Y3 ) )
% 8.90/5.13 = ( divide_divide_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9228_div__dvd__sgn__abs,axiom,
% 8.90/5.13 ! [L: int,K: int] :
% 8.90/5.13 ( ( dvd_dvd_int @ L @ K )
% 8.90/5.13 => ( ( divide_divide_int @ K @ L )
% 8.90/5.13 = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9229_nat__power__eq,axiom,
% 8.90/5.13 ! [Z: int,N: nat] :
% 8.90/5.13 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 8.90/5.13 => ( ( nat2 @ ( power_power_int @ Z @ N ) )
% 8.90/5.13 = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9230_nat__floor__neg,axiom,
% 8.90/5.13 ! [X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 8.90/5.13 => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
% 8.90/5.13 = zero_zero_nat ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9231_nat__mod__distrib,axiom,
% 8.90/5.13 ! [X2: int,Y3: int] :
% 8.90/5.13 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.90/5.13 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.90/5.13 => ( ( nat2 @ ( modulo_modulo_int @ X2 @ Y3 ) )
% 8.90/5.13 = ( modulo_modulo_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9232_div__abs__eq__div__nat,axiom,
% 8.90/5.13 ! [K: int,L: int] :
% 8.90/5.13 ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
% 8.90/5.13 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9233_floor__eq3,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 8.90/5.13 => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
% 8.90/5.13 = N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9234_le__nat__floor,axiom,
% 8.90/5.13 ! [X2: nat,A: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ A )
% 8.90/5.13 => ( ord_less_eq_nat @ X2 @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9235_divide__int__def,axiom,
% 8.90/5.13 ( divide_divide_int
% 8.90/5.13 = ( ^ [K2: int,L2: int] :
% 8.90/5.13 ( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
% 8.90/5.13 @ ( if_int
% 8.90/5.13 @ ( ( sgn_sgn_int @ K2 )
% 8.90/5.13 = ( sgn_sgn_int @ L2 ) )
% 8.90/5.13 @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
% 8.90/5.13 @ ( uminus_uminus_int
% 8.90/5.13 @ ( semiri1314217659103216013at_int
% 8.90/5.13 @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
% 8.90/5.13 @ ( zero_n2687167440665602831ol_nat
% 8.90/5.13 @ ~ ( dvd_dvd_int @ L2 @ K2 ) ) ) ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9236_nat__2,axiom,
% 8.90/5.13 ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.13 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9237_Suc__nat__eq__nat__zadd1,axiom,
% 8.90/5.13 ! [Z: int] :
% 8.90/5.13 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 8.90/5.13 => ( ( suc @ ( nat2 @ Z ) )
% 8.90/5.13 = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9238_nat__less__iff,axiom,
% 8.90/5.13 ! [W: int,M: nat] :
% 8.90/5.13 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 8.90/5.13 => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
% 8.90/5.13 = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9239_nat__mult__distrib__neg,axiom,
% 8.90/5.13 ! [Z: int,Z7: int] :
% 8.90/5.13 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 8.90/5.13 => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 8.90/5.13 = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9240_nat__abs__int__diff,axiom,
% 8.90/5.13 ! [A: nat,B: nat] :
% 8.90/5.13 ( ( ( ord_less_eq_nat @ A @ B )
% 8.90/5.13 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 8.90/5.13 = ( minus_minus_nat @ B @ A ) ) )
% 8.90/5.13 & ( ~ ( ord_less_eq_nat @ A @ B )
% 8.90/5.13 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 8.90/5.13 = ( minus_minus_nat @ A @ B ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9241_floor__eq4,axiom,
% 8.90/5.13 ! [N: nat,X2: real] :
% 8.90/5.13 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X2 )
% 8.90/5.13 => ( ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 8.90/5.13 => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
% 8.90/5.13 = N ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9242_diff__nat__eq__if,axiom,
% 8.90/5.13 ! [Z7: int,Z: int] :
% 8.90/5.13 ( ( ( ord_less_int @ Z7 @ zero_zero_int )
% 8.90/5.13 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
% 8.90/5.13 = ( nat2 @ Z ) ) )
% 8.90/5.13 & ( ~ ( ord_less_int @ Z7 @ zero_zero_int )
% 8.90/5.13 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
% 8.90/5.13 = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z7 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z7 ) ) ) ) ) ) ).
% 8.90/5.13
% 8.90/5.13 thf(fact_9243_Suc__mask__eq__exp,axiom,
% 8.90/5.13 ! [N: nat] :
% 8.90/5.13 ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
% 8.90/5.13 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9244_mask__nat__less__exp,axiom,
% 8.90/5.14 ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9245_div__noneq__sgn__abs,axiom,
% 8.90/5.14 ! [L: int,K: int] :
% 8.90/5.14 ( ( L != zero_zero_int )
% 8.90/5.14 => ( ( ( sgn_sgn_int @ K )
% 8.90/5.14 != ( sgn_sgn_int @ L ) )
% 8.90/5.14 => ( ( divide_divide_int @ K @ L )
% 8.90/5.14 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) )
% 8.90/5.14 @ ( zero_n2684676970156552555ol_int
% 8.90/5.14 @ ~ ( dvd_dvd_int @ L @ K ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9246_nat__dvd__iff,axiom,
% 8.90/5.14 ! [Z: int,M: nat] :
% 8.90/5.14 ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
% 8.90/5.14 = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 8.90/5.14 => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
% 8.90/5.14 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 8.90/5.14 => ( M = zero_zero_nat ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9247_eucl__rel__int__remainderI,axiom,
% 8.90/5.14 ! [R2: int,L: int,K: int,Q3: int] :
% 8.90/5.14 ( ( ( sgn_sgn_int @ R2 )
% 8.90/5.14 = ( sgn_sgn_int @ L ) )
% 8.90/5.14 => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
% 8.90/5.14 => ( ( K
% 8.90/5.14 = ( plus_plus_int @ ( times_times_int @ Q3 @ L ) @ R2 ) )
% 8.90/5.14 => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9248_mask__nat__def,axiom,
% 8.90/5.14 ( bit_se2002935070580805687sk_nat
% 8.90/5.14 = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9249_mask__half__int,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.14 = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9250_mask__int__def,axiom,
% 8.90/5.14 ( bit_se2000444600071755411sk_int
% 8.90/5.14 = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9251_eucl__rel__int_Ocases,axiom,
% 8.90/5.14 ! [A12: int,A23: int,A32: product_prod_int_int] :
% 8.90/5.14 ( ( eucl_rel_int @ A12 @ A23 @ A32 )
% 8.90/5.14 => ( ( ( A23 = zero_zero_int )
% 8.90/5.14 => ( A32
% 8.90/5.14 != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
% 8.90/5.14 => ( ! [Q2: int] :
% 8.90/5.14 ( ( A32
% 8.90/5.14 = ( product_Pair_int_int @ Q2 @ zero_zero_int ) )
% 8.90/5.14 => ( ( A23 != zero_zero_int )
% 8.90/5.14 => ( A12
% 8.90/5.14 != ( times_times_int @ Q2 @ A23 ) ) ) )
% 8.90/5.14 => ~ ! [R3: int,Q2: int] :
% 8.90/5.14 ( ( A32
% 8.90/5.14 = ( product_Pair_int_int @ Q2 @ R3 ) )
% 8.90/5.14 => ( ( ( sgn_sgn_int @ R3 )
% 8.90/5.14 = ( sgn_sgn_int @ A23 ) )
% 8.90/5.14 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A23 ) )
% 8.90/5.14 => ( A12
% 8.90/5.14 != ( plus_plus_int @ ( times_times_int @ Q2 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9252_eucl__rel__int_Osimps,axiom,
% 8.90/5.14 ( eucl_rel_int
% 8.90/5.14 = ( ^ [A1: int,A22: int,A33: product_prod_int_int] :
% 8.90/5.14 ( ? [K2: int] :
% 8.90/5.14 ( ( A1 = K2 )
% 8.90/5.14 & ( A22 = zero_zero_int )
% 8.90/5.14 & ( A33
% 8.90/5.14 = ( product_Pair_int_int @ zero_zero_int @ K2 ) ) )
% 8.90/5.14 | ? [L2: int,K2: int,Q4: int] :
% 8.90/5.14 ( ( A1 = K2 )
% 8.90/5.14 & ( A22 = L2 )
% 8.90/5.14 & ( A33
% 8.90/5.14 = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
% 8.90/5.14 & ( L2 != zero_zero_int )
% 8.90/5.14 & ( K2
% 8.90/5.14 = ( times_times_int @ Q4 @ L2 ) ) )
% 8.90/5.14 | ? [R5: int,L2: int,K2: int,Q4: int] :
% 8.90/5.14 ( ( A1 = K2 )
% 8.90/5.14 & ( A22 = L2 )
% 8.90/5.14 & ( A33
% 8.90/5.14 = ( product_Pair_int_int @ Q4 @ R5 ) )
% 8.90/5.14 & ( ( sgn_sgn_int @ R5 )
% 8.90/5.14 = ( sgn_sgn_int @ L2 ) )
% 8.90/5.14 & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
% 8.90/5.14 & ( K2
% 8.90/5.14 = ( plus_plus_int @ ( times_times_int @ Q4 @ L2 ) @ R5 ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9253_even__nat__iff,axiom,
% 8.90/5.14 ! [K: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.90/5.14 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
% 8.90/5.14 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9254_set__encode__def,axiom,
% 8.90/5.14 ( nat_set_encode
% 8.90/5.14 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9255_powr__real__of__int,axiom,
% 8.90/5.14 ! [X2: real,N: int] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.14 => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
% 8.90/5.14 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N ) )
% 8.90/5.14 = ( power_power_real @ X2 @ ( nat2 @ N ) ) ) )
% 8.90/5.14 & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
% 8.90/5.14 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N ) )
% 8.90/5.14 = ( inverse_inverse_real @ ( power_power_real @ X2 @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9256_divide__int__unfold,axiom,
% 8.90/5.14 ! [L: int,K: int,N: nat,M: nat] :
% 8.90/5.14 ( ( ( ( ( sgn_sgn_int @ L )
% 8.90/5.14 = zero_zero_int )
% 8.90/5.14 | ( ( sgn_sgn_int @ K )
% 8.90/5.14 = zero_zero_int )
% 8.90/5.14 | ( N = zero_zero_nat ) )
% 8.90/5.14 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 8.90/5.14 = zero_zero_int ) )
% 8.90/5.14 & ( ~ ( ( ( sgn_sgn_int @ L )
% 8.90/5.14 = zero_zero_int )
% 8.90/5.14 | ( ( sgn_sgn_int @ K )
% 8.90/5.14 = zero_zero_int )
% 8.90/5.14 | ( N = zero_zero_nat ) )
% 8.90/5.14 => ( ( ( ( sgn_sgn_int @ K )
% 8.90/5.14 = ( sgn_sgn_int @ L ) )
% 8.90/5.14 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 8.90/5.14 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
% 8.90/5.14 & ( ( ( sgn_sgn_int @ K )
% 8.90/5.14 != ( sgn_sgn_int @ L ) )
% 8.90/5.14 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 8.90/5.14 = ( uminus_uminus_int
% 8.90/5.14 @ ( semiri1314217659103216013at_int
% 8.90/5.14 @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
% 8.90/5.14 @ ( zero_n2687167440665602831ol_nat
% 8.90/5.14 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9257_num_Osize__gen_I3_J,axiom,
% 8.90/5.14 ! [X32: num] :
% 8.90/5.14 ( ( size_num @ ( bit1 @ X32 ) )
% 8.90/5.14 = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9258_num_Osize__gen_I2_J,axiom,
% 8.90/5.14 ! [X22: num] :
% 8.90/5.14 ( ( size_num @ ( bit0 @ X22 ) )
% 8.90/5.14 = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9259_zero__le__sgn__iff,axiom,
% 8.90/5.14 ! [X2: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X2 ) )
% 8.90/5.14 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9260_sgn__le__0__iff,axiom,
% 8.90/5.14 ! [X2: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ ( sgn_sgn_real @ X2 ) @ zero_zero_real )
% 8.90/5.14 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9261_concat__bit__of__zero__2,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ( ( bit_concat_bit @ N @ K @ zero_zero_int )
% 8.90/5.14 = ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9262_take__bit__of__Suc__0,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
% 8.90/5.14 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9263_nat__take__bit__eq,axiom,
% 8.90/5.14 ! [K: int,N: nat] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.90/5.14 => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 8.90/5.14 = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9264_take__bit__nat__eq,axiom,
% 8.90/5.14 ! [K: int,N: nat] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.90/5.14 => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
% 8.90/5.14 = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9265_take__bit__mult,axiom,
% 8.90/5.14 ! [N: nat,K: int,L: int] :
% 8.90/5.14 ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
% 8.90/5.14 = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9266_take__bit__diff,axiom,
% 8.90/5.14 ! [N: nat,K: int,L: int] :
% 8.90/5.14 ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
% 8.90/5.14 = ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9267_take__bit__minus,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 8.90/5.14 = ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9268_take__bit__nat__less__eq__self,axiom,
% 8.90/5.14 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9269_take__bit__tightened__less__eq__nat,axiom,
% 8.90/5.14 ! [M: nat,N: nat,Q3: nat] :
% 8.90/5.14 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.14 => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q3 ) @ ( bit_se2925701944663578781it_nat @ N @ Q3 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9270_concat__bit__eq__iff,axiom,
% 8.90/5.14 ! [N: nat,K: int,L: int,R2: int,S: int] :
% 8.90/5.14 ( ( ( bit_concat_bit @ N @ K @ L )
% 8.90/5.14 = ( bit_concat_bit @ N @ R2 @ S ) )
% 8.90/5.14 = ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 8.90/5.14 = ( bit_se2923211474154528505it_int @ N @ R2 ) )
% 8.90/5.14 & ( L = S ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9271_concat__bit__take__bit__eq,axiom,
% 8.90/5.14 ! [N: nat,B: int] :
% 8.90/5.14 ( ( bit_concat_bit @ N @ ( bit_se2923211474154528505it_int @ N @ B ) )
% 8.90/5.14 = ( bit_concat_bit @ N @ B ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9272_take__bit__tightened__less__eq__int,axiom,
% 8.90/5.14 ! [M: nat,N: nat,K: int] :
% 8.90/5.14 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.14 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9273_take__bit__nonnegative,axiom,
% 8.90/5.14 ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9274_take__bit__int__less__eq__self__iff,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 8.90/5.14 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9275_not__take__bit__negative,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9276_take__bit__int__greater__self__iff,axiom,
% 8.90/5.14 ! [K: int,N: nat] :
% 8.90/5.14 ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 8.90/5.14 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9277_sgn__root,axiom,
% 8.90/5.14 ! [N: nat,X2: real] :
% 8.90/5.14 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( ( sgn_sgn_real @ ( root @ N @ X2 ) )
% 8.90/5.14 = ( sgn_sgn_real @ X2 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9278_take__bit__eq__mask__iff,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 8.90/5.14 = ( bit_se2000444600071755411sk_int @ N ) )
% 8.90/5.14 = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 8.90/5.14 = zero_zero_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9279_take__bit__decr__eq,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 8.90/5.14 != zero_zero_int )
% 8.90/5.14 => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
% 8.90/5.14 = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9280_sgn__real__def,axiom,
% 8.90/5.14 ( sgn_sgn_real
% 8.90/5.14 = ( ^ [A3: real] : ( if_real @ ( A3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9281_take__bit__nat__eq__self,axiom,
% 8.90/5.14 ! [M: nat,N: nat] :
% 8.90/5.14 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.14 => ( ( bit_se2925701944663578781it_nat @ N @ M )
% 8.90/5.14 = M ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9282_take__bit__nat__less__exp,axiom,
% 8.90/5.14 ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9283_take__bit__nat__eq__self__iff,axiom,
% 8.90/5.14 ! [N: nat,M: nat] :
% 8.90/5.14 ( ( ( bit_se2925701944663578781it_nat @ N @ M )
% 8.90/5.14 = M )
% 8.90/5.14 = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9284_take__bit__nat__def,axiom,
% 8.90/5.14 ( bit_se2925701944663578781it_nat
% 8.90/5.14 = ( ^ [N3: nat,M3: nat] : ( modulo_modulo_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9285_sgn__power__injE,axiom,
% 8.90/5.14 ! [A: real,N: nat,X2: real,B: real] :
% 8.90/5.14 ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 8.90/5.14 = X2 )
% 8.90/5.14 => ( ( X2
% 8.90/5.14 = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
% 8.90/5.14 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( A = B ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9286_take__bit__int__less__exp,axiom,
% 8.90/5.14 ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9287_take__bit__int__def,axiom,
% 8.90/5.14 ( bit_se2923211474154528505it_int
% 8.90/5.14 = ( ^ [N3: nat,K2: int] : ( modulo_modulo_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9288_num_Osize__gen_I1_J,axiom,
% 8.90/5.14 ( ( size_num @ one )
% 8.90/5.14 = zero_zero_nat ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9289_take__bit__nat__less__self__iff,axiom,
% 8.90/5.14 ! [N: nat,M: nat] :
% 8.90/5.14 ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
% 8.90/5.14 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9290_take__bit__Suc__minus__bit0,axiom,
% 8.90/5.14 ! [N: nat,K: num] :
% 8.90/5.14 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 8.90/5.14 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9291_sgn__power__root,axiom,
% 8.90/5.14 ! [N: nat,X2: real] :
% 8.90/5.14 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X2 ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X2 ) ) @ N ) )
% 8.90/5.14 = X2 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9292_root__sgn__power,axiom,
% 8.90/5.14 ! [N: nat,Y3: real] :
% 8.90/5.14 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) ) )
% 8.90/5.14 = Y3 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9293_take__bit__int__greater__eq__self__iff,axiom,
% 8.90/5.14 ! [K: int,N: nat] :
% 8.90/5.14 ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 8.90/5.14 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9294_take__bit__int__less__self__iff,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 8.90/5.14 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9295_cis__Arg__unique,axiom,
% 8.90/5.14 ! [Z: complex,X2: real] :
% 8.90/5.14 ( ( ( sgn_sgn_complex @ Z )
% 8.90/5.14 = ( cis @ X2 ) )
% 8.90/5.14 => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X2 )
% 8.90/5.14 => ( ( ord_less_eq_real @ X2 @ pi )
% 8.90/5.14 => ( ( arg @ Z )
% 8.90/5.14 = X2 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9296_split__root,axiom,
% 8.90/5.14 ! [P: real > $o,N: nat,X2: real] :
% 8.90/5.14 ( ( P @ ( root @ N @ X2 ) )
% 8.90/5.14 = ( ( ( N = zero_zero_nat )
% 8.90/5.14 => ( P @ zero_zero_real ) )
% 8.90/5.14 & ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ! [Y: real] :
% 8.90/5.14 ( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 8.90/5.14 = X2 )
% 8.90/5.14 => ( P @ Y ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9297_take__bit__int__eq__self,axiom,
% 8.90/5.14 ! [K: int,N: nat] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.90/5.14 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.14 => ( ( bit_se2923211474154528505it_int @ N @ K )
% 8.90/5.14 = K ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9298_take__bit__int__eq__self__iff,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 8.90/5.14 = K )
% 8.90/5.14 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.90/5.14 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9299_take__bit__numeral__minus__bit0,axiom,
% 8.90/5.14 ! [L: num,K: num] :
% 8.90/5.14 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 8.90/5.14 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9300_take__bit__incr__eq,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 8.90/5.14 != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 8.90/5.14 => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 8.90/5.14 = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9301_take__bit__eq__mask__iff__exp__dvd,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 8.90/5.14 = ( bit_se2000444600071755411sk_int @ N ) )
% 8.90/5.14 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9302_Arg__correct,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( Z != zero_zero_complex )
% 8.90/5.14 => ( ( ( sgn_sgn_complex @ Z )
% 8.90/5.14 = ( cis @ ( arg @ Z ) ) )
% 8.90/5.14 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 8.90/5.14 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9303_take__bit__int__less__eq,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 8.90/5.14 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9304_take__bit__int__greater__eq,axiom,
% 8.90/5.14 ! [K: int,N: nat] :
% 8.90/5.14 ( ( ord_less_int @ K @ zero_zero_int )
% 8.90/5.14 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9305_signed__take__bit__eq__take__bit__shift,axiom,
% 8.90/5.14 ( bit_ri631733984087533419it_int
% 8.90/5.14 = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9306_take__bit__minus__small__eq,axiom,
% 8.90/5.14 ! [K: int,N: nat] :
% 8.90/5.14 ( ( ord_less_int @ zero_zero_int @ K )
% 8.90/5.14 => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.14 => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
% 8.90/5.14 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9307_arctan__inverse,axiom,
% 8.90/5.14 ! [X2: real] :
% 8.90/5.14 ( ( X2 != zero_zero_real )
% 8.90/5.14 => ( ( arctan @ ( divide_divide_real @ one_one_real @ X2 ) )
% 8.90/5.14 = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X2 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X2 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9308_take__bit__numeral__minus__bit1,axiom,
% 8.90/5.14 ! [L: num,K: num] :
% 8.90/5.14 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 8.90/5.14 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9309_take__bit__Suc__minus__bit1,axiom,
% 8.90/5.14 ! [N: nat,K: num] :
% 8.90/5.14 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 8.90/5.14 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9310_and__int__unfold,axiom,
% 8.90/5.14 ( bit_se725231765392027082nd_int
% 8.90/5.14 = ( ^ [K2: int,L2: int] :
% 8.90/5.14 ( if_int
% 8.90/5.14 @ ( ( K2 = zero_zero_int )
% 8.90/5.14 | ( L2 = zero_zero_int ) )
% 8.90/5.14 @ zero_zero_int
% 8.90/5.14 @ ( if_int
% 8.90/5.14 @ ( K2
% 8.90/5.14 = ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.14 @ L2
% 8.90/5.14 @ ( if_int
% 8.90/5.14 @ ( L2
% 8.90/5.14 = ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.14 @ K2
% 8.90/5.14 @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9311_and__nonnegative__int__iff,axiom,
% 8.90/5.14 ! [K: int,L: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 8.90/5.14 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.90/5.14 | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9312_and__negative__int__iff,axiom,
% 8.90/5.14 ! [K: int,L: int] :
% 8.90/5.14 ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
% 8.90/5.14 = ( ( ord_less_int @ K @ zero_zero_int )
% 8.90/5.14 & ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9313_pred__numeral__inc,axiom,
% 8.90/5.14 ! [K: num] :
% 8.90/5.14 ( ( pred_numeral @ ( inc @ K ) )
% 8.90/5.14 = ( numeral_numeral_nat @ K ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9314_and__minus__numerals_I6_J,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 8.90/5.14 = one_one_int ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9315_and__minus__numerals_I2_J,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.90/5.14 = one_one_int ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9316_and__minus__numerals_I5_J,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 8.90/5.14 = zero_zero_int ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9317_and__minus__numerals_I1_J,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.90/5.14 = zero_zero_int ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9318_num__induct,axiom,
% 8.90/5.14 ! [P: num > $o,X2: num] :
% 8.90/5.14 ( ( P @ one )
% 8.90/5.14 => ( ! [X5: num] :
% 8.90/5.14 ( ( P @ X5 )
% 8.90/5.14 => ( P @ ( inc @ X5 ) ) )
% 8.90/5.14 => ( P @ X2 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9319_add__inc,axiom,
% 8.90/5.14 ! [X2: num,Y3: num] :
% 8.90/5.14 ( ( plus_plus_num @ X2 @ ( inc @ Y3 ) )
% 8.90/5.14 = ( inc @ ( plus_plus_num @ X2 @ Y3 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9320_AND__upper2_H,axiom,
% 8.90/5.14 ! [Y3: int,Z: int,X2: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.90/5.14 => ( ( ord_less_eq_int @ Y3 @ Z )
% 8.90/5.14 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ Z ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9321_AND__upper1_H,axiom,
% 8.90/5.14 ! [Y3: int,Z: int,Ya: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.90/5.14 => ( ( ord_less_eq_int @ Y3 @ Z )
% 8.90/5.14 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9322_AND__upper2,axiom,
% 8.90/5.14 ! [Y3: int,X2: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.90/5.14 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ Y3 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9323_AND__upper1,axiom,
% 8.90/5.14 ! [X2: int,Y3: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.90/5.14 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ X2 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9324_AND__lower,axiom,
% 8.90/5.14 ! [X2: int,Y3: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.90/5.14 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9325_pow_Osimps_I1_J,axiom,
% 8.90/5.14 ! [X2: num] :
% 8.90/5.14 ( ( pow @ X2 @ one )
% 8.90/5.14 = X2 ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9326_inc_Osimps_I1_J,axiom,
% 8.90/5.14 ( ( inc @ one )
% 8.90/5.14 = ( bit0 @ one ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9327_inc_Osimps_I2_J,axiom,
% 8.90/5.14 ! [X2: num] :
% 8.90/5.14 ( ( inc @ ( bit0 @ X2 ) )
% 8.90/5.14 = ( bit1 @ X2 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9328_inc_Osimps_I3_J,axiom,
% 8.90/5.14 ! [X2: num] :
% 8.90/5.14 ( ( inc @ ( bit1 @ X2 ) )
% 8.90/5.14 = ( bit0 @ ( inc @ X2 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9329_add__One,axiom,
% 8.90/5.14 ! [X2: num] :
% 8.90/5.14 ( ( plus_plus_num @ X2 @ one )
% 8.90/5.14 = ( inc @ X2 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9330_and__less__eq,axiom,
% 8.90/5.14 ! [L: int,K: int] :
% 8.90/5.14 ( ( ord_less_int @ L @ zero_zero_int )
% 8.90/5.14 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9331_AND__upper1_H_H,axiom,
% 8.90/5.14 ! [Y3: int,Z: int,Ya: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.90/5.14 => ( ( ord_less_int @ Y3 @ Z )
% 8.90/5.14 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9332_AND__upper2_H_H,axiom,
% 8.90/5.14 ! [Y3: int,Z: int,X2: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.90/5.14 => ( ( ord_less_int @ Y3 @ Z )
% 8.90/5.14 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ Z ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9333_inc__BitM__eq,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( inc @ ( bitM @ N ) )
% 8.90/5.14 = ( bit0 @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9334_BitM__inc__eq,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( bitM @ ( inc @ N ) )
% 8.90/5.14 = ( bit1 @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9335_mult__inc,axiom,
% 8.90/5.14 ! [X2: num,Y3: num] :
% 8.90/5.14 ( ( times_times_num @ X2 @ ( inc @ Y3 ) )
% 8.90/5.14 = ( plus_plus_num @ ( times_times_num @ X2 @ Y3 ) @ X2 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9336_even__and__iff__int,axiom,
% 8.90/5.14 ! [K: int,L: int] :
% 8.90/5.14 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 8.90/5.14 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 8.90/5.14 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9337_and__int__rec,axiom,
% 8.90/5.14 ( bit_se725231765392027082nd_int
% 8.90/5.14 = ( ^ [K2: int,L2: int] :
% 8.90/5.14 ( plus_plus_int
% 8.90/5.14 @ ( zero_n2684676970156552555ol_int
% 8.90/5.14 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 8.90/5.14 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 8.90/5.14 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9338_and__int_Oelims,axiom,
% 8.90/5.14 ! [X2: int,Xa2: int,Y3: int] :
% 8.90/5.14 ( ( ( bit_se725231765392027082nd_int @ X2 @ Xa2 )
% 8.90/5.14 = Y3 )
% 8.90/5.14 => ( ( ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 8.90/5.14 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 8.90/5.14 => ( Y3
% 8.90/5.14 = ( uminus_uminus_int
% 8.90/5.14 @ ( zero_n2684676970156552555ol_int
% 8.90/5.14 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
% 8.90/5.14 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 8.90/5.14 & ( ~ ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 8.90/5.14 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 8.90/5.14 => ( Y3
% 8.90/5.14 = ( plus_plus_int
% 8.90/5.14 @ ( zero_n2684676970156552555ol_int
% 8.90/5.14 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
% 8.90/5.14 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 8.90/5.14 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9339_and__int_Osimps,axiom,
% 8.90/5.14 ( bit_se725231765392027082nd_int
% 8.90/5.14 = ( ^ [K2: int,L2: int] :
% 8.90/5.14 ( if_int
% 8.90/5.14 @ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 8.90/5.14 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 8.90/5.14 @ ( uminus_uminus_int
% 8.90/5.14 @ ( zero_n2684676970156552555ol_int
% 8.90/5.14 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 8.90/5.14 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 8.90/5.14 @ ( plus_plus_int
% 8.90/5.14 @ ( zero_n2684676970156552555ol_int
% 8.90/5.14 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 8.90/5.14 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 8.90/5.14 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9340_Arg__def,axiom,
% 8.90/5.14 ( arg
% 8.90/5.14 = ( ^ [Z3: complex] :
% 8.90/5.14 ( if_real @ ( Z3 = zero_zero_complex ) @ zero_zero_real
% 8.90/5.14 @ ( fChoice_real
% 8.90/5.14 @ ^ [A3: real] :
% 8.90/5.14 ( ( ( sgn_sgn_complex @ Z3 )
% 8.90/5.14 = ( cis @ A3 ) )
% 8.90/5.14 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A3 )
% 8.90/5.14 & ( ord_less_eq_real @ A3 @ pi ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9341_and__int_Opelims,axiom,
% 8.90/5.14 ! [X2: int,Xa2: int,Y3: int] :
% 8.90/5.14 ( ( ( bit_se725231765392027082nd_int @ X2 @ Xa2 )
% 8.90/5.14 = Y3 )
% 8.90/5.14 => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) )
% 8.90/5.14 => ~ ( ( ( ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 8.90/5.14 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 8.90/5.14 => ( Y3
% 8.90/5.14 = ( uminus_uminus_int
% 8.90/5.14 @ ( zero_n2684676970156552555ol_int
% 8.90/5.14 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
% 8.90/5.14 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 8.90/5.14 & ( ~ ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 8.90/5.14 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 8.90/5.14 => ( Y3
% 8.90/5.14 = ( plus_plus_int
% 8.90/5.14 @ ( zero_n2684676970156552555ol_int
% 8.90/5.14 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
% 8.90/5.14 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 8.90/5.14 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 8.90/5.14 => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9342_and__nat__numerals_I3_J,axiom,
% 8.90/5.14 ! [X2: num] :
% 8.90/5.14 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 8.90/5.14 = zero_zero_nat ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9343_and__nat__numerals_I1_J,axiom,
% 8.90/5.14 ! [Y3: num] :
% 8.90/5.14 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 8.90/5.14 = zero_zero_nat ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9344_and__nat__numerals_I2_J,axiom,
% 8.90/5.14 ! [Y3: num] :
% 8.90/5.14 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 8.90/5.14 = one_one_nat ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9345_and__nat__numerals_I4_J,axiom,
% 8.90/5.14 ! [X2: num] :
% 8.90/5.14 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 8.90/5.14 = one_one_nat ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9346_Suc__0__and__eq,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.90/5.14 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9347_and__Suc__0__eq,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
% 8.90/5.14 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9348_and__nat__def,axiom,
% 8.90/5.14 ( bit_se727722235901077358nd_nat
% 8.90/5.14 = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9349_atLeastAtMostPlus1__int__conv,axiom,
% 8.90/5.14 ! [M: int,N: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 8.90/5.14 => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 8.90/5.14 = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9350_simp__from__to,axiom,
% 8.90/5.14 ( set_or1266510415728281911st_int
% 8.90/5.14 = ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9351_and__int_Opinduct,axiom,
% 8.90/5.14 ! [A0: int,A12: int,P: int > int > $o] :
% 8.90/5.14 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 8.90/5.14 => ( ! [K3: int,L4: int] :
% 8.90/5.14 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L4 ) )
% 8.90/5.14 => ( ( ~ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 8.90/5.14 & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 8.90/5.14 => ( P @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 8.90/5.14 => ( P @ K3 @ L4 ) ) )
% 8.90/5.14 => ( P @ A0 @ A12 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9352_and__nat__unfold,axiom,
% 8.90/5.14 ( bit_se727722235901077358nd_nat
% 8.90/5.14 = ( ^ [M3: nat,N3: nat] :
% 8.90/5.14 ( if_nat
% 8.90/5.14 @ ( ( M3 = zero_zero_nat )
% 8.90/5.14 | ( N3 = zero_zero_nat ) )
% 8.90/5.14 @ zero_zero_nat
% 8.90/5.14 @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9353_and__nat__rec,axiom,
% 8.90/5.14 ( bit_se727722235901077358nd_nat
% 8.90/5.14 = ( ^ [M3: nat,N3: nat] :
% 8.90/5.14 ( plus_plus_nat
% 8.90/5.14 @ ( zero_n2687167440665602831ol_nat
% 8.90/5.14 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 8.90/5.14 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 8.90/5.14 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9354_and__int_Opsimps,axiom,
% 8.90/5.14 ! [K: int,L: int] :
% 8.90/5.14 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
% 8.90/5.14 => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 8.90/5.14 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 8.90/5.14 => ( ( bit_se725231765392027082nd_int @ K @ L )
% 8.90/5.14 = ( uminus_uminus_int
% 8.90/5.14 @ ( zero_n2684676970156552555ol_int
% 8.90/5.14 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 8.90/5.14 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
% 8.90/5.14 & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 8.90/5.14 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 8.90/5.14 => ( ( bit_se725231765392027082nd_int @ K @ L )
% 8.90/5.14 = ( plus_plus_int
% 8.90/5.14 @ ( zero_n2684676970156552555ol_int
% 8.90/5.14 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 8.90/5.14 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 8.90/5.14 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9355_signed__take__bit__eq__take__bit__minus,axiom,
% 8.90/5.14 ( bit_ri631733984087533419it_int
% 8.90/5.14 = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ K2 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9356_or__int__unfold,axiom,
% 8.90/5.14 ( bit_se1409905431419307370or_int
% 8.90/5.14 = ( ^ [K2: int,L2: int] :
% 8.90/5.14 ( if_int
% 8.90/5.14 @ ( ( K2
% 8.90/5.14 = ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.14 | ( L2
% 8.90/5.14 = ( uminus_uminus_int @ one_one_int ) ) )
% 8.90/5.14 @ ( uminus_uminus_int @ one_one_int )
% 8.90/5.14 @ ( if_int @ ( K2 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9357_cis__multiple__2pi,axiom,
% 8.90/5.14 ! [N: real] :
% 8.90/5.14 ( ( member_real @ N @ ring_1_Ints_real )
% 8.90/5.14 => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 8.90/5.14 = one_one_complex ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9358_or__nonnegative__int__iff,axiom,
% 8.90/5.14 ! [K: int,L: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
% 8.90/5.14 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.90/5.14 & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9359_or__negative__int__iff,axiom,
% 8.90/5.14 ! [K: int,L: int] :
% 8.90/5.14 ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
% 8.90/5.14 = ( ( ord_less_int @ K @ zero_zero_int )
% 8.90/5.14 | ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9360_signed__take__bit__nonnegative__iff,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 8.90/5.14 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9361_signed__take__bit__negative__iff,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
% 8.90/5.14 = ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9362_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 8.90/5.14 ! [W: num,N: nat] :
% 8.90/5.14 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
% 8.90/5.14 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9363_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 8.90/5.14 ! [W: num,N: nat] :
% 8.90/5.14 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
% 8.90/5.14 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9364_or__minus__numerals_I6_J,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 8.90/5.14 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9365_or__minus__numerals_I2_J,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.90/5.14 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9366_bit__minus__numeral__int_I1_J,axiom,
% 8.90/5.14 ! [W: num,N: num] :
% 8.90/5.14 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 8.90/5.14 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9367_bit__minus__numeral__int_I2_J,axiom,
% 8.90/5.14 ! [W: num,N: num] :
% 8.90/5.14 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 8.90/5.14 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9368_set__encode__insert,axiom,
% 8.90/5.14 ! [A2: set_nat,N: nat] :
% 8.90/5.14 ( ( finite_finite_nat @ A2 )
% 8.90/5.14 => ( ~ ( member_nat @ N @ A2 )
% 8.90/5.14 => ( ( nat_set_encode @ ( insert_nat @ N @ A2 ) )
% 8.90/5.14 = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9369_bit__and__int__iff,axiom,
% 8.90/5.14 ! [K: int,L: int,N: nat] :
% 8.90/5.14 ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ N )
% 8.90/5.14 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 8.90/5.14 & ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9370_bit__or__int__iff,axiom,
% 8.90/5.14 ! [K: int,L: int,N: nat] :
% 8.90/5.14 ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ N )
% 8.90/5.14 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 8.90/5.14 | ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9371_or__greater__eq,axiom,
% 8.90/5.14 ! [L: int,K: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ L )
% 8.90/5.14 => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9372_OR__lower,axiom,
% 8.90/5.14 ! [X2: int,Y3: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.90/5.14 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.90/5.14 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X2 @ Y3 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9373_plus__and__or,axiom,
% 8.90/5.14 ! [X2: int,Y3: int] :
% 8.90/5.14 ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ ( bit_se1409905431419307370or_int @ X2 @ Y3 ) )
% 8.90/5.14 = ( plus_plus_int @ X2 @ Y3 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9374_lessThan__Suc,axiom,
% 8.90/5.14 ! [K: nat] :
% 8.90/5.14 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 8.90/5.14 = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9375_atMost__Suc,axiom,
% 8.90/5.14 ! [K: nat] :
% 8.90/5.14 ( ( set_ord_atMost_nat @ ( suc @ K ) )
% 8.90/5.14 = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9376_bit__not__int__iff_H,axiom,
% 8.90/5.14 ! [K: int,N: nat] :
% 8.90/5.14 ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
% 8.90/5.14 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9377_atLeast0__atMost__Suc,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 8.90/5.14 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9378_Icc__eq__insert__lb__nat,axiom,
% 8.90/5.14 ! [M: nat,N: nat] :
% 8.90/5.14 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.14 => ( ( set_or1269000886237332187st_nat @ M @ N )
% 8.90/5.14 = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9379_atLeastAtMostSuc__conv,axiom,
% 8.90/5.14 ! [M: nat,N: nat] :
% 8.90/5.14 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 8.90/5.14 => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
% 8.90/5.14 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9380_atLeastAtMost__insertL,axiom,
% 8.90/5.14 ! [M: nat,N: nat] :
% 8.90/5.14 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.14 => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 8.90/5.14 = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9381_lessThan__nat__numeral,axiom,
% 8.90/5.14 ! [K: num] :
% 8.90/5.14 ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
% 8.90/5.14 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9382_atMost__nat__numeral,axiom,
% 8.90/5.14 ! [K: num] :
% 8.90/5.14 ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
% 8.90/5.14 = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9383_bit__imp__take__bit__positive,axiom,
% 8.90/5.14 ! [N: nat,M: nat,K: int] :
% 8.90/5.14 ( ( ord_less_nat @ N @ M )
% 8.90/5.14 => ( ( bit_se1146084159140164899it_int @ K @ N )
% 8.90/5.14 => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9384_bit__concat__bit__iff,axiom,
% 8.90/5.14 ! [M: nat,K: int,L: int,N: nat] :
% 8.90/5.14 ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
% 8.90/5.14 = ( ( ( ord_less_nat @ N @ M )
% 8.90/5.14 & ( bit_se1146084159140164899it_int @ K @ N ) )
% 8.90/5.14 | ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.14 & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9385_sin__times__pi__eq__0,axiom,
% 8.90/5.14 ! [X2: real] :
% 8.90/5.14 ( ( ( sin_real @ ( times_times_real @ X2 @ pi ) )
% 8.90/5.14 = zero_zero_real )
% 8.90/5.14 = ( member_real @ X2 @ ring_1_Ints_real ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9386_signed__take__bit__eq__concat__bit,axiom,
% 8.90/5.14 ( bit_ri631733984087533419it_int
% 8.90/5.14 = ( ^ [N3: nat,K2: int] : ( bit_concat_bit @ N3 @ K2 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9387_int__bit__bound,axiom,
% 8.90/5.14 ! [K: int] :
% 8.90/5.14 ~ ! [N2: nat] :
% 8.90/5.14 ( ! [M2: nat] :
% 8.90/5.14 ( ( ord_less_eq_nat @ N2 @ M2 )
% 8.90/5.14 => ( ( bit_se1146084159140164899it_int @ K @ M2 )
% 8.90/5.14 = ( bit_se1146084159140164899it_int @ K @ N2 ) ) )
% 8.90/5.14 => ~ ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 8.90/5.14 => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 8.90/5.14 = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9388_atLeast1__atMost__eq__remove0,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.90/5.14 = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9389_bit__int__def,axiom,
% 8.90/5.14 ( bit_se1146084159140164899it_int
% 8.90/5.14 = ( ^ [K2: int,N3: nat] :
% 8.90/5.14 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9390_set__decode__plus__power__2,axiom,
% 8.90/5.14 ! [N: nat,Z: nat] :
% 8.90/5.14 ( ~ ( member_nat @ N @ ( nat_set_decode @ Z ) )
% 8.90/5.14 => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z ) )
% 8.90/5.14 = ( insert_nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9391_OR__upper,axiom,
% 8.90/5.14 ! [X2: int,N: nat,Y3: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.90/5.14 => ( ( ord_less_int @ X2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.14 => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.14 => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X2 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9392_sin__integer__2pi,axiom,
% 8.90/5.14 ! [N: real] :
% 8.90/5.14 ( ( member_real @ N @ ring_1_Ints_real )
% 8.90/5.14 => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 8.90/5.14 = zero_zero_real ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9393_cos__integer__2pi,axiom,
% 8.90/5.14 ! [N: real] :
% 8.90/5.14 ( ( member_real @ N @ ring_1_Ints_real )
% 8.90/5.14 => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 8.90/5.14 = one_one_real ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9394_or__int__rec,axiom,
% 8.90/5.14 ( bit_se1409905431419307370or_int
% 8.90/5.14 = ( ^ [K2: int,L2: int] :
% 8.90/5.14 ( plus_plus_int
% 8.90/5.14 @ ( zero_n2684676970156552555ol_int
% 8.90/5.14 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 8.90/5.14 | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 8.90/5.14 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9395_set__bit__eq,axiom,
% 8.90/5.14 ( bit_se7879613467334960850it_int
% 8.90/5.14 = ( ^ [N3: nat,K2: int] :
% 8.90/5.14 ( plus_plus_int @ K2
% 8.90/5.14 @ ( times_times_int
% 8.90/5.14 @ ( zero_n2684676970156552555ol_int
% 8.90/5.14 @ ~ ( bit_se1146084159140164899it_int @ K2 @ N3 ) )
% 8.90/5.14 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9396_unset__bit__eq,axiom,
% 8.90/5.14 ( bit_se4203085406695923979it_int
% 8.90/5.14 = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ K2 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9397_take__bit__Suc__from__most,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
% 8.90/5.14 = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N ) ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9398_or__minus__numerals_I5_J,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 8.90/5.14 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9399_or__minus__numerals_I1_J,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.90/5.14 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9400_xor__Suc__0__eq,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
% 8.90/5.14 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.90/5.14 @ ( zero_n2687167440665602831ol_nat
% 8.90/5.14 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9401_or__nat__numerals_I2_J,axiom,
% 8.90/5.14 ! [Y3: num] :
% 8.90/5.14 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 8.90/5.14 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9402_or__nat__numerals_I4_J,axiom,
% 8.90/5.14 ! [X2: num] :
% 8.90/5.14 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 8.90/5.14 = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9403_or__nat__numerals_I1_J,axiom,
% 8.90/5.14 ! [Y3: num] :
% 8.90/5.14 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 8.90/5.14 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9404_or__nat__numerals_I3_J,axiom,
% 8.90/5.14 ! [X2: num] :
% 8.90/5.14 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 8.90/5.14 = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9405_xor__nat__numerals_I1_J,axiom,
% 8.90/5.14 ! [Y3: num] :
% 8.90/5.14 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 8.90/5.14 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9406_xor__nat__numerals_I2_J,axiom,
% 8.90/5.14 ! [Y3: num] :
% 8.90/5.14 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 8.90/5.14 = ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9407_xor__nat__numerals_I3_J,axiom,
% 8.90/5.14 ! [X2: num] :
% 8.90/5.14 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 8.90/5.14 = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9408_xor__nat__numerals_I4_J,axiom,
% 8.90/5.14 ! [X2: num] :
% 8.90/5.14 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 8.90/5.14 = ( numeral_numeral_nat @ ( bit0 @ X2 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9409_or__minus__numerals_I4_J,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.90/5.14 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9410_or__minus__numerals_I8_J,axiom,
% 8.90/5.14 ! [N: num,M: num] :
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 8.90/5.14 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9411_or__minus__numerals_I3_J,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.90/5.14 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9412_or__minus__numerals_I7_J,axiom,
% 8.90/5.14 ! [N: num,M: num] :
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 8.90/5.14 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9413_or__not__num__neg_Osimps_I1_J,axiom,
% 8.90/5.14 ( ( bit_or_not_num_neg @ one @ one )
% 8.90/5.14 = one ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9414_bit__Suc__0__iff,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.90/5.14 = ( N = zero_zero_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9415_not__bit__Suc__0__Suc,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9416_or__not__num__neg_Osimps_I4_J,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
% 8.90/5.14 = ( bit0 @ one ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9417_or__not__num__neg_Osimps_I6_J,axiom,
% 8.90/5.14 ! [N: num,M: num] :
% 8.90/5.14 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
% 8.90/5.14 = ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9418_or__not__num__neg_Osimps_I7_J,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
% 8.90/5.14 = one ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9419_or__not__num__neg_Osimps_I3_J,axiom,
% 8.90/5.14 ! [M: num] :
% 8.90/5.14 ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
% 8.90/5.14 = ( bit1 @ M ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9420_or__not__num__neg_Osimps_I5_J,axiom,
% 8.90/5.14 ! [N: num,M: num] :
% 8.90/5.14 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
% 8.90/5.14 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9421_not__bit__Suc__0__numeral,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9422_or__not__num__neg_Osimps_I9_J,axiom,
% 8.90/5.14 ! [N: num,M: num] :
% 8.90/5.14 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
% 8.90/5.14 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9423_or__nat__def,axiom,
% 8.90/5.14 ( bit_se1412395901928357646or_nat
% 8.90/5.14 = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9424_or__not__num__neg_Osimps_I2_J,axiom,
% 8.90/5.14 ! [M: num] :
% 8.90/5.14 ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
% 8.90/5.14 = ( bit1 @ M ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9425_or__not__num__neg_Osimps_I8_J,axiom,
% 8.90/5.14 ! [N: num,M: num] :
% 8.90/5.14 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
% 8.90/5.14 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9426_bit__nat__iff,axiom,
% 8.90/5.14 ! [K: int,N: nat] :
% 8.90/5.14 ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
% 8.90/5.14 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.90/5.14 & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9427_or__not__num__neg_Oelims,axiom,
% 8.90/5.14 ! [X2: num,Xa2: num,Y3: num] :
% 8.90/5.14 ( ( ( bit_or_not_num_neg @ X2 @ Xa2 )
% 8.90/5.14 = Y3 )
% 8.90/5.14 => ( ( ( X2 = one )
% 8.90/5.14 => ( ( Xa2 = one )
% 8.90/5.14 => ( Y3 != one ) ) )
% 8.90/5.14 => ( ( ( X2 = one )
% 8.90/5.14 => ! [M4: num] :
% 8.90/5.14 ( ( Xa2
% 8.90/5.14 = ( bit0 @ M4 ) )
% 8.90/5.14 => ( Y3
% 8.90/5.14 != ( bit1 @ M4 ) ) ) )
% 8.90/5.14 => ( ( ( X2 = one )
% 8.90/5.14 => ! [M4: num] :
% 8.90/5.14 ( ( Xa2
% 8.90/5.14 = ( bit1 @ M4 ) )
% 8.90/5.14 => ( Y3
% 8.90/5.14 != ( bit1 @ M4 ) ) ) )
% 8.90/5.14 => ( ( ? [N2: num] :
% 8.90/5.14 ( X2
% 8.90/5.14 = ( bit0 @ N2 ) )
% 8.90/5.14 => ( ( Xa2 = one )
% 8.90/5.14 => ( Y3
% 8.90/5.14 != ( bit0 @ one ) ) ) )
% 8.90/5.14 => ( ! [N2: num] :
% 8.90/5.14 ( ( X2
% 8.90/5.14 = ( bit0 @ N2 ) )
% 8.90/5.14 => ! [M4: num] :
% 8.90/5.14 ( ( Xa2
% 8.90/5.14 = ( bit0 @ M4 ) )
% 8.90/5.14 => ( Y3
% 8.90/5.14 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
% 8.90/5.14 => ( ! [N2: num] :
% 8.90/5.14 ( ( X2
% 8.90/5.14 = ( bit0 @ N2 ) )
% 8.90/5.14 => ! [M4: num] :
% 8.90/5.14 ( ( Xa2
% 8.90/5.14 = ( bit1 @ M4 ) )
% 8.90/5.14 => ( Y3
% 8.90/5.14 != ( bit0 @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
% 8.90/5.14 => ( ( ? [N2: num] :
% 8.90/5.14 ( X2
% 8.90/5.14 = ( bit1 @ N2 ) )
% 8.90/5.14 => ( ( Xa2 = one )
% 8.90/5.14 => ( Y3 != one ) ) )
% 8.90/5.14 => ( ! [N2: num] :
% 8.90/5.14 ( ( X2
% 8.90/5.14 = ( bit1 @ N2 ) )
% 8.90/5.14 => ! [M4: num] :
% 8.90/5.14 ( ( Xa2
% 8.90/5.14 = ( bit0 @ M4 ) )
% 8.90/5.14 => ( Y3
% 8.90/5.14 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
% 8.90/5.14 => ~ ! [N2: num] :
% 8.90/5.14 ( ( X2
% 8.90/5.14 = ( bit1 @ N2 ) )
% 8.90/5.14 => ! [M4: num] :
% 8.90/5.14 ( ( Xa2
% 8.90/5.14 = ( bit1 @ M4 ) )
% 8.90/5.14 => ( Y3
% 8.90/5.14 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9428_bit__nat__def,axiom,
% 8.90/5.14 ( bit_se1148574629649215175it_nat
% 8.90/5.14 = ( ^ [M3: nat,N3: nat] :
% 8.90/5.14 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9429_Suc__0__or__eq,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.90/5.14 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9430_or__Suc__0__eq,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
% 8.90/5.14 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9431_or__nat__rec,axiom,
% 8.90/5.14 ( bit_se1412395901928357646or_nat
% 8.90/5.14 = ( ^ [M3: nat,N3: nat] :
% 8.90/5.14 ( plus_plus_nat
% 8.90/5.14 @ ( zero_n2687167440665602831ol_nat
% 8.90/5.14 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 8.90/5.14 | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 8.90/5.14 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9432_xor__nat__unfold,axiom,
% 8.90/5.14 ( bit_se6528837805403552850or_nat
% 8.90/5.14 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9433_xor__nat__rec,axiom,
% 8.90/5.14 ( bit_se6528837805403552850or_nat
% 8.90/5.14 = ( ^ [M3: nat,N3: nat] :
% 8.90/5.14 ( plus_plus_nat
% 8.90/5.14 @ ( zero_n2687167440665602831ol_nat
% 8.90/5.14 @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 8.90/5.14 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 8.90/5.14 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9434_or__nat__unfold,axiom,
% 8.90/5.14 ( bit_se1412395901928357646or_nat
% 8.90/5.14 = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9435_Suc__0__xor__eq,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.90/5.14 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.90/5.14 @ ( zero_n2687167440665602831ol_nat
% 8.90/5.14 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9436_or__not__num__neg_Opelims,axiom,
% 8.90/5.14 ! [X2: num,Xa2: num,Y3: num] :
% 8.90/5.14 ( ( ( bit_or_not_num_neg @ X2 @ Xa2 )
% 8.90/5.14 = Y3 )
% 8.90/5.14 => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X2 @ Xa2 ) )
% 8.90/5.14 => ( ( ( X2 = one )
% 8.90/5.14 => ( ( Xa2 = one )
% 8.90/5.14 => ( ( Y3 = one )
% 8.90/5.14 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 8.90/5.14 => ( ( ( X2 = one )
% 8.90/5.14 => ! [M4: num] :
% 8.90/5.14 ( ( Xa2
% 8.90/5.14 = ( bit0 @ M4 ) )
% 8.90/5.14 => ( ( Y3
% 8.90/5.14 = ( bit1 @ M4 ) )
% 8.90/5.14 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M4 ) ) ) ) ) )
% 8.90/5.14 => ( ( ( X2 = one )
% 8.90/5.14 => ! [M4: num] :
% 8.90/5.14 ( ( Xa2
% 8.90/5.14 = ( bit1 @ M4 ) )
% 8.90/5.14 => ( ( Y3
% 8.90/5.14 = ( bit1 @ M4 ) )
% 8.90/5.14 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M4 ) ) ) ) ) )
% 8.90/5.14 => ( ! [N2: num] :
% 8.90/5.14 ( ( X2
% 8.90/5.14 = ( bit0 @ N2 ) )
% 8.90/5.14 => ( ( Xa2 = one )
% 8.90/5.14 => ( ( Y3
% 8.90/5.14 = ( bit0 @ one ) )
% 8.90/5.14 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ one ) ) ) ) )
% 8.90/5.14 => ( ! [N2: num] :
% 8.90/5.14 ( ( X2
% 8.90/5.14 = ( bit0 @ N2 ) )
% 8.90/5.14 => ! [M4: num] :
% 8.90/5.14 ( ( Xa2
% 8.90/5.14 = ( bit0 @ M4 ) )
% 8.90/5.14 => ( ( Y3
% 8.90/5.14 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
% 8.90/5.14 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 8.90/5.14 => ( ! [N2: num] :
% 8.90/5.14 ( ( X2
% 8.90/5.14 = ( bit0 @ N2 ) )
% 8.90/5.14 => ! [M4: num] :
% 8.90/5.14 ( ( Xa2
% 8.90/5.14 = ( bit1 @ M4 ) )
% 8.90/5.14 => ( ( Y3
% 8.90/5.14 = ( bit0 @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
% 8.90/5.14 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ ( bit1 @ M4 ) ) ) ) ) )
% 8.90/5.14 => ( ! [N2: num] :
% 8.90/5.14 ( ( X2
% 8.90/5.14 = ( bit1 @ N2 ) )
% 8.90/5.14 => ( ( Xa2 = one )
% 8.90/5.14 => ( ( Y3 = one )
% 8.90/5.14 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ one ) ) ) ) )
% 8.90/5.14 => ( ! [N2: num] :
% 8.90/5.14 ( ( X2
% 8.90/5.14 = ( bit1 @ N2 ) )
% 8.90/5.14 => ! [M4: num] :
% 8.90/5.14 ( ( Xa2
% 8.90/5.14 = ( bit0 @ M4 ) )
% 8.90/5.14 => ( ( Y3
% 8.90/5.14 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
% 8.90/5.14 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 8.90/5.14 => ~ ! [N2: num] :
% 8.90/5.14 ( ( X2
% 8.90/5.14 = ( bit1 @ N2 ) )
% 8.90/5.14 => ! [M4: num] :
% 8.90/5.14 ( ( Xa2
% 8.90/5.14 = ( bit1 @ M4 ) )
% 8.90/5.14 => ( ( Y3
% 8.90/5.14 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) )
% 8.90/5.14 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ ( bit1 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9437_setceilmax,axiom,
% 8.90/5.14 ! [S: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,N: nat] :
% 8.90/5.14 ( ( vEBT_invar_vebt @ S @ M )
% 8.90/5.14 => ( ! [X5: vEBT_VEBT] :
% 8.90/5.14 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Listy ) )
% 8.90/5.14 => ( vEBT_invar_vebt @ X5 @ N ) )
% 8.90/5.14 => ( ( M
% 8.90/5.14 = ( suc @ N ) )
% 8.90/5.14 => ( ! [X5: vEBT_VEBT] :
% 8.90/5.14 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Listy ) )
% 8.90/5.14 => ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ X5 ) )
% 8.90/5.14 = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 8.90/5.14 => ( ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ S ) )
% 8.90/5.14 = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) )
% 8.90/5.14 => ( ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ S @ ( set_VEBT_VEBT2 @ Listy ) ) ) ) )
% 8.90/5.14 = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9438_horner__sum__of__bool__2__less,axiom,
% 8.90/5.14 ! [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 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9439_height__compose__list,axiom,
% 8.90/5.14 ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 8.90/5.14 ( ( member_VEBT_VEBT @ T @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.90/5.14 => ( ord_less_eq_nat @ ( vEBT_VEBT_height @ T ) @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary @ ( set_VEBT_VEBT2 @ TreeList ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9440_max__ins__scaled,axiom,
% 8.90/5.14 ! [N: nat,X14: vEBT_VEBT,M: nat,X13: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( vEBT_VEBT_height @ X14 ) ) @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ ( lattic8265883725875713057ax_nat @ ( insert_nat @ ( vEBT_VEBT_height @ X14 ) @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9441_height__i__max,axiom,
% 8.90/5.14 ! [I2: nat,X13: list_VEBT_VEBT,Foo: nat] :
% 8.90/5.14 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ X13 ) )
% 8.90/5.14 => ( ord_less_eq_nat @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ X13 @ I2 ) ) @ ( ord_max_nat @ Foo @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9442_max__idx__list,axiom,
% 8.90/5.14 ! [I2: nat,X13: list_VEBT_VEBT,N: nat,X14: vEBT_VEBT] :
% 8.90/5.14 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ X13 ) )
% 8.90/5.14 => ( ord_less_eq_nat @ ( times_times_nat @ N @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ X13 @ I2 ) ) ) @ ( suc @ ( suc @ ( times_times_nat @ N @ ( ord_max_nat @ ( vEBT_VEBT_height @ X14 ) @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9443_xor__nonnegative__int__iff,axiom,
% 8.90/5.14 ! [K: int,L: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 8.90/5.14 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 8.90/5.14 = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9444_xor__negative__int__iff,axiom,
% 8.90/5.14 ! [K: int,L: int] :
% 8.90/5.14 ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
% 8.90/5.14 = ( ( ord_less_int @ K @ zero_zero_int )
% 8.90/5.14 != ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9445_bit__xor__int__iff,axiom,
% 8.90/5.14 ! [K: int,L: int,N: nat] :
% 8.90/5.14 ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ N )
% 8.90/5.14 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 8.90/5.14 != ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9446_XOR__lower,axiom,
% 8.90/5.14 ! [X2: int,Y3: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.90/5.14 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.90/5.14 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X2 @ Y3 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9447_xor__nat__def,axiom,
% 8.90/5.14 ( bit_se6528837805403552850or_nat
% 8.90/5.14 = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9448_VEBT__internal_Oheight_Osimps_I2_J,axiom,
% 8.90/5.14 ! [Uu: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 8.90/5.14 ( ( vEBT_VEBT_height @ ( vEBT_Node @ Uu @ Deg @ TreeList @ Summary ) )
% 8.90/5.14 = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary @ ( set_VEBT_VEBT2 @ TreeList ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9449_VEBT__internal_Oheight_Oelims,axiom,
% 8.90/5.14 ! [X2: vEBT_VEBT,Y3: nat] :
% 8.90/5.14 ( ( ( vEBT_VEBT_height @ X2 )
% 8.90/5.14 = Y3 )
% 8.90/5.14 => ( ( ? [A4: $o,B4: $o] :
% 8.90/5.14 ( X2
% 8.90/5.14 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.90/5.14 => ( Y3 != zero_zero_nat ) )
% 8.90/5.14 => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.90/5.14 ( ( X2
% 8.90/5.14 = ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) )
% 8.90/5.14 => ( Y3
% 8.90/5.14 != ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary2 @ ( set_VEBT_VEBT2 @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9450_divide__nat__def,axiom,
% 8.90/5.14 ( divide_divide_nat
% 8.90/5.14 = ( ^ [M3: nat,N3: nat] :
% 8.90/5.14 ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat
% 8.90/5.14 @ ( lattic8265883725875713057ax_nat
% 8.90/5.14 @ ( collect_nat
% 8.90/5.14 @ ^ [K2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K2 @ N3 ) @ M3 ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9451_XOR__upper,axiom,
% 8.90/5.14 ! [X2: int,N: nat,Y3: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.90/5.14 => ( ( ord_less_int @ X2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.14 => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 8.90/5.14 => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X2 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9452_xor__int__rec,axiom,
% 8.90/5.14 ( bit_se6526347334894502574or_int
% 8.90/5.14 = ( ^ [K2: int,L2: int] :
% 8.90/5.14 ( plus_plus_int
% 8.90/5.14 @ ( zero_n2684676970156552555ol_int
% 8.90/5.14 @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) )
% 8.90/5.14 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 8.90/5.14 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9453_VEBT__internal_Oheight_Opelims,axiom,
% 8.90/5.14 ! [X2: vEBT_VEBT,Y3: nat] :
% 8.90/5.14 ( ( ( vEBT_VEBT_height @ X2 )
% 8.90/5.14 = Y3 )
% 8.90/5.14 => ( ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ X2 )
% 8.90/5.14 => ( ! [A4: $o,B4: $o] :
% 8.90/5.14 ( ( X2
% 8.90/5.14 = ( vEBT_Leaf @ A4 @ B4 ) )
% 8.90/5.14 => ( ( Y3 = zero_zero_nat )
% 8.90/5.14 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
% 8.90/5.14 => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.90/5.14 ( ( X2
% 8.90/5.14 = ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) )
% 8.90/5.14 => ( ( Y3
% 8.90/5.14 = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary2 @ ( set_VEBT_VEBT2 @ TreeList2 ) ) ) ) ) )
% 8.90/5.14 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Node @ Uu2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9454_xor__int__unfold,axiom,
% 8.90/5.14 ( bit_se6526347334894502574or_int
% 8.90/5.14 = ( ^ [K2: int,L2: int] :
% 8.90/5.14 ( if_int
% 8.90/5.14 @ ( K2
% 8.90/5.14 = ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.14 @ ( bit_ri7919022796975470100ot_int @ L2 )
% 8.90/5.14 @ ( if_int
% 8.90/5.14 @ ( L2
% 8.90/5.14 = ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.14 @ ( bit_ri7919022796975470100ot_int @ K2 )
% 8.90/5.14 @ ( if_int @ ( K2 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9455_bij__betw__Suc,axiom,
% 8.90/5.14 ! [M7: set_nat,N5: set_nat] :
% 8.90/5.14 ( ( bij_betw_nat_nat @ suc @ M7 @ N5 )
% 8.90/5.14 = ( ( image_nat_nat @ suc @ M7 )
% 8.90/5.14 = N5 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9456_image__Suc__atLeastAtMost,axiom,
% 8.90/5.14 ! [I2: nat,J: nat] :
% 8.90/5.14 ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I2 @ J ) )
% 8.90/5.14 = ( set_or1269000886237332187st_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9457_not__nonnegative__int__iff,axiom,
% 8.90/5.14 ! [K: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 8.90/5.14 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9458_not__negative__int__iff,axiom,
% 8.90/5.14 ! [K: int] :
% 8.90/5.14 ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 8.90/5.14 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9459_or__minus__minus__numerals,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.14 = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9460_and__minus__minus__numerals,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.14 = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9461_bit__not__int__iff,axiom,
% 8.90/5.14 ! [K: int,N: nat] :
% 8.90/5.14 ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N )
% 8.90/5.14 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9462_zero__notin__Suc__image,axiom,
% 8.90/5.14 ! [A2: set_nat] :
% 8.90/5.14 ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9463_or__int__def,axiom,
% 8.90/5.14 ( bit_se1409905431419307370or_int
% 8.90/5.14 = ( ^ [K2: int,L2: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ ( bit_ri7919022796975470100ot_int @ L2 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9464_finite__int__iff__bounded,axiom,
% 8.90/5.14 ( finite_finite_int
% 8.90/5.14 = ( ^ [S5: set_int] :
% 8.90/5.14 ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S5 ) @ ( set_ord_lessThan_int @ K2 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9465_finite__int__iff__bounded__le,axiom,
% 8.90/5.14 ( finite_finite_int
% 8.90/5.14 = ( ^ [S5: set_int] :
% 8.90/5.14 ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S5 ) @ ( set_ord_atMost_int @ K2 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9466_not__int__def,axiom,
% 8.90/5.14 ( bit_ri7919022796975470100ot_int
% 8.90/5.14 = ( ^ [K2: int] : ( minus_minus_int @ ( uminus_uminus_int @ K2 ) @ one_one_int ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9467_and__not__numerals_I1_J,axiom,
% 8.90/5.14 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 8.90/5.14 = zero_zero_int ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9468_or__not__numerals_I1_J,axiom,
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 8.90/5.14 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9469_xor__int__def,axiom,
% 8.90/5.14 ( bit_se6526347334894502574or_int
% 8.90/5.14 = ( ^ [K2: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ L2 ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ L2 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9470_not__int__div__2,axiom,
% 8.90/5.14 ! [K: int] :
% 8.90/5.14 ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.14 = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9471_even__not__iff__int,axiom,
% 8.90/5.14 ! [K: int] :
% 8.90/5.14 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
% 8.90/5.14 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9472_and__not__numerals_I2_J,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.90/5.14 = one_one_int ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9473_and__not__numerals_I4_J,axiom,
% 8.90/5.14 ! [M: num] :
% 8.90/5.14 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 8.90/5.14 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9474_or__not__numerals_I2_J,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.90/5.14 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9475_or__not__numerals_I4_J,axiom,
% 8.90/5.14 ! [M: num] :
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 8.90/5.14 = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9476_bit__minus__int__iff,axiom,
% 8.90/5.14 ! [K: int,N: nat] :
% 8.90/5.14 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
% 8.90/5.14 = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9477_image__Suc__lessThan,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.14 = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9478_image__Suc__atMost,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
% 8.90/5.14 = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9479_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 8.90/5.14 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9480_lessThan__Suc__eq__insert__0,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( set_ord_lessThan_nat @ ( suc @ N ) )
% 8.90/5.14 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9481_numeral__or__not__num__eq,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
% 8.90/5.14 = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9482_int__numeral__not__or__num__neg,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.90/5.14 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9483_int__numeral__or__not__num__neg,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.14 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9484_atMost__Suc__eq__insert__0,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( set_ord_atMost_nat @ ( suc @ N ) )
% 8.90/5.14 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9485_and__not__numerals_I5_J,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.90/5.14 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9486_and__not__numerals_I7_J,axiom,
% 8.90/5.14 ! [M: num] :
% 8.90/5.14 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 8.90/5.14 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9487_or__not__numerals_I3_J,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.90/5.14 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9488_and__not__numerals_I3_J,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.90/5.14 = zero_zero_int ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9489_or__not__numerals_I7_J,axiom,
% 8.90/5.14 ! [M: num] :
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 8.90/5.14 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9490_and__not__numerals_I6_J,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.90/5.14 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9491_and__not__numerals_I9_J,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.90/5.14 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9492_or__not__numerals_I6_J,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.90/5.14 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9493_or__not__numerals_I5_J,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.90/5.14 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9494_and__not__numerals_I8_J,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.90/5.14 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9495_or__not__numerals_I8_J,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.90/5.14 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9496_or__not__numerals_I9_J,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.90/5.14 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9497_not__int__rec,axiom,
% 8.90/5.14 ( bit_ri7919022796975470100ot_int
% 8.90/5.14 = ( ^ [K2: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9498_push__bit__nonnegative__int__iff,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
% 8.90/5.14 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9499_push__bit__negative__int__iff,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
% 8.90/5.14 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9500_concat__bit__of__zero__1,axiom,
% 8.90/5.14 ! [N: nat,L: int] :
% 8.90/5.14 ( ( bit_concat_bit @ N @ zero_zero_int @ L )
% 8.90/5.14 = ( bit_se545348938243370406it_int @ N @ L ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9501_push__bit__of__Suc__0,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
% 8.90/5.14 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9502_push__bit__nat__eq,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K ) )
% 8.90/5.14 = ( nat2 @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9503_flip__bit__nat__def,axiom,
% 8.90/5.14 ( bit_se2161824704523386999it_nat
% 8.90/5.14 = ( ^ [M3: nat,N3: nat] : ( bit_se6528837805403552850or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9504_set__bit__nat__def,axiom,
% 8.90/5.14 ( bit_se7882103937844011126it_nat
% 8.90/5.14 = ( ^ [M3: nat,N3: nat] : ( bit_se1412395901928357646or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9505_bit__push__bit__iff__int,axiom,
% 8.90/5.14 ! [M: nat,K: int,N: nat] :
% 8.90/5.14 ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
% 8.90/5.14 = ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.14 & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9506_bit__push__bit__iff__nat,axiom,
% 8.90/5.14 ! [M: nat,Q3: nat,N: nat] :
% 8.90/5.14 ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q3 ) @ N )
% 8.90/5.14 = ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.14 & ( bit_se1148574629649215175it_nat @ Q3 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9507_concat__bit__eq,axiom,
% 8.90/5.14 ( bit_concat_bit
% 8.90/5.14 = ( ^ [N3: nat,K2: int,L2: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N3 @ K2 ) @ ( bit_se545348938243370406it_int @ N3 @ L2 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9508_concat__bit__def,axiom,
% 8.90/5.14 ( bit_concat_bit
% 8.90/5.14 = ( ^ [N3: nat,K2: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N3 @ K2 ) @ ( bit_se545348938243370406it_int @ N3 @ L2 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9509_set__bit__int__def,axiom,
% 8.90/5.14 ( bit_se7879613467334960850it_int
% 8.90/5.14 = ( ^ [N3: nat,K2: int] : ( bit_se1409905431419307370or_int @ K2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9510_flip__bit__int__def,axiom,
% 8.90/5.14 ( bit_se2159334234014336723it_int
% 8.90/5.14 = ( ^ [N3: nat,K2: int] : ( bit_se6526347334894502574or_int @ K2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9511_unset__bit__int__def,axiom,
% 8.90/5.14 ( bit_se4203085406695923979it_int
% 8.90/5.14 = ( ^ [N3: nat,K2: int] : ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9512_push__bit__int__def,axiom,
% 8.90/5.14 ( bit_se545348938243370406it_int
% 8.90/5.14 = ( ^ [N3: nat,K2: int] : ( times_times_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9513_push__bit__nat__def,axiom,
% 8.90/5.14 ( bit_se547839408752420682it_nat
% 8.90/5.14 = ( ^ [N3: nat,M3: nat] : ( times_times_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9514_push__bit__minus__one,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.14 = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9515_Cauchy__iff2,axiom,
% 8.90/5.14 ( topolo4055970368930404560y_real
% 8.90/5.14 = ( ^ [X4: nat > real] :
% 8.90/5.14 ! [J3: nat] :
% 8.90/5.14 ? [M8: nat] :
% 8.90/5.14 ! [M3: nat] :
% 8.90/5.14 ( ( ord_less_eq_nat @ M8 @ M3 )
% 8.90/5.14 => ! [N3: nat] :
% 8.90/5.14 ( ( ord_less_eq_nat @ M8 @ N3 )
% 8.90/5.14 => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X4 @ M3 ) @ ( X4 @ N3 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9516_Sum__Ico__nat,axiom,
% 8.90/5.14 ! [M: nat,N: nat] :
% 8.90/5.14 ( ( groups3542108847815614940at_nat
% 8.90/5.14 @ ^ [X: nat] : X
% 8.90/5.14 @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 8.90/5.14 = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9517_image__Suc__atLeastLessThan,axiom,
% 8.90/5.14 ! [I2: nat,J: nat] :
% 8.90/5.14 ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I2 @ J ) )
% 8.90/5.14 = ( set_or4665077453230672383an_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9518_atLeastLessThan__singleton,axiom,
% 8.90/5.14 ! [M: nat] :
% 8.90/5.14 ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
% 8.90/5.14 = ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9519_all__nat__less__eq,axiom,
% 8.90/5.14 ! [N: nat,P: nat > $o] :
% 8.90/5.14 ( ( ! [M3: nat] :
% 8.90/5.14 ( ( ord_less_nat @ M3 @ N )
% 8.90/5.14 => ( P @ M3 ) ) )
% 8.90/5.14 = ( ! [X: nat] :
% 8.90/5.14 ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 8.90/5.14 => ( P @ X ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9520_ex__nat__less__eq,axiom,
% 8.90/5.14 ! [N: nat,P: nat > $o] :
% 8.90/5.14 ( ( ? [M3: nat] :
% 8.90/5.14 ( ( ord_less_nat @ M3 @ N )
% 8.90/5.14 & ( P @ M3 ) ) )
% 8.90/5.14 = ( ? [X: nat] :
% 8.90/5.14 ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 8.90/5.14 & ( P @ X ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9521_atLeastLessThanSuc__atLeastAtMost,axiom,
% 8.90/5.14 ! [L: nat,U: nat] :
% 8.90/5.14 ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
% 8.90/5.14 = ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9522_atLeast0__lessThan__Suc,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 8.90/5.14 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9523_subset__eq__atLeast0__lessThan__finite,axiom,
% 8.90/5.14 ! [N5: set_nat,N: nat] :
% 8.90/5.14 ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 8.90/5.14 => ( finite_finite_nat @ N5 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9524_atLeastLessThanSuc,axiom,
% 8.90/5.14 ! [M: nat,N: nat] :
% 8.90/5.14 ( ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.14 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 8.90/5.14 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
% 8.90/5.14 & ( ~ ( ord_less_eq_nat @ M @ N )
% 8.90/5.14 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 8.90/5.14 = bot_bot_set_nat ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9525_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 8.90/5.14 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9526_prod__Suc__Suc__fact,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 8.90/5.14 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9527_prod__Suc__fact,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 8.90/5.14 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9528_atLeastLessThan__nat__numeral,axiom,
% 8.90/5.14 ! [M: nat,K: num] :
% 8.90/5.14 ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 8.90/5.14 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 8.90/5.14 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 8.90/5.14 & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 8.90/5.14 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 8.90/5.14 = bot_bot_set_nat ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9529_image__minus__const__atLeastLessThan__nat,axiom,
% 8.90/5.14 ! [C: nat,Y3: nat,X2: nat] :
% 8.90/5.14 ( ( ( ord_less_nat @ C @ Y3 )
% 8.90/5.14 => ( ( image_nat_nat
% 8.90/5.14 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 8.90/5.14 @ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
% 8.90/5.14 = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X2 @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
% 8.90/5.14 & ( ~ ( ord_less_nat @ C @ Y3 )
% 8.90/5.14 => ( ( ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.14 => ( ( image_nat_nat
% 8.90/5.14 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 8.90/5.14 @ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
% 8.90/5.14 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 8.90/5.14 & ( ~ ( ord_less_nat @ X2 @ Y3 )
% 8.90/5.14 => ( ( image_nat_nat
% 8.90/5.14 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 8.90/5.14 @ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
% 8.90/5.14 = bot_bot_set_nat ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9530_atLeast1__lessThan__eq__remove0,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
% 8.90/5.14 = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9531_sum__power2,axiom,
% 8.90/5.14 ! [K: nat] :
% 8.90/5.14 ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 8.90/5.14 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9532_Chebyshev__sum__upper__nat,axiom,
% 8.90/5.14 ! [N: nat,A: nat > nat,B: nat > nat] :
% 8.90/5.14 ( ! [I3: nat,J2: nat] :
% 8.90/5.14 ( ( ord_less_eq_nat @ I3 @ J2 )
% 8.90/5.14 => ( ( ord_less_nat @ J2 @ N )
% 8.90/5.14 => ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
% 8.90/5.14 => ( ! [I3: nat,J2: nat] :
% 8.90/5.14 ( ( ord_less_eq_nat @ I3 @ J2 )
% 8.90/5.14 => ( ( ord_less_nat @ J2 @ N )
% 8.90/5.14 => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I3 ) ) ) )
% 8.90/5.14 => ( ord_less_eq_nat
% 8.90/5.14 @ ( times_times_nat @ N
% 8.90/5.14 @ ( groups3542108847815614940at_nat
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( B @ I4 ) )
% 8.90/5.14 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 8.90/5.14 @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9533_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
% 8.90/5.14 ! [L: int,U: int] :
% 8.90/5.14 ( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
% 8.90/5.14 = ( set_or1266510415728281911st_int @ L @ U ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9534_image__add__int__atLeastLessThan,axiom,
% 8.90/5.14 ! [L: int,U: int] :
% 8.90/5.14 ( ( image_int_int
% 8.90/5.14 @ ^ [X: int] : ( plus_plus_int @ X @ L )
% 8.90/5.14 @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
% 8.90/5.14 = ( set_or4662586982721622107an_int @ L @ U ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9535_image__atLeastZeroLessThan__int,axiom,
% 8.90/5.14 ! [U: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ U )
% 8.90/5.14 => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
% 8.90/5.14 = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9536_VEBT_Osize_I3_J,axiom,
% 8.90/5.14 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 8.90/5.14 ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 8.90/5.14 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9537_valid__eq2,axiom,
% 8.90/5.14 ! [T: vEBT_VEBT,D: nat] :
% 8.90/5.14 ( ( vEBT_VEBT_valid @ T @ D )
% 8.90/5.14 => ( vEBT_invar_vebt @ T @ D ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9538_valid__eq1,axiom,
% 8.90/5.14 ! [T: vEBT_VEBT,D: nat] :
% 8.90/5.14 ( ( vEBT_invar_vebt @ T @ D )
% 8.90/5.14 => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9539_valid__eq,axiom,
% 8.90/5.14 vEBT_VEBT_valid = vEBT_invar_vebt ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9540_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 8.90/5.14 ! [Uu: $o,Uv: $o,D: nat] :
% 8.90/5.14 ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
% 8.90/5.14 = ( D = one_one_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9541_VEBT_Osize__gen_I1_J,axiom,
% 8.90/5.14 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 8.90/5.14 ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 8.90/5.14 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9542_VEBT_Osize__gen_I2_J,axiom,
% 8.90/5.14 ! [X21: $o,X222: $o] :
% 8.90/5.14 ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 8.90/5.14 = zero_zero_nat ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9543_Code__Target__Int_Opositive__def,axiom,
% 8.90/5.14 code_Target_positive = numeral_numeral_int ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9544_csqrt_Osimps_I1_J,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( re @ ( csqrt @ Z ) )
% 8.90/5.14 = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9545_complex__Re__numeral,axiom,
% 8.90/5.14 ! [V: num] :
% 8.90/5.14 ( ( re @ ( numera6690914467698888265omplex @ V ) )
% 8.90/5.14 = ( numeral_numeral_real @ V ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9546_Re__divide__numeral,axiom,
% 8.90/5.14 ! [Z: complex,W: num] :
% 8.90/5.14 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 8.90/5.14 = ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9547_complex__Re__le__cmod,axiom,
% 8.90/5.14 ! [X2: complex] : ( ord_less_eq_real @ ( re @ X2 ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9548_one__complex_Osimps_I1_J,axiom,
% 8.90/5.14 ( ( re @ one_one_complex )
% 8.90/5.14 = one_one_real ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9549_scaleR__complex_Osimps_I1_J,axiom,
% 8.90/5.14 ! [R2: real,X2: complex] :
% 8.90/5.14 ( ( re @ ( real_V2046097035970521341omplex @ R2 @ X2 ) )
% 8.90/5.14 = ( times_times_real @ R2 @ ( re @ X2 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9550_minus__complex_Osimps_I1_J,axiom,
% 8.90/5.14 ! [X2: complex,Y3: complex] :
% 8.90/5.14 ( ( re @ ( minus_minus_complex @ X2 @ Y3 ) )
% 8.90/5.14 = ( minus_minus_real @ ( re @ X2 ) @ ( re @ Y3 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9551_abs__Re__le__cmod,axiom,
% 8.90/5.14 ! [X2: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X2 ) ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9552_Re__csqrt,axiom,
% 8.90/5.14 ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9553_cmod__plus__Re__le__0__iff,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
% 8.90/5.14 = ( ( re @ Z )
% 8.90/5.14 = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9554_cos__n__Re__cis__pow__n,axiom,
% 8.90/5.14 ! [N: nat,A: real] :
% 8.90/5.14 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 8.90/5.14 = ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9555_csqrt_Ocode,axiom,
% 8.90/5.14 ( csqrt
% 8.90/5.14 = ( ^ [Z3: complex] :
% 8.90/5.14 ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.14 @ ( times_times_real
% 8.90/5.14 @ ( if_real
% 8.90/5.14 @ ( ( im @ Z3 )
% 8.90/5.14 = zero_zero_real )
% 8.90/5.14 @ one_one_real
% 8.90/5.14 @ ( sgn_sgn_real @ ( im @ Z3 ) ) )
% 8.90/5.14 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9556_csqrt_Osimps_I2_J,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( im @ ( csqrt @ Z ) )
% 8.90/5.14 = ( times_times_real
% 8.90/5.14 @ ( if_real
% 8.90/5.14 @ ( ( im @ Z )
% 8.90/5.14 = zero_zero_real )
% 8.90/5.14 @ one_one_real
% 8.90/5.14 @ ( sgn_sgn_real @ ( im @ Z ) ) )
% 8.90/5.14 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9557_csqrt__of__real__nonpos,axiom,
% 8.90/5.14 ! [X2: complex] :
% 8.90/5.14 ( ( ( im @ X2 )
% 8.90/5.14 = zero_zero_real )
% 8.90/5.14 => ( ( ord_less_eq_real @ ( re @ X2 ) @ zero_zero_real )
% 8.90/5.14 => ( ( csqrt @ X2 )
% 8.90/5.14 = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X2 ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9558_Im__power__real,axiom,
% 8.90/5.14 ! [X2: complex,N: nat] :
% 8.90/5.14 ( ( ( im @ X2 )
% 8.90/5.14 = zero_zero_real )
% 8.90/5.14 => ( ( im @ ( power_power_complex @ X2 @ N ) )
% 8.90/5.14 = zero_zero_real ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9559_complex__Im__numeral,axiom,
% 8.90/5.14 ! [V: num] :
% 8.90/5.14 ( ( im @ ( numera6690914467698888265omplex @ V ) )
% 8.90/5.14 = zero_zero_real ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9560_Im__i__times,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( im @ ( times_times_complex @ imaginary_unit @ Z ) )
% 8.90/5.14 = ( re @ Z ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9561_Re__power__real,axiom,
% 8.90/5.14 ! [X2: complex,N: nat] :
% 8.90/5.14 ( ( ( im @ X2 )
% 8.90/5.14 = zero_zero_real )
% 8.90/5.14 => ( ( re @ ( power_power_complex @ X2 @ N ) )
% 8.90/5.14 = ( power_power_real @ ( re @ X2 ) @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9562_Re__i__times,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( re @ ( times_times_complex @ imaginary_unit @ Z ) )
% 8.90/5.14 = ( uminus_uminus_real @ ( im @ Z ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9563_Im__divide__numeral,axiom,
% 8.90/5.14 ! [Z: complex,W: num] :
% 8.90/5.14 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 8.90/5.14 = ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9564_csqrt__of__real__nonneg,axiom,
% 8.90/5.14 ! [X2: complex] :
% 8.90/5.14 ( ( ( im @ X2 )
% 8.90/5.14 = zero_zero_real )
% 8.90/5.14 => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X2 ) )
% 8.90/5.14 => ( ( csqrt @ X2 )
% 8.90/5.14 = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X2 ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9565_csqrt__minus,axiom,
% 8.90/5.14 ! [X2: complex] :
% 8.90/5.14 ( ( ( ord_less_real @ ( im @ X2 ) @ zero_zero_real )
% 8.90/5.14 | ( ( ( im @ X2 )
% 8.90/5.14 = zero_zero_real )
% 8.90/5.14 & ( ord_less_eq_real @ zero_zero_real @ ( re @ X2 ) ) ) )
% 8.90/5.14 => ( ( csqrt @ ( uminus1482373934393186551omplex @ X2 ) )
% 8.90/5.14 = ( times_times_complex @ imaginary_unit @ ( csqrt @ X2 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9566_imaginary__unit_Osimps_I2_J,axiom,
% 8.90/5.14 ( ( im @ imaginary_unit )
% 8.90/5.14 = one_one_real ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9567_one__complex_Osimps_I2_J,axiom,
% 8.90/5.14 ( ( im @ one_one_complex )
% 8.90/5.14 = zero_zero_real ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9568_scaleR__complex_Osimps_I2_J,axiom,
% 8.90/5.14 ! [R2: real,X2: complex] :
% 8.90/5.14 ( ( im @ ( real_V2046097035970521341omplex @ R2 @ X2 ) )
% 8.90/5.14 = ( times_times_real @ R2 @ ( im @ X2 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9569_minus__complex_Osimps_I2_J,axiom,
% 8.90/5.14 ! [X2: complex,Y3: complex] :
% 8.90/5.14 ( ( im @ ( minus_minus_complex @ X2 @ Y3 ) )
% 8.90/5.14 = ( minus_minus_real @ ( im @ X2 ) @ ( im @ Y3 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9570_abs__Im__le__cmod,axiom,
% 8.90/5.14 ! [X2: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X2 ) ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9571_times__complex_Osimps_I2_J,axiom,
% 8.90/5.14 ! [X2: complex,Y3: complex] :
% 8.90/5.14 ( ( im @ ( times_times_complex @ X2 @ Y3 ) )
% 8.90/5.14 = ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y3 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y3 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9572_cmod__Im__le__iff,axiom,
% 8.90/5.14 ! [X2: complex,Y3: complex] :
% 8.90/5.14 ( ( ( re @ X2 )
% 8.90/5.14 = ( re @ Y3 ) )
% 8.90/5.14 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 8.90/5.14 = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X2 ) ) @ ( abs_abs_real @ ( im @ Y3 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9573_cmod__Re__le__iff,axiom,
% 8.90/5.14 ! [X2: complex,Y3: complex] :
% 8.90/5.14 ( ( ( im @ X2 )
% 8.90/5.14 = ( im @ Y3 ) )
% 8.90/5.14 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 8.90/5.14 = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X2 ) ) @ ( abs_abs_real @ ( re @ Y3 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9574_times__complex_Osimps_I1_J,axiom,
% 8.90/5.14 ! [X2: complex,Y3: complex] :
% 8.90/5.14 ( ( re @ ( times_times_complex @ X2 @ Y3 ) )
% 8.90/5.14 = ( minus_minus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y3 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9575_scaleR__complex_Ocode,axiom,
% 8.90/5.14 ( real_V2046097035970521341omplex
% 8.90/5.14 = ( ^ [R5: real,X: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X ) ) @ ( times_times_real @ R5 @ ( im @ X ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9576_minus__complex_Ocode,axiom,
% 8.90/5.14 ( minus_minus_complex
% 8.90/5.14 = ( ^ [X: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( re @ X ) @ ( re @ Y ) ) @ ( minus_minus_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9577_csqrt__principal,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
% 8.90/5.14 | ( ( ( re @ ( csqrt @ Z ) )
% 8.90/5.14 = zero_zero_real )
% 8.90/5.14 & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9578_cmod__le,axiom,
% 8.90/5.14 ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9579_sin__n__Im__cis__pow__n,axiom,
% 8.90/5.14 ! [N: nat,A: real] :
% 8.90/5.14 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 8.90/5.14 = ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9580_Re__exp,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( re @ ( exp_complex @ Z ) )
% 8.90/5.14 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9581_Im__exp,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( im @ ( exp_complex @ Z ) )
% 8.90/5.14 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9582_complex__eq,axiom,
% 8.90/5.14 ! [A: complex] :
% 8.90/5.14 ( A
% 8.90/5.14 = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9583_times__complex_Ocode,axiom,
% 8.90/5.14 ( times_times_complex
% 8.90/5.14 = ( ^ [X: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9584_exp__eq__polar,axiom,
% 8.90/5.14 ( exp_complex
% 8.90/5.14 = ( ^ [Z3: complex] : ( times_times_complex @ ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z3 ) ) ) @ ( cis @ ( im @ Z3 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9585_cmod__power2,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.14 = ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9586_Im__power2,axiom,
% 8.90/5.14 ! [X2: complex] :
% 8.90/5.14 ( ( im @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.14 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X2 ) ) @ ( im @ X2 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9587_Re__power2,axiom,
% 8.90/5.14 ! [X2: complex] :
% 8.90/5.14 ( ( re @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.14 = ( minus_minus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9588_complex__eq__0,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( Z = zero_zero_complex )
% 8.90/5.14 = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.14 = zero_zero_real ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9589_norm__complex__def,axiom,
% 8.90/5.14 ( real_V1022390504157884413omplex
% 8.90/5.14 = ( ^ [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 ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9590_inverse__complex_Osimps_I1_J,axiom,
% 8.90/5.14 ! [X2: complex] :
% 8.90/5.14 ( ( re @ ( invers8013647133539491842omplex @ X2 ) )
% 8.90/5.14 = ( 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 ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9591_complex__neq__0,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( Z != zero_zero_complex )
% 8.90/5.14 = ( 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 ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9592_Re__divide,axiom,
% 8.90/5.14 ! [X2: complex,Y3: complex] :
% 8.90/5.14 ( ( re @ ( divide1717551699836669952omplex @ X2 @ Y3 ) )
% 8.90/5.14 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9593_csqrt__unique,axiom,
% 8.90/5.14 ! [W: complex,Z: complex] :
% 8.90/5.14 ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 8.90/5.14 = Z )
% 8.90/5.14 => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
% 8.90/5.14 | ( ( ( re @ W )
% 8.90/5.14 = zero_zero_real )
% 8.90/5.14 & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
% 8.90/5.14 => ( ( csqrt @ Z )
% 8.90/5.14 = W ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9594_csqrt__square,axiom,
% 8.90/5.14 ! [B: complex] :
% 8.90/5.14 ( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
% 8.90/5.14 | ( ( ( re @ B )
% 8.90/5.14 = zero_zero_real )
% 8.90/5.14 & ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
% 8.90/5.14 => ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.14 = B ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9595_inverse__complex_Osimps_I2_J,axiom,
% 8.90/5.14 ! [X2: complex] :
% 8.90/5.14 ( ( im @ ( invers8013647133539491842omplex @ X2 ) )
% 8.90/5.14 = ( 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 ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9596_Im__divide,axiom,
% 8.90/5.14 ! [X2: complex,Y3: complex] :
% 8.90/5.14 ( ( im @ ( divide1717551699836669952omplex @ X2 @ Y3 ) )
% 8.90/5.14 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9597_complex__abs__le__norm,axiom,
% 8.90/5.14 ! [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 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9598_complex__unit__circle,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( Z != zero_zero_complex )
% 8.90/5.14 => ( ( 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 ) ) ) )
% 8.90/5.14 = one_one_real ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9599_inverse__complex_Ocode,axiom,
% 8.90/5.14 ( invers8013647133539491842omplex
% 8.90/5.14 = ( ^ [X: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9600_Complex__divide,axiom,
% 8.90/5.14 ( divide1717551699836669952omplex
% 8.90/5.14 = ( ^ [X: complex,Y: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9601_Im__Reals__divide,axiom,
% 8.90/5.14 ! [R2: complex,Z: complex] :
% 8.90/5.14 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 8.90/5.14 => ( ( im @ ( divide1717551699836669952omplex @ R2 @ Z ) )
% 8.90/5.14 = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R2 ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9602_Re__Reals__divide,axiom,
% 8.90/5.14 ! [R2: complex,Z: complex] :
% 8.90/5.14 ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
% 8.90/5.14 => ( ( re @ ( divide1717551699836669952omplex @ R2 @ Z ) )
% 8.90/5.14 = ( divide_divide_real @ ( times_times_real @ ( re @ R2 ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9603_complex__diff__cnj,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
% 8.90/5.14 = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9604_complex__cnj__mult,axiom,
% 8.90/5.14 ! [X2: complex,Y3: complex] :
% 8.90/5.14 ( ( cnj @ ( times_times_complex @ X2 @ Y3 ) )
% 8.90/5.14 = ( times_times_complex @ ( cnj @ X2 ) @ ( cnj @ Y3 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9605_complex__cnj__one,axiom,
% 8.90/5.14 ( ( cnj @ one_one_complex )
% 8.90/5.14 = one_one_complex ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9606_complex__cnj__one__iff,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( ( cnj @ Z )
% 8.90/5.14 = one_one_complex )
% 8.90/5.14 = ( Z = one_one_complex ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9607_complex__cnj__power,axiom,
% 8.90/5.14 ! [X2: complex,N: nat] :
% 8.90/5.14 ( ( cnj @ ( power_power_complex @ X2 @ N ) )
% 8.90/5.14 = ( power_power_complex @ ( cnj @ X2 ) @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9608_complex__cnj__numeral,axiom,
% 8.90/5.14 ! [W: num] :
% 8.90/5.14 ( ( cnj @ ( numera6690914467698888265omplex @ W ) )
% 8.90/5.14 = ( numera6690914467698888265omplex @ W ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9609_complex__cnj__diff,axiom,
% 8.90/5.14 ! [X2: complex,Y3: complex] :
% 8.90/5.14 ( ( cnj @ ( minus_minus_complex @ X2 @ Y3 ) )
% 8.90/5.14 = ( minus_minus_complex @ ( cnj @ X2 ) @ ( cnj @ Y3 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9610_complex__cnj__neg__numeral,axiom,
% 8.90/5.14 ! [W: num] :
% 8.90/5.14 ( ( cnj @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 8.90/5.14 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9611_complex__In__mult__cnj__zero,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( im @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 8.90/5.14 = zero_zero_real ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9612_real__eq__imaginary__iff,axiom,
% 8.90/5.14 ! [Y3: complex,X2: complex] :
% 8.90/5.14 ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
% 8.90/5.14 => ( ( member_complex @ X2 @ real_V2521375963428798218omplex )
% 8.90/5.14 => ( ( X2
% 8.90/5.14 = ( times_times_complex @ imaginary_unit @ Y3 ) )
% 8.90/5.14 = ( ( X2 = zero_zero_complex )
% 8.90/5.14 & ( Y3 = zero_zero_complex ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9613_imaginary__eq__real__iff,axiom,
% 8.90/5.14 ! [Y3: complex,X2: complex] :
% 8.90/5.14 ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
% 8.90/5.14 => ( ( member_complex @ X2 @ real_V2521375963428798218omplex )
% 8.90/5.14 => ( ( ( times_times_complex @ imaginary_unit @ Y3 )
% 8.90/5.14 = X2 )
% 8.90/5.14 = ( ( X2 = zero_zero_complex )
% 8.90/5.14 & ( Y3 = zero_zero_complex ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9614_Re__complex__div__eq__0,axiom,
% 8.90/5.14 ! [A: complex,B: complex] :
% 8.90/5.14 ( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.90/5.14 = zero_zero_real )
% 8.90/5.14 = ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 8.90/5.14 = zero_zero_real ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9615_Im__complex__div__eq__0,axiom,
% 8.90/5.14 ! [A: complex,B: complex] :
% 8.90/5.14 ( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
% 8.90/5.14 = zero_zero_real )
% 8.90/5.14 = ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 8.90/5.14 = zero_zero_real ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9616_complex__mod__sqrt__Re__mult__cnj,axiom,
% 8.90/5.14 ( real_V1022390504157884413omplex
% 8.90/5.14 = ( ^ [Z3: complex] : ( sqrt @ ( re @ ( times_times_complex @ Z3 @ ( cnj @ Z3 ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9617_Re__complex__div__lt__0,axiom,
% 8.90/5.14 ! [A: complex,B: complex] :
% 8.90/5.14 ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 8.90/5.14 = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9618_Re__complex__div__gt__0,axiom,
% 8.90/5.14 ! [A: complex,B: complex] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 8.90/5.14 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9619_Re__complex__div__ge__0,axiom,
% 8.90/5.14 ! [A: complex,B: complex] :
% 8.90/5.14 ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 8.90/5.14 = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9620_Re__complex__div__le__0,axiom,
% 8.90/5.14 ! [A: complex,B: complex] :
% 8.90/5.14 ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 8.90/5.14 = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9621_Im__complex__div__lt__0,axiom,
% 8.90/5.14 ! [A: complex,B: complex] :
% 8.90/5.14 ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 8.90/5.14 = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9622_Im__complex__div__gt__0,axiom,
% 8.90/5.14 ! [A: complex,B: complex] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 8.90/5.14 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9623_Im__complex__div__ge__0,axiom,
% 8.90/5.14 ! [A: complex,B: complex] :
% 8.90/5.14 ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 8.90/5.14 = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9624_Im__complex__div__le__0,axiom,
% 8.90/5.14 ! [A: complex,B: complex] :
% 8.90/5.14 ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 8.90/5.14 = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9625_complex__mod__mult__cnj,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 8.90/5.14 = ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9626_complex__div__gt__0,axiom,
% 8.90/5.14 ! [A: complex,B: complex] :
% 8.90/5.14 ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 8.90/5.14 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
% 8.90/5.14 & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 8.90/5.14 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9627_complex__norm__square,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 8.90/5.14 = ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9628_complex__add__cnj,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
% 8.90/5.14 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9629_complex__div__cnj,axiom,
% 8.90/5.14 ( divide1717551699836669952omplex
% 8.90/5.14 = ( ^ [A3: complex,B2: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ ( cnj @ B2 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9630_cnj__add__mult__eq__Re,axiom,
% 8.90/5.14 ! [Z: complex,W: complex] :
% 8.90/5.14 ( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
% 8.90/5.14 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9631_complex__mult__cnj,axiom,
% 8.90/5.14 ! [Z: complex] :
% 8.90/5.14 ( ( times_times_complex @ Z @ ( cnj @ Z ) )
% 8.90/5.14 = ( 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 ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9632_divmod__step__integer__def,axiom,
% 8.90/5.14 ( unique4921790084139445826nteger
% 8.90/5.14 = ( ^ [L2: num] :
% 8.90/5.14 ( produc6916734918728496179nteger
% 8.90/5.14 @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9633_card__Collect__less__nat,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( finite_card_nat
% 8.90/5.14 @ ( collect_nat
% 8.90/5.14 @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
% 8.90/5.14 = N ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9634_card__atMost,axiom,
% 8.90/5.14 ! [U: nat] :
% 8.90/5.14 ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
% 8.90/5.14 = ( suc @ U ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9635_card__atLeastLessThan,axiom,
% 8.90/5.14 ! [L: nat,U: nat] :
% 8.90/5.14 ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
% 8.90/5.14 = ( minus_minus_nat @ U @ L ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9636_card__Collect__le__nat,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( finite_card_nat
% 8.90/5.14 @ ( collect_nat
% 8.90/5.14 @ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
% 8.90/5.14 = ( suc @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9637_card__atLeastAtMost,axiom,
% 8.90/5.14 ! [L: nat,U: nat] :
% 8.90/5.14 ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 8.90/5.14 = ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9638_card__atLeastLessThan__int,axiom,
% 8.90/5.14 ! [L: int,U: int] :
% 8.90/5.14 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L @ U ) )
% 8.90/5.14 = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9639_card__atLeastAtMost__int,axiom,
% 8.90/5.14 ! [L: int,U: int] :
% 8.90/5.14 ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
% 8.90/5.14 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9640_minus__integer__code_I1_J,axiom,
% 8.90/5.14 ! [K: code_integer] :
% 8.90/5.14 ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
% 8.90/5.14 = K ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9641_minus__integer__code_I2_J,axiom,
% 8.90/5.14 ! [L: code_integer] :
% 8.90/5.14 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
% 8.90/5.14 = ( uminus1351360451143612070nteger @ L ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9642_times__integer__code_I1_J,axiom,
% 8.90/5.14 ! [K: code_integer] :
% 8.90/5.14 ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
% 8.90/5.14 = zero_z3403309356797280102nteger ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9643_times__integer__code_I2_J,axiom,
% 8.90/5.14 ! [L: code_integer] :
% 8.90/5.14 ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
% 8.90/5.14 = zero_z3403309356797280102nteger ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9644_divmod__integer_H__def,axiom,
% 8.90/5.14 ( unique3479559517661332726nteger
% 8.90/5.14 = ( ^ [M3: num,N3: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N3 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N3 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9645_sgn__integer__code,axiom,
% 8.90/5.14 ( sgn_sgn_Code_integer
% 8.90/5.14 = ( ^ [K2: code_integer] : ( if_Code_integer @ ( K2 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9646_less__eq__integer__code_I1_J,axiom,
% 8.90/5.14 ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9647_card__less__Suc2,axiom,
% 8.90/5.14 ! [M7: set_nat,I2: nat] :
% 8.90/5.14 ( ~ ( member_nat @ zero_zero_nat @ M7 )
% 8.90/5.14 => ( ( finite_card_nat
% 8.90/5.14 @ ( collect_nat
% 8.90/5.14 @ ^ [K2: nat] :
% 8.90/5.14 ( ( member_nat @ ( suc @ K2 ) @ M7 )
% 8.90/5.14 & ( ord_less_nat @ K2 @ I2 ) ) ) )
% 8.90/5.14 = ( finite_card_nat
% 8.90/5.14 @ ( collect_nat
% 8.90/5.14 @ ^ [K2: nat] :
% 8.90/5.14 ( ( member_nat @ K2 @ M7 )
% 8.90/5.14 & ( ord_less_nat @ K2 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9648_card__less__Suc,axiom,
% 8.90/5.14 ! [M7: set_nat,I2: nat] :
% 8.90/5.14 ( ( member_nat @ zero_zero_nat @ M7 )
% 8.90/5.14 => ( ( suc
% 8.90/5.14 @ ( finite_card_nat
% 8.90/5.14 @ ( collect_nat
% 8.90/5.14 @ ^ [K2: nat] :
% 8.90/5.14 ( ( member_nat @ ( suc @ K2 ) @ M7 )
% 8.90/5.14 & ( ord_less_nat @ K2 @ I2 ) ) ) ) )
% 8.90/5.14 = ( finite_card_nat
% 8.90/5.14 @ ( collect_nat
% 8.90/5.14 @ ^ [K2: nat] :
% 8.90/5.14 ( ( member_nat @ K2 @ M7 )
% 8.90/5.14 & ( ord_less_nat @ K2 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9649_card__less,axiom,
% 8.90/5.14 ! [M7: set_nat,I2: nat] :
% 8.90/5.14 ( ( member_nat @ zero_zero_nat @ M7 )
% 8.90/5.14 => ( ( finite_card_nat
% 8.90/5.14 @ ( collect_nat
% 8.90/5.14 @ ^ [K2: nat] :
% 8.90/5.14 ( ( member_nat @ K2 @ M7 )
% 8.90/5.14 & ( ord_less_nat @ K2 @ ( suc @ I2 ) ) ) ) )
% 8.90/5.14 != zero_zero_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9650_one__natural_Orsp,axiom,
% 8.90/5.14 one_one_nat = one_one_nat ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9651_one__integer_Orsp,axiom,
% 8.90/5.14 one_one_int = one_one_int ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9652_subset__card__intvl__is__intvl,axiom,
% 8.90/5.14 ! [A2: set_nat,K: nat] :
% 8.90/5.14 ( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
% 8.90/5.14 => ( A2
% 8.90/5.14 = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9653_card__le__Suc__Max,axiom,
% 8.90/5.14 ! [S3: set_nat] :
% 8.90/5.14 ( ( finite_finite_nat @ S3 )
% 8.90/5.14 => ( ord_less_eq_nat @ ( finite_card_nat @ S3 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S3 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9654_subset__eq__atLeast0__lessThan__card,axiom,
% 8.90/5.14 ! [N5: set_nat,N: nat] :
% 8.90/5.14 ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 8.90/5.14 => ( ord_less_eq_nat @ ( finite_card_nat @ N5 ) @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9655_card__sum__le__nat__sum,axiom,
% 8.90/5.14 ! [S3: set_nat] :
% 8.90/5.14 ( ord_less_eq_nat
% 8.90/5.14 @ ( groups3542108847815614940at_nat
% 8.90/5.14 @ ^ [X: nat] : X
% 8.90/5.14 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S3 ) ) )
% 8.90/5.14 @ ( groups3542108847815614940at_nat
% 8.90/5.14 @ ^ [X: nat] : X
% 8.90/5.14 @ S3 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9656_card__nth__roots,axiom,
% 8.90/5.14 ! [C: complex,N: nat] :
% 8.90/5.14 ( ( C != zero_zero_complex )
% 8.90/5.14 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( ( finite_card_complex
% 8.90/5.14 @ ( collect_complex
% 8.90/5.14 @ ^ [Z3: complex] :
% 8.90/5.14 ( ( power_power_complex @ Z3 @ N )
% 8.90/5.14 = C ) ) )
% 8.90/5.14 = N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9657_card__roots__unity__eq,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( ( finite_card_complex
% 8.90/5.14 @ ( collect_complex
% 8.90/5.14 @ ^ [Z3: complex] :
% 8.90/5.14 ( ( power_power_complex @ Z3 @ N )
% 8.90/5.14 = one_one_complex ) ) )
% 8.90/5.14 = N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9658_integer__of__int__code,axiom,
% 8.90/5.14 ( code_integer_of_int
% 8.90/5.14 = ( ^ [K2: int] :
% 8.90/5.14 ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K2 ) ) )
% 8.90/5.14 @ ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 8.90/5.14 @ ( if_Code_integer
% 8.90/5.14 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 8.90/5.14 = zero_zero_int )
% 8.90/5.14 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 8.90/5.14 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9659_times__integer_Oabs__eq,axiom,
% 8.90/5.14 ! [Xa2: int,X2: int] :
% 8.90/5.14 ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X2 ) )
% 8.90/5.14 = ( code_integer_of_int @ ( times_times_int @ Xa2 @ X2 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9660_one__integer__def,axiom,
% 8.90/5.14 ( one_one_Code_integer
% 8.90/5.14 = ( code_integer_of_int @ one_one_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9661_less__eq__integer_Oabs__eq,axiom,
% 8.90/5.14 ! [Xa2: int,X2: int] :
% 8.90/5.14 ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X2 ) )
% 8.90/5.14 = ( ord_less_eq_int @ Xa2 @ X2 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9662_minus__integer_Oabs__eq,axiom,
% 8.90/5.14 ! [Xa2: int,X2: int] :
% 8.90/5.14 ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X2 ) )
% 8.90/5.14 = ( code_integer_of_int @ ( minus_minus_int @ Xa2 @ X2 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9663_Code__Numeral_Opositive__def,axiom,
% 8.90/5.14 code_positive = numera6620942414471956472nteger ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9664_integer__of__num_I3_J,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( code_integer_of_num @ ( bit1 @ N ) )
% 8.90/5.14 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9665_int__of__integer__code,axiom,
% 8.90/5.14 ( code_int_of_integer
% 8.90/5.14 = ( ^ [K2: code_integer] :
% 8.90/5.14 ( if_int @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K2 ) ) )
% 8.90/5.14 @ ( if_int @ ( K2 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 8.90/5.14 @ ( produc1553301316500091796er_int
% 8.90/5.14 @ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
% 8.90/5.14 @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9666_int__of__integer__numeral,axiom,
% 8.90/5.14 ! [K: num] :
% 8.90/5.14 ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 8.90/5.14 = ( numeral_numeral_int @ K ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9667_times__integer_Orep__eq,axiom,
% 8.90/5.14 ! [X2: code_integer,Xa2: code_integer] :
% 8.90/5.14 ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X2 @ Xa2 ) )
% 8.90/5.14 = ( times_times_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9668_one__integer_Orep__eq,axiom,
% 8.90/5.14 ( ( code_int_of_integer @ one_one_Code_integer )
% 8.90/5.14 = one_one_int ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9669_minus__integer_Orep__eq,axiom,
% 8.90/5.14 ! [X2: code_integer,Xa2: code_integer] :
% 8.90/5.14 ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X2 @ Xa2 ) )
% 8.90/5.14 = ( minus_minus_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9670_less__eq__integer_Orep__eq,axiom,
% 8.90/5.14 ( ord_le3102999989581377725nteger
% 8.90/5.14 = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9671_integer__less__eq__iff,axiom,
% 8.90/5.14 ( ord_le3102999989581377725nteger
% 8.90/5.14 = ( ^ [K2: code_integer,L2: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9672_integer__of__num__def,axiom,
% 8.90/5.14 code_integer_of_num = numera6620942414471956472nteger ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9673_integer__of__num__triv_I1_J,axiom,
% 8.90/5.14 ( ( code_integer_of_num @ one )
% 8.90/5.14 = one_one_Code_integer ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9674_integer__of__num_I2_J,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( code_integer_of_num @ ( bit0 @ N ) )
% 8.90/5.14 = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9675_integer__of__num__triv_I2_J,axiom,
% 8.90/5.14 ( ( code_integer_of_num @ ( bit0 @ one ) )
% 8.90/5.14 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9676_num__of__integer__code,axiom,
% 8.90/5.14 ( code_num_of_integer
% 8.90/5.14 = ( ^ [K2: code_integer] :
% 8.90/5.14 ( if_num @ ( ord_le3102999989581377725nteger @ K2 @ one_one_Code_integer ) @ one
% 8.90/5.14 @ ( produc7336495610019696514er_num
% 8.90/5.14 @ ^ [L2: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one ) )
% 8.90/5.14 @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9677_nat__of__integer__code,axiom,
% 8.90/5.14 ( code_nat_of_integer
% 8.90/5.14 = ( ^ [K2: code_integer] :
% 8.90/5.14 ( if_nat @ ( ord_le3102999989581377725nteger @ K2 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 8.90/5.14 @ ( produc1555791787009142072er_nat
% 8.90/5.14 @ ^ [L2: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ one_one_nat ) )
% 8.90/5.14 @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9678_bit__cut__integer__def,axiom,
% 8.90/5.14 ( code_bit_cut_integer
% 8.90/5.14 = ( ^ [K2: code_integer] :
% 8.90/5.14 ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 8.90/5.14 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K2 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9679_nat__of__integer__non__positive,axiom,
% 8.90/5.14 ! [K: code_integer] :
% 8.90/5.14 ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
% 8.90/5.14 => ( ( code_nat_of_integer @ K )
% 8.90/5.14 = zero_zero_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9680_nat__of__integer__code__post_I3_J,axiom,
% 8.90/5.14 ! [K: num] :
% 8.90/5.14 ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 8.90/5.14 = ( numeral_numeral_nat @ K ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9681_nat__of__integer__code__post_I2_J,axiom,
% 8.90/5.14 ( ( code_nat_of_integer @ one_one_Code_integer )
% 8.90/5.14 = one_one_nat ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9682_bit__cut__integer__code,axiom,
% 8.90/5.14 ( code_bit_cut_integer
% 8.90/5.14 = ( ^ [K2: code_integer] :
% 8.90/5.14 ( if_Pro5737122678794959658eger_o @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 8.90/5.14 @ ( produc9125791028180074456eger_o
% 8.90/5.14 @ ^ [R5: code_integer,S6: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S6 ) ) @ ( S6 = one_one_Code_integer ) )
% 8.90/5.14 @ ( code_divmod_abs @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9683_divmod__integer__code,axiom,
% 8.90/5.14 ( code_divmod_integer
% 8.90/5.14 = ( ^ [K2: code_integer,L2: code_integer] :
% 8.90/5.14 ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 8.90/5.14 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
% 8.90/5.14 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ ( code_divmod_abs @ K2 @ L2 )
% 8.90/5.14 @ ( produc6916734918728496179nteger
% 8.90/5.14 @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S6 ) ) )
% 8.90/5.14 @ ( code_divmod_abs @ K2 @ L2 ) ) )
% 8.90/5.14 @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 8.90/5.14 @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 8.90/5.14 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K2 @ L2 )
% 8.90/5.14 @ ( produc6916734918728496179nteger
% 8.90/5.14 @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S6 ) ) )
% 8.90/5.14 @ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9684_floor__real__def,axiom,
% 8.90/5.14 ( archim6058952711729229775r_real
% 8.90/5.14 = ( ^ [X: real] :
% 8.90/5.14 ( the_int
% 8.90/5.14 @ ^ [Z3: int] :
% 8.90/5.14 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X )
% 8.90/5.14 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9685_floor__rat__def,axiom,
% 8.90/5.14 ( archim3151403230148437115or_rat
% 8.90/5.14 = ( ^ [X: rat] :
% 8.90/5.14 ( the_int
% 8.90/5.14 @ ^ [Z3: int] :
% 8.90/5.14 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X )
% 8.90/5.14 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9686_less__eq__rat__def,axiom,
% 8.90/5.14 ( ord_less_eq_rat
% 8.90/5.14 = ( ^ [X: rat,Y: rat] :
% 8.90/5.14 ( ( ord_less_rat @ X @ Y )
% 8.90/5.14 | ( X = Y ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9687_obtain__pos__sum,axiom,
% 8.90/5.14 ! [R2: rat] :
% 8.90/5.14 ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 8.90/5.14 => ~ ! [S2: rat] :
% 8.90/5.14 ( ( ord_less_rat @ zero_zero_rat @ S2 )
% 8.90/5.14 => ! [T5: rat] :
% 8.90/5.14 ( ( ord_less_rat @ zero_zero_rat @ T5 )
% 8.90/5.14 => ( R2
% 8.90/5.14 != ( plus_plus_rat @ S2 @ T5 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9688_sgn__rat__def,axiom,
% 8.90/5.14 ( sgn_sgn_rat
% 8.90/5.14 = ( ^ [A3: rat] : ( if_rat @ ( A3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9689_nat_Odisc__eq__case_I1_J,axiom,
% 8.90/5.14 ! [Nat: nat] :
% 8.90/5.14 ( ( Nat = zero_zero_nat )
% 8.90/5.14 = ( case_nat_o @ $true
% 8.90/5.14 @ ^ [Uu3: nat] : $false
% 8.90/5.14 @ Nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9690_nat_Odisc__eq__case_I2_J,axiom,
% 8.90/5.14 ! [Nat: nat] :
% 8.90/5.14 ( ( Nat != zero_zero_nat )
% 8.90/5.14 = ( case_nat_o @ $false
% 8.90/5.14 @ ^ [Uu3: nat] : $true
% 8.90/5.14 @ Nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9691_less__eq__nat_Osimps_I2_J,axiom,
% 8.90/5.14 ! [M: nat,N: nat] :
% 8.90/5.14 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 8.90/5.14 = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9692_max__Suc2,axiom,
% 8.90/5.14 ! [M: nat,N: nat] :
% 8.90/5.14 ( ( ord_max_nat @ M @ ( suc @ N ) )
% 8.90/5.14 = ( case_nat_nat @ ( suc @ N )
% 8.90/5.14 @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ M5 @ N ) )
% 8.90/5.14 @ M ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9693_max__Suc1,axiom,
% 8.90/5.14 ! [N: nat,M: nat] :
% 8.90/5.14 ( ( ord_max_nat @ ( suc @ N ) @ M )
% 8.90/5.14 = ( case_nat_nat @ ( suc @ N )
% 8.90/5.14 @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ N @ M5 ) )
% 8.90/5.14 @ M ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9694_diff__Suc,axiom,
% 8.90/5.14 ! [M: nat,N: nat] :
% 8.90/5.14 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 8.90/5.14 = ( case_nat_nat @ zero_zero_nat
% 8.90/5.14 @ ^ [K2: nat] : K2
% 8.90/5.14 @ ( minus_minus_nat @ M @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9695_rat__inverse__code,axiom,
% 8.90/5.14 ! [P6: rat] :
% 8.90/5.14 ( ( quotient_of @ ( inverse_inverse_rat @ P6 ) )
% 8.90/5.14 = ( produc4245557441103728435nt_int
% 8.90/5.14 @ ^ [A3: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( A3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A3 ) @ B2 ) @ ( abs_abs_int @ A3 ) ) )
% 8.90/5.14 @ ( quotient_of @ P6 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9696_quotient__of__number_I3_J,axiom,
% 8.90/5.14 ! [K: num] :
% 8.90/5.14 ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
% 8.90/5.14 = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9697_rat__one__code,axiom,
% 8.90/5.14 ( ( quotient_of @ one_one_rat )
% 8.90/5.14 = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9698_rat__zero__code,axiom,
% 8.90/5.14 ( ( quotient_of @ zero_zero_rat )
% 8.90/5.14 = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9699_quotient__of__number_I5_J,axiom,
% 8.90/5.14 ! [K: num] :
% 8.90/5.14 ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 8.90/5.14 = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9700_quotient__of__number_I4_J,axiom,
% 8.90/5.14 ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
% 8.90/5.14 = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9701_diff__rat__def,axiom,
% 8.90/5.14 ( minus_minus_rat
% 8.90/5.14 = ( ^ [Q4: rat,R5: rat] : ( plus_plus_rat @ Q4 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9702_divide__rat__def,axiom,
% 8.90/5.14 ( divide_divide_rat
% 8.90/5.14 = ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9703_pred__def,axiom,
% 8.90/5.14 ( pred
% 8.90/5.14 = ( case_nat_nat @ zero_zero_nat
% 8.90/5.14 @ ^ [X24: nat] : X24 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9704_rat__less__code,axiom,
% 8.90/5.14 ( ord_less_rat
% 8.90/5.14 = ( ^ [P4: rat,Q4: rat] :
% 8.90/5.14 ( produc4947309494688390418_int_o
% 8.90/5.14 @ ^ [A3: int,C2: int] :
% 8.90/5.14 ( produc4947309494688390418_int_o
% 8.90/5.14 @ ^ [B2: int,D2: int] : ( ord_less_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C2 @ B2 ) )
% 8.90/5.14 @ ( quotient_of @ Q4 ) )
% 8.90/5.14 @ ( quotient_of @ P4 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9705_rat__less__eq__code,axiom,
% 8.90/5.14 ( ord_less_eq_rat
% 8.90/5.14 = ( ^ [P4: rat,Q4: rat] :
% 8.90/5.14 ( produc4947309494688390418_int_o
% 8.90/5.14 @ ^ [A3: int,C2: int] :
% 8.90/5.14 ( produc4947309494688390418_int_o
% 8.90/5.14 @ ^ [B2: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C2 @ B2 ) )
% 8.90/5.14 @ ( quotient_of @ Q4 ) )
% 8.90/5.14 @ ( quotient_of @ P4 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9706_quotient__of__int,axiom,
% 8.90/5.14 ! [A: int] :
% 8.90/5.14 ( ( quotient_of @ ( of_int @ A ) )
% 8.90/5.14 = ( product_Pair_int_int @ A @ one_one_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9707_rat__plus__code,axiom,
% 8.90/5.14 ! [P6: rat,Q3: rat] :
% 8.90/5.14 ( ( quotient_of @ ( plus_plus_rat @ P6 @ Q3 ) )
% 8.90/5.14 = ( produc4245557441103728435nt_int
% 8.90/5.14 @ ^ [A3: int,C2: int] :
% 8.90/5.14 ( produc4245557441103728435nt_int
% 8.90/5.14 @ ^ [B2: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ B2 @ C2 ) ) @ ( times_times_int @ C2 @ D2 ) ) )
% 8.90/5.14 @ ( quotient_of @ Q3 ) )
% 8.90/5.14 @ ( quotient_of @ P6 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9708_rat__minus__code,axiom,
% 8.90/5.14 ! [P6: rat,Q3: rat] :
% 8.90/5.14 ( ( quotient_of @ ( minus_minus_rat @ P6 @ Q3 ) )
% 8.90/5.14 = ( produc4245557441103728435nt_int
% 8.90/5.14 @ ^ [A3: int,C2: int] :
% 8.90/5.14 ( produc4245557441103728435nt_int
% 8.90/5.14 @ ^ [B2: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ B2 @ C2 ) ) @ ( times_times_int @ C2 @ D2 ) ) )
% 8.90/5.14 @ ( quotient_of @ Q3 ) )
% 8.90/5.14 @ ( quotient_of @ P6 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9709_normalize__denom__zero,axiom,
% 8.90/5.14 ! [P6: int] :
% 8.90/5.14 ( ( normalize @ ( product_Pair_int_int @ P6 @ zero_zero_int ) )
% 8.90/5.14 = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9710_normalize__crossproduct,axiom,
% 8.90/5.14 ! [Q3: int,S: int,P6: int,R2: int] :
% 8.90/5.14 ( ( Q3 != zero_zero_int )
% 8.90/5.14 => ( ( S != zero_zero_int )
% 8.90/5.14 => ( ( ( normalize @ ( product_Pair_int_int @ P6 @ Q3 ) )
% 8.90/5.14 = ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
% 8.90/5.14 => ( ( times_times_int @ P6 @ S )
% 8.90/5.14 = ( times_times_int @ R2 @ Q3 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9711_rat__divide__code,axiom,
% 8.90/5.14 ! [P6: rat,Q3: rat] :
% 8.90/5.14 ( ( quotient_of @ ( divide_divide_rat @ P6 @ Q3 ) )
% 8.90/5.14 = ( produc4245557441103728435nt_int
% 8.90/5.14 @ ^ [A3: int,C2: int] :
% 8.90/5.14 ( produc4245557441103728435nt_int
% 8.90/5.14 @ ^ [B2: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C2 @ B2 ) ) )
% 8.90/5.14 @ ( quotient_of @ Q3 ) )
% 8.90/5.14 @ ( quotient_of @ P6 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9712_rat__times__code,axiom,
% 8.90/5.14 ! [P6: rat,Q3: rat] :
% 8.90/5.14 ( ( quotient_of @ ( times_times_rat @ P6 @ Q3 ) )
% 8.90/5.14 = ( produc4245557441103728435nt_int
% 8.90/5.14 @ ^ [A3: int,C2: int] :
% 8.90/5.14 ( produc4245557441103728435nt_int
% 8.90/5.14 @ ^ [B2: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ B2 ) @ ( times_times_int @ C2 @ D2 ) ) )
% 8.90/5.14 @ ( quotient_of @ Q3 ) )
% 8.90/5.14 @ ( quotient_of @ P6 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9713_Frct__code__post_I5_J,axiom,
% 8.90/5.14 ! [K: num] :
% 8.90/5.14 ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 8.90/5.14 = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9714_Frct__code__post_I6_J,axiom,
% 8.90/5.14 ! [K: num,L: num] :
% 8.90/5.14 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
% 8.90/5.14 = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9715_bezw__0,axiom,
% 8.90/5.14 ! [X2: nat] :
% 8.90/5.14 ( ( bezw @ X2 @ zero_zero_nat )
% 8.90/5.14 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9716_Frct__code__post_I3_J,axiom,
% 8.90/5.14 ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
% 8.90/5.14 = one_one_rat ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9717_Frct__code__post_I4_J,axiom,
% 8.90/5.14 ! [K: num] :
% 8.90/5.14 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
% 8.90/5.14 = ( numeral_numeral_rat @ K ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9718_drop__bit__numeral__minus__bit1,axiom,
% 8.90/5.14 ! [L: num,K: num] :
% 8.90/5.14 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 8.90/5.14 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9719_prod__decode__aux_Osimps,axiom,
% 8.90/5.14 ( nat_prod_decode_aux
% 8.90/5.14 = ( ^ [K2: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K2 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K2 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M3 @ ( suc @ K2 ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9720_prod__decode__aux_Oelims,axiom,
% 8.90/5.14 ! [X2: nat,Xa2: nat,Y3: product_prod_nat_nat] :
% 8.90/5.14 ( ( ( nat_prod_decode_aux @ X2 @ Xa2 )
% 8.90/5.14 = Y3 )
% 8.90/5.14 => ( ( ( ord_less_eq_nat @ Xa2 @ X2 )
% 8.90/5.14 => ( Y3
% 8.90/5.14 = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X2 @ Xa2 ) ) ) )
% 8.90/5.14 & ( ~ ( ord_less_eq_nat @ Xa2 @ X2 )
% 8.90/5.14 => ( Y3
% 8.90/5.14 = ( nat_prod_decode_aux @ ( suc @ X2 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X2 ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9721_drop__bit__nonnegative__int__iff,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
% 8.90/5.14 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9722_drop__bit__negative__int__iff,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
% 8.90/5.14 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9723_drop__bit__minus__one,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 8.90/5.14 = ( uminus_uminus_int @ one_one_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9724_drop__bit__Suc__minus__bit0,axiom,
% 8.90/5.14 ! [N: nat,K: num] :
% 8.90/5.14 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 8.90/5.14 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9725_drop__bit__numeral__minus__bit0,axiom,
% 8.90/5.14 ! [L: num,K: num] :
% 8.90/5.14 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 8.90/5.14 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9726_drop__bit__Suc__minus__bit1,axiom,
% 8.90/5.14 ! [N: nat,K: num] :
% 8.90/5.14 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 8.90/5.14 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9727_drop__bit__push__bit__int,axiom,
% 8.90/5.14 ! [M: nat,N: nat,K: int] :
% 8.90/5.14 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N @ K ) )
% 8.90/5.14 = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M ) @ K ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9728_drop__bit__int__def,axiom,
% 8.90/5.14 ( bit_se8568078237143864401it_int
% 8.90/5.14 = ( ^ [N3: nat,K2: int] : ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9729_prod__decode__aux_Opelims,axiom,
% 8.90/5.14 ! [X2: nat,Xa2: nat,Y3: product_prod_nat_nat] :
% 8.90/5.14 ( ( ( nat_prod_decode_aux @ X2 @ Xa2 )
% 8.90/5.14 = Y3 )
% 8.90/5.14 => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) )
% 8.90/5.14 => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X2 )
% 8.90/5.14 => ( Y3
% 8.90/5.14 = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X2 @ Xa2 ) ) ) )
% 8.90/5.14 & ( ~ ( ord_less_eq_nat @ Xa2 @ X2 )
% 8.90/5.14 => ( Y3
% 8.90/5.14 = ( nat_prod_decode_aux @ ( suc @ X2 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X2 ) ) ) ) ) )
% 8.90/5.14 => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9730_Suc__0__mod__numeral,axiom,
% 8.90/5.14 ! [K: num] :
% 8.90/5.14 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 8.90/5.14 = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9731_Suc__0__div__numeral,axiom,
% 8.90/5.14 ! [K: num] :
% 8.90/5.14 ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 8.90/5.14 = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9732_drop__bit__of__Suc__0,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
% 8.90/5.14 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9733_fst__divmod__nat,axiom,
% 8.90/5.14 ! [M: nat,N: nat] :
% 8.90/5.14 ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
% 8.90/5.14 = ( divide_divide_nat @ M @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9734_drop__bit__nat__eq,axiom,
% 8.90/5.14 ! [N: nat,K: int] :
% 8.90/5.14 ( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K ) )
% 8.90/5.14 = ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9735_drop__bit__nat__def,axiom,
% 8.90/5.14 ( bit_se8570568707652914677it_nat
% 8.90/5.14 = ( ^ [N3: nat,M3: nat] : ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9736_bezw__non__0,axiom,
% 8.90/5.14 ! [Y3: nat,X2: nat] :
% 8.90/5.14 ( ( ord_less_nat @ zero_zero_nat @ Y3 )
% 8.90/5.14 => ( ( bezw @ X2 @ Y3 )
% 8.90/5.14 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9737_bezw_Osimps,axiom,
% 8.90/5.14 ( bezw
% 8.90/5.14 = ( ^ [X: nat,Y: nat] : ( if_Pro3027730157355071871nt_int @ ( Y = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9738_bezw_Oelims,axiom,
% 8.90/5.14 ! [X2: nat,Xa2: nat,Y3: product_prod_int_int] :
% 8.90/5.14 ( ( ( bezw @ X2 @ Xa2 )
% 8.90/5.14 = Y3 )
% 8.90/5.14 => ( ( ( Xa2 = zero_zero_nat )
% 8.90/5.14 => ( Y3
% 8.90/5.14 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 8.90/5.14 & ( ( Xa2 != zero_zero_nat )
% 8.90/5.14 => ( Y3
% 8.90/5.14 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Xa2 ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9739_rat__sgn__code,axiom,
% 8.90/5.14 ! [P6: rat] :
% 8.90/5.14 ( ( quotient_of @ ( sgn_sgn_rat @ P6 ) )
% 8.90/5.14 = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P6 ) ) ) @ one_one_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9740_bezw_Opelims,axiom,
% 8.90/5.14 ! [X2: nat,Xa2: nat,Y3: product_prod_int_int] :
% 8.90/5.14 ( ( ( bezw @ X2 @ Xa2 )
% 8.90/5.14 = Y3 )
% 8.90/5.14 => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) )
% 8.90/5.14 => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 8.90/5.14 => ( Y3
% 8.90/5.14 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 8.90/5.14 & ( ( Xa2 != zero_zero_nat )
% 8.90/5.14 => ( Y3
% 8.90/5.14 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Xa2 ) ) ) ) ) ) ) )
% 8.90/5.14 => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9741_one__mod__minus__numeral,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.14 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9742_minus__one__mod__numeral,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 8.90/5.14 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9743_minus__numeral__mod__numeral,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.90/5.14 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9744_numeral__mod__minus__numeral,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.14 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9745_Divides_Oadjust__mod__def,axiom,
% 8.90/5.14 ( adjust_mod
% 8.90/5.14 = ( ^ [L2: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L2 @ R5 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9746_normalize__def,axiom,
% 8.90/5.14 ( normalize
% 8.90/5.14 = ( ^ [P4: product_prod_int_int] :
% 8.90/5.14 ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P4 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P4 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P4 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) )
% 8.90/5.14 @ ( if_Pro3027730157355071871nt_int
% 8.90/5.14 @ ( ( product_snd_int_int @ P4 )
% 8.90/5.14 = zero_zero_int )
% 8.90/5.14 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 8.90/5.14 @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P4 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P4 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9747_finite__enumerate,axiom,
% 8.90/5.14 ! [S3: set_nat] :
% 8.90/5.14 ( ( finite_finite_nat @ S3 )
% 8.90/5.14 => ? [R3: nat > nat] :
% 8.90/5.14 ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S3 ) ) )
% 8.90/5.14 & ! [N9: nat] :
% 8.90/5.14 ( ( ord_less_nat @ N9 @ ( finite_card_nat @ S3 ) )
% 8.90/5.14 => ( member_nat @ ( R3 @ N9 ) @ S3 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9748_gcd__1__int,axiom,
% 8.90/5.14 ! [M: int] :
% 8.90/5.14 ( ( gcd_gcd_int @ M @ one_one_int )
% 8.90/5.14 = one_one_int ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9749_gcd__neg__numeral__1__int,axiom,
% 8.90/5.14 ! [N: num,X2: int] :
% 8.90/5.14 ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ X2 )
% 8.90/5.14 = ( gcd_gcd_int @ ( numeral_numeral_int @ N ) @ X2 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9750_gcd__neg__numeral__2__int,axiom,
% 8.90/5.14 ! [X2: int,N: num] :
% 8.90/5.14 ( ( gcd_gcd_int @ X2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.14 = ( gcd_gcd_int @ X2 @ ( numeral_numeral_int @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9751_gcd__ge__0__int,axiom,
% 8.90/5.14 ! [X2: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X2 @ Y3 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9752_bezout__int,axiom,
% 8.90/5.14 ! [X2: int,Y3: int] :
% 8.90/5.14 ? [U3: int,V2: int] :
% 8.90/5.14 ( ( plus_plus_int @ ( times_times_int @ U3 @ X2 ) @ ( times_times_int @ V2 @ Y3 ) )
% 8.90/5.14 = ( gcd_gcd_int @ X2 @ Y3 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9753_gcd__mult__distrib__int,axiom,
% 8.90/5.14 ! [K: int,M: int,N: int] :
% 8.90/5.14 ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
% 8.90/5.14 = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9754_gcd__le2__int,axiom,
% 8.90/5.14 ! [B: int,A: int] :
% 8.90/5.14 ( ( ord_less_int @ zero_zero_int @ B )
% 8.90/5.14 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9755_gcd__le1__int,axiom,
% 8.90/5.14 ! [A: int,B: int] :
% 8.90/5.14 ( ( ord_less_int @ zero_zero_int @ A )
% 8.90/5.14 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9756_gcd__cases__int,axiom,
% 8.90/5.14 ! [X2: int,Y3: int,P: int > $o] :
% 8.90/5.14 ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.90/5.14 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.90/5.14 => ( P @ ( gcd_gcd_int @ X2 @ Y3 ) ) ) )
% 8.90/5.14 => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 8.90/5.14 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 8.90/5.14 => ( P @ ( gcd_gcd_int @ X2 @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 8.90/5.14 => ( ( ( ord_less_eq_int @ X2 @ zero_zero_int )
% 8.90/5.14 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 8.90/5.14 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X2 ) @ Y3 ) ) ) )
% 8.90/5.14 => ( ( ( ord_less_eq_int @ X2 @ zero_zero_int )
% 8.90/5.14 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 8.90/5.14 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X2 ) @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 8.90/5.14 => ( P @ ( gcd_gcd_int @ X2 @ Y3 ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9757_gcd__unique__int,axiom,
% 8.90/5.14 ! [D: int,A: int,B: int] :
% 8.90/5.14 ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 8.90/5.14 & ( dvd_dvd_int @ D @ A )
% 8.90/5.14 & ( dvd_dvd_int @ D @ B )
% 8.90/5.14 & ! [E3: int] :
% 8.90/5.14 ( ( ( dvd_dvd_int @ E3 @ A )
% 8.90/5.14 & ( dvd_dvd_int @ E3 @ B ) )
% 8.90/5.14 => ( dvd_dvd_int @ E3 @ D ) ) )
% 8.90/5.14 = ( D
% 8.90/5.14 = ( gcd_gcd_int @ A @ B ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9758_nat__descend__induct,axiom,
% 8.90/5.14 ! [N: nat,P: nat > $o,M: nat] :
% 8.90/5.14 ( ! [K3: nat] :
% 8.90/5.14 ( ( ord_less_nat @ N @ K3 )
% 8.90/5.14 => ( P @ K3 ) )
% 8.90/5.14 => ( ! [K3: nat] :
% 8.90/5.14 ( ( ord_less_eq_nat @ K3 @ N )
% 8.90/5.14 => ( ! [I: nat] :
% 8.90/5.14 ( ( ord_less_nat @ K3 @ I )
% 8.90/5.14 => ( P @ I ) )
% 8.90/5.14 => ( P @ K3 ) ) )
% 8.90/5.14 => ( P @ M ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9759_gcd__1__nat,axiom,
% 8.90/5.14 ! [M: nat] :
% 8.90/5.14 ( ( gcd_gcd_nat @ M @ one_one_nat )
% 8.90/5.14 = one_one_nat ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9760_gcd__Suc__0,axiom,
% 8.90/5.14 ! [M: nat] :
% 8.90/5.14 ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
% 8.90/5.14 = ( suc @ zero_zero_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9761_gcd__pos__nat,axiom,
% 8.90/5.14 ! [M: nat,N: nat] :
% 8.90/5.14 ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
% 8.90/5.14 = ( ( M != zero_zero_nat )
% 8.90/5.14 | ( N != zero_zero_nat ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9762_gcd__le2__nat,axiom,
% 8.90/5.14 ! [B: nat,A: nat] :
% 8.90/5.14 ( ( B != zero_zero_nat )
% 8.90/5.14 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9763_gcd__le1__nat,axiom,
% 8.90/5.14 ! [A: nat,B: nat] :
% 8.90/5.14 ( ( A != zero_zero_nat )
% 8.90/5.14 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9764_gcd__mult__distrib__nat,axiom,
% 8.90/5.14 ! [K: nat,M: nat,N: nat] :
% 8.90/5.14 ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
% 8.90/5.14 = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9765_gcd__diff1__nat,axiom,
% 8.90/5.14 ! [N: nat,M: nat] :
% 8.90/5.14 ( ( ord_less_eq_nat @ N @ M )
% 8.90/5.14 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 8.90/5.14 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9766_gcd__diff2__nat,axiom,
% 8.90/5.14 ! [M: nat,N: nat] :
% 8.90/5.14 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.14 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
% 8.90/5.14 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9767_bezout__nat,axiom,
% 8.90/5.14 ! [A: nat,B: nat] :
% 8.90/5.14 ( ( A != zero_zero_nat )
% 8.90/5.14 => ? [X5: nat,Y4: nat] :
% 8.90/5.14 ( ( times_times_nat @ A @ X5 )
% 8.90/5.14 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9768_bezout__gcd__nat_H,axiom,
% 8.90/5.14 ! [B: nat,A: nat] :
% 8.90/5.14 ? [X5: nat,Y4: nat] :
% 8.90/5.14 ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y4 ) @ ( times_times_nat @ A @ X5 ) )
% 8.90/5.14 & ( ( minus_minus_nat @ ( times_times_nat @ A @ X5 ) @ ( times_times_nat @ B @ Y4 ) )
% 8.90/5.14 = ( gcd_gcd_nat @ A @ B ) ) )
% 8.90/5.14 | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B @ X5 ) )
% 8.90/5.14 & ( ( minus_minus_nat @ ( times_times_nat @ B @ X5 ) @ ( times_times_nat @ A @ Y4 ) )
% 8.90/5.14 = ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9769_gcd__is__Max__divisors__nat,axiom,
% 8.90/5.14 ! [N: nat,M: nat] :
% 8.90/5.14 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( ( gcd_gcd_nat @ M @ N )
% 8.90/5.14 = ( lattic8265883725875713057ax_nat
% 8.90/5.14 @ ( collect_nat
% 8.90/5.14 @ ^ [D2: nat] :
% 8.90/5.14 ( ( dvd_dvd_nat @ D2 @ M )
% 8.90/5.14 & ( dvd_dvd_nat @ D2 @ N ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9770_bezw__aux,axiom,
% 8.90/5.14 ! [X2: nat,Y3: nat] :
% 8.90/5.14 ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X2 @ Y3 ) )
% 8.90/5.14 = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X2 @ Y3 ) ) @ ( semiri1314217659103216013at_int @ X2 ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X2 @ Y3 ) ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9771_gcd__nat_Opelims,axiom,
% 8.90/5.14 ! [X2: nat,Xa2: nat,Y3: nat] :
% 8.90/5.14 ( ( ( gcd_gcd_nat @ X2 @ Xa2 )
% 8.90/5.14 = Y3 )
% 8.90/5.14 => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) )
% 8.90/5.14 => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 8.90/5.14 => ( Y3 = X2 ) )
% 8.90/5.14 & ( ( Xa2 != zero_zero_nat )
% 8.90/5.14 => ( Y3
% 8.90/5.14 = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) ) )
% 8.90/5.14 => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9772_card__greaterThanLessThan__int,axiom,
% 8.90/5.14 ! [L: int,U: int] :
% 8.90/5.14 ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
% 8.90/5.14 = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9773_xor__minus__numerals_I1_J,axiom,
% 8.90/5.14 ! [N: num,K: int] :
% 8.90/5.14 ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
% 8.90/5.14 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9774_xor__minus__numerals_I2_J,axiom,
% 8.90/5.14 ! [K: int,N: num] :
% 8.90/5.14 ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.14 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9775_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
% 8.90/5.14 ! [L: int,U: int] :
% 8.90/5.14 ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
% 8.90/5.14 = ( set_or5832277885323065728an_int @ L @ U ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9776_sub__BitM__One__eq,axiom,
% 8.90/5.14 ! [N: num] :
% 8.90/5.14 ( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
% 8.90/5.14 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9777_card__greaterThanLessThan,axiom,
% 8.90/5.14 ! [L: nat,U: nat] :
% 8.90/5.14 ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
% 8.90/5.14 = ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9778_atLeastSucLessThan__greaterThanLessThan,axiom,
% 8.90/5.14 ! [L: nat,U: nat] :
% 8.90/5.14 ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
% 8.90/5.14 = ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9779_tanh__real__bounds,axiom,
% 8.90/5.14 ! [X2: real] : ( member_real @ ( tanh_real @ X2 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9780_Suc__funpow,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( compow_nat_nat @ N @ suc )
% 8.90/5.14 = ( plus_plus_nat @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9781_divmod__integer__eq__cases,axiom,
% 8.90/5.14 ( code_divmod_integer
% 8.90/5.14 = ( ^ [K2: code_integer,L2: code_integer] :
% 8.90/5.14 ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 8.90/5.14 @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 8.90/5.14 @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
% 8.90/5.14 @ ( if_Pro6119634080678213985nteger
% 8.90/5.14 @ ( ( sgn_sgn_Code_integer @ K2 )
% 8.90/5.14 = ( sgn_sgn_Code_integer @ L2 ) )
% 8.90/5.14 @ ( code_divmod_abs @ K2 @ L2 )
% 8.90/5.14 @ ( produc6916734918728496179nteger
% 8.90/5.14 @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S6 ) ) )
% 8.90/5.14 @ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9782_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 8.90/5.14 ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 8.90/5.14 @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
% 8.90/5.14 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9783_card_Ocomp__fun__commute__on,axiom,
% 8.90/5.14 ( ( comp_nat_nat_nat @ suc @ suc )
% 8.90/5.14 = ( comp_nat_nat_nat @ suc @ suc ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9784_Inf__real__def,axiom,
% 8.90/5.14 ( comple4887499456419720421f_real
% 8.90/5.14 = ( ^ [X4: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X4 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9785_Code__Numeral_Onegative__def,axiom,
% 8.90/5.14 ( code_negative
% 8.90/5.14 = ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9786_Code__Target__Int_Onegative__def,axiom,
% 8.90/5.14 ( code_Target_negative
% 8.90/5.14 = ( comp_int_int_num @ uminus_uminus_int @ numeral_numeral_int ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9787_suminf__eq__SUP__real,axiom,
% 8.90/5.14 ! [X8: nat > real] :
% 8.90/5.14 ( ( summable_real @ X8 )
% 8.90/5.14 => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X8 @ I3 ) )
% 8.90/5.14 => ( ( suminf_real @ X8 )
% 8.90/5.14 = ( comple1385675409528146559p_real
% 8.90/5.14 @ ( image_nat_real
% 8.90/5.14 @ ^ [I4: nat] : ( groups6591440286371151544t_real @ X8 @ ( set_ord_lessThan_nat @ I4 ) )
% 8.90/5.14 @ top_top_set_nat ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9788_range__mod,axiom,
% 8.90/5.14 ! [N: nat] :
% 8.90/5.14 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( ( image_nat_nat
% 8.90/5.14 @ ^ [M3: nat] : ( modulo_modulo_nat @ M3 @ N )
% 8.90/5.14 @ top_top_set_nat )
% 8.90/5.14 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9789_UNIV__nat__eq,axiom,
% 8.90/5.14 ( top_top_set_nat
% 8.90/5.14 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9790_card__UNIV__unit,axiom,
% 8.90/5.14 ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
% 8.90/5.14 = one_one_nat ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9791_card__UNIV__bool,axiom,
% 8.90/5.14 ( ( finite_card_o @ top_top_set_o )
% 8.90/5.14 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9792_range__mult,axiom,
% 8.90/5.14 ! [A: real] :
% 8.90/5.14 ( ( ( A = zero_zero_real )
% 8.90/5.14 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 8.90/5.14 = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 8.90/5.14 & ( ( A != zero_zero_real )
% 8.90/5.14 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 8.90/5.14 = top_top_set_real ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9793_root__def,axiom,
% 8.90/5.14 ( root
% 8.90/5.14 = ( ^ [N3: nat,X: real] :
% 8.90/5.14 ( if_real @ ( N3 = zero_zero_nat ) @ zero_zero_real
% 8.90/5.14 @ ( the_in5290026491893676941l_real @ top_top_set_real
% 8.90/5.14 @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N3 ) )
% 8.90/5.14 @ X ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9794_card__UNIV__char,axiom,
% 8.90/5.14 ( ( finite_card_char @ top_top_set_char )
% 8.90/5.14 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9795_UNIV__char__of__nat,axiom,
% 8.90/5.14 ( top_top_set_char
% 8.90/5.14 = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9796_nat__of__char__less__256,axiom,
% 8.90/5.14 ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9797_range__nat__of__char,axiom,
% 8.90/5.14 ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
% 8.90/5.14 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9798_integer__of__char__code,axiom,
% 8.90/5.14 ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 8.90/5.14 ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 8.90/5.14 = ( 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 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9799_String_Ochar__of__ascii__of,axiom,
% 8.90/5.14 ! [C: char] :
% 8.90/5.14 ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
% 8.90/5.14 = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9800_DERIV__real__root__generic,axiom,
% 8.90/5.14 ! [N: nat,X2: real,D4: real] :
% 8.90/5.14 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( ( X2 != zero_zero_real )
% 8.90/5.14 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.14 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.14 => ( D4
% 8.90/5.14 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 8.90/5.14 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.14 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 8.90/5.14 => ( D4
% 8.90/5.14 = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 8.90/5.14 => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.14 => ( D4
% 8.90/5.14 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9801_deriv__nonneg__imp__mono,axiom,
% 8.90/5.14 ! [A: real,B: real,G: real > real,G2: real > real] :
% 8.90/5.14 ( ! [X5: real] :
% 8.90/5.14 ( ( member_real @ X5 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 8.90/5.14 => ( ! [X5: real] :
% 8.90/5.14 ( ( member_real @ X5 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 8.90/5.14 => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X5 ) ) )
% 8.90/5.14 => ( ( ord_less_eq_real @ A @ B )
% 8.90/5.14 => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9802_DERIV__nonpos__imp__nonincreasing,axiom,
% 8.90/5.14 ! [A: real,B: real,F: real > real] :
% 8.90/5.14 ( ( ord_less_eq_real @ A @ B )
% 8.90/5.14 => ( ! [X5: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ A @ X5 )
% 8.90/5.14 => ( ( ord_less_eq_real @ X5 @ B )
% 8.90/5.14 => ? [Y5: real] :
% 8.90/5.14 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 8.90/5.14 & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
% 8.90/5.14 => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9803_DERIV__nonneg__imp__nondecreasing,axiom,
% 8.90/5.14 ! [A: real,B: real,F: real > real] :
% 8.90/5.14 ( ( ord_less_eq_real @ A @ B )
% 8.90/5.14 => ( ! [X5: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ A @ X5 )
% 8.90/5.14 => ( ( ord_less_eq_real @ X5 @ B )
% 8.90/5.14 => ? [Y5: real] :
% 8.90/5.14 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 8.90/5.14 & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
% 8.90/5.14 => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9804_DERIV__const__ratio__const,axiom,
% 8.90/5.14 ! [A: real,B: real,F: real > real,K: real] :
% 8.90/5.14 ( ( A != B )
% 8.90/5.14 => ( ! [X5: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 8.90/5.14 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 8.90/5.14 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ K ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9805_DERIV__const__ratio__const2,axiom,
% 8.90/5.14 ! [A: real,B: real,F: real > real,K: real] :
% 8.90/5.14 ( ( A != B )
% 8.90/5.14 => ( ! [X5: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 8.90/5.14 => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
% 8.90/5.14 = K ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9806_DERIV__neg__dec__left,axiom,
% 8.90/5.14 ! [F: real > real,L: real,X2: real] :
% 8.90/5.14 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 8.90/5.14 => ( ( ord_less_real @ L @ zero_zero_real )
% 8.90/5.14 => ? [D3: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ D3 )
% 8.90/5.14 & ! [H4: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ H4 )
% 8.90/5.14 => ( ( ord_less_real @ H4 @ D3 )
% 8.90/5.14 => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9807_DERIV__pos__inc__left,axiom,
% 8.90/5.14 ! [F: real > real,L: real,X2: real] :
% 8.90/5.14 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 8.90/5.14 => ( ( ord_less_real @ zero_zero_real @ L )
% 8.90/5.14 => ? [D3: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ D3 )
% 8.90/5.14 & ! [H4: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ H4 )
% 8.90/5.14 => ( ( ord_less_real @ H4 @ D3 )
% 8.90/5.14 => ( ord_less_real @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9808_has__real__derivative__neg__dec__left,axiom,
% 8.90/5.14 ! [F: real > real,L: real,X2: real,S3: set_real] :
% 8.90/5.14 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S3 ) )
% 8.90/5.14 => ( ( ord_less_real @ L @ zero_zero_real )
% 8.90/5.14 => ? [D3: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ D3 )
% 8.90/5.14 & ! [H4: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ H4 )
% 8.90/5.14 => ( ( member_real @ ( minus_minus_real @ X2 @ H4 ) @ S3 )
% 8.90/5.14 => ( ( ord_less_real @ H4 @ D3 )
% 8.90/5.14 => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9809_has__real__derivative__pos__inc__left,axiom,
% 8.90/5.14 ! [F: real > real,L: real,X2: real,S3: set_real] :
% 8.90/5.14 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S3 ) )
% 8.90/5.14 => ( ( ord_less_real @ zero_zero_real @ L )
% 8.90/5.14 => ? [D3: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ D3 )
% 8.90/5.14 & ! [H4: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ H4 )
% 8.90/5.14 => ( ( member_real @ ( minus_minus_real @ X2 @ H4 ) @ S3 )
% 8.90/5.14 => ( ( ord_less_real @ H4 @ D3 )
% 8.90/5.14 => ( ord_less_real @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9810_DERIV__neg__imp__decreasing,axiom,
% 8.90/5.14 ! [A: real,B: real,F: real > real] :
% 8.90/5.14 ( ( ord_less_real @ A @ B )
% 8.90/5.14 => ( ! [X5: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ A @ X5 )
% 8.90/5.14 => ( ( ord_less_eq_real @ X5 @ B )
% 8.90/5.14 => ? [Y5: real] :
% 8.90/5.14 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 8.90/5.14 & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 8.90/5.14 => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9811_DERIV__pos__imp__increasing,axiom,
% 8.90/5.14 ! [A: real,B: real,F: real > real] :
% 8.90/5.14 ( ( ord_less_real @ A @ B )
% 8.90/5.14 => ( ! [X5: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ A @ X5 )
% 8.90/5.14 => ( ( ord_less_eq_real @ X5 @ B )
% 8.90/5.14 => ? [Y5: real] :
% 8.90/5.14 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 8.90/5.14 & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 8.90/5.14 => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9812_MVT2,axiom,
% 8.90/5.14 ! [A: real,B: real,F: real > real,F4: real > real] :
% 8.90/5.14 ( ( ord_less_real @ A @ B )
% 8.90/5.14 => ( ! [X5: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ A @ X5 )
% 8.90/5.14 => ( ( ord_less_eq_real @ X5 @ B )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 8.90/5.14 => ? [Z2: real] :
% 8.90/5.14 ( ( ord_less_real @ A @ Z2 )
% 8.90/5.14 & ( ord_less_real @ Z2 @ B )
% 8.90/5.14 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 8.90/5.14 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F4 @ Z2 ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9813_DERIV__local__const,axiom,
% 8.90/5.14 ! [F: real > real,L: real,X2: real,D: real] :
% 8.90/5.14 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 8.90/5.14 => ( ( ord_less_real @ zero_zero_real @ D )
% 8.90/5.14 => ( ! [Y4: real] :
% 8.90/5.14 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ D )
% 8.90/5.14 => ( ( F @ X2 )
% 8.90/5.14 = ( F @ Y4 ) ) )
% 8.90/5.14 => ( L = zero_zero_real ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9814_DERIV__const__average,axiom,
% 8.90/5.14 ! [A: real,B: real,V: real > real,K: real] :
% 8.90/5.14 ( ( A != B )
% 8.90/5.14 => ( ! [X5: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 8.90/5.14 => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 8.90/5.14 = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9815_DERIV__local__min,axiom,
% 8.90/5.14 ! [F: real > real,L: real,X2: real,D: real] :
% 8.90/5.14 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 8.90/5.14 => ( ( ord_less_real @ zero_zero_real @ D )
% 8.90/5.14 => ( ! [Y4: real] :
% 8.90/5.14 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ D )
% 8.90/5.14 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 8.90/5.14 => ( L = zero_zero_real ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9816_DERIV__local__max,axiom,
% 8.90/5.14 ! [F: real > real,L: real,X2: real,D: real] :
% 8.90/5.14 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 8.90/5.14 => ( ( ord_less_real @ zero_zero_real @ D )
% 8.90/5.14 => ( ! [Y4: real] :
% 8.90/5.14 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ D )
% 8.90/5.14 => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X2 ) ) )
% 8.90/5.14 => ( L = zero_zero_real ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9817_DERIV__ln__divide,axiom,
% 8.90/5.14 ! [X2: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9818_DERIV__pow,axiom,
% 8.90/5.14 ! [N: nat,X2: real,S: set_real] :
% 8.90/5.14 ( has_fi5821293074295781190e_real
% 8.90/5.14 @ ^ [X: real] : ( power_power_real @ X @ N )
% 8.90/5.14 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 8.90/5.14 @ ( topolo2177554685111907308n_real @ X2 @ S ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9819_DERIV__fun__pow,axiom,
% 8.90/5.14 ! [G: real > real,M: real,X2: real,N: nat] :
% 8.90/5.14 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 8.90/5.14 => ( has_fi5821293074295781190e_real
% 8.90/5.14 @ ^ [X: real] : ( power_power_real @ ( G @ X ) @ N )
% 8.90/5.14 @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X2 ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
% 8.90/5.14 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9820_has__real__derivative__powr,axiom,
% 8.90/5.14 ! [Z: real,R2: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ Z )
% 8.90/5.14 => ( has_fi5821293074295781190e_real
% 8.90/5.14 @ ^ [Z3: real] : ( powr_real @ Z3 @ R2 )
% 8.90/5.14 @ ( times_times_real @ R2 @ ( powr_real @ Z @ ( minus_minus_real @ R2 @ one_one_real ) ) )
% 8.90/5.14 @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9821_DERIV__log,axiom,
% 8.90/5.14 ! [X2: real,B: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X2 ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9822_DERIV__fun__powr,axiom,
% 8.90/5.14 ! [G: real > real,M: real,X2: real,R2: real] :
% 8.90/5.14 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 8.90/5.14 => ( ( ord_less_real @ zero_zero_real @ ( G @ X2 ) )
% 8.90/5.14 => ( has_fi5821293074295781190e_real
% 8.90/5.14 @ ^ [X: real] : ( powr_real @ ( G @ X ) @ R2 )
% 8.90/5.14 @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X2 ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 8.90/5.14 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9823_DERIV__powr,axiom,
% 8.90/5.14 ! [G: real > real,M: real,X2: real,F: real > real,R2: real] :
% 8.90/5.14 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 8.90/5.14 => ( ( ord_less_real @ zero_zero_real @ ( G @ X2 ) )
% 8.90/5.14 => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 8.90/5.14 => ( has_fi5821293074295781190e_real
% 8.90/5.14 @ ^ [X: real] : ( powr_real @ ( G @ X ) @ ( F @ X ) )
% 8.90/5.14 @ ( times_times_real @ ( powr_real @ ( G @ X2 ) @ ( F @ X2 ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X2 ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X2 ) ) @ ( G @ X2 ) ) ) )
% 8.90/5.14 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9824_DERIV__real__sqrt,axiom,
% 8.90/5.14 ! [X2: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9825_DERIV__series_H,axiom,
% 8.90/5.14 ! [F: real > nat > real,F4: real > nat > real,X0: real,A: real,B: real,L5: nat > real] :
% 8.90/5.14 ( ! [N2: nat] :
% 8.90/5.14 ( has_fi5821293074295781190e_real
% 8.90/5.14 @ ^ [X: real] : ( F @ X @ N2 )
% 8.90/5.14 @ ( F4 @ X0 @ N2 )
% 8.90/5.14 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
% 8.90/5.14 => ( ! [X5: real] :
% 8.90/5.14 ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 8.90/5.14 => ( summable_real @ ( F @ X5 ) ) )
% 8.90/5.14 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 8.90/5.14 => ( ( summable_real @ ( F4 @ X0 ) )
% 8.90/5.14 => ( ( summable_real @ L5 )
% 8.90/5.14 => ( ! [N2: nat,X5: real,Y4: real] :
% 8.90/5.14 ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 8.90/5.14 => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 8.90/5.14 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X5 @ N2 ) @ ( F @ Y4 @ N2 ) ) ) @ ( times_times_real @ ( L5 @ N2 ) @ ( abs_abs_real @ ( minus_minus_real @ X5 @ Y4 ) ) ) ) ) )
% 8.90/5.14 => ( has_fi5821293074295781190e_real
% 8.90/5.14 @ ^ [X: real] : ( suminf_real @ ( F @ X ) )
% 8.90/5.14 @ ( suminf_real @ ( F4 @ X0 ) )
% 8.90/5.14 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9826_DERIV__arctan,axiom,
% 8.90/5.14 ! [X2: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9827_arsinh__real__has__field__derivative,axiom,
% 8.90/5.14 ! [X2: real,A2: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ A2 ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9828_DERIV__real__sqrt__generic,axiom,
% 8.90/5.14 ! [X2: real,D4: real] :
% 8.90/5.14 ( ( X2 != zero_zero_real )
% 8.90/5.14 => ( ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.14 => ( D4
% 8.90/5.14 = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 8.90/5.14 => ( ( ( ord_less_real @ X2 @ zero_zero_real )
% 8.90/5.14 => ( D4
% 8.90/5.14 = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9829_arcosh__real__has__field__derivative,axiom,
% 8.90/5.14 ! [X2: real,A2: set_real] :
% 8.90/5.14 ( ( ord_less_real @ one_one_real @ X2 )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ A2 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9830_artanh__real__has__field__derivative,axiom,
% 8.90/5.14 ! [X2: real,A2: set_real] :
% 8.90/5.14 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ A2 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9831_DERIV__power__series_H,axiom,
% 8.90/5.14 ! [R: real,F: nat > real,X0: real] :
% 8.90/5.14 ( ! [X5: real] :
% 8.90/5.14 ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 8.90/5.14 => ( summable_real
% 8.90/5.14 @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X5 @ N3 ) ) ) )
% 8.90/5.14 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 8.90/5.14 => ( ( ord_less_real @ zero_zero_real @ R )
% 8.90/5.14 => ( has_fi5821293074295781190e_real
% 8.90/5.14 @ ^ [X: real] :
% 8.90/5.14 ( suminf_real
% 8.90/5.14 @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X @ ( suc @ N3 ) ) ) )
% 8.90/5.14 @ ( suminf_real
% 8.90/5.14 @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X0 @ N3 ) ) )
% 8.90/5.14 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9832_DERIV__real__root,axiom,
% 8.90/5.14 ! [N: nat,X2: real] :
% 8.90/5.14 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9833_DERIV__arccos,axiom,
% 8.90/5.14 ! [X2: real] :
% 8.90/5.14 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.14 => ( ( ord_less_real @ X2 @ one_one_real )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9834_DERIV__arcsin,axiom,
% 8.90/5.14 ! [X2: real] :
% 8.90/5.14 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.14 => ( ( ord_less_real @ X2 @ one_one_real )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9835_Maclaurin__all__le__objl,axiom,
% 8.90/5.14 ! [Diff: nat > real > real,F: real > real,X2: real,N: nat] :
% 8.90/5.14 ( ( ( ( Diff @ zero_zero_nat )
% 8.90/5.14 = F )
% 8.90/5.14 & ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 8.90/5.14 => ? [T5: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X2 ) )
% 8.90/5.14 & ( ( F @ X2 )
% 8.90/5.14 = ( plus_plus_real
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.14 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9836_Maclaurin__all__le,axiom,
% 8.90/5.14 ! [Diff: nat > real > real,F: real > real,X2: real,N: nat] :
% 8.90/5.14 ( ( ( Diff @ zero_zero_nat )
% 8.90/5.14 = F )
% 8.90/5.14 => ( ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 8.90/5.14 => ? [T5: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X2 ) )
% 8.90/5.14 & ( ( F @ X2 )
% 8.90/5.14 = ( plus_plus_real
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.14 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9837_DERIV__odd__real__root,axiom,
% 8.90/5.14 ! [N: nat,X2: real] :
% 8.90/5.14 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.14 => ( ( X2 != zero_zero_real )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9838_Maclaurin__minus,axiom,
% 8.90/5.14 ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 8.90/5.14 ( ( ord_less_real @ H2 @ zero_zero_real )
% 8.90/5.14 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( ( ( Diff @ zero_zero_nat )
% 8.90/5.14 = F )
% 8.90/5.14 => ( ! [M4: nat,T5: real] :
% 8.90/5.14 ( ( ( ord_less_nat @ M4 @ N )
% 8.90/5.14 & ( ord_less_eq_real @ H2 @ T5 )
% 8.90/5.14 & ( ord_less_eq_real @ T5 @ zero_zero_real ) )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 8.90/5.14 => ? [T5: real] :
% 8.90/5.14 ( ( ord_less_real @ H2 @ T5 )
% 8.90/5.14 & ( ord_less_real @ T5 @ zero_zero_real )
% 8.90/5.14 & ( ( F @ H2 )
% 8.90/5.14 = ( plus_plus_real
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.14 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9839_Maclaurin2,axiom,
% 8.90/5.14 ! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ H2 )
% 8.90/5.14 => ( ( ( Diff @ zero_zero_nat )
% 8.90/5.14 = F )
% 8.90/5.14 => ( ! [M4: nat,T5: real] :
% 8.90/5.14 ( ( ( ord_less_nat @ M4 @ N )
% 8.90/5.14 & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 8.90/5.14 & ( ord_less_eq_real @ T5 @ H2 ) )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 8.90/5.14 => ? [T5: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ T5 )
% 8.90/5.14 & ( ord_less_eq_real @ T5 @ H2 )
% 8.90/5.14 & ( ( F @ H2 )
% 8.90/5.14 = ( plus_plus_real
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.14 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9840_Maclaurin,axiom,
% 8.90/5.14 ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ H2 )
% 8.90/5.14 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( ( ( Diff @ zero_zero_nat )
% 8.90/5.14 = F )
% 8.90/5.14 => ( ! [M4: nat,T5: real] :
% 8.90/5.14 ( ( ( ord_less_nat @ M4 @ N )
% 8.90/5.14 & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 8.90/5.14 & ( ord_less_eq_real @ T5 @ H2 ) )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 8.90/5.14 => ? [T5: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ T5 )
% 8.90/5.14 & ( ord_less_real @ T5 @ H2 )
% 8.90/5.14 & ( ( F @ H2 )
% 8.90/5.14 = ( plus_plus_real
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.14 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9841_Maclaurin__all__lt,axiom,
% 8.90/5.14 ! [Diff: nat > real > real,F: real > real,N: nat,X2: real] :
% 8.90/5.14 ( ( ( Diff @ zero_zero_nat )
% 8.90/5.14 = F )
% 8.90/5.14 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( ( X2 != zero_zero_real )
% 8.90/5.14 => ( ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 8.90/5.14 => ? [T5: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
% 8.90/5.14 & ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X2 ) )
% 8.90/5.14 & ( ( F @ X2 )
% 8.90/5.14 = ( plus_plus_real
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.14 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9842_Maclaurin__bi__le,axiom,
% 8.90/5.14 ! [Diff: nat > real > real,F: real > real,N: nat,X2: real] :
% 8.90/5.14 ( ( ( Diff @ zero_zero_nat )
% 8.90/5.14 = F )
% 8.90/5.14 => ( ! [M4: nat,T5: real] :
% 8.90/5.14 ( ( ( ord_less_nat @ M4 @ N )
% 8.90/5.14 & ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X2 ) ) )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 8.90/5.14 => ? [T5: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X2 ) )
% 8.90/5.14 & ( ( F @ X2 )
% 8.90/5.14 = ( plus_plus_real
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.14 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9843_Taylor__down,axiom,
% 8.90/5.14 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 8.90/5.14 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( ( ( Diff @ zero_zero_nat )
% 8.90/5.14 = F )
% 8.90/5.14 => ( ! [M4: nat,T5: real] :
% 8.90/5.14 ( ( ( ord_less_nat @ M4 @ N )
% 8.90/5.14 & ( ord_less_eq_real @ A @ T5 )
% 8.90/5.14 & ( ord_less_eq_real @ T5 @ B ) )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 8.90/5.14 => ( ( ord_less_real @ A @ C )
% 8.90/5.14 => ( ( ord_less_eq_real @ C @ B )
% 8.90/5.14 => ? [T5: real] :
% 8.90/5.14 ( ( ord_less_real @ A @ T5 )
% 8.90/5.14 & ( ord_less_real @ T5 @ C )
% 8.90/5.14 & ( ( F @ A )
% 8.90/5.14 = ( plus_plus_real
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M3 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.14 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9844_Taylor__up,axiom,
% 8.90/5.14 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 8.90/5.14 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( ( ( Diff @ zero_zero_nat )
% 8.90/5.14 = F )
% 8.90/5.14 => ( ! [M4: nat,T5: real] :
% 8.90/5.14 ( ( ( ord_less_nat @ M4 @ N )
% 8.90/5.14 & ( ord_less_eq_real @ A @ T5 )
% 8.90/5.14 & ( ord_less_eq_real @ T5 @ B ) )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 8.90/5.14 => ( ( ord_less_eq_real @ A @ C )
% 8.90/5.14 => ( ( ord_less_real @ C @ B )
% 8.90/5.14 => ? [T5: real] :
% 8.90/5.14 ( ( ord_less_real @ C @ T5 )
% 8.90/5.14 & ( ord_less_real @ T5 @ B )
% 8.90/5.14 & ( ( F @ B )
% 8.90/5.14 = ( plus_plus_real
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M3 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.14 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9845_Taylor,axiom,
% 8.90/5.14 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X2: real] :
% 8.90/5.14 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( ( ( Diff @ zero_zero_nat )
% 8.90/5.14 = F )
% 8.90/5.14 => ( ! [M4: nat,T5: real] :
% 8.90/5.14 ( ( ( ord_less_nat @ M4 @ N )
% 8.90/5.14 & ( ord_less_eq_real @ A @ T5 )
% 8.90/5.14 & ( ord_less_eq_real @ T5 @ B ) )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 8.90/5.14 => ( ( ord_less_eq_real @ A @ C )
% 8.90/5.14 => ( ( ord_less_eq_real @ C @ B )
% 8.90/5.14 => ( ( ord_less_eq_real @ A @ X2 )
% 8.90/5.14 => ( ( ord_less_eq_real @ X2 @ B )
% 8.90/5.14 => ( ( X2 != C )
% 8.90/5.14 => ? [T5: real] :
% 8.90/5.14 ( ( ( ord_less_real @ X2 @ C )
% 8.90/5.14 => ( ( ord_less_real @ X2 @ T5 )
% 8.90/5.14 & ( ord_less_real @ T5 @ C ) ) )
% 8.90/5.14 & ( ~ ( ord_less_real @ X2 @ C )
% 8.90/5.14 => ( ( ord_less_real @ C @ T5 )
% 8.90/5.14 & ( ord_less_real @ T5 @ X2 ) ) )
% 8.90/5.14 & ( ( F @ X2 )
% 8.90/5.14 = ( plus_plus_real
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ X2 @ C ) @ M3 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ N ) )
% 8.90/5.14 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X2 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9846_Maclaurin__lemma2,axiom,
% 8.90/5.14 ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B3: real] :
% 8.90/5.14 ( ! [M4: nat,T5: real] :
% 8.90/5.14 ( ( ( ord_less_nat @ M4 @ N )
% 8.90/5.14 & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 8.90/5.14 & ( ord_less_eq_real @ T5 @ H2 ) )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 8.90/5.14 => ( ( N
% 8.90/5.14 = ( suc @ K ) )
% 8.90/5.14 => ! [M2: nat,T6: real] :
% 8.90/5.14 ( ( ( ord_less_nat @ M2 @ N )
% 8.90/5.14 & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 8.90/5.14 & ( ord_less_eq_real @ T6 @ H2 ) )
% 8.90/5.14 => ( has_fi5821293074295781190e_real
% 8.90/5.14 @ ^ [U2: real] :
% 8.90/5.14 ( minus_minus_real @ ( Diff @ M2 @ U2 )
% 8.90/5.14 @ ( plus_plus_real
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [P4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P4 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ U2 @ P4 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M2 ) ) )
% 8.90/5.14 @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) )
% 8.90/5.14 @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T6 )
% 8.90/5.14 @ ( plus_plus_real
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [P4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P4 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ T6 @ P4 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) )
% 8.90/5.14 @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ T6 @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) ) ) ) )
% 8.90/5.14 @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9847_DERIV__arctan__series,axiom,
% 8.90/5.14 ! [X2: real] :
% 8.90/5.14 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 8.90/5.14 => ( has_fi5821293074295781190e_real
% 8.90/5.14 @ ^ [X9: real] :
% 8.90/5.14 ( suminf_real
% 8.90/5.14 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 8.90/5.14 @ ( suminf_real
% 8.90/5.14 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( power_power_real @ X2 @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.90/5.14 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9848_DERIV__even__real__root,axiom,
% 8.90/5.14 ! [N: nat,X2: real] :
% 8.90/5.14 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.14 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9849_isCont__Lb__Ub,axiom,
% 8.90/5.14 ! [A: real,B: real,F: real > real] :
% 8.90/5.14 ( ( ord_less_eq_real @ A @ B )
% 8.90/5.14 => ( ! [X5: real] :
% 8.90/5.14 ( ( ( ord_less_eq_real @ A @ X5 )
% 8.90/5.14 & ( ord_less_eq_real @ X5 @ B ) )
% 8.90/5.14 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ F ) )
% 8.90/5.14 => ? [L6: real,M9: real] :
% 8.90/5.14 ( ! [X3: real] :
% 8.90/5.14 ( ( ( ord_less_eq_real @ A @ X3 )
% 8.90/5.14 & ( ord_less_eq_real @ X3 @ B ) )
% 8.90/5.14 => ( ( ord_less_eq_real @ L6 @ ( F @ X3 ) )
% 8.90/5.14 & ( ord_less_eq_real @ ( F @ X3 ) @ M9 ) ) )
% 8.90/5.14 & ! [Y5: real] :
% 8.90/5.14 ( ( ( ord_less_eq_real @ L6 @ Y5 )
% 8.90/5.14 & ( ord_less_eq_real @ Y5 @ M9 ) )
% 8.90/5.14 => ? [X5: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ A @ X5 )
% 8.90/5.14 & ( ord_less_eq_real @ X5 @ B )
% 8.90/5.14 & ( ( F @ X5 )
% 8.90/5.14 = Y5 ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9850_isCont__real__sqrt,axiom,
% 8.90/5.14 ! [X2: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ sqrt ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9851_isCont__real__root,axiom,
% 8.90/5.14 ! [X2: real,N: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ ( root @ N ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9852_isCont__inverse__function2,axiom,
% 8.90/5.14 ! [A: real,X2: real,B: real,G: real > real,F: real > real] :
% 8.90/5.14 ( ( ord_less_real @ A @ X2 )
% 8.90/5.14 => ( ( ord_less_real @ X2 @ B )
% 8.90/5.14 => ( ! [Z2: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ A @ Z2 )
% 8.90/5.14 => ( ( ord_less_eq_real @ Z2 @ B )
% 8.90/5.14 => ( ( G @ ( F @ Z2 ) )
% 8.90/5.14 = Z2 ) ) )
% 8.90/5.14 => ( ! [Z2: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ A @ Z2 )
% 8.90/5.14 => ( ( ord_less_eq_real @ Z2 @ B )
% 8.90/5.14 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
% 8.90/5.14 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X2 ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9853_isCont__arcosh,axiom,
% 8.90/5.14 ! [X2: real] :
% 8.90/5.14 ( ( ord_less_real @ one_one_real @ X2 )
% 8.90/5.14 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arcosh_real ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9854_isCont__arccos,axiom,
% 8.90/5.14 ! [X2: real] :
% 8.90/5.14 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.14 => ( ( ord_less_real @ X2 @ one_one_real )
% 8.90/5.14 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arccos ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9855_isCont__arcsin,axiom,
% 8.90/5.14 ! [X2: real] :
% 8.90/5.14 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.14 => ( ( ord_less_real @ X2 @ one_one_real )
% 8.90/5.14 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arcsin ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9856_LIM__less__bound,axiom,
% 8.90/5.14 ! [B: real,X2: real,F: real > real] :
% 8.90/5.14 ( ( ord_less_real @ B @ X2 )
% 8.90/5.14 => ( ! [X5: real] :
% 8.90/5.14 ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ B @ X2 ) )
% 8.90/5.14 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 8.90/5.14 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ F )
% 8.90/5.14 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9857_isCont__artanh,axiom,
% 8.90/5.14 ! [X2: real] :
% 8.90/5.14 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 8.90/5.14 => ( ( ord_less_real @ X2 @ one_one_real )
% 8.90/5.14 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ artanh_real ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9858_isCont__inverse__function,axiom,
% 8.90/5.14 ! [D: real,X2: real,G: real > real,F: real > real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ D )
% 8.90/5.14 => ( ! [Z2: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X2 ) ) @ D )
% 8.90/5.14 => ( ( G @ ( F @ Z2 ) )
% 8.90/5.14 = Z2 ) )
% 8.90/5.14 => ( ! [Z2: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X2 ) ) @ D )
% 8.90/5.14 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) )
% 8.90/5.14 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X2 ) @ top_top_set_real ) @ G ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9859_GMVT_H,axiom,
% 8.90/5.14 ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F4: real > real] :
% 8.90/5.14 ( ( ord_less_real @ A @ B )
% 8.90/5.14 => ( ! [Z2: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ A @ Z2 )
% 8.90/5.14 => ( ( ord_less_eq_real @ Z2 @ B )
% 8.90/5.14 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
% 8.90/5.14 => ( ! [Z2: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ A @ Z2 )
% 8.90/5.14 => ( ( ord_less_eq_real @ Z2 @ B )
% 8.90/5.14 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ G ) ) )
% 8.90/5.14 => ( ! [Z2: real] :
% 8.90/5.14 ( ( ord_less_real @ A @ Z2 )
% 8.90/5.14 => ( ( ord_less_real @ Z2 @ B )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
% 8.90/5.14 => ( ! [Z2: real] :
% 8.90/5.14 ( ( ord_less_real @ A @ Z2 )
% 8.90/5.14 => ( ( ord_less_real @ Z2 @ B )
% 8.90/5.14 => ( has_fi5821293074295781190e_real @ F @ ( F4 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
% 8.90/5.14 => ? [C3: real] :
% 8.90/5.14 ( ( ord_less_real @ A @ C3 )
% 8.90/5.14 & ( ord_less_real @ C3 @ B )
% 8.90/5.14 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C3 ) )
% 8.90/5.14 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F4 @ C3 ) ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9860_upto__aux__rec,axiom,
% 8.90/5.14 ( upto_aux
% 8.90/5.14 = ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9861_LIM__fun__gt__zero,axiom,
% 8.90/5.14 ! [F: real > real,L: real,C: real] :
% 8.90/5.14 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 8.90/5.14 => ( ( ord_less_real @ zero_zero_real @ L )
% 8.90/5.14 => ? [R3: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ R3 )
% 8.90/5.14 & ! [X3: real] :
% 8.90/5.14 ( ( ( X3 != C )
% 8.90/5.14 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X3 ) ) @ R3 ) )
% 8.90/5.14 => ( ord_less_real @ zero_zero_real @ ( F @ X3 ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9862_LIM__fun__not__zero,axiom,
% 8.90/5.14 ! [F: real > real,L: real,C: real] :
% 8.90/5.14 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 8.90/5.14 => ( ( L != zero_zero_real )
% 8.90/5.14 => ? [R3: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ R3 )
% 8.90/5.14 & ! [X3: real] :
% 8.90/5.14 ( ( ( X3 != C )
% 8.90/5.14 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X3 ) ) @ R3 ) )
% 8.90/5.14 => ( ( F @ X3 )
% 8.90/5.14 != zero_zero_real ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9863_LIM__fun__less__zero,axiom,
% 8.90/5.14 ! [F: real > real,L: real,C: real] :
% 8.90/5.14 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 8.90/5.14 => ( ( ord_less_real @ L @ zero_zero_real )
% 8.90/5.14 => ? [R3: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ R3 )
% 8.90/5.14 & ! [X3: real] :
% 8.90/5.14 ( ( ( X3 != C )
% 8.90/5.14 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X3 ) ) @ R3 ) )
% 8.90/5.14 => ( ord_less_real @ ( F @ X3 ) @ zero_zero_real ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9864_LIM__cos__div__sin,axiom,
% 8.90/5.14 ( filterlim_real_real
% 8.90/5.14 @ ^ [X: real] : ( divide_divide_real @ ( cos_real @ X ) @ ( sin_real @ X ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.90/5.14 @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9865_summable__Leibniz_I3_J,axiom,
% 8.90/5.14 ! [A: nat > real] :
% 8.90/5.14 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.90/5.14 => ( ( topolo6980174941875973593q_real @ A )
% 8.90/5.14 => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 8.90/5.14 => ! [N9: nat] :
% 8.90/5.14 ( member_real
% 8.90/5.14 @ ( suminf_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 8.90/5.14 @ ( set_or1222579329274155063t_real
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) )
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9866_summable__Leibniz_I2_J,axiom,
% 8.90/5.14 ! [A: nat > real] :
% 8.90/5.14 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.90/5.14 => ( ( topolo6980174941875973593q_real @ A )
% 8.90/5.14 => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 8.90/5.14 => ! [N9: nat] :
% 8.90/5.14 ( member_real
% 8.90/5.14 @ ( suminf_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 8.90/5.14 @ ( set_or1222579329274155063t_real
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9867_filterlim__Suc,axiom,
% 8.90/5.14 filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9868_mult__nat__left__at__top,axiom,
% 8.90/5.14 ! [C: nat] :
% 8.90/5.14 ( ( ord_less_nat @ zero_zero_nat @ C )
% 8.90/5.14 => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9869_mult__nat__right__at__top,axiom,
% 8.90/5.14 ! [C: nat] :
% 8.90/5.14 ( ( ord_less_nat @ zero_zero_nat @ C )
% 8.90/5.14 => ( filterlim_nat_nat
% 8.90/5.14 @ ^ [X: nat] : ( times_times_nat @ X @ C )
% 8.90/5.14 @ at_top_nat
% 8.90/5.14 @ at_top_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9870_monoseq__convergent,axiom,
% 8.90/5.14 ! [X8: nat > real,B3: real] :
% 8.90/5.14 ( ( topolo6980174941875973593q_real @ X8 )
% 8.90/5.14 => ( ! [I3: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X8 @ I3 ) ) @ B3 )
% 8.90/5.14 => ~ ! [L6: real] :
% 8.90/5.14 ~ ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9871_LIMSEQ__root,axiom,
% 8.90/5.14 ( filterlim_nat_real
% 8.90/5.14 @ ^ [N3: nat] : ( root @ N3 @ ( semiri5074537144036343181t_real @ N3 ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ one_one_real )
% 8.90/5.14 @ at_top_nat ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9872_nested__sequence__unique,axiom,
% 8.90/5.14 ! [F: nat > real,G: nat > real] :
% 8.90/5.14 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.90/5.14 => ( ! [N2: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N2 ) ) @ ( G @ N2 ) )
% 8.90/5.14 => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 8.90/5.14 => ( ( filterlim_nat_real
% 8.90/5.14 @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.90/5.14 @ at_top_nat )
% 8.90/5.14 => ? [L4: real] :
% 8.90/5.14 ( ! [N9: nat] : ( ord_less_eq_real @ ( F @ N9 ) @ L4 )
% 8.90/5.14 & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
% 8.90/5.14 & ! [N9: nat] : ( ord_less_eq_real @ L4 @ ( G @ N9 ) )
% 8.90/5.14 & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9873_LIMSEQ__inverse__zero,axiom,
% 8.90/5.14 ! [X8: nat > real] :
% 8.90/5.14 ( ! [R3: real] :
% 8.90/5.14 ? [N7: nat] :
% 8.90/5.14 ! [N2: nat] :
% 8.90/5.14 ( ( ord_less_eq_nat @ N7 @ N2 )
% 8.90/5.14 => ( ord_less_real @ R3 @ ( X8 @ N2 ) ) )
% 8.90/5.14 => ( filterlim_nat_real
% 8.90/5.14 @ ^ [N3: nat] : ( inverse_inverse_real @ ( X8 @ N3 ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.90/5.14 @ at_top_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9874_lim__inverse__n_H,axiom,
% 8.90/5.14 ( filterlim_nat_real
% 8.90/5.14 @ ^ [N3: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N3 ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.90/5.14 @ at_top_nat ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9875_LIMSEQ__root__const,axiom,
% 8.90/5.14 ! [C: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ C )
% 8.90/5.14 => ( filterlim_nat_real
% 8.90/5.14 @ ^ [N3: nat] : ( root @ N3 @ C )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ one_one_real )
% 8.90/5.14 @ at_top_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9876_LIMSEQ__inverse__real__of__nat,axiom,
% 8.90/5.14 ( filterlim_nat_real
% 8.90/5.14 @ ^ [N3: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.90/5.14 @ at_top_nat ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9877_LIMSEQ__inverse__real__of__nat__add,axiom,
% 8.90/5.14 ! [R2: real] :
% 8.90/5.14 ( filterlim_nat_real
% 8.90/5.14 @ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ R2 )
% 8.90/5.14 @ at_top_nat ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9878_increasing__LIMSEQ,axiom,
% 8.90/5.14 ! [F: nat > real,L: real] :
% 8.90/5.14 ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 8.90/5.14 => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ L )
% 8.90/5.14 => ( ! [E2: real] :
% 8.90/5.14 ( ( ord_less_real @ zero_zero_real @ E2 )
% 8.90/5.14 => ? [N9: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N9 ) @ E2 ) ) )
% 8.90/5.14 => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9879_LIMSEQ__realpow__zero,axiom,
% 8.90/5.14 ! [X2: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.14 => ( ( ord_less_real @ X2 @ one_one_real )
% 8.90/5.14 => ( filterlim_nat_real @ ( power_power_real @ X2 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9880_LIMSEQ__divide__realpow__zero,axiom,
% 8.90/5.14 ! [X2: real,A: real] :
% 8.90/5.14 ( ( ord_less_real @ one_one_real @ X2 )
% 8.90/5.14 => ( filterlim_nat_real
% 8.90/5.14 @ ^ [N3: nat] : ( divide_divide_real @ A @ ( power_power_real @ X2 @ N3 ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.90/5.14 @ at_top_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9881_LIMSEQ__abs__realpow__zero,axiom,
% 8.90/5.14 ! [C: real] :
% 8.90/5.14 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 8.90/5.14 => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9882_LIMSEQ__abs__realpow__zero2,axiom,
% 8.90/5.14 ! [C: real] :
% 8.90/5.14 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 8.90/5.14 => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9883_LIMSEQ__inverse__realpow__zero,axiom,
% 8.90/5.14 ! [X2: real] :
% 8.90/5.14 ( ( ord_less_real @ one_one_real @ X2 )
% 8.90/5.14 => ( filterlim_nat_real
% 8.90/5.14 @ ^ [N3: nat] : ( inverse_inverse_real @ ( power_power_real @ X2 @ N3 ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.90/5.14 @ at_top_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9884_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
% 8.90/5.14 ! [R2: real] :
% 8.90/5.14 ( filterlim_nat_real
% 8.90/5.14 @ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ R2 )
% 8.90/5.14 @ at_top_nat ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9885_tendsto__exp__limit__sequentially,axiom,
% 8.90/5.14 ! [X2: real] :
% 8.90/5.14 ( filterlim_nat_real
% 8.90/5.14 @ ^ [N3: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N3 ) ) ) @ N3 )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ ( exp_real @ X2 ) )
% 8.90/5.14 @ at_top_nat ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9886_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
% 8.90/5.14 ! [R2: real] :
% 8.90/5.14 ( filterlim_nat_real
% 8.90/5.14 @ ^ [N3: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ R2 )
% 8.90/5.14 @ at_top_nat ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9887_summable__Leibniz_I1_J,axiom,
% 8.90/5.14 ! [A: nat > real] :
% 8.90/5.14 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.90/5.14 => ( ( topolo6980174941875973593q_real @ A )
% 8.90/5.14 => ( summable_real
% 8.90/5.14 @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9888_summable,axiom,
% 8.90/5.14 ! [A: nat > real] :
% 8.90/5.14 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.90/5.14 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 8.90/5.14 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 8.90/5.14 => ( summable_real
% 8.90/5.14 @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9889_cos__diff__limit__1,axiom,
% 8.90/5.14 ! [Theta: nat > real,Theta2: real] :
% 8.90/5.14 ( ( filterlim_nat_real
% 8.90/5.14 @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ one_one_real )
% 8.90/5.14 @ at_top_nat )
% 8.90/5.14 => ~ ! [K3: nat > int] :
% 8.90/5.14 ~ ( filterlim_nat_real
% 8.90/5.14 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ Theta2 )
% 8.90/5.14 @ at_top_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9890_cos__limit__1,axiom,
% 8.90/5.14 ! [Theta: nat > real] :
% 8.90/5.14 ( ( filterlim_nat_real
% 8.90/5.14 @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ one_one_real )
% 8.90/5.14 @ at_top_nat )
% 8.90/5.14 => ? [K3: nat > int] :
% 8.90/5.14 ( filterlim_nat_real
% 8.90/5.14 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.90/5.14 @ at_top_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9891_summable__Leibniz_I4_J,axiom,
% 8.90/5.14 ! [A: nat > real] :
% 8.90/5.14 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.90/5.14 => ( ( topolo6980174941875973593q_real @ A )
% 8.90/5.14 => ( filterlim_nat_real
% 8.90/5.14 @ ^ [N3: nat] :
% 8.90/5.14 ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real
% 8.90/5.14 @ ( suminf_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 8.90/5.14 @ at_top_nat ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9892_zeroseq__arctan__series,axiom,
% 8.90/5.14 ! [X2: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 8.90/5.14 => ( filterlim_nat_real
% 8.90/5.14 @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.90/5.14 @ at_top_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9893_summable__Leibniz_H_I3_J,axiom,
% 8.90/5.14 ! [A: nat > real] :
% 8.90/5.14 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.90/5.14 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 8.90/5.14 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 8.90/5.14 => ( filterlim_nat_real
% 8.90/5.14 @ ^ [N3: nat] :
% 8.90/5.14 ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real
% 8.90/5.14 @ ( suminf_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 8.90/5.14 @ at_top_nat ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9894_summable__Leibniz_H_I2_J,axiom,
% 8.90/5.14 ! [A: nat > real,N: nat] :
% 8.90/5.14 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.90/5.14 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 8.90/5.14 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 8.90/5.14 => ( ord_less_eq_real
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 8.90/5.14 @ ( suminf_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9895_sums__alternating__upper__lower,axiom,
% 8.90/5.14 ! [A: nat > real] :
% 8.90/5.14 ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 8.90/5.14 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 8.90/5.14 => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.90/5.14 => ? [L4: real] :
% 8.90/5.14 ( ! [N9: nat] :
% 8.90/5.14 ( ord_less_eq_real
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
% 8.90/5.14 @ L4 )
% 8.90/5.14 & ( filterlim_nat_real
% 8.90/5.14 @ ^ [N3: nat] :
% 8.90/5.14 ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ L4 )
% 8.90/5.14 @ at_top_nat )
% 8.90/5.14 & ! [N9: nat] :
% 8.90/5.14 ( ord_less_eq_real @ L4
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) )
% 8.90/5.14 & ( filterlim_nat_real
% 8.90/5.14 @ ^ [N3: nat] :
% 8.90/5.14 ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ L4 )
% 8.90/5.14 @ at_top_nat ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9896_summable__Leibniz_I5_J,axiom,
% 8.90/5.14 ! [A: nat > real] :
% 8.90/5.14 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.90/5.14 => ( ( topolo6980174941875973593q_real @ A )
% 8.90/5.14 => ( filterlim_nat_real
% 8.90/5.14 @ ^ [N3: nat] :
% 8.90/5.14 ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real
% 8.90/5.14 @ ( suminf_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 8.90/5.14 @ at_top_nat ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9897_summable__Leibniz_H_I5_J,axiom,
% 8.90/5.14 ! [A: nat > real] :
% 8.90/5.14 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.90/5.14 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 8.90/5.14 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 8.90/5.14 => ( filterlim_nat_real
% 8.90/5.14 @ ^ [N3: nat] :
% 8.90/5.14 ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real
% 8.90/5.14 @ ( suminf_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 8.90/5.14 @ at_top_nat ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9898_summable__Leibniz_H_I4_J,axiom,
% 8.90/5.14 ! [A: nat > real,N: nat] :
% 8.90/5.14 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 8.90/5.14 => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
% 8.90/5.14 => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
% 8.90/5.14 => ( ord_less_eq_real
% 8.90/5.14 @ ( suminf_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 8.90/5.14 @ ( groups6591440286371151544t_real
% 8.90/5.14 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 8.90/5.14 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9899_real__bounded__linear,axiom,
% 8.90/5.14 ( real_V5970128139526366754l_real
% 8.90/5.14 = ( ^ [F3: real > real] :
% 8.90/5.14 ? [C2: real] :
% 8.90/5.14 ( F3
% 8.90/5.14 = ( ^ [X: real] : ( times_times_real @ X @ C2 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9900_tendsto__exp__limit__at__right,axiom,
% 8.90/5.14 ! [X2: real] :
% 8.90/5.14 ( filterlim_real_real
% 8.90/5.14 @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X2 @ Y ) ) @ ( divide_divide_real @ one_one_real @ Y ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ ( exp_real @ X2 ) )
% 8.90/5.14 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9901_dist__real__def,axiom,
% 8.90/5.14 ( real_V975177566351809787t_real
% 8.90/5.14 = ( ^ [X: real,Y: real] : ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9902_dist__complex__def,axiom,
% 8.90/5.14 ( real_V3694042436643373181omplex
% 8.90/5.14 = ( ^ [X: complex,Y: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9903_tendsto__arcosh__at__left__1,axiom,
% 8.90/5.14 filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9904_greaterThan__0,axiom,
% 8.90/5.14 ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
% 8.90/5.14 = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9905_greaterThan__Suc,axiom,
% 8.90/5.14 ! [K: nat] :
% 8.90/5.14 ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 8.90/5.14 = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9906_filterlim__tan__at__right,axiom,
% 8.90/5.14 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 ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9907_tendsto__arctan__at__bot,axiom,
% 8.90/5.14 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9908_tanh__real__at__bot,axiom,
% 8.90/5.14 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ one_one_real ) ) @ at_bot_real ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9909_artanh__real__at__right__1,axiom,
% 8.90/5.14 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 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9910_DERIV__pos__imp__increasing__at__bot,axiom,
% 8.90/5.14 ! [B: real,F: real > real,Flim: real] :
% 8.90/5.14 ( ! [X5: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ X5 @ B )
% 8.90/5.14 => ? [Y5: real] :
% 8.90/5.14 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 8.90/5.14 & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
% 8.90/5.14 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 8.90/5.14 => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9911_filterlim__pow__at__bot__odd,axiom,
% 8.90/5.14 ! [N: nat,F: real > real,F5: filter_real] :
% 8.90/5.14 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
% 8.90/5.14 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.14 => ( filterlim_real_real
% 8.90/5.14 @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
% 8.90/5.14 @ at_bot_real
% 8.90/5.14 @ F5 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9912_filterlim__pow__at__bot__even,axiom,
% 8.90/5.14 ! [N: nat,F: real > real,F5: filter_real] :
% 8.90/5.14 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.14 => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
% 8.90/5.14 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 8.90/5.14 => ( filterlim_real_real
% 8.90/5.14 @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
% 8.90/5.14 @ at_top_real
% 8.90/5.14 @ F5 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9913_at__bot__le__at__infinity,axiom,
% 8.90/5.14 ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9914_at__top__le__at__infinity,axiom,
% 8.90/5.14 ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9915_sqrt__at__top,axiom,
% 8.90/5.14 filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9916_tanh__real__at__top,axiom,
% 8.90/5.14 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9917_artanh__real__at__left__1,axiom,
% 8.90/5.14 filterlim_real_real @ artanh_real @ at_top_real @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5984915006950818249n_real @ one_one_real ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9918_tendsto__power__div__exp__0,axiom,
% 8.90/5.14 ! [K: nat] :
% 8.90/5.14 ( filterlim_real_real
% 8.90/5.14 @ ^ [X: real] : ( divide_divide_real @ ( power_power_real @ X @ K ) @ ( exp_real @ X ) )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 8.90/5.14 @ at_top_real ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9919_tendsto__exp__limit__at__top,axiom,
% 8.90/5.14 ! [X2: real] :
% 8.90/5.14 ( filterlim_real_real
% 8.90/5.14 @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X2 @ Y ) ) @ Y )
% 8.90/5.14 @ ( topolo2815343760600316023s_real @ ( exp_real @ X2 ) )
% 8.90/5.14 @ at_top_real ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9920_filterlim__tan__at__left,axiom,
% 8.90/5.14 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 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9921_DERIV__neg__imp__decreasing__at__top,axiom,
% 8.90/5.14 ! [B: real,F: real > real,Flim: real] :
% 8.90/5.14 ( ! [X5: real] :
% 8.90/5.14 ( ( ord_less_eq_real @ B @ X5 )
% 8.90/5.14 => ? [Y5: real] :
% 8.90/5.14 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 8.90/5.14 & ( ord_less_real @ Y5 @ zero_zero_real ) ) )
% 8.90/5.14 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 8.90/5.14 => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9922_tendsto__arctan__at__top,axiom,
% 8.90/5.14 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9923_upto_Opelims,axiom,
% 8.90/5.14 ! [X2: int,Xa2: int,Y3: list_int] :
% 8.90/5.14 ( ( ( upto @ X2 @ Xa2 )
% 8.90/5.14 = Y3 )
% 8.90/5.14 => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) )
% 8.90/5.14 => ~ ( ( ( ( ord_less_eq_int @ X2 @ Xa2 )
% 8.90/5.14 => ( Y3
% 8.90/5.14 = ( cons_int @ X2 @ ( upto @ ( plus_plus_int @ X2 @ one_one_int ) @ Xa2 ) ) ) )
% 8.90/5.14 & ( ~ ( ord_less_eq_int @ X2 @ Xa2 )
% 8.90/5.14 => ( Y3 = nil_int ) ) )
% 8.90/5.14 => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9924_eventually__sequentially__Suc,axiom,
% 8.90/5.14 ! [P: nat > $o] :
% 8.90/5.14 ( ( eventually_nat
% 8.90/5.14 @ ^ [I4: nat] : ( P @ ( suc @ I4 ) )
% 8.90/5.14 @ at_top_nat )
% 8.90/5.14 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9925_eventually__sequentially__seg,axiom,
% 8.90/5.14 ! [P: nat > $o,K: nat] :
% 8.90/5.14 ( ( eventually_nat
% 8.90/5.14 @ ^ [N3: nat] : ( P @ ( plus_plus_nat @ N3 @ K ) )
% 8.90/5.14 @ at_top_nat )
% 8.90/5.14 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9926_nth__upto,axiom,
% 8.90/5.14 ! [I2: int,K: nat,J: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 8.90/5.14 => ( ( nth_int @ ( upto @ I2 @ J ) @ K )
% 8.90/5.14 = ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9927_length__upto,axiom,
% 8.90/5.14 ! [I2: int,J: int] :
% 8.90/5.14 ( ( size_size_list_int @ ( upto @ I2 @ J ) )
% 8.90/5.14 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I2 ) @ one_one_int ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9928_upto__rec__numeral_I1_J,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 8.90/5.14 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 8.90/5.14 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 8.90/5.14 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 8.90/5.14 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 8.90/5.14 = nil_int ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9929_upto__rec__numeral_I4_J,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.14 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.14 = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 8.90/5.14 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.14 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.14 = nil_int ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9930_upto__rec__numeral_I3_J,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.90/5.14 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.90/5.14 = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 8.90/5.14 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.90/5.14 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.90/5.14 = nil_int ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9931_upto__rec__numeral_I2_J,axiom,
% 8.90/5.14 ! [M: num,N: num] :
% 8.90/5.14 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.14 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.14 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 8.90/5.14 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.14 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.14 = nil_int ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9932_sequentially__offset,axiom,
% 8.90/5.14 ! [P: nat > $o,K: nat] :
% 8.90/5.14 ( ( eventually_nat @ P @ at_top_nat )
% 8.90/5.14 => ( eventually_nat
% 8.90/5.14 @ ^ [I4: nat] : ( P @ ( plus_plus_nat @ I4 @ K ) )
% 8.90/5.14 @ at_top_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9933_eventually__sequentially,axiom,
% 8.90/5.14 ! [P: nat > $o] :
% 8.90/5.14 ( ( eventually_nat @ P @ at_top_nat )
% 8.90/5.14 = ( ? [N6: nat] :
% 8.90/5.14 ! [N3: nat] :
% 8.90/5.14 ( ( ord_less_eq_nat @ N6 @ N3 )
% 8.90/5.14 => ( P @ N3 ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9934_eventually__sequentiallyI,axiom,
% 8.90/5.14 ! [C: nat,P: nat > $o] :
% 8.90/5.14 ( ! [X5: nat] :
% 8.90/5.14 ( ( ord_less_eq_nat @ C @ X5 )
% 8.90/5.14 => ( P @ X5 ) )
% 8.90/5.14 => ( eventually_nat @ P @ at_top_nat ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9935_le__sequentially,axiom,
% 8.90/5.14 ! [F5: filter_nat] :
% 8.90/5.14 ( ( ord_le2510731241096832064er_nat @ F5 @ at_top_nat )
% 8.90/5.14 = ( ! [N6: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N6 ) @ F5 ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9936_upto__split2,axiom,
% 8.90/5.14 ! [I2: int,J: int,K: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ I2 @ J )
% 8.90/5.14 => ( ( ord_less_eq_int @ J @ K )
% 8.90/5.14 => ( ( upto @ I2 @ K )
% 8.90/5.14 = ( append_int @ ( upto @ I2 @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9937_upto__split1,axiom,
% 8.90/5.14 ! [I2: int,J: int,K: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ I2 @ J )
% 8.90/5.14 => ( ( ord_less_eq_int @ J @ K )
% 8.90/5.14 => ( ( upto @ I2 @ K )
% 8.90/5.14 = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9938_atLeastLessThan__upto,axiom,
% 8.90/5.14 ( set_or4662586982721622107an_int
% 8.90/5.14 = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9939_upto_Osimps,axiom,
% 8.90/5.14 ( upto
% 8.90/5.14 = ( ^ [I4: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J3 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9940_upto_Oelims,axiom,
% 8.90/5.14 ! [X2: int,Xa2: int,Y3: list_int] :
% 8.90/5.14 ( ( ( upto @ X2 @ Xa2 )
% 8.90/5.14 = Y3 )
% 8.90/5.14 => ( ( ( ord_less_eq_int @ X2 @ Xa2 )
% 8.90/5.14 => ( Y3
% 8.90/5.14 = ( cons_int @ X2 @ ( upto @ ( plus_plus_int @ X2 @ one_one_int ) @ Xa2 ) ) ) )
% 8.90/5.14 & ( ~ ( ord_less_eq_int @ X2 @ Xa2 )
% 8.90/5.14 => ( Y3 = nil_int ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9941_upto__rec1,axiom,
% 8.90/5.14 ! [I2: int,J: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ I2 @ J )
% 8.90/5.14 => ( ( upto @ I2 @ J )
% 8.90/5.14 = ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9942_upto__rec2,axiom,
% 8.90/5.14 ! [I2: int,J: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ I2 @ J )
% 8.90/5.14 => ( ( upto @ I2 @ J )
% 8.90/5.14 = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9943_greaterThanLessThan__upto,axiom,
% 8.90/5.14 ( set_or5832277885323065728an_int
% 8.90/5.14 = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9944_upto__split3,axiom,
% 8.90/5.14 ! [I2: int,J: int,K: int] :
% 8.90/5.14 ( ( ord_less_eq_int @ I2 @ J )
% 8.90/5.14 => ( ( ord_less_eq_int @ J @ K )
% 8.90/5.14 => ( ( upto @ I2 @ K )
% 8.90/5.14 = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 8.90/5.14
% 8.90/5.14 thf(fact_9945_upto_Opsimps,axiom,
% 8.90/5.14 ! [I2: int,J: int] :
% 8.90/5.14 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J ) )
% 8.90/5.15 => ( ( ( ord_less_eq_int @ I2 @ J )
% 8.90/5.15 => ( ( upto @ I2 @ J )
% 8.90/5.15 = ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) )
% 8.90/5.15 & ( ~ ( ord_less_eq_int @ I2 @ J )
% 8.90/5.15 => ( ( upto @ I2 @ J )
% 8.90/5.15 = nil_int ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9946_Bseq__eq__bounded,axiom,
% 8.90/5.15 ! [F: nat > real,A: real,B: real] :
% 8.90/5.15 ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A @ B ) )
% 8.90/5.15 => ( bfun_nat_real @ F @ at_top_nat ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9947_Bseq__realpow,axiom,
% 8.90/5.15 ! [X2: real] :
% 8.90/5.15 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.15 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 8.90/5.15 => ( bfun_nat_real @ ( power_power_real @ X2 ) @ at_top_nat ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9948_card__greaterThanAtMost,axiom,
% 8.90/5.15 ! [L: nat,U: nat] :
% 8.90/5.15 ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
% 8.90/5.15 = ( minus_minus_nat @ U @ L ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9949_GreatestI__ex__nat,axiom,
% 8.90/5.15 ! [P: nat > $o,B: nat] :
% 8.90/5.15 ( ? [X_1: nat] : ( P @ X_1 )
% 8.90/5.15 => ( ! [Y4: nat] :
% 8.90/5.15 ( ( P @ Y4 )
% 8.90/5.15 => ( ord_less_eq_nat @ Y4 @ B ) )
% 8.90/5.15 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9950_Greatest__le__nat,axiom,
% 8.90/5.15 ! [P: nat > $o,K: nat,B: nat] :
% 8.90/5.15 ( ( P @ K )
% 8.90/5.15 => ( ! [Y4: nat] :
% 8.90/5.15 ( ( P @ Y4 )
% 8.90/5.15 => ( ord_less_eq_nat @ Y4 @ B ) )
% 8.90/5.15 => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9951_GreatestI__nat,axiom,
% 8.90/5.15 ! [P: nat > $o,K: nat,B: nat] :
% 8.90/5.15 ( ( P @ K )
% 8.90/5.15 => ( ! [Y4: nat] :
% 8.90/5.15 ( ( P @ Y4 )
% 8.90/5.15 => ( ord_less_eq_nat @ Y4 @ B ) )
% 8.90/5.15 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9952_atLeastSucAtMost__greaterThanAtMost,axiom,
% 8.90/5.15 ! [L: nat,U: nat] :
% 8.90/5.15 ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
% 8.90/5.15 = ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9953_card__greaterThanAtMost__int,axiom,
% 8.90/5.15 ! [L: int,U: int] :
% 8.90/5.15 ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L @ U ) )
% 8.90/5.15 = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9954_atLeast__Suc__greaterThan,axiom,
% 8.90/5.15 ! [K: nat] :
% 8.90/5.15 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 8.90/5.15 = ( set_or1210151606488870762an_nat @ K ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9955_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
% 8.90/5.15 ! [L: int,U: int] :
% 8.90/5.15 ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
% 8.90/5.15 = ( set_or6656581121297822940st_int @ L @ U ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9956_decseq__bounded,axiom,
% 8.90/5.15 ! [X8: nat > real,B3: real] :
% 8.90/5.15 ( ( order_9091379641038594480t_real @ X8 )
% 8.90/5.15 => ( ! [I3: nat] : ( ord_less_eq_real @ B3 @ ( X8 @ I3 ) )
% 8.90/5.15 => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9957_greaterThanAtMost__upto,axiom,
% 8.90/5.15 ( set_or6656581121297822940st_int
% 8.90/5.15 = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9958_decseq__convergent,axiom,
% 8.90/5.15 ! [X8: nat > real,B3: real] :
% 8.90/5.15 ( ( order_9091379641038594480t_real @ X8 )
% 8.90/5.15 => ( ! [I3: nat] : ( ord_less_eq_real @ B3 @ ( X8 @ I3 ) )
% 8.90/5.15 => ~ ! [L6: real] :
% 8.90/5.15 ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 8.90/5.15 => ~ ! [I: nat] : ( ord_less_eq_real @ L6 @ ( X8 @ I ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9959_atLeast__Suc,axiom,
% 8.90/5.15 ! [K: nat] :
% 8.90/5.15 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 8.90/5.15 = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9960_GMVT,axiom,
% 8.90/5.15 ! [A: real,B: real,F: real > real,G: real > real] :
% 8.90/5.15 ( ( ord_less_real @ A @ B )
% 8.90/5.15 => ( ! [X5: real] :
% 8.90/5.15 ( ( ( ord_less_eq_real @ A @ X5 )
% 8.90/5.15 & ( ord_less_eq_real @ X5 @ B ) )
% 8.90/5.15 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ F ) )
% 8.90/5.15 => ( ! [X5: real] :
% 8.90/5.15 ( ( ( ord_less_real @ A @ X5 )
% 8.90/5.15 & ( ord_less_real @ X5 @ B ) )
% 8.90/5.15 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 8.90/5.15 => ( ! [X5: real] :
% 8.90/5.15 ( ( ( ord_less_eq_real @ A @ X5 )
% 8.90/5.15 & ( ord_less_eq_real @ X5 @ B ) )
% 8.90/5.15 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ G ) )
% 8.90/5.15 => ( ! [X5: real] :
% 8.90/5.15 ( ( ( ord_less_real @ A @ X5 )
% 8.90/5.15 & ( ord_less_real @ X5 @ B ) )
% 8.90/5.15 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 8.90/5.15 => ? [G_c: real,F_c: real,C3: real] :
% 8.90/5.15 ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 8.90/5.15 & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 8.90/5.15 & ( ord_less_real @ A @ C3 )
% 8.90/5.15 & ( ord_less_real @ C3 @ B )
% 8.90/5.15 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
% 8.90/5.15 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9961_MVT,axiom,
% 8.90/5.15 ! [A: real,B: real,F: real > real] :
% 8.90/5.15 ( ( ord_less_real @ A @ B )
% 8.90/5.15 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 8.90/5.15 => ( ! [X5: real] :
% 8.90/5.15 ( ( ord_less_real @ A @ X5 )
% 8.90/5.15 => ( ( ord_less_real @ X5 @ B )
% 8.90/5.15 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 8.90/5.15 => ? [L4: real,Z2: real] :
% 8.90/5.15 ( ( ord_less_real @ A @ Z2 )
% 8.90/5.15 & ( ord_less_real @ Z2 @ B )
% 8.90/5.15 & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) )
% 8.90/5.15 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 8.90/5.15 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9962_continuous__on__arcosh_H,axiom,
% 8.90/5.15 ! [A2: set_real,F: real > real] :
% 8.90/5.15 ( ( topolo5044208981011980120l_real @ A2 @ F )
% 8.90/5.15 => ( ! [X5: real] :
% 8.90/5.15 ( ( member_real @ X5 @ A2 )
% 8.90/5.15 => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
% 8.90/5.15 => ( topolo5044208981011980120l_real @ A2
% 8.90/5.15 @ ^ [X: real] : ( arcosh_real @ ( F @ X ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9963_continuous__image__closed__interval,axiom,
% 8.90/5.15 ! [A: real,B: real,F: real > real] :
% 8.90/5.15 ( ( ord_less_eq_real @ A @ B )
% 8.90/5.15 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 8.90/5.15 => ? [C3: real,D3: real] :
% 8.90/5.15 ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
% 8.90/5.15 = ( set_or1222579329274155063t_real @ C3 @ D3 ) )
% 8.90/5.15 & ( ord_less_eq_real @ C3 @ D3 ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9964_continuous__on__arcosh,axiom,
% 8.90/5.15 ! [A2: set_real] :
% 8.90/5.15 ( ( ord_less_eq_set_real @ A2 @ ( set_ord_atLeast_real @ one_one_real ) )
% 8.90/5.15 => ( topolo5044208981011980120l_real @ A2 @ arcosh_real ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9965_continuous__on__arccos_H,axiom,
% 8.90/5.15 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arccos ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9966_continuous__on__arcsin_H,axiom,
% 8.90/5.15 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arcsin ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9967_continuous__on__artanh_H,axiom,
% 8.90/5.15 ! [A2: set_real,F: real > real] :
% 8.90/5.15 ( ( topolo5044208981011980120l_real @ A2 @ F )
% 8.90/5.15 => ( ! [X5: real] :
% 8.90/5.15 ( ( member_real @ X5 @ A2 )
% 8.90/5.15 => ( member_real @ ( F @ X5 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
% 8.90/5.15 => ( topolo5044208981011980120l_real @ A2
% 8.90/5.15 @ ^ [X: real] : ( artanh_real @ ( F @ X ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9968_mvt,axiom,
% 8.90/5.15 ! [A: real,B: real,F: real > real,F4: real > real > real] :
% 8.90/5.15 ( ( ord_less_real @ A @ B )
% 8.90/5.15 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 8.90/5.15 => ( ! [X5: real] :
% 8.90/5.15 ( ( ord_less_real @ A @ X5 )
% 8.90/5.15 => ( ( ord_less_real @ X5 @ B )
% 8.90/5.15 => ( has_de1759254742604945161l_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 8.90/5.15 => ~ ! [Xi: real] :
% 8.90/5.15 ( ( ord_less_real @ A @ Xi )
% 8.90/5.15 => ( ( ord_less_real @ Xi @ B )
% 8.90/5.15 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 8.90/5.15 != ( F4 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9969_continuous__on__artanh,axiom,
% 8.90/5.15 ! [A2: set_real] :
% 8.90/5.15 ( ( ord_less_eq_set_real @ A2 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
% 8.90/5.15 => ( topolo5044208981011980120l_real @ A2 @ artanh_real ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9970_DERIV__isconst2,axiom,
% 8.90/5.15 ! [A: real,B: real,F: real > real,X2: real] :
% 8.90/5.15 ( ( ord_less_real @ A @ B )
% 8.90/5.15 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 8.90/5.15 => ( ! [X5: real] :
% 8.90/5.15 ( ( ord_less_real @ A @ X5 )
% 8.90/5.15 => ( ( ord_less_real @ X5 @ B )
% 8.90/5.15 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 8.90/5.15 => ( ( ord_less_eq_real @ A @ X2 )
% 8.90/5.15 => ( ( ord_less_eq_real @ X2 @ B )
% 8.90/5.15 => ( ( F @ X2 )
% 8.90/5.15 = ( F @ A ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9971_mono__times__nat,axiom,
% 8.90/5.15 ! [N: nat] :
% 8.90/5.15 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.15 => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9972_mono__Suc,axiom,
% 8.90/5.15 order_mono_nat_nat @ suc ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9973_incseq__bounded,axiom,
% 8.90/5.15 ! [X8: nat > real,B3: real] :
% 8.90/5.15 ( ( order_mono_nat_real @ X8 )
% 8.90/5.15 => ( ! [I3: nat] : ( ord_less_eq_real @ ( X8 @ I3 ) @ B3 )
% 8.90/5.15 => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9974_incseq__convergent,axiom,
% 8.90/5.15 ! [X8: nat > real,B3: real] :
% 8.90/5.15 ( ( order_mono_nat_real @ X8 )
% 8.90/5.15 => ( ! [I3: nat] : ( ord_less_eq_real @ ( X8 @ I3 ) @ B3 )
% 8.90/5.15 => ~ ! [L6: real] :
% 8.90/5.15 ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 8.90/5.15 => ~ ! [I: nat] : ( ord_less_eq_real @ ( X8 @ I ) @ L6 ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9975_mono__ge2__power__minus__self,axiom,
% 8.90/5.15 ! [K: nat] :
% 8.90/5.15 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 8.90/5.15 => ( order_mono_nat_nat
% 8.90/5.15 @ ^ [M3: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M3 ) @ M3 ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9976_nonneg__incseq__Bseq__subseq__iff,axiom,
% 8.90/5.15 ! [F: nat > real,G: nat > nat] :
% 8.90/5.15 ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 8.90/5.15 => ( ( order_mono_nat_real @ F )
% 8.90/5.15 => ( ( order_5726023648592871131at_nat @ G )
% 8.90/5.15 => ( ( bfun_nat_real
% 8.90/5.15 @ ^ [X: nat] : ( F @ ( G @ X ) )
% 8.90/5.15 @ at_top_nat )
% 8.90/5.15 = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9977_inj__sgn__power,axiom,
% 8.90/5.15 ! [N: nat] :
% 8.90/5.15 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.15 => ( inj_on_real_real
% 8.90/5.15 @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 8.90/5.15 @ top_top_set_real ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9978_strict__mono__imp__increasing,axiom,
% 8.90/5.15 ! [F: nat > nat,N: nat] :
% 8.90/5.15 ( ( order_5726023648592871131at_nat @ F )
% 8.90/5.15 => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9979_log__inj,axiom,
% 8.90/5.15 ! [B: real] :
% 8.90/5.15 ( ( ord_less_real @ one_one_real @ B )
% 8.90/5.15 => ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9980_inj__on__diff__nat,axiom,
% 8.90/5.15 ! [N5: set_nat,K: nat] :
% 8.90/5.15 ( ! [N2: nat] :
% 8.90/5.15 ( ( member_nat @ N2 @ N5 )
% 8.90/5.15 => ( ord_less_eq_nat @ K @ N2 ) )
% 8.90/5.15 => ( inj_on_nat_nat
% 8.90/5.15 @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K )
% 8.90/5.15 @ N5 ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9981_inj__Suc,axiom,
% 8.90/5.15 ! [N5: set_nat] : ( inj_on_nat_nat @ suc @ N5 ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9982_summable__reindex,axiom,
% 8.90/5.15 ! [F: nat > real,G: nat > nat] :
% 8.90/5.15 ( ( summable_real @ F )
% 8.90/5.15 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 8.90/5.15 => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 8.90/5.15 => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9983_suminf__reindex__mono,axiom,
% 8.90/5.15 ! [F: nat > real,G: nat > nat] :
% 8.90/5.15 ( ( summable_real @ F )
% 8.90/5.15 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 8.90/5.15 => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 8.90/5.15 => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9984_inj__on__char__of__nat,axiom,
% 8.90/5.15 inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9985_suminf__reindex,axiom,
% 8.90/5.15 ! [F: nat > real,G: nat > nat] :
% 8.90/5.15 ( ( summable_real @ F )
% 8.90/5.15 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 8.90/5.15 => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 8.90/5.15 => ( ! [X5: nat] :
% 8.90/5.15 ( ~ ( member_nat @ X5 @ ( image_nat_nat @ G @ top_top_set_nat ) )
% 8.90/5.15 => ( ( F @ X5 )
% 8.90/5.15 = zero_zero_real ) )
% 8.90/5.15 => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
% 8.90/5.15 = ( suminf_real @ F ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9986_times__int_Oabs__eq,axiom,
% 8.90/5.15 ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
% 8.90/5.15 ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
% 8.90/5.15 = ( abs_Integ
% 8.90/5.15 @ ( produc27273713700761075at_nat
% 8.90/5.15 @ ^ [X: nat,Y: nat] :
% 8.90/5.15 ( produc2626176000494625587at_nat
% 8.90/5.15 @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) )
% 8.90/5.15 @ Xa2
% 8.90/5.15 @ X2 ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9987_Gcd__remove0__nat,axiom,
% 8.90/5.15 ! [M7: set_nat] :
% 8.90/5.15 ( ( finite_finite_nat @ M7 )
% 8.90/5.15 => ( ( gcd_Gcd_nat @ M7 )
% 8.90/5.15 = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M7 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9988_nat_Oabs__eq,axiom,
% 8.90/5.15 ! [X2: product_prod_nat_nat] :
% 8.90/5.15 ( ( nat2 @ ( abs_Integ @ X2 ) )
% 8.90/5.15 = ( produc6842872674320459806at_nat @ minus_minus_nat @ X2 ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9989_eq__Abs__Integ,axiom,
% 8.90/5.15 ! [Z: int] :
% 8.90/5.15 ~ ! [X5: nat,Y4: nat] :
% 8.90/5.15 ( Z
% 8.90/5.15 != ( abs_Integ @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9990_Gcd__nat__eq__one,axiom,
% 8.90/5.15 ! [N5: set_nat] :
% 8.90/5.15 ( ( member_nat @ one_one_nat @ N5 )
% 8.90/5.15 => ( ( gcd_Gcd_nat @ N5 )
% 8.90/5.15 = one_one_nat ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9991_zero__int__def,axiom,
% 8.90/5.15 ( zero_zero_int
% 8.90/5.15 = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9992_int__def,axiom,
% 8.90/5.15 ( semiri1314217659103216013at_int
% 8.90/5.15 = ( ^ [N3: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N3 @ zero_zero_nat ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9993_uminus__int_Oabs__eq,axiom,
% 8.90/5.15 ! [X2: product_prod_nat_nat] :
% 8.90/5.15 ( ( uminus_uminus_int @ ( abs_Integ @ X2 ) )
% 8.90/5.15 = ( abs_Integ
% 8.90/5.15 @ ( produc2626176000494625587at_nat
% 8.90/5.15 @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X )
% 8.90/5.15 @ X2 ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9994_one__int__def,axiom,
% 8.90/5.15 ( one_one_int
% 8.90/5.15 = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9995_less__int_Oabs__eq,axiom,
% 8.90/5.15 ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
% 8.90/5.15 ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
% 8.90/5.15 = ( produc8739625826339149834_nat_o
% 8.90/5.15 @ ^ [X: nat,Y: nat] :
% 8.90/5.15 ( produc6081775807080527818_nat_o
% 8.90/5.15 @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
% 8.90/5.15 @ Xa2
% 8.90/5.15 @ X2 ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9996_less__eq__int_Oabs__eq,axiom,
% 8.90/5.15 ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
% 8.90/5.15 ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
% 8.90/5.15 = ( produc8739625826339149834_nat_o
% 8.90/5.15 @ ^ [X: nat,Y: nat] :
% 8.90/5.15 ( produc6081775807080527818_nat_o
% 8.90/5.15 @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
% 8.90/5.15 @ Xa2
% 8.90/5.15 @ X2 ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9997_plus__int_Oabs__eq,axiom,
% 8.90/5.15 ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
% 8.90/5.15 ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
% 8.90/5.15 = ( abs_Integ
% 8.90/5.15 @ ( produc27273713700761075at_nat
% 8.90/5.15 @ ^ [X: nat,Y: nat] :
% 8.90/5.15 ( produc2626176000494625587at_nat
% 8.90/5.15 @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) )
% 8.90/5.15 @ Xa2
% 8.90/5.15 @ X2 ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9998_minus__int_Oabs__eq,axiom,
% 8.90/5.15 ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
% 8.90/5.15 ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
% 8.90/5.15 = ( abs_Integ
% 8.90/5.15 @ ( produc27273713700761075at_nat
% 8.90/5.15 @ ^ [X: nat,Y: nat] :
% 8.90/5.15 ( produc2626176000494625587at_nat
% 8.90/5.15 @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) )
% 8.90/5.15 @ Xa2
% 8.90/5.15 @ X2 ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_9999_Gcd__int__greater__eq__0,axiom,
% 8.90/5.15 ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10000_powr__real__of__int_H,axiom,
% 8.90/5.15 ! [X2: real,N: int] :
% 8.90/5.15 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 8.90/5.15 => ( ( ( X2 != zero_zero_real )
% 8.90/5.15 | ( ord_less_int @ zero_zero_int @ N ) )
% 8.90/5.15 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N ) )
% 8.90/5.15 = ( power_int_real @ X2 @ N ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10001_less__eq__int_Orep__eq,axiom,
% 8.90/5.15 ( ord_less_eq_int
% 8.90/5.15 = ( ^ [X: int,Xa4: int] :
% 8.90/5.15 ( produc8739625826339149834_nat_o
% 8.90/5.15 @ ^ [Y: nat,Z3: nat] :
% 8.90/5.15 ( produc6081775807080527818_nat_o
% 8.90/5.15 @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
% 8.90/5.15 @ ( rep_Integ @ X )
% 8.90/5.15 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10002_less__int_Orep__eq,axiom,
% 8.90/5.15 ( ord_less_int
% 8.90/5.15 = ( ^ [X: int,Xa4: int] :
% 8.90/5.15 ( produc8739625826339149834_nat_o
% 8.90/5.15 @ ^ [Y: nat,Z3: nat] :
% 8.90/5.15 ( produc6081775807080527818_nat_o
% 8.90/5.15 @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
% 8.90/5.15 @ ( rep_Integ @ X )
% 8.90/5.15 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10003_nat_Orep__eq,axiom,
% 8.90/5.15 ( nat2
% 8.90/5.15 = ( ^ [X: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10004_prod__encode__def,axiom,
% 8.90/5.15 ( nat_prod_encode
% 8.90/5.15 = ( produc6842872674320459806at_nat
% 8.90/5.15 @ ^ [M3: nat,N3: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M3 @ N3 ) ) @ M3 ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10005_le__prod__encode__1,axiom,
% 8.90/5.15 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10006_le__prod__encode__2,axiom,
% 8.90/5.15 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10007_prod__encode__prod__decode__aux,axiom,
% 8.90/5.15 ! [K: nat,M: nat] :
% 8.90/5.15 ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
% 8.90/5.15 = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10008_uminus__int__def,axiom,
% 8.90/5.15 ( uminus_uminus_int
% 8.90/5.15 = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
% 8.90/5.15 @ ( produc2626176000494625587at_nat
% 8.90/5.15 @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10009_times__int__def,axiom,
% 8.90/5.15 ( times_times_int
% 8.90/5.15 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 8.90/5.15 @ ( produc27273713700761075at_nat
% 8.90/5.15 @ ^ [X: nat,Y: nat] :
% 8.90/5.15 ( produc2626176000494625587at_nat
% 8.90/5.15 @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10010_minus__int__def,axiom,
% 8.90/5.15 ( minus_minus_int
% 8.90/5.15 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 8.90/5.15 @ ( produc27273713700761075at_nat
% 8.90/5.15 @ ^ [X: nat,Y: nat] :
% 8.90/5.15 ( produc2626176000494625587at_nat
% 8.90/5.15 @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10011_plus__int__def,axiom,
% 8.90/5.15 ( plus_plus_int
% 8.90/5.15 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 8.90/5.15 @ ( produc27273713700761075at_nat
% 8.90/5.15 @ ^ [X: nat,Y: nat] :
% 8.90/5.15 ( produc2626176000494625587at_nat
% 8.90/5.15 @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10012_list__encode_Oelims,axiom,
% 8.90/5.15 ! [X2: list_nat,Y3: nat] :
% 8.90/5.15 ( ( ( nat_list_encode @ X2 )
% 8.90/5.15 = Y3 )
% 8.90/5.15 => ( ( ( X2 = nil_nat )
% 8.90/5.15 => ( Y3 != zero_zero_nat ) )
% 8.90/5.15 => ~ ! [X5: nat,Xs3: list_nat] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( cons_nat @ X5 @ Xs3 ) )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10013_list__encode_Osimps_I2_J,axiom,
% 8.90/5.15 ! [X2: nat,Xs2: list_nat] :
% 8.90/5.15 ( ( nat_list_encode @ ( cons_nat @ X2 @ Xs2 ) )
% 8.90/5.15 = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X2 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10014_list__encode_Opelims,axiom,
% 8.90/5.15 ! [X2: list_nat,Y3: nat] :
% 8.90/5.15 ( ( ( nat_list_encode @ X2 )
% 8.90/5.15 = Y3 )
% 8.90/5.15 => ( ( accp_list_nat @ nat_list_encode_rel @ X2 )
% 8.90/5.15 => ( ( ( X2 = nil_nat )
% 8.90/5.15 => ( ( Y3 = zero_zero_nat )
% 8.90/5.15 => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
% 8.90/5.15 => ~ ! [X5: nat,Xs3: list_nat] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( cons_nat @ X5 @ Xs3 ) )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) )
% 8.90/5.15 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X5 @ Xs3 ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10015_pred__nat__def,axiom,
% 8.90/5.15 ( pred_nat
% 8.90/5.15 = ( collec3392354462482085612at_nat
% 8.90/5.15 @ ( produc6081775807080527818_nat_o
% 8.90/5.15 @ ^ [M3: nat,N3: nat] :
% 8.90/5.15 ( N3
% 8.90/5.15 = ( suc @ M3 ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10016_num__of__nat_Osimps_I2_J,axiom,
% 8.90/5.15 ! [N: nat] :
% 8.90/5.15 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.15 => ( ( num_of_nat @ ( suc @ N ) )
% 8.90/5.15 = ( inc @ ( num_of_nat @ N ) ) ) )
% 8.90/5.15 & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.15 => ( ( num_of_nat @ ( suc @ N ) )
% 8.90/5.15 = one ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10017_num__of__nat__numeral__eq,axiom,
% 8.90/5.15 ! [Q3: num] :
% 8.90/5.15 ( ( num_of_nat @ ( numeral_numeral_nat @ Q3 ) )
% 8.90/5.15 = Q3 ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10018_num__of__nat_Osimps_I1_J,axiom,
% 8.90/5.15 ( ( num_of_nat @ zero_zero_nat )
% 8.90/5.15 = one ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10019_numeral__num__of__nat,axiom,
% 8.90/5.15 ! [N: nat] :
% 8.90/5.15 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.15 => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
% 8.90/5.15 = N ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10020_num__of__nat__One,axiom,
% 8.90/5.15 ! [N: nat] :
% 8.90/5.15 ( ( ord_less_eq_nat @ N @ one_one_nat )
% 8.90/5.15 => ( ( num_of_nat @ N )
% 8.90/5.15 = one ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10021_num__of__nat__double,axiom,
% 8.90/5.15 ! [N: nat] :
% 8.90/5.15 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.15 => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
% 8.90/5.15 = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10022_num__of__nat__plus__distrib,axiom,
% 8.90/5.15 ! [M: nat,N: nat] :
% 8.90/5.15 ( ( ord_less_nat @ zero_zero_nat @ M )
% 8.90/5.15 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.90/5.15 => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
% 8.90/5.15 = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10023_pow_Osimps_I3_J,axiom,
% 8.90/5.15 ! [X2: num,Y3: num] :
% 8.90/5.15 ( ( pow @ X2 @ ( bit1 @ Y3 ) )
% 8.90/5.15 = ( times_times_num @ ( sqr @ ( pow @ X2 @ Y3 ) ) @ X2 ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10024_sorted__list__of__set__lessThan__Suc,axiom,
% 8.90/5.15 ! [K: nat] :
% 8.90/5.15 ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
% 8.90/5.15 = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10025_sorted__list__of__set__atMost__Suc,axiom,
% 8.90/5.15 ! [K: nat] :
% 8.90/5.15 ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
% 8.90/5.15 = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10026_sqr__conv__mult,axiom,
% 8.90/5.15 ( sqr
% 8.90/5.15 = ( ^ [X: num] : ( times_times_num @ X @ X ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10027_sqr_Osimps_I1_J,axiom,
% 8.90/5.15 ( ( sqr @ one )
% 8.90/5.15 = one ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10028_sqr_Osimps_I2_J,axiom,
% 8.90/5.15 ! [N: num] :
% 8.90/5.15 ( ( sqr @ ( bit0 @ N ) )
% 8.90/5.15 = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10029_pow_Osimps_I2_J,axiom,
% 8.90/5.15 ! [X2: num,Y3: num] :
% 8.90/5.15 ( ( pow @ X2 @ ( bit0 @ Y3 ) )
% 8.90/5.15 = ( sqr @ ( pow @ X2 @ Y3 ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10030_sorted__list__of__set__greaterThanAtMost,axiom,
% 8.90/5.15 ! [I2: nat,J: nat] :
% 8.90/5.15 ( ( ord_less_eq_nat @ ( suc @ I2 ) @ J )
% 8.90/5.15 => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J ) )
% 8.90/5.15 = ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10031_sorted__list__of__set__greaterThanLessThan,axiom,
% 8.90/5.15 ! [I2: nat,J: nat] :
% 8.90/5.15 ( ( ord_less_nat @ ( suc @ I2 ) @ J )
% 8.90/5.15 => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J ) )
% 8.90/5.15 = ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10032_sqr_Osimps_I3_J,axiom,
% 8.90/5.15 ! [N: num] :
% 8.90/5.15 ( ( sqr @ ( bit1 @ N ) )
% 8.90/5.15 = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10033_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 8.90/5.15 ! [N: nat,J: nat,I2: nat] :
% 8.90/5.15 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I2 ) )
% 8.90/5.15 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J ) ) @ N )
% 8.90/5.15 = ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10034_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 8.90/5.15 ! [N: nat,J: nat,I2: nat] :
% 8.90/5.15 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I2 ) ) )
% 8.90/5.15 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J ) ) @ N )
% 8.90/5.15 = ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10035_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 8.90/5.15 ! [X2: vEBT_VEBT,Xa2: nat] :
% 8.90/5.15 ( ~ ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 8.90/5.15 => ( ( ? [Uu2: $o,Uv2: $o] :
% 8.90/5.15 ( X2
% 8.90/5.15 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.90/5.15 => ( Xa2 = one_one_nat ) )
% 8.90/5.15 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 8.90/5.15 => ( ( Deg2 = Xa2 )
% 8.90/5.15 & ! [X5: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.90/5.15 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.15 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.15 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 8.90/5.15 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.90/5.15 & ( case_o184042715313410164at_nat
% 8.90/5.15 @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X4 )
% 8.90/5.15 & ! [X: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.90/5.15 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 8.90/5.15 @ ( produc6081775807080527818_nat_o
% 8.90/5.15 @ ^ [Mi3: nat,Ma3: nat] :
% 8.90/5.15 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 8.90/5.15 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 8.90/5.15 & ! [I4: nat] :
% 8.90/5.15 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.90/5.15 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X4 ) )
% 8.90/5.15 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 8.90/5.15 & ( ( Mi3 = Ma3 )
% 8.90/5.15 => ! [X: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.90/5.15 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 8.90/5.15 & ( ( Mi3 != Ma3 )
% 8.90/5.15 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 8.90/5.15 & ! [X: nat] :
% 8.90/5.15 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 8.90/5.15 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 8.90/5.15 => ( ( ord_less_nat @ Mi3 @ X )
% 8.90/5.15 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 8.90/5.15 @ Mima ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10036_atLeastLessThan__add__Un,axiom,
% 8.90/5.15 ! [I2: nat,J: nat,K: nat] :
% 8.90/5.15 ( ( ord_less_eq_nat @ I2 @ J )
% 8.90/5.15 => ( ( set_or4665077453230672383an_nat @ I2 @ ( plus_plus_nat @ J @ K ) )
% 8.90/5.15 = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10037_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 8.90/5.15 ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
% 8.90/5.15 ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg4 )
% 8.90/5.15 = ( ( Deg = Deg4 )
% 8.90/5.15 & ! [X: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.90/5.15 => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.15 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.15 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 8.90/5.15 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.90/5.15 & ( case_o184042715313410164at_nat
% 8.90/5.15 @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X4 )
% 8.90/5.15 & ! [X: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.90/5.15 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 8.90/5.15 @ ( produc6081775807080527818_nat_o
% 8.90/5.15 @ ^ [Mi3: nat,Ma3: nat] :
% 8.90/5.15 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 8.90/5.15 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 8.90/5.15 & ! [I4: nat] :
% 8.90/5.15 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.90/5.15 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X4 ) )
% 8.90/5.15 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 8.90/5.15 & ( ( Mi3 = Ma3 )
% 8.90/5.15 => ! [X: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 8.90/5.15 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 8.90/5.15 & ( ( Mi3 != Ma3 )
% 8.90/5.15 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
% 8.90/5.15 & ! [X: nat] :
% 8.90/5.15 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 8.90/5.15 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
% 8.90/5.15 => ( ( ord_less_nat @ Mi3 @ X )
% 8.90/5.15 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 8.90/5.15 @ Mima2 ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10038_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 8.90/5.15 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 8.90/5.15 ( ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 8.90/5.15 = Y3 )
% 8.90/5.15 => ( ( ? [Uu2: $o,Uv2: $o] :
% 8.90/5.15 ( X2
% 8.90/5.15 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.90/5.15 => ( Y3
% 8.90/5.15 = ( Xa2 != one_one_nat ) ) )
% 8.90/5.15 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 8.90/5.15 => ( Y3
% 8.90/5.15 = ( ~ ( ( Deg2 = Xa2 )
% 8.90/5.15 & ! [X: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.90/5.15 => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.15 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.15 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 8.90/5.15 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.90/5.15 & ( case_o184042715313410164at_nat
% 8.90/5.15 @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X4 )
% 8.90/5.15 & ! [X: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.90/5.15 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 8.90/5.15 @ ( produc6081775807080527818_nat_o
% 8.90/5.15 @ ^ [Mi3: nat,Ma3: nat] :
% 8.90/5.15 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 8.90/5.15 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 8.90/5.15 & ! [I4: nat] :
% 8.90/5.15 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.90/5.15 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X4 ) )
% 8.90/5.15 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 8.90/5.15 & ( ( Mi3 = Ma3 )
% 8.90/5.15 => ! [X: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.90/5.15 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 8.90/5.15 & ( ( Mi3 != Ma3 )
% 8.90/5.15 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 8.90/5.15 & ! [X: nat] :
% 8.90/5.15 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 8.90/5.15 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 8.90/5.15 => ( ( ord_less_nat @ Mi3 @ X )
% 8.90/5.15 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 8.90/5.15 @ Mima ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10039_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 8.90/5.15 ! [X2: vEBT_VEBT,Xa2: nat] :
% 8.90/5.15 ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 8.90/5.15 => ( ( ? [Uu2: $o,Uv2: $o] :
% 8.90/5.15 ( X2
% 8.90/5.15 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.90/5.15 => ( Xa2 != one_one_nat ) )
% 8.90/5.15 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 8.90/5.15 => ~ ( ( Deg2 = Xa2 )
% 8.90/5.15 & ! [X3: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.90/5.15 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.15 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.15 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 8.90/5.15 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.90/5.15 & ( case_o184042715313410164at_nat
% 8.90/5.15 @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X4 )
% 8.90/5.15 & ! [X: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.90/5.15 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 8.90/5.15 @ ( produc6081775807080527818_nat_o
% 8.90/5.15 @ ^ [Mi3: nat,Ma3: nat] :
% 8.90/5.15 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 8.90/5.15 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 8.90/5.15 & ! [I4: nat] :
% 8.90/5.15 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.90/5.15 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X4 ) )
% 8.90/5.15 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 8.90/5.15 & ( ( Mi3 = Ma3 )
% 8.90/5.15 => ! [X: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.90/5.15 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 8.90/5.15 & ( ( Mi3 != Ma3 )
% 8.90/5.15 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 8.90/5.15 & ! [X: nat] :
% 8.90/5.15 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 8.90/5.15 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 8.90/5.15 => ( ( ord_less_nat @ Mi3 @ X )
% 8.90/5.15 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 8.90/5.15 @ Mima ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10040_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 8.90/5.15 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 8.90/5.15 ( ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 8.90/5.15 = Y3 )
% 8.90/5.15 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 8.90/5.15 => ( ! [Uu2: $o,Uv2: $o] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( Xa2 = one_one_nat ) )
% 8.90/5.15 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 8.90/5.15 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( ( Deg2 = Xa2 )
% 8.90/5.15 & ! [X: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.90/5.15 => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.15 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.15 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 8.90/5.15 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.90/5.15 & ( case_o184042715313410164at_nat
% 8.90/5.15 @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X4 )
% 8.90/5.15 & ! [X: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.90/5.15 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 8.90/5.15 @ ( produc6081775807080527818_nat_o
% 8.90/5.15 @ ^ [Mi3: nat,Ma3: nat] :
% 8.90/5.15 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 8.90/5.15 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 8.90/5.15 & ! [I4: nat] :
% 8.90/5.15 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.90/5.15 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X4 ) )
% 8.90/5.15 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 8.90/5.15 & ( ( Mi3 = Ma3 )
% 8.90/5.15 => ! [X: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.90/5.15 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 8.90/5.15 & ( ( Mi3 != Ma3 )
% 8.90/5.15 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 8.90/5.15 & ! [X: nat] :
% 8.90/5.15 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 8.90/5.15 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 8.90/5.15 => ( ( ord_less_nat @ Mi3 @ X )
% 8.90/5.15 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 8.90/5.15 @ Mima ) ) )
% 8.90/5.15 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10041_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 8.90/5.15 ! [X2: vEBT_VEBT,Xa2: nat] :
% 8.90/5.15 ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 8.90/5.15 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 8.90/5.15 => ( ! [Uu2: $o,Uv2: $o] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.90/5.15 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 8.90/5.15 => ( Xa2 != one_one_nat ) ) )
% 8.90/5.15 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 8.90/5.15 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) )
% 8.90/5.15 => ~ ( ( Deg2 = Xa2 )
% 8.90/5.15 & ! [X3: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.90/5.15 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.15 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.15 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 8.90/5.15 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.90/5.15 & ( case_o184042715313410164at_nat
% 8.90/5.15 @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X4 )
% 8.90/5.15 & ! [X: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.90/5.15 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 8.90/5.15 @ ( produc6081775807080527818_nat_o
% 8.90/5.15 @ ^ [Mi3: nat,Ma3: nat] :
% 8.90/5.15 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 8.90/5.15 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 8.90/5.15 & ! [I4: nat] :
% 8.90/5.15 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.90/5.15 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X4 ) )
% 8.90/5.15 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 8.90/5.15 & ( ( Mi3 = Ma3 )
% 8.90/5.15 => ! [X: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.90/5.15 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 8.90/5.15 & ( ( Mi3 != Ma3 )
% 8.90/5.15 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 8.90/5.15 & ! [X: nat] :
% 8.90/5.15 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 8.90/5.15 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 8.90/5.15 => ( ( ord_less_nat @ Mi3 @ X )
% 8.90/5.15 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 8.90/5.15 @ Mima ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10042_sup__nat__def,axiom,
% 8.90/5.15 sup_sup_nat = ord_max_nat ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10043_sup__enat__def,axiom,
% 8.90/5.15 sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10044_Sup__int__def,axiom,
% 8.90/5.15 ( complete_Sup_Sup_int
% 8.90/5.15 = ( ^ [X4: set_int] :
% 8.90/5.15 ( the_int
% 8.90/5.15 @ ^ [X: int] :
% 8.90/5.15 ( ( member_int @ X @ X4 )
% 8.90/5.15 & ! [Y: int] :
% 8.90/5.15 ( ( member_int @ Y @ X4 )
% 8.90/5.15 => ( ord_less_eq_int @ Y @ X ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10045_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 8.90/5.15 ! [X2: vEBT_VEBT,Xa2: nat] :
% 8.90/5.15 ( ~ ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 8.90/5.15 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 8.90/5.15 => ( ! [Uu2: $o,Uv2: $o] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 8.90/5.15 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 8.90/5.15 => ( Xa2 = one_one_nat ) ) )
% 8.90/5.15 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
% 8.90/5.15 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) )
% 8.90/5.15 => ( ( Deg2 = Xa2 )
% 8.90/5.15 & ! [X5: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.90/5.15 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.15 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 8.90/5.15 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 8.90/5.15 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.90/5.15 & ( case_o184042715313410164at_nat
% 8.90/5.15 @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X4 )
% 8.90/5.15 & ! [X: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.90/5.15 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 8.90/5.15 @ ( produc6081775807080527818_nat_o
% 8.90/5.15 @ ^ [Mi3: nat,Ma3: nat] :
% 8.90/5.15 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 8.90/5.15 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 8.90/5.15 & ! [I4: nat] :
% 8.90/5.15 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 8.90/5.15 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X4 ) )
% 8.90/5.15 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 8.90/5.15 & ( ( Mi3 = Ma3 )
% 8.90/5.15 => ! [X: vEBT_VEBT] :
% 8.90/5.15 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 8.90/5.15 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X @ X4 ) ) )
% 8.90/5.15 & ( ( Mi3 != Ma3 )
% 8.90/5.15 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 8.90/5.15 & ! [X: nat] :
% 8.90/5.15 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 8.90/5.15 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 8.90/5.15 => ( ( ord_less_nat @ Mi3 @ X )
% 8.90/5.15 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 8.90/5.15 @ Mima ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10046_take__bit__numeral__minus__numeral__int,axiom,
% 8.90/5.15 ! [M: num,N: num] :
% 8.90/5.15 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.15 = ( case_option_int_num @ zero_zero_int
% 8.90/5.15 @ ^ [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 ) ) )
% 8.90/5.15 @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10047_take__bit__num__simps_I1_J,axiom,
% 8.90/5.15 ! [M: num] :
% 8.90/5.15 ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 8.90/5.15 = none_num ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10048_take__bit__num__simps_I2_J,axiom,
% 8.90/5.15 ! [N: nat] :
% 8.90/5.15 ( ( bit_take_bit_num @ ( suc @ N ) @ one )
% 8.90/5.15 = ( some_num @ one ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10049_take__bit__num__simps_I5_J,axiom,
% 8.90/5.15 ! [R2: num] :
% 8.90/5.15 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ one )
% 8.90/5.15 = ( some_num @ one ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10050_take__bit__num__simps_I3_J,axiom,
% 8.90/5.15 ! [N: nat,M: num] :
% 8.90/5.15 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
% 8.90/5.15 = ( case_o6005452278849405969um_num @ none_num
% 8.90/5.15 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 8.90/5.15 @ ( bit_take_bit_num @ N @ M ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10051_take__bit__num__simps_I4_J,axiom,
% 8.90/5.15 ! [N: nat,M: num] :
% 8.90/5.15 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
% 8.90/5.15 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10052_take__bit__num__simps_I6_J,axiom,
% 8.90/5.15 ! [R2: num,M: num] :
% 8.90/5.15 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
% 8.90/5.15 = ( case_o6005452278849405969um_num @ none_num
% 8.90/5.15 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 8.90/5.15 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10053_take__bit__num__simps_I7_J,axiom,
% 8.90/5.15 ! [R2: num,M: num] :
% 8.90/5.15 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M ) )
% 8.90/5.15 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10054_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 8.90/5.15 ! [N: nat,M: num] :
% 8.90/5.15 ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
% 8.90/5.15 = ( case_nat_option_num @ none_num
% 8.90/5.15 @ ^ [N3: nat] :
% 8.90/5.15 ( case_o6005452278849405969um_num @ none_num
% 8.90/5.15 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 8.90/5.15 @ ( bit_take_bit_num @ N3 @ M ) )
% 8.90/5.15 @ N ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10055_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 8.90/5.15 ! [N: nat] :
% 8.90/5.15 ( ( bit_take_bit_num @ N @ one )
% 8.90/5.15 = ( case_nat_option_num @ none_num
% 8.90/5.15 @ ^ [N3: nat] : ( some_num @ one )
% 8.90/5.15 @ N ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10056_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 8.90/5.15 ! [N: nat,M: num] :
% 8.90/5.15 ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
% 8.90/5.15 = ( case_nat_option_num @ none_num
% 8.90/5.15 @ ^ [N3: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N3 @ M ) ) )
% 8.90/5.15 @ N ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10057_take__bit__num__def,axiom,
% 8.90/5.15 ( bit_take_bit_num
% 8.90/5.15 = ( ^ [N3: nat,M3: num] :
% 8.90/5.15 ( if_option_num
% 8.90/5.15 @ ( ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M3 ) )
% 8.90/5.15 = zero_zero_nat )
% 8.90/5.15 @ none_num
% 8.90/5.15 @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M3 ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10058_and__minus__numerals_I3_J,axiom,
% 8.90/5.15 ! [M: num,N: num] :
% 8.90/5.15 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 8.90/5.15 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10059_and__minus__numerals_I8_J,axiom,
% 8.90/5.15 ! [N: num,M: num] :
% 8.90/5.15 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 8.90/5.15 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10060_and__minus__numerals_I4_J,axiom,
% 8.90/5.15 ! [M: num,N: num] :
% 8.90/5.15 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 8.90/5.15 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10061_and__minus__numerals_I7_J,axiom,
% 8.90/5.15 ! [N: num,M: num] :
% 8.90/5.15 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 8.90/5.15 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10062_and__not__num_Osimps_I1_J,axiom,
% 8.90/5.15 ( ( bit_and_not_num @ one @ one )
% 8.90/5.15 = none_num ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10063_and__not__num_Osimps_I3_J,axiom,
% 8.90/5.15 ! [N: num] :
% 8.90/5.15 ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
% 8.90/5.15 = none_num ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10064_and__not__num_Osimps_I2_J,axiom,
% 8.90/5.15 ! [N: num] :
% 8.90/5.15 ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
% 8.90/5.15 = ( some_num @ one ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10065_and__not__num_Osimps_I4_J,axiom,
% 8.90/5.15 ! [M: num] :
% 8.90/5.15 ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
% 8.90/5.15 = ( some_num @ ( bit0 @ M ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10066_and__not__num_Osimps_I7_J,axiom,
% 8.90/5.15 ! [M: num] :
% 8.90/5.15 ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
% 8.90/5.15 = ( some_num @ ( bit0 @ M ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10067_and__not__num__eq__Some__iff,axiom,
% 8.90/5.15 ! [M: num,N: num,Q3: num] :
% 8.90/5.15 ( ( ( bit_and_not_num @ M @ N )
% 8.90/5.15 = ( some_num @ Q3 ) )
% 8.90/5.15 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.15 = ( numeral_numeral_int @ Q3 ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10068_and__not__num_Osimps_I8_J,axiom,
% 8.90/5.15 ! [M: num,N: num] :
% 8.90/5.15 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.90/5.15 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 8.90/5.15 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 8.90/5.15 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10069_and__not__num__eq__None__iff,axiom,
% 8.90/5.15 ! [M: num,N: num] :
% 8.90/5.15 ( ( ( bit_and_not_num @ M @ N )
% 8.90/5.15 = none_num )
% 8.90/5.15 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.15 = zero_zero_int ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10070_int__numeral__and__not__num,axiom,
% 8.90/5.15 ! [M: num,N: num] :
% 8.90/5.15 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 8.90/5.15 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10071_int__numeral__not__and__num,axiom,
% 8.90/5.15 ! [M: num,N: num] :
% 8.90/5.15 ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 8.90/5.15 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10072_Bit__Operations_Otake__bit__num__code,axiom,
% 8.90/5.15 ( bit_take_bit_num
% 8.90/5.15 = ( ^ [N3: nat,M3: num] :
% 8.90/5.15 ( produc478579273971653890on_num
% 8.90/5.15 @ ^ [A3: nat,X: num] :
% 8.90/5.15 ( case_nat_option_num @ none_num
% 8.90/5.15 @ ^ [O: nat] :
% 8.90/5.15 ( case_num_option_num @ ( some_num @ one )
% 8.90/5.15 @ ^ [P4: num] :
% 8.90/5.15 ( case_o6005452278849405969um_num @ none_num
% 8.90/5.15 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 8.90/5.15 @ ( bit_take_bit_num @ O @ P4 ) )
% 8.90/5.15 @ ^ [P4: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P4 ) ) )
% 8.90/5.15 @ X )
% 8.90/5.15 @ A3 )
% 8.90/5.15 @ ( product_Pair_nat_num @ N3 @ M3 ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10073_and__not__num_Oelims,axiom,
% 8.90/5.15 ! [X2: num,Xa2: num,Y3: option_num] :
% 8.90/5.15 ( ( ( bit_and_not_num @ X2 @ Xa2 )
% 8.90/5.15 = Y3 )
% 8.90/5.15 => ( ( ( X2 = one )
% 8.90/5.15 => ( ( Xa2 = one )
% 8.90/5.15 => ( Y3 != none_num ) ) )
% 8.90/5.15 => ( ( ( X2 = one )
% 8.90/5.15 => ( ? [N2: num] :
% 8.90/5.15 ( Xa2
% 8.90/5.15 = ( bit0 @ N2 ) )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( some_num @ one ) ) ) )
% 8.90/5.15 => ( ( ( X2 = one )
% 8.90/5.15 => ( ? [N2: num] :
% 8.90/5.15 ( Xa2
% 8.90/5.15 = ( bit1 @ N2 ) )
% 8.90/5.15 => ( Y3 != none_num ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit0 @ M4 ) )
% 8.90/5.15 => ( ( Xa2 = one )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit0 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit0 @ N2 ) )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit0 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit1 @ N2 ) )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit1 @ M4 ) )
% 8.90/5.15 => ( ( Xa2 = one )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit1 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit0 @ N2 ) )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 8.90/5.15 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 8.90/5.15 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) )
% 8.90/5.15 => ~ ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit1 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit1 @ N2 ) )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10074_and__not__num_Osimps_I9_J,axiom,
% 8.90/5.15 ! [M: num,N: num] :
% 8.90/5.15 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.90/5.15 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10075_and__not__num_Osimps_I6_J,axiom,
% 8.90/5.15 ! [M: num,N: num] :
% 8.90/5.15 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.90/5.15 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10076_and__not__num_Osimps_I5_J,axiom,
% 8.90/5.15 ! [M: num,N: num] :
% 8.90/5.15 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.90/5.15 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10077_and__not__num_Opelims,axiom,
% 8.90/5.15 ! [X2: num,Xa2: num,Y3: option_num] :
% 8.90/5.15 ( ( ( bit_and_not_num @ X2 @ Xa2 )
% 8.90/5.15 = Y3 )
% 8.90/5.15 => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X2 @ Xa2 ) )
% 8.90/5.15 => ( ( ( X2 = one )
% 8.90/5.15 => ( ( Xa2 = one )
% 8.90/5.15 => ( ( Y3 = none_num )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 8.90/5.15 => ( ( ( X2 = one )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit0 @ N2 ) )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( some_num @ one ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 8.90/5.15 => ( ( ( X2 = one )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit1 @ N2 ) )
% 8.90/5.15 => ( ( Y3 = none_num )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit0 @ M4 ) )
% 8.90/5.15 => ( ( Xa2 = one )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( some_num @ ( bit0 @ M4 ) ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit0 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit0 @ N2 ) )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit0 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit1 @ N2 ) )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit1 @ M4 ) )
% 8.90/5.15 => ( ( Xa2 = one )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( some_num @ ( bit0 @ M4 ) ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit1 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit0 @ N2 ) )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 8.90/5.15 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 8.90/5.15 @ ( bit_and_not_num @ M4 @ N2 ) ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 8.90/5.15 => ~ ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit1 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit1 @ N2 ) )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10078_and__num_Oelims,axiom,
% 8.90/5.15 ! [X2: num,Xa2: num,Y3: option_num] :
% 8.90/5.15 ( ( ( bit_un7362597486090784418nd_num @ X2 @ Xa2 )
% 8.90/5.15 = Y3 )
% 8.90/5.15 => ( ( ( X2 = one )
% 8.90/5.15 => ( ( Xa2 = one )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( some_num @ one ) ) ) )
% 8.90/5.15 => ( ( ( X2 = one )
% 8.90/5.15 => ( ? [N2: num] :
% 8.90/5.15 ( Xa2
% 8.90/5.15 = ( bit0 @ N2 ) )
% 8.90/5.15 => ( Y3 != none_num ) ) )
% 8.90/5.15 => ( ( ( X2 = one )
% 8.90/5.15 => ( ? [N2: num] :
% 8.90/5.15 ( Xa2
% 8.90/5.15 = ( bit1 @ N2 ) )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( some_num @ one ) ) ) )
% 8.90/5.15 => ( ( ? [M4: num] :
% 8.90/5.15 ( X2
% 8.90/5.15 = ( bit0 @ M4 ) )
% 8.90/5.15 => ( ( Xa2 = one )
% 8.90/5.15 => ( Y3 != none_num ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit0 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit0 @ N2 ) )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit0 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit1 @ N2 ) )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) )
% 8.90/5.15 => ( ( ? [M4: num] :
% 8.90/5.15 ( X2
% 8.90/5.15 = ( bit1 @ M4 ) )
% 8.90/5.15 => ( ( Xa2 = one )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( some_num @ one ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit1 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit0 @ N2 ) )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) )
% 8.90/5.15 => ~ ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit1 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit1 @ N2 ) )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 8.90/5.15 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 8.90/5.15 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10079_xor__num_Oelims,axiom,
% 8.90/5.15 ! [X2: num,Xa2: num,Y3: option_num] :
% 8.90/5.15 ( ( ( bit_un2480387367778600638or_num @ X2 @ Xa2 )
% 8.90/5.15 = Y3 )
% 8.90/5.15 => ( ( ( X2 = one )
% 8.90/5.15 => ( ( Xa2 = one )
% 8.90/5.15 => ( Y3 != none_num ) ) )
% 8.90/5.15 => ( ( ( X2 = one )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit0 @ N2 ) )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( some_num @ ( bit1 @ N2 ) ) ) ) )
% 8.90/5.15 => ( ( ( X2 = one )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit1 @ N2 ) )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( some_num @ ( bit0 @ N2 ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit0 @ M4 ) )
% 8.90/5.15 => ( ( Xa2 = one )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( some_num @ ( bit1 @ M4 ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit0 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit0 @ N2 ) )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit0 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit1 @ N2 ) )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit1 @ M4 ) )
% 8.90/5.15 => ( ( Xa2 = one )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit1 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit0 @ N2 ) )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) ) )
% 8.90/5.15 => ~ ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit1 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit1 @ N2 ) )
% 8.90/5.15 => ( Y3
% 8.90/5.15 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10080_and__num_Osimps_I1_J,axiom,
% 8.90/5.15 ( ( bit_un7362597486090784418nd_num @ one @ one )
% 8.90/5.15 = ( some_num @ one ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10081_xor__num_Osimps_I1_J,axiom,
% 8.90/5.15 ( ( bit_un2480387367778600638or_num @ one @ one )
% 8.90/5.15 = none_num ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10082_and__num_Osimps_I5_J,axiom,
% 8.90/5.15 ! [M: num,N: num] :
% 8.90/5.15 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.90/5.15 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10083_xor__num_Osimps_I5_J,axiom,
% 8.90/5.15 ! [M: num,N: num] :
% 8.90/5.15 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 8.90/5.15 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10084_and__num_Osimps_I7_J,axiom,
% 8.90/5.15 ! [M: num] :
% 8.90/5.15 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
% 8.90/5.15 = ( some_num @ one ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10085_and__num_Osimps_I3_J,axiom,
% 8.90/5.15 ! [N: num] :
% 8.90/5.15 ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
% 8.90/5.15 = ( some_num @ one ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10086_and__num_Osimps_I2_J,axiom,
% 8.90/5.15 ! [N: num] :
% 8.90/5.15 ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
% 8.90/5.15 = none_num ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10087_and__num_Osimps_I4_J,axiom,
% 8.90/5.15 ! [M: num] :
% 8.90/5.15 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 8.90/5.15 = none_num ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10088_xor__num_Osimps_I9_J,axiom,
% 8.90/5.15 ! [M: num,N: num] :
% 8.90/5.15 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.90/5.15 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10089_and__num_Osimps_I6_J,axiom,
% 8.90/5.15 ! [M: num,N: num] :
% 8.90/5.15 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.90/5.15 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10090_and__num_Osimps_I8_J,axiom,
% 8.90/5.15 ! [M: num,N: num] :
% 8.90/5.15 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.90/5.15 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10091_xor__num_Osimps_I7_J,axiom,
% 8.90/5.15 ! [M: num] :
% 8.90/5.15 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
% 8.90/5.15 = ( some_num @ ( bit0 @ M ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10092_xor__num_Osimps_I4_J,axiom,
% 8.90/5.15 ! [M: num] :
% 8.90/5.15 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
% 8.90/5.15 = ( some_num @ ( bit1 @ M ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10093_xor__num_Osimps_I3_J,axiom,
% 8.90/5.15 ! [N: num] :
% 8.90/5.15 ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
% 8.90/5.15 = ( some_num @ ( bit0 @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10094_xor__num_Osimps_I2_J,axiom,
% 8.90/5.15 ! [N: num] :
% 8.90/5.15 ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
% 8.90/5.15 = ( some_num @ ( bit1 @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10095_and__num_Osimps_I9_J,axiom,
% 8.90/5.15 ! [M: num,N: num] :
% 8.90/5.15 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 8.90/5.15 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 8.90/5.15 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 8.90/5.15 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10096_xor__num_Osimps_I6_J,axiom,
% 8.90/5.15 ! [M: num,N: num] :
% 8.90/5.15 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 8.90/5.15 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10097_xor__num_Osimps_I8_J,axiom,
% 8.90/5.15 ! [M: num,N: num] :
% 8.90/5.15 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 8.90/5.15 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10098_and__num_Opelims,axiom,
% 8.90/5.15 ! [X2: num,Xa2: num,Y3: option_num] :
% 8.90/5.15 ( ( ( bit_un7362597486090784418nd_num @ X2 @ Xa2 )
% 8.90/5.15 = Y3 )
% 8.90/5.15 => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X2 @ Xa2 ) )
% 8.90/5.15 => ( ( ( X2 = one )
% 8.90/5.15 => ( ( Xa2 = one )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( some_num @ one ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 8.90/5.15 => ( ( ( X2 = one )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit0 @ N2 ) )
% 8.90/5.15 => ( ( Y3 = none_num )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 8.90/5.15 => ( ( ( X2 = one )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit1 @ N2 ) )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( some_num @ one ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit0 @ M4 ) )
% 8.90/5.15 => ( ( Xa2 = one )
% 8.90/5.15 => ( ( Y3 = none_num )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit0 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit0 @ N2 ) )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit0 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit1 @ N2 ) )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit1 @ M4 ) )
% 8.90/5.15 => ( ( Xa2 = one )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( some_num @ one ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit1 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit0 @ N2 ) )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 8.90/5.15 => ~ ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit1 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit1 @ N2 ) )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 8.90/5.15 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 8.90/5.15 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10099_xor__num_Opelims,axiom,
% 8.90/5.15 ! [X2: num,Xa2: num,Y3: option_num] :
% 8.90/5.15 ( ( ( bit_un2480387367778600638or_num @ X2 @ Xa2 )
% 8.90/5.15 = Y3 )
% 8.90/5.15 => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X2 @ Xa2 ) )
% 8.90/5.15 => ( ( ( X2 = one )
% 8.90/5.15 => ( ( Xa2 = one )
% 8.90/5.15 => ( ( Y3 = none_num )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 8.90/5.15 => ( ( ( X2 = one )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit0 @ N2 ) )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( some_num @ ( bit1 @ N2 ) ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
% 8.90/5.15 => ( ( ( X2 = one )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit1 @ N2 ) )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( some_num @ ( bit0 @ N2 ) ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit0 @ M4 ) )
% 8.90/5.15 => ( ( Xa2 = one )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( some_num @ ( bit1 @ M4 ) ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit0 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit0 @ N2 ) )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit0 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit1 @ N2 ) )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit1 @ M4 ) )
% 8.90/5.15 => ( ( Xa2 = one )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( some_num @ ( bit0 @ M4 ) ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 8.90/5.15 => ( ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit1 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit0 @ N2 ) )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
% 8.90/5.15 => ~ ! [M4: num] :
% 8.90/5.15 ( ( X2
% 8.90/5.15 = ( bit1 @ M4 ) )
% 8.90/5.15 => ! [N2: num] :
% 8.90/5.15 ( ( Xa2
% 8.90/5.15 = ( bit1 @ N2 ) )
% 8.90/5.15 => ( ( Y3
% 8.90/5.15 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N2 ) ) )
% 8.90/5.15 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10100_and__num__rel__dict,axiom,
% 8.90/5.15 bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10101_xor__num__rel__dict,axiom,
% 8.90/5.15 bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10102_xor__num__dict,axiom,
% 8.90/5.15 bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10103_and__num__dict,axiom,
% 8.90/5.15 bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10104_less__eq,axiom,
% 8.90/5.15 ! [M: nat,N: nat] :
% 8.90/5.15 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
% 8.90/5.15 = ( ord_less_nat @ M @ N ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10105_pred__nat__trancl__eq__le,axiom,
% 8.90/5.15 ! [M: nat,N: nat] :
% 8.90/5.15 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
% 8.90/5.15 = ( ord_less_eq_nat @ M @ N ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10106_Rats__eq__int__div__nat,axiom,
% 8.90/5.15 ( field_5140801741446780682s_real
% 8.90/5.15 = ( collect_real
% 8.90/5.15 @ ^ [Uu3: real] :
% 8.90/5.15 ? [I4: int,N3: nat] :
% 8.90/5.15 ( ( Uu3
% 8.90/5.15 = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
% 8.90/5.15 & ( N3 != zero_zero_nat ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10107_Rats__abs__iff,axiom,
% 8.90/5.15 ! [X2: real] :
% 8.90/5.15 ( ( member_real @ ( abs_abs_real @ X2 ) @ field_5140801741446780682s_real )
% 8.90/5.15 = ( member_real @ X2 @ field_5140801741446780682s_real ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10108_Rats__no__bot__less,axiom,
% 8.90/5.15 ! [X2: real] :
% 8.90/5.15 ? [X5: real] :
% 8.90/5.15 ( ( member_real @ X5 @ field_5140801741446780682s_real )
% 8.90/5.15 & ( ord_less_real @ X5 @ X2 ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10109_Rats__dense__in__real,axiom,
% 8.90/5.15 ! [X2: real,Y3: real] :
% 8.90/5.15 ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.15 => ? [X5: real] :
% 8.90/5.15 ( ( member_real @ X5 @ field_5140801741446780682s_real )
% 8.90/5.15 & ( ord_less_real @ X2 @ X5 )
% 8.90/5.15 & ( ord_less_real @ X5 @ Y3 ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10110_Rats__no__top__le,axiom,
% 8.90/5.15 ! [X2: real] :
% 8.90/5.15 ? [X5: real] :
% 8.90/5.15 ( ( member_real @ X5 @ field_5140801741446780682s_real )
% 8.90/5.15 & ( ord_less_eq_real @ X2 @ X5 ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10111_Rats__eq__int__div__int,axiom,
% 8.90/5.15 ( field_5140801741446780682s_real
% 8.90/5.15 = ( collect_real
% 8.90/5.15 @ ^ [Uu3: real] :
% 8.90/5.15 ? [I4: int,J3: int] :
% 8.90/5.15 ( ( Uu3
% 8.90/5.15 = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( ring_1_of_int_real @ J3 ) ) )
% 8.90/5.15 & ( J3 != zero_zero_int ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10112_rat__floor__lemma,axiom,
% 8.90/5.15 ! [A: int,B: int] :
% 8.90/5.15 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
% 8.90/5.15 & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10113_mult__rat,axiom,
% 8.90/5.15 ! [A: int,B: int,C: int,D: int] :
% 8.90/5.15 ( ( times_times_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 8.90/5.15 = ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10114_divide__rat,axiom,
% 8.90/5.15 ! [A: int,B: int,C: int,D: int] :
% 8.90/5.15 ( ( divide_divide_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 8.90/5.15 = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10115_less__rat,axiom,
% 8.90/5.15 ! [B: int,D: int,A: int,C: int] :
% 8.90/5.15 ( ( B != zero_zero_int )
% 8.90/5.15 => ( ( D != zero_zero_int )
% 8.90/5.15 => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 8.90/5.15 = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10116_add__rat,axiom,
% 8.90/5.15 ! [B: int,D: int,A: int,C: int] :
% 8.90/5.15 ( ( B != zero_zero_int )
% 8.90/5.15 => ( ( D != zero_zero_int )
% 8.90/5.15 => ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 8.90/5.15 = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10117_le__rat,axiom,
% 8.90/5.15 ! [B: int,D: int,A: int,C: int] :
% 8.90/5.15 ( ( B != zero_zero_int )
% 8.90/5.15 => ( ( D != zero_zero_int )
% 8.90/5.15 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 8.90/5.15 = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10118_diff__rat,axiom,
% 8.90/5.15 ! [B: int,D: int,A: int,C: int] :
% 8.90/5.15 ( ( B != zero_zero_int )
% 8.90/5.15 => ( ( D != zero_zero_int )
% 8.90/5.15 => ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 8.90/5.15 = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10119_sgn__rat,axiom,
% 8.90/5.15 ! [A: int,B: int] :
% 8.90/5.15 ( ( sgn_sgn_rat @ ( fract @ A @ B ) )
% 8.90/5.15 = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10120_Zero__rat__def,axiom,
% 8.90/5.15 ( zero_zero_rat
% 8.90/5.15 = ( fract @ zero_zero_int @ one_one_int ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10121_Fract__of__int__eq,axiom,
% 8.90/5.15 ! [K: int] :
% 8.90/5.15 ( ( fract @ K @ one_one_int )
% 8.90/5.15 = ( ring_1_of_int_rat @ K ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10122_Fract__of__nat__eq,axiom,
% 8.90/5.15 ! [K: nat] :
% 8.90/5.15 ( ( fract @ ( semiri1314217659103216013at_int @ K ) @ one_one_int )
% 8.90/5.15 = ( semiri681578069525770553at_rat @ K ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10123_mult__rat__cancel,axiom,
% 8.90/5.15 ! [C: int,A: int,B: int] :
% 8.90/5.15 ( ( C != zero_zero_int )
% 8.90/5.15 => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 8.90/5.15 = ( fract @ A @ B ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10124_eq__rat_I1_J,axiom,
% 8.90/5.15 ! [B: int,D: int,A: int,C: int] :
% 8.90/5.15 ( ( B != zero_zero_int )
% 8.90/5.15 => ( ( D != zero_zero_int )
% 8.90/5.15 => ( ( ( fract @ A @ B )
% 8.90/5.15 = ( fract @ C @ D ) )
% 8.90/5.15 = ( ( times_times_int @ A @ D )
% 8.90/5.15 = ( times_times_int @ C @ B ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10125_eq__rat_I2_J,axiom,
% 8.90/5.15 ! [A: int] :
% 8.90/5.15 ( ( fract @ A @ zero_zero_int )
% 8.90/5.15 = ( fract @ zero_zero_int @ one_one_int ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10126_One__rat__def,axiom,
% 8.90/5.15 ( one_one_rat
% 8.90/5.15 = ( fract @ one_one_int @ one_one_int ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10127_rat__number__expand_I3_J,axiom,
% 8.90/5.15 ( numeral_numeral_rat
% 8.90/5.15 = ( ^ [K2: num] : ( fract @ ( numeral_numeral_int @ K2 ) @ one_one_int ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10128_rat__number__collapse_I3_J,axiom,
% 8.90/5.15 ! [W: num] :
% 8.90/5.15 ( ( fract @ ( numeral_numeral_int @ W ) @ one_one_int )
% 8.90/5.15 = ( numeral_numeral_rat @ W ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10129_one__less__Fract__iff,axiom,
% 8.90/5.15 ! [B: int,A: int] :
% 8.90/5.15 ( ( ord_less_int @ zero_zero_int @ B )
% 8.90/5.15 => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
% 8.90/5.15 = ( ord_less_int @ B @ A ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10130_Fract__less__one__iff,axiom,
% 8.90/5.15 ! [B: int,A: int] :
% 8.90/5.15 ( ( ord_less_int @ zero_zero_int @ B )
% 8.90/5.15 => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
% 8.90/5.15 = ( ord_less_int @ A @ B ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10131_rat__number__collapse_I5_J,axiom,
% 8.90/5.15 ( ( fract @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 8.90/5.15 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10132_Fract__add__one,axiom,
% 8.90/5.15 ! [N: int,M: int] :
% 8.90/5.15 ( ( N != zero_zero_int )
% 8.90/5.15 => ( ( fract @ ( plus_plus_int @ M @ N ) @ N )
% 8.90/5.15 = ( plus_plus_rat @ ( fract @ M @ N ) @ one_one_rat ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10133_Fract__le__zero__iff,axiom,
% 8.90/5.15 ! [B: int,A: int] :
% 8.90/5.15 ( ( ord_less_int @ zero_zero_int @ B )
% 8.90/5.15 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 8.90/5.15 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10134_zero__le__Fract__iff,axiom,
% 8.90/5.15 ! [B: int,A: int] :
% 8.90/5.15 ( ( ord_less_int @ zero_zero_int @ B )
% 8.90/5.15 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 8.90/5.15 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10135_Fract__le__one__iff,axiom,
% 8.90/5.15 ! [B: int,A: int] :
% 8.90/5.15 ( ( ord_less_int @ zero_zero_int @ B )
% 8.90/5.15 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
% 8.90/5.15 = ( ord_less_eq_int @ A @ B ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10136_one__le__Fract__iff,axiom,
% 8.90/5.15 ! [B: int,A: int] :
% 8.90/5.15 ( ( ord_less_int @ zero_zero_int @ B )
% 8.90/5.15 => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
% 8.90/5.15 = ( ord_less_eq_int @ B @ A ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10137_rat__number__expand_I5_J,axiom,
% 8.90/5.15 ! [K: num] :
% 8.90/5.15 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) )
% 8.90/5.15 = ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10138_rat__number__collapse_I4_J,axiom,
% 8.90/5.15 ! [W: num] :
% 8.90/5.15 ( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ one_one_int )
% 8.90/5.15 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10139_of__nat__eq__id,axiom,
% 8.90/5.15 semiri1316708129612266289at_nat = id_nat ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10140_less__eq__int__def,axiom,
% 8.90/5.15 ( ord_less_eq_int
% 8.90/5.15 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 8.90/5.15 @ ( produc8739625826339149834_nat_o
% 8.90/5.15 @ ^ [X: nat,Y: nat] :
% 8.90/5.15 ( produc6081775807080527818_nat_o
% 8.90/5.15 @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10141_less__int__def,axiom,
% 8.90/5.15 ( ord_less_int
% 8.90/5.15 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 8.90/5.15 @ ( produc8739625826339149834_nat_o
% 8.90/5.15 @ ^ [X: nat,Y: nat] :
% 8.90/5.15 ( produc6081775807080527818_nat_o
% 8.90/5.15 @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10142_nat__def,axiom,
% 8.90/5.15 ( nat2
% 8.90/5.15 = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10143_positive__rat,axiom,
% 8.90/5.15 ! [A: int,B: int] :
% 8.90/5.15 ( ( positive @ ( fract @ A @ B ) )
% 8.90/5.15 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10144_Rat_Opositive__mult,axiom,
% 8.90/5.15 ! [X2: rat,Y3: rat] :
% 8.90/5.15 ( ( positive @ X2 )
% 8.90/5.15 => ( ( positive @ Y3 )
% 8.90/5.15 => ( positive @ ( times_times_rat @ X2 @ Y3 ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10145_less__rat__def,axiom,
% 8.90/5.15 ( ord_less_rat
% 8.90/5.15 = ( ^ [X: rat,Y: rat] : ( positive @ ( minus_minus_rat @ Y @ X ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10146_Rat_Opositive_Orep__eq,axiom,
% 8.90/5.15 ( positive
% 8.90/5.15 = ( ^ [X: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X ) ) @ ( product_snd_int_int @ ( rep_Rat @ X ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10147_Rat_Opositive__def,axiom,
% 8.90/5.15 ( positive
% 8.90/5.15 = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
% 8.90/5.15 @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10148_of__rat__dense,axiom,
% 8.90/5.15 ! [X2: real,Y3: real] :
% 8.90/5.15 ( ( ord_less_real @ X2 @ Y3 )
% 8.90/5.15 => ? [Q2: rat] :
% 8.90/5.15 ( ( ord_less_real @ X2 @ ( field_7254667332652039916t_real @ Q2 ) )
% 8.90/5.15 & ( ord_less_real @ ( field_7254667332652039916t_real @ Q2 ) @ Y3 ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10149_plus__rat__def,axiom,
% 8.90/5.15 ( plus_plus_rat
% 8.90/5.15 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 8.90/5.15 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10150_inverse__rat__def,axiom,
% 8.90/5.15 ( inverse_inverse_rat
% 8.90/5.15 = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
% 8.90/5.15 @ ^ [X: product_prod_int_int] :
% 8.90/5.15 ( if_Pro3027730157355071871nt_int
% 8.90/5.15 @ ( ( product_fst_int_int @ X )
% 8.90/5.15 = zero_zero_int )
% 8.90/5.15 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 8.90/5.15 @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10151_one__rat__def,axiom,
% 8.90/5.15 ( one_one_rat
% 8.90/5.15 = ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10152_Fract_Oabs__eq,axiom,
% 8.90/5.15 ( fract
% 8.90/5.15 = ( ^ [Xa4: int,X: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10153_zero__rat__def,axiom,
% 8.90/5.15 ( zero_zero_rat
% 8.90/5.15 = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10154_times__rat__def,axiom,
% 8.90/5.15 ( times_times_rat
% 8.90/5.15 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 8.90/5.15 @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10155_plus__rat_Oabs__eq,axiom,
% 8.90/5.15 ! [Xa2: product_prod_int_int,X2: product_prod_int_int] :
% 8.90/5.15 ( ( ratrel @ Xa2 @ Xa2 )
% 8.90/5.15 => ( ( ratrel @ X2 @ X2 )
% 8.90/5.15 => ( ( plus_plus_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X2 ) )
% 8.90/5.15 = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_snd_int_int @ X2 ) ) @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10156_ratrel__iff,axiom,
% 8.90/5.15 ( ratrel
% 8.90/5.15 = ( ^ [X: product_prod_int_int,Y: product_prod_int_int] :
% 8.90/5.15 ( ( ( product_snd_int_int @ X )
% 8.90/5.15 != zero_zero_int )
% 8.90/5.15 & ( ( product_snd_int_int @ Y )
% 8.90/5.15 != zero_zero_int )
% 8.90/5.15 & ( ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) )
% 8.90/5.15 = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10157_one__rat_Orsp,axiom,
% 8.90/5.15 ratrel @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10158_zero__rat_Orsp,axiom,
% 8.90/5.15 ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10159_ratrel__def,axiom,
% 8.90/5.15 ( ratrel
% 8.90/5.15 = ( ^ [X: product_prod_int_int,Y: product_prod_int_int] :
% 8.90/5.15 ( ( ( product_snd_int_int @ X )
% 8.90/5.15 != zero_zero_int )
% 8.90/5.15 & ( ( product_snd_int_int @ Y )
% 8.90/5.15 != zero_zero_int )
% 8.90/5.15 & ( ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) )
% 8.90/5.15 = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10160_times__rat_Oabs__eq,axiom,
% 8.90/5.15 ! [Xa2: product_prod_int_int,X2: product_prod_int_int] :
% 8.90/5.15 ( ( ratrel @ Xa2 @ Xa2 )
% 8.90/5.15 => ( ( ratrel @ X2 @ X2 )
% 8.90/5.15 => ( ( times_times_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X2 ) )
% 8.90/5.15 = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_fst_int_int @ X2 ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10161_Rat_Opositive_Oabs__eq,axiom,
% 8.90/5.15 ! [X2: product_prod_int_int] :
% 8.90/5.15 ( ( ratrel @ X2 @ X2 )
% 8.90/5.15 => ( ( positive @ ( abs_Rat @ X2 ) )
% 8.90/5.15 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10162_inverse__rat_Oabs__eq,axiom,
% 8.90/5.15 ! [X2: product_prod_int_int] :
% 8.90/5.15 ( ( ratrel @ X2 @ X2 )
% 8.90/5.15 => ( ( inverse_inverse_rat @ ( abs_Rat @ X2 ) )
% 8.90/5.15 = ( abs_Rat
% 8.90/5.15 @ ( if_Pro3027730157355071871nt_int
% 8.90/5.15 @ ( ( product_fst_int_int @ X2 )
% 8.90/5.15 = zero_zero_int )
% 8.90/5.15 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 8.90/5.15 @ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10163_min__Suc__Suc,axiom,
% 8.90/5.15 ! [M: nat,N: nat] :
% 8.90/5.15 ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
% 8.90/5.15 = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10164_min__0R,axiom,
% 8.90/5.15 ! [N: nat] :
% 8.90/5.15 ( ( ord_min_nat @ N @ zero_zero_nat )
% 8.90/5.15 = zero_zero_nat ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10165_min__0L,axiom,
% 8.90/5.15 ! [N: nat] :
% 8.90/5.15 ( ( ord_min_nat @ zero_zero_nat @ N )
% 8.90/5.15 = zero_zero_nat ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10166_min__Suc__numeral,axiom,
% 8.90/5.15 ! [N: nat,K: num] :
% 8.90/5.15 ( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 8.90/5.15 = ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10167_min__numeral__Suc,axiom,
% 8.90/5.15 ! [K: num,N: nat] :
% 8.90/5.15 ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 8.90/5.15 = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10168_nat__mult__min__left,axiom,
% 8.90/5.15 ! [M: nat,N: nat,Q3: nat] :
% 8.90/5.15 ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q3 )
% 8.90/5.15 = ( ord_min_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10169_nat__mult__min__right,axiom,
% 8.90/5.15 ! [M: nat,N: nat,Q3: nat] :
% 8.90/5.15 ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q3 ) )
% 8.90/5.15 = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q3 ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10170_inf__nat__def,axiom,
% 8.90/5.15 inf_inf_nat = ord_min_nat ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10171_min__diff,axiom,
% 8.90/5.15 ! [M: nat,I2: nat,N: nat] :
% 8.90/5.15 ( ( ord_min_nat @ ( minus_minus_nat @ M @ I2 ) @ ( minus_minus_nat @ N @ I2 ) )
% 8.90/5.15 = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I2 ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10172_concat__bit__assoc__sym,axiom,
% 8.90/5.15 ! [M: nat,N: nat,K: int,L: int,R2: int] :
% 8.90/5.15 ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L ) @ R2 )
% 8.90/5.15 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N ) @ L @ R2 ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10173_take__bit__concat__bit__eq,axiom,
% 8.90/5.15 ! [M: nat,N: nat,K: int,L: int] :
% 8.90/5.15 ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N @ K @ L ) )
% 8.90/5.15 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ L ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10174_min__Suc2,axiom,
% 8.90/5.15 ! [M: nat,N: nat] :
% 8.90/5.15 ( ( ord_min_nat @ M @ ( suc @ N ) )
% 8.90/5.15 = ( case_nat_nat @ zero_zero_nat
% 8.90/5.15 @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ M5 @ N ) )
% 8.90/5.15 @ M ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10175_min__Suc1,axiom,
% 8.90/5.15 ! [N: nat,M: nat] :
% 8.90/5.15 ( ( ord_min_nat @ ( suc @ N ) @ M )
% 8.90/5.15 = ( case_nat_nat @ zero_zero_nat
% 8.90/5.15 @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ N @ M5 ) )
% 8.90/5.15 @ M ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10176_min__enat__simps_I2_J,axiom,
% 8.90/5.15 ! [Q3: extended_enat] :
% 8.90/5.15 ( ( ord_mi8085742599997312461d_enat @ Q3 @ zero_z5237406670263579293d_enat )
% 8.90/5.15 = zero_z5237406670263579293d_enat ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10177_min__enat__simps_I3_J,axiom,
% 8.90/5.15 ! [Q3: extended_enat] :
% 8.90/5.15 ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q3 )
% 8.90/5.15 = zero_z5237406670263579293d_enat ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10178_inf__enat__def,axiom,
% 8.90/5.15 inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10179_tl__upt,axiom,
% 8.90/5.15 ! [M: nat,N: nat] :
% 8.90/5.15 ( ( tl_nat @ ( upt @ M @ N ) )
% 8.90/5.15 = ( upt @ ( suc @ M ) @ N ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10180_hd__upt,axiom,
% 8.90/5.15 ! [I2: nat,J: nat] :
% 8.90/5.15 ( ( ord_less_nat @ I2 @ J )
% 8.90/5.15 => ( ( hd_nat @ ( upt @ I2 @ J ) )
% 8.90/5.15 = I2 ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10181_drop__upt,axiom,
% 8.90/5.15 ! [M: nat,I2: nat,J: nat] :
% 8.90/5.15 ( ( drop_nat @ M @ ( upt @ I2 @ J ) )
% 8.90/5.15 = ( upt @ ( plus_plus_nat @ I2 @ M ) @ J ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10182_length__upt,axiom,
% 8.90/5.15 ! [I2: nat,J: nat] :
% 8.90/5.15 ( ( size_size_list_nat @ ( upt @ I2 @ J ) )
% 8.90/5.15 = ( minus_minus_nat @ J @ I2 ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10183_take__upt,axiom,
% 8.90/5.15 ! [I2: nat,M: nat,N: nat] :
% 8.90/5.15 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ M ) @ N )
% 8.90/5.15 => ( ( take_nat @ M @ ( upt @ I2 @ N ) )
% 8.90/5.15 = ( upt @ I2 @ ( plus_plus_nat @ I2 @ M ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10184_upt__conv__Nil,axiom,
% 8.90/5.15 ! [J: nat,I2: nat] :
% 8.90/5.15 ( ( ord_less_eq_nat @ J @ I2 )
% 8.90/5.15 => ( ( upt @ I2 @ J )
% 8.90/5.15 = nil_nat ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10185_upt__eq__Nil__conv,axiom,
% 8.90/5.15 ! [I2: nat,J: nat] :
% 8.90/5.15 ( ( ( upt @ I2 @ J )
% 8.90/5.15 = nil_nat )
% 8.90/5.15 = ( ( J = zero_zero_nat )
% 8.90/5.15 | ( ord_less_eq_nat @ J @ I2 ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10186_nth__upt,axiom,
% 8.90/5.15 ! [I2: nat,K: nat,J: nat] :
% 8.90/5.15 ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J )
% 8.90/5.15 => ( ( nth_nat @ ( upt @ I2 @ J ) @ K )
% 8.90/5.15 = ( plus_plus_nat @ I2 @ K ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10187_upt__rec__numeral,axiom,
% 8.90/5.15 ! [M: num,N: num] :
% 8.90/5.15 ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 8.90/5.15 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 8.90/5.15 = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
% 8.90/5.15 & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 8.90/5.15 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 8.90/5.15 = nil_nat ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10188_map__add__upt,axiom,
% 8.90/5.15 ! [N: nat,M: nat] :
% 8.90/5.15 ( ( map_nat_nat
% 8.90/5.15 @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
% 8.90/5.15 @ ( upt @ zero_zero_nat @ M ) )
% 8.90/5.15 = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10189_map__Suc__upt,axiom,
% 8.90/5.15 ! [M: nat,N: nat] :
% 8.90/5.15 ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
% 8.90/5.15 = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10190_map__decr__upt,axiom,
% 8.90/5.15 ! [M: nat,N: nat] :
% 8.90/5.15 ( ( map_nat_nat
% 8.90/5.15 @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) )
% 8.90/5.15 @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
% 8.90/5.15 = ( upt @ M @ N ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10191_upt__conv__Cons__Cons,axiom,
% 8.90/5.15 ! [M: nat,N: nat,Ns: list_nat,Q3: nat] :
% 8.90/5.15 ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
% 8.90/5.15 = ( upt @ M @ Q3 ) )
% 8.90/5.15 = ( ( cons_nat @ N @ Ns )
% 8.90/5.15 = ( upt @ ( suc @ M ) @ Q3 ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10192_atMost__upto,axiom,
% 8.90/5.15 ( set_ord_atMost_nat
% 8.90/5.15 = ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N3 ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10193_atLeastAtMost__upt,axiom,
% 8.90/5.15 ( set_or1269000886237332187st_nat
% 8.90/5.15 = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ N3 @ ( suc @ M3 ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10194_greaterThanLessThan__upt,axiom,
% 8.90/5.15 ( set_or5834768355832116004an_nat
% 8.90/5.15 = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ M3 ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10195_greaterThanAtMost__upt,axiom,
% 8.90/5.15 ( set_or6659071591806873216st_nat
% 8.90/5.15 = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ ( suc @ M3 ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10196_upt__conv__Cons,axiom,
% 8.90/5.15 ! [I2: nat,J: nat] :
% 8.90/5.15 ( ( ord_less_nat @ I2 @ J )
% 8.90/5.15 => ( ( upt @ I2 @ J )
% 8.90/5.15 = ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10197_upt__add__eq__append,axiom,
% 8.90/5.15 ! [I2: nat,J: nat,K: nat] :
% 8.90/5.15 ( ( ord_less_eq_nat @ I2 @ J )
% 8.90/5.15 => ( ( upt @ I2 @ ( plus_plus_nat @ J @ K ) )
% 8.90/5.15 = ( append_nat @ ( upt @ I2 @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10198_upt__eq__Cons__conv,axiom,
% 8.90/5.15 ! [I2: nat,J: nat,X2: nat,Xs2: list_nat] :
% 8.90/5.15 ( ( ( upt @ I2 @ J )
% 8.90/5.15 = ( cons_nat @ X2 @ Xs2 ) )
% 8.90/5.15 = ( ( ord_less_nat @ I2 @ J )
% 8.90/5.15 & ( I2 = X2 )
% 8.90/5.15 & ( ( upt @ ( plus_plus_nat @ I2 @ one_one_nat ) @ J )
% 8.90/5.15 = Xs2 ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10199_upt__rec,axiom,
% 8.90/5.15 ( upt
% 8.90/5.15 = ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10200_upt__Suc,axiom,
% 8.90/5.15 ! [I2: nat,J: nat] :
% 8.90/5.15 ( ( ( ord_less_eq_nat @ I2 @ J )
% 8.90/5.15 => ( ( upt @ I2 @ ( suc @ J ) )
% 8.90/5.15 = ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
% 8.90/5.15 & ( ~ ( ord_less_eq_nat @ I2 @ J )
% 8.90/5.15 => ( ( upt @ I2 @ ( suc @ J ) )
% 8.90/5.15 = nil_nat ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10201_upt__Suc__append,axiom,
% 8.90/5.15 ! [I2: nat,J: nat] :
% 8.90/5.15 ( ( ord_less_eq_nat @ I2 @ J )
% 8.90/5.15 => ( ( upt @ I2 @ ( suc @ J ) )
% 8.90/5.15 = ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10202_sum__list__upt,axiom,
% 8.90/5.15 ! [M: nat,N: nat] :
% 8.90/5.15 ( ( ord_less_eq_nat @ M @ N )
% 8.90/5.15 => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
% 8.90/5.15 = ( groups3542108847815614940at_nat
% 8.90/5.15 @ ^ [X: nat] : X
% 8.90/5.15 @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10203_card__length__sum__list__rec,axiom,
% 8.90/5.15 ! [M: nat,N5: nat] :
% 8.90/5.15 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 8.90/5.15 => ( ( finite_card_list_nat
% 8.90/5.15 @ ( collect_list_nat
% 8.90/5.15 @ ^ [L2: list_nat] :
% 8.90/5.15 ( ( ( size_size_list_nat @ L2 )
% 8.90/5.15 = M )
% 8.90/5.15 & ( ( groups4561878855575611511st_nat @ L2 )
% 8.90/5.15 = N5 ) ) ) )
% 8.90/5.15 = ( plus_plus_nat
% 8.90/5.15 @ ( finite_card_list_nat
% 8.90/5.15 @ ( collect_list_nat
% 8.90/5.15 @ ^ [L2: list_nat] :
% 8.90/5.15 ( ( ( size_size_list_nat @ L2 )
% 8.90/5.15 = ( minus_minus_nat @ M @ one_one_nat ) )
% 8.90/5.15 & ( ( groups4561878855575611511st_nat @ L2 )
% 8.90/5.15 = N5 ) ) ) )
% 8.90/5.15 @ ( finite_card_list_nat
% 8.90/5.15 @ ( collect_list_nat
% 8.90/5.15 @ ^ [L2: list_nat] :
% 8.90/5.15 ( ( ( size_size_list_nat @ L2 )
% 8.90/5.15 = M )
% 8.90/5.15 & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
% 8.90/5.15 = N5 ) ) ) ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10204_card__length__sum__list,axiom,
% 8.90/5.15 ! [M: nat,N5: nat] :
% 8.90/5.15 ( ( finite_card_list_nat
% 8.90/5.15 @ ( collect_list_nat
% 8.90/5.15 @ ^ [L2: list_nat] :
% 8.90/5.15 ( ( ( size_size_list_nat @ L2 )
% 8.90/5.15 = M )
% 8.90/5.15 & ( ( groups4561878855575611511st_nat @ L2 )
% 8.90/5.15 = N5 ) ) ) )
% 8.90/5.15 = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N5 @ M ) @ one_one_nat ) @ N5 ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10205_sorted__upto,axiom,
% 8.90/5.15 ! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10206_sorted__upt,axiom,
% 8.90/5.15 ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10207_sorted__wrt__upt,axiom,
% 8.90/5.15 ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(fact_10208_sorted__wrt__less__idx,axiom,
% 8.90/5.15 ! [Ns: list_nat,I2: nat] :
% 8.90/5.15 ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
% 8.90/5.15 => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ns ) )
% 8.90/5.15 => ( ord_less_eq_nat @ I2 @ ( nth_nat @ Ns @ I2 ) ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 8.90/5.15 ! [X2: int,Y3: int] :
% 8.90/5.15 ( ( if_int @ $false @ X2 @ Y3 )
% 8.90/5.15 = Y3 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 8.90/5.15 ! [X2: int,Y3: int] :
% 8.90/5.15 ( ( if_int @ $true @ X2 @ Y3 )
% 8.90/5.15 = X2 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 8.90/5.15 ! [X2: nat,Y3: nat] :
% 8.90/5.15 ( ( if_nat @ $false @ X2 @ Y3 )
% 8.90/5.15 = Y3 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 8.90/5.15 ! [X2: nat,Y3: nat] :
% 8.90/5.15 ( ( if_nat @ $true @ X2 @ Y3 )
% 8.90/5.15 = X2 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
% 8.90/5.15 ! [X2: num,Y3: num] :
% 8.90/5.15 ( ( if_num @ $false @ X2 @ Y3 )
% 8.90/5.15 = Y3 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
% 8.90/5.15 ! [X2: num,Y3: num] :
% 8.90/5.15 ( ( if_num @ $true @ X2 @ Y3 )
% 8.90/5.15 = X2 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 8.90/5.15 ! [X2: rat,Y3: rat] :
% 8.90/5.15 ( ( if_rat @ $false @ X2 @ Y3 )
% 8.90/5.15 = Y3 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 8.90/5.15 ! [X2: rat,Y3: rat] :
% 8.90/5.15 ( ( if_rat @ $true @ X2 @ Y3 )
% 8.90/5.15 = X2 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 8.90/5.15 ! [X2: real,Y3: real] :
% 8.90/5.15 ( ( if_real @ $false @ X2 @ Y3 )
% 8.90/5.15 = Y3 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 8.90/5.15 ! [X2: real,Y3: real] :
% 8.90/5.15 ( ( if_real @ $true @ X2 @ Y3 )
% 8.90/5.15 = X2 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 8.90/5.15 ! [P: real > $o] :
% 8.90/5.15 ( ( P @ ( fChoice_real @ P ) )
% 8.90/5.15 = ( ? [X4: real] : ( P @ X4 ) ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 8.90/5.15 ! [X2: complex,Y3: complex] :
% 8.90/5.15 ( ( if_complex @ $false @ X2 @ Y3 )
% 8.90/5.15 = Y3 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 8.90/5.15 ! [X2: complex,Y3: complex] :
% 8.90/5.15 ( ( if_complex @ $true @ X2 @ Y3 )
% 8.90/5.15 = X2 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 8.90/5.15 ! [X2: extended_enat,Y3: extended_enat] :
% 8.90/5.15 ( ( if_Extended_enat @ $false @ X2 @ Y3 )
% 8.90/5.15 = Y3 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 8.90/5.15 ! [X2: extended_enat,Y3: extended_enat] :
% 8.90/5.15 ( ( if_Extended_enat @ $true @ X2 @ Y3 )
% 8.90/5.15 = X2 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 8.90/5.15 ! [X2: code_integer,Y3: code_integer] :
% 8.90/5.15 ( ( if_Code_integer @ $false @ X2 @ Y3 )
% 8.90/5.15 = Y3 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 8.90/5.15 ! [X2: code_integer,Y3: code_integer] :
% 8.90/5.15 ( ( if_Code_integer @ $true @ X2 @ Y3 )
% 8.90/5.15 = X2 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 8.90/5.15 ! [X2: set_int,Y3: set_int] :
% 8.90/5.15 ( ( if_set_int @ $false @ X2 @ Y3 )
% 8.90/5.15 = Y3 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 8.90/5.15 ! [X2: set_int,Y3: set_int] :
% 8.90/5.15 ( ( if_set_int @ $true @ X2 @ Y3 )
% 8.90/5.15 = X2 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 8.90/5.15 ! [X2: vEBT_VEBT,Y3: vEBT_VEBT] :
% 8.90/5.15 ( ( if_VEBT_VEBT @ $false @ X2 @ Y3 )
% 8.90/5.15 = Y3 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 8.90/5.15 ! [X2: vEBT_VEBT,Y3: vEBT_VEBT] :
% 8.90/5.15 ( ( if_VEBT_VEBT @ $true @ X2 @ Y3 )
% 8.90/5.15 = X2 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 8.90/5.15 ! [X2: list_int,Y3: list_int] :
% 8.90/5.15 ( ( if_list_int @ $false @ X2 @ Y3 )
% 8.90/5.15 = Y3 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 8.90/5.15 ! [X2: list_int,Y3: list_int] :
% 8.90/5.15 ( ( if_list_int @ $true @ X2 @ Y3 )
% 8.90/5.15 = X2 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 8.90/5.15 ! [X2: list_nat,Y3: list_nat] :
% 8.90/5.15 ( ( if_list_nat @ $false @ X2 @ Y3 )
% 8.90/5.15 = Y3 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 8.90/5.15 ! [X2: list_nat,Y3: list_nat] :
% 8.90/5.15 ( ( if_list_nat @ $true @ X2 @ Y3 )
% 8.90/5.15 = X2 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 8.90/5.15 ! [X2: option_nat,Y3: option_nat] :
% 8.90/5.15 ( ( if_option_nat @ $false @ X2 @ Y3 )
% 8.90/5.15 = Y3 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 8.90/5.15 ! [X2: option_nat,Y3: option_nat] :
% 8.90/5.15 ( ( if_option_nat @ $true @ X2 @ Y3 )
% 8.90/5.15 = X2 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 8.90/5.15 ! [X2: option_num,Y3: option_num] :
% 8.90/5.15 ( ( if_option_num @ $false @ X2 @ Y3 )
% 8.90/5.15 = Y3 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 8.90/5.15 ! [X2: option_num,Y3: option_num] :
% 8.90/5.15 ( ( if_option_num @ $true @ X2 @ Y3 )
% 8.90/5.15 = X2 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 8.90/5.15 ! [X2: product_prod_int_int,Y3: product_prod_int_int] :
% 8.90/5.15 ( ( if_Pro3027730157355071871nt_int @ $false @ X2 @ Y3 )
% 8.90/5.15 = Y3 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 8.90/5.15 ! [X2: product_prod_int_int,Y3: product_prod_int_int] :
% 8.90/5.15 ( ( if_Pro3027730157355071871nt_int @ $true @ X2 @ Y3 )
% 8.90/5.15 = X2 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 8.90/5.15 ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 8.90/5.15 ( ( if_Pro6206227464963214023at_nat @ $false @ X2 @ Y3 )
% 8.90/5.15 = Y3 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 8.90/5.15 ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 8.90/5.15 ( ( if_Pro6206227464963214023at_nat @ $true @ X2 @ Y3 )
% 8.90/5.15 = X2 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 8.90/5.15 ! [X2: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 8.90/5.15 ( ( if_Pro5737122678794959658eger_o @ $false @ X2 @ Y3 )
% 8.90/5.15 = Y3 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 8.90/5.15 ! [X2: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 8.90/5.15 ( ( if_Pro5737122678794959658eger_o @ $true @ X2 @ Y3 )
% 8.90/5.15 = X2 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 8.90/5.15 ! [P: $o] :
% 8.90/5.15 ( ( P = $true )
% 8.90/5.15 | ( P = $false ) ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 8.90/5.15 ! [X2: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 8.90/5.15 ( ( if_Pro6119634080678213985nteger @ $false @ X2 @ Y3 )
% 8.90/5.15 = Y3 ) ).
% 8.90/5.15
% 8.90/5.15 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 8.90/5.15 ! [X2: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 8.90/5.15 ( ( if_Pro6119634080678213985nteger @ $true @ X2 @ Y3 )
% 8.90/5.15 = X2 ) ).
% 8.90/5.15
% 8.90/5.15 thf(conj_0,conjecture,
% 8.90/5.15 ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) ) ) ).
% 8.90/5.15 ------------------------
% 8.92/5.65 % (16216)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_16103 for (2995ds/18Mi)
% 8.92/5.66 % (16210)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_16103 for (2995ds/183Mi)
% 8.92/5.66 % (16212)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_16103 for (2995ds/27Mi)
% 8.92/5.66 % (16211)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_16103 for (2995ds/4Mi)
% 8.92/5.66 % (16213)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_16103 for (2995ds/2Mi)
% 8.92/5.66 % (16214)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_16103 for (2995ds/2Mi)
% 8.92/5.66 % (16215)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_16103 for (2995ds/275Mi)
% 8.92/5.66 % (16213)Instruction limit reached!
% 8.92/5.66 % (16213)------------------------------
% 8.92/5.66 % (16213)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.66 % (16213)Termination reason: Unknown
% 8.92/5.66 % (16213)Termination phase: shuffling
% 8.92/5.66
% 8.92/5.66 % (16213)Memory used [KB]: 12792
% 8.92/5.66 % (16213)Time elapsed: 0.003 s
% 8.92/5.66 % (16213)Instructions burned: 2 (million)
% 8.92/5.66 % (16213)------------------------------
% 8.92/5.66 % (16213)------------------------------
% 8.92/5.66 % (16214)Instruction limit reached!
% 8.92/5.66 % (16214)------------------------------
% 8.92/5.66 % (16214)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.66 % (16214)Termination reason: Unknown
% 8.92/5.66 % (16214)Termination phase: shuffling
% 8.92/5.66
% 8.92/5.66 % (16214)Memory used [KB]: 12792
% 8.92/5.66 % (16214)Time elapsed: 0.003 s
% 8.92/5.66 % (16214)Instructions burned: 2 (million)
% 8.92/5.66 % (16214)------------------------------
% 8.92/5.66 % (16214)------------------------------
% 8.92/5.66 % (16211)Instruction limit reached!
% 8.92/5.66 % (16211)------------------------------
% 8.92/5.66 % (16211)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.66 % (16211)Termination reason: Unknown
% 8.92/5.66 % (16211)Termination phase: shuffling
% 8.92/5.66
% 8.92/5.66 % (16211)Memory used [KB]: 12920
% 8.92/5.66 % (16211)Time elapsed: 0.006 s
% 8.92/5.66 % (16211)Instructions burned: 4 (million)
% 8.92/5.66 % (16211)------------------------------
% 8.92/5.66 % (16211)------------------------------
% 8.92/5.67 % (16216)Instruction limit reached!
% 8.92/5.67 % (16216)------------------------------
% 8.92/5.67 % (16216)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.67 % (16216)Termination reason: Unknown
% 8.92/5.67 % (16216)Termination phase: shuffling
% 8.92/5.67
% 8.92/5.67 % (16216)Memory used [KB]: 13688
% 8.92/5.67 % (16216)Time elapsed: 0.015 s
% 8.92/5.67 % (16216)Instructions burned: 19 (million)
% 8.92/5.67 % (16216)------------------------------
% 8.92/5.67 % (16216)------------------------------
% 8.92/5.68 % (16220)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_16103 for (2995ds/3Mi)
% 8.92/5.68 % (16220)Instruction limit reached!
% 8.92/5.68 % (16220)------------------------------
% 8.92/5.68 % (16220)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.68 % (16220)Termination reason: Unknown
% 8.92/5.68 % (16220)Termination phase: shuffling
% 8.92/5.68
% 8.92/5.68 % (16220)Memory used [KB]: 12920
% 8.92/5.68 % (16220)Time elapsed: 0.004 s
% 8.92/5.68 % (16220)Instructions burned: 3 (million)
% 8.92/5.68 % (16220)------------------------------
% 8.92/5.68 % (16220)------------------------------
% 8.92/5.68 % (16217)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_16103 for (2995ds/3Mi)
% 8.92/5.68 % (16218)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_16103 for (2995ds/37Mi)
% 8.92/5.68 % (16219)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_16103 for (2995ds/15Mi)
% 8.92/5.68 % (16217)Instruction limit reached!
% 8.92/5.68 % (16217)------------------------------
% 8.92/5.68 % (16217)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.68 % (16217)Termination reason: Unknown
% 8.92/5.68 % (16217)Termination phase: shuffling
% 8.92/5.68
% 8.92/5.68 % (16217)Memory used [KB]: 12920
% 8.92/5.68 % (16217)Time elapsed: 0.005 s
% 8.92/5.68 % (16217)Instructions burned: 3 (million)
% 8.92/5.68 % (16217)------------------------------
% 8.92/5.68 % (16217)------------------------------
% 8.92/5.69 % (16212)Instruction limit reached!
% 8.92/5.69 % (16212)------------------------------
% 8.92/5.69 % (16212)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.69 % (16212)Termination reason: Unknown
% 8.92/5.69 % (16212)Termination phase: shuffling
% 8.92/5.69
% 8.92/5.69 % (16212)Memory used [KB]: 13816
% 8.92/5.69 % (16212)Time elapsed: 0.029 s
% 8.92/5.69 % (16212)Instructions burned: 27 (million)
% 8.92/5.69 % (16212)------------------------------
% 8.92/5.69 % (16212)------------------------------
% 8.92/5.69 % (16221)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_16103 for (2994ds/1041Mi)
% 8.92/5.70 % (16219)Instruction limit reached!
% 8.92/5.70 % (16219)------------------------------
% 8.92/5.70 % (16219)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.70 % (16219)Termination reason: Unknown
% 8.92/5.70 % (16219)Termination phase: shuffling
% 8.92/5.70
% 8.92/5.70 % (16219)Memory used [KB]: 13688
% 8.92/5.70 % (16219)Time elapsed: 0.021 s
% 8.92/5.70 % (16219)Instructions burned: 15 (million)
% 8.92/5.70 % (16219)------------------------------
% 8.92/5.70 % (16219)------------------------------
% 8.92/5.70 % (16222)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_16103 for (2994ds/7Mi)
% 8.92/5.71 % (16223)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_16103 for (2994ds/16Mi)
% 8.92/5.71 % (16218)Instruction limit reached!
% 8.92/5.71 % (16218)------------------------------
% 8.92/5.71 % (16218)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.71 % (16222)Instruction limit reached!
% 8.92/5.71 % (16222)------------------------------
% 8.92/5.71 % (16222)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.71 % (16222)Termination reason: Unknown
% 8.92/5.71 % (16222)Termination phase: shuffling
% 8.92/5.71
% 8.92/5.71 % (16222)Memory used [KB]: 13176
% 8.92/5.71 % (16222)Time elapsed: 0.011 s
% 8.92/5.71 % (16222)Instructions burned: 7 (million)
% 8.92/5.71 % (16222)------------------------------
% 8.92/5.71 % (16222)------------------------------
% 8.92/5.71 % (16218)Termination reason: Unknown
% 8.92/5.71 % (16218)Termination phase: shuffling
% 8.92/5.71
% 8.92/5.71 % (16218)Memory used [KB]: 13944
% 8.92/5.71 % (16218)Time elapsed: 0.033 s
% 8.92/5.71 % (16218)Instructions burned: 38 (million)
% 8.92/5.71 % (16218)------------------------------
% 8.92/5.71 % (16218)------------------------------
% 8.92/5.72 % (16224)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_16103 for (2994ds/3Mi)
% 8.92/5.73 % (16224)Instruction limit reached!
% 8.92/5.73 % (16224)------------------------------
% 8.92/5.73 % (16224)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.73 % (16224)Termination reason: Unknown
% 8.92/5.73 % (16224)Termination phase: shuffling
% 8.92/5.73
% 8.92/5.73 % (16224)Memory used [KB]: 12920
% 8.92/5.73 % (16224)Time elapsed: 0.005 s
% 8.92/5.73 % (16224)Instructions burned: 3 (million)
% 8.92/5.73 % (16224)------------------------------
% 8.92/5.73 % (16224)------------------------------
% 8.92/5.73 % (16223)Instruction limit reached!
% 8.92/5.73 % (16223)------------------------------
% 8.92/5.73 % (16223)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.73 % (16223)Termination reason: Unknown
% 8.92/5.73 % (16223)Termination phase: shuffling
% 8.92/5.73
% 8.92/5.73 % (16223)Memory used [KB]: 13688
% 8.92/5.73 % (16223)Time elapsed: 0.023 s
% 8.92/5.73 % (16223)Instructions burned: 16 (million)
% 8.92/5.73 % (16223)------------------------------
% 8.92/5.73 % (16223)------------------------------
% 8.92/5.73 % (16225)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_16103 for (2994ds/3Mi)
% 8.92/5.73 % (16226)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_16103 for (2994ds/7Mi)
% 8.92/5.73 % (16225)Instruction limit reached!
% 8.92/5.73 % (16225)------------------------------
% 8.92/5.73 % (16225)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.73 % (16225)Termination reason: Unknown
% 8.92/5.73 % (16225)Termination phase: shuffling
% 8.92/5.73
% 8.92/5.73 % (16225)Memory used [KB]: 12920
% 8.92/5.73 % (16225)Time elapsed: 0.005 s
% 8.92/5.73 % (16225)Instructions burned: 3 (million)
% 8.92/5.73 % (16225)------------------------------
% 8.92/5.73 % (16225)------------------------------
% 8.92/5.74 % (16226)Instruction limit reached!
% 8.92/5.74 % (16226)------------------------------
% 8.92/5.74 % (16226)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.74 % (16226)Termination reason: Unknown
% 8.92/5.74 % (16226)Termination phase: shuffling
% 8.92/5.74
% 8.92/5.74 % (16226)Memory used [KB]: 13176
% 8.92/5.74 % (16226)Time elapsed: 0.011 s
% 8.92/5.74 % (16226)Instructions burned: 7 (million)
% 8.92/5.74 % (16226)------------------------------
% 8.92/5.74 % (16226)------------------------------
% 8.92/5.74 % (16227)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_16103 for (2994ds/3Mi)
% 8.92/5.75 % (16227)Instruction limit reached!
% 8.92/5.75 % (16227)------------------------------
% 8.92/5.75 % (16227)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.75 % (16227)Termination reason: Unknown
% 8.92/5.75 % (16227)Termination phase: shuffling
% 8.92/5.75
% 8.92/5.75 % (16227)Memory used [KB]: 12920
% 8.92/5.75 % (16227)Time elapsed: 0.005 s
% 8.92/5.75 % (16227)Instructions burned: 3 (million)
% 8.92/5.75 % (16227)------------------------------
% 8.92/5.75 % (16227)------------------------------
% 8.92/5.75 % (16228)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_16103 for (2994ds/4Mi)
% 8.92/5.75 % (16228)Instruction limit reached!
% 8.92/5.75 % (16228)------------------------------
% 8.92/5.75 % (16228)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.75 % (16228)Termination reason: Unknown
% 8.92/5.75 % (16228)Termination phase: shuffling
% 8.92/5.75
% 8.92/5.75 % (16228)Memory used [KB]: 12920
% 8.92/5.75 % (16228)Time elapsed: 0.006 s
% 8.92/5.75 % (16228)Instructions burned: 4 (million)
% 8.92/5.75 % (16228)------------------------------
% 8.92/5.75 % (16228)------------------------------
% 8.92/5.75 % (16229)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_16103 for (2994ds/18Mi)
% 8.92/5.76 % (16230)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_16103 for (2994ds/710Mi)
% 8.92/5.77 % (16231)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_16103 for (2994ds/6Mi)
% 8.92/5.77 % (16232)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_16103 for (2993ds/902Mi)
% 8.92/5.77 % (16229)Instruction limit reached!
% 8.92/5.77 % (16229)------------------------------
% 8.92/5.77 % (16229)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.77 % (16229)Termination reason: Unknown
% 8.92/5.77 % (16229)Termination phase: shuffling
% 8.92/5.77
% 8.92/5.77 % (16229)Memory used [KB]: 13688
% 8.92/5.77 % (16229)Time elapsed: 0.020 s
% 8.92/5.77 % (16229)Instructions burned: 18 (million)
% 8.92/5.77 % (16229)------------------------------
% 8.92/5.77 % (16229)------------------------------
% 8.92/5.77 % (16231)Instruction limit reached!
% 8.92/5.77 % (16231)------------------------------
% 8.92/5.77 % (16231)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.77 % (16231)Termination reason: Unknown
% 8.92/5.77 % (16231)Termination phase: shuffling
% 8.92/5.77
% 8.92/5.77 % (16231)Memory used [KB]: 13048
% 8.92/5.77 % (16231)Time elapsed: 0.008 s
% 8.92/5.77 % (16231)Instructions burned: 6 (million)
% 8.92/5.77 % (16231)------------------------------
% 8.92/5.77 % (16231)------------------------------
% 8.92/5.77 % (16210)Instruction limit reached!
% 8.92/5.77 % (16210)------------------------------
% 8.92/5.77 % (16210)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.77 % (16210)Termination reason: Unknown
% 8.92/5.77 % (16210)Termination phase: shuffling
% 8.92/5.77
% 8.92/5.77 % (16210)Memory used [KB]: 16119
% 8.92/5.77 % (16210)Time elapsed: 0.117 s
% 8.92/5.77 % (16210)Instructions burned: 183 (million)
% 8.92/5.77 % (16210)------------------------------
% 8.92/5.77 % (16210)------------------------------
% 8.92/5.78 % (16233)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_16103 for (2993ds/21Mi)
% 8.92/5.78 % (16234)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_16103 for (2993ds/5Mi)
% 8.92/5.79 % (16235)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_16103 for (2993ds/6Mi)
% 8.92/5.79 % (16234)Instruction limit reached!
% 8.92/5.79 % (16234)------------------------------
% 8.92/5.79 % (16234)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.79 % (16234)Termination reason: Unknown
% 8.92/5.79 % (16234)Termination phase: shuffling
% 8.92/5.79
% 8.92/5.79 % (16234)Memory used [KB]: 13048
% 8.92/5.79 % (16234)Time elapsed: 0.005 s
% 8.92/5.79 % (16234)Instructions burned: 5 (million)
% 8.92/5.79 % (16234)------------------------------
% 8.92/5.79 % (16234)------------------------------
% 8.92/5.79 % (16235)Instruction limit reached!
% 8.92/5.79 % (16235)------------------------------
% 8.92/5.79 % (16235)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.79 % (16235)Termination reason: Unknown
% 8.92/5.79 % (16235)Termination phase: shuffling
% 8.92/5.79
% 8.92/5.79 % (16235)Memory used [KB]: 13176
% 8.92/5.79 % (16235)Time elapsed: 0.006 s
% 8.92/5.79 % (16235)Instructions burned: 6 (million)
% 8.92/5.79 % (16235)------------------------------
% 8.92/5.79 % (16235)------------------------------
% 8.92/5.80 % (16233)Instruction limit reached!
% 8.92/5.80 % (16233)------------------------------
% 8.92/5.80 % (16233)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.80 % (16233)Termination reason: Unknown
% 8.92/5.80 % (16233)Termination phase: shuffling
% 8.92/5.80
% 8.92/5.80 % (16233)Memory used [KB]: 13688
% 8.92/5.80 % (16233)Time elapsed: 0.017 s
% 8.92/5.80 % (16233)Instructions burned: 22 (million)
% 8.92/5.80 % (16233)------------------------------
% 8.92/5.80 % (16233)------------------------------
% 8.92/5.80 % (16236)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_16103 for (2993ds/377Mi)
% 8.92/5.80 % (16237)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_16103 for (2993ds/779Mi)
% 8.92/5.81 % (16238)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_16103 for (2993ds/19Mi)
% 8.92/5.81 % (16215)Instruction limit reached!
% 8.92/5.81 % (16215)------------------------------
% 8.92/5.81 % (16215)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.81 % (16215)Termination reason: Unknown
% 8.92/5.81 % (16215)Termination phase: shuffling
% 8.92/5.81
% 8.92/5.81 % (16215)Memory used [KB]: 17398
% 8.92/5.81 % (16215)Time elapsed: 0.154 s
% 8.92/5.81 % (16215)Instructions burned: 275 (million)
% 8.92/5.81 % (16215)------------------------------
% 8.92/5.81 % (16215)------------------------------
% 8.92/5.82 % (16239)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_16103 for (2993ds/879Mi)
% 8.92/5.83 % (16238)Instruction limit reached!
% 8.92/5.83 % (16238)------------------------------
% 8.92/5.83 % (16238)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.83 % (16238)Termination reason: Unknown
% 8.92/5.83 % (16238)Termination phase: shuffling
% 8.92/5.83
% 8.92/5.83 % (16238)Memory used [KB]: 13688
% 8.92/5.83 % (16238)Time elapsed: 0.016 s
% 8.92/5.83 % (16238)Instructions burned: 21 (million)
% 8.92/5.83 % (16238)------------------------------
% 8.92/5.83 % (16238)------------------------------
% 8.92/5.84 % (16240)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_16103 for (2993ds/17Mi)
% 8.92/5.85 % (16240)Instruction limit reached!
% 8.92/5.85 % (16240)------------------------------
% 8.92/5.85 % (16240)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.85 % (16240)Termination reason: Unknown
% 8.92/5.85 % (16240)Termination phase: shuffling
% 8.92/5.85
% 8.92/5.85 % (16240)Memory used [KB]: 13688
% 8.92/5.85 % (16240)Time elapsed: 0.015 s
% 8.92/5.85 % (16240)Instructions burned: 17 (million)
% 8.92/5.85 % (16240)------------------------------
% 8.92/5.85 % (16240)------------------------------
% 8.92/5.86 % (16241)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_16103 for (2993ds/3Mi)
% 8.92/5.87 % (16241)Instruction limit reached!
% 8.92/5.87 % (16241)------------------------------
% 8.92/5.87 % (16241)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.87 % (16241)Termination reason: Unknown
% 8.92/5.87 % (16241)Termination phase: shuffling
% 8.92/5.87
% 8.92/5.87 % (16241)Memory used [KB]: 12920
% 8.92/5.87 % (16241)Time elapsed: 0.004 s
% 8.92/5.87 % (16241)Instructions burned: 3 (million)
% 8.92/5.87 % (16241)------------------------------
% 8.92/5.87 % (16241)------------------------------
% 8.92/5.88 % (16242)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_16103 for (2992ds/30Mi)
% 8.92/5.90 % (16242)Instruction limit reached!
% 8.92/5.90 % (16242)------------------------------
% 8.92/5.90 % (16242)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.90 % (16242)Termination reason: Unknown
% 8.92/5.90 % (16242)Termination phase: shuffling
% 8.92/5.90
% 8.92/5.90 % (16242)Memory used [KB]: 13944
% 8.92/5.90 % (16242)Time elapsed: 0.020 s
% 8.92/5.90 % (16242)Instructions burned: 32 (million)
% 8.92/5.90 % (16242)------------------------------
% 8.92/5.90 % (16242)------------------------------
% 8.92/5.91 % (16243)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_16103 for (2992ds/127Mi)
% 8.92/5.94 % (16236)Instruction limit reached!
% 8.92/5.94 % (16236)------------------------------
% 8.92/5.94 % (16236)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.94 % (16236)Termination reason: Unknown
% 8.92/5.94 % (16236)Termination phase: shuffling
% 8.92/5.94
% 8.92/5.94 % (16236)Memory used [KB]: 18677
% 8.92/5.94 % (16236)Time elapsed: 0.144 s
% 8.92/5.94 % (16236)Instructions burned: 378 (million)
% 8.92/5.94 % (16236)------------------------------
% 8.92/5.94 % (16236)------------------------------
% 8.92/5.96 % (16244)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_16103 for (2992ds/100Mi)
% 8.92/5.96 % (16243)Instruction limit reached!
% 8.92/5.96 % (16243)------------------------------
% 8.92/5.96 % (16243)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.96 % (16243)Termination reason: Unknown
% 8.92/5.96 % (16243)Termination phase: shuffling
% 8.92/5.96
% 8.92/5.96 % (16243)Memory used [KB]: 15351
% 8.92/5.96 % (16243)Time elapsed: 0.055 s
% 8.92/5.96 % (16243)Instructions burned: 129 (million)
% 8.92/5.96 % (16243)------------------------------
% 8.92/5.96 % (16243)------------------------------
% 8.92/5.97 % (16245)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_16103 for (2991ds/3Mi)
% 8.92/5.98 % (16245)Instruction limit reached!
% 8.92/5.98 % (16245)------------------------------
% 8.92/5.98 % (16245)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.98 % (16245)Termination reason: Unknown
% 8.92/5.98 % (16245)Termination phase: shuffling
% 8.92/5.98
% 8.92/5.98 % (16245)Memory used [KB]: 12920
% 8.92/5.98 % (16245)Time elapsed: 0.004 s
% 8.92/5.98 % (16245)Instructions burned: 3 (million)
% 8.92/5.98 % (16245)------------------------------
% 8.92/5.98 % (16245)------------------------------
% 8.92/5.99 % (16246)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_16103 for (2991ds/20Mi)
% 8.92/5.99 % (16221)Instruction limit reached!
% 8.92/5.99 % (16221)------------------------------
% 8.92/5.99 % (16221)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/5.99 % (16221)Termination reason: Unknown
% 8.92/5.99 % (16221)Termination phase: shuffling
% 8.92/5.99
% 8.92/5.99 % (16221)Memory used [KB]: 34157
% 8.92/5.99 % (16221)Time elapsed: 0.305 s
% 8.92/5.99 % (16221)Instructions burned: 1043 (million)
% 8.92/5.99 % (16221)------------------------------
% 8.92/5.99 % (16221)------------------------------
% 8.92/6.00 % (16244)Instruction limit reached!
% 8.92/6.00 % (16244)------------------------------
% 8.92/6.00 % (16244)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/6.00 % (16244)Termination reason: Unknown
% 8.92/6.00 % (16244)Termination phase: shuffling
% 8.92/6.00
% 8.92/6.00 % (16244)Memory used [KB]: 14967
% 8.92/6.00 % (16244)Time elapsed: 0.041 s
% 8.92/6.00 % (16244)Instructions burned: 102 (million)
% 8.92/6.00 % (16244)------------------------------
% 8.92/6.00 % (16244)------------------------------
% 8.92/6.00 % (16246)Instruction limit reached!
% 8.92/6.00 % (16246)------------------------------
% 8.92/6.00 % (16246)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.92/6.00 % (16246)Termination reason: Unknown
% 8.92/6.00 % (16246)Termination phase: shuffling
% 8.92/6.00
% 8.92/6.00 % (16246)Memory used [KB]: 13688
% 8.92/6.00 % (16246)Time elapsed: 0.017 s
% 8.92/6.00 % (16246)Instructions burned: 20 (million)
% 8.92/6.00 % (16246)------------------------------
% 8.92/6.00 % (16246)------------------------------
% 8.92/6.01 % (16247)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_16103 for (2991ds/86Mi)
% 8.92/6.01 % (16248)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_16103 for (2991ds/107Mi)
% 11.93/6.02 % (16249)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_16103 for (2991ds/39Mi)
% 11.93/6.03 % (16237)Instruction limit reached!
% 11.93/6.03 % (16237)------------------------------
% 11.93/6.03 % (16237)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.03 % (16237)Termination reason: Unknown
% 11.93/6.03 % (16237)Termination phase: shuffling
% 11.93/6.03
% 11.93/6.03 % (16237)Memory used [KB]: 26481
% 11.93/6.03 % (16237)Time elapsed: 0.231 s
% 11.93/6.03 % (16237)Instructions burned: 780 (million)
% 11.93/6.03 % (16237)------------------------------
% 11.93/6.03 % (16237)------------------------------
% 11.93/6.03 % (16230)Instruction limit reached!
% 11.93/6.03 % (16230)------------------------------
% 11.93/6.03 % (16230)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.03 % (16230)Termination reason: Unknown
% 11.93/6.03 % (16230)Termination phase: shuffling
% 11.93/6.03
% 11.93/6.03 % (16230)Memory used [KB]: 25202
% 11.93/6.03 % (16230)Time elapsed: 0.274 s
% 11.93/6.03 % (16230)Instructions burned: 710 (million)
% 11.93/6.03 % (16230)------------------------------
% 11.93/6.03 % (16230)------------------------------
% 11.93/6.04 % (16249)Instruction limit reached!
% 11.93/6.04 % (16249)------------------------------
% 11.93/6.04 % (16249)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.04 % (16249)Termination reason: Unknown
% 11.93/6.04 % (16249)Termination phase: shuffling
% 11.93/6.04
% 11.93/6.04 % (16249)Memory used [KB]: 14072
% 11.93/6.04 % (16249)Time elapsed: 0.023 s
% 11.93/6.04 % (16249)Instructions burned: 41 (million)
% 11.93/6.04 % (16249)------------------------------
% 11.93/6.04 % (16249)------------------------------
% 11.93/6.04 % (16247)Instruction limit reached!
% 11.93/6.04 % (16247)------------------------------
% 11.93/6.04 % (16247)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.04 % (16247)Termination reason: Unknown
% 11.93/6.04 % (16247)Termination phase: shuffling
% 11.93/6.04
% 11.93/6.04 % (16247)Memory used [KB]: 14711
% 11.93/6.04 % (16247)Time elapsed: 0.038 s
% 11.93/6.04 % (16247)Instructions burned: 87 (million)
% 11.93/6.04 % (16247)------------------------------
% 11.93/6.04 % (16247)------------------------------
% 11.93/6.04 % (16250)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_16103 for (2991ds/448Mi)
% 11.93/6.04 % (16251)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_16103 for (2991ds/46Mi)
% 11.93/6.05 % (16252)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_16103 for (2991ds/98Mi)
% 11.93/6.05 % (16253)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_16103 for (2991ds/507Mi)
% 11.93/6.07 % (16251)Instruction limit reached!
% 11.93/6.07 % (16251)------------------------------
% 11.93/6.07 % (16251)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.07 % (16251)Termination reason: Unknown
% 11.93/6.07 % (16251)Termination phase: shuffling
% 11.93/6.07
% 11.93/6.07 % (16251)Memory used [KB]: 14072
% 11.93/6.07 % (16251)Time elapsed: 0.026 s
% 11.93/6.07 % (16251)Instructions burned: 47 (million)
% 11.93/6.07 % (16251)------------------------------
% 11.93/6.07 % (16251)------------------------------
% 11.93/6.07 % (16248)Instruction limit reached!
% 11.93/6.07 % (16248)------------------------------
% 11.93/6.07 % (16248)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.07 % (16248)Termination reason: Unknown
% 11.93/6.07 % (16248)Termination phase: shuffling
% 11.93/6.07
% 11.93/6.07 % (16248)Memory used [KB]: 15095
% 11.93/6.07 % (16248)Time elapsed: 0.060 s
% 11.93/6.07 % (16248)Instructions burned: 108 (million)
% 11.93/6.07 % (16248)------------------------------
% 11.93/6.07 % (16248)------------------------------
% 11.93/6.08 % (16254)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_16103 for (2990ds/149Mi)
% 11.93/6.08 % (16255)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_16103 for (2990ds/39Mi)
% 11.93/6.09 % (16252)Instruction limit reached!
% 11.93/6.09 % (16252)------------------------------
% 11.93/6.09 % (16252)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.09 % (16252)Termination reason: Unknown
% 11.93/6.09 % (16252)Termination phase: shuffling
% 11.93/6.09
% 11.93/6.09 % (16252)Memory used [KB]: 14967
% 11.93/6.09 % (16252)Time elapsed: 0.040 s
% 11.93/6.09 % (16252)Instructions burned: 100 (million)
% 11.93/6.09 % (16252)------------------------------
% 11.93/6.09 % (16252)------------------------------
% 11.93/6.10 % (16256)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_16103 for (2990ds/6Mi)
% 11.93/6.10 % (16255)Instruction limit reached!
% 11.93/6.10 % (16255)------------------------------
% 11.93/6.10 % (16255)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.10 % (16255)Termination reason: Unknown
% 11.93/6.10 % (16255)Termination phase: shuffling
% 11.93/6.10
% 11.93/6.10 % (16255)Memory used [KB]: 14072
% 11.93/6.10 % (16255)Time elapsed: 0.023 s
% 11.93/6.10 % (16255)Instructions burned: 39 (million)
% 11.93/6.10 % (16255)------------------------------
% 11.93/6.10 % (16255)------------------------------
% 11.93/6.11 % (16232)Instruction limit reached!
% 11.93/6.11 % (16232)------------------------------
% 11.93/6.11 % (16232)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.11 % (16232)Termination reason: Unknown
% 11.93/6.11 % (16232)Termination phase: Property scanning
% 11.93/6.11
% 11.93/6.11 % (16232)Memory used [KB]: 26353
% 11.93/6.11 % (16232)Time elapsed: 0.338 s
% 11.93/6.11 % (16232)Instructions burned: 905 (million)
% 11.93/6.11 % (16232)------------------------------
% 11.93/6.11 % (16232)------------------------------
% 11.93/6.11 % (16256)Instruction limit reached!
% 11.93/6.11 % (16256)------------------------------
% 11.93/6.11 % (16256)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.11 % (16256)Termination reason: Unknown
% 11.93/6.11 % (16256)Termination phase: shuffling
% 11.93/6.11
% 11.93/6.11 % (16256)Memory used [KB]: 13176
% 11.93/6.11 % (16256)Time elapsed: 0.007 s
% 11.93/6.11 % (16256)Instructions burned: 6 (million)
% 11.93/6.11 % (16256)------------------------------
% 11.93/6.11 % (16256)------------------------------
% 11.93/6.12 % (16257)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_16103 for (2990ds/1087Mi)
% 12.46/6.12 % (16258)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_16103 for (2990ds/47Mi)
% 12.46/6.12 % (16259)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_16103 for (2990ds/44Mi)
% 12.46/6.14 % (16259)Instruction limit reached!
% 12.46/6.14 % (16259)------------------------------
% 12.46/6.14 % (16259)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/6.14 % (16259)Termination reason: Unknown
% 12.46/6.14 % (16259)Termination phase: shuffling
% 12.46/6.14
% 12.46/6.14 % (16259)Memory used [KB]: 14072
% 12.46/6.14 % (16259)Time elapsed: 0.021 s
% 12.46/6.14 % (16259)Instructions burned: 45 (million)
% 12.46/6.14 % (16259)------------------------------
% 12.46/6.14 % (16259)------------------------------
% 12.46/6.14 % (16258)Instruction limit reached!
% 12.46/6.14 % (16258)------------------------------
% 12.46/6.14 % (16258)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/6.14 % (16258)Termination reason: Unknown
% 12.46/6.14 % (16258)Termination phase: shuffling
% 12.46/6.14
% 12.46/6.14 % (16258)Memory used [KB]: 14200
% 12.46/6.14 % (16258)Time elapsed: 0.024 s
% 12.46/6.14 % (16258)Instructions burned: 47 (million)
% 12.46/6.14 % (16258)------------------------------
% 12.46/6.14 % (16258)------------------------------
% 12.46/6.14 % (16254)Instruction limit reached!
% 12.46/6.14 % (16254)------------------------------
% 12.46/6.14 % (16254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/6.14 % (16254)Termination reason: Unknown
% 12.46/6.14 % (16254)Termination phase: shuffling
% 12.46/6.14
% 12.46/6.14 % (16254)Memory used [KB]: 15607
% 12.46/6.14 % (16254)Time elapsed: 0.062 s
% 12.46/6.14 % (16254)Instructions burned: 149 (million)
% 12.46/6.14 % (16254)------------------------------
% 12.46/6.14 % (16254)------------------------------
% 12.46/6.15 % (16239)Instruction limit reached!
% 12.46/6.15 % (16239)------------------------------
% 12.46/6.15 % (16239)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/6.15 % (16239)Termination reason: Unknown
% 12.46/6.15 % (16239)Termination phase: Property scanning
% 12.46/6.15
% 12.46/6.15 % (16239)Memory used [KB]: 26225
% 12.46/6.15 % (16239)Time elapsed: 0.322 s
% 12.46/6.15 % (16239)Instructions burned: 881 (million)
% 12.46/6.15 % (16239)------------------------------
% 12.46/6.15 % (16239)------------------------------
% 12.46/6.15 % (16260)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_16103 for (2990ds/432Mi)
% 12.46/6.15 % (16261)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_16103 for (2990ds/62Mi)
% 12.46/6.15 % (16262)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_16103 for (2990ds/31Mi)
% 12.46/6.16 % (16263)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_16103 for (2990ds/96Mi)
% 12.46/6.17 % (16262)Instruction limit reached!
% 12.46/6.17 % (16262)------------------------------
% 12.46/6.17 % (16262)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/6.17 % (16262)Termination reason: Unknown
% 12.46/6.17 % (16262)Termination phase: shuffling
% 12.46/6.17
% 12.46/6.17 % (16262)Memory used [KB]: 13944
% 12.46/6.17 % (16262)Time elapsed: 0.019 s
% 12.46/6.17 % (16262)Instructions burned: 33 (million)
% 12.46/6.17 % (16262)------------------------------
% 12.46/6.17 % (16262)------------------------------
% 12.46/6.18 % (16261)Instruction limit reached!
% 12.46/6.18 % (16261)------------------------------
% 12.46/6.18 % (16261)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/6.18 % (16261)Termination reason: Unknown
% 12.46/6.18 % (16261)Termination phase: shuffling
% 12.46/6.18
% 12.46/6.18 % (16261)Memory used [KB]: 14328
% 12.46/6.18 % (16261)Time elapsed: 0.026 s
% 12.46/6.18 % (16261)Instructions burned: 63 (million)
% 12.46/6.18 % (16261)------------------------------
% 12.46/6.18 % (16261)------------------------------
% 12.46/6.18 % (16250)Instruction limit reached!
% 12.46/6.18 % (16250)------------------------------
% 12.46/6.18 % (16250)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.46/6.18 % (16250)Termination reason: Unknown
% 12.46/6.18 % (16250)Termination phase: shuffling
% 12.46/6.18
% 12.46/6.18 % (16250)Memory used [KB]: 21875
% 12.46/6.18 % (16250)Time elapsed: 0.136 s
% 12.46/6.18 % (16250)Instructions burned: 448 (million)
% 12.46/6.18 % (16250)------------------------------
% 12.46/6.18 % (16250)------------------------------
% 12.46/6.18 % (16264)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_16103 for (2990ds/453Mi)
% 12.46/6.19 % (16265)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_16103 for (2989ds/71Mi)
% 12.46/6.19 % (16266)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_16103 for (2989ds/83Mi)
% 12.85/6.20 % (16263)Instruction limit reached!
% 12.85/6.20 % (16263)------------------------------
% 12.85/6.20 % (16263)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/6.20 % (16263)Termination reason: Unknown
% 12.85/6.20 % (16263)Termination phase: shuffling
% 12.85/6.20
% 12.85/6.20 % (16263)Memory used [KB]: 14839
% 12.85/6.20 % (16263)Time elapsed: 0.041 s
% 12.85/6.20 % (16263)Instructions burned: 98 (million)
% 12.85/6.20 % (16263)------------------------------
% 12.85/6.20 % (16263)------------------------------
% 12.85/6.21 % (16267)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_16103 for (2989ds/11Mi)
% 12.85/6.22 % (16267)Instruction limit reached!
% 12.85/6.22 % (16267)------------------------------
% 12.85/6.22 % (16267)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/6.22 % (16265)Instruction limit reached!
% 12.85/6.22 % (16265)------------------------------
% 12.85/6.22 % (16265)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/6.22 % (16265)Termination reason: Unknown
% 12.85/6.22 % (16265)Termination phase: shuffling
% 12.85/6.22
% 12.85/6.22 % (16265)Memory used [KB]: 14456
% 12.85/6.22 % (16265)Time elapsed: 0.032 s
% 12.85/6.22 % (16265)Instructions burned: 73 (million)
% 12.85/6.22 % (16265)------------------------------
% 12.85/6.22 % (16265)------------------------------
% 12.85/6.22 % (16267)Termination reason: Unknown
% 12.85/6.22 % (16267)Termination phase: shuffling
% 12.85/6.22
% 12.85/6.22 % (16267)Memory used [KB]: 13688
% 12.85/6.22 % (16267)Time elapsed: 0.012 s
% 12.85/6.22 % (16267)Instructions burned: 12 (million)
% 12.85/6.22 % (16267)------------------------------
% 12.85/6.22 % (16267)------------------------------
% 12.85/6.22 % (16266)Instruction limit reached!
% 12.85/6.22 % (16266)------------------------------
% 12.85/6.22 % (16266)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/6.22 % (16266)Termination reason: Unknown
% 12.85/6.22 % (16266)Termination phase: shuffling
% 12.85/6.22
% 12.85/6.22 % (16266)Memory used [KB]: 14711
% 12.85/6.22 % (16266)Time elapsed: 0.035 s
% 12.85/6.22 % (16266)Instructions burned: 84 (million)
% 12.85/6.22 % (16266)------------------------------
% 12.85/6.22 % (16266)------------------------------
% 12.85/6.23 % (16253)Instruction limit reached!
% 12.85/6.23 % (16253)------------------------------
% 12.85/6.23 % (16253)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/6.23 % (16253)Termination reason: Unknown
% 12.85/6.23 % (16253)Termination phase: shuffling
% 12.85/6.23
% 12.85/6.23 % (16253)Memory used [KB]: 20212
% 12.85/6.23 % (16253)Time elapsed: 0.175 s
% 12.85/6.23 % (16253)Instructions burned: 509 (million)
% 12.85/6.23 % (16253)------------------------------
% 12.85/6.23 % (16253)------------------------------
% 12.85/6.23 % (16268)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_16103 for (2989ds/41Mi)
% 12.85/6.23 % (16269)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_16103 for (2989ds/6Mi)
% 12.85/6.23 % (16269)Instruction limit reached!
% 12.85/6.23 % (16269)------------------------------
% 12.85/6.23 % (16269)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/6.23 % (16270)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_16103 for (2989ds/150Mi)
% 12.85/6.23 % (16269)Termination reason: Unknown
% 12.85/6.23 % (16269)Termination phase: shuffling
% 12.85/6.23
% 12.85/6.23 % (16269)Memory used [KB]: 13048
% 12.85/6.23 % (16269)Time elapsed: 0.006 s
% 12.85/6.23 % (16269)Instructions burned: 6 (million)
% 12.85/6.23 % (16269)------------------------------
% 12.85/6.23 % (16269)------------------------------
% 12.85/6.24 % (16271)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_16103 for (2989ds/343Mi)
% 12.85/6.24 % (16272)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_16103 for (2989ds/109Mi)
% 12.85/6.25 % (16268)Instruction limit reached!
% 12.85/6.25 % (16268)------------------------------
% 12.85/6.25 % (16268)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/6.25 % (16268)Termination reason: Unknown
% 12.85/6.25 % (16268)Termination phase: shuffling
% 12.85/6.25
% 12.85/6.25 % (16268)Memory used [KB]: 14072
% 12.85/6.25 % (16268)Time elapsed: 0.021 s
% 12.85/6.25 % (16268)Instructions burned: 41 (million)
% 12.85/6.25 % (16268)------------------------------
% 12.85/6.25 % (16268)------------------------------
% 12.85/6.26 % (16273)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_16103 for (2989ds/253Mi)
% 12.85/6.28 % (16260)Instruction limit reached!
% 12.85/6.28 % (16260)------------------------------
% 12.85/6.28 % (16260)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/6.28 % (16260)Termination reason: Unknown
% 12.85/6.28 % (16260)Termination phase: shuffling
% 12.85/6.28
% 12.85/6.28 % (16260)Memory used [KB]: 21748
% 12.85/6.28 % (16260)Time elapsed: 0.132 s
% 12.85/6.28 % (16260)Instructions burned: 433 (million)
% 12.85/6.28 % (16260)------------------------------
% 12.85/6.28 % (16260)------------------------------
% 12.85/6.29 % (16272)Instruction limit reached!
% 12.85/6.29 % (16272)------------------------------
% 12.85/6.29 % (16272)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/6.29 % (16272)Termination reason: Unknown
% 12.85/6.29 % (16272)Termination phase: shuffling
% 12.85/6.29
% 12.85/6.29 % (16272)Memory used [KB]: 14967
% 12.85/6.29 % (16272)Time elapsed: 0.044 s
% 12.85/6.29 % (16272)Instructions burned: 109 (million)
% 12.85/6.29 % (16272)------------------------------
% 12.85/6.29 % (16272)------------------------------
% 12.85/6.29 % (16270)Instruction limit reached!
% 12.85/6.29 % (16270)------------------------------
% 12.85/6.29 % (16270)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.85/6.29 % (16270)Termination reason: Unknown
% 12.85/6.29 % (16270)Termination phase: shuffling
% 12.85/6.29
% 12.85/6.29 % (16270)Memory used [KB]: 15607
% 12.85/6.29 % (16270)Time elapsed: 0.058 s
% 12.85/6.29 % (16270)Instructions burned: 152 (million)
% 12.85/6.29 % (16270)------------------------------
% 12.85/6.29 % (16270)------------------------------
% 12.85/6.29 % (16274)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_16103 for (2988ds/140Mi)
% 13.50/6.30 % (16275)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_16103 for (2988ds/614Mi)
% 13.50/6.30 % (16276)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_16103 for (2988ds/202Mi)
% 13.50/6.32 % (16264)Instruction limit reached!
% 13.50/6.32 % (16264)------------------------------
% 13.50/6.32 % (16264)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.50/6.32 % (16264)Termination reason: Unknown
% 13.50/6.32 % (16264)Termination phase: shuffling
% 13.50/6.32
% 13.50/6.32 % (16264)Memory used [KB]: 22003
% 13.50/6.32 % (16264)Time elapsed: 0.140 s
% 13.50/6.32 % (16264)Instructions burned: 454 (million)
% 13.50/6.32 % (16264)------------------------------
% 13.50/6.32 % (16264)------------------------------
% 13.50/6.33 % (16277)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_16103 for (2988ds/339Mi)
% 13.50/6.34 % (16274)Instruction limit reached!
% 13.50/6.34 % (16274)------------------------------
% 13.50/6.34 % (16274)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.50/6.34 % (16274)Termination reason: Unknown
% 13.50/6.34 % (16274)Termination phase: shuffling
% 13.50/6.34
% 13.50/6.34 % (16274)Memory used [KB]: 15479
% 13.50/6.34 % (16274)Time elapsed: 0.054 s
% 13.50/6.34 % (16274)Instructions burned: 140 (million)
% 13.50/6.34 % (16274)------------------------------
% 13.50/6.34 % (16274)------------------------------
% 13.50/6.35 % (16273)Instruction limit reached!
% 13.50/6.35 % (16273)------------------------------
% 13.50/6.35 % (16273)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.50/6.35 % (16273)Termination reason: Unknown
% 13.50/6.35 % (16273)Termination phase: shuffling
% 13.50/6.35
% 13.50/6.35 % (16273)Memory used [KB]: 17142
% 13.50/6.35 % (16273)Time elapsed: 0.093 s
% 13.50/6.35 % (16273)Instructions burned: 254 (million)
% 13.50/6.35 % (16273)------------------------------
% 13.50/6.35 % (16273)------------------------------
% 13.50/6.36 % (16278)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_16103 for (2988ds/283Mi)
% 13.50/6.36 % (16271)Instruction limit reached!
% 13.50/6.36 % (16271)------------------------------
% 13.50/6.36 % (16271)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.50/6.36 % (16271)Termination reason: Unknown
% 13.50/6.36 % (16271)Termination phase: shuffling
% 13.50/6.36
% 13.50/6.36 % (16271)Memory used [KB]: 18293
% 13.50/6.36 % (16271)Time elapsed: 0.121 s
% 13.50/6.36 % (16271)Instructions burned: 344 (million)
% 13.50/6.36 % (16271)------------------------------
% 13.50/6.36 % (16271)------------------------------
% 13.50/6.36 % (16279)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_16103 for (2988ds/134Mi)
% 13.50/6.37 % (16280)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_16103 for (2987ds/298Mi)
% 13.50/6.38 % (16276)Instruction limit reached!
% 13.50/6.38 % (16276)------------------------------
% 13.50/6.38 % (16276)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.50/6.38 % (16276)Termination reason: Unknown
% 13.50/6.38 % (16276)Termination phase: shuffling
% 13.50/6.38
% 13.50/6.38 % (16276)Memory used [KB]: 16375
% 13.50/6.38 % (16276)Time elapsed: 0.076 s
% 13.50/6.38 % (16276)Instructions burned: 202 (million)
% 13.50/6.38 % (16276)------------------------------
% 13.50/6.38 % (16276)------------------------------
% 13.50/6.39 % (16281)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_16103 for (2987ds/192Mi)
% 14.03/6.41 % (16279)Instruction limit reached!
% 14.03/6.41 % (16279)------------------------------
% 14.03/6.41 % (16279)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.03/6.41 % (16279)Termination reason: Unknown
% 14.03/6.41 % (16279)Termination phase: shuffling
% 14.03/6.41
% 14.03/6.41 % (16279)Memory used [KB]: 15351
% 14.03/6.41 % (16279)Time elapsed: 0.052 s
% 14.03/6.41 % (16279)Instructions burned: 134 (million)
% 14.03/6.41 % (16279)------------------------------
% 14.03/6.41 % (16279)------------------------------
% 14.03/6.42 % (16282)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_16103 for (2987ds/258Mi)
% 14.03/6.44 % (16278)Instruction limit reached!
% 14.03/6.44 % (16278)------------------------------
% 14.03/6.44 % (16278)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.03/6.44 % (16278)Termination reason: Unknown
% 14.03/6.44 % (16278)Termination phase: shuffling
% 14.03/6.44
% 14.03/6.44 % (16278)Memory used [KB]: 19701
% 14.03/6.44 % (16278)Time elapsed: 0.089 s
% 14.03/6.44 % (16278)Instructions burned: 284 (million)
% 14.03/6.44 % (16278)------------------------------
% 14.03/6.44 % (16278)------------------------------
% 14.03/6.45 % (16277)Instruction limit reached!
% 14.03/6.45 % (16277)------------------------------
% 14.03/6.45 % (16277)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.03/6.45 % (16277)Termination reason: Unknown
% 14.03/6.45 % (16277)Termination phase: shuffling
% 14.03/6.45
% 14.03/6.45 % (16277)Memory used [KB]: 18166
% 14.03/6.45 % (16277)Time elapsed: 0.121 s
% 14.03/6.45 % (16277)Instructions burned: 339 (million)
% 14.03/6.45 % (16277)------------------------------
% 14.03/6.45 % (16277)------------------------------
% 14.03/6.46 % (16283)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_16103 for (2987ds/90Mi)
% 14.03/6.46 % (16281)Instruction limit reached!
% 14.03/6.46 % (16281)------------------------------
% 14.03/6.46 % (16281)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.03/6.46 % (16281)Termination reason: Unknown
% 14.03/6.46 % (16281)Termination phase: shuffling
% 14.03/6.46
% 14.03/6.46 % (16281)Memory used [KB]: 16119
% 14.03/6.46 % (16281)Time elapsed: 0.072 s
% 14.03/6.46 % (16281)Instructions burned: 194 (million)
% 14.03/6.46 % (16281)------------------------------
% 14.03/6.46 % (16281)------------------------------
% 14.03/6.46 % (16280)Instruction limit reached!
% 14.03/6.46 % (16280)------------------------------
% 14.03/6.46 % (16280)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.03/6.46 % (16280)Termination reason: Unknown
% 14.03/6.46 % (16280)Termination phase: shuffling
% 14.03/6.46
% 14.03/6.46 % (16280)Memory used [KB]: 19829
% 14.03/6.46 % (16280)Time elapsed: 0.093 s
% 14.03/6.46 % (16280)Instructions burned: 300 (million)
% 14.03/6.46 % (16280)------------------------------
% 14.03/6.46 % (16280)------------------------------
% 14.03/6.46 % (16284)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_16103 for (2987ds/231Mi)
% 14.03/6.47 % (16257)Instruction limit reached!
% 14.03/6.47 % (16257)------------------------------
% 14.03/6.47 % (16257)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.03/6.47 % (16257)Termination reason: Unknown
% 14.03/6.47 % (16257)Termination phase: Property scanning
% 14.03/6.47
% 14.03/6.47 % (16257)Memory used [KB]: 26353
% 14.03/6.47 % (16257)Time elapsed: 0.354 s
% 14.03/6.47 % (16257)Instructions burned: 1087 (million)
% 14.03/6.47 % (16257)------------------------------
% 14.03/6.47 % (16257)------------------------------
% 14.03/6.47 % (16285)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_16103 for (2987ds/458Mi)
% 14.03/6.47 % (16286)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_16103 for (2987ds/31Mi)
% 14.03/6.48 % (16287)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_16103 for (2986ds/297Mi)
% 14.03/6.49 % (16286)Instruction limit reached!
% 14.03/6.49 % (16286)------------------------------
% 14.03/6.49 % (16286)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.03/6.49 % (16286)Termination reason: Unknown
% 14.03/6.49 % (16286)Termination phase: shuffling
% 14.03/6.49
% 14.03/6.49 % (16286)Memory used [KB]: 13944
% 14.03/6.49 % (16286)Time elapsed: 0.019 s
% 14.03/6.49 % (16286)Instructions burned: 32 (million)
% 14.03/6.49 % (16286)------------------------------
% 14.03/6.49 % (16286)------------------------------
% 14.03/6.49 % (16283)Instruction limit reached!
% 14.03/6.49 % (16283)------------------------------
% 14.03/6.49 % (16283)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.03/6.49 % (16283)Termination reason: Unknown
% 14.03/6.49 % (16283)Termination phase: shuffling
% 14.03/6.49
% 14.03/6.49 % (16283)Memory used [KB]: 14839
% 14.03/6.49 % (16283)Time elapsed: 0.039 s
% 14.03/6.49 % (16283)Instructions burned: 92 (million)
% 14.03/6.49 % (16283)------------------------------
% 14.03/6.49 % (16283)------------------------------
% 14.95/6.50 % (16288)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_16103 for (2986ds/987Mi)
% 14.95/6.50 % (16289)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_16103 for (2986ds/182Mi)
% 14.95/6.51 % (16275)Instruction limit reached!
% 14.95/6.51 % (16275)------------------------------
% 14.95/6.51 % (16275)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.95/6.51 % (16275)Termination reason: Unknown
% 14.95/6.51 % (16275)Termination phase: shuffling
% 14.95/6.51
% 14.95/6.51 % (16275)Memory used [KB]: 23794
% 14.95/6.51 % (16275)Time elapsed: 0.213 s
% 14.95/6.51 % (16275)Instructions burned: 614 (million)
% 14.95/6.51 % (16275)------------------------------
% 14.95/6.51 % (16275)------------------------------
% 14.95/6.52 % (16282)Instruction limit reached!
% 14.95/6.52 % (16282)------------------------------
% 14.95/6.52 % (16282)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.95/6.52 % (16282)Termination reason: Unknown
% 14.95/6.52 % (16282)Termination phase: shuffling
% 14.95/6.52
% 14.95/6.52 % (16282)Memory used [KB]: 17142
% 14.95/6.52 % (16282)Time elapsed: 0.094 s
% 14.95/6.52 % (16282)Instructions burned: 258 (million)
% 14.95/6.52 % (16282)------------------------------
% 14.95/6.52 % (16282)------------------------------
% 14.95/6.52 % (16290)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_16103 for (2986ds/182Mi)
% 14.95/6.53 % (16291)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_16103 for (2986ds/154Mi)
% 14.95/6.54 % (16284)Instruction limit reached!
% 14.95/6.54 % (16284)------------------------------
% 14.95/6.54 % (16284)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.95/6.54 % (16284)Termination reason: Unknown
% 14.95/6.54 % (16284)Termination phase: shuffling
% 14.95/6.54
% 14.95/6.54 % (16284)Memory used [KB]: 16886
% 14.95/6.54 % (16284)Time elapsed: 0.075 s
% 14.95/6.54 % (16284)Instructions burned: 232 (million)
% 14.95/6.54 % (16284)------------------------------
% 14.95/6.54 % (16284)------------------------------
% 14.95/6.55 % (16292)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_16103 for (2986ds/79Mi)
% 14.95/6.57 % (16289)Instruction limit reached!
% 14.95/6.57 % (16289)------------------------------
% 14.95/6.57 % (16289)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.95/6.57 % (16289)Termination reason: Unknown
% 14.95/6.57 % (16289)Termination phase: shuffling
% 14.95/6.57
% 14.95/6.57 % (16289)Memory used [KB]: 16119
% 14.95/6.57 % (16289)Time elapsed: 0.068 s
% 14.95/6.57 % (16289)Instructions burned: 182 (million)
% 14.95/6.57 % (16289)------------------------------
% 14.95/6.57 % (16289)------------------------------
% 14.95/6.57 % (16287)Instruction limit reached!
% 14.95/6.57 % (16287)------------------------------
% 14.95/6.57 % (16287)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.95/6.57 % (16287)Termination reason: Unknown
% 14.95/6.57 % (16287)Termination phase: shuffling
% 14.95/6.57
% 14.95/6.57 % (16287)Memory used [KB]: 19829
% 14.95/6.57 % (16287)Time elapsed: 0.093 s
% 14.95/6.57 % (16287)Instructions burned: 297 (million)
% 14.95/6.57 % (16287)------------------------------
% 14.95/6.57 % (16287)------------------------------
% 14.95/6.58 % (16291)Instruction limit reached!
% 14.95/6.58 % (16291)------------------------------
% 14.95/6.58 % (16291)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.95/6.58 % (16291)Termination reason: Unknown
% 14.95/6.58 % (16291)Termination phase: shuffling
% 14.95/6.58
% 14.95/6.58 % (16291)Memory used [KB]: 15735
% 14.95/6.58 % (16291)Time elapsed: 0.052 s
% 14.95/6.58 % (16291)Instructions burned: 155 (million)
% 14.95/6.58 % (16291)------------------------------
% 14.95/6.58 % (16291)------------------------------
% 14.95/6.58 % (16292)Instruction limit reached!
% 14.95/6.58 % (16292)------------------------------
% 14.95/6.58 % (16292)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.95/6.58 % (16292)Termination reason: Unknown
% 14.95/6.58 % (16292)Termination phase: shuffling
% 14.95/6.58
% 14.95/6.58 % (16292)Memory used [KB]: 14583
% 14.95/6.58 % (16292)Time elapsed: 0.035 s
% 14.95/6.58 % (16292)Instructions burned: 79 (million)
% 14.95/6.58 % (16292)------------------------------
% 14.95/6.58 % (16292)------------------------------
% 14.95/6.58 % (16293)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_16103 for (2985ds/216Mi)
% 14.95/6.58 % (16294)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_16103 for (2985ds/114Mi)
% 14.95/6.59 % (16290)Instruction limit reached!
% 14.95/6.59 % (16290)------------------------------
% 14.95/6.59 % (16290)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.95/6.59 % (16290)Termination reason: Unknown
% 14.95/6.59 % (16290)Termination phase: shuffling
% 14.95/6.59
% 14.95/6.59 % (16290)Memory used [KB]: 16119
% 14.95/6.59 % (16290)Time elapsed: 0.069 s
% 14.95/6.59 % (16290)Instructions burned: 182 (million)
% 14.95/6.59 % (16290)------------------------------
% 14.95/6.59 % (16290)------------------------------
% 14.95/6.59 % (16295)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_16103 for (2985ds/922Mi)
% 14.95/6.59 % (16296)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_16103 for (2985ds/182Mi)
% 15.73/6.60 % (16297)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_16103 for (2985ds/164Mi)
% 15.73/6.63 % (16294)Instruction limit reached!
% 15.73/6.63 % (16294)------------------------------
% 15.73/6.63 % (16294)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.73/6.63 % (16294)Termination reason: Unknown
% 15.73/6.63 % (16294)Termination phase: shuffling
% 15.73/6.63
% 15.73/6.63 % (16294)Memory used [KB]: 15095
% 15.73/6.63 % (16294)Time elapsed: 0.045 s
% 15.73/6.63 % (16294)Instructions burned: 114 (million)
% 15.73/6.63 % (16294)------------------------------
% 15.73/6.63 % (16294)------------------------------
% 15.73/6.63 % (16285)Instruction limit reached!
% 15.73/6.63 % (16285)------------------------------
% 15.73/6.63 % (16285)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.73/6.63 % (16285)Termination reason: Unknown
% 15.73/6.63 % (16285)Termination phase: shuffling
% 15.73/6.63
% 15.73/6.63 % (16285)Memory used [KB]: 19701
% 15.73/6.63 % (16285)Time elapsed: 0.160 s
% 15.73/6.63 % (16285)Instructions burned: 458 (million)
% 15.73/6.63 % (16285)------------------------------
% 15.73/6.63 % (16285)------------------------------
% 15.73/6.64 % (16298)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_16103 for (2985ds/50Mi)
% 15.73/6.64 % (16299)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_16103 for (2985ds/368Mi)
% 15.73/6.65 % (16293)Instruction limit reached!
% 15.73/6.65 % (16293)------------------------------
% 15.73/6.65 % (16293)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.73/6.65 % (16293)Termination reason: Unknown
% 15.73/6.65 % (16293)Termination phase: shuffling
% 15.73/6.65
% 15.73/6.65 % (16293)Memory used [KB]: 16630
% 15.73/6.65 % (16293)Time elapsed: 0.070 s
% 15.73/6.65 % (16293)Instructions burned: 218 (million)
% 15.73/6.65 % (16293)------------------------------
% 15.73/6.65 % (16293)------------------------------
% 15.73/6.65 % (16296)Instruction limit reached!
% 15.73/6.65 % (16296)------------------------------
% 15.73/6.65 % (16296)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.73/6.65 % (16296)Termination reason: Unknown
% 15.73/6.65 % (16296)Termination phase: shuffling
% 15.73/6.65
% 15.73/6.65 % (16296)Memory used [KB]: 16119
% 15.73/6.65 % (16296)Time elapsed: 0.060 s
% 15.73/6.65 % (16296)Instructions burned: 182 (million)
% 15.73/6.65 % (16296)------------------------------
% 15.73/6.65 % (16296)------------------------------
% 15.73/6.65 % (16297)Instruction limit reached!
% 15.73/6.65 % (16297)------------------------------
% 15.73/6.65 % (16297)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.73/6.65 % (16297)Termination reason: Unknown
% 15.73/6.65 % (16297)Termination phase: shuffling
% 15.73/6.65
% 15.73/6.65 % (16297)Memory used [KB]: 15991
% 15.73/6.65 % (16297)Time elapsed: 0.055 s
% 15.73/6.65 % (16297)Instructions burned: 164 (million)
% 15.73/6.65 % (16297)------------------------------
% 15.73/6.65 % (16297)------------------------------
% 15.73/6.66 % (16298)Instruction limit reached!
% 15.73/6.66 % (16298)------------------------------
% 15.73/6.66 % (16298)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.73/6.66 % (16298)Termination reason: Unknown
% 15.73/6.66 % (16298)Termination phase: shuffling
% 15.73/6.66
% 15.73/6.66 % (16298)Memory used [KB]: 14328
% 15.73/6.66 % (16298)Time elapsed: 0.024 s
% 15.73/6.66 % (16298)Instructions burned: 53 (million)
% 15.73/6.66 % (16298)------------------------------
% 15.73/6.66 % (16298)------------------------------
% 15.73/6.66 % (16300)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_16103 for (2985ds/1021Mi)
% 15.73/6.66 % (16301)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_16103 for (2985ds/267Mi)
% 15.73/6.66 % (16302)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_16103 for (2985ds/204Mi)
% 15.73/6.67 % (16303)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_16103 for (2985ds/26Mi)
% 15.73/6.69 % (16303)Instruction limit reached!
% 15.73/6.69 % (16303)------------------------------
% 15.73/6.69 % (16303)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.73/6.69 % (16303)Termination reason: Unknown
% 15.73/6.69 % (16303)Termination phase: shuffling
% 15.73/6.69
% 15.73/6.69 % (16303)Memory used [KB]: 13816
% 15.73/6.69 % (16303)Time elapsed: 0.017 s
% 15.73/6.69 % (16303)Instructions burned: 27 (million)
% 15.73/6.69 % (16303)------------------------------
% 15.73/6.69 % (16303)------------------------------
% 16.23/6.70 % (16304)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_16103 for (2984ds/53Mi)
% 16.23/6.72 % (16304)Instruction limit reached!
% 16.23/6.72 % (16304)------------------------------
% 16.23/6.72 % (16304)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.23/6.72 % (16304)Termination reason: Unknown
% 16.23/6.72 % (16304)Termination phase: shuffling
% 16.23/6.72
% 16.23/6.72 % (16304)Memory used [KB]: 14200
% 16.23/6.72 % (16304)Time elapsed: 0.024 s
% 16.23/6.72 % (16304)Instructions burned: 53 (million)
% 16.23/6.72 % (16304)------------------------------
% 16.23/6.72 % (16304)------------------------------
% 16.23/6.73 % (16305)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_16103 for (2984ds/4436Mi)
% 16.23/6.74 % (16302)Instruction limit reached!
% 16.23/6.74 % (16302)------------------------------
% 16.23/6.74 % (16302)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.23/6.74 % (16302)Termination reason: Unknown
% 16.23/6.74 % (16302)Termination phase: shuffling
% 16.23/6.74
% 16.23/6.74 % (16302)Memory used [KB]: 16502
% 16.23/6.74 % (16302)Time elapsed: 0.075 s
% 16.23/6.74 % (16302)Instructions burned: 205 (million)
% 16.23/6.74 % (16302)------------------------------
% 16.23/6.74 % (16302)------------------------------
% 16.23/6.75 % (16301)Instruction limit reached!
% 16.23/6.75 % (16301)------------------------------
% 16.23/6.75 % (16301)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.23/6.75 % (16301)Termination reason: Unknown
% 16.23/6.75 % (16301)Termination phase: shuffling
% 16.23/6.75
% 16.23/6.75 % (16301)Memory used [KB]: 19445
% 16.23/6.75 % (16301)Time elapsed: 0.085 s
% 16.23/6.75 % (16301)Instructions burned: 268 (million)
% 16.23/6.75 % (16301)------------------------------
% 16.23/6.75 % (16301)------------------------------
% 16.23/6.75 % (16306)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_16103 for (2984ds/1539Mi)
% 16.23/6.76 % (16307)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_16103 for (2984ds/408Mi)
% 16.23/6.77 % (16299)Instruction limit reached!
% 16.23/6.77 % (16299)------------------------------
% 16.23/6.77 % (16299)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.23/6.77 % (16299)Termination reason: Unknown
% 16.23/6.77 % (16299)Termination phase: shuffling
% 16.23/6.77
% 16.23/6.77 % (16299)Memory used [KB]: 18549
% 16.23/6.77 % (16299)Time elapsed: 0.129 s
% 16.23/6.77 % (16299)Instructions burned: 368 (million)
% 16.23/6.77 % (16299)------------------------------
% 16.23/6.77 % (16299)------------------------------
% 16.23/6.78 % (16308)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_16103 for (2983ds/5205Mi)
% 16.23/6.79 % (16288)Instruction limit reached!
% 16.23/6.79 % (16288)------------------------------
% 16.23/6.79 % (16288)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.23/6.79 % (16288)Termination reason: Unknown
% 16.23/6.79 % (16288)Termination phase: shuffling
% 16.23/6.79
% 16.23/6.79 % (16288)Memory used [KB]: 33389
% 16.23/6.79 % (16288)Time elapsed: 0.293 s
% 16.23/6.79 % (16288)Instructions burned: 989 (million)
% 16.23/6.79 % (16288)------------------------------
% 16.23/6.79 % (16288)------------------------------
% 16.88/6.80 % (16309)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_16103 for (2983ds/1913Mi)
% 16.88/6.90 % (16295)Instruction limit reached!
% 16.88/6.90 % (16295)------------------------------
% 16.88/6.90 % (16295)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.88/6.90 % (16295)Termination reason: Unknown
% 16.88/6.90 % (16295)Termination phase: Property scanning
% 16.88/6.90
% 16.88/6.90 % (16295)Memory used [KB]: 26353
% 16.88/6.90 % (16295)Time elapsed: 0.310 s
% 16.88/6.90 % (16295)Instructions burned: 925 (million)
% 16.88/6.90 % (16295)------------------------------
% 16.88/6.90 % (16295)------------------------------
% 16.88/6.90 % (16307)Instruction limit reached!
% 16.88/6.90 % (16307)------------------------------
% 16.88/6.90 % (16307)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.88/6.90 % (16307)Termination reason: Unknown
% 16.88/6.90 % (16307)Termination phase: shuffling
% 16.88/6.90
% 16.88/6.90 % (16307)Memory used [KB]: 19317
% 16.88/6.90 % (16307)Time elapsed: 0.146 s
% 16.88/6.90 % (16307)Instructions burned: 410 (million)
% 16.88/6.90 % (16307)------------------------------
% 16.88/6.90 % (16307)------------------------------
% 16.88/6.91 % (16310)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_16103 for (2982ds/4350Mi)
% 16.88/6.91 % (16311)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_16103 for (2982ds/68Mi)
% 16.88/6.94 % (16311)Instruction limit reached!
% 16.88/6.94 % (16311)------------------------------
% 16.88/6.94 % (16311)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.88/6.94 % (16311)Termination reason: Unknown
% 16.88/6.94 % (16311)Termination phase: shuffling
% 16.88/6.94
% 16.88/6.94 % (16311)Memory used [KB]: 14456
% 16.88/6.94 % (16311)Time elapsed: 0.031 s
% 16.88/6.94 % (16311)Instructions burned: 68 (million)
% 16.88/6.94 % (16311)------------------------------
% 16.88/6.94 % (16311)------------------------------
% 16.88/6.95 % (16312)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_16103 for (2982ds/358Mi)
% 16.88/6.99 % (16300)Instruction limit reached!
% 16.88/6.99 % (16300)------------------------------
% 16.88/6.99 % (16300)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.88/6.99 % (16300)Termination reason: Unknown
% 16.88/6.99 % (16300)Termination phase: Property scanning
% 16.88/6.99
% 16.88/6.99 % (16300)Memory used [KB]: 26353
% 16.88/6.99 % (16300)Time elapsed: 0.335 s
% 16.88/6.99 % (16300)Instructions burned: 1024 (million)
% 16.88/6.99 % (16300)------------------------------
% 16.88/6.99 % (16300)------------------------------
% 16.88/7.01 % (16313)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_16103 for (2981ds/31Mi)
% 18.87/7.03 % (16313)Instruction limit reached!
% 18.87/7.03 % (16313)------------------------------
% 18.87/7.03 % (16313)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.87/7.03 % (16313)Termination reason: Unknown
% 18.87/7.03 % (16313)Termination phase: shuffling
% 18.87/7.03
% 18.87/7.03 % (16313)Memory used [KB]: 13944
% 18.87/7.03 % (16313)Time elapsed: 0.020 s
% 18.87/7.03 % (16313)Instructions burned: 33 (million)
% 18.87/7.03 % (16313)------------------------------
% 18.87/7.03 % (16313)------------------------------
% 18.87/7.04 % (16314)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_16103 for (2981ds/279Mi)
% 18.87/7.09 % (16312)Instruction limit reached!
% 18.87/7.09 % (16312)------------------------------
% 18.87/7.09 % (16312)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.87/7.09 % (16312)Termination reason: Unknown
% 18.87/7.09 % (16312)Termination phase: shuffling
% 18.87/7.09
% 18.87/7.09 % (16312)Memory used [KB]: 18421
% 18.87/7.09 % (16312)Time elapsed: 0.128 s
% 18.87/7.09 % (16312)Instructions burned: 358 (million)
% 18.87/7.09 % (16312)------------------------------
% 18.87/7.09 % (16312)------------------------------
% 18.87/7.10 % (16315)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_16103 for (2980ds/258Mi)
% 19.51/7.12 % (16314)Instruction limit reached!
% 19.51/7.12 % (16314)------------------------------
% 19.51/7.12 % (16314)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.51/7.12 % (16314)Termination reason: Unknown
% 19.51/7.12 % (16314)Termination phase: shuffling
% 19.51/7.12
% 19.51/7.12 % (16314)Memory used [KB]: 19573
% 19.51/7.12 % (16314)Time elapsed: 0.088 s
% 19.51/7.12 % (16314)Instructions burned: 281 (million)
% 19.51/7.12 % (16314)------------------------------
% 19.51/7.12 % (16314)------------------------------
% 19.51/7.14 % (16316)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_16103 for (2980ds/26Mi)
% 19.51/7.15 % (16316)Instruction limit reached!
% 19.51/7.15 % (16316)------------------------------
% 19.51/7.15 % (16316)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.51/7.15 % (16316)Termination reason: Unknown
% 19.51/7.15 % (16316)Termination phase: shuffling
% 19.51/7.15
% 19.51/7.15 % (16316)Memory used [KB]: 13816
% 19.51/7.15 % (16316)Time elapsed: 0.017 s
% 19.51/7.15 % (16316)Instructions burned: 27 (million)
% 19.51/7.15 % (16316)------------------------------
% 19.51/7.15 % (16316)------------------------------
% 19.51/7.16 % (16317)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_16103 for (2980ds/995Mi)
% 19.51/7.19 % (16315)Instruction limit reached!
% 19.51/7.19 % (16315)------------------------------
% 19.51/7.19 % (16315)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.51/7.19 % (16315)Termination reason: Unknown
% 19.51/7.19 % (16315)Termination phase: shuffling
% 19.51/7.19
% 19.51/7.19 % (16315)Memory used [KB]: 17142
% 19.51/7.19 % (16315)Time elapsed: 0.093 s
% 19.51/7.19 % (16315)Instructions burned: 258 (million)
% 19.51/7.19 % (16315)------------------------------
% 19.51/7.19 % (16315)------------------------------
% 20.14/7.20 % (16318)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_16103 for (2979ds/373Mi)
% 20.14/7.22 % (16306)Instruction limit reached!
% 20.14/7.22 % (16306)------------------------------
% 20.14/7.22 % (16306)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.14/7.22 % (16306)Termination reason: Unknown
% 20.14/7.22 % (16306)Termination phase: SInE selection
% 20.14/7.22
% 20.14/7.22 % (16306)Memory used [KB]: 26353
% 20.14/7.22 % (16306)Time elapsed: 0.474 s
% 20.14/7.22 % (16306)Instructions burned: 1539 (million)
% 20.14/7.22 % (16306)------------------------------
% 20.14/7.22 % (16306)------------------------------
% 20.14/7.23 % (16319)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_16103 for (2979ds/253Mi)
% 20.58/7.31 % (16319)Instruction limit reached!
% 20.58/7.31 % (16319)------------------------------
% 20.58/7.31 % (16319)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.58/7.31 % (16319)Termination reason: Unknown
% 20.58/7.31 % (16319)Termination phase: shuffling
% 20.58/7.31
% 20.58/7.31 % (16319)Memory used [KB]: 19189
% 20.58/7.31 % (16319)Time elapsed: 0.081 s
% 20.58/7.31 % (16319)Instructions burned: 254 (million)
% 20.58/7.31 % (16319)------------------------------
% 20.58/7.31 % (16319)------------------------------
% 20.58/7.33 % (16320)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_16103 for (2978ds/832Mi)
% 20.58/7.33 % (16318)Instruction limit reached!
% 20.58/7.33 % (16318)------------------------------
% 20.58/7.33 % (16318)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.58/7.33 % (16318)Termination reason: Unknown
% 20.58/7.33 % (16318)Termination phase: shuffling
% 20.58/7.33
% 20.58/7.33 % (16318)Memory used [KB]: 18677
% 20.58/7.33 % (16318)Time elapsed: 0.131 s
% 20.58/7.33 % (16318)Instructions burned: 375 (million)
% 20.58/7.33 % (16318)------------------------------
% 20.58/7.33 % (16318)------------------------------
% 20.58/7.34 % (16321)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_16103 for (2978ds/668Mi)
% 20.58/7.40 % (16309)Instruction limit reached!
% 20.58/7.40 % (16309)------------------------------
% 20.58/7.40 % (16309)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.58/7.40 % (16309)Termination reason: Unknown
% 20.58/7.40 % (16309)Termination phase: SInE selection
% 20.58/7.40
% 20.58/7.40 % (16309)Memory used [KB]: 27121
% 20.58/7.40 % (16309)Time elapsed: 0.593 s
% 20.58/7.40 % (16309)Instructions burned: 1915 (million)
% 20.58/7.40 % (16309)------------------------------
% 20.58/7.40 % (16309)------------------------------
% 20.61/7.41 % (16322)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_16103 for (2977ds/40Mi)
% 20.61/7.43 % (16322)Instruction limit reached!
% 20.61/7.43 % (16322)------------------------------
% 20.61/7.43 % (16322)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.61/7.43 % (16322)Termination reason: Unknown
% 20.61/7.43 % (16322)Termination phase: shuffling
% 20.61/7.43
% 20.61/7.43 % (16322)Memory used [KB]: 14072
% 20.61/7.43 % (16322)Time elapsed: 0.020 s
% 20.61/7.43 % (16322)Instructions burned: 40 (million)
% 20.61/7.43 % (16322)------------------------------
% 20.61/7.43 % (16322)------------------------------
% 20.61/7.44 % (16323)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_16103 for (2977ds/294Mi)
% 20.61/7.45 % (16317)Instruction limit reached!
% 20.61/7.45 % (16317)------------------------------
% 20.61/7.45 % (16317)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.61/7.45 % (16317)Termination reason: Unknown
% 20.61/7.45 % (16317)Termination phase: shuffling
% 20.61/7.45
% 20.61/7.45 % (16317)Memory used [KB]: 33389
% 20.61/7.45 % (16317)Time elapsed: 0.291 s
% 20.61/7.45 % (16317)Instructions burned: 996 (million)
% 20.61/7.45 % (16317)------------------------------
% 20.61/7.45 % (16317)------------------------------
% 20.61/7.46 % (16324)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_16103 for (2977ds/187Mi)
% 21.85/7.53 % (16323)Instruction limit reached!
% 21.85/7.53 % (16323)------------------------------
% 21.85/7.53 % (16323)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.85/7.53 % (16323)Termination reason: Unknown
% 21.85/7.53 % (16323)Termination phase: shuffling
% 21.85/7.53
% 21.85/7.53 % (16323)Memory used [KB]: 19829
% 21.85/7.53 % (16323)Time elapsed: 0.091 s
% 21.85/7.53 % (16323)Instructions burned: 295 (million)
% 21.85/7.53 % (16323)------------------------------
% 21.85/7.53 % (16323)------------------------------
% 21.85/7.53 % (16324)Instruction limit reached!
% 21.85/7.53 % (16324)------------------------------
% 21.85/7.53 % (16324)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.85/7.53 % (16324)Termination reason: Unknown
% 21.85/7.53 % (16324)Termination phase: shuffling
% 21.85/7.53
% 21.85/7.53 % (16324)Memory used [KB]: 16119
% 21.85/7.53 % (16324)Time elapsed: 0.070 s
% 21.85/7.53 % (16324)Instructions burned: 188 (million)
% 21.85/7.53 % (16324)------------------------------
% 21.85/7.53 % (16324)------------------------------
% 21.85/7.54 % (16321)Instruction limit reached!
% 21.85/7.54 % (16321)------------------------------
% 21.85/7.54 % (16321)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.85/7.54 % (16321)Termination reason: Unknown
% 21.85/7.54 % (16321)Termination phase: shuffling
% 21.85/7.54
% 21.85/7.54 % (16321)Memory used [KB]: 25074
% 21.85/7.54 % (16321)Time elapsed: 0.198 s
% 21.85/7.54 % (16321)Instructions burned: 670 (million)
% 21.85/7.54 % (16321)------------------------------
% 21.85/7.54 % (16321)------------------------------
% 21.85/7.54 % (16325)lrs+1010_2:3_au=on:fsr=off:sd=4:ss=axioms:st=1.5:i=371:si=on:rtra=on_0 on DTF2THF_16103 for (2976ds/371Mi)
% 21.85/7.54 % (16326)dis+1003_1:1_au=on:bd=preordered:fde=none:ixr=off:nwc=10.0:i=1827:si=on:rtra=on_0 on DTF2THF_16103 for (2976ds/1827Mi)
% 21.85/7.55 % (16327)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_16103 for (2976ds/70Mi)
% 21.85/7.58 % (16327)Instruction limit reached!
% 21.85/7.58 % (16327)------------------------------
% 21.85/7.58 % (16327)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.85/7.58 % (16327)Termination reason: Unknown
% 21.85/7.58 % (16327)Termination phase: shuffling
% 21.85/7.58
% 21.85/7.58 % (16327)Memory used [KB]: 14583
% 21.85/7.58 % (16327)Time elapsed: 0.029 s
% 21.85/7.58 % (16327)Instructions burned: 72 (million)
% 21.85/7.58 % (16327)------------------------------
% 21.85/7.58 % (16327)------------------------------
% 21.85/7.59 % (16328)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_16103 for (2975ds/316Mi)
% 22.57/7.61 % (16320)Instruction limit reached!
% 22.57/7.61 % (16320)------------------------------
% 22.57/7.61 % (16320)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.57/7.61 % (16320)Termination reason: Unknown
% 22.57/7.61 % (16320)Termination phase: Property scanning
% 22.57/7.61
% 22.57/7.61 % (16320)Memory used [KB]: 26353
% 22.57/7.61 % (16320)Time elapsed: 0.286 s
% 22.57/7.61 % (16320)Instructions burned: 835 (million)
% 22.57/7.61 % (16320)------------------------------
% 22.57/7.61 % (16320)------------------------------
% 22.57/7.62 % (16329)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_16103 for (2975ds/932Mi)
% 22.57/7.67 % (16325)Instruction limit reached!
% 22.57/7.67 % (16325)------------------------------
% 22.57/7.67 % (16325)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.57/7.67 % (16325)Termination reason: Unknown
% 22.57/7.67 % (16325)Termination phase: shuffling
% 22.57/7.67
% 22.57/7.67 % (16325)Memory used [KB]: 18677
% 22.57/7.67 % (16325)Time elapsed: 0.129 s
% 22.57/7.67 % (16325)Instructions burned: 371 (million)
% 22.57/7.67 % (16325)------------------------------
% 22.57/7.67 % (16325)------------------------------
% 22.57/7.68 % (16330)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_16103 for (2975ds/4714Mi)
% 22.57/7.68 % (16328)Instruction limit reached!
% 22.57/7.68 % (16328)------------------------------
% 22.57/7.68 % (16328)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.57/7.68 % (16328)Termination reason: Unknown
% 22.57/7.68 % (16328)Termination phase: shuffling
% 22.57/7.68
% 22.57/7.68 % (16328)Memory used [KB]: 20084
% 22.57/7.68 % (16328)Time elapsed: 0.097 s
% 22.57/7.68 % (16328)Instructions burned: 316 (million)
% 22.57/7.68 % (16328)------------------------------
% 22.57/7.68 % (16328)------------------------------
% 22.57/7.70 % (16331)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_16103 for (2974ds/8250Mi)
% 23.72/7.89 % (16329)Instruction limit reached!
% 23.72/7.89 % (16329)------------------------------
% 23.72/7.89 % (16329)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.72/7.89 % (16329)Termination reason: Unknown
% 23.72/7.89 % (16329)Termination phase: shuffling
% 23.72/7.89
% 23.72/7.89 % (16329)Memory used [KB]: 32494
% 23.72/7.89 % (16329)Time elapsed: 0.271 s
% 23.72/7.89 % (16329)Instructions burned: 932 (million)
% 23.72/7.89 % (16329)------------------------------
% 23.72/7.89 % (16329)------------------------------
% 24.30/7.90 % (16332)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_16103 for (2972ds/770Mi)
% 25.91/8.10 % (16308)Instruction limit reached!
% 25.91/8.10 % (16308)------------------------------
% 25.91/8.10 % (16308)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.91/8.10 % (16308)Termination reason: Unknown
% 25.91/8.10 % (16308)Termination phase: Function definition elimination
% 25.91/8.10
% 25.91/8.10 % (16308)Memory used [KB]: 55265
% 25.91/8.10 % (16308)Time elapsed: 1.326 s
% 25.91/8.10 % (16308)Instructions burned: 5208 (million)
% 25.91/8.10 % (16308)------------------------------
% 25.91/8.10 % (16308)------------------------------
% 25.91/8.11 % (16305)Instruction limit reached!
% 25.91/8.11 % (16305)------------------------------
% 25.91/8.11 % (16305)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.91/8.11 % (16305)Termination reason: Unknown
% 25.91/8.11 % (16305)Termination phase: Saturation
% 25.91/8.11
% 25.91/8.11 % (16305)Memory used [KB]: 34796
% 25.91/8.11 % (16305)Time elapsed: 1.379 s
% 25.91/8.11 % (16305)Instructions burned: 4439 (million)
% 25.91/8.11 % (16305)------------------------------
% 25.91/8.11 % (16305)------------------------------
% 25.91/8.12 % (16333)ott+1010_5:1_av=off:fe=off:ins=1:kws=precedence:sp=reverse_arity:i=794:si=on:rtra=on_0 on DTF2THF_16103 for (2970ds/794Mi)
% 25.91/8.12 % (16326)Instruction limit reached!
% 25.91/8.12 % (16326)------------------------------
% 25.91/8.12 % (16326)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.91/8.12 % (16326)Termination reason: Unknown
% 25.91/8.12 % (16326)Termination phase: Preprocessing 3
% 25.91/8.12
% 25.91/8.12 % (16326)Memory used [KB]: 34413
% 25.91/8.12 % (16326)Time elapsed: 0.575 s
% 25.91/8.12 % (16326)Instructions burned: 1829 (million)
% 25.91/8.12 % (16326)------------------------------
% 25.91/8.12 % (16326)------------------------------
% 25.91/8.12 % (16334)dis+1002_1:1_au=on:avsq=on:avsqr=1,16:cond=on:sp=frequency:i=464:si=on:rtra=on_0 on DTF2THF_16103 for (2970ds/464Mi)
% 26.61/8.13 % (16335)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_16103 for (2970ds/1376Mi)
% 26.61/8.16 % (16332)Instruction limit reached!
% 26.61/8.16 % (16332)------------------------------
% 26.61/8.16 % (16332)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.61/8.16 % (16332)Termination reason: Unknown
% 26.61/8.16 % (16332)Termination phase: shuffling
% 26.61/8.16
% 26.61/8.16 % (16332)Memory used [KB]: 25458
% 26.61/8.16 % (16332)Time elapsed: 0.263 s
% 26.61/8.16 % (16332)Instructions burned: 770 (million)
% 26.61/8.16 % (16332)------------------------------
% 26.61/8.16 % (16332)------------------------------
% 26.61/8.18 % (16336)dis+10_1:1_etr=on:plsq=on:plsqc=1:plsqr=32,1:i=906:si=on:rtra=on_0 on DTF2THF_16103 for (2969ds/906Mi)
% 26.88/8.26 % (16310)Instruction limit reached!
% 26.88/8.26 % (16310)------------------------------
% 26.88/8.26 % (16310)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.88/8.26 % (16310)Termination reason: Unknown
% 26.88/8.26 % (16310)Termination phase: Saturation
% 26.88/8.26
% 26.88/8.26 % (16310)Memory used [KB]: 34029
% 26.88/8.26 % (16310)Time elapsed: 1.358 s
% 26.88/8.26 % (16310)Instructions burned: 4351 (million)
% 26.88/8.26 % (16310)------------------------------
% 26.88/8.26 % (16310)------------------------------
% 26.88/8.27 % (16337)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_16103 for (2968ds/434Mi)
% 26.88/8.28 % (16334)Instruction limit reached!
% 26.88/8.28 % (16334)------------------------------
% 26.88/8.28 % (16334)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.88/8.28 % (16334)Termination reason: Unknown
% 26.88/8.28 % (16334)Termination phase: shuffling
% 26.88/8.28
% 26.88/8.28 % (16334)Memory used [KB]: 19829
% 26.88/8.28 % (16334)Time elapsed: 0.161 s
% 26.88/8.28 % (16334)Instructions burned: 464 (million)
% 26.88/8.28 % (16334)------------------------------
% 26.88/8.28 % (16334)------------------------------
% 26.88/8.29 % (16338)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_16103 for (2968ds/865Mi)
% 27.34/8.39 % (16333)Instruction limit reached!
% 27.34/8.39 % (16333)------------------------------
% 27.34/8.39 % (16333)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.34/8.39 % (16333)Termination reason: Unknown
% 27.34/8.39 % (16333)Termination phase: shuffling
% 27.34/8.39
% 27.34/8.39 % (16333)Memory used [KB]: 26097
% 27.34/8.39 % (16333)Time elapsed: 0.274 s
% 27.34/8.39 % (16333)Instructions burned: 795 (million)
% 27.34/8.39 % (16333)------------------------------
% 27.34/8.39 % (16333)------------------------------
% 27.84/8.40 % (16339)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_16103 for (2967ds/593Mi)
% 27.84/8.43 % (16337)Instruction limit reached!
% 27.84/8.43 % (16337)------------------------------
% 27.84/8.43 % (16337)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.84/8.43 % (16337)Termination reason: Unknown
% 27.84/8.43 % (16337)Termination phase: shuffling
% 27.84/8.43
% 27.84/8.43 % (16337)Memory used [KB]: 19573
% 27.84/8.43 % (16337)Time elapsed: 0.155 s
% 27.84/8.43 % (16337)Instructions burned: 435 (million)
% 27.84/8.43 % (16337)------------------------------
% 27.84/8.43 % (16337)------------------------------
% 27.84/8.44 % (16340)dis+10_1:1_sd=2:sgt=20:slsq=on:ss=axioms:st=6.0:i=443:si=on:rtra=on_0 on DTF2THF_16103 for (2967ds/443Mi)
% 27.84/8.48 % (16336)Instruction limit reached!
% 27.84/8.48 % (16336)------------------------------
% 27.84/8.48 % (16336)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.84/8.48 % (16336)Termination reason: Unknown
% 27.84/8.48 % (16336)Termination phase: Property scanning
% 27.84/8.48
% 27.84/8.48 % (16336)Memory used [KB]: 26225
% 27.84/8.48 % (16336)Time elapsed: 0.303 s
% 27.84/8.48 % (16336)Instructions burned: 907 (million)
% 27.84/8.48 % (16336)------------------------------
% 27.84/8.48 % (16336)------------------------------
% 27.84/8.49 % (16341)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_16103 for (2966ds/594Mi)
% 28.51/8.53 % (16335)Instruction limit reached!
% 28.51/8.53 % (16335)------------------------------
% 28.51/8.53 % (16335)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.51/8.53 % (16335)Termination reason: Unknown
% 28.51/8.53 % (16335)Termination phase: shuffling
% 28.51/8.53
% 28.51/8.53 % (16335)Memory used [KB]: 38506
% 28.51/8.53 % (16335)Time elapsed: 0.400 s
% 28.51/8.53 % (16335)Instructions burned: 1377 (million)
% 28.51/8.53 % (16335)------------------------------
% 28.51/8.53 % (16335)------------------------------
% 28.51/8.54 % (16342)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_16103 for (2966ds/327Mi)
% 28.51/8.55 % (16338)Instruction limit reached!
% 28.51/8.55 % (16338)------------------------------
% 28.51/8.55 % (16338)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.51/8.55 % (16338)Termination reason: Unknown
% 28.51/8.55 % (16338)Termination phase: shuffling
% 28.51/8.55
% 28.51/8.55 % (16338)Memory used [KB]: 31598
% 28.51/8.55 % (16338)Time elapsed: 0.255 s
% 28.51/8.55 % (16338)Instructions burned: 865 (million)
% 28.51/8.55 % (16338)------------------------------
% 28.51/8.55 % (16338)------------------------------
% 28.51/8.56 % (16343)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_16103 for (2966ds/5411Mi)
% 28.51/8.58 % (16339)Instruction limit reached!
% 28.51/8.58 % (16339)------------------------------
% 28.51/8.58 % (16339)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.51/8.58 % (16339)Termination reason: Unknown
% 28.51/8.58 % (16339)Termination phase: shuffling
% 28.51/8.58
% 28.51/8.58 % (16339)Memory used [KB]: 23922
% 28.51/8.58 % (16339)Time elapsed: 0.177 s
% 28.51/8.58 % (16339)Instructions burned: 595 (million)
% 28.51/8.58 % (16339)------------------------------
% 28.51/8.58 % (16339)------------------------------
% 28.51/8.59 % (16344)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_16103 for (2965ds/1762Mi)
% 28.51/8.59 % (16340)Instruction limit reached!
% 28.51/8.59 % (16340)------------------------------
% 28.51/8.59 % (16340)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.51/8.59 % (16340)Termination reason: Unknown
% 28.51/8.59 % (16340)Termination phase: shuffling
% 28.51/8.59
% 28.51/8.59 % (16340)Memory used [KB]: 19573
% 28.51/8.59 % (16340)Time elapsed: 0.154 s
% 28.51/8.59 % (16340)Instructions burned: 445 (million)
% 28.51/8.59 % (16340)------------------------------
% 28.51/8.59 % (16340)------------------------------
% 29.76/8.60 % (16345)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_16103 for (2965ds/96Mi)
% 29.76/8.64 % (16345)Instruction limit reached!
% 29.76/8.64 % (16345)------------------------------
% 29.76/8.64 % (16345)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.76/8.64 % (16345)Termination reason: Unknown
% 29.76/8.64 % (16345)Termination phase: shuffling
% 29.76/8.64
% 29.76/8.64 % (16345)Memory used [KB]: 14839
% 29.76/8.64 % (16345)Time elapsed: 0.035 s
% 29.76/8.64 % (16345)Instructions burned: 96 (million)
% 29.76/8.64 % (16345)------------------------------
% 29.76/8.64 % (16345)------------------------------
% 29.76/8.65 % (16346)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_16103 for (2965ds/503Mi)
% 29.76/8.66 % (16342)Instruction limit reached!
% 29.76/8.66 % (16342)------------------------------
% 29.76/8.66 % (16342)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.76/8.66 % (16342)Termination reason: Unknown
% 29.76/8.66 % (16342)Termination phase: shuffling
% 29.76/8.66
% 29.76/8.66 % (16342)Memory used [KB]: 18166
% 29.76/8.66 % (16342)Time elapsed: 0.117 s
% 29.76/8.66 % (16342)Instructions burned: 327 (million)
% 29.76/8.66 % (16342)------------------------------
% 29.76/8.66 % (16342)------------------------------
% 29.76/8.67 % (16347)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_16103 for (2965ds/90Mi)
% 29.76/8.70 % (16341)Instruction limit reached!
% 29.76/8.70 % (16341)------------------------------
% 29.76/8.70 % (16341)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.76/8.70 % (16341)Termination reason: Unknown
% 29.76/8.70 % (16341)Termination phase: shuffling
% 29.76/8.70
% 29.76/8.70 % (16341)Memory used [KB]: 23539
% 29.76/8.70 % (16341)Time elapsed: 0.206 s
% 29.76/8.70 % (16341)Instructions burned: 594 (million)
% 29.76/8.70 % (16341)------------------------------
% 29.76/8.70 % (16341)------------------------------
% 30.12/8.71 % (16347)Instruction limit reached!
% 30.12/8.71 % (16347)------------------------------
% 30.12/8.71 % (16347)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.12/8.71 % (16347)Termination reason: Unknown
% 30.12/8.71 % (16347)Termination phase: shuffling
% 30.12/8.71
% 30.12/8.71 % (16347)Memory used [KB]: 14839
% 30.12/8.71 % (16347)Time elapsed: 0.039 s
% 30.12/8.71 % (16347)Instructions burned: 91 (million)
% 30.12/8.71 % (16347)------------------------------
% 30.12/8.71 % (16347)------------------------------
% 30.12/8.71 % (16348)lrs+10_1:64_hud=5:plsq=on:plsqr=2011171,524288:i=593:si=on:rtra=on_0 on DTF2THF_16103 for (2964ds/593Mi)
% 30.12/8.72 % (16349)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_16103 for (2964ds/1512Mi)
% 30.59/8.82 % (16346)Instruction limit reached!
% 30.59/8.82 % (16346)------------------------------
% 30.59/8.82 % (16346)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.59/8.82 % (16346)Termination reason: Unknown
% 30.59/8.82 % (16346)Termination phase: shuffling
% 30.59/8.82
% 30.59/8.82 % (16346)Memory used [KB]: 20340
% 30.59/8.82 % (16346)Time elapsed: 0.175 s
% 30.59/8.82 % (16346)Instructions burned: 504 (million)
% 30.59/8.82 % (16346)------------------------------
% 30.59/8.82 % (16346)------------------------------
% 30.59/8.83 % (16350)lrs+1002_1:128_cbe=off:cnfonf=lazy_not_gen_be_off:ins=1:i=569:si=on:rtra=on_0 on DTF2THF_16103 for (2963ds/569Mi)
% 30.97/8.91 % (16348)Instruction limit reached!
% 30.97/8.91 % (16348)------------------------------
% 30.97/8.91 % (16348)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.97/8.91 % (16348)Termination reason: Unknown
% 30.97/8.91 % (16348)Termination phase: shuffling
% 30.97/8.91
% 30.97/8.91 % (16348)Memory used [KB]: 23539
% 30.97/8.91 % (16348)Time elapsed: 0.204 s
% 30.97/8.91 % (16348)Instructions burned: 593 (million)
% 30.97/8.91 % (16348)------------------------------
% 30.97/8.91 % (16348)------------------------------
% 30.97/8.92 % (16351)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_16103 for (2962ds/501Mi)
% 30.97/8.93 % (16330)Instruction limit reached!
% 30.97/8.93 % (16330)------------------------------
% 30.97/8.93 % (16330)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.97/8.93 % (16330)Termination reason: Unknown
% 30.97/8.93 % (16330)Termination phase: Function definition elimination
% 30.97/8.93
% 30.97/8.93 % (16330)Memory used [KB]: 56544
% 30.97/8.93 % (16330)Time elapsed: 1.257 s
% 30.97/8.93 % (16330)Instructions burned: 4716 (million)
% 30.97/8.93 % (16330)------------------------------
% 30.97/8.93 % (16330)------------------------------
% 30.97/8.95 % (16352)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_16103 for (2962ds/562Mi)
% 30.97/9.01 % (16350)Instruction limit reached!
% 30.97/9.01 % (16350)------------------------------
% 30.97/9.01 % (16350)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.97/9.01 % (16350)Termination reason: Unknown
% 30.97/9.01 % (16350)Termination phase: shuffling
% 30.97/9.01
% 30.97/9.01 % (16350)Memory used [KB]: 23539
% 30.97/9.01 % (16350)Time elapsed: 0.174 s
% 30.97/9.01 % (16350)Instructions burned: 569 (million)
% 30.97/9.01 % (16350)------------------------------
% 30.97/9.01 % (16350)------------------------------
% 32.85/9.03 % (16353)lrs+10_1:1_au=on:cnfonf=off:fde=unused:kws=precedence:tgt=full:i=406:si=on:rtra=on_0 on DTF2THF_16103 for (2961ds/406Mi)
% 32.85/9.09 % (16344)Instruction limit reached!
% 32.85/9.09 % (16344)------------------------------
% 32.85/9.09 % (16344)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.85/9.09 % (16344)Termination reason: Unknown
% 32.85/9.09 % (16344)Termination phase: Property scanning
% 32.85/9.09
% 32.85/9.09 % (16344)Memory used [KB]: 41577
% 32.85/9.09 % (16344)Time elapsed: 0.506 s
% 32.85/9.09 % (16344)Instructions burned: 1762 (million)
% 32.85/9.09 % (16344)------------------------------
% 32.85/9.09 % (16344)------------------------------
% 32.85/9.10 % (16351)Instruction limit reached!
% 32.85/9.10 % (16351)------------------------------
% 32.85/9.10 % (16351)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.85/9.10 % (16351)Termination reason: Unknown
% 32.85/9.10 % (16351)Termination phase: shuffling
% 32.85/9.10
% 32.85/9.10 % (16351)Memory used [KB]: 20212
% 32.85/9.10 % (16351)Time elapsed: 0.175 s
% 32.85/9.10 % (16351)Instructions burned: 503 (million)
% 32.85/9.10 % (16351)------------------------------
% 32.85/9.10 % (16351)------------------------------
% 32.85/9.11 % (16354)dis+1004_1:128_au=on:cbe=off:e2e=on:sac=on:i=582:si=on:rtra=on_0 on DTF2THF_16103 for (2960ds/582Mi)
% 32.85/9.11 % (16355)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_16103 for (2960ds/456Mi)
% 33.47/9.11 % (16352)Instruction limit reached!
% 33.47/9.11 % (16352)------------------------------
% 33.47/9.11 % (16352)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.47/9.11 % (16352)Termination reason: Unknown
% 33.47/9.11 % (16352)Termination phase: shuffling
% 33.47/9.11
% 33.47/9.12 % (16352)Memory used [KB]: 23539
% 33.47/9.12 % (16352)Time elapsed: 0.169 s
% 33.47/9.12 % (16352)Instructions burned: 562 (million)
% 33.47/9.12 % (16352)------------------------------
% 33.47/9.12 % (16352)------------------------------
% 33.47/9.13 % (16356)lrs+1010_1:1_au=on:cbe=off:cnfonf=conj_eager:tgt=ground:i=393:si=on:rtra=on_0 on DTF2THF_16103 for (2960ds/393Mi)
% 33.47/9.19 % (16349)Instruction limit reached!
% 33.47/9.19 % (16349)------------------------------
% 33.47/9.19 % (16349)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.47/9.19 % (16349)Termination reason: Unknown
% 33.47/9.19 % (16349)Termination phase: Property scanning
% 33.47/9.19
% 33.47/9.19 % (16349)Memory used [KB]: 29040
% 33.47/9.19 % (16349)Time elapsed: 0.474 s
% 33.47/9.19 % (16349)Instructions burned: 1513 (million)
% 33.47/9.19 % (16349)------------------------------
% 33.47/9.19 % (16349)------------------------------
% 33.47/9.19 % (16353)Instruction limit reached!
% 33.47/9.19 % (16353)------------------------------
% 33.47/9.19 % (16353)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.47/9.19 % (16353)Termination reason: Unknown
% 33.47/9.19 % (16353)Termination phase: shuffling
% 33.47/9.19
% 33.47/9.19 % (16353)Memory used [KB]: 18933
% 33.47/9.19 % (16353)Time elapsed: 0.165 s
% 33.47/9.19 % (16353)Instructions burned: 408 (million)
% 33.47/9.19 % (16353)------------------------------
% 33.47/9.19 % (16353)------------------------------
% 33.93/9.20 % (16357)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_16103 for (2959ds/1335Mi)
% 33.93/9.20 % (16358)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_16103 for (2959ds/3373Mi)
% 33.93/9.25 % (16356)Instruction limit reached!
% 33.93/9.25 % (16356)------------------------------
% 33.93/9.25 % (16356)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.93/9.25 % (16356)Termination reason: Unknown
% 33.93/9.25 % (16356)Termination phase: shuffling
% 33.93/9.25
% 33.93/9.25 % (16356)Memory used [KB]: 21108
% 33.93/9.25 % (16356)Time elapsed: 0.121 s
% 33.93/9.25 % (16356)Instructions burned: 396 (million)
% 33.93/9.25 % (16356)------------------------------
% 33.93/9.25 % (16356)------------------------------
% 33.93/9.26 % (16359)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_16103 for (2959ds/474Mi)
% 33.93/9.26 % (16355)Instruction limit reached!
% 33.93/9.26 % (16355)------------------------------
% 33.93/9.26 % (16355)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.93/9.26 % (16355)Termination reason: Unknown
% 33.93/9.26 % (16355)Termination phase: shuffling
% 33.93/9.26
% 33.93/9.26 % (16355)Memory used [KB]: 19701
% 33.93/9.26 % (16355)Time elapsed: 0.158 s
% 33.93/9.26 % (16355)Instructions burned: 457 (million)
% 33.93/9.26 % (16355)------------------------------
% 33.93/9.26 % (16355)------------------------------
% 33.93/9.28 % (16360)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_16103 for (2958ds/157Mi)
% 34.42/9.31 % (16354)Instruction limit reached!
% 34.42/9.31 % (16354)------------------------------
% 34.42/9.31 % (16354)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.42/9.31 % (16354)Termination reason: Unknown
% 34.42/9.31 % (16354)Termination phase: shuffling
% 34.42/9.31
% 34.42/9.31 % (16354)Memory used [KB]: 21620
% 34.42/9.31 % (16354)Time elapsed: 0.204 s
% 34.42/9.31 % (16354)Instructions burned: 583 (million)
% 34.42/9.31 % (16354)------------------------------
% 34.42/9.31 % (16354)------------------------------
% 34.42/9.32 % (16361)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_16103 for (2958ds/590Mi)
% 34.42/9.33 % (16360)Instruction limit reached!
% 34.42/9.33 % (16360)------------------------------
% 34.42/9.33 % (16360)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.42/9.33 % (16360)Termination reason: Unknown
% 34.42/9.33 % (16360)Termination phase: shuffling
% 34.42/9.33
% 34.42/9.33 % (16360)Memory used [KB]: 15735
% 34.42/9.33 % (16360)Time elapsed: 0.053 s
% 34.42/9.33 % (16360)Instructions burned: 158 (million)
% 34.42/9.33 % (16360)------------------------------
% 34.42/9.33 % (16360)------------------------------
% 34.42/9.34 % (16362)dis+10_1:1024_kws=precedence:prag=on:rawr=on:sos=all:sp=unary_first:i=104:si=on:rtra=on_0 on DTF2THF_16103 for (2958ds/104Mi)
% 34.42/9.38 % (16362)Instruction limit reached!
% 34.42/9.38 % (16362)------------------------------
% 34.42/9.38 % (16362)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.42/9.38 % (16362)Termination reason: Unknown
% 34.42/9.38 % (16362)Termination phase: shuffling
% 34.42/9.38
% 34.42/9.38 % (16362)Memory used [KB]: 14967
% 34.42/9.38 % (16362)Time elapsed: 0.043 s
% 34.42/9.38 % (16362)Instructions burned: 104 (million)
% 34.42/9.38 % (16362)------------------------------
% 34.42/9.38 % (16362)------------------------------
% 34.42/9.39 % (16363)lrs+1010_1:10_au=on:bd=off:cnfonf=lazy_pi_sigma_gen:sp=occurrence:i=414:si=on:rtra=on_0 on DTF2THF_16103 for (2957ds/414Mi)
% 34.95/9.40 % (16359)Instruction limit reached!
% 34.95/9.40 % (16359)------------------------------
% 34.95/9.40 % (16359)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.95/9.40 % (16359)Termination reason: Unknown
% 34.95/9.40 % (16359)Termination phase: shuffling
% 34.95/9.40
% 34.95/9.40 % (16359)Memory used [KB]: 22259
% 34.95/9.40 % (16359)Time elapsed: 0.143 s
% 34.95/9.40 % (16359)Instructions burned: 475 (million)
% 34.95/9.40 % (16359)------------------------------
% 34.95/9.40 % (16359)------------------------------
% 34.95/9.41 % (16364)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_16103 for (2957ds/715Mi)
% 35.30/9.51 % (16363)Instruction limit reached!
% 35.30/9.51 % (16363)------------------------------
% 35.30/9.51 % (16363)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.30/9.51 % (16363)Termination reason: Unknown
% 35.30/9.51 % (16363)Termination phase: shuffling
% 35.30/9.51
% 35.30/9.51 % (16363)Memory used [KB]: 21364
% 35.30/9.51 % (16363)Time elapsed: 0.125 s
% 35.30/9.51 % (16363)Instructions burned: 414 (million)
% 35.30/9.51 % (16363)------------------------------
% 35.30/9.51 % (16363)------------------------------
% 35.30/9.52 % (16361)Instruction limit reached!
% 35.30/9.52 % (16361)------------------------------
% 35.30/9.52 % (16361)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.30/9.52 % (16361)Termination reason: Unknown
% 35.30/9.52 % (16361)Termination phase: shuffling
% 35.30/9.52
% 35.30/9.52 % (16361)Memory used [KB]: 23539
% 35.30/9.52 % (16361)Time elapsed: 0.203 s
% 35.30/9.52 % (16361)Instructions burned: 590 (million)
% 35.30/9.52 % (16361)------------------------------
% 35.30/9.52 % (16361)------------------------------
% 35.30/9.53 % (16365)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_16103 for (2956ds/961Mi)
% 35.30/9.53 % (16366)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_16103 for (2956ds/1213Mi)
% 35.30/9.59 % (16357)Instruction limit reached!
% 35.30/9.59 % (16357)------------------------------
% 35.30/9.59 % (16357)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.30/9.59 % (16357)Termination reason: Unknown
% 35.30/9.59 % (16357)Termination phase: shuffling
% 35.30/9.59
% 35.30/9.59 % (16357)Memory used [KB]: 37995
% 35.30/9.59 % (16357)Time elapsed: 0.389 s
% 35.30/9.59 % (16357)Instructions burned: 1336 (million)
% 35.30/9.59 % (16357)------------------------------
% 35.30/9.59 % (16357)------------------------------
% 36.25/9.60 % (16367)lrs+1002_1:12_bd=off:hud=1:ntd=on:piset=equals:prag=on:i=1979:si=on:rtra=on_0 on DTF2THF_16103 for (2955ds/1979Mi)
% 36.25/9.66 % (16364)Instruction limit reached!
% 36.25/9.66 % (16364)------------------------------
% 36.25/9.66 % (16364)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 36.25/9.66 % (16364)Termination reason: Unknown
% 36.25/9.66 % (16364)Termination phase: shuffling
% 36.25/9.66
% 36.25/9.66 % (16364)Memory used [KB]: 25074
% 36.25/9.66 % (16364)Time elapsed: 0.247 s
% 36.25/9.66 % (16364)Instructions burned: 717 (million)
% 36.25/9.66 % (16364)------------------------------
% 36.25/9.66 % (16364)------------------------------
% 36.25/9.67 % (16368)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_16103 for (2954ds/481Mi)
% 36.90/9.81 % (16368)Instruction limit reached!
% 36.90/9.81 % (16368)------------------------------
% 36.90/9.81 % (16368)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 36.90/9.81 % (16368)Termination reason: Unknown
% 36.90/9.82 % (16368)Termination phase: shuffling
% 36.90/9.82
% 36.90/9.82 % (16368)Memory used [KB]: 22387
% 36.90/9.82 % (16368)Time elapsed: 0.146 s
% 36.90/9.82 % (16368)Instructions burned: 482 (million)
% 36.90/9.82 % (16368)------------------------------
% 36.90/9.82 % (16368)------------------------------
% 36.90/9.83 % (16369)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_16103 for (2953ds/783Mi)
% 36.90/9.84 % (16365)Instruction limit reached!
% 36.90/9.84 % (16365)------------------------------
% 36.90/9.84 % (16365)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 36.90/9.84 % (16365)Termination reason: Unknown
% 36.90/9.84 % (16365)Termination phase: Property scanning
% 36.90/9.84
% 36.90/9.84 % (16365)Memory used [KB]: 26225
% 36.90/9.84 % (16365)Time elapsed: 0.315 s
% 36.90/9.84 % (16365)Instructions burned: 963 (million)
% 36.90/9.84 % (16365)------------------------------
% 36.90/9.84 % (16365)------------------------------
% 36.90/9.85 % (16370)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_16103 for (2953ds/432Mi)
% 37.59/9.92 % (16366)Instruction limit reached!
% 37.59/9.92 % (16366)------------------------------
% 37.59/9.92 % (16366)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 37.59/9.92 % (16366)Termination reason: Unknown
% 37.59/9.92 % (16366)Termination phase: Property scanning
% 37.59/9.92
% 37.59/9.92 % (16366)Memory used [KB]: 26353
% 37.59/9.92 % (16366)Time elapsed: 0.384 s
% 37.59/9.92 % (16366)Instructions burned: 1216 (million)
% 37.59/9.92 % (16366)------------------------------
% 37.59/9.92 % (16366)------------------------------
% 37.59/9.93 % (16371)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_16103 for (2952ds/1670Mi)
% 39.86/10.03 % (16370)Instruction limit reached!
% 39.86/10.03 % (16370)------------------------------
% 39.86/10.03 % (16370)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.86/10.03 % (16370)Termination reason: Unknown
% 39.86/10.03 % (16370)Termination phase: shuffling
% 39.86/10.03
% 39.86/10.03 % (16370)Memory used [KB]: 19317
% 39.86/10.03 % (16370)Time elapsed: 0.179 s
% 39.86/10.03 % (16370)Instructions burned: 432 (million)
% 39.86/10.03 % (16370)------------------------------
% 39.86/10.03 % (16370)------------------------------
% 39.86/10.04 % (16372)dis+2_1:1_au=on:hfaw=0:hfsq=on:sp=weighted_frequency:i=90:si=on:rtra=on_0 on DTF2THF_16103 for (2951ds/90Mi)
% 39.86/10.08 % (16372)Instruction limit reached!
% 39.86/10.08 % (16372)------------------------------
% 39.86/10.08 % (16372)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.86/10.08 % (16372)Termination reason: Unknown
% 39.86/10.08 % (16372)Termination phase: shuffling
% 39.86/10.08
% 39.86/10.08 % (16372)Memory used [KB]: 14839
% 39.86/10.08 % (16372)Time elapsed: 0.039 s
% 39.86/10.08 % (16372)Instructions burned: 92 (million)
% 39.86/10.08 % (16372)------------------------------
% 39.86/10.08 % (16372)------------------------------
% 39.86/10.09 % (16373)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_16103 for (2950ds/346Mi)
% 39.86/10.10 % (16369)Instruction limit reached!
% 39.86/10.10 % (16369)------------------------------
% 39.86/10.10 % (16369)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.86/10.10 % (16369)Termination reason: Unknown
% 39.86/10.10 % (16369)Termination phase: shuffling
% 39.86/10.10
% 39.86/10.10 % (16369)Memory used [KB]: 25841
% 39.86/10.10 % (16369)Time elapsed: 0.270 s
% 39.86/10.10 % (16369)Instructions burned: 785 (million)
% 39.86/10.10 % (16369)------------------------------
% 39.86/10.10 % (16369)------------------------------
% 39.86/10.11 % (16374)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_16103 for (2950ds/401Mi)
% 40.55/10.18 % (16343)Instruction limit reached!
% 40.55/10.18 % (16343)------------------------------
% 40.55/10.18 % (16343)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.55/10.18 % (16343)Termination reason: Unknown
% 40.55/10.18 % (16343)Termination phase: Saturation
% 40.55/10.18
% 40.55/10.18 % (16343)Memory used [KB]: 38123
% 40.55/10.18 % (16343)Time elapsed: 1.624 s
% 40.55/10.18 % (16343)Instructions burned: 5411 (million)
% 40.55/10.18 % (16343)------------------------------
% 40.55/10.18 % (16343)------------------------------
% 40.55/10.19 % (16375)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=6454:si=on:rtra=on_0 on DTF2THF_16103 for (2949ds/6454Mi)
% 40.55/10.20 % (16373)Instruction limit reached!
% 40.55/10.20 % (16373)------------------------------
% 40.55/10.20 % (16373)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.55/10.20 % (16373)Termination reason: Unknown
% 40.55/10.20 % (16373)Termination phase: shuffling
% 40.55/10.20
% 40.55/10.20 % (16373)Memory used [KB]: 20468
% 40.55/10.20 % (16373)Time elapsed: 0.107 s
% 40.55/10.20 % (16373)Instructions burned: 347 (million)
% 40.55/10.20 % (16373)------------------------------
% 40.55/10.20 % (16373)------------------------------
% 40.96/10.21 % (16376)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_16103 for (2949ds/283Mi)
% 40.96/10.22 % (16358)Instruction limit reached!
% 40.96/10.22 % (16358)------------------------------
% 40.96/10.22 % (16358)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.96/10.22 % (16358)Termination reason: Unknown
% 40.96/10.22 % (16358)Termination phase: Function definition elimination
% 40.96/10.22
% 40.96/10.22 % (16358)Memory used [KB]: 38506
% 40.96/10.22 % (16358)Time elapsed: 1.015 s
% 40.96/10.22 % (16358)Instructions burned: 3373 (million)
% 40.96/10.22 % (16358)------------------------------
% 40.96/10.22 % (16358)------------------------------
% 40.96/10.23 % (16374)Instruction limit reached!
% 40.96/10.23 % (16374)------------------------------
% 40.96/10.23 % (16374)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.96/10.23 % (16374)Termination reason: Unknown
% 40.96/10.23 % (16374)Termination phase: shuffling
% 40.96/10.23
% 40.96/10.23 % (16374)Memory used [KB]: 21236
% 40.96/10.23 % (16374)Time elapsed: 0.122 s
% 40.96/10.23 % (16374)Instructions burned: 401 (million)
% 40.96/10.23 % (16374)------------------------------
% 40.96/10.23 % (16374)------------------------------
% 40.96/10.23 % (16367)Instruction limit reached!
% 40.96/10.23 % (16367)------------------------------
% 40.96/10.23 % (16367)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.96/10.23 % (16367)Termination reason: Unknown
% 40.96/10.23 % (16367)Termination phase: Property scanning
% 40.96/10.23
% 40.96/10.23 % (16367)Memory used [KB]: 37355
% 40.96/10.23 % (16367)Time elapsed: 0.629 s
% 40.96/10.23 % (16367)Instructions burned: 1982 (million)
% 40.96/10.23 % (16367)------------------------------
% 40.96/10.23 % (16367)------------------------------
% 40.96/10.23 % (16377)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_16103 for (2949ds/865Mi)
% 40.96/10.24 % (16378)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_16103 for (2949ds/4528Mi)
% 40.96/10.24 % (16379)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_16103 for (2949ds/1001Mi)
% 40.96/10.27 % (16331)Instruction limit reached!
% 40.96/10.27 % (16331)------------------------------
% 40.96/10.27 % (16331)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.96/10.27 % (16331)Termination reason: Unknown
% 40.96/10.27 % (16331)Termination phase: Saturation
% 40.96/10.27
% 40.96/10.27 % (16331)Memory used [KB]: 58591
% 40.96/10.27 % (16331)Time elapsed: 2.602 s
% 40.96/10.27 % (16331)Instructions burned: 8251 (million)
% 40.96/10.27 % (16331)------------------------------
% 40.96/10.27 % (16331)------------------------------
% 40.96/10.29 % (16380)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=102:si=on:rtra=on_0 on DTF2THF_16103 for (2948ds/102Mi)
% 41.47/10.31 % (16376)Instruction limit reached!
% 41.47/10.31 % (16376)------------------------------
% 41.47/10.31 % (16376)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.47/10.31 % (16376)Termination reason: Unknown
% 41.47/10.31 % (16376)Termination phase: shuffling
% 41.47/10.31
% 41.47/10.31 % (16376)Memory used [KB]: 17526
% 41.47/10.31 % (16376)Time elapsed: 0.102 s
% 41.47/10.31 % (16376)Instructions burned: 283 (million)
% 41.47/10.31 % (16376)------------------------------
% 41.47/10.31 % (16376)------------------------------
% 41.47/10.32 % (16381)ott+1010_3:1_nm=20:s2a=on:s2at=3.0:sp=const_frequency:i=952:si=on:rtra=on_0 on DTF2THF_16103 for (2948ds/952Mi)
% 41.47/10.33 % (16380)Instruction limit reached!
% 41.47/10.33 % (16380)------------------------------
% 41.47/10.33 % (16380)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.47/10.33 % (16380)Termination reason: Unknown
% 41.47/10.33 % (16380)Termination phase: shuffling
% 41.47/10.33
% 41.47/10.33 % (16380)Memory used [KB]: 14967
% 41.47/10.33 % (16380)Time elapsed: 0.038 s
% 41.47/10.33 % (16380)Instructions burned: 104 (million)
% 41.47/10.33 % (16380)------------------------------
% 41.47/10.33 % (16380)------------------------------
% 41.47/10.34 % (16382)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_16103 for (2948ds/578Mi)
% 41.73/10.41 % (16371)Instruction limit reached!
% 41.73/10.41 % (16371)------------------------------
% 41.73/10.41 % (16371)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.73/10.41 % (16371)Termination reason: Unknown
% 41.73/10.41 % (16371)Termination phase: Property scanning
% 41.73/10.41
% 41.73/10.41 % (16371)Memory used [KB]: 41577
% 41.73/10.41 % (16371)Time elapsed: 0.486 s
% 41.73/10.41 % (16371)Instructions burned: 1673 (million)
% 41.73/10.41 % (16371)------------------------------
% 41.73/10.41 % (16371)------------------------------
% 41.73/10.43 % (16383)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_16103 for (2947ds/643Mi)
% 41.76/10.51 % (16382)Instruction limit reached!
% 41.76/10.51 % (16382)------------------------------
% 41.76/10.51 % (16382)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.76/10.51 % (16382)Termination reason: Unknown
% 41.76/10.51 % (16382)Termination phase: shuffling
% 41.76/10.51
% 41.76/10.51 % (16382)Memory used [KB]: 23666
% 41.76/10.51 % (16382)Time elapsed: 0.173 s
% 41.76/10.51 % (16382)Instructions burned: 581 (million)
% 41.76/10.51 % (16382)------------------------------
% 41.76/10.51 % (16382)------------------------------
% 41.76/10.52 % (16384)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_16103 for (2946ds/865Mi)
% 41.76/10.52 % (16377)Instruction limit reached!
% 41.76/10.52 % (16377)------------------------------
% 41.76/10.52 % (16377)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.76/10.52 % (16377)Termination reason: Unknown
% 41.76/10.52 % (16377)Termination phase: shuffling
% 41.76/10.52
% 41.76/10.52 % (16377)Memory used [KB]: 26609
% 41.76/10.52 % (16377)Time elapsed: 0.296 s
% 41.76/10.52 % (16377)Instructions burned: 867 (million)
% 41.76/10.52 % (16377)------------------------------
% 41.76/10.52 % (16377)------------------------------
% 41.76/10.54 % (16385)lrs+10_1:1_au=on:erd=off:s2a=on:s2at=3.0:sos=all:i=1174:si=on:rtra=on_0 on DTF2THF_16103 for (2946ds/1174Mi)
% 41.76/10.57 % (16379)Instruction limit reached!
% 41.76/10.57 % (16379)------------------------------
% 41.76/10.57 % (16379)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.76/10.57 % (16379)Termination reason: Unknown
% 41.76/10.57 % (16379)Termination phase: Property scanning
% 41.76/10.57
% 41.76/10.57 % (16379)Memory used [KB]: 26225
% 41.76/10.57 % (16379)Time elapsed: 0.328 s
% 41.76/10.57 % (16379)Instructions burned: 1004 (million)
% 41.76/10.57 % (16379)------------------------------
% 41.76/10.57 % (16379)------------------------------
% 41.76/10.58 % (16386)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_16103 for (2945ds/644Mi)
% 42.59/10.62 % (16383)Instruction limit reached!
% 42.59/10.62 % (16383)------------------------------
% 42.59/10.62 % (16383)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.59/10.62 % (16383)Termination reason: Unknown
% 42.59/10.62 % (16383)Termination phase: shuffling
% 42.59/10.62
% 42.59/10.62 % (16383)Memory used [KB]: 24690
% 42.59/10.62 % (16383)Time elapsed: 0.191 s
% 42.59/10.62 % (16383)Instructions burned: 646 (million)
% 42.59/10.62 % (16383)------------------------------
% 42.59/10.62 % (16383)------------------------------
% 42.59/10.63 % (16387)lrs+10_1:1_alpa=true:au=on:cs=on:tgt=ground:i=1201:si=on:rtra=on_0 on DTF2THF_16103 for (2945ds/1201Mi)
% 42.59/10.63 % (16381)Instruction limit reached!
% 42.59/10.63 % (16381)------------------------------
% 42.59/10.63 % (16381)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.59/10.63 % (16381)Termination reason: Unknown
% 42.59/10.63 % (16381)Termination phase: Property scanning
% 42.59/10.63
% 42.59/10.63 % (16381)Memory used [KB]: 26225
% 42.59/10.63 % (16381)Time elapsed: 0.314 s
% 42.59/10.63 % (16381)Instructions burned: 954 (million)
% 42.59/10.63 % (16381)------------------------------
% 42.59/10.63 % (16381)------------------------------
% 42.59/10.64 % (16388)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_16103 for (2945ds/1046Mi)
% 43.12/10.80 % (16386)Instruction limit reached!
% 43.12/10.80 % (16386)------------------------------
% 43.12/10.80 % (16386)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.12/10.80 % (16386)Termination reason: Unknown
% 43.12/10.80 % (16386)Termination phase: shuffling
% 43.12/10.80
% 43.12/10.80 % (16386)Memory used [KB]: 23922
% 43.12/10.80 % (16386)Time elapsed: 0.220 s
% 43.12/10.80 % (16386)Instructions burned: 646 (million)
% 43.12/10.80 % (16386)------------------------------
% 43.12/10.80 % (16386)------------------------------
% 43.12/10.81 % (16384)Instruction limit reached!
% 43.12/10.81 % (16384)------------------------------
% 43.12/10.81 % (16384)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.12/10.81 % (16384)Termination reason: Unknown
% 43.12/10.81 % (16384)Termination phase: shuffling
% 43.12/10.81
% 43.12/10.81 % (16384)Memory used [KB]: 26737
% 43.12/10.81 % (16384)Time elapsed: 0.293 s
% 43.12/10.81 % (16384)Instructions burned: 867 (million)
% 43.12/10.81 % (16384)------------------------------
% 43.12/10.81 % (16384)------------------------------
% 43.12/10.81 % (16389)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_16103 for (2943ds/222Mi)
% 43.12/10.82 % (16390)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_16103 for (2943ds/68Mi)
% 43.12/10.85 % (16390)Instruction limit reached!
% 43.12/10.85 % (16390)------------------------------
% 43.12/10.85 % (16390)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.12/10.85 % (16390)Termination reason: Unknown
% 43.12/10.85 % (16390)Termination phase: shuffling
% 43.12/10.85
% 43.12/10.85 % (16390)Memory used [KB]: 14456
% 43.12/10.85 % (16390)Time elapsed: 0.032 s
% 43.12/10.85 % (16390)Instructions burned: 71 (million)
% 43.12/10.85 % (16390)------------------------------
% 43.12/10.85 % (16390)------------------------------
% 43.12/10.87 % (16391)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_16103 for (2943ds/454Mi)
% 43.12/10.88 % (16389)Instruction limit reached!
% 43.12/10.88 % (16389)------------------------------
% 43.12/10.88 % (16389)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.12/10.88 % (16389)Termination reason: Unknown
% 43.12/10.88 % (16389)Termination phase: shuffling
% 43.12/10.88
% 43.12/10.88 % (16389)Memory used [KB]: 16758
% 43.12/10.88 % (16389)Time elapsed: 0.071 s
% 43.12/10.88 % (16389)Instructions burned: 225 (million)
% 43.12/10.88 % (16389)------------------------------
% 43.12/10.88 % (16389)------------------------------
% 43.12/10.89 % (16392)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_16103 for (2942ds/360Mi)
% 43.78/10.91 % (16385)Instruction limit reached!
% 43.78/10.91 % (16385)------------------------------
% 43.78/10.91 % (16385)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.78/10.91 % (16385)Termination reason: Unknown
% 43.78/10.91 % (16385)Termination phase: Property scanning
% 43.78/10.91
% 43.78/10.91 % (16385)Memory used [KB]: 26225
% 43.78/10.91 % (16385)Time elapsed: 0.373 s
% 43.78/10.91 % (16385)Instructions burned: 1175 (million)
% 43.78/10.91 % (16385)------------------------------
% 43.78/10.91 % (16385)------------------------------
% 43.78/10.92 % (16393)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_16103 for (2942ds/157Mi)
% 43.78/10.95 % (16388)Instruction limit reached!
% 43.78/10.95 % (16388)------------------------------
% 43.78/10.95 % (16388)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.78/10.95 % (16388)Termination reason: Unknown
% 43.78/10.95 % (16388)Termination phase: shuffling
% 43.78/10.95
% 43.78/10.95 % (16388)Memory used [KB]: 34157
% 43.78/10.95 % (16388)Time elapsed: 0.309 s
% 43.78/10.95 % (16388)Instructions burned: 1048 (million)
% 43.78/10.95 % (16388)------------------------------
% 43.78/10.95 % (16388)------------------------------
% 43.78/10.96 % (16394)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_16103 for (2942ds/5061Mi)
% 43.78/10.97 % (16393)Instruction limit reached!
% 43.78/10.97 % (16393)------------------------------
% 43.78/10.97 % (16393)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.78/10.97 % (16393)Termination reason: Unknown
% 43.78/10.97 % (16393)Termination phase: shuffling
% 43.78/10.97
% 43.78/10.97 % (16393)Memory used [KB]: 15863
% 43.78/10.97 % (16393)Time elapsed: 0.053 s
% 43.78/10.97 % (16393)Instructions burned: 158 (million)
% 43.78/10.97 % (16393)------------------------------
% 43.78/10.97 % (16393)------------------------------
% 43.78/10.98 % (16395)lrs+1002_1:8_av=off:fd=off:hud=5:tnu=1:i=2401:si=on:rtra=on_0 on DTF2THF_16103 for (2941ds/2401Mi)
% 43.78/11.00 % (16391)Instruction limit reached!
% 43.78/11.00 % (16391)------------------------------
% 43.78/11.00 % (16391)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.78/11.00 % (16391)Termination reason: Unknown
% 43.78/11.00 % (16391)Termination phase: shuffling
% 43.78/11.00
% 43.78/11.00 % (16391)Memory used [KB]: 22003
% 43.78/11.00 % (16391)Time elapsed: 0.137 s
% 43.78/11.00 % (16391)Instructions burned: 455 (million)
% 43.78/11.00 % (16391)------------------------------
% 43.78/11.00 % (16391)------------------------------
% 43.78/11.00 % (16392)Instruction limit reached!
% 43.78/11.00 % (16392)------------------------------
% 43.78/11.00 % (16392)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.78/11.00 % (16392)Termination reason: Unknown
% 43.78/11.00 % (16392)Termination phase: shuffling
% 43.78/11.00
% 43.78/11.00 % (16392)Memory used [KB]: 20724
% 43.78/11.00 % (16392)Time elapsed: 0.112 s
% 43.78/11.00 % (16392)Instructions burned: 362 (million)
% 43.78/11.00 % (16392)------------------------------
% 43.78/11.00 % (16392)------------------------------
% 43.78/11.01 % (16387)Instruction limit reached!
% 43.78/11.01 % (16387)------------------------------
% 43.78/11.01 % (16387)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.78/11.01 % (16387)Termination reason: Unknown
% 43.78/11.01 % (16387)Termination phase: Property scanning
% 43.78/11.01
% 43.78/11.01 % (16387)Memory used [KB]: 26225
% 43.78/11.01 % (16387)Time elapsed: 0.378 s
% 43.78/11.01 % (16387)Instructions burned: 1203 (million)
% 43.78/11.01 % (16387)------------------------------
% 43.78/11.01 % (16387)------------------------------
% 46.86/11.02 % (16397)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_16103 for (2941ds/346Mi)
% 46.86/11.02 % (16396)dis+10_1:1_av=off:e2e=on:nm=4:s2a=on:i=5798:si=on:rtra=on_0 on DTF2THF_16103 for (2941ds/5798Mi)
% 46.86/11.03 % (16398)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_16103 for (2941ds/1321Mi)
% 47.27/11.12 % (16397)Instruction limit reached!
% 47.27/11.12 % (16397)------------------------------
% 47.27/11.12 % (16397)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 47.27/11.12 % (16397)Termination reason: Unknown
% 47.27/11.12 % (16397)Termination phase: shuffling
% 47.27/11.12
% 47.27/11.12 % (16397)Memory used [KB]: 20468
% 47.27/11.12 % (16397)Time elapsed: 0.106 s
% 47.27/11.12 % (16397)Instructions burned: 346 (million)
% 47.27/11.12 % (16397)------------------------------
% 47.27/11.12 % (16397)------------------------------
% 47.27/11.13 % (16399)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:i=1312:si=on:rtra=on_0 on DTF2THF_16103 for (2940ds/1312Mi)
% 48.50/11.44 % (16398)Instruction limit reached!
% 48.50/11.44 % (16398)------------------------------
% 48.50/11.44 % (16398)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.50/11.44 % (16398)Termination reason: Unknown
% 48.50/11.44 % (16398)Termination phase: Property scanning
% 48.50/11.44
% 48.50/11.44 % (16398)Memory used [KB]: 26353
% 48.50/11.44 % (16398)Time elapsed: 0.415 s
% 48.50/11.44 % (16398)Instructions burned: 1324 (million)
% 48.50/11.44 % (16398)------------------------------
% 48.50/11.44 % (16398)------------------------------
% 48.50/11.46 % (16400)ott+2_8:1_add=large:c=on:cnfonf=off:e2e=on:er=filter:hud=5:plsq=on:plsqr=1,1:ss=axioms:i=672:si=on:rtra=on_0 on DTF2THF_16103 for (2937ds/672Mi)
% 49.32/11.52 % (16399)Instruction limit reached!
% 49.32/11.52 % (16399)------------------------------
% 49.32/11.52 % (16399)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.32/11.52 % (16399)Termination reason: Unknown
% 49.32/11.52 % (16399)Termination phase: shuffling
% 49.32/11.52
% 49.32/11.52 % (16399)Memory used [KB]: 37739
% 49.32/11.52 % (16399)Time elapsed: 0.386 s
% 49.32/11.52 % (16399)Instructions burned: 1315 (million)
% 49.32/11.52 % (16399)------------------------------
% 49.32/11.52 % (16399)------------------------------
% 49.32/11.53 % (16401)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=1897:si=on:rtra=on_0 on DTF2THF_16103 for (2936ds/1897Mi)
% 49.32/11.57 % (16378)Instruction limit reached!
% 49.32/11.57 % (16378)------------------------------
% 49.32/11.57 % (16378)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.32/11.57 % (16378)Termination reason: Unknown
% 49.32/11.57 % (16378)Termination phase: Saturation
% 49.32/11.57
% 49.32/11.57 % (16378)Memory used [KB]: 31854
% 49.32/11.57 % (16378)Time elapsed: 1.337 s
% 49.32/11.57 % (16378)Instructions burned: 4529 (million)
% 49.32/11.57 % (16378)------------------------------
% 49.32/11.57 % (16378)------------------------------
% 49.32/11.59 % (16402)lrs+10_1:40_av=off:fde=unused:ntd=on:plsq=on:plsqc=2:plsqr=411915,1048576:sp=unary_frequency:tgt=full:i=1202:si=on:rtra=on_0 on DTF2THF_16103 for (2935ds/1202Mi)
% 50.12/11.69 % (16400)Instruction limit reached!
% 50.12/11.69 % (16400)------------------------------
% 50.12/11.69 % (16400)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 50.12/11.69 % (16400)Termination reason: Unknown
% 50.12/11.69 % (16400)Termination phase: shuffling
% 50.12/11.69
% 50.12/11.69 % (16400)Memory used [KB]: 24434
% 50.12/11.69 % (16400)Time elapsed: 0.234 s
% 50.12/11.69 % (16400)Instructions burned: 673 (million)
% 50.12/11.69 % (16400)------------------------------
% 50.12/11.69 % (16400)------------------------------
% 50.66/11.70 % (16403)dis+21_1:1_au=on:cnfonf=lazy_pi_sigma_gen:hfsq=on:hfsqc=40:hfsqr=8,1:ntd=on:sos=on:sp=occurrence:ss=axioms:i=1493:si=on:rtra=on_0 on DTF2THF_16103 for (2934ds/1493Mi)
% 50.66/11.71 % (16395)Instruction limit reached!
% 50.66/11.71 % (16395)------------------------------
% 50.66/11.71 % (16395)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 50.66/11.71 % (16395)Termination reason: Unknown
% 50.66/11.71 % (16395)Termination phase: Property scanning
% 50.66/11.71
% 50.66/11.71 % (16395)Memory used [KB]: 37227
% 50.66/11.71 % (16395)Time elapsed: 0.728 s
% 50.66/11.71 % (16395)Instructions burned: 2404 (million)
% 50.66/11.71 % (16395)------------------------------
% 50.66/11.71 % (16395)------------------------------
% 50.66/11.72 % (16404)dis+1010_3:1_av=off:fde=unused:hud=10:nwc=10.0:s2a=on:ss=axioms:i=7679:si=on:rtra=on_0 on DTF2THF_16103 for (2934ds/7679Mi)
% 51.09/11.96 % (16402)Instruction limit reached!
% 51.09/11.96 % (16402)------------------------------
% 51.09/11.96 % (16402)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 51.09/11.96 % (16402)Termination reason: Unknown
% 51.09/11.96 % (16402)Termination phase: Property scanning
% 51.09/11.96
% 51.09/11.96 % (16402)Memory used [KB]: 26225
% 51.09/11.96 % (16402)Time elapsed: 0.377 s
% 51.09/11.96 % (16402)Instructions burned: 1205 (million)
% 51.09/11.96 % (16402)------------------------------
% 51.09/11.96 % (16402)------------------------------
% 51.09/11.98 % (16405)lrs+2_1:1024_anc=all:cnfonf=off:cs=on:e2e=on:nm=20:s2at=1.5:slsq=on:slsqr=1,2:spb=goal:i=340:si=on:rtra=on_0 on DTF2THF_16103 for (2931ds/340Mi)
% 53.79/12.10 % (16405)Instruction limit reached!
% 53.79/12.10 % (16405)------------------------------
% 53.79/12.10 % (16405)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 53.79/12.10 % (16405)Termination reason: Unknown
% 53.79/12.10 % (16405)Termination phase: shuffling
% 53.79/12.10
% 53.79/12.10 % (16405)Memory used [KB]: 18166
% 53.79/12.10 % (16405)Time elapsed: 0.122 s
% 53.79/12.10 % (16405)Instructions burned: 341 (million)
% 53.79/12.10 % (16405)------------------------------
% 53.79/12.10 % (16405)------------------------------
% 53.79/12.11 % (16406)dis+1002_1:1_acc=model:au=on:nwc=5.0:sd=5:ss=axioms:st=2.5:i=3672:si=on:rtra=on_0 on DTF2THF_16103 for (2930ds/3672Mi)
% 54.53/12.12 % (16401)Instruction limit reached!
% 54.53/12.12 % (16401)------------------------------
% 54.53/12.12 % (16401)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.53/12.12 % (16401)Termination reason: Unknown
% 54.53/12.12 % (16401)Termination phase: Property scanning
% 54.53/12.12
% 54.53/12.12 % (16401)Memory used [KB]: 31470
% 54.53/12.12 % (16401)Time elapsed: 0.593 s
% 54.53/12.12 % (16401)Instructions burned: 1898 (million)
% 54.53/12.12 % (16401)------------------------------
% 54.53/12.12 % (16401)------------------------------
% 54.53/12.13 % (16407)dis-1002_3:1_acc=model:bd=off:bsd=on:bsdmm=1:cnfonf=lazy_not_be_gen:fd=off:fe=off:ntd=on:nwc=10.0:piset=equals:plsq=on:plsqc=1:plsqr=32,1:sp=const_max:uhcvi=on:i=1535:si=on:rtra=on_0 on DTF2THF_16103 for (2930ds/1535Mi)
% 54.53/12.14 % (16403)Instruction limit reached!
% 54.53/12.14 % (16403)------------------------------
% 54.53/12.14 % (16403)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.53/12.14 % (16403)Termination reason: Unknown
% 54.53/12.14 % (16403)Termination phase: shuffling
% 54.53/12.14
% 54.53/12.14 % (16403)Memory used [KB]: 40169
% 54.53/12.14 % (16403)Time elapsed: 0.436 s
% 54.53/12.14 % (16403)Instructions burned: 1494 (million)
% 54.53/12.14 % (16403)------------------------------
% 54.53/12.14 % (16403)------------------------------
% 54.53/12.15 % (16408)dis+10_16:1_hud=1:plsq=on:plsqc=1:plsqr=32,1:prag=on:sd=1:sgt=64:ss=axioms:i=4637:si=on:rtra=on_0 on DTF2THF_16103 for (2930ds/4637Mi)
% 54.92/12.21 % (16375)Instruction limit reached!
% 54.92/12.21 % (16375)------------------------------
% 54.92/12.21 % (16375)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.92/12.21 % (16375)Termination reason: Unknown
% 54.92/12.21 % (16375)Termination phase: Saturation
% 54.92/12.21
% 54.92/12.21 % (16375)Memory used [KB]: 36076
% 54.92/12.21 % (16375)Time elapsed: 2.012 s
% 54.92/12.21 % (16375)Instructions burned: 6455 (million)
% 54.92/12.21 % (16375)------------------------------
% 54.92/12.21 % (16375)------------------------------
% 54.92/12.22 % (16409)lrs+10_1:1_fde=unused:prag=on:sd=2:sgt=32:ss=axioms:i=3174:si=on:rtra=on_0 on DTF2THF_16103 for (2929ds/3174Mi)
% 55.24/12.31 % (16394)Instruction limit reached!
% 55.24/12.31 % (16394)------------------------------
% 55.24/12.31 % (16394)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 55.24/12.31 % (16394)Termination reason: Unknown
% 55.24/12.31 % (16394)Termination phase: Function definition elimination
% 55.24/12.31
% 55.24/12.31 % (16394)Memory used [KB]: 58335
% 55.24/12.31 % (16394)Time elapsed: 1.341 s
% 55.24/12.31 % (16394)Instructions burned: 5061 (million)
% 55.24/12.31 % (16394)------------------------------
% 55.24/12.31 % (16394)------------------------------
% 55.24/12.32 % (16410)lrs+1010_1:10_au=on:bd=off:cnfonf=lazy_pi_sigma_gen:sp=occurrence:i=307:si=on:rtra=on_0 on DTF2THF_16103 for (2928ds/307Mi)
% 55.48/12.42 % (16410)Instruction limit reached!
% 55.48/12.42 % (16410)------------------------------
% 55.48/12.42 % (16410)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 55.48/12.42 % (16410)Termination reason: Unknown
% 55.48/12.42 % (16410)Termination phase: shuffling
% 55.48/12.42
% 55.48/12.42 % (16410)Memory used [KB]: 19957
% 55.48/12.42 % (16410)Time elapsed: 0.096 s
% 55.48/12.42 % (16410)Instructions burned: 307 (million)
% 55.48/12.42 % (16410)------------------------------
% 55.48/12.42 % (16410)------------------------------
% 55.48/12.43 % (16411)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=3619:si=on:rtra=on_0 on DTF2THF_16103 for (2927ds/3619Mi)
% 55.78/12.55 % (16396)Instruction limit reached!
% 55.78/12.55 % (16396)------------------------------
% 55.78/12.55 % (16396)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 55.78/12.55 % (16396)Termination reason: Unknown
% 55.78/12.55 % (16396)Termination phase: Property scanning
% 55.78/12.55
% 55.78/12.55 % (16396)Memory used [KB]: 51683
% 55.78/12.55 % (16396)Time elapsed: 1.537 s
% 55.78/12.55 % (16396)Instructions burned: 5799 (million)
% 55.78/12.55 % (16396)------------------------------
% 55.78/12.55 % (16396)------------------------------
% 55.78/12.57 % (16412)lrs+1002_1:1_au=on:cbe=off:fde=none:ins=1:sp=const_max:i=2325:si=on:rtra=on_0 on DTF2THF_16103 for (2926ds/2325Mi)
% 55.78/12.58 % (16407)Instruction limit reached!
% 55.78/12.58 % (16407)------------------------------
% 55.78/12.58 % (16407)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 55.78/12.58 % (16407)Termination reason: Unknown
% 55.78/12.58 % (16407)Termination phase: shuffling
% 55.78/12.58
% 55.78/12.58 % (16407)Memory used [KB]: 40681
% 55.78/12.58 % (16407)Time elapsed: 0.444 s
% 55.78/12.58 % (16407)Instructions burned: 1537 (million)
% 55.78/12.58 % (16407)------------------------------
% 55.78/12.58 % (16407)------------------------------
% 55.78/12.59 % (16413)dis+1002_1:1_au=on:sac=on:sd=5:slsq=on:slsqr=1,4:ss=axioms:st=2.0:i=2341:si=on:rtra=on_0 on DTF2THF_16103 for (2925ds/2341Mi)
% 61.51/13.17 % (16406)Instruction limit reached!
% 61.51/13.17 % (16406)------------------------------
% 61.51/13.17 % (16406)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 61.51/13.17 % (16406)Termination reason: Unknown
% 61.51/13.17 % (16406)Termination phase: Property scanning
% 61.51/13.17
% 61.51/13.17 % (16406)Memory used [KB]: 37995
% 61.51/13.17 % (16406)Time elapsed: 1.060 s
% 61.51/13.17 % (16406)Instructions burned: 3673 (million)
% 61.51/13.17 % (16406)------------------------------
% 61.51/13.17 % (16406)------------------------------
% 61.51/13.18 % (16414)lrs+1010_1:16_apa=on:bsd=on:cnfonf=off:hud=10:sac=on:sos=on:ss=axioms:i=7257:si=on:rtra=on_0 on DTF2THF_16103 for (2920ds/7257Mi)
% 62.07/13.23 % (16409)Instruction limit reached!
% 62.07/13.23 % (16409)------------------------------
% 62.07/13.23 % (16409)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 62.07/13.23 % (16409)Termination reason: Unknown
% 62.07/13.23 % (16409)Termination phase: Saturation
% 62.07/13.23
% 62.07/13.23 % (16409)Memory used [KB]: 36843
% 62.07/13.23 % (16409)Time elapsed: 1.006 s
% 62.07/13.23 % (16409)Instructions burned: 3174 (million)
% 62.07/13.23 % (16409)------------------------------
% 62.07/13.23 % (16409)------------------------------
% 62.07/13.24 % (16415)ott+1010_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=3548551,65536:s2a=on:i=3710:si=on:rtra=on_0 on DTF2THF_16103 for (2919ds/3710Mi)
% 62.07/13.27 % (16412)Instruction limit reached!
% 62.07/13.27 % (16412)------------------------------
% 62.07/13.27 % (16412)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 62.07/13.27 % (16412)Termination reason: Unknown
% 62.07/13.27 % (16412)Termination phase: Property scanning
% 62.07/13.27
% 62.07/13.27 % (16412)Memory used [KB]: 37227
% 62.07/13.27 % (16412)Time elapsed: 0.704 s
% 62.07/13.27 % (16412)Instructions burned: 2325 (million)
% 62.07/13.27 % (16412)------------------------------
% 62.07/13.27 % (16412)------------------------------
% 62.07/13.28 % (16416)lrs+1010_1:1_avsq=on:avsqr=1,16:kws=inv_arity:nwc=5.0:sd=1:ss=axioms:i=326:si=on:rtra=on_0 on DTF2THF_16103 for (2919ds/326Mi)
% 62.60/13.35 % (16413)Instruction limit reached!
% 62.60/13.35 % (16413)------------------------------
% 62.60/13.35 % (16413)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 62.60/13.35 % (16413)Termination reason: Unknown
% 62.60/13.35 % (16413)Termination phase: Property scanning
% 62.60/13.35
% 62.60/13.35 % (16413)Memory used [KB]: 32750
% 62.60/13.35 % (16413)Time elapsed: 0.758 s
% 62.60/13.35 % (16413)Instructions burned: 2341 (million)
% 62.60/13.35 % (16413)------------------------------
% 62.60/13.35 % (16413)------------------------------
% 62.60/13.36 % (16417)lrs+1010_1:64_au=on:c=on:e2e=on:flr=on:hfsq=on:hfsql=off:hfsqr=4,1:ins=1:ntd=on:sp=occurrence:i=1100:si=on:rtra=on_0 on DTF2THF_16103 for (2918ds/1100Mi)
% 62.60/13.40 % (16416)Instruction limit reached!
% 62.60/13.40 % (16416)------------------------------
% 62.60/13.40 % (16416)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 62.60/13.40 % (16416)Termination reason: Unknown
% 62.60/13.40 % (16416)Termination phase: shuffling
% 62.60/13.40
% 62.60/13.40 % (16416)Memory used [KB]: 18038
% 62.60/13.40 % (16416)Time elapsed: 0.116 s
% 62.60/13.40 % (16416)Instructions burned: 328 (million)
% 62.60/13.40 % (16416)------------------------------
% 62.60/13.40 % (16416)------------------------------
% 62.98/13.41 % (16418)lrs+10_1:1_au=on:cnfonf=off:fde=unused:kws=precedence:tgt=full:i=406:si=on:rtra=on_0 on DTF2THF_16103 for (2917ds/406Mi)
% 62.98/13.49 % (16411)Instruction limit reached!
% 62.98/13.49 % (16411)------------------------------
% 62.98/13.49 % (16411)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 62.98/13.49 % (16411)Termination reason: Unknown
% 62.98/13.49 % (16411)Termination phase: Property scanning
% 62.98/13.49
% 62.98/13.49 % (16411)Memory used [KB]: 41193
% 62.98/13.49 % (16411)Time elapsed: 1.058 s
% 62.98/13.49 % (16411)Instructions burned: 3621 (million)
% 62.98/13.49 % (16411)------------------------------
% 62.98/13.49 % (16411)------------------------------
% 63.38/13.50 % (16419)dis+1010_1:1_aac=none:bd=preordered:e2e=on:fde=none:fe=axiom:nm=16:nwc=10.0:prag=on:i=1871:si=on:rtra=on_0 on DTF2THF_16103 for (2916ds/1871Mi)
% 63.38/13.55 % (16418)Instruction limit reached!
% 63.38/13.55 % (16418)------------------------------
% 63.38/13.55 % (16418)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 63.38/13.55 % (16418)Termination reason: Unknown
% 63.38/13.55 % (16418)Termination phase: shuffling
% 63.38/13.55
% 63.38/13.55 % (16418)Memory used [KB]: 18933
% 63.38/13.55 % (16418)Time elapsed: 0.141 s
% 63.38/13.55 % (16418)Instructions burned: 409 (million)
% 63.38/13.55 % (16418)------------------------------
% 63.38/13.55 % (16418)------------------------------
% 63.38/13.56 % (16420)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1194:si=on:rtra=on_0 on DTF2THF_16103 for (2916ds/1194Mi)
% 63.38/13.58 % (16408)Instruction limit reached!
% 63.38/13.58 % (16408)------------------------------
% 63.38/13.58 % (16408)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 63.38/13.58 % (16408)Termination reason: Unknown
% 63.38/13.59 % (16408)Termination phase: Saturation
% 63.38/13.59
% 63.38/13.59 % (16408)Memory used [KB]: 34668
% 63.38/13.59 % (16408)Time elapsed: 1.437 s
% 63.38/13.59 % (16408)Instructions burned: 4639 (million)
% 63.38/13.59 % (16408)------------------------------
% 63.38/13.59 % (16408)------------------------------
% 63.74/13.60 % (16421)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_16103 for (2915ds/474Mi)
% 63.84/13.72 % (16417)Instruction limit reached!
% 63.84/13.72 % (16417)------------------------------
% 63.84/13.72 % (16417)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 63.84/13.72 % (16417)Termination reason: Unknown
% 63.84/13.72 % (16417)Termination phase: Property scanning
% 63.84/13.72
% 63.84/13.72 % (16417)Memory used [KB]: 26353
% 63.84/13.72 % (16417)Time elapsed: 0.357 s
% 63.84/13.72 % (16417)Instructions burned: 1103 (million)
% 63.84/13.72 % (16417)------------------------------
% 63.84/13.72 % (16417)------------------------------
% 63.84/13.73 % (16422)dis+1010_1:1_au=on:cnfonf=lazy_gen:ins=3:plsq=on:plsqc=1:plsqr=15142123,262144:sp=reverse_arity:i=3115:si=on:rtra=on_0 on DTF2THF_16103 for (2914ds/3115Mi)
% 63.84/13.74 % (16421)Instruction limit reached!
% 63.84/13.74 % (16421)------------------------------
% 63.84/13.74 % (16421)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 63.84/13.74 % (16421)Termination reason: Unknown
% 63.84/13.74 % (16421)Termination phase: shuffling
% 63.84/13.74
% 63.84/13.74 % (16421)Memory used [KB]: 22259
% 63.84/13.74 % (16421)Time elapsed: 0.143 s
% 63.84/13.74 % (16421)Instructions burned: 474 (million)
% 63.84/13.74 % (16421)------------------------------
% 63.84/13.74 % (16421)------------------------------
% 63.84/13.75 % (16423)dis+10_1:1024_kws=precedence:prag=on:rawr=on:sos=all:sp=unary_first:i=831:si=on:rtra=on_0 on DTF2THF_16103 for (2914ds/831Mi)
% 64.56/13.94 % (16420)Instruction limit reached!
% 64.56/13.94 % (16420)------------------------------
% 64.56/13.94 % (16420)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 64.56/13.94 % (16420)Termination reason: Unknown
% 64.56/13.94 % (16420)Termination phase: Property scanning
% 64.56/13.94
% 64.56/13.94 % (16420)Memory used [KB]: 26353
% 64.56/13.94 % (16420)Time elapsed: 0.378 s
% 64.56/13.94 % (16420)Instructions burned: 1194 (million)
% 64.56/13.94 % (16420)------------------------------
% 64.56/13.94 % (16420)------------------------------
% 64.56/13.95 % (16424)dis+1666_14:1_cnfonf=lazy_gen:fe=off:hud=10:piset=all_but_not_eq:prag=on:sp=const_max:i=1188:si=on:rtra=on_0 on DTF2THF_16103 for (2912ds/1188Mi)
% 64.56/13.99 % (16404)Instruction limit reached!
% 64.56/13.99 % (16404)------------------------------
% 64.56/13.99 % (16404)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 64.56/13.99 % (16404)Termination reason: Unknown
% 64.56/13.99 % (16404)Termination phase: Saturation
% 64.56/13.99
% 64.56/13.99 % (16404)Memory used [KB]: 43624
% 64.56/13.99 % (16404)Time elapsed: 2.269 s
% 64.56/13.99 % (16404)Instructions burned: 7679 (million)
% 64.56/13.99 % (16404)------------------------------
% 64.56/13.99 % (16404)------------------------------
% 64.56/14.00 % (16425)ott+1010_1:20_cnfonf=lazy_pi_sigma_gen:fe=axiom:plsq=on:plsqc=2:plsqr=32,1:sac=on:sos=all:i=2868:si=on:rtra=on_0 on DTF2THF_16103 for (2911ds/2868Mi)
% 67.78/14.03 % (16423)Instruction limit reached!
% 67.78/14.03 % (16423)------------------------------
% 67.78/14.03 % (16423)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 67.78/14.03 % (16423)Termination reason: Unknown
% 67.78/14.03 % (16423)Termination phase: Property scanning
% 67.78/14.03
% 67.78/14.03 % (16423)Memory used [KB]: 26225
% 67.78/14.03 % (16423)Time elapsed: 0.284 s
% 67.78/14.03 % (16423)Instructions burned: 831 (million)
% 67.78/14.03 % (16423)------------------------------
% 67.78/14.03 % (16423)------------------------------
% 67.78/14.05 % (16426)dis+10_1:4_atotf=0.1:avsq=on:bsr=unit_only:e2e=on:sd=1:sp=const_max:ss=axioms:i=1516:si=on:rtra=on_0 on DTF2THF_16103 for (2911ds/1516Mi)
% 67.78/14.10 % (16419)Instruction limit reached!
% 67.78/14.10 % (16419)------------------------------
% 67.78/14.10 % (16419)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 67.78/14.10 % (16419)Termination reason: Unknown
% 67.78/14.10 % (16419)Termination phase: Preprocessing 3
% 67.78/14.10
% 67.78/14.10 % (16419)Memory used [KB]: 36715
% 67.78/14.10 % (16419)Time elapsed: 0.600 s
% 67.78/14.10 % (16419)Instructions burned: 1871 (million)
% 67.78/14.10 % (16419)------------------------------
% 67.78/14.10 % (16419)------------------------------
% 67.78/14.11 % (16427)lrs+1002_1:20_afp=1000:au=on:bsd=on:cnfonf=lazy_not_gen_be_off:sos=all:sp=const_max:i=2321:si=on:rtra=on_0 on DTF2THF_16103 for (2910ds/2321Mi)
% 68.84/14.26 % (16415)Instruction limit reached!
% 68.84/14.26 % (16415)------------------------------
% 68.84/14.26 % (16415)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 68.84/14.26 % (16415)Termination reason: Unknown
% 68.84/14.26 % (16415)Termination phase: Property scanning
% 68.84/14.26
% 68.84/14.26 % (16415)Memory used [KB]: 43240
% 68.84/14.26 % (16415)Time elapsed: 1.022 s
% 68.84/14.26 % (16415)Instructions burned: 3712 (million)
% 68.84/14.26 % (16415)------------------------------
% 68.84/14.26 % (16415)------------------------------
% 68.84/14.27 % (16428)dis+1010_1:1_au=on:bd=off:bsd=on:ixr=off:sp=const_max:i=2812:si=on:rtra=on_0 on DTF2THF_16103 for (2908ds/2812Mi)
% 68.84/14.30 % (16424)Instruction limit reached!
% 68.84/14.30 % (16424)------------------------------
% 68.84/14.30 % (16424)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 68.84/14.30 % (16424)Termination reason: Unknown
% 68.84/14.30 % (16424)Termination phase: shuffling
% 68.84/14.30
% 68.84/14.30 % (16424)Memory used [KB]: 36076
% 68.84/14.30 % (16424)Time elapsed: 0.350 s
% 68.84/14.30 % (16424)Instructions burned: 1190 (million)
% 68.84/14.30 % (16424)------------------------------
% 68.84/14.30 % (16424)------------------------------
% 68.97/14.31 % (16429)dis+1002_8:1_bd=off:sd=1:sgt=16:ss=axioms:i=4604:si=on:rtra=on_0 on DTF2THF_16103 for (2908ds/4604Mi)
% 69.94/14.52 % (16426)Instruction limit reached!
% 69.94/14.52 % (16426)------------------------------
% 69.94/14.52 % (16426)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 69.94/14.52 % (16426)Termination reason: Unknown
% 69.94/14.52 % (16426)Termination phase: SInE selection
% 69.94/14.52
% 69.94/14.52 % (16426)Memory used [KB]: 26481
% 69.94/14.52 % (16426)Time elapsed: 0.477 s
% 69.94/14.52 % (16426)Instructions burned: 1518 (million)
% 69.94/14.52 % (16426)------------------------------
% 69.94/14.52 % (16426)------------------------------
% 69.94/14.53 % (16430)ott+1002_1:4_c=on:chr=on:cnfonf=lazy_not_be_gen:hud=10:nm=10:piset=all_but_not_eq:rp=on:spb=goal:uhcvi=on:i=2678:si=on:rtra=on_0 on DTF2THF_16103 for (2906ds/2678Mi)
% 70.72/14.63 % (16422)Instruction limit reached!
% 70.72/14.63 % (16422)------------------------------
% 70.72/14.63 % (16422)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 70.72/14.63 % (16422)Termination reason: Unknown
% 70.72/14.63 % (16422)Termination phase: Property scanning
% 70.72/14.63
% 70.72/14.63 % (16422)Memory used [KB]: 43112
% 70.72/14.63 % (16422)Time elapsed: 0.899 s
% 70.72/14.63 % (16422)Instructions burned: 3116 (million)
% 70.72/14.63 % (16422)------------------------------
% 70.72/14.63 % (16422)------------------------------
% 70.72/14.64 % (16431)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_16103 for (2905ds/643Mi)
% 70.98/14.75 % (16427)Instruction limit reached!
% 70.98/14.75 % (16427)------------------------------
% 70.98/14.75 % (16427)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 70.98/14.75 % (16427)Termination reason: Unknown
% 70.98/14.75 % (16427)Termination phase: Property scanning
% 70.98/14.75
% 70.98/14.75 % (16427)Memory used [KB]: 41577
% 70.98/14.75 % (16427)Time elapsed: 0.645 s
% 70.98/14.75 % (16427)Instructions burned: 2321 (million)
% 70.98/14.75 % (16427)------------------------------
% 70.98/14.75 % (16427)------------------------------
% 70.98/14.77 % (16432)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_be_gen:ntd=on:pe=on:i=458:si=on:rtra=on_0 on DTF2THF_16103 for (2904ds/458Mi)
% 70.98/14.79 % (16425)Instruction limit reached!
% 70.98/14.79 % (16425)------------------------------
% 70.98/14.79 % (16425)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 70.98/14.79 % (16425)Termination reason: Unknown
% 70.98/14.79 % (16425)Termination phase: Property scanning
% 70.98/14.79
% 70.98/14.79 % (16425)Memory used [KB]: 43112
% 70.98/14.79 % (16425)Time elapsed: 0.793 s
% 70.98/14.79 % (16425)Instructions burned: 2869 (million)
% 70.98/14.79 % (16425)------------------------------
% 70.98/14.79 % (16425)------------------------------
% 71.03/14.81 % (16433)lrs+1010_1:1_anc=all_dependent:apa=on:au=on:bd=off:c=on:cnfonf=off:ntd=on:i=2546:si=on:rtra=on_0 on DTF2THF_16103 for (2903ds/2546Mi)
% 71.03/14.84 % (16431)Instruction limit reached!
% 71.03/14.84 % (16431)------------------------------
% 71.03/14.84 % (16431)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 71.03/14.84 % (16431)Termination reason: Unknown
% 71.03/14.84 % (16431)Termination phase: shuffling
% 71.03/14.84
% 71.03/14.84 % (16431)Memory used [KB]: 24690
% 71.03/14.84 % (16431)Time elapsed: 0.197 s
% 71.03/14.84 % (16431)Instructions burned: 645 (million)
% 71.03/14.84 % (16431)------------------------------
% 71.03/14.84 % (16431)------------------------------
% 71.03/14.85 % (16434)dis+21_1:1_acc=model:au=on:cnfonf=lazy_gen:ins=2:plsq=on:plsqc=1:plsqr=32,1:i=1432:si=on:rtra=on_0 on DTF2THF_16103 for (2903ds/1432Mi)
% 72.14/14.90 % (16432)Instruction limit reached!
% 72.14/14.90 % (16432)------------------------------
% 72.14/14.90 % (16432)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 72.14/14.90 % (16432)Termination reason: Unknown
% 72.14/14.90 % (16432)Termination phase: shuffling
% 72.14/14.90
% 72.14/14.90 % (16432)Memory used [KB]: 22131
% 72.14/14.90 % (16432)Time elapsed: 0.139 s
% 72.14/14.90 % (16432)Instructions burned: 460 (million)
% 72.14/14.90 % (16432)------------------------------
% 72.14/14.90 % (16432)------------------------------
% 72.14/14.92 % (16435)dis+1002_1:1_cnfonf=lazy_not_gen:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=386:si=on:rtra=on_0 on DTF2THF_16103 for (2902ds/386Mi)
% 74.82/15.03 % (16435)Instruction limit reached!
% 74.82/15.03 % (16435)------------------------------
% 74.82/15.03 % (16435)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 74.82/15.03 % (16435)Termination reason: Unknown
% 74.82/15.03 % (16435)Termination phase: shuffling
% 74.82/15.03
% 74.82/15.03 % (16435)Memory used [KB]: 21108
% 74.82/15.03 % (16435)Time elapsed: 0.119 s
% 74.82/15.03 % (16435)Instructions burned: 387 (million)
% 74.82/15.03 % (16435)------------------------------
% 74.82/15.03 % (16435)------------------------------
% 74.82/15.05 % (16436)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_16103 for (2901ds/360Mi)
% 75.46/15.13 % (16428)Instruction limit reached!
% 75.46/15.13 % (16428)------------------------------
% 75.46/15.13 % (16428)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 75.46/15.13 % (16428)Termination reason: Unknown
% 75.46/15.13 % (16428)Termination phase: Function definition elimination
% 75.46/15.13
% 75.46/15.13 % (16428)Memory used [KB]: 37867
% 75.46/15.13 % (16428)Time elapsed: 0.849 s
% 75.46/15.13 % (16428)Instructions burned: 2812 (million)
% 75.46/15.13 % (16428)------------------------------
% 75.46/15.13 % (16428)------------------------------
% 75.46/15.14 % (16437)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=5446:si=on:rtra=on_0 on DTF2THF_16103 for (2900ds/5446Mi)
% 75.46/15.15 % (16436)Instruction limit reached!
% 75.46/15.15 % (16436)------------------------------
% 75.46/15.15 % (16436)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 75.46/15.15 % (16436)Termination reason: Unknown
% 75.46/15.15 % (16436)Termination phase: shuffling
% 75.46/15.15
% 75.46/15.15 % (16436)Memory used [KB]: 20724
% 75.46/15.15 % (16436)Time elapsed: 0.111 s
% 75.46/15.15 % (16436)Instructions burned: 363 (million)
% 75.46/15.15 % (16436)------------------------------
% 75.46/15.15 % (16436)------------------------------
% 75.46/15.17 % (16438)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=2573:si=on:rtra=on_0 on DTF2THF_16103 for (2900ds/2573Mi)
% 75.98/15.27 % (16434)Instruction limit reached!
% 75.98/15.27 % (16434)------------------------------
% 75.98/15.27 % (16434)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 75.98/15.27 % (16434)Termination reason: Unknown
% 75.98/15.27 % (16434)Termination phase: shuffling
% 75.98/15.27
% 75.98/15.27 % (16434)Memory used [KB]: 39274
% 75.98/15.27 % (16434)Time elapsed: 0.417 s
% 75.98/15.27 % (16434)Instructions burned: 1432 (million)
% 75.98/15.27 % (16434)------------------------------
% 75.98/15.27 % (16434)------------------------------
% 75.98/15.28 % (16430)Instruction limit reached!
% 75.98/15.28 % (16430)------------------------------
% 75.98/15.28 % (16430)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 75.98/15.28 % (16430)Termination reason: Unknown
% 75.98/15.28 % (16430)Termination phase: Property scanning
% 75.98/15.28
% 75.98/15.28 % (16430)Memory used [KB]: 43112
% 75.98/15.28 % (16430)Time elapsed: 0.746 s
% 75.98/15.28 % (16430)Instructions burned: 2679 (million)
% 75.98/15.28 % (16430)------------------------------
% 75.98/15.28 % (16430)------------------------------
% 75.98/15.29 % (16439)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=2655:si=on:rtra=on_0 on DTF2THF_16103 for (2898ds/2655Mi)
% 75.98/15.29 % (16440)dis+1002_16:1_apa=on:au=on:av=off:cnfonf=off:e2e=on:nm=0:nwc=10.0:i=2781:si=on:rtra=on_0 on DTF2THF_16103 for (2898ds/2781Mi)
% 76.47/15.39 % (16414)Instruction limit reached!
% 76.47/15.39 % (16414)------------------------------
% 76.47/15.39 % (16414)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 76.47/15.39 % (16414)Termination reason: Unknown
% 76.47/15.39 % (16414)Termination phase: Saturation
% 76.47/15.39
% 76.47/15.39 % (16414)Memory used [KB]: 69337
% 76.47/15.39 % (16414)Time elapsed: 2.211 s
% 76.47/15.39 % (16414)Instructions burned: 7257 (million)
% 76.47/15.39 % (16414)------------------------------
% 76.47/15.39 % (16414)------------------------------
% 76.77/15.40 % (16441)lrs+1010_1:10_au=on:bd=off:cnfonf=lazy_pi_sigma_gen:sp=occurrence:i=307:si=on:rtra=on_0 on DTF2THF_16103 for (2897ds/307Mi)
% 76.97/15.50 % (16441)Instruction limit reached!
% 76.97/15.50 % (16441)------------------------------
% 76.97/15.50 % (16441)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 76.97/15.50 % (16441)Termination reason: Unknown
% 76.97/15.50 % (16441)Termination phase: shuffling
% 76.97/15.50
% 76.97/15.50 % (16441)Memory used [KB]: 20084
% 76.97/15.50 % (16441)Time elapsed: 0.097 s
% 76.97/15.50 % (16441)Instructions burned: 309 (million)
% 76.97/15.50 % (16441)------------------------------
% 76.97/15.50 % (16441)------------------------------
% 76.97/15.51 % (16442)lrs+1010_1:1_avsq=on:avsqr=1,16:kws=inv_arity:nwc=5.0:sd=1:ss=axioms:i=1548:si=on:rtra=on_0 on DTF2THF_16103 for (2896ds/1548Mi)
% 76.97/15.58 % (16433)Instruction limit reached!
% 76.97/15.58 % (16433)------------------------------
% 76.97/15.58 % (16433)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 76.97/15.58 % (16433)Termination reason: Unknown
% 76.97/15.58 % (16433)Termination phase: Property scanning
% 76.97/15.58
% 76.97/15.58 % (16433)Memory used [KB]: 38506
% 76.97/15.58 % (16433)Time elapsed: 0.776 s
% 76.97/15.58 % (16433)Instructions burned: 2547 (million)
% 76.97/15.58 % (16433)------------------------------
% 76.97/15.58 % (16433)------------------------------
% 76.97/15.60 % (16443)lrs+10_1:3_alpa=true:hud=3:ins=1:ntd=on:plsq=on:plsqc=1:plsqr=32,1:prag=on:s2a=on:s2agt=10:i=5214:si=on:rtra=on_0 on DTF2THF_16103 for (2895ds/5214Mi)
% 77.38/15.71 % (16429)Instruction limit reached!
% 77.38/15.71 % (16429)------------------------------
% 77.38/15.71 % (16429)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 77.38/15.71 % (16429)Termination reason: Unknown
% 77.38/15.71 % (16429)Termination phase: Saturation
% 77.38/15.71
% 77.38/15.71 % (16429)Memory used [KB]: 35052
% 77.38/15.71 % (16429)Time elapsed: 1.404 s
% 77.38/15.71 % (16429)Instructions burned: 4605 (million)
% 77.38/15.71 % (16429)------------------------------
% 77.38/15.71 % (16429)------------------------------
% 77.38/15.72 % (16444)dis+21_1:1_aac=none:bsr=on:nwc=3.0:s2a=on:i=3201:si=on:rtra=on_0 on DTF2THF_16103 for (2894ds/3201Mi)
% 77.40/15.87 % (16438)Instruction limit reached!
% 77.40/15.87 % (16438)------------------------------
% 77.40/15.87 % (16438)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 77.40/15.87 % (16438)Termination reason: Unknown
% 77.40/15.87 % (16438)Termination phase: Property scanning
% 77.40/15.87
% 77.40/15.87 % (16438)Memory used [KB]: 41577
% 77.40/15.87 % (16438)Time elapsed: 0.705 s
% 77.40/15.87 % (16438)Instructions burned: 2574 (million)
% 77.40/15.87 % (16438)------------------------------
% 77.40/15.87 % (16438)------------------------------
% 77.40/15.88 % (16445)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_16103 for (2892ds/454Mi)
% 78.13/15.99 % (16442)Instruction limit reached!
% 78.13/15.99 % (16442)------------------------------
% 78.13/15.99 % (16442)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 78.13/15.99 % (16442)Termination reason: Unknown
% 78.13/15.99 % (16442)Termination phase: SInE selection
% 78.13/15.99
% 78.13/15.99 % (16442)Memory used [KB]: 26353
% 78.13/15.99 % (16442)Time elapsed: 0.477 s
% 78.13/15.99 % (16442)Instructions burned: 1548 (million)
% 78.13/15.99 % (16442)------------------------------
% 78.13/15.99 % (16442)------------------------------
% 78.13/16.00 % (16446)lrs+1002_1:32_au=on:ins=2:nm=16:ntd=on:piset=all_but_not_eq:sp=occurrence:i=3049:si=on:rtra=on_0 on DTF2THF_16103 for (2891ds/3049Mi)
% 81.74/16.02 % (16445)Instruction limit reached!
% 81.74/16.02 % (16445)------------------------------
% 81.74/16.02 % (16445)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 81.74/16.02 % (16445)Termination reason: Unknown
% 81.74/16.02 % (16445)Termination phase: shuffling
% 81.74/16.02
% 81.74/16.02 % (16445)Memory used [KB]: 22003
% 81.74/16.02 % (16445)Time elapsed: 0.138 s
% 81.74/16.02 % (16445)Instructions burned: 455 (million)
% 81.74/16.02 % (16445)------------------------------
% 81.74/16.02 % (16445)------------------------------
% 81.74/16.02 % (16439)Instruction limit reached!
% 81.74/16.02 % (16439)------------------------------
% 81.74/16.02 % (16439)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 81.74/16.02 % (16439)Termination reason: Unknown
% 81.74/16.02 % (16439)Termination phase: Naming
% 81.74/16.02
% 81.74/16.02 % (16439)Memory used [KB]: 41960
% 81.74/16.02 % (16439)Time elapsed: 0.738 s
% 81.74/16.02 % (16439)Instructions burned: 2656 (million)
% 81.74/16.02 % (16439)------------------------------
% 81.74/16.02 % (16439)------------------------------
% 81.74/16.03 % (16447)lrs+1010_1:1_bd=preordered:cnfonf=off:e2e=on:kws=inv_precedence:lwlo=on:sd=2:ss=axioms:i=7311:si=on:rtra=on_0 on DTF2THF_16103 for (2891ds/7311Mi)
% 81.74/16.04 % (16448)lrs+1002_1:14_au=on:bd=off:e2e=on:fde=unused:sac=on:sos=on:i=6804:si=on:rtra=on_0 on DTF2THF_16103 for (2891ds/6804Mi)
% 82.70/16.25 % (16440)Instruction limit reached!
% 82.70/16.25 % (16440)------------------------------
% 82.70/16.25 % (16440)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 82.70/16.25 % (16440)Termination reason: Unknown
% 82.70/16.25 % (16440)Termination phase: Property scanning
% 82.70/16.25
% 82.70/16.25 % (16440)Memory used [KB]: 295133
% 82.70/16.25 % (16440)Time elapsed: 0.956 s
% 82.70/16.25 % (16440)Instructions burned: 2784 (million)
% 82.70/16.25 % (16440)------------------------------
% 82.70/16.25 % (16440)------------------------------
% 82.70/16.28 % (16449)dis+1003_1:1_au=on:bd=preordered:fde=none:ixr=off:nwc=10.0:i=1683:si=on:rtra=on_0 on DTF2THF_16103 for (2889ds/1683Mi)
% 84.39/16.64 % (16437)Instruction limit reached!
% 84.39/16.64 % (16437)------------------------------
% 84.39/16.64 % (16437)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 84.39/16.64 % (16437)Termination reason: Unknown
% 84.39/16.64 % (16437)Termination phase: Property scanning
% 84.39/16.64
% 84.39/16.64 % (16437)Memory used [KB]: 47973
% 84.39/16.64 % (16437)Time elapsed: 1.495 s
% 84.39/16.64 % (16437)Instructions burned: 5447 (million)
% 84.39/16.64 % (16437)------------------------------
% 84.39/16.64 % (16437)------------------------------
% 84.39/16.66 % (16450)lrs+10_8:1_au=on:av=off:bd=off:cs=on:e2e=on:hfsq=on:hfsqc=50:hfsql=off:nm=0:i=4688:si=on:rtra=on_0 on DTF2THF_16103 for (2885ds/4688Mi)
% 84.39/16.67 % (16444)Instruction limit reached!
% 84.39/16.67 % (16444)------------------------------
% 84.39/16.67 % (16444)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 84.39/16.67 % (16444)Termination reason: Unknown
% 84.39/16.67 % (16444)Termination phase: Function definition elimination
% 84.39/16.67
% 84.39/16.67 % (16444)Memory used [KB]: 37995
% 84.39/16.67 % (16444)Time elapsed: 0.942 s
% 84.39/16.67 % (16444)Instructions burned: 3201 (million)
% 84.39/16.67 % (16444)------------------------------
% 84.39/16.67 % (16444)------------------------------
% 84.39/16.68 % (16451)dis-1_1:1_amm=off:au=on:cbe=off:spb=intro:i=2207:si=on:rtra=on_0 on DTF2THF_16103 for (2884ds/2207Mi)
% 84.93/16.81 % (16449)Instruction limit reached!
% 84.93/16.81 % (16449)------------------------------
% 84.93/16.81 % (16449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 84.93/16.81 % (16449)Termination reason: Unknown
% 84.93/16.81 % (16449)Termination phase: Preprocessing 3
% 84.93/16.81
% 84.93/16.81 % (16449)Memory used [KB]: 32494
% 84.93/16.81 % (16449)Time elapsed: 0.534 s
% 84.93/16.81 % (16449)Instructions burned: 1683 (million)
% 84.93/16.81 % (16449)------------------------------
% 84.93/16.81 % (16449)------------------------------
% 84.93/16.82 % (16452)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=8216:si=on:rtra=on_0 on DTF2THF_16103 for (2883ds/8216Mi)
% 84.93/16.88 % (16446)Instruction limit reached!
% 84.93/16.88 % (16446)------------------------------
% 84.93/16.88 % (16446)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 84.93/16.88 % (16446)Termination reason: Unknown
% 84.93/16.88 % (16446)Termination phase: Function definition elimination
% 84.93/16.88
% 84.93/16.88 % (16446)Memory used [KB]: 38250
% 84.93/16.88 % (16446)Time elapsed: 0.885 s
% 84.93/16.88 % (16446)Instructions burned: 3050 (million)
% 84.93/16.88 % (16446)------------------------------
% 84.93/16.88 % (16446)------------------------------
% 84.93/16.90 % (16453)lrs+1010_1:1_atotf=0.5:au=on:cbe=off:cnfonf=lazy_not_be_gen:fe=off:nwc=5.0:spb=units:i=430:si=on:rtra=on_0 on DTF2THF_16103 for (2882ds/430Mi)
% 88.71/17.02 % (16453)Instruction limit reached!
% 88.71/17.02 % (16453)------------------------------
% 88.71/17.02 % (16453)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 88.71/17.02 % (16453)Termination reason: Unknown
% 88.71/17.02 % (16453)Termination phase: shuffling
% 88.71/17.02
% 88.71/17.02 % (16453)Memory used [KB]: 21748
% 88.71/17.02 % (16453)Time elapsed: 0.131 s
% 88.71/17.02 % (16453)Instructions burned: 432 (million)
% 88.71/17.02 % (16453)------------------------------
% 88.71/17.02 % (16453)------------------------------
% 88.71/17.04 % (16454)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=2012:si=on:rtra=on_0 on DTF2THF_16103 for (2881ds/2012Mi)
% 88.71/17.06 % (16443)Instruction limit reached!
% 88.71/17.06 % (16443)------------------------------
% 88.71/17.06 % (16443)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 88.71/17.06 % (16443)Termination reason: Unknown
% 88.71/17.06 % (16443)Termination phase: Property scanning
% 88.71/17.06
% 88.71/17.06 % (16443)Memory used [KB]: 58463
% 88.71/17.06 % (16443)Time elapsed: 1.465 s
% 88.71/17.06 % (16443)Instructions burned: 5214 (million)
% 88.71/17.06 % (16443)------------------------------
% 88.71/17.06 % (16443)------------------------------
% 88.71/17.08 % (16455)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=631:si=on:rtra=on_0 on DTF2THF_16103 for (2881ds/631Mi)
% 90.24/17.30 % (16455)Instruction limit reached!
% 90.24/17.30 % (16455)------------------------------
% 90.24/17.30 % (16455)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 90.24/17.30 % (16455)Termination reason: Unknown
% 90.24/17.30 % (16455)Termination phase: shuffling
% 90.24/17.30
% 90.24/17.30 % (16455)Memory used [KB]: 23794
% 90.24/17.30 % (16455)Time elapsed: 0.220 s
% 90.24/17.30 % (16455)Instructions burned: 632 (million)
% 90.24/17.30 % (16455)------------------------------
% 90.24/17.30 % (16455)------------------------------
% 90.24/17.31 % (16456)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=8785:si=on:rtra=on_0 on DTF2THF_16103 for (2878ds/8785Mi)
% 90.24/17.37 % (16451)Instruction limit reached!
% 90.24/17.37 % (16451)------------------------------
% 90.24/17.37 % (16451)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 90.24/17.37 % (16451)Termination reason: Unknown
% 90.24/17.37 % (16451)Termination phase: Property scanning
% 90.24/17.37
% 90.24/17.37 % (16451)Memory used [KB]: 37355
% 90.24/17.37 % (16451)Time elapsed: 0.685 s
% 90.24/17.37 % (16451)Instructions burned: 2207 (million)
% 90.24/17.37 % (16451)------------------------------
% 90.24/17.37 % (16451)------------------------------
% 90.24/17.38 % (16457)lrs-1002_1:128_abs=on:bd=preordered:cbe=off:e2e=on:fde=unused:fe=off:tgt=full:i=739:si=on:rtra=on_0 on DTF2THF_16103 for (2877ds/739Mi)
% 91.72/17.61 % (16454)Instruction limit reached!
% 91.72/17.61 % (16454)------------------------------
% 91.72/17.61 % (16454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 91.72/17.61 % (16454)Termination reason: Unknown
% 91.72/17.61 % (16454)Termination phase: Property scanning
% 91.72/17.61
% 91.72/17.61 % (16454)Memory used [KB]: 41577
% 91.72/17.61 % (16454)Time elapsed: 0.572 s
% 91.72/17.61 % (16454)Instructions burned: 2013 (million)
% 91.72/17.61 % (16454)------------------------------
% 91.72/17.61 % (16454)------------------------------
% 91.72/17.62 % (16458)dis+1002_1:1024_fe=off:hud=5:prag=on:i=5473:si=on:rtra=on_0 on DTF2THF_16103 for (2875ds/5473Mi)
% 91.72/17.64 % (16457)Instruction limit reached!
% 91.72/17.64 % (16457)------------------------------
% 91.72/17.64 % (16457)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 91.72/17.64 % (16457)Termination reason: Unknown
% 91.72/17.64 % (16457)Termination phase: shuffling
% 91.72/17.64
% 91.72/17.64 % (16457)Memory used [KB]: 25585
% 91.72/17.64 % (16457)Time elapsed: 0.261 s
% 91.72/17.64 % (16457)Instructions burned: 739 (million)
% 91.72/17.64 % (16457)------------------------------
% 91.72/17.64 % (16457)------------------------------
% 91.72/17.65 % (16459)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=43570:si=on:rtra=on_0 on DTF2THF_16103 for (2875ds/43570Mi)
% 95.69/18.05 % (16450)Instruction limit reached!
% 95.69/18.05 % (16450)------------------------------
% 95.69/18.05 % (16450)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 95.69/18.05 % (16450)Termination reason: Unknown
% 95.69/18.05 % (16450)Termination phase: Property scanning
% 95.69/18.05
% 95.69/18.05 % (16450)Memory used [KB]: 294110
% 95.69/18.05 % (16450)Time elapsed: 1.380 s
% 95.69/18.05 % (16450)Instructions burned: 4690 (million)
% 95.69/18.05 % (16450)------------------------------
% 95.69/18.05 % (16450)------------------------------
% 95.69/18.08 % (16460)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=8382:si=on:rtra=on_0 on DTF2THF_16103 for (2870ds/8382Mi)
% 96.85/18.21 % (16448)Instruction limit reached!
% 96.85/18.21 % (16448)------------------------------
% 96.85/18.21 % (16448)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 96.85/18.21 % (16448)Termination reason: Unknown
% 96.85/18.21 % (16448)Termination phase: Saturation
% 96.85/18.21
% 96.85/18.21 % (16448)Memory used [KB]: 75990
% 96.85/18.21 % (16448)Time elapsed: 2.186 s
% 96.85/18.21 % (16448)Instructions burned: 6806 (million)
% 96.85/18.21 % (16448)------------------------------
% 96.85/18.21 % (16448)------------------------------
% 96.85/18.23 % (16461)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=407:si=on:rtra=on_0 on DTF2THF_16103 for (2869ds/407Mi)
% 96.85/18.29 % (16447)Instruction limit reached!
% 96.85/18.29 % (16447)------------------------------
% 96.85/18.29 % (16447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 96.85/18.29 % (16447)Termination reason: Unknown
% 96.85/18.29 % (16447)Termination phase: Saturation
% 96.85/18.29
% 96.85/18.29 % (16447)Memory used [KB]: 40681
% 96.85/18.29 % (16447)Time elapsed: 2.257 s
% 96.85/18.29 % (16447)Instructions burned: 7313 (million)
% 96.85/18.29 % (16447)------------------------------
% 96.85/18.29 % (16447)------------------------------
% 97.41/18.30 % (16462)lrs+10_1:50_cbe=off:cnfonf=lazy_pi_sigma_gen:fe=off:hud=5:plsq=on:plsqr=1,32:sos=all:i=2971:si=on:rtra=on_0 on DTF2THF_16103 for (2868ds/2971Mi)
% 97.41/18.37 % (16461)Instruction limit reached!
% 97.41/18.37 % (16461)------------------------------
% 97.41/18.37 % (16461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 97.41/18.37 % (16461)Termination reason: Unknown
% 97.41/18.37 % (16461)Termination phase: shuffling
% 97.41/18.37
% 97.41/18.37 % (16461)Memory used [KB]: 19061
% 97.41/18.37 % (16461)Time elapsed: 0.143 s
% 97.41/18.37 % (16461)Instructions burned: 409 (million)
% 97.41/18.37 % (16461)------------------------------
% 97.41/18.37 % (16461)------------------------------
% 97.41/18.38 % (16463)lrs+1002_1:1_aac=none:bd=off:e2e=on:fde=none:fe=off:piset=or:prag=on:rawr=on:sos=on:updr=off:i=9822:si=on:rtra=on_0 on DTF2THF_16103 for (2867ds/9822Mi)
% 99.96/19.00 % (16452)Instruction limit reached!
% 99.96/19.00 % (16452)------------------------------
% 99.96/19.00 % (16452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 99.96/19.00 % (16452)Termination reason: Unknown
% 99.96/19.00 % (16452)Termination phase: Property scanning
% 99.96/19.00
% 99.96/19.00 % (16452)Memory used [KB]: 56289
% 99.96/19.00 % (16452)Time elapsed: 2.169 s
% 99.96/19.00 % (16452)Instructions burned: 8218 (million)
% 99.96/19.00 % (16452)------------------------------
% 99.96/19.00 % (16452)------------------------------
% 99.96/19.01 % (16464)dis+10_5:1_av=off:ntd=on:s2a=on:s2at=3.0:sgt=16:sp=unary_frequency:ss=axioms:i=8425:si=on:rtra=on_0 on DTF2THF_16103 for (2861ds/8425Mi)
% 102.67/19.07 % (16458)Instruction limit reached!
% 102.67/19.07 % (16458)------------------------------
% 102.67/19.07 % (16458)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 102.67/19.07 % (16458)Termination reason: Unknown
% 102.67/19.07 % (16458)Termination phase: Property scanning
% 102.67/19.07
% 102.67/19.07 % (16458)Memory used [KB]: 47078
% 102.67/19.07 % (16458)Time elapsed: 1.452 s
% 102.67/19.07 % (16458)Instructions burned: 5476 (million)
% 102.67/19.07 % (16458)------------------------------
% 102.67/19.07 % (16458)------------------------------
% 102.67/19.09 % (16465)lrs+30_1:1_aac=none:au=on:cbe=off:cs=on:nm=2:rawr=on:sos=on:i=3107:si=on:rtra=on_0 on DTF2THF_16103 for (2860ds/3107Mi)
% 103.23/19.12 % (16462)Instruction limit reached!
% 103.23/19.12 % (16462)------------------------------
% 103.23/19.12 % (16462)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 103.23/19.12 % (16462)Termination reason: Unknown
% 103.23/19.12 % (16462)Termination phase: Property scanning
% 103.23/19.12
% 103.23/19.12 % (16462)Memory used [KB]: 43112
% 103.23/19.12 % (16462)Time elapsed: 0.819 s
% 103.23/19.12 % (16462)Instructions burned: 2974 (million)
% 103.23/19.12 % (16462)------------------------------
% 103.23/19.12 % (16462)------------------------------
% 103.23/19.13 % (16466)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=7967:si=on:rtra=on_0 on DTF2THF_16103 for (2860ds/7967Mi)
% 105.19/19.61 % (16456)Instruction limit reached!
% 105.19/19.61 % (16456)------------------------------
% 105.19/19.61 % (16456)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 105.19/19.61 % (16456)Termination reason: Unknown
% 105.19/19.61 % (16456)Termination phase: Property scanning
% 105.19/19.61
% 105.19/19.61 % (16456)Memory used [KB]: 52962
% 105.19/19.61 % (16456)Time elapsed: 2.299 s
% 105.19/19.61 % (16456)Instructions burned: 8786 (million)
% 105.19/19.61 % (16456)------------------------------
% 105.19/19.61 % (16456)------------------------------
% 105.19/19.62 % (16467)lrs+1010_10:959_abs=on:acc=on:bd=off:ccuc=small_ones:cha=on:cnfonf=lazy_simp:fd=preordered:fe=off:hud=3:ile=on:kws=inv_frequency:nm=2:piset=and:sd=5:sp=frequency:spb=goal_then_units:ss=axioms:st=4.5:i=3629:si=on:rtra=on_0 on DTF2THF_16103 for (2855ds/3629Mi)
% 106.13/19.99 % (16465)Instruction limit reached!
% 106.13/19.99 % (16465)------------------------------
% 106.13/19.99 % (16465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 106.13/19.99 % (16465)Termination reason: Unknown
% 106.13/19.99 % (16465)Termination phase: Function definition elimination
% 106.13/19.99
% 106.13/19.99 % (16465)Memory used [KB]: 44519
% 106.13/19.99 % (16465)Time elapsed: 0.907 s
% 106.13/19.99 % (16465)Instructions burned: 3111 (million)
% 106.13/19.99 % (16465)------------------------------
% 106.13/19.99 % (16465)------------------------------
% 106.13/20.01 % (16468)lrs+1010_1:1024_au=on:fe=axiom:lwlo=on:nm=0:ntd=on:nwc=10.0:spb=non_intro:uhcvi=on:i=939:si=on:rtra=on_0 on DTF2THF_16103 for (2851ds/939Mi)
% 110.95/20.28 % (16460)Instruction limit reached!
% 110.95/20.28 % (16460)------------------------------
% 110.95/20.28 % (16460)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 110.95/20.28 % (16460)Termination reason: Unknown
% 110.95/20.28 % (16460)Termination phase: Property scanning
% 110.95/20.28
% 110.95/20.28 % (16460)Memory used [KB]: 288609
% 110.95/20.28 % (16460)Time elapsed: 2.204 s
% 110.95/20.28 % (16460)Instructions burned: 8384 (million)
% 110.95/20.28 % (16460)------------------------------
% 110.95/20.28 % (16460)------------------------------
% 109.42/20.31 % (16469)dis+1666_14:1_cnfonf=lazy_gen:fe=off:hud=10:piset=all_but_not_eq:prag=on:sp=const_max:i=1188:si=on:rtra=on_0 on DTF2THF_16103 for (2848ds/1188Mi)
% 109.42/20.31 % (16468)Instruction limit reached!
% 109.42/20.31 % (16468)------------------------------
% 109.42/20.31 % (16468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 109.42/20.31 % (16468)Termination reason: Unknown
% 109.42/20.31 % (16468)Termination phase: Property scanning
% 109.42/20.31
% 109.42/20.31 % (16468)Memory used [KB]: 26225
% 109.42/20.31 % (16468)Time elapsed: 0.307 s
% 109.42/20.31 % (16468)Instructions burned: 942 (million)
% 109.42/20.31 % (16468)------------------------------
% 109.42/20.31 % (16468)------------------------------
% 109.42/20.32 % (16470)dis+21_3:2_avsq=on:cbe=off:e2e=on:flr=on:ile=on:nwc=5.0:piset=or:rawr=on:sp=occurrence:i=3205:si=on:rtra=on_0 on DTF2THF_16103 for (2848ds/3205Mi)
% 113.19/20.63 % (16467)Instruction limit reached!
% 113.19/20.63 % (16467)------------------------------
% 113.19/20.63 % (16467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 113.19/20.63 % (16467)Termination reason: Unknown
% 113.19/20.63 % (16467)Termination phase: Property scanning
% 113.19/20.63
% 113.19/20.63 % (16467)Memory used [KB]: 43240
% 113.19/20.63 % (16467)Time elapsed: 1.009 s
% 113.19/20.63 % (16467)Instructions burned: 3632 (million)
% 113.19/20.63 % (16467)------------------------------
% 113.19/20.63 % (16467)------------------------------
% 113.19/20.64 % (16471)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_16103 for (2845ds/865Mi)
% 113.19/20.65 % (16469)Instruction limit reached!
% 113.19/20.65 % (16469)------------------------------
% 113.19/20.65 % (16469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 113.19/20.65 % (16469)Termination reason: Unknown
% 113.19/20.65 % (16469)Termination phase: shuffling
% 113.19/20.65
% 113.19/20.65 % (16469)Memory used [KB]: 36076
% 113.19/20.65 % (16469)Time elapsed: 0.342 s
% 113.19/20.65 % (16469)Instructions burned: 1190 (million)
% 113.19/20.65 % (16469)------------------------------
% 113.19/20.65 % (16469)------------------------------
% 113.19/20.66 % (16472)ott+1002_1:1_au=on:bd=off:cbe=off:cha=on:cnfonf=lazy_pi_sigma_gen:fd=off:fe=off:ntd=on:sp=const_frequency:i=7469:si=on:rtra=on_0 on DTF2THF_16103 for (2845ds/7469Mi)
% 114.79/20.93 % (16471)Instruction limit reached!
% 114.79/20.93 % (16471)------------------------------
% 114.79/20.93 % (16471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 114.79/20.93 % (16471)Termination reason: Unknown
% 114.79/20.93 % (16471)Termination phase: shuffling
% 114.79/20.93
% 114.79/20.93 % (16471)Memory used [KB]: 26737
% 114.79/20.93 % (16471)Time elapsed: 0.290 s
% 114.79/20.93 % (16471)Instructions burned: 865 (million)
% 114.79/20.93 % (16471)------------------------------
% 114.79/20.93 % (16471)------------------------------
% 114.79/20.95 % (16473)dis+10_1:10_bd=preordered:e2e=on:fde=none:fe=axiom:hud=3:kws=precedence:nwc=3.0:prag=on:sp=const_max:tgt=full:i=4165:si=on:rtra=on_0 on DTF2THF_16103 for (2842ds/4165Mi)
% 117.35/21.18 % (16466)Instruction limit reached!
% 117.35/21.18 % (16466)------------------------------
% 117.35/21.18 % (16466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 117.35/21.18 % (16466)Termination reason: Unknown
% 117.35/21.18 % (16466)Termination phase: Property scanning
% 117.35/21.18
% 117.35/21.18 % (16466)Memory used [KB]: 59487
% 117.35/21.18 % (16466)Time elapsed: 2.048 s
% 117.35/21.18 % (16466)Instructions burned: 7970 (million)
% 117.35/21.18 % (16466)------------------------------
% 117.35/21.18 % (16466)------------------------------
% 117.35/21.19 % (16474)dis+1010_1:1_au=on:avsq=on:cnfonf=lazy_simp:fsr=off:ntd=on:nwc=10.0:sp=reverse_frequency:ss=axioms:st=1.5:i=13468:si=on:rtra=on_0 on DTF2THF_16103 for (2839ds/13468Mi)
% 117.91/21.24 % (16470)Instruction limit reached!
% 117.91/21.24 % (16470)------------------------------
% 117.91/21.24 % (16470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 117.91/21.24 % (16470)Termination reason: Unknown
% 117.91/21.24 % (16470)Termination phase: Function definition elimination
% 117.91/21.24
% 117.91/21.24 % (16470)Memory used [KB]: 40553
% 117.91/21.24 % (16470)Time elapsed: 0.919 s
% 117.91/21.24 % (16470)Instructions burned: 3208 (million)
% 117.91/21.24 % (16470)------------------------------
% 117.91/21.24 % (16470)------------------------------
% 117.91/21.26 % (16475)dis+1002_1:1_au=on:bsr=unit_only:cbe=off:fe=off:fsr=off:nm=10:nwc=6.0:plsq=on:plsqc=1:plsqr=32,1:sos=all:i=2635:si=on:rtra=on_0 on DTF2THF_16103 for (2839ds/2635Mi)
% 118.48/21.39 % (16463)Instruction limit reached!
% 118.48/21.39 % (16463)------------------------------
% 118.48/21.39 % (16463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 118.48/21.39 % (16463)Termination reason: Unknown
% 118.48/21.39 % (16463)Termination phase: Saturation
% 118.48/21.39
% 118.48/21.39 % (16463)Memory used [KB]: 80083
% 118.48/21.39 % (16463)Time elapsed: 3.022 s
% 118.48/21.39 % (16463)Instructions burned: 9824 (million)
% 118.48/21.39 % (16463)------------------------------
% 118.48/21.39 % (16463)------------------------------
% 115.98/21.41 % (16476)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=631:si=on:rtra=on_0 on DTF2THF_16103 for (2837ds/631Mi)
% 119.39/21.54 % (16464)Instruction limit reached!
% 119.39/21.54 % (16464)------------------------------
% 119.39/21.54 % (16464)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 119.39/21.54 % (16464)Termination reason: Unknown
% 119.39/21.54 % (16464)Termination phase: Saturation
% 119.39/21.54
% 119.39/21.54 % (16464)Memory used [KB]: 48613
% 119.39/21.54 % (16464)Time elapsed: 2.535 s
% 119.39/21.54 % (16464)Instructions burned: 8426 (million)
% 119.39/21.54 % (16464)------------------------------
% 119.39/21.54 % (16464)------------------------------
% 119.39/21.55 % (16477)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=15557:si=on:rtra=on_0 on DTF2THF_16103 for (2836ds/15557Mi)
% 119.39/21.60 % (16476)Instruction limit reached!
% 119.39/21.60 % (16476)------------------------------
% 119.39/21.60 % (16476)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 119.39/21.60 % (16476)Termination reason: Unknown
% 119.39/21.60 % (16476)Termination phase: shuffling
% 119.39/21.60
% 119.39/21.60 % (16476)Memory used [KB]: 24434
% 119.39/21.60 % (16476)Time elapsed: 0.187 s
% 119.39/21.60 % (16476)Instructions burned: 634 (million)
% 119.39/21.60 % (16476)------------------------------
% 119.39/21.60 % (16476)------------------------------
% 119.74/21.61 % (16478)dis+1002_1:1_au=on:sac=on:sd=5:slsq=on:slsqr=1,4:ss=axioms:st=2.0:i=2142:si=on:rtra=on_0 on DTF2THF_16103 for (2835ds/2142Mi)
% 123.61/22.05 % (16475)Instruction limit reached!
% 123.61/22.05 % (16475)------------------------------
% 123.61/22.05 % (16475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 123.61/22.05 % (16475)Termination reason: Unknown
% 123.61/22.05 % (16475)Termination phase: Property scanning
% 123.61/22.05
% 123.61/22.05 % (16475)Memory used [KB]: 37099
% 123.61/22.05 % (16475)Time elapsed: 0.795 s
% 123.61/22.05 % (16475)Instructions burned: 2638 (million)
% 123.61/22.05 % (16475)------------------------------
% 123.61/22.05 % (16475)------------------------------
% 123.61/22.06 % (16479)dis+1002_1:1_aac=none:au=on:bd=off:cnfonf=conj_eager:nwc=5.0:sgt=5:sp=reverse_arity:ss=axioms:i=5624:si=on:rtra=on_0 on DTF2THF_16103 for (2831ds/5624Mi)
% 124.26/22.18 % (16473)Instruction limit reached!
% 124.26/22.18 % (16473)------------------------------
% 124.26/22.18 % (16473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 124.26/22.18 % (16473)Termination reason: Unknown
% 124.26/22.18 % (16473)Termination phase: Saturation
% 124.26/22.18
% 124.26/22.18 % (16473)Memory used [KB]: 47078
% 124.26/22.18 % (16473)Time elapsed: 1.238 s
% 124.26/22.18 % (16473)Instructions burned: 4165 (million)
% 124.26/22.18 % (16473)------------------------------
% 124.26/22.18 % (16473)------------------------------
% 124.26/22.20 % (16480)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_be_gen:ntd=on:pe=on:i=1665:si=on:rtra=on_0 on DTF2THF_16103 for (2829ds/1665Mi)
% 125.25/22.32 % (16478)Instruction limit reached!
% 125.25/22.32 % (16478)------------------------------
% 125.25/22.32 % (16478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 125.25/22.32 % (16478)Termination reason: Unknown
% 125.25/22.32 % (16478)Termination phase: Preprocessing 3
% 125.25/22.32
% 125.25/22.32 % (16478)Memory used [KB]: 30575
% 125.25/22.32 % (16478)Time elapsed: 0.710 s
% 125.25/22.32 % (16478)Instructions burned: 2142 (million)
% 125.25/22.32 % (16478)------------------------------
% 125.25/22.32 % (16478)------------------------------
% 125.25/22.33 % (16481)dis+1002_1:1_av=off:prag=on:sd=1:sos=on:sp=occurrence:ss=axioms:i=64912:si=on:rtra=on_0 on DTF2THF_16103 for (2828ds/64912Mi)
% 126.08/22.54 % (16472)Instruction limit reached!
% 126.08/22.54 % (16472)------------------------------
% 126.08/22.54 % (16472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 126.08/22.54 % (16472)Termination reason: Unknown
% 126.08/22.54 % (16472)Termination phase: Property scanning
% 126.08/22.54
% 126.08/22.54 % (16472)Memory used [KB]: 59743
% 126.08/22.54 % (16472)Time elapsed: 1.870 s
% 126.08/22.54 % (16472)Instructions burned: 7469 (million)
% 126.08/22.54 % (16472)------------------------------
% 126.08/22.54 % (16472)------------------------------
% 126.08/22.55 % (16482)lrs+10_1:32_atotf=0.1:avsq=on:bsr=on:cnfonf=lazy_gen:fde=unused:i=7801:si=on:rtra=on_0 on DTF2THF_16103 for (2826ds/7801Mi)
% 126.43/22.69 % (16480)Instruction limit reached!
% 126.43/22.69 % (16480)------------------------------
% 126.43/22.69 % (16480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 126.43/22.69 % (16480)Termination reason: Unknown
% 126.43/22.69 % (16480)Termination phase: Property scanning
% 126.43/22.69
% 126.43/22.69 % (16480)Memory used [KB]: 41577
% 126.43/22.69 % (16480)Time elapsed: 0.492 s
% 126.43/22.69 % (16480)Instructions burned: 1667 (million)
% 126.43/22.69 % (16480)------------------------------
% 126.43/22.69 % (16480)------------------------------
% 127.24/22.70 % (16483)ott+2_1:1_au=on:hfsq=on:hfsqc=5:hfsql=off:nwc=5.0:sd=3:ss=axioms:st=5.0:i=15194:si=on:rtra=on_0 on DTF2THF_16103 for (2824ds/15194Mi)
% 134.68/23.60 % (16479)Instruction limit reached!
% 134.68/23.60 % (16479)------------------------------
% 134.68/23.60 % (16479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 134.68/23.60 % (16479)Termination reason: Unknown
% 134.68/23.60 % (16479)Termination phase: Saturation
% 134.68/23.60
% 134.68/23.60 % (16479)Memory used [KB]: 61278
% 134.68/23.60 % (16479)Time elapsed: 1.543 s
% 134.68/23.60 % (16479)Instructions burned: 5626 (million)
% 134.68/23.60 % (16479)------------------------------
% 134.68/23.60 % (16479)------------------------------
% 134.68/23.62 % (16484)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=6753:si=on:rtra=on_0 on DTF2THF_16103 for (2815ds/6753Mi)
% 141.65/24.67 % (16482)Instruction limit reached!
% 141.65/24.67 % (16482)------------------------------
% 141.65/24.67 % (16482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 141.65/24.67 % (16482)Termination reason: Unknown
% 141.65/24.67 % (16482)Termination phase: Saturation
% 141.65/24.67
% 141.65/24.67 % (16482)Memory used [KB]: 50660
% 141.65/24.67 % (16482)Time elapsed: 2.120 s
% 141.65/24.67 % (16482)Instructions burned: 7801 (million)
% 141.65/24.67 % (16482)------------------------------
% 141.65/24.67 % (16482)------------------------------
% 141.65/24.68 % (16485)lrs+1002_1:28_bd=off:cnfonf=lazy_simp:plsq=on:plsqr=1,1:prag=on:i=1003:si=on:rtra=on_0 on DTF2THF_16103 for (2804ds/1003Mi)
% 142.33/24.72 % (16474)Instruction limit reached!
% 142.33/24.72 % (16474)------------------------------
% 142.33/24.72 % (16474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 142.33/24.72 % (16474)Termination reason: Unknown
% 142.33/24.72 % (16474)Termination phase: Saturation
% 142.33/24.72
% 142.33/24.72 % (16474)Memory used [KB]: 72152
% 142.33/24.72 % (16474)Time elapsed: 3.532 s
% 142.33/24.72 % (16474)Instructions burned: 13468 (million)
% 142.33/24.72 % (16474)------------------------------
% 142.33/24.72 % (16474)------------------------------
% 142.33/24.74 % (16486)ott-21_5:4_atotf=0.3:e2e=on:hud=15:ins=1:plsq=on:plsqc=1:plsqr=32,1:sp=unary_frequency:tnu=2:i=13018:si=on:rtra=on_0 on DTF2THF_16103 for (2804ds/13018Mi)
% 143.37/24.98 % (16485)Instruction limit reached!
% 143.37/24.98 % (16485)------------------------------
% 143.37/24.98 % (16485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 143.37/24.98 % (16485)Termination reason: Unknown
% 143.37/24.98 % (16485)Termination phase: shuffling
% 143.37/24.98
% 143.37/24.98 % (16485)Memory used [KB]: 33517
% 143.37/24.98 % (16485)Time elapsed: 0.299 s
% 143.37/24.98 % (16485)Instructions burned: 1005 (million)
% 143.37/24.98 % (16485)------------------------------
% 143.37/24.98 % (16485)------------------------------
% 143.37/24.99 % (16487)dis+10_1:4_atotf=0.1:avsq=on:bsr=unit_only:e2e=on:sd=1:sp=const_max:ss=axioms:i=1267:si=on:rtra=on_0 on DTF2THF_16103 for (2801ds/1267Mi)
% 147.20/25.40 % (16487)Instruction limit reached!
% 147.20/25.40 % (16487)------------------------------
% 147.20/25.40 % (16487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 147.20/25.40 % (16487)Termination reason: Unknown
% 147.20/25.40 % (16487)Termination phase: Property scanning
% 147.20/25.40
% 147.20/25.40 % (16487)Memory used [KB]: 26353
% 147.20/25.40 % (16487)Time elapsed: 0.408 s
% 147.20/25.40 % (16487)Instructions burned: 1267 (million)
% 147.20/25.40 % (16487)------------------------------
% 147.20/25.40 % (16487)------------------------------
% 147.20/25.41 % (16477)Instruction limit reached!
% 147.20/25.41 % (16477)------------------------------
% 147.20/25.41 % (16477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 147.20/25.41 % (16477)Termination reason: Unknown
% 147.20/25.41 % (16477)Termination phase: Saturation
% 147.20/25.41
% 147.20/25.41 % (16477)Memory used [KB]: 64348
% 147.20/25.41 % (16477)Time elapsed: 3.852 s
% 147.20/25.41 % (16477)Instructions burned: 15560 (million)
% 147.20/25.41 % (16477)------------------------------
% 147.20/25.41 % (16477)------------------------------
% 147.20/25.41 % (16488)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=2570:si=on:rtra=on_0 on DTF2THF_16103 for (2797ds/2570Mi)
% 147.20/25.42 % (16489)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=13163:si=on:rtra=on_0 on DTF2THF_16103 for (2797ds/13163Mi)
% 147.75/25.59 % (16481)First to succeed.
% 145.00/25.60 % (16481)Refutation found. Thanks to Tanya!
% 145.00/25.60 % SZS status Theorem for DTF2THF_16103
% 145.00/25.60 % SZS output start Proof for DTF2THF_16103
% 145.00/25.60 thf(type_def_5, type, rat: $tType).
% 145.00/25.60 thf(type_def_6, type, int: $tType).
% 145.00/25.60 thf(type_def_8, type, real: $tType).
% 145.00/25.60 thf(type_def_9, type, nat: $tType).
% 145.00/25.60 thf(type_def_10, type, product_prod_int_int: $tType).
% 145.00/25.60 thf(type_def_11, type, num: $tType).
% 145.00/25.60 thf(type_def_12, type, option_num: $tType).
% 145.00/25.60 thf(type_def_13, type, product_prod_num_num: $tType).
% 145.00/25.60 thf(type_def_14, type, code_integer: $tType).
% 145.00/25.60 thf(type_def_15, type, produc6271795597528267376eger_o: $tType).
% 145.00/25.60 thf(type_def_16, type, produc8923325533196201883nteger: $tType).
% 145.00/25.60 thf(type_def_17, type, set_real: $tType).
% 145.00/25.60 thf(type_def_18, type, set_int: $tType).
% 145.00/25.60 thf(type_def_19, type, complex: $tType).
% 145.00/25.60 thf(type_def_20, type, filter_real: $tType).
% 145.00/25.60 thf(type_def_21, type, product_prod_nat_nat: $tType).
% 145.00/25.60 thf(type_def_22, type, filter_nat: $tType).
% 145.00/25.60 thf(type_def_23, type, set_o: $tType).
% 145.00/25.60 thf(type_def_24, type, set_complex: $tType).
% 145.00/25.60 thf(type_def_25, type, set_list_nat: $tType).
% 145.00/25.60 thf(type_def_26, type, set_nat: $tType).
% 145.00/25.60 thf(type_def_27, type, set_Product_unit: $tType).
% 145.00/25.60 thf(type_def_28, type, set_char: $tType).
% 145.00/25.60 thf(type_def_29, type, set_list_o: $tType).
% 145.00/25.60 thf(type_def_30, type, set_list_complex: $tType).
% 145.00/25.60 thf(type_def_31, type, set_list_int: $tType).
% 145.00/25.60 thf(type_def_32, type, set_list_VEBT_VEBT: $tType).
% 145.00/25.60 thf(type_def_33, type, set_num: $tType).
% 145.00/25.60 thf(type_def_34, type, set_rat: $tType).
% 145.00/25.60 thf(type_def_35, type, set_set_complex: $tType).
% 145.00/25.60 thf(type_def_36, type, set_set_int: $tType).
% 145.00/25.60 thf(type_def_37, type, set_set_nat: $tType).
% 145.00/25.60 thf(type_def_38, type, set_VEBT_VEBT: $tType).
% 145.00/25.60 thf(type_def_39, type, char: $tType).
% 145.00/25.60 thf(type_def_40, type, list_nat: $tType).
% 145.00/25.60 thf(type_def_41, type, extended_enat: $tType).
% 145.00/25.60 thf(type_def_42, type, set_Pr1261947904930325089at_nat: $tType).
% 145.00/25.60 thf(type_def_43, type, list_o: $tType).
% 145.00/25.60 thf(type_def_44, type, list_int: $tType).
% 145.00/25.60 thf(type_def_45, type, option_nat: $tType).
% 145.00/25.60 thf(type_def_46, type, vEBT_VEBT: $tType).
% 145.00/25.60 thf(type_def_47, type, set_Pr958786334691620121nt_int: $tType).
% 145.00/25.60 thf(type_def_48, type, list_complex: $tType).
% 145.00/25.60 thf(type_def_49, type, list_P6011104703257516679at_nat: $tType).
% 145.00/25.60 thf(type_def_50, type, list_real: $tType).
% 145.00/25.60 thf(type_def_51, type, list_VEBT_VEBT: $tType).
% 145.00/25.60 thf(type_def_52, type, list_Code_integer: $tType).
% 145.00/25.60 thf(type_def_53, type, list_num: $tType).
% 145.00/25.60 thf(type_def_54, type, list_P4002435161011370285od_o_o: $tType).
% 145.00/25.60 thf(type_def_55, type, product_prod_o_o: $tType).
% 145.00/25.60 thf(type_def_56, type, list_P3795440434834930179_o_int: $tType).
% 145.00/25.60 thf(type_def_57, type, product_prod_o_int: $tType).
% 145.00/25.60 thf(type_def_58, type, list_P6285523579766656935_o_nat: $tType).
% 145.00/25.60 thf(type_def_59, type, product_prod_o_nat: $tType).
% 145.00/25.60 thf(type_def_60, type, list_P7495141550334521929T_VEBT: $tType).
% 145.00/25.60 thf(type_def_61, type, produc2504756804600209347T_VEBT: $tType).
% 145.00/25.60 thf(type_def_62, type, list_P8526636022914148096eger_o: $tType).
% 145.00/25.60 thf(type_def_63, type, list_P3744719386663036955um_num: $tType).
% 145.00/25.60 thf(type_def_64, type, list_P3126845725202233233VEBT_o: $tType).
% 145.00/25.60 thf(type_def_65, type, produc334124729049499915VEBT_o: $tType).
% 145.00/25.60 thf(type_def_66, type, list_P4547456442757143711BT_int: $tType).
% 145.00/25.60 thf(type_def_67, type, produc4894624898956917775BT_int: $tType).
% 145.00/25.60 thf(type_def_68, type, list_P7037539587688870467BT_nat: $tType).
% 145.00/25.60 thf(type_def_69, type, produc9072475918466114483BT_nat: $tType).
% 145.00/25.60 thf(type_def_70, type, list_P7413028617227757229T_VEBT: $tType).
% 145.00/25.60 thf(type_def_71, type, produc8243902056947475879T_VEBT: $tType).
% 145.00/25.60 thf(type_def_72, type, list_P7333126701944960589_nat_o: $tType).
% 145.00/25.60 thf(type_def_73, type, list_P5647936690300460905T_VEBT: $tType).
% 145.00/25.60 thf(type_def_74, type, option4927543243414619207at_nat: $tType).
% 145.00/25.60 thf(type_def_75, type, set_Pr448751882837621926eger_o: $tType).
% 145.00/25.60 thf(type_def_76, type, set_Pr6200539531224447659at_num: $tType).
% 145.00/25.60 thf(type_def_77, type, set_Pr8218934625190621173um_num: $tType).
% 145.00/25.60 thf(type_def_78, type, set_Code_integer: $tType).
% 145.00/25.60 thf(type_def_79, type, produc4953844613479565601on_nat: $tType).
% 145.00/25.60 thf(type_def_80, type, produc2233624965454879586on_nat: $tType).
% 145.00/25.60 thf(type_def_81, type, produc7248412053542808358at_nat: $tType).
% 145.00/25.60 thf(type_def_82, type, produc4471711990508489141at_nat: $tType).
% 145.00/25.60 thf(type_def_83, type, produc8306885398267862888on_nat: $tType).
% 145.00/25.60 thf(type_def_84, type, produc2963631642982155120at_num: $tType).
% 145.00/25.60 thf(type_def_85, type, produc3368934014287244435at_num: $tType).
% 145.00/25.60 thf(type_def_86, type, produc3447558737645232053on_num: $tType).
% 145.00/25.60 thf(type_def_87, type, produc7036089656553540234on_num: $tType).
% 145.00/25.60 thf(type_def_88, type, produc1193250871479095198on_num: $tType).
% 145.00/25.60 thf(type_def_89, type, produc6121120109295599847at_nat: $tType).
% 145.00/25.60 thf(type_def_90, type, produc5491161045314408544at_nat: $tType).
% 145.00/25.60 thf(type_def_91, type, produc5542196010084753463at_nat: $tType).
% 145.00/25.60 thf(type_def_92, type, product_prod_nat_num: $tType).
% 145.00/25.60 thf(func_def_0, type, produc5542196010084753463at_nat: $tType).
% 145.00/25.60 thf(func_def_1, type, produc5491161045314408544at_nat: $tType).
% 145.00/25.60 thf(func_def_2, type, produc1193250871479095198on_num: $tType).
% 145.00/25.60 thf(func_def_3, type, produc8306885398267862888on_nat: $tType).
% 145.00/25.60 thf(func_def_4, type, produc6121120109295599847at_nat: $tType).
% 145.00/25.60 thf(func_def_5, type, produc3368934014287244435at_num: $tType).
% 145.00/25.60 thf(func_def_6, type, produc4471711990508489141at_nat: $tType).
% 145.00/25.60 thf(func_def_7, type, produc7036089656553540234on_num: $tType).
% 145.00/25.60 thf(func_def_8, type, produc2233624965454879586on_nat: $tType).
% 145.00/25.60 thf(func_def_9, type, list_P7413028617227757229T_VEBT: $tType).
% 145.00/25.60 thf(func_def_10, type, produc3447558737645232053on_num: $tType).
% 145.00/25.60 thf(func_def_11, type, produc4953844613479565601on_nat: $tType).
% 145.00/25.60 thf(func_def_12, type, produc2963631642982155120at_num: $tType).
% 145.00/25.60 thf(func_def_13, type, produc7248412053542808358at_nat: $tType).
% 145.00/25.60 thf(func_def_14, type, list_P7037539587688870467BT_nat: $tType).
% 145.00/25.60 thf(func_def_15, type, list_P4547456442757143711BT_int: $tType).
% 145.00/25.60 thf(func_def_16, type, list_P5647936690300460905T_VEBT: $tType).
% 145.00/25.60 thf(func_def_17, type, produc8243902056947475879T_VEBT: $tType).
% 145.00/25.60 thf(func_def_18, type, produc8923325533196201883nteger: $tType).
% 145.00/25.60 thf(func_def_19, type, list_P3126845725202233233VEBT_o: $tType).
% 145.00/25.60 thf(func_def_20, type, list_P7495141550334521929T_VEBT: $tType).
% 145.00/25.60 thf(func_def_21, type, list_P8526636022914148096eger_o: $tType).
% 145.00/25.60 thf(func_def_22, type, set_Pr448751882837621926eger_o: $tType).
% 145.00/25.60 thf(func_def_23, type, option4927543243414619207at_nat: $tType).
% 145.00/25.60 thf(func_def_24, type, list_P3744719386663036955um_num: $tType).
% 145.00/25.60 thf(func_def_25, type, list_P6011104703257516679at_nat: $tType).
% 145.00/25.60 thf(func_def_26, type, produc9072475918466114483BT_nat: $tType).
% 145.00/25.60 thf(func_def_27, type, produc4894624898956917775BT_int: $tType).
% 145.00/25.60 thf(func_def_28, type, set_Pr8218934625190621173um_num: $tType).
% 145.00/25.60 thf(func_def_29, type, set_Pr6200539531224447659at_num: $tType).
% 145.00/25.60 thf(func_def_30, type, set_Pr1261947904930325089at_nat: $tType).
% 145.00/25.60 thf(func_def_31, type, set_Pr958786334691620121nt_int: $tType).
% 145.00/25.60 thf(func_def_32, type, list_P7333126701944960589_nat_o: $tType).
% 145.00/25.60 thf(func_def_33, type, list_P6285523579766656935_o_nat: $tType).
% 145.00/25.60 thf(func_def_34, type, list_P3795440434834930179_o_int: $tType).
% 145.00/25.60 thf(func_def_35, type, set_list_VEBT_VEBT: $tType).
% 145.00/25.60 thf(func_def_36, type, produc334124729049499915VEBT_o: $tType).
% 145.00/25.60 thf(func_def_37, type, produc2504756804600209347T_VEBT: $tType).
% 145.00/25.60 thf(func_def_38, type, produc6271795597528267376eger_o: $tType).
% 145.00/25.60 thf(func_def_39, type, product_prod_num_num: $tType).
% 145.00/25.60 thf(func_def_40, type, product_prod_nat_num: $tType).
% 145.00/25.60 thf(func_def_41, type, product_prod_nat_nat: $tType).
% 145.00/25.60 thf(func_def_42, type, product_prod_int_int: $tType).
% 145.00/25.60 thf(func_def_43, type, list_P4002435161011370285od_o_o: $tType).
% 145.00/25.60 thf(func_def_44, type, set_list_complex: $tType).
% 145.00/25.60 thf(func_def_45, type, set_set_complex: $tType).
% 145.00/25.60 thf(func_def_46, type, list_VEBT_VEBT: $tType).
% 145.00/25.60 thf(func_def_47, type, set_list_nat: $tType).
% 145.00/25.60 thf(func_def_48, type, set_list_int: $tType).
% 145.00/25.60 thf(func_def_49, type, product_prod_o_nat: $tType).
% 145.00/25.60 thf(func_def_50, type, product_prod_o_int: $tType).
% 145.00/25.60 thf(func_def_51, type, list_Code_integer: $tType).
% 145.00/25.60 thf(func_def_52, type, set_VEBT_VEBT: $tType).
% 145.00/25.60 thf(func_def_53, type, set_set_nat: $tType).
% 145.00/25.60 thf(func_def_54, type, set_set_int: $tType).
% 145.00/25.60 thf(func_def_55, type, set_Code_integer: $tType).
% 145.00/25.60 thf(func_def_56, type, set_Product_unit: $tType).
% 145.00/25.60 thf(func_def_57, type, list_complex: $tType).
% 145.00/25.60 thf(func_def_58, type, set_list_o: $tType).
% 145.00/25.60 thf(func_def_59, type, product_prod_o_o: $tType).
% 145.00/25.60 thf(func_def_60, type, set_complex: $tType).
% 145.00/25.60 thf(func_def_61, type, filter_real: $tType).
% 145.00/25.60 thf(func_def_62, type, option_num: $tType).
% 145.00/25.60 thf(func_def_63, type, option_nat: $tType).
% 145.00/25.60 thf(func_def_64, type, filter_nat: $tType).
% 145.00/25.60 thf(func_def_65, type, set_char: $tType).
% 145.00/25.60 thf(func_def_66, type, list_real: $tType).
% 145.00/25.60 thf(func_def_67, type, set_real: $tType).
% 145.00/25.60 thf(func_def_68, type, list_num: $tType).
% 145.00/25.60 thf(func_def_69, type, list_nat: $tType).
% 145.00/25.60 thf(func_def_70, type, list_int: $tType).
% 145.00/25.60 thf(func_def_71, type, vEBT_VEBT: $tType).
% 145.00/25.60 thf(func_def_72, type, set_rat: $tType).
% 145.00/25.60 thf(func_def_73, type, set_num: $tType).
% 145.00/25.60 thf(func_def_74, type, set_nat: $tType).
% 145.00/25.60 thf(func_def_75, type, set_int: $tType).
% 145.00/25.60 thf(func_def_76, type, code_integer: $tType).
% 145.00/25.60 thf(func_def_77, type, extended_enat: $tType).
% 145.00/25.60 thf(func_def_78, type, list_o: $tType).
% 145.00/25.60 thf(func_def_79, type, complex: $tType).
% 145.00/25.60 thf(func_def_80, type, set_o: $tType).
% 145.00/25.60 thf(func_def_81, type, char: $tType).
% 145.00/25.60 thf(func_def_82, type, real: $tType).
% 145.00/25.60 thf(func_def_83, type, rat: $tType).
% 145.00/25.60 thf(func_def_84, type, num: $tType).
% 145.00/25.60 thf(func_def_85, type, nat: $tType).
% 145.00/25.60 thf(func_def_86, type, int: $tType).
% 145.00/25.60 thf(func_def_87, type, archim2889992004027027881ng_rat: rat > int).
% 145.00/25.60 thf(func_def_88, type, archim7802044766580827645g_real: real > int).
% 145.00/25.60 thf(func_def_89, type, archim3151403230148437115or_rat: rat > int).
% 145.00/25.60 thf(func_def_90, type, archim6058952711729229775r_real: real > int).
% 145.00/25.60 thf(func_def_91, type, archim7778729529865785530nd_rat: rat > int).
% 145.00/25.60 thf(func_def_92, type, archim8280529875227126926d_real: real > int).
% 145.00/25.60 thf(func_def_93, type, binomial: nat > nat > nat).
% 145.00/25.60 thf(func_def_94, type, bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o).
% 145.00/25.60 thf(func_def_95, type, bit_and_not_num: num > num > option_num).
% 145.00/25.60 thf(func_def_96, type, bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o).
% 145.00/25.60 thf(func_def_97, type, bit_concat_bit: nat > int > int > int).
% 145.00/25.60 thf(func_def_98, type, bit_or_not_num_neg: num > num > num).
% 145.00/25.60 thf(func_def_99, type, bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o).
% 145.00/25.60 thf(func_def_100, type, bit_ri7919022796975470100ot_int: int > int).
% 145.00/25.60 thf(func_def_101, type, bit_ri6519982836138164636nteger: nat > code_integer > code_integer).
% 145.00/25.60 thf(func_def_102, type, bit_ri631733984087533419it_int: nat > int > int).
% 145.00/25.60 thf(func_def_103, type, bit_se725231765392027082nd_int: int > int > int).
% 145.00/25.60 thf(func_def_104, type, bit_se727722235901077358nd_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_105, type, bit_se8568078237143864401it_int: nat > int > int).
% 145.00/25.60 thf(func_def_106, type, bit_se8570568707652914677it_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_107, type, bit_se1345352211410354436nteger: nat > code_integer > code_integer).
% 145.00/25.60 thf(func_def_108, type, bit_se2159334234014336723it_int: nat > int > int).
% 145.00/25.60 thf(func_def_109, type, bit_se2161824704523386999it_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_110, type, bit_se2000444600071755411sk_int: nat > int).
% 145.00/25.60 thf(func_def_111, type, bit_se2002935070580805687sk_nat: nat > nat).
% 145.00/25.60 thf(func_def_112, type, bit_se1409905431419307370or_int: int > int > int).
% 145.00/25.60 thf(func_def_113, type, bit_se1412395901928357646or_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_114, type, bit_se545348938243370406it_int: nat > int > int).
% 145.00/25.60 thf(func_def_115, type, bit_se547839408752420682it_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_116, type, bit_se2793503036327961859nteger: nat > code_integer > code_integer).
% 145.00/25.60 thf(func_def_117, type, bit_se7879613467334960850it_int: nat > int > int).
% 145.00/25.60 thf(func_def_118, type, bit_se7882103937844011126it_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_119, type, bit_se2923211474154528505it_int: nat > int > int).
% 145.00/25.60 thf(func_def_120, type, bit_se2925701944663578781it_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_121, type, bit_se8260200283734997820nteger: nat > code_integer > code_integer).
% 145.00/25.60 thf(func_def_122, type, bit_se4203085406695923979it_int: nat > int > int).
% 145.00/25.60 thf(func_def_123, type, bit_se4205575877204974255it_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_124, type, bit_se6526347334894502574or_int: int > int > int).
% 145.00/25.60 thf(func_def_125, type, bit_se6528837805403552850or_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_126, type, bit_se1146084159140164899it_int: int > nat > $o).
% 145.00/25.60 thf(func_def_127, type, bit_se1148574629649215175it_nat: nat > nat > $o).
% 145.00/25.60 thf(func_def_128, type, bit_take_bit_num: nat > num > option_num).
% 145.00/25.60 thf(func_def_129, type, bit_un1837492267222099188nd_num: num > num > option_num).
% 145.00/25.60 thf(func_def_130, type, bit_un5425074673868309765um_rel: product_prod_num_num > product_prod_num_num > $o).
% 145.00/25.60 thf(func_def_131, type, bit_un6178654185764691216or_num: num > num > option_num).
% 145.00/25.60 thf(func_def_132, type, bit_un3595099601533988841um_rel: product_prod_num_num > product_prod_num_num > $o).
% 145.00/25.60 thf(func_def_133, type, bit_un7362597486090784418nd_num: num > num > option_num).
% 145.00/25.60 thf(func_def_134, type, bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o).
% 145.00/25.60 thf(func_def_135, type, bit_un2480387367778600638or_num: num > num > option_num).
% 145.00/25.60 thf(func_def_136, type, bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o).
% 145.00/25.60 thf(func_def_137, type, code_bit_cut_integer: code_integer > produc6271795597528267376eger_o).
% 145.00/25.60 thf(func_def_138, type, code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger).
% 145.00/25.60 thf(func_def_139, type, code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger).
% 145.00/25.60 thf(func_def_140, type, code_int_of_integer: code_integer > int).
% 145.00/25.60 thf(func_def_141, type, code_integer_of_int: int > code_integer).
% 145.00/25.60 thf(func_def_142, type, code_integer_of_num: num > code_integer).
% 145.00/25.60 thf(func_def_143, type, code_nat_of_integer: code_integer > nat).
% 145.00/25.60 thf(func_def_144, type, code_negative: num > code_integer).
% 145.00/25.60 thf(func_def_145, type, code_num_of_integer: code_integer > num).
% 145.00/25.60 thf(func_def_146, type, code_positive: num > code_integer).
% 145.00/25.60 thf(func_def_147, type, code_Target_negative: num > int).
% 145.00/25.60 thf(func_def_148, type, code_Target_positive: num > int).
% 145.00/25.60 thf(func_def_149, type, comple4887499456419720421f_real: set_real > real).
% 145.00/25.60 thf(func_def_150, type, complete_Sup_Sup_int: set_int > int).
% 145.00/25.60 thf(func_def_151, type, comple1385675409528146559p_real: set_real > real).
% 145.00/25.60 thf(func_def_152, type, arg: complex > real).
% 145.00/25.60 thf(func_def_153, type, cis: real > complex).
% 145.00/25.60 thf(func_def_154, type, cnj: complex > complex).
% 145.00/25.60 thf(func_def_155, type, complex2: real > real > complex).
% 145.00/25.60 thf(func_def_156, type, im: complex > real).
% 145.00/25.60 thf(func_def_157, type, re: complex > real).
% 145.00/25.60 thf(func_def_158, type, csqrt: complex > complex).
% 145.00/25.60 thf(func_def_159, type, imaginary_unit: complex).
% 145.00/25.60 thf(func_def_160, type, differ6690327859849518006l_real: (real > real) > filter_real > $o).
% 145.00/25.60 thf(func_def_161, type, has_de1759254742604945161l_real: (real > real) > (real > real) > filter_real > $o).
% 145.00/25.60 thf(func_def_162, type, has_fi5821293074295781190e_real: (real > real) > real > filter_real > $o).
% 145.00/25.60 thf(func_def_163, type, adjust_div: product_prod_int_int > int).
% 145.00/25.60 thf(func_def_164, type, adjust_mod: int > int > int).
% 145.00/25.60 thf(func_def_165, type, divmod_nat: nat > nat > product_prod_nat_nat).
% 145.00/25.60 thf(func_def_166, type, eucl_rel_int: int > int > product_prod_int_int > $o).
% 145.00/25.60 thf(func_def_167, type, unique5706413561485394159nteger: produc8923325533196201883nteger > $o).
% 145.00/25.60 thf(func_def_168, type, unique6319869463603278526ux_int: product_prod_int_int > $o).
% 145.00/25.60 thf(func_def_169, type, unique6322359934112328802ux_nat: product_prod_nat_nat > $o).
% 145.00/25.60 thf(func_def_170, type, unique3479559517661332726nteger: num > num > produc8923325533196201883nteger).
% 145.00/25.60 thf(func_def_171, type, unique5052692396658037445od_int: num > num > product_prod_int_int).
% 145.00/25.60 thf(func_def_172, type, unique5055182867167087721od_nat: num > num > product_prod_nat_nat).
% 145.00/25.60 thf(func_def_173, type, unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 145.00/25.60 thf(func_def_174, type, unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int).
% 145.00/25.60 thf(func_def_175, type, unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat).
% 145.00/25.60 thf(func_def_176, type, comm_s8582702949713902594nteger: code_integer > nat > code_integer).
% 145.00/25.60 thf(func_def_177, type, comm_s2602460028002588243omplex: complex > nat > complex).
% 145.00/25.60 thf(func_def_178, type, comm_s4660882817536571857er_int: int > nat > int).
% 145.00/25.60 thf(func_def_179, type, comm_s4663373288045622133er_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_180, type, comm_s4028243227959126397er_rat: rat > nat > rat).
% 145.00/25.60 thf(func_def_181, type, comm_s7457072308508201937r_real: real > nat > real).
% 145.00/25.60 thf(func_def_182, type, semiri3624122377584611663nteger: nat > code_integer).
% 145.00/25.60 thf(func_def_183, type, semiri5044797733671781792omplex: nat > complex).
% 145.00/25.60 thf(func_def_184, type, semiri1406184849735516958ct_int: nat > int).
% 145.00/25.60 thf(func_def_185, type, semiri1408675320244567234ct_nat: nat > nat).
% 145.00/25.60 thf(func_def_186, type, semiri773545260158071498ct_rat: nat > rat).
% 145.00/25.60 thf(func_def_187, type, semiri2265585572941072030t_real: nat > real).
% 145.00/25.60 thf(func_def_188, type, invers8013647133539491842omplex: complex > complex).
% 145.00/25.60 thf(func_def_189, type, inverse_inverse_rat: rat > rat).
% 145.00/25.60 thf(func_def_190, type, inverse_inverse_real: real > real).
% 145.00/25.60 thf(func_def_191, type, at_bot_real: filter_real).
% 145.00/25.60 thf(func_def_192, type, at_top_nat: filter_nat).
% 145.00/25.60 thf(func_def_193, type, at_top_real: filter_real).
% 145.00/25.60 thf(func_def_194, type, eventually_nat: (nat > $o) > filter_nat > $o).
% 145.00/25.60 thf(func_def_195, type, filterlim_nat_nat: (nat > nat) > filter_nat > filter_nat > $o).
% 145.00/25.60 thf(func_def_196, type, filterlim_nat_real: (nat > real) > filter_real > filter_nat > $o).
% 145.00/25.60 thf(func_def_197, type, filterlim_real_real: (real > real) > filter_real > filter_real > $o).
% 145.00/25.60 thf(func_def_198, type, finite_card_o: set_o > nat).
% 145.00/25.60 thf(func_def_199, type, finite_card_complex: set_complex > nat).
% 145.00/25.60 thf(func_def_200, type, finite_card_int: set_int > nat).
% 145.00/25.60 thf(func_def_201, type, finite_card_list_nat: set_list_nat > nat).
% 145.00/25.60 thf(func_def_202, type, finite_card_nat: set_nat > nat).
% 145.00/25.60 thf(func_def_203, type, finite410649719033368117t_unit: set_Product_unit > nat).
% 145.00/25.60 thf(func_def_204, type, finite_card_char: set_char > nat).
% 145.00/25.60 thf(func_def_205, type, finite_finite_o: set_o > $o).
% 145.00/25.60 thf(func_def_206, type, finite3207457112153483333omplex: set_complex > $o).
% 145.00/25.60 thf(func_def_207, type, finite_finite_int: set_int > $o).
% 145.00/25.60 thf(func_def_208, type, finite_finite_list_o: set_list_o > $o).
% 145.00/25.60 thf(func_def_209, type, finite8712137658972009173omplex: set_list_complex > $o).
% 145.00/25.60 thf(func_def_210, type, finite3922522038869484883st_int: set_list_int > $o).
% 145.00/25.60 thf(func_def_211, type, finite8100373058378681591st_nat: set_list_nat > $o).
% 145.00/25.60 thf(func_def_212, type, finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o).
% 145.00/25.60 thf(func_def_213, type, finite_finite_nat: set_nat > $o).
% 145.00/25.60 thf(func_def_214, type, finite_finite_num: set_num > $o).
% 145.00/25.60 thf(func_def_215, type, finite_finite_rat: set_rat > $o).
% 145.00/25.60 thf(func_def_216, type, finite_finite_real: set_real > $o).
% 145.00/25.60 thf(func_def_217, type, finite6551019134538273531omplex: set_set_complex > $o).
% 145.00/25.60 thf(func_def_218, type, finite6197958912794628473et_int: set_set_int > $o).
% 145.00/25.60 thf(func_def_219, type, finite1152437895449049373et_nat: set_set_nat > $o).
% 145.00/25.60 thf(func_def_220, type, finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o).
% 145.00/25.60 thf(func_def_221, type, bij_be1856998921033663316omplex: (complex > complex) > set_complex > set_complex > $o).
% 145.00/25.60 thf(func_def_222, type, bij_betw_nat_complex: (nat > complex) > set_nat > set_complex > $o).
% 145.00/25.60 thf(func_def_223, type, bij_betw_nat_nat: (nat > nat) > set_nat > set_nat > $o).
% 145.00/25.60 thf(func_def_224, type, comp_C8797469213163452608nteger: ((code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 145.00/25.60 thf(func_def_225, type, comp_C1593894019821074884nteger: (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 145.00/25.60 thf(func_def_226, type, comp_C3531382070062128313er_num: (code_integer > code_integer) > (num > code_integer) > num > code_integer).
% 145.00/25.60 thf(func_def_227, type, comp_int_int_num: (int > int) > (num > int) > num > int).
% 145.00/25.60 thf(func_def_228, type, comp_nat_nat_nat: (nat > nat) > (nat > nat) > nat > nat).
% 145.00/25.60 thf(func_def_229, type, comp_nat_real_nat: (nat > real) > (nat > nat) > nat > real).
% 145.00/25.60 thf(func_def_230, type, id_o: $o > $o).
% 145.00/25.60 thf(func_def_231, type, id_nat: nat > nat).
% 145.00/25.60 thf(func_def_232, type, inj_on_nat_nat: (nat > nat) > set_nat > $o).
% 145.00/25.60 thf(func_def_233, type, inj_on_nat_char: (nat > char) > set_nat > $o).
% 145.00/25.60 thf(func_def_234, type, inj_on_real_real: (real > real) > set_real > $o).
% 145.00/25.60 thf(func_def_235, type, map_fu434086159418415080_int_o: (int > product_prod_nat_nat) > ((product_prod_nat_nat > $o) > int > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > int > int > $o).
% 145.00/25.60 thf(func_def_236, type, map_fu4960017516451851995nt_int: (int > product_prod_nat_nat) > ((product_prod_nat_nat > product_prod_nat_nat) > int > int) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > int > int > int).
% 145.00/25.60 thf(func_def_237, type, map_fu4826362097070443709at_o_o: (int > product_prod_nat_nat) > ($o > $o) > (product_prod_nat_nat > $o) > int > $o).
% 145.00/25.60 thf(func_def_238, type, map_fu2345160673673942751at_nat: (int > product_prod_nat_nat) > (nat > nat) > (product_prod_nat_nat > nat) > int > nat).
% 145.00/25.60 thf(func_def_239, type, map_fu3667384564859982768at_int: (int > product_prod_nat_nat) > (product_prod_nat_nat > int) > (product_prod_nat_nat > product_prod_nat_nat) > int > int).
% 145.00/25.60 thf(func_def_240, type, map_fu4333342158222067775at_rat: (rat > product_prod_int_int) > ((product_prod_int_int > product_prod_int_int) > rat > rat) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > rat > rat > rat).
% 145.00/25.60 thf(func_def_241, type, map_fu898904425404107465nt_o_o: (rat > product_prod_int_int) > ($o > $o) > (product_prod_int_int > $o) > rat > $o).
% 145.00/25.60 thf(func_def_242, type, map_fu5673905371560938248nt_rat: (rat > product_prod_int_int) > (product_prod_int_int > rat) > (product_prod_int_int > product_prod_int_int) > rat > rat).
% 145.00/25.60 thf(func_def_243, type, strict1292158309912662752at_nat: (nat > nat) > set_nat > $o).
% 145.00/25.60 thf(func_def_244, type, the_in5290026491893676941l_real: set_real > (real > real) > real > real).
% 145.00/25.60 thf(func_def_245, type, gcd_Gcd_int: set_int > int).
% 145.00/25.60 thf(func_def_246, type, gcd_Gcd_nat: set_nat > nat).
% 145.00/25.60 thf(func_def_247, type, bezw: nat > nat > product_prod_int_int).
% 145.00/25.60 thf(func_def_248, type, bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 145.00/25.60 thf(func_def_249, type, gcd_gcd_int: int > int > int).
% 145.00/25.60 thf(func_def_250, type, gcd_gcd_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_251, type, gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 145.00/25.60 thf(func_def_252, type, abs_abs_Code_integer: code_integer > code_integer).
% 145.00/25.60 thf(func_def_253, type, abs_abs_complex: complex > complex).
% 145.00/25.60 thf(func_def_254, type, abs_abs_int: int > int).
% 145.00/25.60 thf(func_def_255, type, abs_abs_rat: rat > rat).
% 145.00/25.60 thf(func_def_256, type, abs_abs_real: real > real).
% 145.00/25.60 thf(func_def_257, type, minus_8727706125548526216plex_o: (complex > $o) > (complex > $o) > complex > $o).
% 145.00/25.60 thf(func_def_258, type, minus_minus_int_o: (int > $o) > (int > $o) > int > $o).
% 145.00/25.60 thf(func_def_259, type, minus_1139252259498527702_nat_o: (list_nat > $o) > (list_nat > $o) > list_nat > $o).
% 145.00/25.60 thf(func_def_260, type, minus_minus_nat_o: (nat > $o) > (nat > $o) > nat > $o).
% 145.00/25.60 thf(func_def_261, type, minus_2270307095948843157_nat_o: (product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > product_prod_nat_nat > $o).
% 145.00/25.60 thf(func_def_262, type, minus_minus_real_o: (real > $o) > (real > $o) > real > $o).
% 145.00/25.60 thf(func_def_263, type, minus_8373710615458151222nteger: code_integer > code_integer > code_integer).
% 145.00/25.60 thf(func_def_264, type, minus_minus_complex: complex > complex > complex).
% 145.00/25.60 thf(func_def_265, type, minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat).
% 145.00/25.60 thf(func_def_266, type, minus_minus_int: int > int > int).
% 145.00/25.60 thf(func_def_267, type, minus_minus_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_268, type, minus_minus_rat: rat > rat > rat).
% 145.00/25.60 thf(func_def_269, type, minus_minus_real: real > real > real).
% 145.00/25.60 thf(func_def_270, type, minus_811609699411566653omplex: set_complex > set_complex > set_complex).
% 145.00/25.60 thf(func_def_271, type, minus_minus_set_int: set_int > set_int > set_int).
% 145.00/25.60 thf(func_def_272, type, minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat).
% 145.00/25.60 thf(func_def_273, type, minus_minus_set_nat: set_nat > set_nat > set_nat).
% 145.00/25.60 thf(func_def_274, type, minus_1356011639430497352at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 145.00/25.60 thf(func_def_275, type, minus_minus_set_real: set_real > set_real > set_real).
% 145.00/25.60 thf(func_def_276, type, one_one_Code_integer: code_integer).
% 145.00/25.60 thf(func_def_277, type, one_one_complex: complex).
% 145.00/25.60 thf(func_def_278, type, one_on7984719198319812577d_enat: extended_enat).
% 145.00/25.60 thf(func_def_279, type, one_one_int: int).
% 145.00/25.60 thf(func_def_280, type, one_one_nat: nat).
% 145.00/25.60 thf(func_def_281, type, one_one_rat: rat).
% 145.00/25.60 thf(func_def_282, type, one_one_real: real).
% 145.00/25.60 thf(func_def_283, type, plus_p5714425477246183910nteger: code_integer > code_integer > code_integer).
% 145.00/25.60 thf(func_def_284, type, plus_plus_complex: complex > complex > complex).
% 145.00/25.60 thf(func_def_285, type, plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat).
% 145.00/25.60 thf(func_def_286, type, plus_plus_int: int > int > int).
% 145.00/25.60 thf(func_def_287, type, plus_plus_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_288, type, plus_plus_num: num > num > num).
% 145.00/25.60 thf(func_def_289, type, plus_plus_rat: rat > rat > rat).
% 145.00/25.60 thf(func_def_290, type, plus_plus_real: real > real > real).
% 145.00/25.60 thf(func_def_291, type, sgn_sgn_Code_integer: code_integer > code_integer).
% 145.00/25.60 thf(func_def_292, type, sgn_sgn_complex: complex > complex).
% 145.00/25.60 thf(func_def_293, type, sgn_sgn_int: int > int).
% 145.00/25.60 thf(func_def_294, type, sgn_sgn_rat: rat > rat).
% 145.00/25.60 thf(func_def_295, type, sgn_sgn_real: real > real).
% 145.00/25.60 thf(func_def_296, type, times_3573771949741848930nteger: code_integer > code_integer > code_integer).
% 145.00/25.60 thf(func_def_297, type, times_times_complex: complex > complex > complex).
% 145.00/25.60 thf(func_def_298, type, times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat).
% 145.00/25.60 thf(func_def_299, type, times_times_int: int > int > int).
% 145.00/25.60 thf(func_def_300, type, times_times_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_301, type, times_times_num: num > num > num).
% 145.00/25.60 thf(func_def_302, type, times_times_rat: rat > rat > rat).
% 145.00/25.60 thf(func_def_303, type, times_times_real: real > real > real).
% 145.00/25.60 thf(func_def_304, type, uminus1351360451143612070nteger: code_integer > code_integer).
% 145.00/25.60 thf(func_def_305, type, uminus1482373934393186551omplex: complex > complex).
% 145.00/25.60 thf(func_def_306, type, uminus_uminus_int: int > int).
% 145.00/25.60 thf(func_def_307, type, uminus_uminus_rat: rat > rat).
% 145.00/25.60 thf(func_def_308, type, uminus_uminus_real: real > real).
% 145.00/25.60 thf(func_def_309, type, uminus1532241313380277803et_int: set_int > set_int).
% 145.00/25.60 thf(func_def_310, type, uminus5710092332889474511et_nat: set_nat > set_nat).
% 145.00/25.60 thf(func_def_311, type, uminus612125837232591019t_real: set_real > set_real).
% 145.00/25.60 thf(func_def_312, type, zero_z3403309356797280102nteger: code_integer).
% 145.00/25.60 thf(func_def_313, type, zero_zero_complex: complex).
% 145.00/25.60 thf(func_def_314, type, zero_z5237406670263579293d_enat: extended_enat).
% 145.00/25.60 thf(func_def_315, type, zero_zero_int: int).
% 145.00/25.60 thf(func_def_316, type, zero_zero_nat: nat).
% 145.00/25.60 thf(func_def_317, type, zero_zero_rat: rat).
% 145.00/25.60 thf(func_def_318, type, zero_zero_real: real).
% 145.00/25.60 thf(func_def_319, type, groups6621422865394947399nteger: (complex > code_integer) > set_complex > code_integer).
% 145.00/25.60 thf(func_def_320, type, groups7754918857620584856omplex: (complex > complex) > set_complex > complex).
% 145.00/25.60 thf(func_def_321, type, groups5690904116761175830ex_int: (complex > int) > set_complex > int).
% 145.00/25.60 thf(func_def_322, type, groups5693394587270226106ex_nat: (complex > nat) > set_complex > nat).
% 145.00/25.60 thf(func_def_323, type, groups5058264527183730370ex_rat: (complex > rat) > set_complex > rat).
% 145.00/25.60 thf(func_def_324, type, groups5808333547571424918x_real: (complex > real) > set_complex > real).
% 145.00/25.60 thf(func_def_325, type, groups7873554091576472773nteger: (int > code_integer) > set_int > code_integer).
% 145.00/25.60 thf(func_def_326, type, groups3049146728041665814omplex: (int > complex) > set_int > complex).
% 145.00/25.60 thf(func_def_327, type, groups4538972089207619220nt_int: (int > int) > set_int > int).
% 145.00/25.60 thf(func_def_328, type, groups4541462559716669496nt_nat: (int > nat) > set_int > nat).
% 145.00/25.60 thf(func_def_329, type, groups3906332499630173760nt_rat: (int > rat) > set_int > rat).
% 145.00/25.60 thf(func_def_330, type, groups8778361861064173332t_real: (int > real) > set_int > real).
% 145.00/25.60 thf(func_def_331, type, groups7501900531339628137nteger: (nat > code_integer) > set_nat > code_integer).
% 145.00/25.60 thf(func_def_332, type, groups2073611262835488442omplex: (nat > complex) > set_nat > complex).
% 145.00/25.60 thf(func_def_333, type, groups3539618377306564664at_int: (nat > int) > set_nat > int).
% 145.00/25.60 thf(func_def_334, type, groups3542108847815614940at_nat: (nat > nat) > set_nat > nat).
% 145.00/25.60 thf(func_def_335, type, groups2906978787729119204at_rat: (nat > rat) > set_nat > rat).
% 145.00/25.60 thf(func_def_336, type, groups6591440286371151544t_real: (nat > real) > set_nat > real).
% 145.00/25.60 thf(func_def_337, type, groups6381953495645901045omplex: (product_prod_nat_nat > complex) > set_Pr1261947904930325089at_nat > complex).
% 145.00/25.60 thf(func_def_338, type, groups977919841031483927at_nat: (product_prod_nat_nat > nat) > set_Pr1261947904930325089at_nat > nat).
% 145.00/25.60 thf(func_def_339, type, groups4567486121110086003t_real: (product_prod_nat_nat > real) > set_Pr1261947904930325089at_nat > real).
% 145.00/25.60 thf(func_def_340, type, groups7713935264441627589nteger: (real > code_integer) > set_real > code_integer).
% 145.00/25.60 thf(func_def_341, type, groups5754745047067104278omplex: (real > complex) > set_real > complex).
% 145.00/25.60 thf(func_def_342, type, groups1932886352136224148al_int: (real > int) > set_real > int).
% 145.00/25.60 thf(func_def_343, type, groups1935376822645274424al_nat: (real > nat) > set_real > nat).
% 145.00/25.60 thf(func_def_344, type, groups1300246762558778688al_rat: (real > rat) > set_real > rat).
% 145.00/25.60 thf(func_def_345, type, groups8097168146408367636l_real: (real > real) > set_real > real).
% 145.00/25.60 thf(func_def_346, type, groups1705073143266064639nt_int: (int > int) > set_int > int).
% 145.00/25.60 thf(func_def_347, type, groups705719431365010083at_int: (nat > int) > set_nat > int).
% 145.00/25.60 thf(func_def_348, type, groups708209901874060359at_nat: (nat > nat) > set_nat > nat).
% 145.00/25.60 thf(func_def_349, type, groups9116527308978886569_o_int: ($o > int) > int > list_o > int).
% 145.00/25.60 thf(func_def_350, type, groups4561878855575611511st_nat: list_nat > nat).
% 145.00/25.60 thf(func_def_351, type, the_int: (int > $o) > int).
% 145.00/25.60 thf(func_def_352, type, the_real: (real > $o) > real).
% 145.00/25.60 thf(func_def_353, type, if_Code_integer: $o > code_integer > code_integer > code_integer).
% 145.00/25.60 thf(func_def_354, type, if_complex: $o > complex > complex > complex).
% 145.00/25.60 thf(func_def_355, type, if_Extended_enat: $o > extended_enat > extended_enat > extended_enat).
% 145.00/25.60 thf(func_def_356, type, if_int: $o > int > int > int).
% 145.00/25.60 thf(func_def_357, type, if_list_int: $o > list_int > list_int > list_int).
% 145.00/25.60 thf(func_def_358, type, if_list_nat: $o > list_nat > list_nat > list_nat).
% 145.00/25.60 thf(func_def_359, type, if_nat: $o > nat > nat > nat).
% 145.00/25.60 thf(func_def_360, type, if_num: $o > num > num > num).
% 145.00/25.60 thf(func_def_361, type, if_option_nat: $o > option_nat > option_nat > option_nat).
% 145.00/25.60 thf(func_def_362, type, if_option_num: $o > option_num > option_num > option_num).
% 145.00/25.60 thf(func_def_363, type, if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o).
% 145.00/25.60 thf(func_def_364, type, if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 145.00/25.60 thf(func_def_365, type, if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int).
% 145.00/25.60 thf(func_def_366, type, if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat).
% 145.00/25.60 thf(func_def_367, type, if_rat: $o > rat > rat > rat).
% 145.00/25.60 thf(func_def_368, type, if_real: $o > real > real > real).
% 145.00/25.60 thf(func_def_369, type, if_set_int: $o > set_int > set_int > set_int).
% 145.00/25.60 thf(func_def_370, type, if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 145.00/25.60 thf(func_def_371, type, abs_Integ: product_prod_nat_nat > int).
% 145.00/25.60 thf(func_def_372, type, rep_Integ: int > product_prod_nat_nat).
% 145.00/25.60 thf(func_def_373, type, int_ge_less_than: int > set_Pr958786334691620121nt_int).
% 145.00/25.60 thf(func_def_374, type, int_ge_less_than2: int > set_Pr958786334691620121nt_int).
% 145.00/25.60 thf(func_def_375, type, nat2: int > nat).
% 145.00/25.60 thf(func_def_376, type, power_int_real: real > int > real).
% 145.00/25.60 thf(func_def_377, type, ring_1_Ints_real: set_real).
% 145.00/25.60 thf(func_def_378, type, ring_18347121197199848620nteger: int > code_integer).
% 145.00/25.60 thf(func_def_379, type, ring_17405671764205052669omplex: int > complex).
% 145.00/25.60 thf(func_def_380, type, ring_1_of_int_int: int > int).
% 145.00/25.60 thf(func_def_381, type, ring_1_of_int_rat: int > rat).
% 145.00/25.60 thf(func_def_382, type, ring_1_of_int_real: int > real).
% 145.00/25.60 thf(func_def_383, type, inf_in1870772243966228564d_enat: extended_enat > extended_enat > extended_enat).
% 145.00/25.60 thf(func_def_384, type, inf_inf_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_385, type, semila1623282765462674594er_nat: (nat > nat > nat) > nat > (nat > nat > $o) > (nat > nat > $o) > $o).
% 145.00/25.60 thf(func_def_386, type, sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat).
% 145.00/25.60 thf(func_def_387, type, sup_sup_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_388, type, sup_sup_set_nat: set_nat > set_nat > set_nat).
% 145.00/25.60 thf(func_def_389, type, lattic8265883725875713057ax_nat: set_nat > nat).
% 145.00/25.60 thf(func_def_390, type, bfun_nat_real: (nat > real) > filter_nat > $o).
% 145.00/25.60 thf(func_def_391, type, at_infinity_real: filter_real).
% 145.00/25.60 thf(func_def_392, type, append_int: list_int > list_int > list_int).
% 145.00/25.60 thf(func_def_393, type, append_nat: list_nat > list_nat > list_nat).
% 145.00/25.60 thf(func_def_394, type, drop_nat: nat > list_nat > list_nat).
% 145.00/25.60 thf(func_def_395, type, linord2614967742042102400et_nat: set_nat > list_nat).
% 145.00/25.60 thf(func_def_396, type, cons_int: int > list_int > list_int).
% 145.00/25.60 thf(func_def_397, type, cons_nat: nat > list_nat > list_nat).
% 145.00/25.60 thf(func_def_398, type, nil_int: list_int).
% 145.00/25.60 thf(func_def_399, type, nil_nat: list_nat).
% 145.00/25.60 thf(func_def_400, type, hd_nat: list_nat > nat).
% 145.00/25.60 thf(func_def_401, type, map_nat_nat: (nat > nat) > list_nat > list_nat).
% 145.00/25.60 thf(func_def_402, type, set_o2: list_o > set_o).
% 145.00/25.60 thf(func_def_403, type, set_complex2: list_complex > set_complex).
% 145.00/25.60 thf(func_def_404, type, set_int2: list_int > set_int).
% 145.00/25.60 thf(func_def_405, type, set_nat2: list_nat > set_nat).
% 145.00/25.60 thf(func_def_406, type, set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat).
% 145.00/25.60 thf(func_def_407, type, set_real2: list_real > set_real).
% 145.00/25.60 thf(func_def_408, type, set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT).
% 145.00/25.60 thf(func_def_409, type, size_list_VEBT_VEBT: (vEBT_VEBT > nat) > list_VEBT_VEBT > nat).
% 145.00/25.60 thf(func_def_410, type, tl_nat: list_nat > list_nat).
% 145.00/25.60 thf(func_def_411, type, list_update_o: list_o > nat > $o > list_o).
% 145.00/25.60 thf(func_def_412, type, list_update_complex: list_complex > nat > complex > list_complex).
% 145.00/25.60 thf(func_def_413, type, list_update_int: list_int > nat > int > list_int).
% 145.00/25.60 thf(func_def_414, type, list_update_nat: list_nat > nat > nat > list_nat).
% 145.00/25.60 thf(func_def_415, type, list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat).
% 145.00/25.60 thf(func_def_416, type, list_update_real: list_real > nat > real > list_real).
% 145.00/25.60 thf(func_def_417, type, list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT).
% 145.00/25.60 thf(func_def_418, type, nth_o: list_o > nat > $o).
% 145.00/25.60 thf(func_def_419, type, nth_Code_integer: list_Code_integer > nat > code_integer).
% 145.00/25.60 thf(func_def_420, type, nth_complex: list_complex > nat > complex).
% 145.00/25.60 thf(func_def_421, type, nth_int: list_int > nat > int).
% 145.00/25.60 thf(func_def_422, type, nth_nat: list_nat > nat > nat).
% 145.00/25.60 thf(func_def_423, type, nth_num: list_num > nat > num).
% 145.00/25.60 thf(func_def_424, type, nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o).
% 145.00/25.60 thf(func_def_425, type, nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int).
% 145.00/25.60 thf(func_def_426, type, nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat).
% 145.00/25.60 thf(func_def_427, type, nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT).
% 145.00/25.60 thf(func_def_428, type, nth_Pr8522763379788166057eger_o: list_P8526636022914148096eger_o > nat > produc6271795597528267376eger_o).
% 145.00/25.60 thf(func_def_429, type, nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat).
% 145.00/25.60 thf(func_def_430, type, nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num).
% 145.00/25.60 thf(func_def_431, type, nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o).
% 145.00/25.60 thf(func_def_432, type, nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int).
% 145.00/25.60 thf(func_def_433, type, nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat).
% 145.00/25.60 thf(func_def_434, type, nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT).
% 145.00/25.60 thf(func_def_435, type, nth_real: list_real > nat > real).
% 145.00/25.60 thf(func_def_436, type, nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT).
% 145.00/25.60 thf(func_def_437, type, product_o_o: list_o > list_o > list_P4002435161011370285od_o_o).
% 145.00/25.60 thf(func_def_438, type, product_o_int: list_o > list_int > list_P3795440434834930179_o_int).
% 145.00/25.60 thf(func_def_439, type, product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat).
% 145.00/25.60 thf(func_def_440, type, product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT).
% 145.00/25.60 thf(func_def_441, type, produc3607205314601156340eger_o: list_Code_integer > list_o > list_P8526636022914148096eger_o).
% 145.00/25.60 thf(func_def_442, type, product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o).
% 145.00/25.60 thf(func_def_443, type, produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT).
% 145.00/25.60 thf(func_def_444, type, product_num_num: list_num > list_num > list_P3744719386663036955um_num).
% 145.00/25.60 thf(func_def_445, type, product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o).
% 145.00/25.60 thf(func_def_446, type, produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int).
% 145.00/25.60 thf(func_def_447, type, produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat).
% 145.00/25.60 thf(func_def_448, type, produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT).
% 145.00/25.60 thf(func_def_449, type, replicate_o: nat > $o > list_o).
% 145.00/25.60 thf(func_def_450, type, replicate_complex: nat > complex > list_complex).
% 145.00/25.60 thf(func_def_451, type, replicate_int: nat > int > list_int).
% 145.00/25.60 thf(func_def_452, type, replicate_nat: nat > nat > list_nat).
% 145.00/25.60 thf(func_def_453, type, replic4235873036481779905at_nat: nat > product_prod_nat_nat > list_P6011104703257516679at_nat).
% 145.00/25.60 thf(func_def_454, type, replicate_real: nat > real > list_real).
% 145.00/25.60 thf(func_def_455, type, replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT).
% 145.00/25.60 thf(func_def_456, type, sorted_wrt_int: (int > int > $o) > list_int > $o).
% 145.00/25.60 thf(func_def_457, type, sorted_wrt_nat: (nat > nat > $o) > list_nat > $o).
% 145.00/25.60 thf(func_def_458, type, take_nat: nat > list_nat > list_nat).
% 145.00/25.60 thf(func_def_459, type, upt: nat > nat > list_nat).
% 145.00/25.60 thf(func_def_460, type, upto: int > int > list_int).
% 145.00/25.60 thf(func_def_461, type, upto_aux: int > int > list_int > list_int).
% 145.00/25.60 thf(func_def_462, type, upto_rel: product_prod_int_int > product_prod_int_int > $o).
% 145.00/25.60 thf(func_def_463, type, suc: nat > nat).
% 145.00/25.60 thf(func_def_464, type, compow_nat_nat: nat > (nat > nat) > nat > nat).
% 145.00/25.60 thf(func_def_465, type, case_nat_o: $o > (nat > $o) > nat > $o).
% 145.00/25.60 thf(func_def_466, type, case_nat_nat: nat > (nat > nat) > nat > nat).
% 145.00/25.60 thf(func_def_467, type, case_nat_option_num: option_num > (nat > option_num) > nat > option_num).
% 145.00/25.60 thf(func_def_468, type, pred: nat > nat).
% 145.00/25.60 thf(func_def_469, type, semiri4939895301339042750nteger: nat > code_integer).
% 145.00/25.60 thf(func_def_470, type, semiri8010041392384452111omplex: nat > complex).
% 145.00/25.60 thf(func_def_471, type, semiri4216267220026989637d_enat: nat > extended_enat).
% 145.00/25.60 thf(func_def_472, type, semiri1314217659103216013at_int: nat > int).
% 145.00/25.60 thf(func_def_473, type, semiri1316708129612266289at_nat: nat > nat).
% 145.00/25.60 thf(func_def_474, type, semiri681578069525770553at_rat: nat > rat).
% 145.00/25.60 thf(func_def_475, type, semiri5074537144036343181t_real: nat > real).
% 145.00/25.60 thf(func_def_476, type, semiri2816024913162550771omplex: (complex > complex) > nat > complex > complex).
% 145.00/25.60 thf(func_def_477, type, semiri8420488043553186161ux_int: (int > int) > nat > int > int).
% 145.00/25.60 thf(func_def_478, type, semiri8422978514062236437ux_nat: (nat > nat) > nat > nat > nat).
% 145.00/25.60 thf(func_def_479, type, semiri7787848453975740701ux_rat: (rat > rat) > nat > rat > rat).
% 145.00/25.60 thf(func_def_480, type, semiri7260567687927622513x_real: (real > real) > nat > real > real).
% 145.00/25.60 thf(func_def_481, type, size_size_list_o: list_o > nat).
% 145.00/25.60 thf(func_def_482, type, size_s3445333598471063425nteger: list_Code_integer > nat).
% 145.00/25.60 thf(func_def_483, type, size_s3451745648224563538omplex: list_complex > nat).
% 145.00/25.60 thf(func_def_484, type, size_size_list_int: list_int > nat).
% 145.00/25.60 thf(func_def_485, type, size_size_list_nat: list_nat > nat).
% 145.00/25.60 thf(func_def_486, type, size_size_list_num: list_num > nat).
% 145.00/25.60 thf(func_def_487, type, size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat).
% 145.00/25.60 thf(func_def_488, type, size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat).
% 145.00/25.60 thf(func_def_489, type, size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat).
% 145.00/25.60 thf(func_def_490, type, size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat).
% 145.00/25.60 thf(func_def_491, type, size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat).
% 145.00/25.60 thf(func_def_492, type, size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat).
% 145.00/25.60 thf(func_def_493, type, size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat).
% 145.00/25.60 thf(func_def_494, type, size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat).
% 145.00/25.60 thf(func_def_495, type, size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat).
% 145.00/25.60 thf(func_def_496, type, size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat).
% 145.00/25.60 thf(func_def_497, type, size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat).
% 145.00/25.60 thf(func_def_498, type, size_size_list_real: list_real > nat).
% 145.00/25.60 thf(func_def_499, type, size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat).
% 145.00/25.60 thf(func_def_500, type, size_size_num: num > nat).
% 145.00/25.60 thf(func_def_501, type, size_size_option_nat: option_nat > nat).
% 145.00/25.60 thf(func_def_502, type, size_size_option_num: option_num > nat).
% 145.00/25.60 thf(func_def_503, type, size_s170228958280169651at_nat: option4927543243414619207at_nat > nat).
% 145.00/25.60 thf(func_def_504, type, size_size_VEBT_VEBT: vEBT_VEBT > nat).
% 145.00/25.60 thf(func_def_505, type, nat_list_encode: list_nat > nat).
% 145.00/25.60 thf(func_def_506, type, nat_list_encode_rel: list_nat > list_nat > $o).
% 145.00/25.60 thf(func_def_507, type, nat_prod_decode_aux: nat > nat > product_prod_nat_nat).
% 145.00/25.60 thf(func_def_508, type, nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 145.00/25.60 thf(func_def_509, type, nat_prod_encode: product_prod_nat_nat > nat).
% 145.00/25.60 thf(func_def_510, type, nat_set_decode: nat > set_nat).
% 145.00/25.60 thf(func_def_511, type, nat_set_encode: set_nat > nat).
% 145.00/25.60 thf(func_def_512, type, nat_triangle: nat > nat).
% 145.00/25.60 thf(func_def_513, type, root: nat > real > real).
% 145.00/25.60 thf(func_def_514, type, sqrt: real > real).
% 145.00/25.60 thf(func_def_515, type, bitM: num > num).
% 145.00/25.60 thf(func_def_516, type, inc: num > num).
% 145.00/25.60 thf(func_def_517, type, neg_nu8804712462038260780nteger: code_integer > code_integer).
% 145.00/25.60 thf(func_def_518, type, neg_nu7009210354673126013omplex: complex > complex).
% 145.00/25.60 thf(func_def_519, type, neg_numeral_dbl_int: int > int).
% 145.00/25.60 thf(func_def_520, type, neg_numeral_dbl_rat: rat > rat).
% 145.00/25.60 thf(func_def_521, type, neg_numeral_dbl_real: real > real).
% 145.00/25.60 thf(func_def_522, type, neg_nu7757733837767384882nteger: code_integer > code_integer).
% 145.00/25.60 thf(func_def_523, type, neg_nu6511756317524482435omplex: complex > complex).
% 145.00/25.60 thf(func_def_524, type, neg_nu3811975205180677377ec_int: int > int).
% 145.00/25.60 thf(func_def_525, type, neg_nu3179335615603231917ec_rat: rat > rat).
% 145.00/25.60 thf(func_def_526, type, neg_nu6075765906172075777c_real: real > real).
% 145.00/25.60 thf(func_def_527, type, neg_nu5831290666863070958nteger: code_integer > code_integer).
% 145.00/25.60 thf(func_def_528, type, neg_nu8557863876264182079omplex: complex > complex).
% 145.00/25.60 thf(func_def_529, type, neg_nu5851722552734809277nc_int: int > int).
% 145.00/25.60 thf(func_def_530, type, neg_nu5219082963157363817nc_rat: rat > rat).
% 145.00/25.60 thf(func_def_531, type, neg_nu8295874005876285629c_real: real > real).
% 145.00/25.60 thf(func_def_532, type, neg_numeral_sub_int: num > num > int).
% 145.00/25.60 thf(func_def_533, type, bit0: num > num).
% 145.00/25.60 thf(func_def_534, type, bit1: num > num).
% 145.00/25.60 thf(func_def_535, type, one: num).
% 145.00/25.60 thf(func_def_536, type, case_num_option_num: option_num > (num > option_num) > (num > option_num) > num > option_num).
% 145.00/25.60 thf(func_def_537, type, size_num: num > nat).
% 145.00/25.60 thf(func_def_538, type, num_of_nat: nat > num).
% 145.00/25.60 thf(func_def_539, type, numera6620942414471956472nteger: num > code_integer).
% 145.00/25.60 thf(func_def_540, type, numera6690914467698888265omplex: num > complex).
% 145.00/25.60 thf(func_def_541, type, numera1916890842035813515d_enat: num > extended_enat).
% 145.00/25.60 thf(func_def_542, type, numeral_numeral_int: num > int).
% 145.00/25.60 thf(func_def_543, type, numeral_numeral_nat: num > nat).
% 145.00/25.60 thf(func_def_544, type, numeral_numeral_rat: num > rat).
% 145.00/25.60 thf(func_def_545, type, numeral_numeral_real: num > real).
% 145.00/25.60 thf(func_def_546, type, pow: num > num > num).
% 145.00/25.60 thf(func_def_547, type, pred_numeral: num > nat).
% 145.00/25.60 thf(func_def_548, type, sqr: num > num).
% 145.00/25.60 thf(func_def_549, type, none_nat: option_nat).
% 145.00/25.60 thf(func_def_550, type, none_num: option_num).
% 145.00/25.60 thf(func_def_551, type, none_P5556105721700978146at_nat: option4927543243414619207at_nat).
% 145.00/25.60 thf(func_def_552, type, some_nat: nat > option_nat).
% 145.00/25.60 thf(func_def_553, type, some_num: num > option_num).
% 145.00/25.60 thf(func_def_554, type, some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat).
% 145.00/25.60 thf(func_def_555, type, case_o184042715313410164at_nat: $o > (product_prod_nat_nat > $o) > option4927543243414619207at_nat > $o).
% 145.00/25.60 thf(func_def_556, type, case_option_int_num: int > (num > int) > option_num > int).
% 145.00/25.60 thf(func_def_557, type, case_option_num_num: num > (num > num) > option_num > num).
% 145.00/25.60 thf(func_def_558, type, case_o6005452278849405969um_num: option_num > (num > option_num) > option_num > option_num).
% 145.00/25.60 thf(func_def_559, type, map_option_num_num: (num > num) > option_num > option_num).
% 145.00/25.60 thf(func_def_560, type, size_option_nat: (nat > nat) > option_nat > nat).
% 145.00/25.60 thf(func_def_561, type, size_option_num: (num > nat) > option_num > nat).
% 145.00/25.60 thf(func_def_562, type, size_o8335143837870341156at_nat: (product_prod_nat_nat > nat) > option4927543243414619207at_nat > nat).
% 145.00/25.60 thf(func_def_563, type, the_nat: option_nat > nat).
% 145.00/25.60 thf(func_def_564, type, the_num: option_num > num).
% 145.00/25.60 thf(func_def_565, type, the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat).
% 145.00/25.60 thf(func_def_566, type, bot_bo4731626569425807221er_o_o: code_integer > $o > $o).
% 145.00/25.60 thf(func_def_567, type, bot_bot_int_int_o: int > int > $o).
% 145.00/25.60 thf(func_def_568, type, bot_bot_nat_nat_o: nat > nat > $o).
% 145.00/25.60 thf(func_def_569, type, bot_bot_nat_num_o: nat > num > $o).
% 145.00/25.60 thf(func_def_570, type, bot_bot_num_num_o: num > num > $o).
% 145.00/25.60 thf(func_def_571, type, bot_bo4199563552545308370d_enat: extended_enat).
% 145.00/25.60 thf(func_def_572, type, bot_bot_nat: nat).
% 145.00/25.60 thf(func_def_573, type, bot_bot_set_complex: set_complex).
% 145.00/25.60 thf(func_def_574, type, bot_bot_set_int: set_int).
% 145.00/25.60 thf(func_def_575, type, bot_bot_set_nat: set_nat).
% 145.00/25.60 thf(func_def_576, type, bot_bot_set_num: set_num).
% 145.00/25.60 thf(func_def_577, type, bot_bo5379713665208646970eger_o: set_Pr448751882837621926eger_o).
% 145.00/25.60 thf(func_def_578, type, bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int).
% 145.00/25.60 thf(func_def_579, type, bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat).
% 145.00/25.60 thf(func_def_580, type, bot_bo7038385379056416535at_num: set_Pr6200539531224447659at_num).
% 145.00/25.60 thf(func_def_581, type, bot_bo9056780473022590049um_num: set_Pr8218934625190621173um_num).
% 145.00/25.60 thf(func_def_582, type, bot_bot_set_rat: set_rat).
% 145.00/25.60 thf(func_def_583, type, bot_bot_set_real: set_real).
% 145.00/25.60 thf(func_def_584, type, bot_bot_set_set_int: set_set_int).
% 145.00/25.60 thf(func_def_585, type, ord_le6747313008572928689nteger: code_integer > code_integer > $o).
% 145.00/25.60 thf(func_def_586, type, ord_le72135733267957522d_enat: extended_enat > extended_enat > $o).
% 145.00/25.60 thf(func_def_587, type, ord_less_int: int > int > $o).
% 145.00/25.60 thf(func_def_588, type, ord_less_nat: nat > nat > $o).
% 145.00/25.60 thf(func_def_589, type, ord_less_num: num > num > $o).
% 145.00/25.60 thf(func_def_590, type, ord_less_rat: rat > rat > $o).
% 145.00/25.60 thf(func_def_591, type, ord_less_real: real > real > $o).
% 145.00/25.60 thf(func_def_592, type, ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o).
% 145.00/25.60 thf(func_def_593, type, ord_less_set_complex: set_complex > set_complex > $o).
% 145.00/25.60 thf(func_def_594, type, ord_less_set_int: set_int > set_int > $o).
% 145.00/25.60 thf(func_def_595, type, ord_less_set_nat: set_nat > set_nat > $o).
% 145.00/25.60 thf(func_def_596, type, ord_less_set_num: set_num > set_num > $o).
% 145.00/25.60 thf(func_def_597, type, ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o).
% 145.00/25.60 thf(func_def_598, type, ord_less_set_rat: set_rat > set_rat > $o).
% 145.00/25.60 thf(func_def_599, type, ord_less_set_real: set_real > set_real > $o).
% 145.00/25.60 thf(func_def_600, type, ord_less_set_set_int: set_set_int > set_set_int > $o).
% 145.00/25.60 thf(func_def_601, type, ord_le2162486998276636481er_o_o: (code_integer > $o > $o) > (code_integer > $o > $o) > $o).
% 145.00/25.60 thf(func_def_602, type, ord_le4573692005234683329plex_o: (complex > $o) > (complex > $o) > $o).
% 145.00/25.60 thf(func_def_603, type, ord_le6741204236512500942_int_o: (int > int > $o) > (int > int > $o) > $o).
% 145.00/25.60 thf(func_def_604, type, ord_less_eq_int_o: (int > $o) > (int > $o) > $o).
% 145.00/25.60 thf(func_def_605, type, ord_le6558929396352911974_nat_o: (list_nat > list_nat > $o) > (list_nat > list_nat > $o) > $o).
% 145.00/25.60 thf(func_def_606, type, ord_le1520216061033275535_nat_o: (list_nat > $o) > (list_nat > $o) > $o).
% 145.00/25.60 thf(func_def_607, type, ord_le2646555220125990790_nat_o: (nat > nat > $o) > (nat > nat > $o) > $o).
% 145.00/25.60 thf(func_def_608, type, ord_le3404735783095501756_num_o: (nat > num > $o) > (nat > num > $o) > $o).
% 145.00/25.60 thf(func_def_609, type, ord_less_eq_nat_o: (nat > $o) > (nat > $o) > $o).
% 145.00/25.60 thf(func_def_610, type, ord_le6124364862034508274_num_o: (num > num > $o) > (num > num > $o) > $o).
% 145.00/25.60 thf(func_def_611, type, ord_le1598226405681992910_int_o: (product_prod_int_int > product_prod_int_int > $o) > (product_prod_int_int > product_prod_int_int > $o) > $o).
% 145.00/25.60 thf(func_def_612, type, ord_le8369615600986905444_int_o: (product_prod_int_int > $o) > (product_prod_int_int > $o) > $o).
% 145.00/25.60 thf(func_def_613, type, ord_le5604493270027003598_nat_o: (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > $o).
% 145.00/25.60 thf(func_def_614, type, ord_le704812498762024988_nat_o: (product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > $o).
% 145.00/25.60 thf(func_def_615, type, ord_le2556027599737686990_num_o: (product_prod_num_num > product_prod_num_num > $o) > (product_prod_num_num > product_prod_num_num > $o) > $o).
% 145.00/25.60 thf(func_def_616, type, ord_le2239182809043710856_num_o: (product_prod_num_num > $o) > (product_prod_num_num > $o) > $o).
% 145.00/25.60 thf(func_def_617, type, ord_less_eq_real_o: (real > $o) > (real > $o) > $o).
% 145.00/25.60 thf(func_def_618, type, ord_le3102999989581377725nteger: code_integer > code_integer > $o).
% 145.00/25.60 thf(func_def_619, type, ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o).
% 145.00/25.60 thf(func_def_620, type, ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o).
% 145.00/25.60 thf(func_def_621, type, ord_le4104064031414453916r_real: filter_real > filter_real > $o).
% 145.00/25.60 thf(func_def_622, type, ord_less_eq_int: int > int > $o).
% 145.00/25.60 thf(func_def_623, type, ord_less_eq_nat: nat > nat > $o).
% 145.00/25.60 thf(func_def_624, type, ord_less_eq_num: num > num > $o).
% 145.00/25.60 thf(func_def_625, type, ord_less_eq_rat: rat > rat > $o).
% 145.00/25.60 thf(func_def_626, type, ord_less_eq_real: real > real > $o).
% 145.00/25.60 thf(func_def_627, type, ord_less_eq_set_o: set_o > set_o > $o).
% 145.00/25.60 thf(func_def_628, type, ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o).
% 145.00/25.60 thf(func_def_629, type, ord_le211207098394363844omplex: set_complex > set_complex > $o).
% 145.00/25.60 thf(func_def_630, type, ord_less_eq_set_int: set_int > set_int > $o).
% 145.00/25.60 thf(func_def_631, type, ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o).
% 145.00/25.60 thf(func_def_632, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 145.00/25.60 thf(func_def_633, type, ord_less_eq_set_num: set_num > set_num > $o).
% 145.00/25.60 thf(func_def_634, type, ord_le8980329558974975238eger_o: set_Pr448751882837621926eger_o > set_Pr448751882837621926eger_o > $o).
% 145.00/25.60 thf(func_def_635, type, ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o).
% 145.00/25.60 thf(func_def_636, type, ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o).
% 145.00/25.60 thf(func_def_637, type, ord_le8085105155179020875at_num: set_Pr6200539531224447659at_num > set_Pr6200539531224447659at_num > $o).
% 145.00/25.60 thf(func_def_638, type, ord_le880128212290418581um_num: set_Pr8218934625190621173um_num > set_Pr8218934625190621173um_num > $o).
% 145.00/25.60 thf(func_def_639, type, ord_less_eq_set_rat: set_rat > set_rat > $o).
% 145.00/25.60 thf(func_def_640, type, ord_less_eq_set_real: set_real > set_real > $o).
% 145.00/25.60 thf(func_def_641, type, ord_le4403425263959731960et_int: set_set_int > set_set_int > $o).
% 145.00/25.60 thf(func_def_642, type, ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o).
% 145.00/25.60 thf(func_def_643, type, ord_max_Code_integer: code_integer > code_integer > code_integer).
% 145.00/25.60 thf(func_def_644, type, ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat).
% 145.00/25.60 thf(func_def_645, type, ord_max_int: int > int > int).
% 145.00/25.60 thf(func_def_646, type, ord_max_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_647, type, ord_max_num: num > num > num).
% 145.00/25.60 thf(func_def_648, type, ord_max_rat: rat > rat > rat).
% 145.00/25.60 thf(func_def_649, type, ord_max_real: real > real > real).
% 145.00/25.60 thf(func_def_650, type, ord_max_set_int: set_int > set_int > set_int).
% 145.00/25.60 thf(func_def_651, type, ord_mi8085742599997312461d_enat: extended_enat > extended_enat > extended_enat).
% 145.00/25.60 thf(func_def_652, type, ord_min_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_653, type, order_Greatest_nat: (nat > $o) > nat).
% 145.00/25.60 thf(func_def_654, type, order_9091379641038594480t_real: (nat > real) > $o).
% 145.00/25.60 thf(func_def_655, type, order_mono_nat_nat: (nat > nat) > $o).
% 145.00/25.60 thf(func_def_656, type, order_mono_nat_real: (nat > real) > $o).
% 145.00/25.60 thf(func_def_657, type, order_5726023648592871131at_nat: (nat > nat) > $o).
% 145.00/25.60 thf(func_def_658, type, top_top_set_o: set_o).
% 145.00/25.60 thf(func_def_659, type, top_top_set_nat: set_nat).
% 145.00/25.60 thf(func_def_660, type, top_to1996260823553986621t_unit: set_Product_unit).
% 145.00/25.60 thf(func_def_661, type, top_top_set_real: set_real).
% 145.00/25.60 thf(func_def_662, type, top_top_set_char: set_char).
% 145.00/25.60 thf(func_def_663, type, power_8256067586552552935nteger: code_integer > nat > code_integer).
% 145.00/25.60 thf(func_def_664, type, power_power_complex: complex > nat > complex).
% 145.00/25.60 thf(func_def_665, type, power_power_int: int > nat > int).
% 145.00/25.60 thf(func_def_666, type, power_power_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_667, type, power_power_rat: rat > nat > rat).
% 145.00/25.60 thf(func_def_668, type, power_power_real: real > nat > real).
% 145.00/25.60 thf(func_def_669, type, produc4035269172776083154on_nat: (nat > nat > $o) > produc4953844613479565601on_nat > produc2233624965454879586on_nat).
% 145.00/25.60 thf(func_def_670, type, produc3209952032786966637at_nat: (nat > nat > nat) > produc7248412053542808358at_nat > produc4471711990508489141at_nat).
% 145.00/25.60 thf(func_def_671, type, produc8929957630744042906on_nat: (nat > nat > nat) > produc4953844613479565601on_nat > produc8306885398267862888on_nat).
% 145.00/25.60 thf(func_def_672, type, produc851828971589881931at_num: (nat > num > num) > produc2963631642982155120at_num > produc3368934014287244435at_num).
% 145.00/25.60 thf(func_def_673, type, produc3576312749637752826on_num: (num > num > $o) > produc3447558737645232053on_num > produc7036089656553540234on_num).
% 145.00/25.60 thf(func_def_674, type, produc5778274026573060048on_num: (num > num > num) > produc3447558737645232053on_num > produc1193250871479095198on_num).
% 145.00/25.60 thf(func_def_675, type, produc3994169339658061776at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > produc6121120109295599847at_nat > produc5491161045314408544at_nat).
% 145.00/25.60 thf(func_def_676, type, produc2899441246263362727at_nat: (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > produc6121120109295599847at_nat > produc5542196010084753463at_nat).
% 145.00/25.60 thf(func_def_677, type, product_Pair_o_o: $o > $o > product_prod_o_o).
% 145.00/25.60 thf(func_def_678, type, product_Pair_o_int: $o > int > product_prod_o_int).
% 145.00/25.60 thf(func_def_679, type, product_Pair_o_nat: $o > nat > product_prod_o_nat).
% 145.00/25.60 thf(func_def_680, type, produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT).
% 145.00/25.60 thf(func_def_681, type, produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o).
% 145.00/25.60 thf(func_def_682, type, produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger).
% 145.00/25.60 thf(func_def_683, type, product_Pair_int_int: int > int > product_prod_int_int).
% 145.00/25.60 thf(func_def_684, type, product_Pair_nat_nat: nat > nat > product_prod_nat_nat).
% 145.00/25.60 thf(func_def_685, type, product_Pair_nat_num: nat > num > product_prod_nat_num).
% 145.00/25.60 thf(func_def_686, type, produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat).
% 145.00/25.60 thf(func_def_687, type, produc1195630363706982562at_num: nat > product_prod_nat_num > produc2963631642982155120at_num).
% 145.00/25.60 thf(func_def_688, type, product_Pair_num_num: num > num > product_prod_num_num).
% 145.00/25.60 thf(func_def_689, type, produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat).
% 145.00/25.60 thf(func_def_690, type, produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num).
% 145.00/25.60 thf(func_def_691, type, produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat).
% 145.00/25.60 thf(func_def_692, type, produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o).
% 145.00/25.60 thf(func_def_693, type, produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int).
% 145.00/25.60 thf(func_def_694, type, produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat).
% 145.00/25.60 thf(func_def_695, type, produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT).
% 145.00/25.60 thf(func_def_696, type, produc6499014454317279255nteger: (code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 145.00/25.60 thf(func_def_697, type, produc7828578312038201481er_o_o: (code_integer > $o > $o) > produc6271795597528267376eger_o > $o).
% 145.00/25.60 thf(func_def_698, type, produc1043322548047392435omplex: (code_integer > $o > set_complex) > produc6271795597528267376eger_o > set_complex).
% 145.00/25.60 thf(func_def_699, type, produc1253318751659547953et_int: (code_integer > $o > set_int) > produc6271795597528267376eger_o > set_int).
% 145.00/25.60 thf(func_def_700, type, produc5431169771168744661et_nat: (code_integer > $o > set_nat) > produc6271795597528267376eger_o > set_nat).
% 145.00/25.60 thf(func_def_701, type, produc242741666403216561t_real: (code_integer > $o > set_real) > produc6271795597528267376eger_o > set_real).
% 145.00/25.60 thf(func_def_702, type, produc1553301316500091796er_int: (code_integer > code_integer > int) > produc8923325533196201883nteger > int).
% 145.00/25.60 thf(func_def_703, type, produc1555791787009142072er_nat: (code_integer > code_integer > nat) > produc8923325533196201883nteger > nat).
% 145.00/25.60 thf(func_def_704, type, produc7336495610019696514er_num: (code_integer > code_integer > num) > produc8923325533196201883nteger > num).
% 145.00/25.60 thf(func_def_705, type, produc9125791028180074456eger_o: (code_integer > code_integer > produc6271795597528267376eger_o) > produc8923325533196201883nteger > produc6271795597528267376eger_o).
% 145.00/25.60 thf(func_def_706, type, produc6916734918728496179nteger: (code_integer > code_integer > produc8923325533196201883nteger) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 145.00/25.60 thf(func_def_707, type, produc4947309494688390418_int_o: (int > int > $o) > product_prod_int_int > $o).
% 145.00/25.60 thf(func_def_708, type, produc4245557441103728435nt_int: (int > int > product_prod_int_int) > product_prod_int_int > product_prod_int_int).
% 145.00/25.60 thf(func_def_709, type, produc8739625826339149834_nat_o: (nat > nat > product_prod_nat_nat > $o) > product_prod_nat_nat > product_prod_nat_nat > $o).
% 145.00/25.60 thf(func_def_710, 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).
% 145.00/25.60 thf(func_def_711, type, produc6081775807080527818_nat_o: (nat > nat > $o) > product_prod_nat_nat > $o).
% 145.00/25.60 thf(func_def_712, type, produc1917071388513777916omplex: (nat > nat > complex) > product_prod_nat_nat > complex).
% 145.00/25.60 thf(func_def_713, type, produc6840382203811409530at_int: (nat > nat > int) > product_prod_nat_nat > int).
% 145.00/25.60 thf(func_def_714, type, produc6842872674320459806at_nat: (nat > nat > nat) > product_prod_nat_nat > nat).
% 145.00/25.60 thf(func_def_715, type, produc2626176000494625587at_nat: (nat > nat > product_prod_nat_nat) > product_prod_nat_nat > product_prod_nat_nat).
% 145.00/25.60 thf(func_def_716, type, produc6207742614233964070at_rat: (nat > nat > rat) > product_prod_nat_nat > rat).
% 145.00/25.60 thf(func_def_717, type, produc1703576794950452218t_real: (nat > nat > real) > product_prod_nat_nat > real).
% 145.00/25.60 thf(func_def_718, type, produc4927758841916487424_num_o: (nat > num > $o) > product_prod_nat_num > $o).
% 145.00/25.60 thf(func_def_719, type, produc478579273971653890on_num: (nat > num > option_num) > product_prod_nat_num > option_num).
% 145.00/25.60 thf(func_def_720, type, produc6231982587499038204omplex: (nat > num > set_complex) > product_prod_nat_num > set_complex).
% 145.00/25.60 thf(func_def_721, type, produc1435849484188172666t_real: (nat > num > set_real) > product_prod_nat_num > set_real).
% 145.00/25.60 thf(func_def_722, type, produc5703948589228662326_num_o: (num > num > $o) > product_prod_num_num > $o).
% 145.00/25.60 thf(func_def_723, type, produc2866383454006189126omplex: (num > num > set_complex) > product_prod_num_num > set_complex).
% 145.00/25.60 thf(func_def_724, type, produc6406642877701697732et_int: (num > num > set_int) > product_prod_num_num > set_int).
% 145.00/25.60 thf(func_def_725, type, produc1361121860356118632et_nat: (num > num > set_nat) > product_prod_num_num > set_nat).
% 145.00/25.60 thf(func_def_726, type, produc8296048397933160132t_real: (num > num > set_real) > product_prod_num_num > set_real).
% 145.00/25.60 thf(func_def_727, type, product_fst_int_int: product_prod_int_int > int).
% 145.00/25.60 thf(func_def_728, type, product_fst_nat_nat: product_prod_nat_nat > nat).
% 145.00/25.60 thf(func_def_729, type, product_snd_int_int: product_prod_int_int > int).
% 145.00/25.60 thf(func_def_730, type, product_snd_nat_nat: product_prod_nat_nat > nat).
% 145.00/25.60 thf(func_def_731, type, abs_Rat: product_prod_int_int > rat).
% 145.00/25.60 thf(func_def_732, type, fract: int > int > rat).
% 145.00/25.60 thf(func_def_733, type, frct: product_prod_int_int > rat).
% 145.00/25.60 thf(func_def_734, type, rep_Rat: rat > product_prod_int_int).
% 145.00/25.60 thf(func_def_735, type, field_5140801741446780682s_real: set_real).
% 145.00/25.60 thf(func_def_736, type, field_7254667332652039916t_real: rat > real).
% 145.00/25.60 thf(func_def_737, type, normalize: product_prod_int_int > product_prod_int_int).
% 145.00/25.60 thf(func_def_738, type, of_int: int > rat).
% 145.00/25.60 thf(func_def_739, type, positive: rat > $o).
% 145.00/25.60 thf(func_def_740, type, quotient_of: rat > product_prod_int_int).
% 145.00/25.60 thf(func_def_741, type, ratrel: product_prod_int_int > product_prod_int_int > $o).
% 145.00/25.60 thf(func_def_742, type, real_V2521375963428798218omplex: set_complex).
% 145.00/25.60 thf(func_def_743, type, real_V5970128139526366754l_real: (real > real) > $o).
% 145.00/25.60 thf(func_def_744, type, real_V3694042436643373181omplex: complex > complex > real).
% 145.00/25.60 thf(func_def_745, type, real_V975177566351809787t_real: real > real > real).
% 145.00/25.60 thf(func_def_746, type, real_V1022390504157884413omplex: complex > real).
% 145.00/25.60 thf(func_def_747, type, real_V7735802525324610683m_real: real > real).
% 145.00/25.60 thf(func_def_748, type, real_V4546457046886955230omplex: real > complex).
% 145.00/25.60 thf(func_def_749, type, real_V1803761363581548252l_real: real > real).
% 145.00/25.60 thf(func_def_750, type, real_V2046097035970521341omplex: real > complex > complex).
% 145.00/25.60 thf(func_def_751, type, real_V1485227260804924795R_real: real > real > real).
% 145.00/25.60 thf(func_def_752, type, divide6298287555418463151nteger: code_integer > code_integer > code_integer).
% 145.00/25.60 thf(func_def_753, type, divide1717551699836669952omplex: complex > complex > complex).
% 145.00/25.60 thf(func_def_754, type, divide_divide_int: int > int > int).
% 145.00/25.60 thf(func_def_755, type, divide_divide_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_756, type, divide_divide_rat: rat > rat > rat).
% 145.00/25.60 thf(func_def_757, type, divide_divide_real: real > real > real).
% 145.00/25.60 thf(func_def_758, type, dvd_dvd_Code_integer: code_integer > code_integer > $o).
% 145.00/25.60 thf(func_def_759, type, dvd_dvd_complex: complex > complex > $o).
% 145.00/25.60 thf(func_def_760, type, dvd_dvd_int: int > int > $o).
% 145.00/25.60 thf(func_def_761, type, dvd_dvd_nat: nat > nat > $o).
% 145.00/25.60 thf(func_def_762, type, dvd_dvd_rat: rat > rat > $o).
% 145.00/25.60 thf(func_def_763, type, dvd_dvd_real: real > real > $o).
% 145.00/25.60 thf(func_def_764, type, modulo364778990260209775nteger: code_integer > code_integer > code_integer).
% 145.00/25.60 thf(func_def_765, type, modulo_modulo_int: int > int > int).
% 145.00/25.60 thf(func_def_766, type, modulo_modulo_nat: nat > nat > nat).
% 145.00/25.60 thf(func_def_767, type, zero_n356916108424825756nteger: $o > code_integer).
% 145.00/25.60 thf(func_def_768, type, zero_n1201886186963655149omplex: $o > complex).
% 145.00/25.60 thf(func_def_769, type, zero_n2684676970156552555ol_int: $o > int).
% 145.00/25.60 thf(func_def_770, type, zero_n2687167440665602831ol_nat: $o > nat).
% 145.00/25.60 thf(func_def_771, type, zero_n2052037380579107095ol_rat: $o > rat).
% 145.00/25.60 thf(func_def_772, type, zero_n3304061248610475627l_real: $o > real).
% 145.00/25.60 thf(func_def_773, type, suminf_complex: (nat > complex) > complex).
% 145.00/25.60 thf(func_def_774, type, suminf_int: (nat > int) > int).
% 145.00/25.60 thf(func_def_775, type, suminf_nat: (nat > nat) > nat).
% 145.00/25.60 thf(func_def_776, type, suminf_real: (nat > real) > real).
% 145.00/25.60 thf(func_def_777, type, summable_complex: (nat > complex) > $o).
% 145.00/25.60 thf(func_def_778, type, summable_int: (nat > int) > $o).
% 145.00/25.60 thf(func_def_779, type, summable_nat: (nat > nat) > $o).
% 145.00/25.60 thf(func_def_780, type, summable_real: (nat > real) > $o).
% 145.00/25.60 thf(func_def_781, type, sums_complex: (nat > complex) > complex > $o).
% 145.00/25.60 thf(func_def_782, type, sums_int: (nat > int) > int > $o).
% 145.00/25.60 thf(func_def_783, type, sums_nat: (nat > nat) > nat > $o).
% 145.00/25.60 thf(func_def_784, type, sums_real: (nat > real) > real > $o).
% 145.00/25.60 thf(func_def_785, type, collect_Code_integer: (code_integer > $o) > set_Code_integer).
% 145.00/25.60 thf(func_def_786, type, collect_complex: (complex > $o) > set_complex).
% 145.00/25.60 thf(func_def_787, type, collect_int: (int > $o) > set_int).
% 145.00/25.60 thf(func_def_788, type, collect_list_o: (list_o > $o) > set_list_o).
% 145.00/25.60 thf(func_def_789, type, collect_list_complex: (list_complex > $o) > set_list_complex).
% 145.00/25.60 thf(func_def_790, type, collect_list_int: (list_int > $o) > set_list_int).
% 145.00/25.60 thf(func_def_791, type, collect_list_nat: (list_nat > $o) > set_list_nat).
% 145.00/25.60 thf(func_def_792, type, collec5608196760682091941T_VEBT: (list_VEBT_VEBT > $o) > set_list_VEBT_VEBT).
% 145.00/25.60 thf(func_def_793, type, collect_nat: (nat > $o) > set_nat).
% 145.00/25.60 thf(func_def_794, type, collect_num: (num > $o) > set_num).
% 145.00/25.60 thf(func_def_795, type, collec213857154873943460nt_int: (product_prod_int_int > $o) > set_Pr958786334691620121nt_int).
% 145.00/25.60 thf(func_def_796, type, collec3392354462482085612at_nat: (product_prod_nat_nat > $o) > set_Pr1261947904930325089at_nat).
% 145.00/25.60 thf(func_def_797, type, collect_rat: (rat > $o) > set_rat).
% 145.00/25.60 thf(func_def_798, type, collect_real: (real > $o) > set_real).
% 145.00/25.60 thf(func_def_799, type, collect_set_complex: (set_complex > $o) > set_set_complex).
% 145.00/25.60 thf(func_def_800, type, collect_set_int: (set_int > $o) > set_set_int).
% 145.00/25.60 thf(func_def_801, type, collect_set_nat: (set_nat > $o) > set_set_nat).
% 145.00/25.60 thf(func_def_802, type, image_int_int: (int > int) > set_int > set_int).
% 145.00/25.60 thf(func_def_803, type, image_nat_int: (nat > int) > set_nat > set_int).
% 145.00/25.60 thf(func_def_804, type, image_nat_nat: (nat > nat) > set_nat > set_nat).
% 145.00/25.60 thf(func_def_805, type, image_nat_real: (nat > real) > set_nat > set_real).
% 145.00/25.60 thf(func_def_806, type, image_nat_char: (nat > char) > set_nat > set_char).
% 145.00/25.60 thf(func_def_807, type, image_real_real: (real > real) > set_real > set_real).
% 145.00/25.60 thf(func_def_808, type, image_char_nat: (char > nat) > set_char > set_nat).
% 145.00/25.60 thf(func_def_809, type, image_VEBT_VEBT_nat: (vEBT_VEBT > nat) > set_VEBT_VEBT > set_nat).
% 145.00/25.60 thf(func_def_810, type, insert_int: int > set_int > set_int).
% 145.00/25.60 thf(func_def_811, type, insert_nat: nat > set_nat > set_nat).
% 145.00/25.60 thf(func_def_812, type, insert_real: real > set_real > set_real).
% 145.00/25.60 thf(func_def_813, type, insert_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT).
% 145.00/25.60 thf(func_def_814, type, set_fo1517530859248394432omplex: (nat > complex > complex) > nat > nat > complex > complex).
% 145.00/25.60 thf(func_def_815, type, set_fo2581907887559384638at_int: (nat > int > int) > nat > nat > int > int).
% 145.00/25.60 thf(func_def_816, type, set_fo2584398358068434914at_nat: (nat > nat > nat) > nat > nat > nat > nat).
% 145.00/25.60 thf(func_def_817, type, set_fo8365102181078989356at_num: (nat > num > num) > nat > nat > num > num).
% 145.00/25.60 thf(func_def_818, type, set_fo1949268297981939178at_rat: (nat > rat > rat) > nat > nat > rat > rat).
% 145.00/25.60 thf(func_def_819, type, set_fo3111899725591712190t_real: (nat > real > real) > nat > nat > real > real).
% 145.00/25.60 thf(func_def_820, type, set_fo3699595496184130361el_nat: produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o).
% 145.00/25.60 thf(func_def_821, type, set_fo256927282339908995el_num: produc3368934014287244435at_num > produc3368934014287244435at_num > $o).
% 145.00/25.60 thf(func_def_822, type, set_or1266510415728281911st_int: int > int > set_int).
% 145.00/25.60 thf(func_def_823, type, set_or1269000886237332187st_nat: nat > nat > set_nat).
% 145.00/25.60 thf(func_def_824, type, set_or7049704709247886629st_num: num > num > set_num).
% 145.00/25.60 thf(func_def_825, type, set_or633870826150836451st_rat: rat > rat > set_rat).
% 145.00/25.60 thf(func_def_826, type, set_or1222579329274155063t_real: real > real > set_real).
% 145.00/25.60 thf(func_def_827, type, set_or370866239135849197et_int: set_int > set_int > set_set_int).
% 145.00/25.60 thf(func_def_828, type, set_or4662586982721622107an_int: int > int > set_int).
% 145.00/25.60 thf(func_def_829, type, set_or4665077453230672383an_nat: nat > nat > set_nat).
% 145.00/25.60 thf(func_def_830, type, set_ord_atLeast_nat: nat > set_nat).
% 145.00/25.60 thf(func_def_831, type, set_ord_atLeast_real: real > set_real).
% 145.00/25.60 thf(func_def_832, type, set_ord_atMost_int: int > set_int).
% 145.00/25.60 thf(func_def_833, type, set_ord_atMost_nat: nat > set_nat).
% 145.00/25.60 thf(func_def_834, type, set_ord_atMost_num: num > set_num).
% 145.00/25.60 thf(func_def_835, type, set_ord_atMost_rat: rat > set_rat).
% 145.00/25.60 thf(func_def_836, type, set_ord_atMost_real: real > set_real).
% 145.00/25.60 thf(func_def_837, type, set_or58775011639299419et_int: set_int > set_set_int).
% 145.00/25.60 thf(func_def_838, type, set_or6656581121297822940st_int: int > int > set_int).
% 145.00/25.60 thf(func_def_839, type, set_or6659071591806873216st_nat: nat > nat > set_nat).
% 145.00/25.60 thf(func_def_840, type, set_or5832277885323065728an_int: int > int > set_int).
% 145.00/25.60 thf(func_def_841, type, set_or5834768355832116004an_nat: nat > nat > set_nat).
% 145.00/25.60 thf(func_def_842, type, set_or1633881224788618240n_real: real > real > set_real).
% 145.00/25.60 thf(func_def_843, type, set_or1210151606488870762an_nat: nat > set_nat).
% 145.00/25.60 thf(func_def_844, type, set_or5849166863359141190n_real: real > set_real).
% 145.00/25.60 thf(func_def_845, type, set_ord_lessThan_int: int > set_int).
% 145.00/25.60 thf(func_def_846, type, set_ord_lessThan_nat: nat > set_nat).
% 145.00/25.60 thf(func_def_847, type, set_ord_lessThan_num: num > set_num).
% 145.00/25.60 thf(func_def_848, type, set_ord_lessThan_rat: rat > set_rat).
% 145.00/25.60 thf(func_def_849, type, set_or5984915006950818249n_real: real > set_real).
% 145.00/25.60 thf(func_def_850, type, ascii_of: char > char).
% 145.00/25.60 thf(func_def_851, type, char2: $o > $o > $o > $o > $o > $o > $o > $o > char).
% 145.00/25.60 thf(func_def_852, type, comm_s629917340098488124ar_nat: char > nat).
% 145.00/25.60 thf(func_def_853, type, integer_of_char: char > code_integer).
% 145.00/25.60 thf(func_def_854, type, unique3096191561947761185of_nat: nat > char).
% 145.00/25.60 thf(func_def_855, type, topolo4422821103128117721l_real: filter_real > (real > real) > $o).
% 145.00/25.60 thf(func_def_856, type, topolo5044208981011980120l_real: set_real > (real > real) > $o).
% 145.00/25.60 thf(func_def_857, type, topolo4899668324122417113eq_int: (nat > int) > $o).
% 145.00/25.60 thf(func_def_858, type, topolo4902158794631467389eq_nat: (nat > nat) > $o).
% 145.00/25.60 thf(func_def_859, type, topolo1459490580787246023eq_num: (nat > num) > $o).
% 145.00/25.60 thf(func_def_860, type, topolo4267028734544971653eq_rat: (nat > rat) > $o).
% 145.00/25.60 thf(func_def_861, type, topolo6980174941875973593q_real: (nat > real) > $o).
% 145.00/25.60 thf(func_def_862, type, topolo3100542954746470799et_int: (nat > set_int) > $o).
% 145.00/25.60 thf(func_def_863, type, topolo2177554685111907308n_real: real > set_real > filter_real).
% 145.00/25.60 thf(func_def_864, type, topolo2815343760600316023s_real: real > filter_real).
% 145.00/25.60 thf(func_def_865, type, topolo4055970368930404560y_real: (nat > real) > $o).
% 145.00/25.60 thf(func_def_866, type, arccos: real > real).
% 145.00/25.60 thf(func_def_867, type, arcosh_real: real > real).
% 145.00/25.60 thf(func_def_868, type, arcsin: real > real).
% 145.00/25.60 thf(func_def_869, type, arctan: real > real).
% 145.00/25.60 thf(func_def_870, type, arsinh_real: real > real).
% 145.00/25.60 thf(func_def_871, type, artanh_real: real > real).
% 145.00/25.60 thf(func_def_872, type, cos_complex: complex > complex).
% 145.00/25.60 thf(func_def_873, type, cos_real: real > real).
% 145.00/25.60 thf(func_def_874, type, cos_coeff: nat > real).
% 145.00/25.60 thf(func_def_875, type, cosh_real: real > real).
% 145.00/25.60 thf(func_def_876, type, cot_real: real > real).
% 145.00/25.60 thf(func_def_877, type, diffs_complex: (nat > complex) > nat > complex).
% 145.00/25.60 thf(func_def_878, type, diffs_int: (nat > int) > nat > int).
% 145.00/25.60 thf(func_def_879, type, diffs_rat: (nat > rat) > nat > rat).
% 145.00/25.60 thf(func_def_880, type, diffs_real: (nat > real) > nat > real).
% 145.00/25.60 thf(func_def_881, type, exp_complex: complex > complex).
% 145.00/25.60 thf(func_def_882, type, exp_real: real > real).
% 145.00/25.60 thf(func_def_883, type, ln_ln_real: real > real).
% 145.00/25.60 thf(func_def_884, type, log: real > real > real).
% 145.00/25.60 thf(func_def_885, type, pi: real).
% 145.00/25.60 thf(func_def_886, type, powr_real: real > real > real).
% 145.00/25.60 thf(func_def_887, type, sin_complex: complex > complex).
% 145.00/25.60 thf(func_def_888, type, sin_real: real > real).
% 145.00/25.60 thf(func_def_889, type, sin_coeff: nat > real).
% 145.00/25.60 thf(func_def_890, type, sinh_real: real > real).
% 145.00/25.60 thf(func_def_891, type, tan_complex: complex > complex).
% 145.00/25.60 thf(func_def_892, type, tan_real: real > real).
% 145.00/25.60 thf(func_def_893, type, tanh_complex: complex > complex).
% 145.00/25.60 thf(func_def_894, type, tanh_real: real > real).
% 145.00/25.60 thf(func_def_895, type, transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 145.00/25.60 thf(func_def_896, type, transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 145.00/25.60 thf(func_def_897, type, vEBT_T_i_n_s_e_r_t: vEBT_VEBT > nat > nat).
% 145.00/25.60 thf(func_def_898, type, vEBT_T_i_n_s_e_r_t2: vEBT_VEBT > nat > nat).
% 145.00/25.60 thf(func_def_899, type, vEBT_T5076183648494686801_t_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 145.00/25.60 thf(func_def_900, type, vEBT_T9217963907923527482_t_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 145.00/25.60 thf(func_def_901, type, vEBT_T_m_a_x_t: vEBT_VEBT > nat).
% 145.00/25.60 thf(func_def_902, type, vEBT_T_m_a_x_t_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 145.00/25.60 thf(func_def_903, type, vEBT_T_m_e_m_b_e_r: vEBT_VEBT > nat > nat).
% 145.00/25.60 thf(func_def_904, type, vEBT_T_m_e_m_b_e_r2: vEBT_VEBT > nat > nat).
% 145.00/25.60 thf(func_def_905, type, vEBT_T8099345112685741742_r_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 145.00/25.60 thf(func_def_906, type, vEBT_T5837161174952499735_r_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 145.00/25.60 thf(func_def_907, type, vEBT_T_m_i_n_N_u_l_l: vEBT_VEBT > nat).
% 145.00/25.60 thf(func_def_908, type, vEBT_T5462971552011256508_l_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 145.00/25.60 thf(func_def_909, type, vEBT_T_m_i_n_t: vEBT_VEBT > nat).
% 145.00/25.60 thf(func_def_910, type, vEBT_T_m_i_n_t_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 145.00/25.60 thf(func_def_911, type, vEBT_Leaf: $o > $o > vEBT_VEBT).
% 145.00/25.60 thf(func_def_912, type, vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 145.00/25.60 thf(func_def_913, type, vEBT_size_VEBT: vEBT_VEBT > nat).
% 145.00/25.60 thf(func_def_914, type, vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o).
% 145.00/25.60 thf(func_def_915, type, vEBT_VEBT_high: nat > nat > nat).
% 145.00/25.60 thf(func_def_916, type, vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o).
% 145.00/25.60 thf(func_def_917, type, vEBT_VEBT_low: nat > nat > nat).
% 145.00/25.60 thf(func_def_918, type, vEBT_VEBT_membermima: vEBT_VEBT > nat > $o).
% 145.00/25.60 thf(func_def_919, type, vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 145.00/25.60 thf(func_def_920, type, vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o).
% 145.00/25.60 thf(func_def_921, type, vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 145.00/25.60 thf(func_def_922, type, vEBT_VEBT_valid: vEBT_VEBT > nat > $o).
% 145.00/25.60 thf(func_def_923, type, vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 145.00/25.60 thf(func_def_924, type, vEBT_invar_vebt: vEBT_VEBT > nat > $o).
% 145.00/25.60 thf(func_def_925, type, vEBT_set_vebt: vEBT_VEBT > set_nat).
% 145.00/25.60 thf(func_def_926, type, vEBT_vebt_buildup: nat > vEBT_VEBT).
% 145.00/25.60 thf(func_def_927, type, vEBT_v4011308405150292612up_rel: nat > nat > $o).
% 145.00/25.60 thf(func_def_928, type, vEBT_VEBT_height: vEBT_VEBT > nat).
% 145.00/25.60 thf(func_def_929, type, vEBT_VEBT_height_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 145.00/25.60 thf(func_def_930, type, vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT).
% 145.00/25.60 thf(func_def_931, type, vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 145.00/25.60 thf(func_def_932, type, vEBT_VEBT_bit_concat: nat > nat > nat > nat).
% 145.00/25.60 thf(func_def_933, type, vEBT_VEBT_minNull: vEBT_VEBT > $o).
% 145.00/25.60 thf(func_def_934, type, vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 145.00/25.60 thf(func_def_935, type, vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat).
% 145.00/25.60 thf(func_def_936, type, vEBT_vebt_member: vEBT_VEBT > nat > $o).
% 145.00/25.60 thf(func_def_937, type, vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 145.00/25.60 thf(func_def_938, type, vEBT_VEBT_add: option_nat > option_nat > option_nat).
% 145.00/25.60 thf(func_def_939, type, vEBT_VEBT_greater: option_nat > option_nat > $o).
% 145.00/25.60 thf(func_def_940, type, vEBT_VEBT_less: option_nat > option_nat > $o).
% 145.00/25.60 thf(func_def_941, type, vEBT_VEBT_lesseq: option_nat > option_nat > $o).
% 145.00/25.60 thf(func_def_942, type, vEBT_VEBT_max_in_set: set_nat > nat > $o).
% 145.00/25.60 thf(func_def_943, type, vEBT_VEBT_min_in_set: set_nat > nat > $o).
% 145.00/25.60 thf(func_def_944, type, vEBT_VEBT_mul: option_nat > option_nat > option_nat).
% 145.00/25.60 thf(func_def_945, type, vEBT_V4262088993061758097ft_nat: (nat > nat > nat) > option_nat > option_nat > option_nat).
% 145.00/25.60 thf(func_def_946, type, vEBT_V819420779217536731ft_num: (num > num > num) > option_num > option_num > option_num).
% 145.00/25.60 thf(func_def_947, type, vEBT_V1502963449132264192at_nat: (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat).
% 145.00/25.60 thf(func_def_948, type, vEBT_V3895251965096974666el_nat: produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o).
% 145.00/25.60 thf(func_def_949, type, vEBT_V452583751252753300el_num: produc1193250871479095198on_num > produc1193250871479095198on_num > $o).
% 145.00/25.60 thf(func_def_950, type, vEBT_V7235779383477046023at_nat: produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o).
% 145.00/25.60 thf(func_def_951, type, vEBT_VEBT_power: option_nat > option_nat > option_nat).
% 145.00/25.60 thf(func_def_952, type, vEBT_vebt_maxt: vEBT_VEBT > option_nat).
% 145.00/25.60 thf(func_def_953, type, vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 145.00/25.60 thf(func_def_954, type, vEBT_vebt_mint: vEBT_VEBT > option_nat).
% 145.00/25.60 thf(func_def_955, type, vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 145.00/25.60 thf(func_def_956, type, vEBT_is_pred_in_set: set_nat > nat > nat > $o).
% 145.00/25.60 thf(func_def_957, type, vEBT_vebt_pred: vEBT_VEBT > nat > option_nat).
% 145.00/25.60 thf(func_def_958, type, vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 145.00/25.60 thf(func_def_959, type, vEBT_is_succ_in_set: set_nat > nat > nat > $o).
% 145.00/25.60 thf(func_def_960, type, vEBT_vebt_succ: vEBT_VEBT > nat > option_nat).
% 145.00/25.60 thf(func_def_961, type, vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 145.00/25.60 thf(func_def_962, type, accp_list_nat: (list_nat > list_nat > $o) > list_nat > $o).
% 145.00/25.60 thf(func_def_963, type, accp_nat: (nat > nat > $o) > nat > $o).
% 145.00/25.60 thf(func_def_964, type, accp_P6019419558468335806at_nat: (produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o) > produc4471711990508489141at_nat > $o).
% 145.00/25.60 thf(func_def_965, type, accp_P5496254298877145759on_nat: (produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o) > produc8306885398267862888on_nat > $o).
% 145.00/25.60 thf(func_def_966, type, accp_P4916641582247091100at_num: (produc3368934014287244435at_num > produc3368934014287244435at_num > $o) > produc3368934014287244435at_num > $o).
% 145.00/25.60 thf(func_def_967, type, accp_P7605991808943153877on_num: (produc1193250871479095198on_num > produc1193250871479095198on_num > $o) > produc1193250871479095198on_num > $o).
% 145.00/25.60 thf(func_def_968, type, accp_P3267385326087170368at_nat: (produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o) > produc5542196010084753463at_nat > $o).
% 145.00/25.60 thf(func_def_969, type, accp_P1096762738010456898nt_int: (product_prod_int_int > product_prod_int_int > $o) > product_prod_int_int > $o).
% 145.00/25.60 thf(func_def_970, type, accp_P4275260045618599050at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > product_prod_nat_nat > $o).
% 145.00/25.60 thf(func_def_971, type, accp_P3113834385874906142um_num: (product_prod_num_num > product_prod_num_num > $o) > product_prod_num_num > $o).
% 145.00/25.60 thf(func_def_972, type, accp_P2887432264394892906BT_nat: (produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o) > produc9072475918466114483BT_nat > $o).
% 145.00/25.60 thf(func_def_973, type, accp_VEBT_VEBT: (vEBT_VEBT > vEBT_VEBT > $o) > vEBT_VEBT > $o).
% 145.00/25.60 thf(func_def_974, type, measure_int: (int > nat) > set_Pr958786334691620121nt_int).
% 145.00/25.60 thf(func_def_975, type, measure_nat: (nat > nat) > set_Pr1261947904930325089at_nat).
% 145.00/25.60 thf(func_def_976, type, measure_num: (num > nat) > set_Pr8218934625190621173um_num).
% 145.00/25.60 thf(func_def_977, type, pred_nat: set_Pr1261947904930325089at_nat).
% 145.00/25.60 thf(func_def_978, type, fChoice_real: (real > $o) > real).
% 145.00/25.60 thf(func_def_979, type, member_o: $o > set_o > $o).
% 145.00/25.60 thf(func_def_980, type, member_complex: complex > set_complex > $o).
% 145.00/25.60 thf(func_def_981, type, member_int: int > set_int > $o).
% 145.00/25.60 thf(func_def_982, type, member_list_o: list_o > set_list_o > $o).
% 145.00/25.60 thf(func_def_983, type, member_list_int: list_int > set_list_int > $o).
% 145.00/25.60 thf(func_def_984, type, member_list_nat: list_nat > set_list_nat > $o).
% 145.00/25.60 thf(func_def_985, type, member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o).
% 145.00/25.60 thf(func_def_986, type, member_nat: nat > set_nat > $o).
% 145.00/25.60 thf(func_def_987, type, member_num: num > set_num > $o).
% 145.00/25.60 thf(func_def_988, type, member1379723562493234055eger_o: produc6271795597528267376eger_o > set_Pr448751882837621926eger_o > $o).
% 145.00/25.60 thf(func_def_989, type, member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o).
% 145.00/25.60 thf(func_def_990, type, member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o).
% 145.00/25.60 thf(func_def_991, type, member9148766508732265716at_num: product_prod_nat_num > set_Pr6200539531224447659at_num > $o).
% 145.00/25.60 thf(func_def_992, type, member7279096912039735102um_num: product_prod_num_num > set_Pr8218934625190621173um_num > $o).
% 145.00/25.60 thf(func_def_993, type, member_rat: rat > set_rat > $o).
% 145.00/25.60 thf(func_def_994, type, member_real: real > set_real > $o).
% 145.00/25.60 thf(func_def_995, type, member_set_int: set_int > set_set_int > $o).
% 145.00/25.60 thf(func_def_996, type, member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o).
% 145.00/25.60 thf(func_def_997, type, deg: nat).
% 145.00/25.60 thf(func_def_998, type, m: nat).
% 145.00/25.60 thf(func_def_999, type, ma: nat).
% 145.00/25.60 thf(func_def_1000, type, mi: nat).
% 145.00/25.60 thf(func_def_1001, type, na: nat).
% 145.00/25.60 thf(func_def_1002, type, summary: vEBT_VEBT).
% 145.00/25.60 thf(func_def_1003, type, treeList: list_VEBT_VEBT).
% 145.00/25.60 thf(func_def_1004, type, xa: nat).
% 145.00/25.60 thf(func_def_1023, type, sP0: nat > nat > nat > $o).
% 145.00/25.60 thf(func_def_1024, type, sP1: nat > nat > nat > $o).
% 145.00/25.60 thf(func_def_1025, type, sP2: nat > nat > nat > $o).
% 145.00/25.60 thf(func_def_1026, type, sP3: vEBT_VEBT > nat > $o).
% 145.00/25.60 thf(func_def_1027, type, sP4: vEBT_VEBT > nat > $o).
% 145.00/25.60 thf(func_def_1028, type, sK5: (nat > nat > $o) > nat).
% 145.00/25.60 thf(func_def_1029, type, sK6: (nat > nat > $o) > nat).
% 145.00/25.60 thf(func_def_1030, type, sK7: (nat > nat > $o) > nat).
% 145.00/25.60 thf(func_def_1031, type, sK8: (nat > nat > $o) > nat).
% 145.00/25.60 thf(func_def_1032, type, sK9: (nat > nat > $o) > nat).
% 145.00/25.60 thf(func_def_1033, type, sK10: nat > nat > nat).
% 145.00/25.60 thf(func_def_1034, type, sK11: (nat > $o) > nat).
% 145.00/25.60 thf(func_def_1035, type, sK12: (nat > nat) > nat).
% 145.00/25.60 thf(func_def_1036, type, sK13: (nat > nat) > nat).
% 145.00/25.60 thf(func_def_1037, type, sK14: product_prod_nat_nat > nat).
% 145.00/25.60 thf(func_def_1038, type, sK15: product_prod_nat_nat > nat).
% 145.00/25.60 thf(func_def_1039, type, sK16: (product_prod_nat_nat > $o) > nat).
% 145.00/25.60 thf(func_def_1040, type, sK17: (product_prod_nat_nat > $o) > nat).
% 145.00/25.60 thf(func_def_1041, type, sK18: nat > nat > nat).
% 145.00/25.60 thf(func_def_1042, type, sK19: vEBT_VEBT > vEBT_VEBT).
% 145.00/25.60 thf(func_def_1043, type, sK20: vEBT_VEBT > list_VEBT_VEBT).
% 145.00/25.60 thf(func_def_1044, type, sK21: vEBT_VEBT > option4927543243414619207at_nat).
% 145.00/25.60 thf(func_def_1045, type, sK22: vEBT_VEBT > vEBT_VEBT).
% 145.00/25.60 thf(func_def_1046, type, sK23: vEBT_VEBT > option4927543243414619207at_nat).
% 145.00/25.60 thf(func_def_1047, type, sK24: vEBT_VEBT > list_VEBT_VEBT).
% 145.00/25.60 thf(func_def_1048, type, sK25: vEBT_VEBT > nat).
% 145.00/25.60 thf(func_def_1049, type, sK26: vEBT_VEBT > vEBT_VEBT).
% 145.00/25.60 thf(func_def_1050, type, sK27: vEBT_VEBT > list_VEBT_VEBT).
% 145.00/25.60 thf(func_def_1051, type, sK28: vEBT_VEBT > nat > nat > nat).
% 145.00/25.60 thf(func_def_1052, type, sK29: vEBT_VEBT > nat > nat > nat).
% 145.00/25.60 thf(func_def_1053, type, sK30: vEBT_VEBT > nat > nat > list_VEBT_VEBT).
% 145.00/25.60 thf(func_def_1054, type, sK31: vEBT_VEBT > nat > nat > nat).
% 145.00/25.60 thf(func_def_1055, type, sK32: vEBT_VEBT > nat > nat > vEBT_VEBT).
% 145.00/25.60 thf(func_def_1056, type, sK33: vEBT_VEBT > $o).
% 145.00/25.60 thf(func_def_1057, type, sK34: vEBT_VEBT > $o).
% 145.00/25.60 thf(func_def_1058, type, sK35: nat > (nat > $o) > nat).
% 145.00/25.60 thf(func_def_1059, type, sK36: nat > nat > nat).
% 145.00/25.60 thf(func_def_1060, type, sK37: (nat > $o) > nat).
% 145.00/25.60 thf(func_def_1061, type, sK38: nat > (nat > $o) > nat).
% 145.00/25.60 thf(func_def_1062, type, sK39: product_prod_nat_nat > nat).
% 145.00/25.60 thf(func_def_1063, type, sK40: product_prod_nat_nat > nat).
% 145.00/25.60 thf(func_def_1064, type, sK41: nat > nat).
% 145.00/25.60 thf(func_def_1065, type, sK42: (nat > $o) > nat).
% 145.00/25.60 thf(func_def_1066, type, sK43: vEBT_VEBT > nat).
% 145.00/25.60 thf(func_def_1067, type, sK44: nat > nat > nat).
% 145.00/25.60 thf(func_def_1068, type, sK45: product_prod_nat_nat > nat).
% 145.00/25.60 thf(func_def_1069, type, sK46: product_prod_nat_nat > nat).
% 145.00/25.60 thf(func_def_1070, type, sK47: (nat > nat > $o) > nat).
% 145.00/25.60 thf(func_def_1071, type, sK48: (nat > nat > $o) > nat).
% 145.00/25.60 thf(func_def_1072, type, sK49: (nat > nat > $o) > nat).
% 145.00/25.60 thf(func_def_1073, type, sK50: (nat > nat > $o) > nat).
% 145.00/25.60 thf(func_def_1074, type, sK51: (nat > nat) > nat).
% 145.00/25.60 thf(func_def_1075, type, sK52: (nat > nat) > nat).
% 145.00/25.60 thf(func_def_1076, type, sK53: nat > (nat > $o) > nat).
% 145.00/25.60 thf(func_def_1077, type, sK54: nat > (nat > $o) > nat).
% 145.00/25.60 thf(func_def_1078, type, sK55: nat > nat).
% 145.00/25.60 thf(func_def_1079, type, sK56: (nat > $o) > nat > nat).
% 145.00/25.60 thf(func_def_1080, type, sK57: (nat > $o) > nat).
% 145.00/25.60 thf(func_def_1081, type, sK58: nat > nat).
% 145.00/25.60 thf(func_def_1082, type, sK59: nat > nat).
% 145.00/25.60 thf(func_def_1083, type, sK60: nat > nat).
% 145.00/25.60 thf(func_def_1084, type, sK61: nat > nat > nat).
% 145.00/25.60 thf(func_def_1085, type, sK62: nat > nat).
% 145.00/25.60 thf(func_def_1086, type, sK63: nat > nat > nat).
% 145.00/25.60 thf(func_def_1087, type, ph64: !>[X0: $tType]:(X0)).
% 145.00/25.60 thf(f37737,plain,(
% 145.00/25.60 $false),
% 145.00/25.60 inference(subsumption_resolution,[status(thm)],[f37729,f24653])).
% 145.00/25.60 thf(f24653,plain,(
% 145.00/25.60 ($true = (member_VEBT_VEBT @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (if_nat @ (ord_less_eq_nat @ (plus_plus_nat @ xa @ (numeral_numeral_nat @ one)) @ mi) @ mi @ xa) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ (set_VEBT_VEBT2 @ treeList)))),
% 145.00/25.60 inference(beta_eta_normalization,[status(thm)],[f24499])).
% 145.00/25.60 thf(f24499,plain,(
% 145.00/25.60 ($true = (member_VEBT_VEBT @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (if_nat @ ((^[Y0 : nat]: (ord_less_eq_nat @ (plus_plus_nat @ Y0 @ (numeral_numeral_nat @ one)))) @ xa @ mi) @ mi @ xa) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ (set_VEBT_VEBT2 @ treeList)))),
% 145.00/25.60 inference(definition_unfolding,[status(thm)],[f24293,f24391])).
% 145.00/25.60 thf(f24391,plain,(
% 145.00/25.60 (ord_less_nat = (^[Y0 : nat]: (ord_less_eq_nat @ (plus_plus_nat @ Y0 @ (numeral_numeral_nat @ one)))))),
% 145.00/25.60 inference(definition_unfolding,[status(thm)],[f24279,f24091])).
% 145.00/25.60 thf(f24091,plain,(
% 145.00/25.60 (one_one_nat = (numeral_numeral_nat @ one))),
% 145.00/25.60 inference(cnf_transformation,[status(thm)],[f463])).
% 145.00/25.60 thf(f463,axiom,(
% 145.00/25.60 (one_one_nat = (numeral_numeral_nat @ one))),
% 145.00/25.60 file('/export/starexec/sandbox/tmp/tmp.jDIxpPE1PX/DTF2THF_16103.p',fact_462_numerals_I1_J)).
% 145.00/25.60 thf(f24279,plain,(
% 145.00/25.60 (ord_less_nat = (^[Y0 : nat]: (ord_less_eq_nat @ (plus_plus_nat @ Y0 @ one_one_nat))))),
% 145.00/25.60 inference(cnf_transformation,[status(thm)],[f13162])).
% 145.00/25.60 thf(f13162,plain,(
% 145.00/25.60 (ord_less_nat = (^[Y0 : nat]: (ord_less_eq_nat @ (plus_plus_nat @ Y0 @ one_one_nat))))),
% 145.00/25.60 inference(fool_elimination,[status(thm)],[f13161])).
% 145.00/25.60 thf(f13161,plain,(
% 145.00/25.60 (ord_less_nat = (^[X0 : nat] : (ord_less_eq_nat @ (plus_plus_nat @ X0 @ one_one_nat))))),
% 145.00/25.60 inference(rectify,[status(thm)],[f2639])).
% 145.00/25.60 thf(f2639,axiom,(
% 145.00/25.60 (ord_less_nat = (^[X52 : nat] : (ord_less_eq_nat @ (plus_plus_nat @ X52 @ one_one_nat))))),
% 145.00/25.60 file('/export/starexec/sandbox/tmp/tmp.jDIxpPE1PX/DTF2THF_16103.p',fact_2638_discrete)).
% 145.00/25.60 thf(f24293,plain,(
% 145.00/25.60 ((member_VEBT_VEBT @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (if_nat @ (ord_less_nat @ xa @ mi) @ mi @ xa) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ (set_VEBT_VEBT2 @ treeList)) = $true)),
% 145.00/25.60 inference(cnf_transformation,[status(thm)],[f18526])).
% 145.00/25.60 thf(f18526,plain,(
% 145.00/25.60 ((member_VEBT_VEBT @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (if_nat @ (ord_less_nat @ xa @ mi) @ mi @ xa) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ (set_VEBT_VEBT2 @ treeList)) = $true)),
% 145.00/25.60 inference(fool_elimination,[status(thm)],[f18525])).
% 145.00/25.60 thf(f18525,plain,(
% 145.00/25.60 (member_VEBT_VEBT @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (if_nat @ (ord_less_nat @ xa @ mi) @ mi @ xa) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ (set_VEBT_VEBT2 @ treeList))),
% 145.00/25.60 inference(rectify,[status(thm)],[f10])).
% 145.00/25.60 thf(f10,axiom,(
% 145.00/25.60 (member_VEBT_VEBT @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (if_nat @ (ord_less_nat @ xa @ mi) @ mi @ xa) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ (set_VEBT_VEBT2 @ treeList))),
% 145.00/25.60 file('/export/starexec/sandbox/tmp/tmp.jDIxpPE1PX/DTF2THF_16103.p',fact_9__092_060open_062treeList_A_B_Ahigh_A_Iif_Ax_A_060_Ami_Athen_Ami_Aelse_Ax_J_A_Ideg_Adiv_A2_J_A_092_060in_062_Aset_AtreeList_092_060close_062)).
% 145.00/25.60 thf(f37729,plain,(
% 145.00/25.60 ($true != (member_VEBT_VEBT @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (if_nat @ (ord_less_eq_nat @ (plus_plus_nat @ xa @ (numeral_numeral_nat @ one)) @ mi) @ mi @ xa) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ (set_VEBT_VEBT2 @ treeList)))),
% 145.00/25.60 inference(trivial_inequality_removal,[status(thm)],[f37712])).
% 145.00/25.60 thf(f37712,plain,(
% 145.00/25.60 ($true != (member_VEBT_VEBT @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (if_nat @ (ord_less_eq_nat @ (plus_plus_nat @ xa @ (numeral_numeral_nat @ one)) @ mi) @ mi @ xa) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ (set_VEBT_VEBT2 @ treeList))) | ($true != $true)),
% 145.00/25.60 inference(superposition,[status(thm)],[f25572,f24394])).
% 145.00/25.60 thf(f24394,plain,(
% 145.00/25.60 ( ! [X2 : vEBT_VEBT,X3 : list_VEBT_VEBT,X0 : vEBT_VEBT,X1 : nat,X4 : option4927543243414619207at_nat] : (($true = (ord_less_eq_nat @ (plus_plus_nat @ (numeral_numeral_nat @ one) @ (vEBT_VEBT_height @ X0)) @ (vEBT_VEBT_height @ (vEBT_Node @ X4 @ X1 @ X3 @ X2)))) | ($true != (member_VEBT_VEBT @ X0 @ (set_VEBT_VEBT2 @ X3)))) )),
% 145.00/25.60 inference(definition_unfolding,[status(thm)],[f24079,f24091])).
% 145.00/25.60 thf(f24079,plain,(
% 145.00/25.60 ( ! [X2 : vEBT_VEBT,X3 : list_VEBT_VEBT,X0 : vEBT_VEBT,X1 : nat,X4 : option4927543243414619207at_nat] : (((ord_less_eq_nat @ (plus_plus_nat @ one_one_nat @ (vEBT_VEBT_height @ X0)) @ (vEBT_VEBT_height @ (vEBT_Node @ X4 @ X1 @ X3 @ X2))) = $true) | ($true != (member_VEBT_VEBT @ X0 @ (set_VEBT_VEBT2 @ X3)))) )),
% 145.00/25.60 inference(cnf_transformation,[status(thm)],[f23797])).
% 145.00/25.60 thf(f23797,plain,(
% 145.00/25.60 ! [X0 : vEBT_VEBT,X1 : nat,X2 : vEBT_VEBT,X3 : list_VEBT_VEBT,X4 : option4927543243414619207at_nat] : (((ord_less_eq_nat @ (plus_plus_nat @ one_one_nat @ (vEBT_VEBT_height @ X0)) @ (vEBT_VEBT_height @ (vEBT_Node @ X4 @ X1 @ X3 @ X2))) = $true) | ($true != (member_VEBT_VEBT @ X0 @ (set_VEBT_VEBT2 @ X3))))),
% 145.00/25.60 inference(ennf_transformation,[status(thm)],[f16090])).
% 145.00/25.60 thf(f16090,plain,(
% 145.00/25.60 ! [X0 : vEBT_VEBT,X1 : nat,X2 : vEBT_VEBT,X3 : list_VEBT_VEBT,X4 : option4927543243414619207at_nat] : (($true = (member_VEBT_VEBT @ X0 @ (set_VEBT_VEBT2 @ X3))) => ((ord_less_eq_nat @ (plus_plus_nat @ one_one_nat @ (vEBT_VEBT_height @ X0)) @ (vEBT_VEBT_height @ (vEBT_Node @ X4 @ X1 @ X3 @ X2))) = $true))),
% 145.00/25.60 inference(fool_elimination,[status(thm)],[f16089])).
% 145.00/25.60 thf(f16089,plain,(
% 145.00/25.60 ! [X0 : vEBT_VEBT,X1 : nat,X2 : vEBT_VEBT,X3 : list_VEBT_VEBT,X4 : option4927543243414619207at_nat] : ((member_VEBT_VEBT @ X0 @ (set_VEBT_VEBT2 @ X3)) => (ord_less_eq_nat @ (plus_plus_nat @ one_one_nat @ (vEBT_VEBT_height @ X0)) @ (vEBT_VEBT_height @ (vEBT_Node @ X4 @ X1 @ X3 @ X2))))),
% 145.00/25.60 inference(rectify,[status(thm)],[f6])).
% 145.00/25.60 thf(f6,axiom,(
% 145.00/25.60 ! [X8 : vEBT_VEBT,X5 : nat,X3 : vEBT_VEBT,X6 : list_VEBT_VEBT,X4 : option4927543243414619207at_nat] : ((member_VEBT_VEBT @ X8 @ (set_VEBT_VEBT2 @ X6)) => (ord_less_eq_nat @ (plus_plus_nat @ one_one_nat @ (vEBT_VEBT_height @ X8)) @ (vEBT_VEBT_height @ (vEBT_Node @ X4 @ X5 @ X6 @ X3))))),
% 145.00/25.60 file('/export/starexec/sandbox/tmp/tmp.jDIxpPE1PX/DTF2THF_16103.p',fact_5_height__compose__child)).
% 145.00/25.60 thf(f25572,plain,(
% 145.00/25.60 ($true != (ord_less_eq_nat @ (plus_plus_nat @ (numeral_numeral_nat @ one) @ (vEBT_VEBT_height @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (if_nat @ (ord_less_eq_nat @ (plus_plus_nat @ xa @ (numeral_numeral_nat @ one)) @ mi) @ mi @ xa) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))))) @ (vEBT_VEBT_height @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary))))),
% 145.00/25.60 inference(trivial_inequality_removal,[status(thm)],[f25547])).
% 145.00/25.60 thf(f25547,plain,(
% 145.00/25.60 ($true != $true) | ($true != (ord_less_eq_nat @ (plus_plus_nat @ (numeral_numeral_nat @ one) @ (vEBT_VEBT_height @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (if_nat @ (ord_less_eq_nat @ (plus_plus_nat @ xa @ (numeral_numeral_nat @ one)) @ mi) @ mi @ xa) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))))) @ (vEBT_VEBT_height @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary))))),
% 145.00/25.60 inference(superposition,[status(thm)],[f24954,f24665])).
% 145.00/25.60 thf(f24665,plain,(
% 145.00/25.60 ($true = (ord_less_eq_nat @ (vEBT_T_i_n_s_e_r_t2 @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary) @ xa) @ (plus_plus_nat @ (numeral_numeral_nat @ one) @ (plus_plus_nat @ (numeral_numeral_nat @ one) @ (vEBT_VEBT_height @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (if_nat @ (ord_less_eq_nat @ (plus_plus_nat @ xa @ (numeral_numeral_nat @ one)) @ mi) @ mi @ xa) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))))))))),
% 145.00/25.60 inference(beta_eta_normalization,[status(thm)],[f24473])).
% 145.00/25.60 thf(f24473,plain,(
% 145.00/25.60 ($true = (ord_less_eq_nat @ (vEBT_T_i_n_s_e_r_t2 @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary) @ xa) @ (plus_plus_nat @ (numeral_numeral_nat @ one) @ (plus_plus_nat @ (numeral_numeral_nat @ one) @ (vEBT_VEBT_height @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (if_nat @ ((^[Y0 : nat]: (ord_less_eq_nat @ (plus_plus_nat @ Y0 @ (numeral_numeral_nat @ one)))) @ xa @ mi) @ mi @ xa) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))))))))),
% 145.00/25.60 inference(definition_unfolding,[status(thm)],[f24233,f24091,f24091,f24391])).
% 145.00/25.60 thf(f24233,plain,(
% 145.00/25.60 ((ord_less_eq_nat @ (vEBT_T_i_n_s_e_r_t2 @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary) @ xa) @ (plus_plus_nat @ one_one_nat @ (plus_plus_nat @ one_one_nat @ (vEBT_VEBT_height @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (if_nat @ (ord_less_nat @ xa @ mi) @ mi @ xa) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))))))) = $true)),
% 145.00/25.60 inference(cnf_transformation,[status(thm)],[f22869])).
% 145.00/25.60 thf(f22869,plain,(
% 145.00/25.60 ((ord_less_eq_nat @ (vEBT_T_i_n_s_e_r_t2 @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary) @ xa) @ (plus_plus_nat @ one_one_nat @ (plus_plus_nat @ one_one_nat @ (vEBT_VEBT_height @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (if_nat @ (ord_less_nat @ xa @ mi) @ mi @ xa) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))))))) = $true)),
% 145.00/25.60 inference(fool_elimination,[status(thm)],[f22868])).
% 145.00/25.60 thf(f22868,plain,(
% 145.00/25.60 (ord_less_eq_nat @ (vEBT_T_i_n_s_e_r_t2 @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary) @ xa) @ (plus_plus_nat @ one_one_nat @ (plus_plus_nat @ one_one_nat @ (vEBT_VEBT_height @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (if_nat @ (ord_less_nat @ xa @ mi) @ mi @ xa) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))))))),
% 145.00/25.60 inference(rectify,[status(thm)],[f7])).
% 145.00/25.60 thf(f7,axiom,(
% 145.00/25.60 (ord_less_eq_nat @ (vEBT_T_i_n_s_e_r_t2 @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary) @ xa) @ (plus_plus_nat @ one_one_nat @ (plus_plus_nat @ one_one_nat @ (vEBT_VEBT_height @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (if_nat @ (ord_less_nat @ xa @ mi) @ mi @ xa) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))))))),
% 145.00/25.60 file('/export/starexec/sandbox/tmp/tmp.jDIxpPE1PX/DTF2THF_16103.p',fact_6__092_060open_062T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_092_060le_062_A1_A_L_A_I1_A_L_Aheight_A_ItreeList_A_B_Ahigh_A_Iif_Ax_A_060_Ami_Athen_Ami_Aelse_Ax_J_A_Ideg_Adiv_A2_J_J_J_092_060close_062)).
% 145.00/25.60 thf(f24954,plain,(
% 145.00/25.60 ( ! [X0 : nat] : (((ord_less_eq_nat @ (vEBT_T_i_n_s_e_r_t2 @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary) @ xa) @ (plus_plus_nat @ (numeral_numeral_nat @ one) @ X0)) != $true) | ((ord_less_eq_nat @ X0 @ (vEBT_VEBT_height @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary))) != $true)) )),
% 145.00/25.60 inference(superposition,[status(thm)],[f24703,f24327])).
% 145.00/25.60 thf(f24327,plain,(
% 145.00/25.60 ( ! [X2 : nat,X0 : nat,X1 : nat] : (((ord_less_eq_nat @ X2 @ X1) = (ord_less_eq_nat @ (plus_plus_nat @ X0 @ X2) @ (plus_plus_nat @ X0 @ X1)))) )),
% 145.00/25.60 inference(cnf_transformation,[status(thm)],[f23747])).
% 145.00/25.60 thf(f23747,plain,(
% 145.00/25.60 ! [X0 : nat,X1 : nat,X2 : nat] : ((ord_less_eq_nat @ X2 @ X1) = (ord_less_eq_nat @ (plus_plus_nat @ X0 @ X2) @ (plus_plus_nat @ X0 @ X1)))),
% 145.00/25.60 inference(rectify,[status(thm)],[f11])).
% 145.00/25.60 thf(f11,axiom,(
% 145.00/25.60 ! [X11 : nat,X13 : nat,X12 : nat] : ((ord_less_eq_nat @ (plus_plus_nat @ X11 @ X12) @ (plus_plus_nat @ X11 @ X13)) = (ord_less_eq_nat @ X12 @ X13))),
% 145.00/25.60 file('/export/starexec/sandbox/tmp/tmp.jDIxpPE1PX/DTF2THF_16103.p',fact_10_nat__add__left__cancel__le)).
% 145.00/25.60 thf(f24703,plain,(
% 145.00/25.60 ( ! [X0 : nat] : (($true != (ord_less_eq_nat @ X0 @ (plus_plus_nat @ (numeral_numeral_nat @ one) @ (vEBT_VEBT_height @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary))))) | ((ord_less_eq_nat @ (vEBT_T_i_n_s_e_r_t2 @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary) @ xa) @ X0) != $true)) )),
% 145.00/25.60 inference(trivial_inequality_removal,[status(thm)],[f24696])).
% 145.00/25.60 thf(f24696,plain,(
% 145.00/25.60 ( ! [X0 : nat] : (($true != (ord_less_eq_nat @ X0 @ (plus_plus_nat @ (numeral_numeral_nat @ one) @ (vEBT_VEBT_height @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary))))) | ($true != $true) | ((ord_less_eq_nat @ (vEBT_T_i_n_s_e_r_t2 @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary) @ xa) @ X0) != $true)) )),
% 145.00/25.60 inference(superposition,[status(thm)],[f24454,f24083])).
% 145.00/25.60 thf(f24083,plain,(
% 145.00/25.60 ( ! [X2 : nat,X0 : nat,X1 : nat] : (($true = (ord_less_eq_nat @ X2 @ X0)) | ((ord_less_eq_nat @ X2 @ X1) != $true) | ((ord_less_eq_nat @ X1 @ X0) != $true)) )),
% 145.00/25.60 inference(cnf_transformation,[status(thm)],[f23985])).
% 145.00/25.60 thf(f23985,plain,(
% 145.00/25.60 ! [X0 : nat,X1 : nat,X2 : nat] : (($true = (ord_less_eq_nat @ X2 @ X0)) | ((ord_less_eq_nat @ X1 @ X0) != $true) | ((ord_less_eq_nat @ X2 @ X1) != $true))),
% 145.00/25.60 inference(flattening,[status(thm)],[f23984])).
% 145.00/25.60 thf(f23984,plain,(
% 145.00/25.60 ! [X0 : nat,X1 : nat,X2 : nat] : ((($true = (ord_less_eq_nat @ X2 @ X0)) | ((ord_less_eq_nat @ X1 @ X0) != $true)) | ((ord_less_eq_nat @ X2 @ X1) != $true))),
% 145.00/25.60 inference(ennf_transformation,[status(thm)],[f13792])).
% 145.00/25.60 thf(f13792,plain,(
% 145.00/25.60 ! [X0 : nat,X1 : nat,X2 : nat] : (((ord_less_eq_nat @ X2 @ X1) = $true) => (((ord_less_eq_nat @ X1 @ X0) = $true) => ($true = (ord_less_eq_nat @ X2 @ X0))))),
% 145.00/25.60 inference(fool_elimination,[status(thm)],[f13791])).
% 145.00/25.60 thf(f13791,plain,(
% 145.00/25.60 ! [X0 : nat,X1 : nat,X2 : nat] : ((ord_less_eq_nat @ X2 @ X1) => ((ord_less_eq_nat @ X1 @ X0) => (ord_less_eq_nat @ X2 @ X0)))),
% 145.00/25.60 inference(rectify,[status(thm)],[f174])).
% 145.00/25.60 thf(f174,axiom,(
% 145.00/25.60 ! [X11 : nat,X45 : nat,X44 : nat] : ((ord_less_eq_nat @ X44 @ X45) => ((ord_less_eq_nat @ X45 @ X11) => (ord_less_eq_nat @ X44 @ X11)))),
% 145.00/25.60 file('/export/starexec/sandbox/tmp/tmp.jDIxpPE1PX/DTF2THF_16103.p',fact_173_le__trans)).
% 145.00/25.60 thf(f24454,plain,(
% 145.00/25.60 ($true != (ord_less_eq_nat @ (vEBT_T_i_n_s_e_r_t2 @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary) @ xa) @ (plus_plus_nat @ (numeral_numeral_nat @ one) @ (vEBT_VEBT_height @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary)))))),
% 145.00/25.60 inference(definition_unfolding,[status(thm)],[f24197,f24091])).
% 145.00/25.60 thf(f24197,plain,(
% 145.00/25.60 ((ord_less_eq_nat @ (vEBT_T_i_n_s_e_r_t2 @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary) @ xa) @ (plus_plus_nat @ one_one_nat @ (vEBT_VEBT_height @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary)))) != $true)),
% 145.00/25.60 inference(cnf_transformation,[status(thm)],[f23708])).
% 145.00/25.60 thf(f23708,plain,(
% 145.00/25.60 ((ord_less_eq_nat @ (vEBT_T_i_n_s_e_r_t2 @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary) @ xa) @ (plus_plus_nat @ one_one_nat @ (vEBT_VEBT_height @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary)))) != $true)),
% 145.00/25.60 inference(flattening,[status(thm)],[f14818])).
% 145.00/25.60 thf(f14818,plain,(
% 145.00/25.60 ~((ord_less_eq_nat @ (vEBT_T_i_n_s_e_r_t2 @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary) @ xa) @ (plus_plus_nat @ one_one_nat @ (vEBT_VEBT_height @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary)))) = $true)),
% 145.00/25.60 inference(fool_elimination,[status(thm)],[f14817])).
% 145.00/25.60 thf(f14817,plain,(
% 145.00/25.60 ~(ord_less_eq_nat @ (vEBT_T_i_n_s_e_r_t2 @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary) @ xa) @ (plus_plus_nat @ one_one_nat @ (vEBT_VEBT_height @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary))))),
% 145.00/25.60 inference(rectify,[status(thm)],[f10249])).
% 145.00/25.60 thf(f10249,negated_conjecture,(
% 145.00/25.60 ~(ord_less_eq_nat @ (vEBT_T_i_n_s_e_r_t2 @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary) @ xa) @ (plus_plus_nat @ one_one_nat @ (vEBT_VEBT_height @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary))))),
% 145.00/25.60 inference(negated_conjecture,[status(cth)],[f10248])).
% 145.00/25.60 thf(f10248,conjecture,(
% 145.00/25.60 (ord_less_eq_nat @ (vEBT_T_i_n_s_e_r_t2 @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary) @ xa) @ (plus_plus_nat @ one_one_nat @ (vEBT_VEBT_height @ (vEBT_Node @ (some_P7363390416028606310at_nat @ (product_Pair_nat_nat @ mi @ ma)) @ deg @ treeList @ summary))))),
% 145.00/25.60 file('/export/starexec/sandbox/tmp/tmp.jDIxpPE1PX/DTF2THF_16103.p',conj_0)).
% 145.00/25.60 % SZS output end Proof for DTF2THF_16103
% 145.00/25.60 % (16481)------------------------------
% 145.00/25.60 % (16481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 145.00/25.60 % (16481)Termination reason: Refutation
% 145.00/25.60
% 145.00/25.60 % (16481)Memory used [KB]: 30191
% 145.00/25.60 % (16481)Time elapsed: 3.251 s
% 145.00/25.60 % (16481)Instructions burned: 10834 (million)
% 145.00/25.60 % (16481)------------------------------
% 145.00/25.60 % (16481)------------------------------
% 145.00/25.60 % (16209)Success in time 20.441 s
%------------------------------------------------------------------------------